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1010.5786	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 158726, "num_imgs": 16, "llama3_tokens_count": 44593 }	[ "content_image/1010.5786/x1.png", "content_image/1010.5786/x2.png", "content_image/1010.5786/x3.png", "content_image/1010.5786/x4.png", "content_image/1010.5786/x5.png", "content_image/1010.5786/x6.png", "content_image/1010.5786/x7.png", "content_image/1010.5786/x8.png", "content_image/1010.5786/x9.png", "content_image/1010.5786/x11.png", "content_image/1010.5786/x12.png", "content_image/1010.5786/x13.png", "content_image/1010.5786/x14.png", "content_image/1010.5786/x15.png", "content_image/1010.5786/x16.png", "content_image/1010.5786/x17.png" ]	# The _Hubble Space Telescope* [FOOTNOTE:][ENDFOOTNOTE]_ Cluster Supernova Survey: II. The Type Ia Supernova Rate in High-Redshift Galaxy Clusters K. Barbary¹² , G. Aldering² , R. Amanullah¹³ , M. Brodwin⁴⁵ , N. Connolly⁶ , K. S. Dawson²⁷ , M. Doi⁸ , P. Eisenhardt⁹ , L. Faccioli² , V. Fadeyev¹⁰ , H. K. Fakhouri¹² , A. S. Fruchter¹¹ , D. G. Gilbank ¹² , M. D. Gladders¹³ , G. Goldhaber¹²¹⁴ , A. Goobar³¹⁵ , T. Hattori¹⁶ , E. Hsiao² , X. Huang¹ , Y. Ihara⁸¹⁷ , N. Kashikawa¹⁸ , B. Koester¹³¹⁹ , K. Konishi²⁰ , M. Kowalski²¹ , C. Lidman²² , L. Lubin²³ , J. Meyers¹² , T. Morokuma⁸¹⁸¹⁷ , T. Oda²⁴ , N. Panagia¹¹ , S. Perlmutter¹² , M. Postman¹¹ , P. Ripoche² , P. Rosati²⁵ , D. Rubin¹² , D. J. Schlegel² , A. L. Spadafora² , S. A. Stanford²³²⁶ , M. Strovink¹² , N. Suzuki² , N. Takanashi¹⁸ , K. Tokita⁸ , N. Yasuda²⁰ (The Supernova Cosmology Project) kbarbary@lbl.gov [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:5][ENDFOOTNOTE] [FOOTNOTE:6][ENDFOOTNOTE] [FOOTNOTE:7][ENDFOOTNOTE] [FOOTNOTE:8][ENDFOOTNOTE] [FOOTNOTE:9][ENDFOOTNOTE] [FOOTNOTE:10][ENDFOOTNOTE] [FOOTNOTE:11][ENDFOOTNOTE] [FOOTNOTE:12][ENDFOOTNOTE] [FOOTNOTE:13][ENDFOOTNOTE] [FOOTNOTE:26][ENDFOOTNOTE] [FOOTNOTE:14][ENDFOOTNOTE] [FOOTNOTE:15][ENDFOOTNOTE] [FOOTNOTE:25][ENDFOOTNOTE] [FOOTNOTE:16][ENDFOOTNOTE] [FOOTNOTE:17][ENDFOOTNOTE] [FOOTNOTE:18][ENDFOOTNOTE] [FOOTNOTE:19][ENDFOOTNOTE] [FOOTNOTE:20][ENDFOOTNOTE] [FOOTNOTE:21][ENDFOOTNOTE] [FOOTNOTE:22][ENDFOOTNOTE] [FOOTNOTE:23][ENDFOOTNOTE] [FOOTNOTE:24][ENDFOOTNOTE] ###### Abstract We report a measurement of the Type Ia supernova (SN Ia) rate in galaxy clusters at $0.9<z<1.46$ from the _Hubble Space Telescope (HST)_ Cluster Supernova Survey. This is the first cluster SN Ia rate measurement with detected $z>0.9$ SNe. Finding $8\pm 1$ cluster SNe Ia, we determine a SN Ia rate of $0.50^{+0.23}_{-0.19}$ (stat) ${}^{+0.10}_{-0.09}$ (sys) $h_{70}^{2}$ SNuB (SNuB $\equiv 10^{-12}$ SNe $L_{\odot,B}^{-1}$ yr${}^{-1}$). In units of stellar mass, this translates to $0.36^{+0.16}_{-0.13}$ (stat) ${}^{+0.07}_{-0.06}$ (sys) $h_{70}^{2}$ SNuM (SNuM $\equiv 10^{-12}$ SNe $M_{\odot}^{-1}$ yr${}^{-1}$). This represents a factor of $\approx 5\pm 2$ increase over measurements of the cluster rate at $z<0.2$. We parameterize the late-time SN Ia delay time distribution with a power law: $\Psi(t)\propto t^{s}$. Under the approximation of a single-burst cluster formation redshift of $z_{f}=3$, our rate measurement in combination with lower-redshift cluster SN Ia rates constrains $s=-1.41^{+0.47}_{-0.40}$, consistent with measurements of the delay time distribution in the field. This measurement is generally consistent with expectations for the “double degenerate” scenario and inconsistent with some models for the “single degenerate” scenario predicting a steeper delay time distribution at large delay times. We check for environmental dependence and the influence of younger stellar populations by calculating the rate specifically in cluster red-sequence galaxies and in morphologically early-type galaxies, finding results similar to the full cluster rate. Finally, the upper limit of one host-less cluster SN Ia detected in the survey implies that the fraction of stars in the intra-cluster medium is less than 0.47 ($95\%$ confidence), consistent with measurements at lower redshifts. Subject headings:Supernovae: general — white dwarfs — cosmology: observations † [FOOTNOTE:†][ENDFOOTNOTE] ## 1. Introduction Type Ia supernovae (SNe Ia) are widely accepted to be the result of the thermonuclear explosion of a carbon-oxygen (CO) white dwarf (WD). The explosion is believed to occur as the WD nears the Chandrasekhar mass by accreting mass from its companion star in a binary system. Despite the confidence in this basic model, many uncertainties remain about the process that leads to SNe Ia (see Livio, 2001, for a review). Chief amongst them is the nature of the companion donor star. The leading models fall into two classes: the _single degenerate_ scenario (SD; Whelan & Iben, 1973), and the _double degenerate_ scenario (DD; Iben & Tutukov, 1984; Webbink, 1984). In the SD scenario the companion is a red giant or main sequence star that overflows its Roche lobe. In the DD scenario, the companion is a second WD which merges with the primary after orbital decay due to the emission of gravitational radiation. A better understanding of the SN Ia progenitor is demanded from both an astrophysical and a cosmological perspective. Astrophysically, SNe Ia dominate the production of iron (e.g., Matteucci & Greggio, 1986; Tsujimoto et al., 1995; Thielemann et al., 1996) and provide energy feedback (Scannapieco et al., 2006) in galaxies. Knowledge of the SN Ia rate is necessary to include these effects in galaxy evolution models. However, an accurate prediction of the SN Ia rate in galaxies of varying ages, masses and star formation histories requires a good understanding of the nature of the progenitor. This is particularly true for higher redshifts where direct SN rate constraints are unavailable. From a cosmological perspective, the progenitor has become a central concern following the use of SNe Ia as standardizable candles in the discovery of dark energy (Riess et al., 1998; Perlmutter et al., 1999). With hundreds of SNe now being used in the precision measurement of cosmological parameters (e.g., Hicken et al., 2009; Amanullah et al., 2010), astrophysical sources of systematic error will soon become significant. While the unknown nature of the SN progenitor system is unlikely to bias measurements at the current level of uncertainty (Yungelson & Livio, 2000; Sarkar et al., 2008), it could become a significant source of uncertainty in the future, as it leaves open the question of whether high-redshift SNe are different than low-redshift SNe in a way that affects the inferred distance. Measuring the SN Ia rate as a function of environment has long been recognized as one of the few available methods for probing the SN Ia progenitor (e.g., Ruiz-Lapuente et al., 1995; Ruiz-Lapuente & Canal, 1998; Yungelson & Livio, 2000). SN Ia rates constrain the progenitor scenario via the delay time distribution (DTD), where “delay time” refers to the time between star formation and SN Ia explosion. The DTD is the distribution of these times for a population of stars, and is equivalent to the SN Ia rate as a function of time after a burst of star formation. The delay time is governed by different physical mechanisms in the different progenitor scenarios. For example, in the SD scenario, when the donor is a red giant star the delay time is set by the time the companion takes to evolve off the main sequence. In the DD scenario, it is dominated by the time the orbit takes to decay due to gravitational radiation. The result is that the shape of the DTD depends on the progenitor scenario. However, the interpretation of the DTD is complicated by its dependence on other factors, not all of which are completely understood. These include the initial mass function (IMF) of the stellar population, the distribution of initial separation and mass ratio in binary systems, and the evolution of the binary through one or more common envelope (CE; see, e.g., Yungelson, 2005) phases. Theoretical delay time distributions were computed analytically following the proposal of both the SD (Greggio & Renzini, 1983) and DD (Tornambe & Matteucci, 1986; Tornambe, 1989) scenarios. Later, theoretical DTDs were extended to include various subclasses of each model and a wider range of parameters (Tutukov & Yungelson, 1994; Yungelson & Livio, 2000; Matteucci & Recchi, 2001; Belczynski et al., 2005; Greggio, 2005). In various recent numerical simulations, different plausible prescriptions for the initial conditions and for the binary evolution have lead to widely ranging DTDs, even within one scenario (Hachisu et al., 2008; Kobayashi & Nomoto, 2009; Ruiter et al., 2009; Mennekens et al., 2010). A measurement of the DTD then must constrain not only the relative contribution of various progenitor scenarios, but also the initial conditions and CE phase, which is particularly poorly constrained. Still, most simulations show a difference in the DTD shape between the SD and DD scenarios. In both scenarios, the SN rate is greatest shortly after star formation and gradually decreases with time. However, the SD scenario typically shows a strong drop off in the SN rate at large delay times not seen in the DD scenario (but see Hachisu et al., 2008). The DTD can be measured empirically from the SN Ia rate in stellar populations of different ages. Measurements correlating SN rate with host star formation rate or star formation history have now confirmed that the delay time spans a wide range, from less than 100 Myr (e.g., Aubourg et al., 2008) to many Gyr (e.g., Schawinski, 2009). Correlations with star formation rates (Mannucci et al., 2005, 2006; Sullivan et al., 2006; Pritchet et al., 2008) show that SNe with progenitor ages $\lesssim$ a few hundred Myr comprise perhaps $\sim$50% of all SNe Ia. Measurements as a function of stellar age (Totani et al., 2008; Brandt et al., 2010), show that the rate declines with delay time as expected. It is more straightforward to extract the DTD in stellar populations with a narrow range of ages (with a single burst of star formation being the ideal). Galaxy clusters, which are dominated by early-type galaxies, provide an ideal environment for constraining the shape of the DTD at large delay times. Early-type galaxies are generally expected to have formed early ($z\gtrsim 2$) with little star formation since (Stanford et al., 1998; van Dokkum et al., 2001). Cluster early-type galaxies in particular form even earlier than those in the field, with most star formation occurring at $z\gtrsim 3$(Thomas et al., 2005; Sánchez-Blázquez et al., 2006; Gobat et al., 2008). Measuring the cluster SN Ia rate over a range of redshifts from $z=0$ to $z>1$ provides a measurement of the SN Ia rate at delay times from $\sim$2 to 11 Gyr. Obtaining an accurate rate at the highest-possible redshift is crucial for constraining the shape of the late-time DTD: a larger redshift range corresponds to a larger lever arm in delay time. In addition to DTD constraints, there are also strong motivations for measuring the cluster SN Ia rate from a perspective of cluster studies. SNe Ia are an important source of iron in the intracluster medium (e.g., Loewenstein, 2006). Cluster SN rates constrain the iron contribution from SNe and, paired with measured iron abundances, can also constrain possible enrichment mechanisms (Maoz & Gal-Yam, 2004). The high-redshift cluster rate is particularly important: measurements show that most of the intracluster iron was produced at high redshift (Calura et al., 2007). The poorly-constrained high-redshift cluster rate is one of the largest sources of uncertainty in constraining the metal-loss fraction from galaxies (Sivanandam et al., 2009). Cluster SNe Ia can also be used to trace the diffuse _intracluster_ stellar component. Intracluster stars, bound to the cluster potential rather than individual galaxies, have been found to account for anywhere from $5\%$ to $50\%$ of the stellar mass in clusters (e.g., Ferguson et al., 1998; Feldmeier et al., 1998; Gonzalez et al., 2000; Feldmeier et al., 2004; Lin & Mohr, 2004; Zibetti et al., 2005; Gonzalez et al., 2005; Krick et al., 2006; Mihos et al., 2005). The use of SNe Ia as tracers of this component was first demonstrated by Gal-Yam et al. (2003) who found two likely host-less SNe Ia out of a total of seven cluster SNe Ia in $0.06<z<0.19$ Abell clusters. After correcting for the greater detection efficiency of host-less SNe, they determined that on average, the intracluster medium contained $20^{+20}_{-12}$% of the total cluster stellar mass. The intrinsic faintness of the light from intracluster stars, combined with $(1+z)^{4}$ surface brightness dimming, makes surface brightness measurements impossible at redshifts much higher than $z=0.3$. Type Ia supernovae, which are detectable up to and beyond $z=1$, provide a way to measure the intracluster stellar component and its possible evolution with redshift. The cluster SN Ia rate has recently been measured at lower redshifts ($z>0.3$) in several studies (Sharon et al., 2007; Mannucci et al., 2008; Dilday et al., 2010), and at intermediate redshift ($z\sim 0.6$) by Sharon et al. (2010). However, at higher redshifts ($z\gtrsim 0.8$), only weak constraints on the high-redshift cluster Ia rate exist, based on 1–2 SNe Ia at $z=0.83$(Gal-Yam et al., 2002). In this paper, we calculate the SN Ia rate in $0.9<z<1.46$ clusters observed in the _HST_ Cluster Supernova Survey. We address the host-less SN Ia fraction, and use our result to place constraints on the late-time DTD in clusters. Maoz et al. (2010, hereafter Maoz10) have already combined our results with iron abundance measurements and rate measurements in other environments to place even tighter constraints on the SN Ia DTD. This paper is organized as follows. In §2 we review the survey, placing particular emphasis on the aspects relevant to the rate calculation. In §3 we describe the selection of supernova candidates used in this rate calculation and the determination of supernova type for these candidates. In §4 we carry out efficiency studies to determine the detection efficiency of our SN selection. In §5 we measure the luminosity of the clusters based on data from the survey. In §6 we present results and characterize systematic errors. We discuss interpretations for the delay time distribution and conclude in §7. Throughout the paper we use a cosmology with $H_{0}=70$ km s${}^{-1}$ Mpc${}^{-1}$, $\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$. Unless otherwise noted, magnitudes are in the Vega system. This paper is one of a series of ten papers that report supernova results from the _HST_ Cluster Supernova Survey (PI: Perlmutter, _HST_ program GO-10496), a survey to discover and follow SNe Ia in very distant clusters. Paper I (Dawson et al., 2009, hereafter Dawson09) describes the survey strategy and discoveries. This work, Paper II, reports on the SN Ia rate in clusters. Paper III (Meyers et al., 2011, hereafter Meyers11) addresses the properties of the galaxies that host SNe Ia. Paper IV (Ripoche et al., 2011) introduces a new technique to calibrate the zeropoint of the NICMOS camera at low counts rates, critical for placing NICMOS-observed SNe Ia on the Hubble diagram. Paper V (Suzuki et al., 2011) reports the SNe Ia lightcurves and cosmology from the _HST_ Cluster SN Survey program. Paper VI (Barbary et al., 2011) reports on the volumetric field SN Ia rate. Melbourne et al. (2007), one of several unnumbered papers in the series, present a Keck adaptive optics observation of a $z=1.31$ SN Ia in $H$-band. Barbary et al. (2009) report the discovery of the extraordinary luminous supernova, SN SCP06F6. Morokuma et al. (2010) presents the spectroscopic follow-up observations for SN candidates. Finally, Hsiao et al. (in preparation) develop techniques to remove problematic artifacts remaining after the standard STScI pipeline. A separate series of papers, ten to date, reports on cluster studies from the survey: Hilton et al. (2007); Eisenhardt et al. (2008); Jee et al. (2009); Hilton et al. (2009); Huang et al. (2009); Rosati et al. (2009); Santos et al. (2009); Strazzullo et al. (2010); Brodwin et al. (2011); Jee et al. (2011). ## 2. The Survey ID \| Cluster \| Redshift \| R.A. (J2000) \| Decl. (J2000) \| Discovery ---\|---\|---\|---\|---\|--- A \| XMMXCS J2215.9-1738 \| 1.46 \| 22h15m59s.0 \| −17∘37′59′′ \| X-ray B \| XMMU J2205.8-0159 \| 1.12 \| 22h05m50s.6 \| −01∘59′30′′ \| X-ray C \| XMMU J1229.4+0151 \| 0.97 \| 12h29m29s.2 \| +01∘51′21′′ \| X-ray D \| RCS J0221.6-0347 \| 1.02 \| 02h21m42s.2 \| −03∘21′52′′ \| Optical E \| WARP J1415.1+3612 \| 1.03 \| 14h15m11s.1 \| +36∘12′03′′ \| X-ray F \| ISCS J1432.4+3332 \| 1.11 \| 14h32m28s.1 \| +33∘33′00′′ \| IR-Spitzer G \| ISCS J1429.3+3437 \| 1.26 \| 14h29m17s.7 \| +34∘37′18′′ \| IR-Spitzer H \| ISCS J1434.4+3426 \| 1.24 \| 14h34m28s.6 \| +34∘26′22′′ \| IR-Spitzer I \| ISCS J1432.6+3436 \| 1.34 \| 14h32m38s.8 \| +34∘36′36′′ \| IR-Spitzer J \| ISCS J1434.7+3519 \| 1.37 \| 14h34m46s.0 \| +35∘19′36′′ \| IR-Spitzer K \| ISCS J1438.1+3414 \| 1.41 \| 14h38m08s.2 \| +34∘14′13′′ \| IR-Spitzer L \| ISCS J1433.8+3325 \| 1.37 \| 14h33m51s.1 \| +33∘25′50′′ \| IR-Spitzer M \| Cl J1604+4304 \| 0.90 \| 16h04m23s.8 \| +43∘04′37′′ \| Optical N \| RCS J0220.9-0333 \| 1.03 \| 02h20m55s.5 \| −03∘33′10′′ \| Optical P \| RCS J0337.8-2844 \| 1.1aaphotometric redshift \| 03h37m51s.2 \| −28∘44′58′′ \| Optical Q \| RCS J0439.6-2904 \| 0.95 \| 04h39m37s.6 \| −29∘05′01′′ \| Optical R \| XLSS J0223.0-0436 \| 1.22 \| 02h23m03s.4 \| −04∘36′14′′ \| X-ray S \| RCS J2156.7-0448 \| 1.07 \| 21h56m42s.2 \| −04∘48′04′′ \| Optical T \| RCS J1511.0+0903 \| 0.97 \| 15h11m03s.5 \| +09∘03′09′′ \| Optical U \| RCS J2345.4-3632 \| 1.04 \| 23h45m27s.2 \| −36∘32′49′′ \| Optical V \| RCS J2319.8+0038 \| 0.90 \| 23h19m53s.4 \| +00∘38′13′′ \| Optical W \| RX J0848.9+4452 \| 1.26 \| 08h48m56s.4 \| +44∘52′00′′ \| X-ray X \| RDCS J0910+5422 \| 1.10 \| 09h10m45s.1 \| +54∘22′07′′ \| X-ray Y \| RDCS J1252.9-2927 \| 1.24 \| 12h52m54s.4 \| −29∘27′17′′ \| X-ray Z \| XMMU J2235.3-2557 \| 1.39 \| 22h35m20s.8 \| −25∘57′39′′ \| X-ray References. – A (Stanford et al., 2006; Hilton et al., 2007); B,C (Bohringer et al., 2005; Santos et al., 2009); D (also known as RzCS 052; Andreon et al., 2008a, b); D, N, U (Gilbank et al. in prep); E (Perlman et al., 2002); F (Elston et al., 2006); G, I, J, L (Eisenhardt et al., 2008); L (Brodwin et al. in prep; Stanford et al. in prep); H (Brodwin et al., 2006); K (Stanford et al., 2005); M (Postman et al., 2001); Q (Cain et al., 2008); R (Andreon et al., 2005; Bremer et al., 2006); S (Hicks et al., 2008); V (Gilbank et al., 2008); W (Rosati et al., 1999); X (Stanford et al., 2002); Y (Rosati et al., 2004); Z (Mullis et al., 2005; Rosati et al., 2009). Note. – Cluster positions differ slightly from those reported in Dawson09 due to the use of an updated algorithm for determining cluster centers. Table 1 Cluster positions and redshifts The details of the _HST_ Cluster SN Survey are described in Dawson09. Here, we briefly summarize the survey and highlight the details relevant to the rate calculation. The survey targeted 25 massive galaxy clusters in a rolling search between July 2005 and December 2006. Clusters were selected from X-ray, optical and IR surveys and cover the redshift range $0.9<z<1.46$. Twenty-four of the clusters have spectroscopically confirmed redshifts and the remaining cluster has a photometric redshift estimate. Cluster positions, redshifts and discovery methods are listed in Table 1. Note that cluster positions differ slightly from those reported in Dawson09 due to the use of an updated algorithm for determining cluster centers. During the survey, each cluster was observed once every 20 to 26 days during its _HST_ visibility window (typically four to seven months). Figure 1 shows the dates of visits to each cluster. Each visit consisted of four exposures in the F850LP filter (hereafter $z_{850}$). Most visits also included a fifth exposure in the F775W filter (hereafter $i_{775}$). We revisited clusters D, N, P, Q, R and Z towards the end of the survey when they became visible again. <figure><img src="content_image/1010.5786/x1.png"><figcaption>Figure 1.— Dates of visits to each cluster. All visits included z850 exposures(usually four). Most visits also included one i775 exposure. Filled circlesindicate “search” visits (used for finding SNe). Open circles indicate“follow-up” visits (contingent on the existence of an active SN candidate).Clusters D, N, P, Q and R were re-visited once towards the end of the survey,with additional follow-up visits devoted to clusters in which promising SNcandidates were found (N, Q, R).</figcaption></figure> Immediately following each visit, the four $z_{850}$ exposures were cosmic ray-rejected and combined using MultiDrizzle(Fruchter & Hook, 2002; Koekemoer et al., 2002) and searched for supernovae. Following the technique employed in the earliest Supernova Cosmology Project searches (Perlmutter et al., 1995, 1997), we used the initial visit as a reference image, flagged candidates with software and then considered them by eye. Likely supernovae were followed up spectroscopically using pre-scheduled time on the Keck, and Subaru telescopes and target-of-opportunity observations on VLT. For nearly all SN candidates, either a live SN spectrum or host galaxy spectrum was obtained. In many cases, spectroscopy of cluster galaxies was obtained contemporaneously using slit masks. Candidates deemed likely to be at higher redshift ($z>1$) were also observed with the NICMOS camera on _HST_, but these data are not used in this work. A number of visits were contingent on the existence of an active SN. At the end of a cluster’s visibility window, the last two scheduled visits were cancelled if there was no live SN previously discovered. This is because a SN discovered on the rise in either of the last two visits could not be followed long enough to obtain a cosmologically useful light curve. In addition, supplementary visits between pre-scheduled visits were occasionally added to provide more complete light curve information for SNe (in the case of clusters A, C, Q, and U). We call all visits contingent on the existence of an active SN “follow-up” visits (designated by open circles in Fig. 1). ## 3. Supernova Selection During the survey, our aim was to find as many supernovae as possible and find them as early as possible in order to trigger spectroscopic and NICMOS follow-up. Thus, software thresholds for flagging candidates for consideration were set very low, and all possible supernovae were carefully considered by a human screener. Over the course of the survey, thresholds were changed and the roster of people scanning the subtractions changed. As a result, the initial candidate selection process was inclusive but heterogeneous, and depended heavily on human selection. This made it difficult to calculate a selection efficiency for the SN candidates selected during the survey (listed in Tables 3 and 4 of Dawson09). In this section, we select an independent SN candidate sample (without regard for the Dawson09 sample) using automated selection wherever possible. Although the remainder of this paper will focus on cluster SNe, candidates are selected without regard for cluster membership (which is only known from follow-up spectroscopy once the candidate has already been found) and we determine SN types for both cluster and non-cluster SNe. The non-cluster SNe are considered further in a second paper deriving the volumetric SN Ia field rate (Barbary et al., in preparation). The automated selection consists of initial detection in pairs of subtracted images (§3.1; 86 candidates selected), and subsequent requirements based on the light curve of each candidate (§3.2; 60 candidates remaining). The selection efficiency for these two steps is later calculated via a Monte Carlo simulation. In §3.3 we assign a type (SN Ia, core-collapse SN, or other) to each of the remaining 60 candidates based on all data available (including triggered follow-up observations). For this last step we do not calculate an efficiency or completeness. Instead we estimate the classification uncertainty of the assigned type for each candidate individually. For most candidates the uncertainty in the type is negligible thanks to ample photometric and spectroscopic data. ### Initial detection For the purpose of initially detecting candidates, we use only “search” visits (filled circles in Fig. 1) and disregard the “follow-up” visits (open circles in Fig. 1). (In the following section we will use any available “follow-up” visits to construct more complete light curves for the candidates discovered in this section.) We use the MultiDrizzle-combined, cosmic ray-rejected, $z_{850}$ image from each “search” visit. We consider only regions in this image that are covered by three or more $z_{850}$ exposures. With less than three exposures, the combined images are too heavily contaminated by cosmic rays to be practically searchable for SNe. Although there are typically four $z_{850}$ exposures, the dither pattern used in the survey means that not all regions of the combined image have four exposures. The ACS camera is a mosaic of two $2048\times 4096$ pixel CCD chips (1 pixel = $0.05^{\prime\prime}$) separated by $2.5^{\prime\prime}$. The $z_{850}$ exposures were dithered to cover this gap, meaning that a $5^{\prime\prime}$ wide region in the center of the image and $2.5^{\prime\prime}$ wide regions on either side of the image are only covered by two exposures and thus are not searchable. Due to orbital constraints, the position angle of _HST_ changes between each visit. This means that the unsearchable “gap” region rotates over the field between visits, and that the outer parts of the field are observed in some visits, but not others (Fig. 2, second row). The regions around bright stars are also considered “not searchable” and are similarly masked. <figure><img src="content_image/1010.5786/x2.png"><figcaption>Figure 2.— An example of image orientation and searchable regions for clusterISCS J1432.4+3332. Each column represents an observation of the cluster. Thefirst row is the z850 image for that visit. The second row is the part of thatimage that is searchable. The third row shows the searchable area of thestacked reference image used in the subtraction for this visit. The fourth rowis the searchable area in the subtraction (the intersection of the second andthird rows).</figcaption></figure> For each “search” visit to each cluster, we follow these four steps: 1. A reference image is made* by combining images from other visits to the cluster. All visits that are either 50 or more days before the search epoch or 80 or more days after the search epoch are included. If there are no epochs outside this 130 day range, the range is narrowed symmetrically until one epoch qualifies. Masked pixels in each visit’s image do not contribute to the stacked reference image (Fig. 2, third row). 2. A subtracted image is made by subtracting the stacked reference image from the search epoch image. A map of the sky noise level in the subtraction is made by considering the noise level of the search epoch image and the noise level of each reference image contributing to a given region. Any area masked in either the search epoch or stacked reference image is masked in the subtracted image (Fig. 2, fourth row). 3. Candidates in the subtraction are identified by software. To be flagged, a candidate must have three contiguous pixels with a flux 3.4 times the local sky noise level in the subtraction (as determined by the sky noise map above). Once flagged, it must fulfill the following four requirements: * MultiDrizzle-combined image: A total signal-to-noise ratio (including sky and Poisson noise) of 5 or more in a 3 pixel radius aperture. * MultiDrizzle-combined image: A total signal-to-noise ratio of 1.5 or more in a 10 pixel radius aperture. * Individual exposures: A signal-to-noise ratio of 1 or greater in a 3 pixel radius aperture in three or more individual exposures. * Individual exposures: A candidate cannot have an individual exposure with a flux more than $20\sigma$ greater than the flux in the lowest flux exposure _and_ a second individual exposure with flux more than $10\sigma$ greater than the flux in the lowest flux exposure. The first requirement is designed to eliminate low significance detections on bright galaxies. The second requirement helps eliminate dipoles on bright galaxy cores caused by slight image misalignment. The third and fourth requirements are aimed at false detections due to cosmic ray coincidence. They require the candidate to be detected in most of the exposures and allow no more than one exposure to be greatly affected by a cosmic ray. On the order of five to ten candidates per subtraction pass all the requirements, resulting in approximately 1000 candidates automatically flagged across the 155 search visits. 4. Each candidate is evaluated by eye in the subtraction. Because the position angle changes between each epoch, the orientation of stellar diffraction spikes changes, causing the majority of the false detections. These are easy to detect and eliminate by eye. Occasionally there are mis-subtractions on the cores of bright galaxies that pass the above requirements. Only completely unambiguous false detections are eliminated in this step. If there is any possibility the candidate is a real SN, it is left in the sample for further consideration. After carrying out the above four steps for all 155 search visit, 86 candidates remain. At this point, candidates have been selected based only on information from a single $z_{850}$ subtraction. Detailed information on each of the 86 candidates is available from the _HST_ Cluster SN Survey website¹. [FOOTNOTE:1][ENDFOOTNOTE] ### Lightcurve Requirements The 86 remaining candidates still include a considerable number of non-SNe. We wish to trim the sample down as much as possible in an automated way, so that we can easily calculate the efficiency of our selection. For each candidate, we now make three further automated requirements based on $i_{775}$ data (if available) and the shape of the $z_{850}$ light curve. The requirements and number of candidates remaining after each requirement are summarized in Table 2. Requirement \| Candidates Remaining ---\|--- Before light curve requirements \| 86 Positive i775 flux (if observed in i775) \| 81 2σ Detection in surrounding epochs \| 73 If declining, Require two 5σ detections \| 60 Table 2 Light Curve Requirements First, we require that if $i_{775}$ data exists for the epoch in which the candidate was detected, there be positive flux in a 2 pixel radius aperture at the candidate location in the $i_{775}$ image. From our SN light curve simulations, we find that virtually all SNe should pass (near maximum light there is typically enough SN flux in the $i_{775}$ filter to result in a positive total flux, even with large negative sky fluctuations). Meanwhile, about half of the cosmic rays located far from galaxies will fail this test (due to negative sky fluctuations). If there is no $i_{775}$ data for the detection epoch, this requirement is not applied. Even though nearly all SNe are expected to pass, we account for any real SNe that would be removed in our Monte Carlo simulation. Second, we require that the light curve does not rise and fall too quickly: if there is a “search” visit less than 60 days before the detection visit and also one less than 60 days after the detection visit, the candidate must be detected at a $2\sigma$ level in at least one of these two visits. SNe Ia have light curves wide enough to be detected at this level in two epochs spaced apart by 60 days. However, cosmic rays in one $z_{850}$ image are unlikely to be repeated in the same spot in two epochs and thus will be removed. This requirement is also included in our Monte Carlo simulation. The third and final requirement aims to eliminate candidates that were significantly detected in only the first epoch and that then faded from view. Such candidates would not have been followed up spectroscopically and it would typically be impossible to tell if such candidates were SNe (and if so, Type Ia or core collapse) on the basis of a single detection. We chose to eliminate any such candidates and account for this elimination in our Monte Carlo simulation, rather than dealing with an “untypeable” candidate. Specifically, if a candidate is found on the decline (in the first search epoch), we require two epochs with $5\sigma$ detections. For high-redshift ($z\sim 1$) SNe Ia, this requirement means that the first epoch will be at approximately maximum light, and most of the SN decline is captured, making it possible to confirm a SN and estimate a type. For candidates that are only significantly detected in the last search epoch, typing is not a problem because additional ACS orbits were typically scheduled in order to follow such candidates. After these requirements 60 candidates remain. The automatic selection means that we can easily calculate the completeness of the selection so far; any real SNe Ia removed will be accounted for in the “effective visibility time” (§4) which is calculated using a Monte Carlo simulation. ### Typing We now use all available information about each candidate (spectroscopic confirmation, host galaxy redshift, all light curve information, as well as host galaxy luminosity and color) to classify each of the 60 remaining candidates as image artifact, active galactic nucleus (AGN), core-collapse SN (SN CC), or SN Ia. #### 3.3.1 Image artifacts Although the automated selections were designed to eliminate image artifacts such as subtraction residuals and cosmic rays, they were made to be somewhat tolerant so that real SNe were not eliminated. The result is that some artifacts slip through. Candidates located close to the cores of relatively bright galaxies that show adjoining negative and positive areas in subtractions are likely to be caused by mis-alignment between the reference and search image. For such candidates, we inspected the full light curve for consistency with the general shape of a SN Ia light curve. For fourteen of these, the light curve is completely inconsistent with that of a SN Ia. Their light curves have either multiple peaks, long flat portions followed by one or two lower points, and/or $i_{775}$ data that shows no change. We classify these fourteen candidates as subtraction residuals with negligible classification uncertainty (very unlikely that any are SNe Ia). Candidates where one or two of the four $z_{850}$ exposures was clearly affected by a cosmic ray or hot pixel may be false detections. These can pass the automated cosmic ray rejection when they occur on a galaxy. For two such candidates, we used the lack of any change in the $i_{775}$ light curve to rule out a SN Ia: fitting SN templates with a range of redshifts and extinctions resulted in observed $i_{775}$ fluxes too low by $4\sigma$ or more, given the $z_{850}$ increase. One other candidate, SCP06W50, is less certain. It was discovered in the last visit to the cluster, making it difficult to constrain a template light curve. There is clearly a hot pixel or cosmic ray in one $z_{850}$ exposure, but there appears to be some excess flux in the other three exposures as well. Also, there is a point-source like detection in $i_{775}$, but offset $\sim$1.2 pixels from the $z_{850}$ detection. While the $i_{775}$ detection may also be a cosmic ray, it is possible that this candidate is a SN caught very early. The elliptical “host” galaxy was not observed spectroscopically, but we estimate its redshift to be $0.60<z<0.85$ based on the color of $i_{775}-z_{850}=0.55$ and stellar population models of Bruzual & Charlot (2003, hereafter BC03). Of the 17 total candidates classified as image artifacts, SCP06W50 is the only one with significant uncertainty. However, this uncertainty does not affect the cluster SN Ia rate as the host galaxy is clearly in the cluster foreground. #### 3.3.2 Agn <figure><img src="content_image/1010.5786/x3.png"><figcaption>Figure 3.— Images and light curves of four of the 14 candidates classified asAGN. For each candidate, the upper left panel shows the two-color stackedimage (i775 and z850) of the host galaxy, with the position of the transientindicated. The three smaller panels below the stacked image show thereference, new, and subtracted images for the discovery visit. The right panelshows the light curve at the SN position (including host galaxy light) in thez850 (top) and i775 (bottom) bands. The y axes have units of counts per secondin a 3 pixel radius aperture. The effective zeropoints are 23.94 and 25.02 forz850 and i775, respectively. The discovery visit is indicated with an arrow inthe z850 plot.</figcaption></figure> Candidates positioned directly on the cores of their host galaxies may be AGN. Four such candidates were spectroscopically confirmed as AGN: SCP06L22 ($z=1.369$), SCP06V6 ($z=0.903$) and SCP05X13 ($z=1.642$) and SCP06U3 ($z=1.534$). A fifth candidate, SCP06F3, is spectroscopically consistent with an AGN at $z=1.21$, but is less certain (see spectroscopy reported in Morokuma et al., 2010). SCP06L22, SCP05X13, SCP06U3 and SCP06F3 also have light curves that are clearly inconsistent with SNe Ia (observer frame rise times of 100 days or more, or declining phases preceding rising phases). Of the “on core” candidates that were not observed spectroscopically, five exhibit light curves that decline before rising or have rise times of 100 days or more. A sixth candidate, SCP06Z51 exhibited slightly varying fluxes that could be due to either subtraction residuals or an AGN. However, its light curve is clearly inconsistent with a SN Ia, especially considering the apparent size, magnitude and color of the host galaxy. Summarizing, there are 11 “on-core” candidates certain not to be SNe Ia. Three other “on-core” candidates are also considered likely AGN on the basis of their light curves: SCP06Z50, SCP06U50 and SCP06D51. These three candidates are shown in Figure 3. SCP06Z50 (Fig. 3, top left), has a rise-fall behavior in the first three $z_{850}$ observations of its light curve that _could_ be consistent with a SN Ia light curve. However, given that the host galaxy is likely at $z\lesssim 1$ based on its magnitude and color, the SN would be fainter than a normal SN Ia by 1 magnitude or more. Considering the proximity to the galaxy core and the additional variability seen in the last two observations, SCP06Z50 is most likely an AGN. The light curve of candidate SCP06U50 (Fig. 3, top right) also exhibits a rise-fall that could be consistent with a supernova light curve. However, its host is morphologically elliptical and likely at $z\lesssim 0.7$ based on its color. At $z\lesssim 0.7$, a SN Ia would have to be very reddened ($E(B-V)\gtrsim 1$) to match the color and magnitude of the SCP06U50 light curve. As this is very unlikely (considering that the elliptical host likely contains little dust), we conclude that SCP06U50 is also most likely an AGN. Finally, SCP06D51 (Fig. 3, bottom left) was discovered in the last visit, on the core of a spiral galaxy. We classify it as an AGN based on the earlier variability in the light curve. As these galaxies are all most likely in the cluster foregrounds, even the small uncertainty in these classifications is not a concern for the cluster rate calculation here. Note that one of the candidates classified here as a clear AGN, SCP06U6, was reported as a SN with unknown redshift by Dawson09, due to the fact that spectroscopy revealed no evidence of an AGN. However, it is on the core of a compact galaxy, and has a clear $\gtrsim 100$ day rise in both $z_{850}$ and $i_{775}$ (Fig. 3, bottom right). While it could possibly be a very peculiar SN with a long rise time, what is important for this analysis is that it is clearly not a SN Ia. #### 3.3.3 Supernovae After removing 17 image artifacts and 14 AGN, 29 candidates remain (listed in Table 3). One of these is the peculiar transient SCP 06F6 (also known as SN SCP06F6) reported by Barbary et al. (2009). Various explanations have been considered by, e.g., Gänsicke et al. (2009), Soker et al. (2010) and Chatzopoulos et al. (2009). It appears that SCP 06F6 may be a rare type of supernova, with redshift $z=1.189$(Quimby et al., 2011; Pastorello et al., 2010). While its precise explanation is still uncertain, the important fact for this analysis is that SCP 06F6 is clearly not a SN Ia. ID \| Nickname \| R.A. (J2000) \| Decl. (J2000) \| z \| SN Type \| Confidence \| Typing ---\|---\|---\|---\|---\|---\|---\|--- _Cluster Members_ SN SCP06C1 \| Midge \| 12h29m33s.012 \| +01∘51′36′′.67 \| 0.98 \| Ia \| secure \| a,c SN SCP05D0 \| Frida \| 02h21m42s.066 \| −03∘21′53′′.12 \| 1.014 \| Ia \| secure \| a,b,c SN SCP06F12 \| Caleb \| 14h32m28s.748 \| +33∘32′10′′.05 \| 1.11 \| Ia \| probable \| c SN SCP06H5 \| Emma \| 14h34m30s.139 \| +34∘26′57′′.29 \| 1.231 \| Ia \| secure \| b,c SN SCP06K18 \| Alexander \| 14h38m10s.663 \| +34∘12′47′′.19 \| 1.412 \| Ia \| probable \| b SN SCP06K0 \| Tomo \| 14h38m08s.366 \| +34∘14′18′′.08 \| 1.416 \| Ia \| secure \| b,c SN SCP06R12 \| Jennie \| 02h23m00s.082 \| −04∘36′03′′.04 \| 1.212 \| Ia \| secure \| b,c SN SCP06U4 \| Julia \| 23h45m29s.429 \| −36∘32′45′′.73 \| 1.05 \| Ia \| secure \| a,c _Cluster Membership Uncertain_ SN SCP06E12 \| Ashley \| 14h15m08s.141 \| +36∘12′42′′.94 \| \| Ia \| plausible \| c SN SCP06N32 \| \| 02h20m52s.368 \| −03∘34′13′′.32 \| \| CC \| plausible \| c _Not Cluster Members_ SN SCP06A4 \| Aki \| 22h16m01s.077 \| −17∘37′22′′.09 \| 1.193 \| Ia \| probable \| c SN SCP06B3 \| Isabella \| 22h05m50s.402 \| −01∘59′13′′.34 \| 0.743 \| CC \| probable \| c SN SCP06C0 \| Noa \| 12h29m25s.654 \| +01∘50′56′′.58 \| 1.092 \| Ia \| secure \| b,c SN SCP06C7 \| \| 12h29m36s.517 \| +01∘52′31′′.47 \| 0.61 \| CC \| probable \| c SN SCP05D6 \| Maggie \| 02h21m46s.484 \| −03∘22′56′′.18 \| 1.314 \| Ia \| secure \| b,c SN SCP06F6 \| \| 14h32m27s.394 \| +33∘32′24′′.83 \| 1.189 \| non-Ia \| secure \| a SN SCP06F8 \| Ayako \| 14h32m24s.525 \| +33∘33′50′′.75 \| 0.789 \| CC \| probable \| c SN SCP06G3 \| Brian \| 14h29m28s.430 \| +34∘37′23′′.13 \| 0.962 \| Ia \| plausible \| c SN SCP06G4 \| Shaya \| 14h29m18s.743 \| +34∘38′37′′.38 \| 1.35 \| Ia \| secure \| a,b,c SN SCP06H3 \| Elizabeth \| 14h34m28s.879 \| +34∘27′26′′.61 \| 0.85 \| Ia \| secure \| a,c SN SCP06L21 \| \| 14h33m58s.990 \| +33∘25′04′′.21 \| \| CC \| plausible \| c SN SCP06M50 \| \| 16h04m25s.300 \| +43∘04′51′′.85 \| \| \| \| SN SCP05N10 \| Tobias \| 02h20m52s.878 \| −03∘33′40′′.20 \| 0.203 \| CC \| plausible \| c SN SCP06N33 \| Naima \| 02h20m57s.699 \| −03∘33′23′′.97 \| 1.188 \| Ia \| probable \| c SN SCP05P1 \| Gabe \| 03h37m50s.352 \| −28∘43′02′′.66 \| 0.926 \| Ia \| probable \| c SN SCP05P9 \| Lauren \| 03h37m44s.512 \| −28∘43′54′′.58 \| 0.821 \| Ia \| secure \| a,c SN SCP06U7 \| Ingvar \| 23h45m33s.867 \| −36∘32′43′′.48 \| 0.892 \| CC \| probable \| c SN SCP06X26 \| Joe \| 09h10m37s.889 \| +54∘22′29′′.07 \| 1.44 \| Ia \| plausible \| c SN SCP06Z5 \| Adrian \| 22h35m24s.966 \| −25∘57′09′′.61 \| 0.623 \| Ia \| secure \| a,c Note. – Typing: (a) Spectroscopic confirmation. (b) Host is morphologically early-type, with no signs of recent star formation. (c) Light curve shape, color, magnitude consistent with type. We do not assign a type for SCP06M50 because there is significant uncertainty that the candidate is a SN at all. Table 3 Supernovae Note that Table 3 contains 10 fewer candidates than the list presented by Dawson09. This is unsurprising; here we have intentionally used a stricter selection than in the original search, the source for the Dawson09 sample. Still, after finalizing our selection method we checked that there were no unexpected discrepancies. Five of the Dawson09 candidates (SCP06B4, SCP06U2, SCP06X18, SCP06Q31, SCP06T1) fell just below either the detection or signal-to-noise thresholds in our selection. These were found in the original search because detection thresholds were set slightly lower, and because the images were sometimes searched in several different ways. For example, in the original search SCP06B4 was only found by searching an $i_{775}$ subtraction. Two Dawson09 candidates (SCP05D55, SCP06Z52) were found too far on the decline and failed the light curve requirements (§3.2). Three Dawson09 candidates (SCP06X27, SCP06Z13, SCP06Z53) were found while searching in “follow-up” visits, which were not searched here. SCP06U6 passed all requirements, but is classified here as an AGN, as noted above. With the exception of SCP06U6, all of these candidates are likely to be supernovae (mostly core collapse). However, the types of candidates that did not pass our requirements are not of concern for this analysis. Finally, SCP06M50 was not reported in Dawson09, but is classified here as a SN, although a highly uncertain one (discussed in detail in §3.3.4). Thanks to the extensive ground-based spectroscopic follow-up campaign, we were able to obtain spectroscopic redshifts for 25 of the 29 SNe. The redshift reported in Table 3 is derived from the SN host galaxy for all but one candidate (SCP06C1) where the redshift is from the SN spectrum itself. Of the 25 candidates with redshifts, eight are in clusters and 17 are in the field. Note that this high spectroscopic completeness is particularly important for determining the cluster or non-cluster status of each SN, which directly affects the determination of the cluster SN Ia rate. The possible cluster memberships of the four candidates lacking redshifts are discussed below. We determine the type of each of the 29 supernovae using a combination of methods in order to take into account all available information for each supernova. This includes (a) spectroscopic confirmation, (b) the host galaxy environment, and (c) the SN light curve. To qualify the confidence of each supernova’s type, we rank the type as “secure,” “probable,” or “plausible”: Secure SN Ia: Has spectroscopic confirmation or _both_ of the following: (1) an early-type host galaxy with no recent star formation and (2) a light curve with shape, color and magnitude consistent with SNe Ia and inconsistent with other types. Probable SN Ia: Fulfills either the host galaxy requirement or the light curve requirement, but not both. Plausible SN Ia: The light curve is indicative of a SN Ia, but there is not enough data to rule out other types. Secure SN CC: Has spectroscopic confirmation (note that there are no such candidates in this sample). Probable SN CC: The light curve is consistent with a core-collapse SN and inconsistent with a SN Ia. Plausible SN CC: Has a light curve indicative of a core-collapse SN, but not inconsistent with a SN Ia. This ranking system is largely comparable to the “gold,” “silver,” “bronze” ranking system of Strolger et al. (2004), except that we do not use their “UV deficit” criterion. This is because our data do not include the bluer F606W filter, and because SNe Ia and CC are only distinct in UV flux for a relatively small window early in the light curve. Below, we discuss in detail the three typing methods used. (a) Spectroscopic confirmation: During the survey, seven candidates were spectroscopically confirmed as SNe Ia (Dawson09, Morokuma et al., 2010). These seven (three of which are in clusters) are designated with an “a” in the “typing” column of Table 3. All seven candidates have a light curve shape, absolute magnitude and color consistent with a SN Ia. Although the spectroscopic typing by itself has some degree of uncertainty, the corroborating evidence from the light curve makes these “secure” SNe Ia. (b) Early-type host galaxy: The progenitors of core-collapse SNe are massive stars ($>8M_{\odot}$) with main sequence lifetimes of $<40$ Myr. Thus, core-collapse SNe only occur in galaxies with recent star formation. Early-type galaxies, having typically long ceased star formation, overwhelmingly host Type Ia SNe (e.g., Cappellaro et al., 1999; Hamuy et al., 2000). In fact, in an extensive literature survey of core-collapse SNe reported in early-type hosts, Hakobyan et al. (2008) found that only three core-collapse SNe have been recorded in early-type hosts, and that the three host galaxies in question had either undergone a recent merger or were actively interacting. In all three cases there are independent indicators of recent star formation. Therefore, in the cases where the host galaxy morphology, photometric color, and spectrum all indicate an early-type galaxy with no signs of recent star formation or interaction, we can be extremely confident that the SN type is Ia. These cases are designated by a “b” in the “typing” column of Table 3. We emphasize that in all of these cases, spectroscopy reveals no signs of recent star formation and there are no visual or morphological signs of interaction. (See Meyers11 for detailed studies of these SN host galaxy properties.) (c) Light curve: SNe Ia can be distinguished from most common types of SNe CC by some combination of light curve shape, color, and absolute magnitude. We compare the light curve of each candidate to template light curves for SN Ia and various SN CC subtypes to test if the candidate could be a SN Ia or a SN CC. For candidates lacking both spectroscopic confirmation and an elliptical host galaxy, if there is sufficient light curve data to rule out all SN CC subtypes, the candidate is considered a “probable” SN Ia. If SN Ia can be ruled out, it is considered a “probable” SN CC. If neither SN Ia nor SN CC can be ruled out, the candidate is considered a “plausible” SN Ia or SN CC based on how typical the candidate’s absolute magnitude and/or color would be of each type. This approach can be viewed as a qualitative version of the pseudo-Bayesian light curve typing approaches of, e.g., Kuznetsova & Connolly (2007); Kuznetsova et al. (2008); Poznanski et al. (98, 99). SNe classified as “probable” here would likely have a Bayesian posterior probability approaching $1$, while “plausible” SNe would have an intermediate probability (likely between 0.5 and 1.0). We consciously avoid the full Bayesian typing approach because it can obscure large uncertainties in the priors such as luminosity distributions, relative rates, light curve shapes, and SN subtype fractions. Also, the majority of our candidates have more available light curve information than those of Kuznetsova et al. (2008) and Poznanski et al. (99), making a calculation of precise classification uncertainty less necessary. In general, classification uncertainty from light curve fitting is not a concern for the cluster rate calculation as most cluster-member candidates are securely typed using methods (a) and/or (b), above. It is more of a concern for the volumetric field rate calculation based on the non-cluster candidates (Barbary et al., in preparation), though the uncertainty in the field rate is still dominated by Poisson error. For each candidate we fit template light curves for SN Ia, Ibc, II-P, II-L, and IIn. We use absolute magnitude and color as a discriminant by limiting the allowed fit ranges according to the known distributions for each subtype. For SN Ia we start with the spectral time series template of Hsiao et al. (2007), while for the core-collapse types we use templates of Nugent et al. (2002)². Each spectral time series is redshifted to the candidate redshift and warped according to the desired color. Observer-frame template light curves are then generated by synthetic photometry in the $i_{775}$ and $z_{850}$ filters. The magnitude, color, date of maximum light, and galaxy flux in $i_{775}$ and $z_{850}$ are allowed to vary to fit the light curve data. For the SN Ia template, the linear timescale or “stretch” (e.g., Perlmutter et al., 1997; Guy et al., 2005) is also allowed to vary within the range $0.6<s<1.3$. We constrain the absolute magnitude for each subtype to the range observed by Li et al. (2011); Our allowed range fully encompasses their observed luminosity functions (uncorrected for extinction) for a magnitude-limited survey for each subtype. We correct from their assumed value of $H_{0}=73$ km s${}^{-1}$ Mpc${}^{-1}$ to our assumed value of $H_{0}=70$ km s${}^{-1}$ Mpc${}^{-1}$ and $K$-correct from $R$ to $B$ band. To avoid placing too strict of an upper limit on SN CC brightness, we use the bluest maximum-light spectrum available when $K$-correcting (e.g., for SN Ibc we use a bluer spectrum than that of Nugent et al. (2002), as bluer SNe Ibc have been observed). The resulting allowed $M_{B}$ range for each subtype is shown in Table 4. Note that the range for Ibc does not include ultra-luminous SNe Ic (such as those in the luminosity functions of Richardson et al. (2002)) as none were discovered by Li et al. (2011). While such SNe can mimic a SN Ia photometrically, the Li et al. (2011) results indicate that they are intrinsically rare, and even Richardson et al. (2002) show that they make up at most $\sim$20% of all SNe Ibc. Still, we keep in mind that even candidates compatible only with our SN Ia template and incompatible with SN CC templates may in fact be ultra-luminous SNe Ic, though the probability is low. This is why any candidate typed based on light curve alone has a confidence of at most “probable,” rather than “secure.” The allowed ranges of “extinction,” $E(B-V)$, are also shown in Table 4. For SN Ia, $E(B-V)$ is the difference in $B-V$ color from the Hsiao et al. (2007) template. As the observed distribution of SNe includes SNe bluer than this template, SNe Ia as blue as $E(B-V)=-0.2$ are allowed. Given an $E(B-V)$, the spectral template is warped according to the salt color law (Guy et al., 2005), with an effective $R_{B}=2.28$(Kowalski et al., 2008). For SN CC templates, extinction as low as $E(B-V)=-0.1$ is allowed to reflect the possibility of SNe that are intrinsically bluer than the Nugent et al. (2002) templates. Templates are then warped using a Cardelli et al. (1989) law with $R_{B}=4.1$. Extinctions are limited to $E(B-V)<0.5$ (implying an extinction of $A_{B}\sim 2$ magnitudes for SNe CC). [FOOTNOTE:2][ENDFOOTNOTE] SN type \| Template \| Observed MB \| E(B−V) \| s ---\|---\|---\|---\|--- Ia \| Hsiao \| −17.5 – −20.1 \| −0.2 – 0.6 \| 0.6 – 1.3 Ibc \| Nugent \| −15.5 – −18.5 \| −0.1 – 0.5 \| 1.0 II-L \| Nugent \| −16.0 – −19.0 \| −0.1 – 0.5 \| 1.0 II-P \| Nugent \| −15.5 – −18.0 \| −0.1 – 0.5 \| 1.0 IIn \| Nugent \| −15.5 – −19.1 \| −0.1 – 0.5 \| 1.0 Table 4 SN light curve template parameter ranges The light curve template with the largest $\chi^{2}$$P$-value is generally taken as the type. We also evaluate each fit by eye to check that the best-fit template adequately describes the light curve. Figure 4 shows the best-fit template for each candidate. For candidates typed on the basis of spectroscopic confirmation or an elliptical host galaxy only the SN Ia template is shown. For candidates typed on the basis of the light curve alone, we show both the best-fit SN Ia and best-fit SN CC templates for comparison. The confidence in the best-fit template is either “probable” or “plausible” depending on how well other templates fit: If the next-best fit has a $P$-value that is smaller than $10^{-3}\times P_{\rm best}$, the best-fit template is considered the only acceptable fit and the confidence is “probable.” If the next-best fit has a $P$-value larger than $10^{-3}\times P_{\rm best}$ the confidence is “plausible.” Finally, note that the photometry used here is simple aperture photometry with fixed aperture corrections. For SN Ia cosmology we use color-dependent aperture corrections, as described in Suzuki et al. (in preparation). <figure><img src="content_image/1010.5786/x4.png"><figcaption>Figure 4.— Images and light curves of the 29 candidates classified assupernovae. For each candidate, the upper left panel shows the 2-color stackedimage (i775 and z850) of the supernova host galaxy, with the SN positionindicated. The three smaller panels below the stacked image show thereference, new, and subtracted images for the discovery visit. The right panelshows the light curve at the SN position (including host galaxy light) in thez850 (top) and i775 (bottom) bands. The y axes have units of counts per secondin a 3 pixel radius aperture. The effective zeropoints are 23.94 and 25.02 forz850 and i775, respectively. The discovery visit is indicated with an arrow inthe z850 plot. The best-fit SN Ia template is shown in blue. For cases wherethe type is SN Ia based on spectroscopic confirmation or host galaxyenvironment, only the best-fit SN Ia template is shown, to demonstrate theconsistency of the light curve with the designation. For cases where the typeis based only on the light curve fit, the best-fit core collapse SN templateis shown in red. Note that the photometry used here is simple aperturephotometry with fixed aperture corrections. For SN Ia cosmology we use color-dependent aperture corrections, as described in Suzuki et al. (inpreparation).</figcaption></figure> <figure><img src="content_image/1010.5786/x5.png"><figcaption>Figure 4.— _Continued_</figcaption></figure> <figure><img src="content_image/1010.5786/x6.png"><figcaption>Figure 4.— _Continued_</figcaption></figure> #### 3.3.4 Comments on individual SN light curves Here we comment in greater detail on a selection of individual candidates, particularly those with the greatest uncertainty in typing. For each candidate, see the corresponding panel of Figure 4 for an illustration of the candidate host galaxy and light curve. _SN SCP06E12_. We were unable to obtain a host galaxy redshift due to the faintness of the host. The color of the host galaxy is consistent with the cluster red sequence. The candidate light curve is consistent with a SN Ia at the cluster redshift of $z=1.03$, but is also consistent with SN II-L at $z=1.03$. Different SN types provide an acceptable fit over a fairly wide range of redshifts. As the SN Ia template provides a good fit with typical parameters, we classify the candidate as a “plausible” SN Ia. However, there is considerable uncertainty due to the uncertain redshift. _SN SCP06N32_ also lacks a host galaxy redshift. If the cluster redshift of $z=1.03$ is assumed, the candidate light curve is best fit by a SN Ibc template. A SN Ia template also yields an acceptable fit, but requires an unusually red color of $E(B-V)\sim 0.6$. Given the best-fit $s$ and $M_{B}$ values, the candidate would have an unusually large Hubble diagram residual of approximately $-0.8$ magnitudes. If the redshift is allowed to float, a SN Ia template with more typical parameters provides an acceptable fit at $z=1.3$. A SN Ibc template still provides a better fit, with the best fit redshift being $z\sim 0.9$. As SN Ibc provides a better fit in both cases, we classify this as a “plausible” SN CC. However, there is considerable uncertainty in both the type and cluster membership of this candidate. _SN SCP06A4_. We note that this candidate was observed spectroscopically, as reported in Dawson09. While the spectrum was consistent with a SN Ia, there was not enough evidence to conclusively assign a type. The host galaxy is morphologically and photometrically consistent with an early-type galaxy, but there is detected [OII], a possible indication of star formation. We therefore rely on light curve typing for this candidate, assigning a confidence of “probable” rather than “secure.” _SN SCP06G3_ has only sparse light curve coverage. The best fit template is a SN Ia with $s=1.3$, $E(B-V)=0.3$ and $M_{B}=-18.5$, although these parameters are poorly constrained. A large stretch and red color would not be surprising given the spiral nature of the host galaxy. It is also consistent with a II-L template, although the best fit color is unusually blue: $E(B-V)=-0.1$. Given that SN Ia yields more “typical” fit parameters and that, at $z\sim 1$ a detected SN is more likely to be Type Ia than II, we classify this as a “plausible” Type Ia, with considerable uncertainty in the type. _SN SCP06L21_ lacks a spectroscopic redshift, but has a distinct slowly-declining light curve that rules out a $z>0.6$ SN Ia light curve. Even the best-fit Ia template at $z=0.55$, shown in Fig. 4), is unusually dim ($M_{B}\approx-17.5$), making it unlikely that the candidate is a lower-redshift SN Ia. The light curve is better fit by a SN II-P template (with the best-fit redshift being $z=0.65$). We therefore classify the candidate as a “probable” SN CC. _SN SCP06M50_ is the most questionable “SN” candidate, having no obvious $i_{775}$ counterpart to the increase seen in $z_{850}$. It may in fact be an image artifact or AGN. However, it appears to be off the core of the galaxy by $\sim$2 pixels (making AGN a less likely explanation), and shows an increase in $z_{850}$ flux in two consecutive visits, with no obvious cosmic rays or hot pixels (making an image artifact less likely as well). The galaxy is likely to be a cluster member: its color and magnitude put it on the cluster red sequence, it is morphologically early-type, and it is only $19^{\prime\prime}$ from the cluster center. Under the assumption that the candidate is a supernova and at the cluster redshift of $z=0.92$, no template provides a good fit due to the lack of an $i_{775}$ detection and the constraints on $E(B-V)$. In particular, a SN Ia template would require $E(B-V)>0.6$. (The best-fit template shown in Fig. 4 is with $E(B-V)=0.6$.) If the redshift is allowed to float, it is possible to obtain a good fit at higher redshift ($z\sim 1.3$), but still with $E(B-V)\gtrsim 0.4$, regardless of the template type. Given the color and early-type morphology of the host galaxy, it is unlikely to contain much dust. There is thus no consistent picture of this candidate as a SN, and we do not assign a type. However, note that the candidate is unlikely to be a cluster SN Ia. _SN SCP05N10_ is the lowest-redshift SN candidate in our sample at $z=0.203$. Its light curve shape is inconsistent with a SN Ia occurring well before the first observation, and its luminosity is too low for a SN Ia with maximum only slightly before the first observation. Therefore, we call this a “probable” SN CC. For all SN types, the best fit requires maximum light to occur well before the first observation, making all fits poorly constrained. _SN SCP06X26_ has a tentative redshift of $z=1.44$, derived from a possible [OII] emission line in its host galaxy. Given this redshift, a Ia template provides an acceptable fit, consistent with a typical SN Ia luminosity and color. However, we consider this a “plausible,” rather than “probable, ” SN Ia, given the uncertain redshift and low signal-to-noise of the light curve data. ### Summary In the previous section we addressed the type of all 29 candidates thought to be SNe. However only the cluster-member SNe Ia are of interest for the remainder of this paper. There are six “secure” cluster-member SNe Ia, and two “probable” SNe Ia, for a total of eight. In addition, SCP06E12 is a “plausible” SN Ia and may be a cluster member. Two other candidates, SCP06N32 and SCP06M50, cannot be definitively ruled out as cluster-member SNe Ia, but are quite unlikely for reasons outlined above. We take eight cluster SNe Ia as the most likely total. It is unlikely that _both_ of the “probable” SNe Ia are in fact SNe CC. We therefore assign a classification error of ${}^{+0.0}_{-0.5}$ for each of these, resulting in a lower limit of seven cluster-member SNe Ia. There is a good chance that SCP06E12 is a cluster-member SN Ia, while there is only a small chance that SCP06N32 and SCP06M50 are either cluster SNe Ia. For these three candidates together, we assign a classification error of ${}^{+1}_{-0}$, for an upper limit of nine. Thus, $8\pm 1$ is the total number of observed cluster SNe Ia. ## 4. Effective Visibility Time With a systematically selected SN Ia sample now in hand, the cluster SN Ia rate is given by \[\mathcal{R}=\frac{N_{\rm SN~{}Ia}}{\sum_{j}T_{j}L_{j}},\] (1) where $N_{\rm SN~{}Ia}$ is the total number of SNe Ia observed in clusters in the survey, and the denominator is the total effective time-luminosity for which the survey is sensitive to SNe Ia in clusters. $L_{j}$ is the luminosity of cluster $j$ visible to the survey in a given band. $T_{j}$ is the “effective visibility time” (also known as the “control time”) for cluster $j$. This is the effective time for which the survey is sensitive to detecting a SN Ia, calculated by integrating the probability of detecting a SN Ia as a function of time over the span of the survey. It depends on the redshift of the SN Ia to be detected and the dates and depths of the survey observations. As each cluster has a different redshift and different observations, the control time is determined separately for each cluster. To calculate a rate per stellar mass, $L_{j}$ is replaced by $M_{j}$. Equation (1) is for the case where the entire observed area for each cluster is observed uniformly, yielding a control time $T$ that applies to the entire area. In practice, different areas of each cluster may have different observation dates and/or depths, resulting in a control time that varies with position. This is particularly true for this survey, due to the rotation of the observed field between visits and the gap between ACS chips. Therefore, we calculate the control time as a function of position in each observed field, $T_{j}(x,y)$. As the cluster luminosity is also a function of position, we weight the control time at each position by the luminosity at that position. In other words, we make the substitution \[T_{j}L_{j}\Rightarrow\int_{x,y}T_{j}(x,y)L_{j}(x,y).\] (2) The effective visibility time $T$ at a position $(x,y)$ on the sky is given by \[T(x,y)=\int_{t=-\infty}^{t=\infty}\eta^{\ast}(x,y,t)\epsilon(x,y,t)dt.\] (3) The integrand here is simply the probability for the survey and our selection method to detect (and keep) a SN Ia at the cluster redshift that explodes at time $t$, and position $(x,y)$. This probability is split into the probability $\eta^{\ast}$ of detecting the supernova and the probability $\epsilon$ that the supernova passes all “light curve” cuts. As each SN has multiple chances for detection, the total probability of detection $\eta^{\ast}$ is a combination of the probabilities of detection in each observation. For example, if we have two search visits at position $(x,y)$, $\eta^{\ast}(t)$ is given by \[\eta^{\ast}(t)=\eta_{1}(t)+(1-\eta_{1}(t))\eta_{2}(t),\] (4) where $\eta_{i}(t)$ is the probability of detecting a SN Ia exploding at time $t$ in visit $i$. In other words, the total probability of finding the SN Ia exploding at time $t$ is the probability of finding it in visit 1 plus the probability that it was _not_ found in visit 1 times the probability of finding it in visit 2. This can be generalized to many search visits: The contribution of each additional visit to the total probability is the probability of not finding the SN in any previous visit times the probability of finding the SN in that visit. In practice, we calculate $T(x,y)$ in two steps: First, we determine the probability $\eta$ of detecting a new point source in a single image as a function of the point source magnitude. This is discussed in §4.1. Second, for each $(x,y)$ position in the observed area we simulate a variety of SN Ia light curves at the cluster redshift occurring at various times during the survey. By considering the dates of the observations made during the survey at that specific position, we calculate the brightness and significance each simulated SN Ia would have in each $z_{850}$ and $i_{775}$ image. We then use our calculation of $\eta$ as a function of magnitude to convert the observed brightness into a probability of detecting the simulated SN in each observation. The light curve simulation is discussed in §4.2. The calculation of cluster luminosities, $L_{j}(x,y)$, is discussed in §5. ### Detection Efficiency Versus Magnitude Here we calculate the probability of detecting a new point source as a function of magnitude in a single subtraction. We use a Monte Carlo simulation in which artificial point sources of various magnitudes are added to each of the individual exposure images from the survey, before they are combined using MultiDrizzle. Starting from the individual exposures allows us to test both the efficiency of the MultiDrizzle process and our cosmic ray rejection (which uses the flux observed in the individual exposures). The point sources are placed on galaxies in positions that follow the distribution of light in each galaxy. Poisson noise is added to each pixel in the point source. The altered images are then run through the full image reduction and SN detection pipeline used in the search, and flagged candidates are compared to the input point sources. <figure><img src="content_image/1010.5786/x7.png"><figcaption>Figure 7.— Point source detection efficiency in a single subtraction, as afunction of the ratio of total point source flux to subtraction noise σ(counts sec−1 pixel−1). The artificial point sources are split into four binsdepending on the underlying galaxy surface brightness μ (mag arcsec−2) at thepoint source position. The efficiency curve is calculated separately for eachbin. In the upper left panel, the four bins are shown, offset for clarity. Inthe lower left panel, the fitted curves are reproduced without offset forcomparison. Approximately 72,000 artificial point sources were used in total.The right panel shows the distribution of the noise level in the subtractions.The noise level differs by a factor of about two from the deepest toshallowest subtractions searched.</figcaption></figure> We parameterize the detection efficiency by the ratio of point source flux to sky noise. This is a good choice because, in most cases, the detection efficiency will depend only on the contrast between the point source and the sky noise. However, there is an additional dependence on the surface brightness at the location of the point source: point sources near the core of galaxies will have a lower detection efficiency due to additional Poisson noise from the galaxy. For $0.6<z<1.5$ galaxies, we estimate that only $\sim$$10\%$ of SNe will fall on regions where galaxy Poisson noise is greater than the sky noise (assuming SNe follow the galaxy light distribution). Still, we take this effect into account by splitting our sample of artificial point sources into four bins in underlying surface brightness. The detection efficiency is calculated separately in each bin (Fig. 7, top left panel). The first two bins, $\mu>22.0$ and $22.0>\mu>20.6$ mag arcsec${}^{-2}$, correspond to lower surface brightnesses where sky noise is dominant. As expected, their efficiency curves are very similar. In the third and fourth bins, corresponding to higher surface brightness, the Poisson noise from the galaxy dominates the sky noise, and the efficiency drops as a result. For reference, the distribution of sky noise in the subtractions is shown in Figure 7 (right panel). Nearly all the searched area has a sky noise level between 0.006 and 0.012 counts sec${}^{-1}$ pixel${}^{-1}$. For a typical value of 0.008, we show the corresponding point source $z_{850}$ magnitude on the top axis of the left panel. We find that the efficiency curve in each bin is well-described by the function \[\eta(x)=\left\{\begin{array}[]{ll}\frac{1}{2}(1+ae^{-bx})[\mathrm{erf}((x-c)/d _{1})+1],&x<c\\ \frac{1}{2}(1+ae^{-bx})[\mathrm{erf}((x-c)/d_{2})+1],&x\geq c\end{array}\right.,\] (5) where $x$ is the ratio of point source flux to sky noise, and $a$, $b$, $c$, $d_{1}$ and $d_{2}$ are free parameters. An error function is the curve one would expect with a constant cut and Gaussian noise, but we find that two different scales ($d_{1}$ and $d_{2}$) in the error function, as well as an additional exponential term, are necessary to describe the slow rise to $\eta=1$ at large $x$. This slow rise is due to rarer occurrences, such as cosmic rays coinciding with new point sources. The fitted functions for the four bins are plotted in the top left of Figure 7 and reproduced in the bottom left of the figure for comparison. We use these fitted functions to calculate the effective visibility time in the following section. ### Simulated Lightcurves We simulate SN Ia light curves with a distribution of shapes, colors and absolute magnitudes. We use the (original) salt(Guy et al., 2005) prescription in which the diversity of SN Ia light curves is characterized as a two-parameter family with an additional intrinsic dispersion in luminosity. The two parameters are the linear timescale of the light curve (“stretch”, $s$) and the $B-V$ color excess, $c$. For each simulated SN, $s$ and $c$ are randomly drawn from the distributions shown in Figure 8 (solid lines). The stretch distribution is based on the observed distribution in passive hosts (Fig. 8, left panel, grey histogram) in the first-year Supernova Legacy Survey (SNLS) sample (Sullivan et al., 2006). Similarly, the color distribution is based on the observed color distribution (Fig. 8, right panel, grey histogram) in the first-year SNLS sample (Astier et al., 2006). The absolute magnitude of each simulated SN is set to \[M_{B}=-19.31-\alpha(s-1)+\beta c+I\] (6) where $-19.31$ is the magnitude of an $s=1$, $c=0$ SN Ia in our assumed cosmology (Astier et al., 2006), $\alpha=1.24$, $\beta=2.28$(Kowalski et al., 2008), and $I$ is an added “intrinsic dispersion”, randomly drawn from a Gaussian distribution centered at zero with $\sigma=0.15$ mag. <figure><img src="content_image/1010.5786/x8.png"><figcaption>Figure 8.— Left panel: stretch distribution used for simulated SNe (_solidline_) and the stretch distribution of first-year SNLS z<0.75 SNe in passivehosts (Sullivan et al., 2006) (_grey histogram_). Note that the distributionis not changed significantly by cutting the sample at z<0.6. Therefore we donot expect the sample to be significantly Malmquist biased. Right panel: colordistribution of the first-year SNLS z<0.6 SNe (Astier et al., 2006) (_greyhistogram_) and the color distribution used for simulated SNe (_solid line_).The _dotted lines_ show alternative color distributions used to assess thepossible systematic error due to varying amounts of SNe being affected bydust.</figcaption></figure> We have chosen distributions that represent as accurately as possible the full distribution of SNe Ia occurring in reality. However, note that the control time is not actually very sensitive to the assumed distributions. This is because, for the majority of cluster redshifts in the survey, the detection efficiency is close to 100% during the time of the survey. Supernovae would thus have to be significantly less luminous in order to change the detection efficiency significantly. In the following section §4.3 we quantify the effect on the control time arising from varying the assumed SN Ia properties and show that they are sub-dominant compared to the Poisson error in the number of SNe observed. All sources of systematic errors are also summarized in §6.2. To generate the simulated light curves in the observed bands, we use the Hsiao et al. (2007) SN Ia spectral time series template. For each simulated SN, the spectral time series is warped to match the selected color $c$ and redshifted to the cluster restframe. Light curves are generated in the observed $i_{775}$ and $z_{850}$ filters using synthetic photometry, and the time axis is scaled according to the chosen value of $s$. For each cluster, we calculate $T(x,y)$ in bins of 50 $\times$ 50 pixels ($2^{\prime\prime}.5\ \times\ 2^{\prime\prime}.5$). In each bin, we simulate 100 SN light curves at random positions within the bin. For each simulated SN light curve, we shift the light curve in time across the entire range of observations, starting with maximum light occurring 50 days before the first observation and ending with maximum light occurring 50 days after the last observation. For each step in time we get the $z_{850}$ and $i_{775}$ magnitude of the SN at every date of observation. From the sky noise maps, we know the noise at the position of the simulated SN in every image. Using the curves in Figure 7, we convert the SN flux-to-noise ratio to the probability of the SN being detected in each $z_{850}$ exposure. (Each simulated SN is also assigned a host galaxy surface brightness chosen from a distribution, in addition to the randomly selected $s$, $c$ and $I$ parameters; we use the Fig. 7 curve that corresponds to this surface brightness.) At the same time, we calculate the probability that the SN passes our light curve cuts (using both $z_{850}$ and $i_{775}$ simulated magnitudes). Multiplying these two probabilities gives the total probability of the simulated SN being included in the sample if it peaks at the given date. Integrating the probability over time (the entire range of dates) gives the control time for each simulated SN. We take the average control time of the 100 SNe as the value for the given bin. The resulting control time map, $T(x,y)$, therefore has a resolution of $2^{\prime\prime}.5\ \times\ 2^{\prime\prime}.5$. $T(x,y)$ is shown for two example clusters in Figure 9. <figure><img src="content_image/1010.5786/x9.png"><figcaption>Figure 9.— Example maps of effective visibility time for clusters ISCSJ1432.4+3332 (F) and ISCS J1438.1+3414 (K). The dot denotes the cluster centerand the inner and outer circles represent 0.5 Mpc and 1.0 Mpc radius,respectively. The “noise” in these maps is due to the finite number (100) ofSNe simulated at each position. At lower redshift nearly all simulated SNe arerecovered at each position, whereas at higher redshift a sizable fraction ofsimulated SNe are missed, resulting in a higher “noise” level.</figcaption></figure> ### Effect of Varying SN Properties If the real distributions of SN Ia properties differs significantly from those assumed in our simulation, the $T(x,y)$ maps we have derived could misrepresent the true efficiency of the survey. Above we argued that the effect is likely to be small because the detection efficiency is close to 100% for most of the survey. Here we quantify the size of the possible effect on the control time by varying the assumed distributions. To first order, changing the assumed distributions of $s$ or $c$ or changing the assumed spectral time series will affect the detection efficiency by increasing or decreasing the luminosity of the simulated SN. To jointly capture these effects, we shift the absolute magnitude of the simulated SNe Ia by ${}^{+0.2}_{-0.2}$ mag and recalculate the control times. To first order, this is equivalent to shifting the $s$ distribution by $\Delta s=0.2/\alpha\sim 0.16$ or shifting the $c$ distribution by $\Delta c=0.2/\beta\sim 0.09$. A $-0.2$ mag shift in absolute magnitude increases the control time, decreasing the inferred SN Ia rate by $6\%$. A $+0.2$ mag shift decreases the control time, increasing the SN Ia rate by $8\%$. These effects are sub-dominant compared to the Poisson error of $\gtrsim 30\%$ in the number of SNe observed. (Sources of error are summarized in §6.2 and Table 8.) For the color distribution, in addition to a simple shift, we also quantify the effect of including a smaller or larger fraction of SNe significantly reddened by dust. In fact, we have good reasons to believe that most cluster SNe Ia will be in dust-free environments. A large fraction of the stellar mass in the clusters ($\sim 80\%$) is contained in red-sequence galaxies expected to have little or no dust. Our spectroscopic and photometric analysis (Meyers11) of the red-sequence galaxies confirms this expectation. Therefore, for our default $c$ distribution (Fig. 8, right panel, solid line), we assumed that $20\%$ of SNe (those occurring in galaxies not on the red sequence) could be affected by dust, and that the extinction of these SNe would be distributed according to $P(A_{V})\propto\exp(-A_{V}/0.33)$ [the inferred underlying $A_{V}$ distribution of the SDSS-II sample (Kessler et al., 2009)]. All SNe are assumed to have an intrinsic dispersion in color to match the observed SNLS distribution at $c<0.3$. It might be the case that even fewer SNe are affected by dust, or (unlikely) more SNe are affected by dust. As extreme examples, we tested two alternative distributions (dotted lines in Fig. 8). In the first, we assumed that the SNLS sample was complete and characterized the full $c$ distribution, with a negligible number of $c>0.4$ SNe. This increases the control time by only $2\%$. In the second, we increase the fraction of dust-affected SNe from $20\%$ to $50\%$. Even though this alternative distribution includes an additional $\sim$$30\%$ more reddened SNe (unlikely to be true in reality), the average control time is only lower by $9\%$ (increasing the rate by $10\%$). We use these values as the systematic error in the assumed dust distribution. ## 5. Cluster Luminosities and Masses In this section, we calculate the total luminosity of each cluster and use the luminosity to infer a stellar mass. Only a small subset of galaxies in each field have known redshifts, making it impossible to cleanly separate cluster galaxies from field galaxies. Therefore, we use a “background subtraction” method to estimate cluster luminosities statistically: we sum the luminosity of all detected galaxies in the field and subtract the average “background luminosity” in a non-cluster field. This approach follows that of Sharon et al. (2007). For the blank field, we use the GOODS³ fields (Giavalisco et al., 2004) as they have similarly deep or deeper observations in both ACS $i_{775}$ and $z_{850}$. In §5.1 we describe the galaxy detection and photometry method. Simply summing the photometry from the detected galaxies would include most of the total cluster light. However, for an unbiased estimate of the total light, several small corrections are necessary: We account for light in the outskirts of each galaxy (§5.2), and light from faint galaxies below the detection threshold (§5.4). These corrections are on the order of 20% and 5% respectively. In §5.3 we convert the observed $z_{850}$ flux to a rest-frame $B$-band flux. In §5.5 we sum the light and subtract background light. In §5.6 we repeat this calculation limiting ourselves to red-sequence and red-sequence early-type subsets of galaxies. Finally, in §5.7 we estimate cluster stellar masses based on the cluster luminosities and stellar mass-to-light ratios. [FOOTNOTE:3][ENDFOOTNOTE] ### Galaxy Selection and Photometry We use the stacked $i_{775}$ and $z_{850}$ band images of each cluster, which have total exposure times in the range 1060 – 4450 seconds and 5440 – 16,935 seconds, respectively. Galaxy catalogs are created using the method described in detail by Meyers11: We run SExtractor(Bertin & Arnouts, 1996) in dual-image mode using the $z_{850}$ image for detection, and use a two-pass Cold/Hot method (Rix et al., 2004) to optimally de-blend galaxies. We remove stars from the catalog based on the CLASS_STAR and FLUX_RADIUS parameters from the $z_{850}$ image. It is notoriously difficult to determine accurate total fluxes for extended sources. However, as we are only concerned with the summed flux of many galaxies, it is not important that the estimate be accurate for each individual galaxy, only that the estimate is unbiased in the aggregate. We use the SExtractor MAG_AUTO photometry (which gives the total flux within a flexible elliptical aperture) and apply a correction determined using the Monte Carlo simulation described below. In order to make the aperture correction as small as possible, we use a relatively large “Kron factor” of 5.0, meaning that the MAG_AUTO aperture is scaled to 5.0 times the Kron radius of the galaxy. MAG_AUTO is only used to determine $z_{850}$ magnitudes; $i_{775}-z_{850}$ colors are determined using PSF matching and a smaller aperture, as described in Meyers11. ### Galaxy Detection Completeness and Magnitude Bias To count all the flux in all cluster galaxies, we must make two corrections: (1) add the galaxy light outside of the MAG_AUTO aperture, and (2) add the luminosity of all cluster galaxies below the detection threshold of our galaxy catalog. We use a Monte Carlo simulation of galaxies placed on our real survey data to determine both the detection efficiency as a function of galaxy magnitude, and the fraction of galaxy light inside the MAG_AUTO aperture. Each simulated galaxy has a Sérsic (1968) profile, with the Sérsic index $n$ simply selected from a flat distribution ranging from $n=0.7$ to $n=4.5$, and the minor to major axis ratio $q$ selected from a flat distribution ranging from $q=0.3$ to $q=1$. The distribution of galaxy angular sizes will also affect the results. For guidance on the size of the galaxies of concern (namely, those at $z\gtrsim 0.9$) we turned to the subsample of the 672 galaxies having spectroscopic redshifts $0.85<z<1.6$. These 672 galaxies were all fit with galfit(Peng et al., 2002), which fits a value for $r_{e}$. Based on the distribution of $r_{e}$ as a function of magnitude for these galaxies, we chose $r_{e}$ for each simulated galaxy (based on its magnitude). A total of 15000 and 12000 simulated galaxies were placed on cluster and GOODS fields respectively. <figure><img src="content_image/1010.5786/x11.png"><figcaption>Figure 10.— Percentage of simulated galaxies recovered by SExtractor as afunction of total galaxy z850 magnitude for simulated galaxies placed oncluster fields (black circles) and GOODS fields (grey squares). The detectionefficiency drops to 80% at z850=24.72 for cluster fields (_vertical line_). Wediscard all galaxies dimmer than this value.</figcaption></figure> The detection efficiency as a function of galaxy magnitude is shown in Figure 10. For the average of all cluster fields, the detection efficiency drops to 80% at $z_{850}=24.72$. We use this magnitude as a cutoff in our selection, discarding all galaxies dimmer than this magnitude. We later correct total cluster luminosities for the uncounted light from these galaxies by using an assumed cluster luminosity function. In reality, the detection efficiency varies slightly from field to field (and even within a field) due to exposure time variations. However, to first order, the variation is accounted for by using the average efficiency in all fields. In addition, the total luminosity of $z_{850}>24.72$ cluster galaxies is expected to be small (as we show below), so slight changes in the cutoff will have a negligible effect on the total luminosity. <figure><img src="content_image/1010.5786/x12.png"><figcaption>Figure 11.— Galaxy MAG_AUTO aperture correction as a function of galaxymagnitude. Black circles: Average correction for the full distribution ofgalaxies simulated, including all Sérsic indices n. The black line is a fit tothese points and is the relation we use. Note that it is not extrapolatedbeyond the range shown. To illustrate the effect of n on the aperturecorrection, we plot the aperture correction for subsets of galaxies withdifferent Sérsic indices (Grey squares and triangles). Galaxies with largerSérsic indices have a larger aperture correction.</figcaption></figure> For each simulated galaxy, we determine the difference ($\Delta M$) between the MAG_AUTO magnitude and the true total magnitude. Binning the simulated galaxies by their MAG_AUTO magnitude, we derive a relation between $\Delta M$ and the galaxy brightness (Fig. 11, black circles). $\Delta M$ generally increases with galaxy magnitude because the outskirts of dimmer galaxies are increasingly buried in noise, causing SExtractor to underestimate the true extent of the galaxy, and thereby underestimate the Kron radius, resulting in a smaller MAG_AUTO aperture. We find that the relation is well-fit by a second-order polynomial (Fig. 11, thick black line), given by \[\Delta M = 0.238+0.081(M_{MAG\_AUTO}-23)+\] (7) \[+0.009(M_{MAG\_AUTO}-23)^{2}.\] We use this to correct the magnitude of each detected galaxy. Note that the correction is not extrapolated beyond the fitted range shown. Because we cannot reliably determine $r_{e}$ or the Sérsic index $n$ for each galaxy, we rely on the simulated distribution of $r_{e}$ and $n$ to accurately represent the true distributions. (The black circles in Fig. 11 include all simulated galaxies.) We have based our distribution of $r_{e}$ on actual galaxies, but $n$ is less well-known. To estimate the effect of varying the $n$ distribution, we show $\Delta M$ for subsets of the simulated galaxies, divided by Sérsic index (Fig. 11, grey points and lines). $\Delta M$ increases with Sérsic index, because a larger Sérsic index implies a larger fraction of light in the outskirts of the galaxy, under the detection threshold. This leads to a smaller estimate of the Kron radius, and a smaller MAG_AUTO aperture. If, instead of the flat $1<n<4$ distribution used, all galaxies had $1<n<2$, the aperture correction would be lower by approximately $0.10$ magnitudes. If instead all galaxies had $3<n<4$, the correction would be higher by approximately $0.07$ magnitudes. We use $0.07$ mag as the systematic uncertainty in the aperture correction. (All systematic uncertainties are summarized in §6.2 and Table 8.) ### $K$-Corrections We use a $K$-correction based on the BC03 stellar population spectral models to convert the observed $z_{850}$ magnitude to a rest-frame $B$ magnitude for each cluster. Rather than using a single $K$-correction for all the light in each cluster, we apply a $K$-correction to each galaxy magnitude based on its $i_{775}-z_{850}$ color. For each cluster’s redshift, we determine the relation between $K$-correction ($M_{B}$ (rest) $-z_{850}$) and $i_{775}-z_{850}$ color, using BC03 spectra with initial metallicities in the range $0.004<Z<0.05$ and ages in the range $1\times 10^{8}-5\times 10^{9}$ yr. For most cluster redshifts in our sample, all of the spectra over this wide range fall along the same line in $K$-correction versus color, meaning that the color determines the $K$-correction, regardless of the metallicity or age assumed. The dispersion of the models about the best-fit line is $<0.03$ mag at redshifts $\lesssim 1.1$ and $\gtrsim 1.4$, and reaches its largest value of 0.09 mag at $z=1.26$. We calculate the $K$-correction for each galaxy using this best-fit relation, effectively assuming that every galaxy is at the cluster redshift. This results in an incorrect luminosity for non-cluster member galaxies, but this is accounted for by performing the same $K$-correction on the galaxies in the GOODS fields prior to subtracting their luminosity. ### Luminosity function correction We estimate the total luminosity of all galaxies below the detection limit of $z_{850}=24.72$ using a Schechter (1976) luminosity function, which gives the number of galaxies in the luminosity interval $[L,L+dL]$ in a given sample, \[\Phi(L)dL=\Phi^{\ast}(L/L^{\ast})^{\alpha}e^{-L/L^{\ast}}d(L/L^{\ast}).\] (8) $\Phi^{\ast}$ is a normalization, $L^{\ast}$ is a characteristic galaxy luminosity, and $\alpha$ is a unit-less constant. The ratio of total to observed luminosity is then \[C=\frac{\int_{0}^{\infty}L\Phi(L)dL}{\int_{L_{lim}}^{\infty}L\Phi(L)dL},\] (9) and we multiply each observed cluster luminosity by C to get the total luminosity. We assume values for $L^{\ast}$ and $\alpha$ determined in other studies and use our data to perform a rough consistency check. For $\alpha$, studies have shown that the value does not evolve much from low redshift, at least for redder galaxies. Analyzing only red galaxies in 28 clusters spanning $0<z<1.3$, Andreon (2008) find $\alpha=-0.91\pm 0.06$ (rest-frame $V$-band) with no discernible trend in redshift (see also Andreon, 3, 4). From five intermediate-redshift clusters ($0.54<z<0.9$), Crawford et al. (2009) find a somewhat flatter faint-end slope $\alpha\sim-0.6$ (rest-frame $B$-band) for the red-sequence luminosity function. Looking at the full luminosity function, Goto et al. (2005) find $\alpha=-0.82\pm 0.10$ in one cluster at $z=0.83$ (rest-frame $B$-band), compared to $\alpha=-1.00\pm 0.06$ in 204 low-redshift clusters (rest-frame $g$-band) (Goto et al., 2002). In redder bands, Strazzullo et al. (2006) find $\alpha\sim-1$ for three clusters at redshifts $1.11<z<1.27$ (in approximately rest-frame $z$ band). Summarizing, most studies find a value consistent with $\alpha\sim-0.9$, and we assume this value in computing $C$. Values for $M^{\ast}$ are also reported in most of the above-mentioned studies. Studies of red galaxies find that the variation of $M^{\ast}$ with redshift is consistent with passive evolution, with $M^{\ast}$ decreasing towards higher redshifts (Andreon, 4; Crawford et al., 2009). Crawford et al. (2009) find $M^{\ast}_{B}=-21.1$ and $M^{\ast}_{B}\sim-21.3$ (with errors of approximately a half magnitude) for two clusters at redshifts 0.75 and 0.83. $K$-correcting from the observed $[3.6]$-band, Andreon (4) find $M^{\ast}_{B}\sim-21.7$ at $z\sim 1.1$, with approximately 0.5 magnitudes of evolution between $z=0.3$ and $z=1.1$. At lower redshift (considering all galaxies) Goto et al. (2002) find $M^{\ast}_{B}\sim-21.6$, compared to $M^{\ast}_{B}\sim-21.0$ for one cluster at $z=0.83$(Goto et al., 2005). On the basis of these measurements, we assume a value of $M^{\ast}_{B}=-21.7$. We have checked our assumed $M^{\ast}_{B}$ and $\alpha$ for consistency with our data. With the set of spectroscopically-confirmed cluster galaxies from our clusters at $z<1.2$, we confirmed that the bright end of the luminosity function is consistent with $M^{\ast}_{B}=-21.7$, and strongly inconsistent with values outside the range $M^{\ast}_{B}=-21.7\pm 0.5$. We also determined the luminosity function using a statistical subtraction of the “background” luminosity function from the GOODS fields, finding excellent agreement with the assumed $M^{\ast}_{B}$ and $\alpha$ values over the range $-24<M_{B}<-19.8$ ($M_{B}=-19.8$ corresponds to the detection limit in the highest-redshift clusters). For each cluster, we calculate $C$ in the observer frame, converting $M^{\ast}_{B}=-21.7$ to the observed $z_{850}$ band, using the cluster redshift and a $K$-correction based on a passive galaxy template. In Table 5 we report the value $z_{850}^{\ast}$ and the resulting correction $C$ for each cluster. The correction is less than $5\%$ for the majority of clusters, rising to a maximum of 14% for the highest-redshift cluster. Because the correction is so small, varying the assumed values of $M^{\ast}_{B}$ and $\alpha$ does not have a large effect on the total luminosity. Varying $M^{\ast}_{B}$ by $\pm 0.5$ mag (a larger range than that allowed by our data) changes the average correction by only ${}^{+4}_{-2}\%$. Varying $\alpha$ by $\pm 0.2$ changes the average correction by ${}^{+5}_{-2}\%$. We conservatively take ${}^{+10}_{-3}\%$ (the full range when varying both concurrently) as the systematic uncertainty in luminosity from the faint-end correction (summarized in §6.2). ID \| z \| Cutoff from \| zbright850 \| z∗850 \| C ---\|---\|---\|---\|---\|--- A \| 1.46 \| Max cD \| 21.09 \| 22.80 \| 1.143 B \| 1.12 \| cD \| 20.11 \| 21.38 \| 1.033 C \| 0.97 \| cD \| 19.87 \| 20.79 \| 1.018 D \| 1.02 \| BCG \| 20.13 \| 20.95 \| 1.021 E \| 1.03 \| cD \| 19.40 \| 20.99 \| 1.022 F \| 1.11 \| Max cD \| 19.63 \| 21.34 \| 1.031 G \| 1.26 \| BCG \| 20.34 \| 22.04 \| 1.064 H \| 1.24 \| BCG \| 20.33 \| 21.95 \| 1.058 I \| 1.34 \| Max cD \| 20.66 \| 22.37 \| 1.092 J \| 1.37 \| Max cD \| 20.77 \| 22.50 \| 1.104 K \| 1.41 \| Max cD \| 20.92 \| 22.65 \| 1.122 L \| 1.37 \| Max cD \| 20.77 \| 22.50 \| 1.104 M \| 0.90 \| Max cD \| 18.69 \| 20.53 \| 1.014 N \| 1.03 \| BCG \| 20.22 \| 20.99 \| 1.022 P \| 1.1 \| Max cD \| 19.58 \| 21.29 \| 1.030 Q \| 0.95 \| cD \| 20.01 \| 20.66 \| 1.015 R \| 1.22 \| Max cD \| 20.15 \| 21.86 \| 1.054 S \| 1.07 \| Max cD \| 19.44 \| 21.16 \| 1.026 T \| 0.97 \| Max cD \| 19.00 \| 20.75 \| 1.017 U \| 1.04 \| Max cD \| 19.31 \| 21.04 \| 1.022 V \| 0.90 \| cD \| 18.89 \| 20.49 \| 1.013 W \| 1.26 \| Max cD \| 20.33 \| 22.04 \| 1.064 X \| 1.10 \| Max cD \| 19.58 \| 21.34 \| 1.031 Y \| 1.24 \| cD \| 20.29 \| 21.90 \| 1.056 Z \| 1.39 \| cD \| 20.85 \| 22.58 \| 1.112 Note. – “Cutoff from” refers to how zbright850 is determined. “cD”: magnitude of visually central dominant galaxy. “BCG”: magnitude of visually classified brightest cluster elliptical (but not central) galaxy. “Max cD”: Cluster does not have obvious cD galaxy or clear BCG. In this case, zbright850 is K-corrected from MB=−23.42, the absolute magnitude of the brightest cD galaxy in the entire sample. Table 5 Bright cutoff magnitudes and luminosity function parameters ### Cluster Luminosities and Aggregate Cluster Profile For each cluster we sum the $K$-corrected $B$-band luminosity of all galaxies brighter than the detection limit $z_{850}=24.72$. To reduce noise, we discard galaxies that are clearly too bright to be cluster members. In clusters with a central dominant (cD) galaxy or dominant (but not central) brightest cluster galaxy(BCG), the bright cutoff magnitude is set to the magnitude of the cD galaxy or BCG. In clusters lacking a clearly dominant galaxy, we conservatively set the cutoff based on the absolute magnitude of the most luminous cD galaxy in any cluster, $M_{B}=-23.42$ (from cluster XMMU J2235.3$-$2557). The bright cutoff magnitude in the observer frame, $z_{850}^{\rm bright}$, is listed for each cluster in Table 5. Because the bright cutoff is chosen so conservatively, we expect that no cluster galaxies are discarded. The effect of being overly conservative is only to add noise, and this is captured in the statistical uncertainty described below. For each cluster we apply the same selection criteria and $K$-corrections to the GOODS fields to determine the “background” specific to that cluster. The error in the luminosity comes from the error in this background determination, which we estimate in the following way: We select 30 non-connected circular regions (15 in each of GOODS North and South) of radius $1.4^{\prime}$, similar to the size of the cluster fields. We determine the luminosity density in each of these fields. The average is taken as the background luminosity for the cluster, and the standard deviation (typically 15 – 20 % of the average) is taken as the error in this “background” luminosity due to variations between fields. We have implicitly assumed that the GOODS average accurately represents the cosmic average. GOODS incorporates only two widely separated fields. As a result, the average luminosity density may differ from the cosmic average due to variations in large scale structure. As a rough estimate of the cosmic variance, we compare the two GOODS fields. The average luminosity density of the GOODS-North regions is consistently higher than that of the GOODS-South regions by 15 – 20%. This means that the “standard deviation” of these two samples of large scale structure is $\sim$8%. We checked this using the cosmic variance calculator made available by Trenti & Stiavelli (2008)⁴. The expected cosmic variance in galaxy number counts in the redshift window $0.7<z<1.7$ for one GOODS field is approximately $\sim$6%, in good agreement with our naïve estimate. Conservatively, we take $8\%$ as the cosmic variance for one GOODS field. For the _average_ of the North and South fields, this implies a cosmic variance of $8\%/\sqrt{2}\sim 6\%$. [FOOTNOTE:4][ENDFOOTNOTE] One might be additionally concerned that the “background” in the cluster fields is biased higher than the cosmic average because clusters form in regions of large-scale overdensities. However, each cluster field is a “pencil-beam” galaxy survey, so the vast majority of non-cluster galaxies will not be associated with the high-density region in which each cluster formed. Ideally one would measure a two-dimensional luminosity density, $L(x,y)$, for each cluster, as in Equation (2). However, the large background makes this difficult. For our purpose (which is to account for variations in control time with radius), it is sufficient to assume the clusters have a circularly symmetric luminosity distribution, $L(r)$. For each cluster, we sum the total luminosity in annuli of width 0.1 Mpc. For nearly all clusters there is a clear overdensity relative to the background out to $r\sim 0.3$ Mpc. Beyond $0.3$ Mpc, the luminosity measurement is dominated by background noise for most clusters. This might appear to be a problem; we wish to characterize the cluster luminosities out to $r\gtrsim 0.7$ Mpc, the area over which we searched for SNe. In fact, it is only necessary to accurately measure the _average_ luminosity profile over the full area (the denominator of Eq. 1 is the sum of the cluster luminosities, weighted by control time). Averaging all 25 clusters, there is a significant measurement of the luminosity profile out to $>0.5$ Mpc (Fig. 12, left panels), and the average cluster luminosity within $r<0.6$ Mpc has an error of $12\%$ (statistical only) and $\sim 20\%$ (statistical $+$ cosmic variance), below the Poisson error in the number of SNe detected. Beyond $r<0.6$ Mpc, the control time is generally small (that is, there are few observations covering the outskirts of the clusters) and the cluster luminosity density is low, meaning that these regions will not contribute greatly to the rate measurement. Still, we include these regions in our rate calculation, using the entirely reasonable prior that the luminosity density is decreasing with radius past $r<0.6$ Mpc. How rapidly the luminosity density decreases will not have a significant impact on the result, but as a convenient analytic description we fit a $\beta$-model of the form \[L(r)=\frac{\Sigma_{0}}{(1+(r/r_{\rm core})^{2})^{\beta}}\] (10) over the range $r<0.6$ Mpc and apply this function at $r>0.6$ Mpc. The data are well-fit by this model, with best-fit parameters $r_{\rm core}=0.074$ Mpc and $\beta=0.91$. Varying this model luminosity by $\Delta\Sigma_{0}=\pm 20\%$ (easily enclosing the allowed range of $L(r)$) only changes our results by $\pm 4\%$. This and other systematic uncertainties are summarized in Table 8. <figure><img src="content_image/1010.5786/x13.png"><figcaption>Figure 12.— Average luminosity profile of the 25 clusters. Top row: Averageluminosity density in the cluster fields in annuli of width 0.1 Mpc extendingout from the cluster center. The grey line and shaded region show theestimated “background” luminosity in each annulus and the error on thatbackground, respectively. The darker grey region is the statistical-onlyerror, while the light grey is the statistical + cosmic variance error, addedin quadrature. Bottom row: The total enclosed luminosity as a function ofradius, derived by subtracting the background from the total luminositydensity in each bin in the top row plot. The left plots include galaxies ofall colors and morphologies, while the center plots include only galaxies withi775−z850 colors within ±0.2 mag of the red sequence in their respectiveclusters. The right plots include only galaxies that satisfy the colorrequirement and also have z850<24 and are morphologically early type. Byexcluding bluer galaxies (center and right plots) the background (and error)is reduced dramatically.</figcaption></figure> Cluster subset \| Nclusters \| ¯z \| All galaxies (1012L⊙,B) \| RS galaxies (1012L⊙,B) \| RSE galaxies (1012L⊙,B) ---\|---\|---\|---\|---\|--- X-ray discovered \| 9 \| 1.20 \| 2.86±0.54±0.45 \| 2.42±0.16±0.05 \| 1.47±0.12±0.02 IR-Spitzer discovered \| 7 \| 1.30 \| 2.85±0.70±0.52 \| 1.83±0.24±0.07 \| 0.96±0.16±0.03 Optical discovered \| 9 \| 1.00 \| 1.99±0.37±0.32 \| 1.75±0.08±0.03 \| 1.29±0.06±0.01 z<1.2 \| 14 \| 1.02 \| 2.14±0.31±0.33 \| 1.79±0.07±0.03 \| 1.28±0.05±0.01 z>1.2 \| 11 \| 1.32 \| 3.06±0.58±0.54 \| 2.31±0.19±0.07 \| 1.23±0.14±0.04 All Clusters \| 25 \| 1.15 \| 2.54±0.31±0.42 \| 2.02±0.09±0.05 \| 1.26±0.07±0.02 Note. – “RS”: galaxies within ±0.2 mag of the cluster red sequence. “RSE”: galaxies fulfilling the “RS” requirement, and also z850<24, and morphologically early-type. The first and second confidence intervals are the statistical error and cosmic variance error, respectively. These luminosities do not include the faint-galaxy correction C. Table 6 Average cluster luminosities within r<0.6 Mpc ### Galaxy subsets In addition to measuring the total luminosity of all galaxies in the clusters, we also measure the total luminosity of only red-sequence galaxies and the total luminosity of only red-sequence, morphologically early-type galaxies. These measurements enable us to compute the cluster SN Ia rate specifically in these galaxy subsets. For the red-sequence-only measurement we follow the same procedure as above, but eliminate from the analysis all galaxies with $i_{775}-z_{850}$ colors more than 0.2 mag from their respective cluster red sequences (galaxy colors and cluster red sequences are determined as in Meyers11). For the red-sequence early-type measurement, we make the same requirement in color, and additionally use the quantitative morphology requirements of Meyers11. Meyers11 use two parameters, asymmetry and Gini coefficient, to automatically divide galaxies into early- and late-type subsets. Here we require the asymmetry to be $<0.10$ and the Gini coefficient to be $>0.40$. We also require the galaxies to be $z_{850}<24$ as the asymmetry and Gini coefficient are somewhat less reliable at fainter magnitudes. The luminosity profiles for these two subsets are shown in the center and right columns of Figure 12. The profiles are broadly consistent with the profile of the full cluster luminosity (left column), but the “subset” profiles are much better measured. This is because by excluding bluer galaxies, we have eliminated much of the background while still retaining the majority of cluster galaxies. The red-sequence subset contains $77\%$ of the luminosity of the full cluster within $0.6$ Mpc (Table 6). The red-sequence early-type subset has $62\%$ of the light contained in the red-sequence subset. However, keep in mind that in the early-type subset we have excluded $z_{850}>24$ galaxies, whereas they are included in the red-sequence subset: In fact $68\%$ of $z_{850}>24$ red-sequence galaxies pass the “early-type” morphology requirements. Note that our definition of “red-sequence” here is a relatively simple one. It is sufficient to select a subsample of “more red” galaxies for the purpose of looking for a dependence of the SN rate with galaxy color within the cluster. However, for measuring the red fraction in clusters (e.g., the Butcher-Oemler effect Butcher & Oemler, 1978, 1984), defining red-sequences with a constant color width for all redshifts is not ideal (Andreon, 2). The luminosity content of the subsets are reported above only to give the relative size of each sample; a full analysis of the cluster content is beyond the scope of this paper. ### Stellar Mass-to-Light Ratio To compare SN rates in clusters of different ages, rate measurements must be normalized by stellar mass rather than stellar luminosity because luminosity changes as stars age. To convert our luminosity measurements to mass measurements we use a mass-to-light ($M/L$) ratio based on a stellar evolution model. There are several available models in the literature. The choice of stellar tracks, metallicity, star formation history, and in particular the assumed IMF, will all affect the derived $M/L$ ratio to some extent. For the purpose of measuring the change in rate with redshift, it is important to use a _consistent_ model and assumptions for determining the $M/L$ ratio for all rate measurements. That is, we are most concerned that the model accurately captures the evolution of stellar luminosity over the redshift range of interest ($0<z<1.46$), and less concerned about the overall normalization of the $M/L$ ratio. To that end, for our main result we will use a model and assumptions that match as closely as possible those used for the $M/L$ ratio in low-redshift cluster rate measurements. As we also give results normalized by luminosity, those wishing to use a different $M/L$ ratio can easily do so. Finally, note that the _initial_ stellar mass formed is the quantity of interest for normalizing rate measurements. However, as most rate measurements and $M/L$ ratios have been reported in terms of current mass, we give our results in these units and simply note the difference between current and initial mass for the purpose of comparing rate measurements. Thus, in the following paragraphs $M$ refers to current stellar mass. #### 5.7.1 $M/l$ ratio in low-redshift cluster rate measurements The lower-redshift cluster rate studies of Sharon et al. (2007), Sharon et al. (2010), and by extension, Dilday et al. (2010) have used the relations between $M/L$ ratio and galaxy color derived by Bell et al. (2003, hereafter Bell03). For example, Sharon et al. (2007) use the relation $\log_{10}(M/L_{z})=-0.052+0.923(r-i)$ and Sharon et al. (2010) use $\log_{10}(M/L_{g})=-0.499+1.519(g-r)$, where $M$, $L_{z}$ and $L_{g}$ are in solar units. In order to use a consistent model, it is important to recognize how these relations were derived. Bell03 fit a grid of pégase2(Fioc & Rocca-Volmerange, 1997) synthetic galaxy spectral energy distributions (SEDs) to actual $ugrizK$ photometry of low-redshift galaxies. The grid covers a range of metallicities and star formation histories. The star formation histories have exponentially-decreasing or -increasing star formation rates, and assume that star formation commenced at $z=4$. For each galaxy, the $M/L$ ratio is that of the best-fit synthetic galaxy SED, consistently evolved to $z=0$. Bell03 use a “diet” Salpeter (1955) IMF (following Bell & de Jong, 2001). This IMF is defined as having the same colors and luminosity as a Salpeter IMF, but with a total mass 30% lower. The difference in mass is attributed to a smaller number of faint low-mass stars relative to a Salpeter IMF. These stars do not contribute significantly to the luminosity of the Salpeter IMF. The diet Salpeter IMF results in $M/L$ ratios 30% lower at a given color than a normal Salpeter IMF. Note that because Bell03 simply take the $M/L$ ratio from the best-fit synthetic SED of each galaxy, the Bell03 relations will generally fall within the grid of $M/L$ versus color covered by the synthetic galaxy SEDs. <figure><img src="content_image/1010.5786/x14.png"><figcaption>Figure 13.— Evolution of M/L ratio versus color with redshift. _Left panel:_M/L ratio as a function of u−g color at z=0 and at z=1.2 (typical redshift inthis study). The grid of points show pégase2 models with exponentially-decreasing star formation rates with e-folding times τ and metallicities Z.For each model, star formation begins at z=4. Models with constant metallicityare connected by solid black lines and models with identical star formationhistories are connected by dotted lines. For example, models with τ=0,corresponding to a simple stellar population, are the rightmost points(corresponding to Z=0.01, 0.02, 0.05) connected by dotted lines. As the modelsare evolved back in time from an observed redshift of z=0 to an observedredshift of z=1.2, the M/L ratio decreases and moves away from the Bell03relation (_solid grey line_). The _dashed grey line_ shows the relation usedin this study for z=1.2. At z=1.2 the offset from the Bell03 relation is −0.36dex, or a factor of 0.43. _Right panel:_ Same as left panel, but for g−r colorand for an observed redshift of z=0.6, the typical redshift in the rate studyof Sharon et al. (2010). The offset here is only −0.14 dex, or a factor of0.72.</figcaption></figure> #### 5.7.2 $M/l$ ratio at $0.9<z<1.46$ Ideally, for consistency with Sharon et al. (2007), Sharon et al. (2010) and Dilday et al. (2010), we would simply use the Bell03 relation for $u-g$ color, which most closely matches our observed color: $\log_{10}(M/L_{g})=-0.221+0.485(u-g)$. However, the Bell03 relations are based on $ugrizK$ photometry of low-redshift galaxies, corrected for evolution to $z=0$. As such, they are specific to $z=0$ and not directly applicable at high redshift. A stellar population passively evolving from age a few Gyr (at $z\sim 1$) to $>10$ Gyr (at $z=0$) will dim significantly while only growing slightly redder (see, e.g. BC03), in a manner that does not follow the Bell03 relations. To estimate the effect of evolution from their $z=0$ relation to higher redshift, we make a similar grid of pégase2-generated SEDs with the same formation redshift, metallicities, IMF, and star formation histories. As expected, when evaluated at $z=0$, the $M/L$ ratios of this grid are consistent with the Bell03 relation (Fig. 13, left panel, upper grid of black points). Evaluating the SEDs at higher redshifts, we find that the $M/L$ ratios are well fit by a relation with the same slope, but smaller normalization. For example, at $z=1.2$, the best-fit offset from the $z=0$ relation is $-0.36$ dex (Fig. 13, left panel, dashed line). At the extremes of the redshift range of interest, the best fit offset is $-0.26$ dex ($z=0.9$) and $-0.44$ dex ($z=1.46$). We therefore use a $M/L$ ratio of \[\log_{10}(M/L_{g})=\left\{\begin{array}[]{ll}-0.48+0.485(u-g),&z=0.9\\ -0.66+0.485(u-g),&z=1.46\end{array}\right.\] (11) and linearly interpolate for intermediate redshifts. Another way to view Equation (11) is that, independently of the relation at $z=0$, we have fit a linear relation to the pégase2 SEDs at the redshift of each cluster, assuming a slope consistent with Bell03. Using Equation (11) we calculate mass on a galaxy-by-galaxy basis: we $K$-correct the observed $i_{775}$ and $z_{850}$ magnitude to rest-frame SDSS $u$ and $g$ magnitudes using the method discussed in §5.3, and obtain the $M/L$ ratio from the $u-g$ color. In all, 66% of the clusters’ luminosity is from galaxies with color in the range $1.3<u-g<1.7$, 27% of the luminosity is distributed roughly equally between galaxies in the range $0.6<u-g<1.3$, and the remainder is in redder galaxies with $u-g>1.7$. Thus, while there is a clear presence of bluer cluster galaxies, the majority of the clusters luminosity is confined to a narrow range in color. This narrow color range means that changes in the assumed slope of Equation (11) will not have a large effect on the resulting total mass. The cumulative $M/L$ ratio (the ratio of the total mass of all 25 clusters to the total luminosity of all 25 clusters) is $M/L_{g}=1.25$ (see Table 7, “denom”). For red-sequence galaxies only, the ratio is higher ($M/L_{g}=1.38$) due to the exclusion of bluer galaxies with a lower inferred $M/L$ ratio. #### 5.7.3 $M/l$ ratio uncertainty As noted above, we are primarily concerned with the accuracy of the evolution in the stellar mass and luminosity over the range $0<z<1.46$, rather than the accuracy of the absolute $M/L$ ratio. As a cross-check of the $M/L$ ratio evolution, we have compared the above results (using pégase2) to the results obtained with the BC03 SEDs. We use the standard Padova 1994 evolution and the same star formation histories as above. In terms of evolution offset from $z=0$ to $z\sim 1.2$, we find results consistent within 0.03 dex. This consistent evolution in BC03 and pégase2 is encouraging. However, to be much more conservative in our estimate of the uncertainty in the $M/L$ ratio evolution, we take the scatter of the models around the best-fit line as our uncertainty. In Figure 13, in the color range of interest, the scatter is approximately $\pm 0.08$ dex (20%) at both low and high redshift. We use this as the systematic uncertainty in the $M/L$ ratio for the purpose of comparing SN rates at low and high redshift in §7.3 and §7.4. The uncertainty in the absolute $M/L$ ratio is much greater, due mainly to the uncertainty in the true IMF. ## 6. Results and Systematic Uncertainties Here we present our results for the full cluster rate and for two galaxy subsets (§6.1) and summarize contributions to the uncertainty (§6.2) in each. In §6.3 we show that the rate result in the subsets are not sensitive to the specific parameters used to select the subset. ### Results Environment \| Unit \| ¯z \| NSN Ia \| Denom \| Rate \| (stat) \| (sys) ---\|---\|---\|---\|---\|---\|---\|--- Full cluster \| SNuB \| 1.14 \| 8.0±1.0 \| 15.87 \| 0.50 \| +0.23−0.19 \| +0.10−0.09 Full cluster \| SNug \| \| \| 15.96 \| 0.50 \| +0.23−0.19 \| +0.10−0.09 Full cluster \| SNuM \| \| \| 22.41 \| 0.36 \| +0.16−0.13 \| +0.07−0.07 Red-sequence \| SNuB \| 1.13 \| 6.5±0.5 \| 11.95 \| 0.54 \| +0.25−0.19 \| +0.07−0.07 Red-sequence \| SNug \| \| \| 12.20 \| 0.53 \| +0.24−0.19 \| +0.07−0.07 Red-sequence \| SNuM \| \| \| 17.61 \| 0.37 \| +0.17−0.13 \| +0.05−0.05 Red-sequence early-type \| SNuB \| 1.10 \| 6.0±0.0 \| 7.29 \| 0.82 \| +0.39−0.30 \| +0.09−0.08 Red-sequence early-type \| SNug \| \| \| 7.59 \| 0.79 \| +0.38−0.29 \| +0.09−0.08 Red-sequence early-type \| SNuM \| \| \| 11.77 \| 0.51 \| +0.24−0.19 \| +0.06−0.05 Note. – “Denom” is the denominator of equation (1) and has units of 1012L⊙,B years, 1012L⊙,g years and 1012M⊙ years for rate units of SNuB, SNug and SNuM respectively. Table 7Results The results are presented in Table 7. We derive a rate in the full cluster, in red-sequence galaxies only, and in red-sequence early-type galaxies only. Each subset includes a different number of SNe: As discussed in §3.4, we have discovered $8\pm 1$ cluster SNe, where the quoted uncertainty is due to classification uncertainty (including uncertainty in both SN type and cluster membership). Limiting the sample to only SNe discovered in galaxies included in the red-sequence subset excludes SN SCP06F12 and SN SCP06C1, leaving $6.5\pm 0.5$ cluster SNe Ia. The uncertainty here comes from the uncertainty in the cluster membership and type of SN SCP06E12, which we count $0.5\pm 0.5$ cluster SNe Ia. Further limiting the sample to only SNe discovered in galaxies included in the red-sequence early-type subset, SN SCP06E12 is eliminated as its host galaxy is dimmer than the $z_{850}=24$ cutoff used for this subset leaving $6$ SNe Ia with negligible classification error. The number of SNe Ia discovered in each subset, including classification error, is summarized in Table 7 under $N_{\rm SN~{}Ia}$. We normalize the rate in three different ways: by $B$-band luminosity, by $g$-band luminosity, and by stellar mass. For each cluster, we use the visibility time map $T(x,y)$ (e.g., Fig. 9) and the measured luminosity (or mass) profile to carry out the integral in equation (2) giving the time-luminosity searched. The sum of these values for all 25 clusters is the denominator of equation (1), the total time-luminosity searched in all clusters. This is shown in Table 7 under “Denom” for each sample. The rate is simply $N_{\rm SN~{}Ia}$ divided by “denom,” as in equation (1). The contributions to the statistical and systematic errors are summarized in Table 8. The weighted-average redshift, $\bar{z}$, for each subsample is given by \[\bar{z}=\frac{\sum_{i}z_{i}\int_{x,y}T_{i}(x,y)L_{i}(x,y)}{\sum_{i}\int_{x,y}T _{i}(x,y)L_{i}(x,y)},\] (12) where $z_{i}$, $L_{i}$ and $T_{i}$ are the redshift, luminosity and effective visibility time of the $i$-th cluster, respectively. The weighted-average redshift is slightly smaller for the red-sequence and red-sequence early-type galaxy subsets. This is because in the higher-redshift clusters, a smaller fraction of galaxies meet the subset requirements (see $z<1.2$ versus $z>1.2$ average cluster luminosity in Table 6). ### Summary of Systematic Uncertainties \| Full \| Red- \| Red-sequence ---\|---\|---\|--- \| cluster \| sequence \| early-type Source of error \| (%) \| (%) \| (%) Statistical Poisson \| +40−32 \| +45−35 \| +47−36 Luminosity (stat) \| ±12 \| ±6 \| ±6 Luminosity (cosmic var.) \| ±16 \| ±4 \| ±3 Total statistical \| +45−38 \| +46−35 \| +48−37 Systematic SN type classification \| ±13 \| ±8 \| Control time: varying MB \| +8−6 \| +8−6 \| +8−6 Control time: dust distribution \| +10−2 \| \| Luminosity: MAG_AUTO corr. \| ±7 \| ±7 \| ±7 Luminosity: K-correction \| ±3 \| ±3 \| ±3 Luminosity: Faint galaxy corr. \| +4−9 \| \| Luminosity: r>0.6(0.8) Mpc \| ±4 \| ±1 \| ±1 Total systematic \| +20−19 \| +14−12 \| +11−10 Total statistical + systematic \| +49−42 \| +48−37 \| +49−38 Table 8Sources of Uncertainty Throughout the paper, we have highlighted and addressed possible sources of systematic uncertainty. Here we summarize these sources. In Table 8 we show the relative contribution of each to the total systematic error, and compare to sources of statistical error. (1) _SN type classification:_ The uncertainty in the number of SNe observed in each galaxy subset was addressed in §6.1. The fractional error in the rate is simply the fractional error in the number observed. (2) _Control time: Varying $M_{B}$:_ In our control time simulations, we assumed a distribution of SN Ia light curve shapes and absolute magnitudes. To first order, the impact of these assumptions on the control time is captured by varying the assumed SN Ia absolute magnitude (§4.3). Variations of $\pm 0.2$ mag resulted in a rate change of ${}^{+8}_{-6}\%$ (3) _Control time: dust distribution:_ In §4.3 we assessed the impact of varying amounts of dust extinction on the control time. Assuming an unrealistically large amount of dust-affected SNe decreased the control time by 9% (increasing the SN rate by $10\%$), while decreasing the amount of dust-affected SNe increased the control time by $2\%$ (decreasing the SN rate by $2\%$). We do not apply this systematic error to the red-sequence or red-sequence early-type subsets, as we have independent evidence that the amount of dust is limited in these environments. (4) _MAG_AUTO correction:_ In computing the total $z_{850}$ luminosity of each galaxy, we made a correction to the MAG_AUTO magnitude ranging from $\sim$10% at $z_{850}=20$ to $\sim$30% at $z_{850}=25$. Varying the range of $n$ used in the simulation by $\pm 1$ affects the correction by $\pm 7\%$. (5) _$K$-correction:_ In §5.3, we noted that the scatter of BC03 templates about the best-fit $K$-correction is typically less than 0.03 mag. We use this value as the systematic error on the $K$-correction. (6) _Faint galaxy correction:_ The average correction $C$ reported in Table 5 is 1.054. Varying $M^{\ast}_{B}$ by $\pm$ 0.5 magnitudes results in an average correction of 1.032 and 1.092 for $-0.5$ and $+0.5$ magnitudes, respectively. Varying $\alpha$ by $\pm 0.2$ results in an average correction of 1.027 and 1.098 for $\alpha=-0.7$ and $-1.1$, respectively. Concurrently varying $M^{\ast}_{B}$ and $\alpha$ within the same ranges results in a minimum average correction of 1.015 ($M^{\ast}_{B}=-22.2$, $\alpha=-0.7$) and a maximum average correction of 1.154 ($M^{\ast}_{B}=-21.2$, $\alpha=-1.1$). Conservatively, we assign ${}^{+4\%}_{-9\%}$ as the systematic error on the rate associated with this correction. This error is not applied to the red-sequence or red-sequence early-type subsets because these subsets do not include light from galaxies below the detection threshold. (7) _Luminosity at large radii:_ In §5.5 we assumed a model for the cluster luminosity profile at $r>0.6$ Mpc (0.8 Mpc for red-sequence and red-sequence early-type subsets). Varying the model luminosity by $\pm 20\%$ resulted in a $\pm 4\%$ change in the full cluster rate. The change is much smaller ($\pm 1\%$) for the red galaxy subsets because the model is only used at $r>0.8$ Mpc. (8) _$M/L$ ratio:_ In §5.7 we used a $M/L$ ratio to translate stellar luminosity to stellar mass. Rather than estimating the absolute uncertainty in the $M/L$ ratio (which is strongly dependent on assumptions), we estimated the uncertainty in the _evolution_ of the $M/L$ ratio from low to high redshift. This is the relevant uncertainty for comparing rates at differerent redshifts in order to derive the SN Ia delay time distribution. We defer discussion of this uncertainty to §7.4 where we discuss uncertainties in the DTD. ### Effect of Varying Subset Requirements In selecting our red-sequence and red-sequence early-type galaxy subsamples, we required red-sequence galaxies to be within $\pm 0.2$ mag of the color of their cluster red sequence. For early-type galaxies, we required the asymmetry parameter to be $<0.1$ and the Gini coefficient to be $>0.40$. It is interesting to test the sensitivity of the results to variations in the requirements. In Figures 14 and 15 we vary the requirements and observe the effect on the rates. As requirements are made more strict (for example, narrowing the red sequence) the total mass of the sample decreases. At the same time, SNe fall out of the sample when their host galaxies are cut. The Poisson error increases as the number of included SNe shrinks. <figure><img src="content_image/1010.5786/x15.png"><figcaption>Figure 14.— The effect of varying the width of the red sequence. The nominalred-sequence rate result corresponds to a half-width of 0.20 mag. The innerand outer error bars represent the statistical and total uncertainty,respectively.</figcaption></figure> <figure><img src="content_image/1010.5786/x16.png"><figcaption>Figure 15.— The effect of varying the morphology parameter requirements.Negative Δ values correspond to a more strict selection and a higher-purityearly-type galaxy sample. The requirements are asymmetry <0.1+Δ and Ginicoefficient >0.40−Δ. The nominal red-sequence early-type rate corresponds toΔ=0. The red-sequence half-width is fixed at 0.2 mag. The inner and outererror bars represent the statistical and total uncertainty, respectively.</figcaption></figure> There is not a strong dependence of the SN Ia rate with galaxy color residual from the red sequence (Fig. 14). Even in cluster galaxies that lie in a tight range around the red-sequence ($\pm 0.08$ mag), we find a SN Ia rate consistent with the full cluster rate. Similarly, there is no significant rate trend with the purity of the early-type sample (Fig. 15). We happened to pick morphology requirements that yield a slightly higher rate than other choices, but such variations are expected with small-number statistics and are accounted for by the Poisson uncertainty in the result (Tables 7 and 8). Even in the most-selective subset ($\Delta=-0.04$), the rate is consistent with the full cluster rate. ## 7. Discussion and Conclusions ### Host-less Cluster SNe Ia As reported by Dawson09, we have discovered one potential host-less cluster SN Ia among the $8\pm 1$ cluster SNe Ia. SN SCP06C1 is projected near two possible host galaxies: A $z_{850}=21.6$ spiral galaxy $1^{\prime\prime}.1$ West of the SN, and a significantly fainter $z_{850}=24.6$ galaxy $0^{\prime\prime}.45$ ($\sim$3.5 kpc at the cluster redshift) Northeast of the SN (See Dawson09, Fig. 2). The galaxy-subtracted SN spectrum clearly shows a SN Ia at redshift $z=0.98$ near maximum light, consistent with the light curve fit. The redshift of $z=0.98\pm 0.01$ is consistent with the cluster redshift of 0.974. The bright spiral galaxy is actually in the background of the cluster, at $z=1.091$. Strong [Oii] emission is visible in the spectrum, along with Ca H & K and H$\delta$ absorption. Unfortunately, the small separation between the main galaxy and the smaller galaxy to the Northeast means that the spectrum of the smaller galaxy is dominated by light from the larger galaxy, making it impossible to assess a redshift. It is thus possible that the small galaxy is at the cluster redshift and is the actual host of the SN. Alternatively, the small galaxy might be at the same redshift as the larger galaxy and physically associated with it (either as a satellite galaxy or as part of the spiral structure of the galaxy). It is interesting to note that the SN is only $20^{\prime\prime}$ (160 kpc) projected radius from the center of the cluster, perhaps giving more weight to the hypothesis that it is associated with a diffuse intracluster stellar component. Not being able to confirm or reject this SN as host-less, we have an upper limit of one host-less SN out of a total of $8\pm 1$. Discovering one host-less SNe Ia out of seven total would imply an intrinsic host-less SN Ia fraction of $14\%^{+18\%}_{-7\%}$ (binomial $68\%$ confidence intervals), and a 95% upper limit of $<47\%$. This is broadly consistent with host-less SN Ia constraints at intermediate redshifts (Sharon et al., 2010) and at low redshift (Gal-Yam et al., 2003; Sand et al., 2011). At low redshift it has been possible to confirm the host-less nature of some SNe using deeper follow-up imaging, leading to better constraints. The upper limit of $<47\%$ is also consistent with direct measurements of intracluster light at low redshift, but does not strongly constrain evolution. A sample twice the size or larger, with deeper follow-up to confirm host-less SNe Ia would begin to place interesting constraints on hypotheses for the formation of the intracluster stellar component from $z>1$ to today. ### Comparison to Other Cluster Rate Measurements Cluster SN Ia rates have been reported at lower redshifts by several groups. In nearby ($z\lesssim 0.2$) clusters, measurements include those of Sharon et al. (2007) at $z\sim 0.14$, Mannucci et al. (2008) at $z\sim 0.02$, and Dilday et al. (2010) at $z\sim 0.09$ and $z\sim 0.22$. At intermediate redshifts, Sharon et al. (2010) recently reported the rate in $0.5<z<0.9$ clusters (median $z\sim 0.6$). At higher redshifts, Gal-Yam et al. (2002) placed the first constraints on the $z\gtrsim 0.8$ cluster rate using a sample of three clusters at $z=0.83$, $0.89$ and $z=1.27$. However, their SN sample included only one firm SN Ia at $z=0.83$. The resulting rate has correspondingly large uncertainties and essentially places only an upper limit on the $z>0.9$ cluster rate. Our result is thus a large step forward in the measurement of the SN rate in the highest-redshift clusters. In Figure 16 we compare our full cluster rate to the lower-redshift rate measurements that have been normalized by stellar mass, permitting a comparison across redshifts. Here we have made an adjustment to the value reported by Sharon et al. (2010). Sharon et al. used the mass-to-light ratio of Bell03 for the SDSS $g$ and $r$ bands, but did not apply a correction for evolution between $z\sim 0.6$ and $z=0$. Using the method described in §5.7 we find that a $-0.14$ dex offset should be applied to the mass to account for evolution from $z=0.6$ to $z=0$ (Fig. 13, right panel). We therefore adjust the reported rate of Sharon et al. upward by 0.14 dex ($38\%$). The rate compilation of Maoz10 reflects this adjustment. Whereas the adjusted Sharon et al. rate shows an indication that the cluster rate is increasing with redshift, for the first time we find an increasing rate with high significance ($>2\sigma$). We point out that the popular “$A+B$” model (Scannapieco & Bildsten, 2005) is insufficient for describing the change in cluster rate with redshift. In the $A+B$ model the SN rate is the sum of a term proportional to the total stellar mass and a term proportional to the recent star formation rate: $\mathcal{R}_{\rm SN~{}Ia}=AM_{\ast}+B\dot{M}_{\ast}$. This simple model is convenient for predicting the SN rate in environments with varying amounts of recent star formation as it accounts for the increased SN Ia rate at short delay times. (In fact, we use this model in Meyers11 to derive limits on the expected ratio of SNe Ia to SNe CC in early-type galaxies.) However, the model lacks theoretical motivation and breaks down in other situations. For example, Greggio et al. (2008) note that it cannot adequately describe the observed contribution from SNe with intermediate delay times (e.g., Totani et al., 2008). This point is reinforced by the observation of a changing cluster rate with redshift: In clusters, the $A$ component is dominant at all redshifts observed. As $M_{\ast}$ is not changing significantly with redshift, the rate would be expected to remain constant under this model. Instead, we require a DTD model wherein the rate decreases at large delay times (as it does in most theoretically motivated models). <figure><img src="content_image/1010.5786/x17.png"><figcaption>Figure 16.— Cluster rate measurements (all galaxy types) from this work andthe literature. The rate of Sharon et al. (2010) shown has been adjustedupward by 38% from the reported rate (see text). The top axis shows the timeelapsed since an assumed cluster formation redshift of zf=3. The _solid greyline_ represents the SN Ia rate for the best-fit power-law DTD: RSNIa(t)=Ψ(t)/m(t), where Ψ(t)∝ts. The _dotted grey lines_ show the range of 1σerror on s.</figcaption></figure> ### The Cluster SN Ia Delay Time Distribution The cluster rates constrain the SN Ia delay time distribution, $\Psi(t)$, over the range of delay times from a few Gyr to $\sim 10$ Gyr. To illustrate the cluster rate constraints, we parameterize the DTD with a power law in time: $\Psi(t)\propto t^{s}$. A power law is not only a convenient parameterization in the face of limited data, but is a theoretically motivated function for the DD scenario, where the late-time ($t\gtrsim 1$ Gyr) DTD shape is set by the distribution of WD separation after the second CE phase and the merger timescale due to gravitational radiation (Greggio, 2005). We make the approximation that all clusters formed in a single burst of star formation at $z_{f}=3$ and that the age of the stellar population therefore corresponds to the elapsed time from $z_{f}$ to the cluster redshift (Fig. 16, top axis). While clearly a simplification, a single star-formation burst captures the idea that the timescale over which star formation occured in cluster early-type galaxies is short compared to the time since star formation ceased. The assumed burst redshift $z_{f}=3$ is consistent with measurements of cluster early-type galaxies showing that star formation was mostly completed by this redshift (e.g., Gobat et al., 2008). Below, we show that the derived DTD is relatively insensitive to the redshift assumed. As noted in §5.7, the DTD is normalized by _initial_ stellar mass, whereas the cluster rate measurements (including ours, for consistency) have been normalized by _current_ stellar mass. The DTD, $\Psi(t)$, is therefore related to the cluster rate by $\Psi(t)=m(t)\mathcal{R}_{\rm SN~{}Ia}(t)$ where $m(t)$ is the fraction of stellar mass remaining at time $t$ after the star formation burst. The specific choice of $m(t)$ does not have a significant impact on the derived DTD: regardless of the model or IMF assumed, the stellar mass declines by only $\sim$10% over the age range of interest, $\sim 3$ to 11 Gyr. For consistency with Maoz10, we use the remaining stellar mass fraction tabulated by BC03, $m_{\rm BC03}(t)$, but corrected to $m(t)=1-(1-m_{\rm BC03}(t))/0.7$ to effectively convert from the Salpeter IMF used in BC03 to a “diet” Salpeter IMF. This correction has only a very small effect on the result (see below). We find a best-fit value of \[s=-1.41^{+0.47}_{-0.40},\] (13) using the statistical$+$systematic error (added in quadrature) reported for each rate measurement. In Figure 16, the solid grey line shows the best-fit cluster rate for this value: $\mathcal{R}_{\rm SN~{}Ia}(t)=\Psi(t)/m(t)$, where $\Psi(t)\propto t^{-1.41}$. Note that the $\chi^{2}$ of the best-fit model is surprisingly small: 0.40 for 4 degrees of freedom. The _a priori_ probability of finding a $\chi^{2}$ smaller than 0.40 is less than $2\%$. This is difficult to understand given that the measurement errors are generally dominated by Poisson noise in the number of SNe observed and are thus unlikely to be overestimated. The best-fit value is consistent with measurements of the late-time DTD in the field (Totani et al., 2008). Most predictions for the SD scenario show a steeper late-time DTD (Greggio, 2005; Ruiter et al., 2009; Mennekens et al., 2010) with an effective value for $s$ ranging from $s\sim-1.6$(Greggio, 2005) to $s<-3$(Mennekens et al., 2010), depending on the details of the scenario and binary evolution. However, some groups have found that the SD scenario could be consistent with a less-steep DTD ($s\sim-1$) given the right combination of main sequence and red giant secondaries (Hachisu et al., 2008). In the DD scenario, the predicted shape of the DTD depends on the distribution of binary separations after the common envelope phase of the WDs, a difficult distribution to predict. However, a slope of $s=-1.4$ (and a range of similar values) would not be surprising in the DD scenario. ### Additional DTD Systematic Uncertainties Variations in the assumed cluster star formation, initial mass normalization and mass-to-light ratio evolution have a small affect on $s$ compared to the measurement error. (1) _Age of clusters’ stellar populations:_ Above, we assumed a single burst of star formation at $z_{f}=3$. Moving this single burst to $z_{f}=4$ results in $s=-1.55$. A more recent burst, $z_{f}=2.5$, results in $s=-1.30$. Maoz10 give a treatment of variations from the single-burst approximation, also finding that the affect on $s$ is small. Our rate measurements in red and early-type galaxies provide a good consistency check that recent star formation does not significantly contribute to the SN Ia rate: if it did, we would observe a higher rate in the full cluster than in these subsamples. Surprisingly, we observe the opposite trend (although the significance is low). The red-sequence early-type subsample includes 53% of the stellar mass of the full cluster sample, and 6 SNe Ia. The remaining 47% of the full cluster sample (which includes bluer galaxies and late-type red-sequence galaxies) accounts for only $2\pm 1$ SNe Ia. At low redshift, Mannucci et al. (2008) found a similar trend between E/S0 galaxies and S0a/b galaxies within $0.5$ Mpc of cluster centers, though also at $<1\sigma$ significance. (2) _Remaining stellar mass:_ Whereas the DTD is normalized by initial stellar mass and cluster rate measurements have been normalized by current stellar mass, we have assumed a remaining stellar mass fraction $m(t)$ to convert from current to initial stellar mass. Although different models and IMFs can yield sigificantly different $m(t)$, we are only concerned here with the change in $m(t)$ between $\sim 3$ Gyr and at $\sim 11$ Gyr. (The absolute value of $m(t)$ affects only the normalization of $\Psi(t)$, with which we are not concerned.) Fortunately, the evolution in $m(t)$ in this age range is small and consistent between models, and so the effect on $s$ is small. For example, using $m_{\rm BC03}(t)$ (assuming a Salpeter IMF) rather than correcting to a diet Salpeter IMF (as we have done) only changes the best-fit value from $s=-1.41$ to $s=-1.38$. If in §5.7 we had used a $M/L$ ratio directly normalized by initial mass, rather than normalizing by current mass and later converting to initial mass, the results would be very similar. (We have not done this for consistency with other rate measurements.) In the pégase2 models in Figure 13 (left panel) evaluted at $z=1.2$, the ratio of current to formed stellar mass varies slightly across the models, but is fully contained in the range $0.66\pm 0.03$. The same models evaluted at $z=0$ have a ratio of $0.59\pm 0.03$. This is consistent with the $\sim 10\%$ evolution in $m(t)$ over this range as tabluated by BC03. (3) _$M/L$ ratio evolution:_ While the overall normalization of the $M/L$ ratio will only affect the normalization of $\Psi(t)$ and not $s$, the evolution of the $M/L$ ratio will affect $s$. In §5.7 we assigned a liberal 20% systematic uncertainty to the evolution of the $M/L$ ratio over the redshift range of interest. To estimate the effect of this systematic uncertainty, we adjust our rate measurement by 20% and that of Sharon et al. (2010) by 10% and refit $s$. The resulting change in $s$ for positive and negative shifts is $-0.15$ and $+0.18$ respectively, less than half of the nominal error in $s$. ### Conclusions In this paper, we have made a measurement of the high-redshift cluster SN Ia rate. Thanks to an unusually complete dataset (particularly for a rate study) the measurement is quite robust, with statistical and systematic uncertainties on par with or better than measurement uncertainties at low redshift. We highlight several important and/or unique aspects of the measurement: * The SN classification approach takes advantage of all relevant information. Thanks to the “rolling search” strategy of the survey and the nearly complete spectroscopic follow-up, most candidates have a full light curve and a host galaxy redshift, greatly reducing classification uncertainty. * The position-dependent control time allows one to calculate a supernova rate given an arbitrary observing pattern and luminosity distribution. * The control time calculation includes a full distribution of SN properties and the systematic uncertainty associated with the assumed distribution is carefully quantified. Thanks to the depth of the observations, the detection efficiency approaches 100% during the period of the survey for most of the clusters, meaning that the systematic uncertainty is low. * Statistical uncertainties associated with the cluster luminosities, including both statistical variations and cosmic variance, are included in the total uncertainty. Also, light in the outskirts of each galaxy (outside the SExtractor MAG_AUTO aperture) is accounted for. This is a significant component of the total cluster luminosity. * Cluster SN Ia rate measurements are normalized consistently across redshifts using a redshift-dependent mass-to-light versus color relation. For the first time our result shows at the $>2\sigma$ level that the cluster SN Ia rate is increasing with redshift. Simply by comparing the low- and high-redshift cluster rate measurements, the shape of the late-time SN Ia delay time distribution can be constrained. The power of the measurement for this purpose comes both from the high redshift and relatively low statistical and systematic uncertainties in the measurement. While we cannot conclusively rule out either the single degenerate or double degenerate class of progenitors via the delay time distribution, the binary evolution that could lead to each model are constrained. The DD scenario is consistent with the measurement under a wide range of plausible binary evolution parameters, while there is a stronger constraint on binary scenarios that could lead to an SD scenario. Finally, this measurement is unique in constraining the delay time distribution at delay times of a few Gyr. In future studies, it can be used in combination with other cluster rates and other delay time distribution measurements (e.g., Maoz10) to place even tighter constraints on models for binary evolution and SN Ia progenitor scenarios. We thank Eric Bell and Dan Maoz for helpful discussion. T. M. is financially supported by the Japan Society for the Promotion of Science (JSPS) through the JSPS Research Fellowship. C. L. is financially supported by the Australian Research Council (ARC) through the ARC Future Fellowship program. Financial support for this work was provided by NASA through program GO-10496 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. This work was also supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the U.S. Department of Energy under Contract No. AC02-05CH11231, as well as a JSPS core-to-core program “International Research Network for Dark Energy” and by a JSPS research grant (20040003). The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. 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1003.3975	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23104, "num_imgs": 0, "llama3_tokens_count": 6328 }	[]	# Wormholes and Child Universes E.I. Guendelman Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Day Month YearDay Month YearDay Month YearDay Month Year ###### Abstract Evidence to the case that classical gravitation provides the clue to make sense out of quantum gravity is presented. The key observation is the existence in classical gravitation of child universe solutions or ”almost” solutions , ”almost” because of some singularity problems. The difficulties of these child universe solutions due to their generic singularity problems will be very likely be cured by quantum effects, just like for example ”almost” instanton solutions are made relevant in gauge theories with breaking of conformal invariance. Some well motivated modifcations of General Relativity where these singularity problems are absent even at the classical level are discussed. High energy density excitations, responsible for UV divergences in quantum field theories, including quantum gravity, are likely to be the source of child universes which carry them out of the original space time. This decoupling could prevent these high UV excitations from having any influence on physical amplitudes. Child universe production could therefore be responsible for UV regularization in quantum field theories which take into account semiclassically gravitational effects. Child universe production in the last stages of black hole evaporation, the prediction of absence of tranplanckian primordial perturbations, connection to the minimum length hypothesis and in particular the connection to the maximal curvature hypothesis are discussed. Some discussion of superexcited states in the case these states are Kaluza Klein excitations is carried out. Finally, the posibility of obtaining ”string like” effects from the wormholes associated with the child universes is discussed. Child Universes; Wormholes; Quantum Gravity. Managing Editor February 26, 2024 ## 1 Introduction Quantum field theory and quantum gravity in particular suffer from UV divergences. While some quantum field theories are of the renormalizable type, quantum gravity is not and the UV divergences cannot be hidden into a finite number of ”counter-terms”. Perturbative renormalizability does not appear to be available for quantum gravity. In an apparently unrelated development, the ”child universe” solutions have been studied [1], [2]. These child universes are regions of space that evolve in such a way that they disconnect from the ambient space time. Inflationary bubbles of false vacuum correspond to this definition [1], [2]. In this case an exponentially expanding inflationary bubble arises from an ambient space time with zero pressure which the false vacuum cannot displace. The inflationary bubbles disconnect from the ambient space generating a child universe. There are difficulties with these child universe solutions due to their generic singularity problems [3] which will very likely be cured by quantum effects, just in the way that for example ”almost” instanton solutions are made relevant in gauge theories even with breaking of conformal invariance (this breaking does not permit the existence of exact classical solutions) [4]. Other avenues for the resolution of the initial singularity problem of these child universe solution include an initial semiclassical tunneling region that replaces the singularity [5], the consideration of violation of energy conditions[6] which itself can originate from quantum effects or the non existence of a Cauchy surface [7]. There are also some well motivated modifications of General Relativity where singularity problems could be avoided. For example, in this conference we have heard in the talk by Walter Greiner on his work with Hess on Pseudo Complex General Relativity [8], which could do this. Here we want to explore the possibility that high energy density excitations, associated to the UV dangerous sector of quantum field theory could be the source of child universes, which will carry the UV excitations out of the original ambient space time. Child universe production could be therefore responsible for UV softening in quantum field theory that takes into account gravitational effects. It implies also the existence of a maximum energy density and curvature. We will now show now, using a simple model, that very high UV excitations have appreciable tendency to disconnect from the ambient space time ## 2 The Super High UV Bubble We describe now the model [12] which we will use to describe a high UV excitation which will be associated with the production of a child universe. This model for high UV excitation will consist of a bubble with very high surface tension and very high value of bulk energy density inside the bubble. The entire space-time region consists of two regions and a boundary: 1) Region I de Sitter space 2) Region II, Schwarzschild space and the domain wall boundary separating regions I and II. In Region I: The de Sitter space. The line element is given by \[ds^{2}=-(1-\chi^{2}r^{2})dt^{2}+(1-\chi^{2}r^{2})^{-1}dr^{2}+r^{ 2}d\Omega^{2}\] (1) where $\chi$ is the Hubble constant which is given by \[\chi^{2}=\frac{8}{3}\pi G\rho_{0}\] (2) $\rho_{0}$ being the vacuum energy density of the child universe and $G=\frac{1}{m_{P}^{2}}$ where $m_{P}=10^{19}$ GeV. In Region II: The Schwarzschild line element is given by \[ds^{2}=-(1-\frac{2GM}{r})dt^{2}+(1-\frac{2GM}{r})^{-1}dr^{2}+r^{ 2}d\Omega^{2}\] (3) The Einstein‘s field equations, \[R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}.\] (4) are satisfied in regions I and II and determine also the domain wall evolution [1], using the methods developed by Israel [13]. Using gaussian normal coordinates, which assigns to any point in space three coordinates on the bubble and considers then a geodesic normal to the bubble which reaches any given point after a distance $\eta$ ( the sign of $\eta$ depends on which side of the bubble the point is found). Then energy momentum tensor $T_{\mu\nu}$ is given by \[T_{\mu\nu}(x) =-\rho_{0}g_{\mu\nu},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}(\eta<0){\rm~{}for~{}the~{}child~{}universe,~{}~{}~{}~{}{\bf Region~{}I},~ {}(negative~{}pressure)}\] (5) \[=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}(\eta>0){\rm for~{}the~{}Schwarzschild~{}~{}region,~{}~{}~{}~{}{\bf RegionII}}\] \[=-\sigma h_{\mu\nu}\delta(\eta)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~ {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm for~{}the~{}domain~{ }wall~{}boundary}.\] where $\sigma$ is the surface tension and $h_{\mu\nu}$ is the metric tensor of the wall, that is $h_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$, $n_{\mu}$ being the normal to the wall. The eq. of motion of the wall give[1] \[M_{\rm cr}=\frac{1}{2G\chi}\frac{\gamma^{3}z_{m}^{6}(1-\frac{1}{ 4}\gamma^{2})^{\frac{1}{2}}}{3\sqrt{3}(z_{m}^{6}-1)^{\frac{3}{2}}}\] (6) where the $M_{\rm cr}$ is the mass at (or above) which there is classically a bubble that expands to infinity into a disconnected space, the child universe. In the above equation \[\gamma=\frac{8\pi G\sigma}{\sqrt{\chi^{2}+16\pi^{2}G^{2}\sigma^{2 }}}\] \[z_{m}^{3}=\frac{1}{2}\sqrt{8+(1-\frac{1}{2}\gamma^{2})^{2}}- \frac{1}{2}(1-\frac{1}{2}\gamma^{2})\] (7) where $z^{3}=\frac{\chi_{+}^{2}r^{3}}{2GM}$ and $\chi_{+}^{2}=\chi^{2}+\kappa^{2}$, $\kappa=4\pi G\sigma$. $r_{m}$ is the location of the maximum of the potential barrier that prevents bubbles with mass less than $M_{\rm cr}$ to turn into child universes. We expect this representation of a high UV excitation to be relevant even for a purely gravitational excitation, which can be associated , after an appropriate averaging procedure, to an effective energy momentum, a procedure that gets more and more accurate in the UV limit. Let us now focus our attention on the limit where $\sigma\rightarrow\infty$ (while $\rho_{0}$ is fixed), which we use as our first model of a super UV excitation. Then, we see that $\gamma\to 2$ and $M_{\rm cr}\to 0$. Alternatively, we could obtain another model of super UV excitation, by considering the energy density inside the bubble, $\rho_{0}\rightarrow\infty$ , while keeping $\sigma$ fixed. This also leads to $M_{\rm cr}\to 0$ as well. Finally, letting both $\sigma\rightarrow\infty$ and $\rho_{0}\rightarrow\infty$ while keeping their ratio fixed, also leads to $M_{\rm cr}\to 0$. In all these limits we also get the the radius of the critical bubble $r_{m}\to 0$ as well. In [1] the above expression for $M_{\rm cr}$ was explored for the case that energy densities scales (bulk and surface) were much smaller than the Planck scale, like the GUT scale. This gave a value for $M_{\rm cr}=56kg>>m_{p}$. Here we take the alternative view that the scale of the excitations are much higher than the Planck scale, giving now an arbitrarily small critical mass. Defining the ”scale of the excitation” through by $\rho_{0}\equiv M_{exc}^{4}$, then the pre-factor $\frac{1}{2G\chi}$ in eq (6), goes like $\left(\frac{m_{p}}{M_{exc}}\right)^{2}m_{p}$. We see that for trans planckian excitations, i.e. if $M_{exc}>>m_{p}$, we obtain a very big reduction for $M_{\rm cr}$. This is a kind of ”see saw mechanism”, since the higher the $M_{exc}$, the smaller $M_{\rm cr}$. This means that in these models for high UV excitations there is no barrier for the high UV excitation to be carried out to a disconnected space by the creation of a child universe. Notice also the interesting ”UV -IR mixing” that takes place here: although we go to very high UV limits in the sense that the energy density in the bulk or the surface energy density are very high, the overall critical mass goes to zero. One should notice that these limits where we take the surface tension or the energy density to very big values can be achieved as we go to early times (corresponding to the time of the creation of the child universes) in models where these quantities are dynamical variables. In this context [14] when considering for example models with dynamical tension one can show the existence of child universe production where the critical mass $M_{\rm cr}$ is indeed zero. The difficulties related to the singularities for the solutions with $M>M_{\rm cr}$ should be solved as $M_{\rm cr}$ approaches zero, because then even a very small quantum fluctuation should be able to wash out this singularity if the mass is very, very small (the strength of the singularity is associated to the mass of the solution). ## 3 The String Gas Shell Example Crucial parameters in the child universe formation models based on the description of vacuum bubbles in terms of thin relativistic shells are, apart from the total mass–energy of the asymptotically flat region, the false vacuum energy density and the shell surface tension. Let us then consider two spacetime domains ${\mathcal{M}}_{\pm}$ of two $(3+1)$-dimensional spacetimes, separated by an infinitesimally thin layer of matter $\Sigma$, a _shell_. We will also assume spherical symmetry: this simplifies the algebra, and is a non-restrictive assumption which almost always appears in the literature. Moreover, as a concrete case [9] we will choose the one in which ${\mathcal{M}}_{-}$ is a part of Minkowskii spacetime and ${\mathcal{M}}_{+}$ is a part of Schwarzschild spacetime. Then, the equations of motion for the shell, i.e. Israel junction conditions, reduce to the single equation \[\epsilon_{-}\sqrt{\dot{R}^{2}+1}-\epsilon_{+}\sqrt{\dot{R}^{2}+1-2GM/R}=Gm(R)/R,\] (8) where $G$ is the gravitational constant and the only remaining degree of freedom is $R(\tau)$, the radius of the spherical shell expressed as a function of the proper time $\tau$ of an observer co-moving with the shell. $\epsilon_{\pm}=\mathrm{sgn}(n^{\mu}\partial_{\mu}r)\rceil_{\mathcal{M}_{\pm}}$ are signs, expressing the fact that the radial coordinate $r$ can increase ($\epsilon_{\pm}=+1$) or decrease ($\epsilon_{\pm}=-1$) along the normal direction, defined by $n^{\mu}\rceil_{{\mathcal{M}}_{\pm}}$ in ${\mathcal{M}}_{\pm}$, respectively (our convention is that the normal is pointing from ${\mathcal{M}}_{-}$ to ${\mathcal{M}}_{+}$). The function $m(R)$ is related to the energy–matter content of the shell, and is what remains of the shell stress–energy tensor in spherical symmetry after relating the pressure $p$ and the surface energy density $\rho$_via_ an equation of state. Let us discuss this point in more detail, since our choice will be slightly unusual compared to the existing literature. We will, in fact, use $p=-\rho/2$, $p$ being the uniform pressure and $\rho$ the uniform energy density on $\Sigma$. A string gas in $n$ spatial dimensions satisfies $p=-\rho/n$, therefore the two dimensional shell $\Sigma$ we are dealing with is a _sphere of strings_. This, gives $\rho=\rho_{0}/R$, where $\rho_{0}$ is a constant, and then, $m(R)=cR$, with $c=4\pi\rho_{0}$. Making the above choice, we can then solve the junction condition to obtain the dynamics of the system. It can be seen that solving (8) is equivalent to solving the equivalent effective classical problem \[\dot{R}^{2}+V(R)=0,\quad V(R)=1-\frac{1}{4c^{2}}\left(\frac{2M}{R}+Gc^{2} \right)^{2},\] (9) with the signs determined as $\epsilon_{-}=+1$ and $\epsilon_{+}=\mathrm{sgn}(2M/R-Gc^{2})$. The potential has the following additional properties: \[\lim_{R\to 0^{+}}V(R)=-\infty,\:\lim_{R\to\infty}V(R)=1-\frac{G^{2}c^{2}}{4}, \:\frac{dV(R)}{dR}>0.\] This shows that i) we can have unbounded trajectories only if $c\geq 2/G$ and ii) this is independent from the choice of $M>0$; moreover, from the result for $\epsilon_{+}$, we have that certainly iii) on all the unbounded trajectories $\epsilon_{+}$ changes sign (being positive for small enough $R$ and negative for large enough $R$; the general property that this change of sign must happen behind an horizon, is also obviously satisfied since $c\geq 2/G$, so that at $R=2GM$ the sign $\epsilon_{+}$ is already negative although for small enough $R$ it is positive). In view of the global spacetime structure associated with the above properties of _all_ the unbounded solutions , it is clearly seen that they realize the formation of a baby universe. This happens for a large enough density of strings and _for any_ positive value of $M$. This simple model shows therefore very neatly the phenomena of ”child universes out of almost empty space”. At this point we should mention some additional evidence that string matter has some peculiar features related to its capability of being responsible to produce universes out of nothing, see for example the interesting arguments of Trevisan presented in a poster in this conference, based on work by Berman and Trevisan [10]. Another interesting fact in this respect is that a gas of string matter does not curve the spacetime in the context of the recently formulated theory with a dynamical time [11]. ## 4 The Conjecture This allows us to formulate the conjecture that the dangerous UV excitations that are the source of the infinities and the non renormalizability of quantum gravity are taken out of the original space by child universe production, that is, the consideration of child universe production in the ultrahigh (trans planckian) sector of the theory could result in a finite quantum gravity, since the super high UV modes, after separating from the original space will not be able to contribute anymore to physical processes. The hope is that in this way, child universes could be a of interest not only in cosmology but could become also an essential element necessary for the consistency of quantum gravity. One situation where all the elements required (high energy densities , since the temperature is very big) necessary for obtaining a child universe appears to be the late stages of Black Hole evaporation. If the ideas explained here are correct, we should not get contributions to primordial density perturbations from the trans planckian sector, since these perturbations would have disconnected from our space time. Also, any attempt to measure distances smaller than the Planck length will be according to this also impossible since such a measurement will involve exciting a high UV excitation that will disconnect. This means that there must be a minimum length that we could measure, of the order of the Planck scale. It appears there is a maximal energy density according to this, since now bubbles with high energy density will be quickly disconnected, being replaced in the observable universe by regions of Schwarzschild space, which has zero energy density, i.e., a very big energy density must decay in the observable universe. The ”maximal curvature”[15] hypothesis (here we focus on scalar curvature) is justified by this maximal energy density result, if we use eq. (4). An effective dynamics that takes into account the effect of child universe production (i.e. integrates out this effect) could resemble indeed that of obtained using the maximal curvature hypothesis [15]. ## 5 Super Excited Kaluza Klein (KK) Modes The gravitational effects on the spectrum of modes of particles with momentum in the direction of some periodic dimension have been analyzed [16]. It was found there that the naive, uniformly spaced KK excitation spectrum gets drastically modified, since the size of the compact dimentions likes to grow near the region where the KK excitation lives. This leads to many orders of magnitude decrease of the energy of these KK excitations. The modification of the spectrum of KK excitations due to the growing of the extra dimensions was studied also in [17]. Furthermore, once a configuration like that has been created, it is energetically possible for these modes to become superheavy by the recolapse of the extra dimension provided a child universe with an associated wormhole region is created [18] . The superheavy modes must then decouple from the ambient universe for this to happen, in agreement with our general picture. ## 6 Child Universes, Wormholes and Strings The creation of a child universe implies the creation of a wormhole region. Static wormhole configurations have been studied since the first construction by Einstein and Rosen [19], which consisted of simply joining two exterior Schwarzschild at the horizon, producing a doubling of the space for $r>2M$, but the elimination of the space $r<2M$. Although not noticed by Einstein and Rosen, the consistency of this construction (that is in order for it to be a solution of Einstein‘s equations) requires the existence of light like matter at $r=2M$[20]. Generally wormholes are considered by joining exterior solutions outside the horizon $r>2M$, for a review see [21]. Tranversable wormholes require in general exotic matter. Electric field lines can go from one universe to the other going through the wormhole and causing the appearence of charge, positive appearence on one universe and negative appearence on the other universe as has been pointed out by Misner and Wheeler [22]. The gauge fields that go from one side of the wormhole to the other can be used to construct wormhole throats which can be very, very long [23], [24]. Then it so happens that these very long wormhole throats have a dynamics that mimics that of string theory[24], which raises the interesting question of whether wormhole theory and in particular child universe theory could be origin of string theory, which could appear as the effective description of the dynamics of these long wormhole throats. ## Acknowledgments I want to thank the organizers of IWARA2009 conference for inviting me to this very interesting event and for support, to Vladimir Dzhunushaliev for reading the manuscript and for interesting comments, to Walter Greiner for interesting conversations concerning avoidance of singularities in his model and relevance to a child universe production, to Marcelo Samuel Berman and Luis Augusto Trevisan for interesting conversations concerning the possibility of creating a universe out of nothing. This manuscript was prepared while I was visiting the Astrophysics and Cosmology Group at the Pontificia Universidad Catolica de Valparaiso, Chile. ## References * [1] S. K. Blau, E. I. Guendelman and A. H. Guth, Phys. Rev. D 35, 1747 (1987). * [2] For a review see S. Ansoldi , E. I. Guendelman, I. Shilon, e-Print: arXiv:0711.2198 [gr-qc] * [3]E. Farhi, A. H. Guth, Phys.Lett.B183:149,1987. * [4]G. ’t Hooft, Phys.Rev.D14:3432-3450,1976, Erratum-ibid.D18:2199,1978, I. Affleck, Nucl.Phys.B191:429,1981. * [5]E. Farhi, A. H. Guth and J.Guven, Nucl.Phys.B339:417,1990. * [6]E. I. Guendelman, N. Sakai, Phys.Rev.D77:125002,2008, Erratum-ibid.D80:049901,2009. (arXiv:0803.0268 [gr-qc]) * [7]N. Sakai, K-ichi Nakao, H. Ishihara, M. Kobayashi, Phys.Rev.D74:024026,2006. (gr-qc/0602084) , A. Borde, M. Trodden, T. Vachaspati, Phys.Rev.D59:043513,1999. (gr-qc/9808069) * [8] P. O. Hess, W. Greiner, Int.J.Mod.Phys.E18:51,2009 ( arXiv:0812.1738 [gr-qc]) . * [9] S. Ansoldi and E. I. Guendelman, Prog.Theor.Phys.120:985-993,2008 (arXiv:0706.1233 [gr-qc]). * [10]M. S. Berman , L. A. Trevisan , gr-qc/0104060 . * [11] E.I. Guendelman, arXiv:0911.0178 [gr-qc] . * [12] E.I. Guendelman, Int.J.Mod.Phys.D17:551,2008 (gr-qc/0703105). * [13]W. Israel, Nuovo Cim.B44S10:1,1966, Erratum-ibid.B48:463,1967, Nuovo Cim.B44:1,1966. * [14] S. Ansoldi and E.I. Guendelman, Prog.Theor.Phys.120:985-993,2008 (arXiv:0706.1233 [gr-qc]). * [15] M.Markov, Pis‘ma Zh. Eksp. Theor. Fiz. 36, 214 (1982); ibid 46, 342 (1987); V. P. Frolov, M.A. Markov, V. F. Mukhanov, Phys.Rev.D41:383,1990; V. F. Mukhanov and R. Brandenberger, Phys. Rev. Lett. 68, 1969 (1992), D. A. Easson, JHEP 0302:037,2003. * [16] E.I.Guendelman, Int.J.Mod.Phys.A4:5119,1989, E.I.Guendelman, Gen.Rel.Grav.22:131,1990. * [17]J. R. Morris, Phys.Rev.D67:025005,2003, (hep-th/0211175), E.I. Guendelman, J. R. Morris, Phys.Rev.D68:045008,2003. (hep-th/0307012) , J. R. Morris, Phys.Rev.D70:025007,2004. (hep-th/0405188) , E.I. Guendelman, John R. Morris, Phys.Lett.B608:194,2005. (hep-ph/0501147) . * [18]E.I. Guendelman and D.A.Owen, Gen. Rel. and Grav. 21, 201, 1989. * [19] A. Einstein, N. Rosen, Phys.Rev.48:73-77,1935. * [20]E. I. Guendelman, A. Kaganovich, E. Nissimov, S. Pacheva, Phys.Lett.B681:457, 2009 ( arXiv:0904.3198 [hep-th]). * [21]Lorentzian wormholes: From Einstein to Hawking. Matt Visser, (Washington U., St. Louis) . 1995. Woodbury, USA: AIP (1995) 412 p. * [22] C. W. Misner, J. A. Wheeler, Annals Phys.2:525,1957. * [23] E.I. Guendelman, Gen.Rel.Grav.23:1415,1991. * [24]V. Dzhunushaliev, Class.Quant.Grav.20:2407,2003 (gr-qc/0301118), V. Dzhunushaliev, Gen.Rel.Grav.35:1481,2003 ( gr-qc/0301046), V. Dzhunushaliev, Class.Quant.Grav.19:4817,2002 (gr-qc/0205055), V. D. Dzhunushaliev , U. Kasper, D. Singleton, Phys.Lett.B479:249,2000 (gr-qc/9910092).
1109.3234	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 20156, "num_imgs": 2, "llama3_tokens_count": 5631 }	[ "content_image/1109.3234/x1.png", "content_image/1109.3234/x2.png" ]	# Lagrangian time correlations of vorticity alignments in isotropic turbulence: observations and model predictions Laurent Chevillard${}^{1}$ Charles Meneveau${}^{2}$ ${}^{1}$Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS, Université de Lyon, 46 allée d’Italie F-69007 Lyon, France ${}^{2}$Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA ###### Abstract Motivated by results from recent particle tracking experiments in turbulence (Xu et al., Nat. Phys. 7, 709 (2011)), we study the Lagrangian time correlations of vorticity alignments with the three eigenvectors of the deformation-rate tensor. We use data from direct numerical simulations (DNS), and explore the predictions of a Lagrangian model for the velocity gradient tensor. We find that the initial increase of correlation of vorticity direction with the most extensive eigen-direction observed by Xu et al. is reproduced accurately using the Lagrangian model, as well as the evolution of correlation with the other two eigendirections. Conversely, time correlations of vorticity direction with the eigen-frame of the pressure Hessian tensor show differences with the model. pacs: 02.50.Fz, 47.53.+n, 47.27.Gs In a recent communication [1], the inertial range vorticity vector defined as the rotational motion of an ensemble of four Lagrangian particles in a turbulent flow (“tetrad” [2]) has been shown, at short times, to display growing alignment and correlation with the eigenvector associated to the most positive eigenvalue of the associated initial deformation tensor. It is well established [3; 4; 5] that at any given time the alignment between vorticity is most likely to be with the intermediate strain-rate eigenvector. However, how said correlation changes as function of time-delay between vorticity and strain-rate is an aspect of turbulence fine-scale structure that had not received much attention before. The time evolution of alignment correlations provides an interesting observable on which turbulence models can be tested. The evolution of alignments between material lines and vorticity, and viscous mechanisms of tilting of vorticity, have been studied in experiments and simulations [6; 7]. In this Letter, we examine the Lagrangian time correlation of vorticity and the various eigenvectors of the deformation rate tensor, as predicted by a Lagrangian stochastic model [8; 5]. Results are compared with the results of [1] and additional data from Direct Numerical Simulation (DNS) of the Navier-Stokes equations in moderate Reynolds number isotropic turbulence. In the same spirit as the analysis proposed in Ref. [1], we also study the short time alignment of vorticity with the pressure Hessian eigen-vector directions. The pressure Hessian tensor is a key quantity in the transport equation of the velocity gradients A, where $A_{ij}=\partial u_{i}/\partial x_{j}$ and u is the velocity vector, given by \[\frac{d\textbf{A}}{dt}=-\textbf{A}^{2}-\textbf{P}+\nu\Delta\textbf{A}\mbox{ ,}\] (1) where $d/dt$ is the Lagrangian time derivative, $P_{ij}=\partial p/\partial x_{i}\partial x_{j}$ the pressure Hessian (divided by fluid density) and $\nu$ the fluid’s kinematic viscosity. The results obtained from DNS are compared with the predictions of the Lagrangian stochastic model of Ref. [8]. As in Ref. [1], we focus now on the Lagrangian time correlation of vorticity ${\mbox{\boldmath$\omega$}}={\mbox{\boldmath$\nabla$}}\wedge\textbf{u}$ with eigen-vectors of the rate of deformation tensor $\textbf{S}=(\textbf{A}+\textbf{A}^{\top})/2$. We define, as in Ref. [1], the unit-norm vorticity vector direction $\textbf{e}_{\omega}(t)={\mbox{\boldmath$\omega$}}(t)/\|{\mbox{\boldmath$\omega$ }}(t)\|$ and the orthonormal eigenframe $(\textbf{e}_{1}(t),\textbf{e}_{2}(t),\textbf{e}_{3}(t))$ of S associated to its 3 ordered eigenvalues $\lambda_{1}(t)>\lambda_{2}(t)>\lambda_{3}(t)$. Incompressible fluids are considered, so that $\lambda_{1}(t)+\lambda_{2}(t)+\lambda_{3}(t)=0$ which implies that $\lambda_{1}(t)\geq 0$ and $\lambda_{3}(t)\leq 0$. Following Ref. [1], who focused on the most extensive eigen-direction ($\textbf{e}_{i}(t)$ with $i=1$), the following Lagrangian correlation function is considered \[C_{i}(\tau)=\langle[\textbf{e}_{i}(t).\textbf{e}_{\omega}(t+\tau)]^{2}\rangle \mbox{ ,}\] (2) for $i\in\{1,2,3\}$ and the temporal displacement $\tau$ (and averaging in time $t$) is performed along Lagrangian trajectories. <figure><img src="content_image/1109.3234/x1.png"><figcaption>Figure 1: (a) and (e) Correlation Ci(τ) (Eq. 2) of the vorticity directioneω(t) and the eigenframe of the deformation rate, ei(t) in DNS flows and fromthe model (Eq. 3): dashed line for maximal straining direction (i.e. i=1),solid line for intermediate straining direction (i=2), and dot-dashed line formost contractive eigendirection (i=3). (b,c,d) and (f,g,h) Probability DensityFunctions (PDFs) of \|ei(t).eω(t+τ)\| at various time lag τ represented bydifferent symbols, i.e. 0=τ∘<τ□<τ⋄<τ▽ and indicated in (a) and (b).</figcaption></figure> We consider predictions from the Lagrangian model developed in Ref. [8] for the velocity gradient tensor in turbulent flows and given explicitly by the following stochastic differential equation \[d\textbf{A}=\left(-\textbf{A}^{2}+\frac{\mbox{Tr}(\textbf{A}^{2})}{\mbox{Tr}( \textbf{C}_{{\tau_{k}}}^{-1})}\textbf{C}_{{\tau_{k}}}^{-1}-\frac{\mbox{Tr}( \textbf{C}_{{\tau_{k}}}^{-1})}{3T}\textbf{A}\right)dt+d\textbf{W}\mbox{ .}\] (3) The second and third terms in the right hand side of Eq. 3 are closures for, respectively, (minus) the pressure Hessian and viscous Laplacian that govern the time evolution of A, as it can be seen by comparing with Eq. 1 (see Ref. [5] for a review of this and other models). The term W is a tensorial delta-correlated noise term that has been added in order to represent possible forcing effects, e.g. from neighboring eddies. The “recent Cauchy-Green tensor” $\textbf{C}_{\tau_{k}}$, which arises after invoking the “recent fluid deformation” approximation [8; 9], can be expressed in terms of matrix exponentials: \[\textbf{C}_{\tau_{k}}=e^{\tau_{k}\textbf{A}}e^{\tau_{k}\textbf{A}^{\top}}\mbox { ,}\] (4) where $\tau_{k}$ is the Kolmogorov time-scale (see Ref. [9] for details). Hence, the modeled pressure Hessian tensor $\textbf{P}=-\frac{\mbox{Tr}(\textbf{A}^{2})}{\mbox{Tr}(\textbf{C}_{{\tau_{k}}} ^{-1})}\textbf{C}_{{\tau_{k}}}^{-1}$ can be highly anisotropic, mirroring the deformation undergone by the fluid as represented by the inverse of $\textbf{C}_{{\tau_{k}}}$. If one is interested in the velocity gradient tensor of a ‘coarse-grained’ velocity field (such as occurs in large-eddy simulations at length-scale $\Delta$, or in the context of “tetrads” when they span a typical scale $\Delta$ in the inertial range), the Lagrangian stochastic model may also be interpreted as a model for the coarse-grained velocity gradient tensor if the Kolmogorov time-scale $\tau_{k}$ is replaced by the corresponding eddy turn-over time at the appropriate scale $\tau_{\Delta}=\epsilon^{-1/3}\Delta^{2/3}$, where $\epsilon$ is the mean dissipation. In Eq. 3, the viscous term includes the time-scale $T$, corresponding to the Lagrangian integral time-scale of the velocity [8]. The model (Eq. 3), consisting of 9 (8 independent) stochastic differential equations, can be run with arbitrary initial conditions and it generates stationary statistics for the velocity gradient tensor elements. In particular, signals corresponding to the time histories of the antisymmetric part of $A_{ij}$ (the vorticity vector and its direction $\textbf{e}_{\omega}(t)$) and of the directions of the strain-rate eigenvectors $\textbf{e}_{i}(t)$ are readily obtained from the model runs. The correlation functions are then evaluated by averaging signals over long ($10^{6}~{}T$) records of model signals. In the following, as comparison we will also make use of data from a standard direct numerical simulation (DNS) of the Navier-Stokes equations. DNS is based on a pseudo-spectral (de-aliased according to the $\frac{3}{2}$-rule) method with 2nd-order accurate Adams-Bashforth time stepping; the computational box is cubic (size $2\pi$) with periodic boundary conditions in the three directions and spatial resolution of $256^{3}$. Statistical stationarity is maintained by an isotropic external force acting at low wavenumbers in order to ensure a constant power injection. It provides, in the units of the simulation, a constant energy injection rate of $\epsilon=0.001$. The kinematic viscosity of the fluid is $\nu=0.0004$. The Kolmogorov scale is $\eta_{K}=0.016$ so that $dx/\eta_{K}\approx 1.5$ (with $dx=2\pi/256$). The Taylor-based Reynolds number is of order $R_{\lambda}\sim 125$. The Lagrangian stochastic model is run with the parameter $\tau_{k}/T=0.1$, which is appropriate for modeling turbulence with $R_{\lambda}\sim 150$[9]. We compare in Fig. 1(a) and (e) the correlation functions $C_{i}(\tau)$ as predicted from the model runs (Eq. 3) and measured from DNS. For both the DNS and the model, time is normalized by the “integral” correlation time $\tau_{\eta}$ of a single tensor component (e.g. $A_{11}$), namely \[\tau_{\eta}=\frac{1}{\langle A_{11}^{2}\rangle}\int_{0}^{+\infty}\langle A_{11 }(t)A_{11}(t+\tau)\rangle d\tau\mbox{ .}\] It is found that in DNS, $\tau_{\eta}\approx 2\tau_{k}$ with $\tau_{k}=\sqrt{\nu/\epsilon}$ and for the model, $\tau_{\eta}\approx 1.2\tau_{k}$ with $\tau_{k}/T=0.1$. We see first that the general trends observed in DNS, and as reported in Ref. [1] from DNS and for inertial-range tetrads, are reproduced quite well by the stochastic model (Eq. 3). More specifically, for time lags $\tau\lesssim\tau_{\eta}$ we see that $\textbf{e}_{1}$ and $\textbf{e}_{\omega}$ are increasingly correlated, meaning that we observe, at short times, a more and more pronounced alignment of vorticity with the most extensive eigen-direction of the strain-rate tensor. This was one of the main observations reported in Ref. [1]. The general trend of initial alignment with the most extensive strain-rate direction is expected from vortex stretching, as discussed (e.g.) in Majda (1991) [10]. Conversely, $C_{3}(\tau)$ decreases at short times, i.e. vorticity decorrelates with the most contractive direction. As far as the intermediate eigen-direction is concerned, $C_{2}(\tau)$ decreases monotonically towards the uncorrelated limiting value $C_{i}(\tau)=1/3$. The model shows a slightly faster decay compared to the DNS, in the units of $\tau_{\eta}$. <figure><img src="content_image/1109.3234/x2.png"><figcaption>Figure 2: (a) and (e) Intercorrelation Ci(τ) (Eq. 2) of the vorticitydirection eω(t) and the eigenframe bi(t) of the deviatoric part of thepressure Hessian Pd in DNS flows and from the model (Eq. 3): dashed line formaximal eigenvalue β1, solid line for intermediate eigenvalue β2), and dot-dashed line for most negative eigenvalue β3. Panels (b,c,d) and (f,g,h) arePDFs of \|bi(t).eω(t+τ)\| at various time lag τ represented by differentsymbols, i.e. 0=τ∘<τ□<τ⋄<τ▽ and indicated in (a) and (b).</figcaption></figure> As proposed in Ref. [1], we also display in Figs. 1(b,c,d) for DNS, and in Figs. 1(f,g,h) for the model, the probability density functions (PDFs) of the cosine of the angles between the direction of vorticity at time $t+\tau$ and the three initial eigen-directions of the deformation at time $t$. Remarkably, the trends and results observed in DNS are reproduced quite well by the model. At vanishing time-lags $\tau=0$ (symbols $\circ$), $\textbf{e}_{\omega}$ is uncorrelated with $\textbf{e}_{1}$, mostly aligned with $\textbf{e}_{2}$ and mostly orthogonal to $\textbf{e}_{3}$. Then, for the first non vanishing time lag $\tau_{\square}$, we observe that $\textbf{e}_{\omega}(t+\tau_{\square})$ becomes more aligned with $\textbf{e}_{1}(t)$, less aligned with $\textbf{e}_{2}(t)$ and more orthogonal to $\textbf{e}_{3}(t)$. For later time lags $\tau_{\diamond}<\tau_{\triangledown}$, alignments of vorticity with the eigenframe relax towards the uncorrelated situation, in which the PDF is flat. We see thus that the alignments of vorticity with the eigen-frame of the deformation rate tensor are well predicted by the stochastic Lagrangian model (Eq. 3). A key reason is that the model contains the exact “velocity gradient self-stretching term” $-{\textbf{A}}^{2}$ in its right hand side. This term includes the familiar vortex stretching mechanism that determines the antisymmetric part of the tensor and hence the evolution of vorticity. Therefore, it may be expected that other Lagrangian stochastic models such as the model with prescribed log-normal dissipation [11], the “tetrad model” [2] or the “Lagrangian linear diffusion model” [12] that include the velocity gradient self-stretching term, should also display the trend of growing alignment of vorticity direction with the most extensive strain eigen-direction. Another important and related question is the time correlation of vorticity with the pressure Hessian eigen-frame. The pressure Hessian P is a key quantity that enters into the Lagrangian dynamics of the velocity gradient tensor A (Eq. 1) [5]. As underlined in the literature [9; 13; 14; 15; 16], understanding the time evolution of the alignments of vorticity with the eigenframe of the deviatoric part of the pressure Hessian $\textbf{P}^{d}=\textbf{P}-\mbox{tr}(\textbf{P})\textbf{I/3}$, in both the Euler and Navier-Stokes equations, is important. It has been reported that in stationary turbulent flows [9], vorticity gets preferentially aligned with the eigenvector associated to the intermediate eigenvalue of $\textbf{P}^{d}$. This property was found well reproduced by the stochastic model. We display in Fig. 2 the Lagrangian time behavior of alignments of vorticity with the eigen-frame of $\textbf{P}^{d}$, in a similar way as in Fig. 1 but now with the deformation-rate tensor S replaced by $\textbf{P}^{d}$. We define the correlation function \[~{}~{}D_{i}(\tau)=\langle\left[\textbf{b}_{i}(t).\textbf{e}_{\omega}(t+\tau) \right]^{2}\rangle\mbox{ ,}\] (5) where $\textbf{b}_{i}(t)$ for $i\in\{1,2,3\}$ are the eigenvectors of $\textbf{P}^{d}$ and the time is, again, understood to be along Lagrangian trajectories. In DNS (Fig. 2(a)), we see that at short time (i.e. $\tau\lesssim\tau_{\eta}$), $D_{1}$ (resp. $D_{3}$) decreases (resp. increases), in the opposite way as it was observed for the deformation-rate tensor (the $C_{i}$’s). As far as the intermediate pressure Hessian eigendirection is concerned, $D_{2}$ increases slightly. For larger time lags, i.e. $\tau\gtrsim\tau_{\eta}$, all the functions $D_{i}(\tau)$ relax towards the uncorrelated value $D_{i}(\tau)=1/3$. In the stochastic model (i.e. in Fig. 2(e)), it seems that even at short times, the functions $D_{i}(\tau)$ tend immediately from their initial values towards the uncorrelated value of $1/3$: there is no intermediate non-trivial short time behavior. In Figs. 2(b,c,d) (resp. Figs. 2(f,g,h)), we display the PDFs of the angle of vorticity with the $\textbf{b}_{i}$ for various time lags. At vanishing time lag $\tau_{\circ}=0$, the PDFs are similar to the ones already displayed in Ref. [9]. In particular, in both DNS and in the model, the preferential alignment of vorticity and $\textbf{b}_{2}$ is confirmed. See Ref. [9] for further interpretations and discussions about this behavior. As seen in Figs. 2(b,d) and Figs. 2(f,h), at short times, vorticity becomes orthogonal to $\textbf{b}_{1}$ and becomes aligned to $\textbf{b}_{3}$. This is not reproduced by the model in which we observe a simple relaxation towards the uncorrelated case (all the PDFs become flat). Recall that the model is based on assuming that the upstream Lagrangian pressure Hessian is isotropic, and that the tensor ${\bf A}$ does not change during the (recent) time $\tau_{K}$. These assumptions are likely to be violated in real turbulence thus causing the observed differences. Note, however, that the evolution of vorticity alignment does not directly depend upon the (symmetric) pressure Hessian tensor, but only indirectly through its dependence on strain-rate. Thus, elucidating the real causes of the observed model limitations is challenging and beyond the scope of this Letter. To conclude, we have shown in this Letter that at short times, vorticity becomes aligned with the most extensive eigendirection. This agrees with the trends observed for tetrads obtained at larger scales in the experimental and DNS investigation in Ref. [1]. This behavior is quite consistent with linear stretching of the vorticity field. The observed behaviors can be successfully predicted by the Lagrangian stochastic model based on a closure of the Pressure Hessian using the Recent Fluid Deformation approximation [8]. We have furthermore extended the study to the alignments of vorticity with the eigen-frame of the pressure Hessian, and we have underlined what is and what is not predicted by the model. We are very thankful to Emmanuel Lévêque for providing us with the DNS data. Computations have been performed by using the local computing facilities (PSMN) at ENS Lyon under grant CPER-CIRA. CM acknowledges support from the National Science Foundation (grant # CBET-1033942) ## References * (1) H. Xu, A. Pumir and E. Bodenschatz, The pirouette effect in turbulent flows, Nat. Phys. 7, 709 (2011). * (2) M. Chertkov, A. Pumir, and B. I. Shraiman, Lagrangian tetrad dynamics and the phenomenology of turbulence Phys. Fluids 11, 2394 (1999). * (3) Wm. T. Ashurst, A. R. Kerstein, R. M. Kerr and C. H. Gibson, Alignment of vorticity and scalar gradient with strain rate in simulated Navier Stokes turbulence, Phys. Fluids 30, 2343 (1987). * (4) B.J. Cantwell, Exact solution of a restricted Euler equation for the velocity gradient tensor. Phys. Fluids A 4, 782 (1992). * (5) C. Meneveau, Lagrangian Dynamics and Models of the Velocity Gradient Tensor in Turbulent Flows, Annu. Rev. Fluid Mech., 43, 219 (2011). * (6) M. Guala, B. Luethi, A. Liberzon, A. Tsinober and W. Kinzelbach On the evolution of material lines and vorticity in homogeneous turbulence J. Fluid Mech. 533, 339 (2005). * (7) M. Holzner, M. Guala, B. Luethi, A. Liberzon, N. Nikitin, W. Kinzelbach, and A. Tsinober Viscous tilting and production of vorticity in homogeneous turbulence Phys. Fluids 22, 061701 (2010). * (8) L. Chevillard and C. Meneveau, Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett. 97, 174501 (2006). * (9) L. Chevillard, C. Meneveau, L. Biferale and F. Toschi, Modeling the pressure Hessian and viscous Laplacian in Turbulence: comparisons with DNS and implications on velocity gradients dynamics, Phys. Fluids 20, 101504 (2008). * (10) A. J. Majda, Vorticity, Turbulence, and Acoustics in Fluid Flow, SIAM Rev. 33, 349 (1991). * (11) S. Girimaji, S.B. Pope. A diffusion model for velocity gradients in turbulence, Phys. Fluids A 2, 242 (1990). * (12) E. Jeong and S. S. Girimaji, Velocity-Gradient Dynamics in Turbulence: Effect of Viscosity and Forcing , Theor. Comput. Fluid Dyn. 16, 421 (2003). * (13) K. Ohkitani, Eigenvalue problems in three-dimensional Euler flows, Phys. Fluids A 5, 2570 (1993). * (14) K. Ohkitani and S. Kishiba, Nonlocal nature of vortex stretching in an inviscid fluid, Phys. Fluids A 7, 411 (1995). * (15) K.K. Nomura and G.K. Post, The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence, J. Fluid Mech. 377, 65 (1998). * (16) K. Horiuti and T. Fujisawa, The multi-mode stretched spiral vortex in homogeneous isotropic turbulence, J. Fluid Mech. 595, 341 (2008).
1507.03148	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35968, "num_imgs": 6, "llama3_tokens_count": 8846 }	[ "content_image/1507.03148/x1.png", "content_image/1507.03148/x2.png", "content_image/1507.03148/x4.png", "content_image/1507.03148/x5.png", "content_image/1507.03148/x8.png", "content_image/1507.03148/x10.png" ]	# Face Alignment Assisted by Head Pose Estimation ###### Abstract In this paper we propose supervised initialisation scheme for cascaded face alignment based on explicit head pose estimation. We first investigate the failure cases of most state of the art face alignment approaches and observe that these failures often share one common global property, i.e. the head pose variation is usually large. Inspired by this, we propose a deep convolutional network model for reliable and accurate head pose estimation. Instead of using a mean face shape, or randomly selected shapes for cascaded face alignment initialisation, we propose two schemes for generating initialisation: the first one relies on projecting a mean 3D face shape (represented by 3D facial landmarks) onto 2D image under the estimated head pose; the second one searches nearest neighbour shapes from a training set according to head pose distance. By doing so, the initialisation gets closer to the actual shape, which enhances the possibility of convergence and in turn improves the face alignment performance. We demonstrate the proposed method on the benchmark 300W dataset and show very competitive performance in both head pose estimation and face alignment. Heng Yangheng.yang@cl.cam.ac.uk1 Wenxuan Mouw.mou@qmul.ac.uk2 Yichi Zhangyichizhang@fas.harvard.edu3 Ioannis Patrasi.patras@qmul.ac.uk2 Hatice Gunesh.gunes@qmul.ac.uk 2 Peter Robinsonpeter.robinson@cl.cam.ac.uk1 Computer Laboratory University of Cambridge Cambridge, UK School of EECS Queen Mary University of London London, UK Faculty of Arts & Sciences Harvard University Cambridge, MA, US Face alignment assisted by Head Pose Estimation ## 1 Introduction Both head pose estimation and face alignment have been well studied in recent years given their wide application in human computer interaction, avatar animation, and face recognition/verification. These two problems are very correlated and putting them together will enable mutual benefits. Head pose estimation from 2D images remains a challenging problem due to the high diversity of face images [13, 18]. Recent methods [10] attempt to estimate the head pose by using depth data. On the contrary, face alignment has made significant progress and several methods [36, 2, 20, 30] have reported good performance on images _in the wild_. However, they also show some failures. When we look into their failures cases, we find that those samples share one significant property, i.e., the head (face) in such images is usually rotated from frontal pose in big angles. The best performing face alignment methods proposed in recent years ([30], [2] and [36]) also share a similar cascaded pose regression framework, i.e., face alignment starts from a raw shape (a vector representation of the landmark locations), and updates the shape in a coarse to fine manner. The methods in this framework are usually initialisation dependent. Therefore, the final output of one cascaded face alignment system might change if a different initialisation is provided to the same input image. Moreover, each model has a convergence radius, i.e., if the initialisation lies within the range of the actual shape, the model will be able to output a reasonable alignment result, otherwise it might lead the shape to a wrong location, as shown in Fig. 1. The methods like [30, 2]perform initialisation using a mean shape within the face bounding box or from a randomly selected shape from training set. There is no guarantee the initialisation lies within the convergence radius, especially when head pose variation is large. <figure><img src="content_image/1507.03148/x1.png"><figcaption>Figure 1: Our proposed head pose based cascaded face alignment procedure (pathin cyan color) vs. conventional cascaded face alignment procedure (path in redcolor).</figcaption></figure> In this paper, we aim to address the above discussed problems and make cascaded face alignment perform better under large head pose variations. The difference between our proposed method and the conventional cascaded method procedure is illustrated in Fig. 1. In contrast to using mean shape or random shapes for initialisation by other methods, our proposed method aims to produce better initialisation schemes for cascaded face alignment based on explicit head pose estimation. This is motivated by two facts: 1) most current methods fail on face images with large head pose variation-as we will demonstrate later; 2) most recent face alignment methods work in a cascaded fashion and perform initialisation with mean shape. More specifically, we first estimate the head pose using a deep Convolutional Network (ConvNet) directly from face image. Given the estimated head pose, we propose two schemes of producing the initialisations. The first scheme projects a canonical 3D face shape under the estimated head pose to the detected face bounding box. The second scheme searches shape(s) for initialisation from the training set by nearest neighbour method in the head pose space. We build on our proposed scheme on the Robust Cascaded Pose Regression (RCPR) to demonstrate the effectiveness of supervised initialisation. We note that the proposed initialisation scheme can be naturally applied to any other cascaded face alignment. In summary, we make the following contributions: * We investigate the failure cases of several state of the art face alignment approaches and find that the head pose variation is a common issue across those methods. * Based on the above observation, we propose a ConvNet framework for explicit head pose estimation. It is able to achieve an accuracy of 4${}^{\circ}$ absolute mean error of head pose estimation for face images acquired in unconstrained environment. * We propose two initialisation schemes based on reliable head pose estimation. They enable face alignment method (RCPR) perform better and reduce large head pose failures by 50% when using only one initialisation. To summarise, we propose better initialisation schemes based on explicit head pose estimation for cascaded face alignment, to improve the performance, especially in the case of large head pose variation. ## 2 Related Work Face alignment has made considerable progress in the past years and a large number of methods have been proposed. There are two different sources of information typically used for face alignment: face appearance (i.e., texture of the face image) and the shape information. Based on how the spatial shape information is used, the methods are usually categorized into local-based methods and holistic-based methods. The methods in the former category usually rely on discriminative local detection and use explicit deformable shape models to regularize the local outputs while the methods in the latter category directly regress the shape (the representation of the facial landmarks) in a holistic way, i.e. the shape and appearance are modelled together. ### Local-based methods Local based methods usually consist of two parts. One is for local facial feature detection, which is also called local experts and the other is for spatial shape models. The former describes how image around each facial landmark looks like in terms of local intensity or color patterns while the latter describes how face shape, that is the relative location of the face parts, varies. This captures variations such as wide forehead, narrow eyes, long nose etc. There are three types of local feature detection. (1) Classification methods include Support Vector Machine (SVM) classifier [19, 4] based on various image features such as Gabor [28], SIFT [15, 30], HOG [31] and multichannel correlation filter responses [11]. (2) Regression-based approaches are also widely used. For instance, Support Vector Regressors (SVRs) are used in [16] with a probabilistic MRF-based shape model and Continuous Conditional Neural Fields (CCNF) are used in [3]. (3) Voting-based approaches are also introduced in recent years, including regression forests based voting methods [6, 8, 32] and exemplar based voting methods [23, 22]. One typical shape model is the Constrained Local Model (CLM) [7]. The CLM steps can be summarised as follows: first, sample a region from the image around the current estimate and project it into a reference frame; second, for each point, generate a “response image" giving a cost for having the point at each pixel; third, searching for a combination of points which optimises the total cost, by manipulating the statistical shape model parameters. The methods built on CLM mainly differ from each other in terms of local experts, for instance CCNF in [3] and the Discriminative Response Map Fitting (DRMF) in [1]. There are many other local based methods either using CLM or other models such as RANSAC in [4], graph-matching in [38], Gaussian Newton Deformable Part Model (GNDPM) [26] and mixture of trees [39]. ### Holistic-based methods Methods \| SDM [Xiong and De la Torre(2013)] \| RCPR [Burgos-Artizzu et al.(2013)Burgos-Artizzu, Perona, and Dollár] \| IFA [Asthana et al.(2014)Asthana, Zafeiriou, Cheng, and Pantic] \| LBF [Ren et al.(2014)Ren, Cao, Wei, and Sun] \| CFAN [Zhang et al.(2014a)Zhang, Shan, Kan, and Chen] \| TCDCN [Zhang et al.(2014b)Zhang, Luo, Loy, and Tang] ---\|---\|---\|---\|---\|---\|--- initialisation \| mean pose \| random \| mean pose \| mean pose \| supervised \| supervised features \| SIFT \| pixel \| HOG \| pixel \| auto-encoder \| ConvNet feature regressor \| linear regression \| random ferns \| linear regression \| random forests \| linear regression \| ConvNet Table 1: Holistic methods and their properties. Holistic methods have gained high popularity in recent years and most of them work in a cascaded way like SDM [30] and RCPR [5]. We list very recent holistic methods as well as their properties in Table 1. The methods following the cascaded framework differ from each other mainly in three aspects. First, how to set up the initial shape; Second, how to calculate the shape-indexed features; Third, what type of regressor is applied at each iteration. For initialisation, there are mainly three strategies are proposed in literature: random, mean pose, and supervised. In order to make it less sensitive to initialisation, previous approaches such as [29, 5] propose to run multiple different initialisations and pick the median of all the predictions as the final output. Each initialisation is treated independently way until the output is calculated. However, such a strategy has several issues, first the theoretical support for selecting the median value is not well understood; second, there is no guidance on how to choose the multiple initialisations; third, using multiple initialisations is computationally expensive. A similar supervised initialisation scheme was proposed in [35] where the initialisation shapes were selected by using an additional regression forest model for sparse facial landmarks estimation. A recent work [33] proposed a re-initialisation scheme based on mirrorability to improve the face alignment performance. ## 3 Data preparation In this section we describe how the data is prepared in order to support our further discussion. More specifically, we discuss how we provide ground truth head pose and face bounding boxes from different face detectors for the benchmark dataset. We use face image data from the benchmark face alignment in the wild dataset, 300W [21]. Since their testing samples are not publicly available, we follow the partition of recent methods [20] to set up the experiments. More specifically, we use face images from AFW [39], HELEN [25], LFPW [4] and iBug [21], which include 3148 training images and 689 test images in total. 3148 training images are from AFW (337 images), HELEN training set (2000 images) and LFPW training set (811 images), and 689 test images are from HELEN test set (330 images), LFPW test set (224 images) and iBug (135 images). It is intractable to get the ground truth 3D head pose for face images collected in unconstrained conditions. In order to generate reasonable head pose (Pitch, Yaw and Roll) values, we use the pose estimator provided by Supervised Descent Method (SDM) [30]. Note that, when calculating the head pose, we feed the ground truth facial landmark locations instead of using the detected landmarks. Technically, head pose is estimated by solving the projection function from an average 3D face model (49 3D points) to the input image, given the 3D to 2D correspondences. We also use the 3D head pose estimator provided by [1] for head pose calculation for evaluating the results. It produces very similar results to [30]. We calculate the head pose for all images in 300W. The benchmark dataset only provides two types of face bounding boxes: one is the ground truth bounding box calculated as the tight box of the annotated facial landmarks; the other is the detection results from model of [39], which is quite similar to the ground truth face bounding box. However, several models like SDM [30] and RCPR[5] are trained with different face bounding boxes, thus their performance deteriorates significantly when using the provided face bounding boxes. We therefore provide different face bounding boxes to the test images by employing Viola-Jones detector [27] and HeadHunter detector [17] for fair comparison. For the input images on which the face detector fails we manually set reasonable bounding boxes. <figure><img src="content_image/1507.03148/x2.png"><figcaption>Figure 2: Distribution of the most erroneous samples.</figcaption></figure> ## 4 Method ### Motivation We first run several state of the art methods, including 6 holistic based methods (SDM [30], IFA [2], LBF [20], CFAN [36], TCDCN [37], RCPR [5]) and 3 local based methods (GNDPM [26], DRMF [1], CCNF [3]) given their good performance and availability of source code. For each method, we provide the _best_ type of face bounding boxes in order to get the best performance. For each method, we select 50 difficult samples out of the 689 test samples that provide the biggest sample-wise alignment error. Then we plot their head poses in Fig. 2 (left). As can be seen, most of the points are far away from the original point, i.e. they have big rotation angle(s). We further plot the histogram of the biggest absolute rotation angles of those samples in Fig. 2 (right). The biggest absolute rotation angle is calculated as the one of the three directions with the biggest absolute value. As can be seen, those samples are distributed at big absolute angles. There are very few samples that have small rotation angles. Based on this observation, we can conclude that, large head pose rotation is one of the main factors that make most of the current face alignments fail. Based on this fact, we develop a head pose based initialisation scheme for improving the performance of face alignment under large head pose variations. <figure><img src="content_image/1507.03148/x4.png"><figcaption>Figure 3: ConvNet model for head pose estimation.</figcaption></figure> ### Head Pose Estimation Giving the training data from 300W with augmented head pose annotation, we train a convolutional network (ConvNet) [14] model for head pose estimation on the training set of 300W with 3148 images. The samples are augmented by 3 times with small permutations on the face bounding box. The ConvNet structure is shown is shown in Fig. 3. The input of the network is 96x96 gray-scale face image , normalised to the range between 0 and 1. The feature extraction stage contains three convolutional layers, three pooling layers, two fully connected layers and three drop-out layers. As we pose it as a regression problem, the output layer is 3x1 representing the head pose pitch, yaw and roll angle respectively. The angles are normalised between -1 and 1. We use Nesterov’s Accelerated Gradient Descent (NAG) method [24] for parameter optimisation and we set the momentum to 0.9 and learning rate to 0.01. The training finishes in two hours on Tesla K40c GPU after around 1300 epochs, controlled by early-stop strategy. The learning curve is shown in Fig. 4 (left). The forward propagation of this network on GPU only takes 0.3ms per image on average. ### Pose based Cascaded Face Alignment #### 4.3.1 General Cascaded Face Alignment In order to make this work stand alone, we first summarise the general framework of cascaded face alignment. Face shape is often represented as a vector of landmark locations, i.e., $S=(\mathrm{x}_{1},...,\mathrm{x}_{k},...,\mathrm{x}_{K})\in\mathbf{R}^{2K}$, where $K$ is the number of landmarks. $\mathrm{x}_{k}\in\mathbf{R}^{2}$ is the 2D coordinates of the $k$-th landmark. Most of the current holistic-based method works in a coarse-to-fine fashion, i.e., shape estimation starts from an initial shape $S^{0}$ and progressively refines the shape by a cascade of $T$ regressors, $R^{1...T}$. Each regressor refines the shape by producing an update, $\Delta S$, which is added on the current shape estimate, that is, \[S^{t}=S^{t-1}+\Delta S.\] (1) The update $\Delta S$ returned from the regressor that takes the previous pose estimation and the image feature $I$ as inputs: \[\Delta S=R^{t}(S^{t-1},I)\] (2) An important aspect that differentiates this framework from the classic boosted approaches is the feature re-sampling process. More specifically, instead of using the fixed features, the input feature for regressor $R^{t}$ is calculated relative to the current pose estimation. This is often called pose-indexed feature as in [9]. This introduces weak geometric invariance into the cascade process and shows good performance in practice. The CPR is summarized in Algorithm 1[9]. ``` 1:Image $I$, initial pose $S^{0}$ 2:Estimated pose $S^{T}$ 3:for $t$=1 to $T$ do 4: $f^{t}=h^{t}(I,S^{t-1})$$\triangleright$ Shape-indexed features 5: $\Delta S=R^{t}(f^{t})$$\triangleright$ Apply regressor $R^{t}$ 6: $S^{t}=S^{t-1}+\Delta S$$\triangleright$ update pose 7:end for ``` Algorithm 1 Cascaded Pose Regression #### 4.3.2 Head Pose based Cascaded Face Alignment In section 4.2 we have presented how a ConvNet model can be used for head pose estimation. We propose two head pose based initialisation schemes for face alignment. One is based on an average 3D face shape projection and the other is based on nearest neighbour searching. Scheme 1: 3D face shape based initialisationGiven a 3D mean face shape, represented by 68 3D facial landmark locations, as shown in Fig. 1, we first project this shape under the estimated head pose to a set of canonical 2D locations. More specifically we use constant translation and focus length in order to get a reasonable projection for all images. Then we re-scale the canonical 2D projection by the face bounding box scale of the test image to get the initialisation. We can represent the initialisation process by function $\mathcal{F}$ as follows. \[S_{0}=\mathcal{F}(\theta,bb,\bar{S}^{3D})\] (3) with $bb$ the face bounding box, $\bar{S}^{3D}$, the 3D mean face shape, $\theta$, the estimated head pose, which can be represented by: \[\theta=\mathcal{G}(I,bb)\] (4) where $\mathcal{G}$ is the deep convolutional model described in section 4.2. Scheme 2: Nearest Neighbour based initialisationWe propose a second scheme for head pose based initialisation by nearest neighbour search. Since we have provided the training samples with head pose information as well, we can easily search samples that are with similar head pose of a test sample. Then we calculate similarity transformation between two face bounding boxes in order to calculate the initialisation shape for the test sample. In this way, we can also provide $K$ initialisations by searching $k$-Nearest Neighbors from the training set. Once we get a reliable initialisation (or several ones), we feed it to Algorithm 1 and apply the cascade of regressors in the same way to the baseline approach. In the case of the multiple initialisations, we calculate the output in a similar fashion to [5, 29], i.e., to pick up the median value of their estimations. We build our proposed head pose based initialisation schemes on top of the popular Cascaded Pose Regression (CPR) method due to its simplicity and popularity. We train its recent variant Robust Cascaded Pose Regression (RCPR) [5] model by using its new interpolated feature extraction, which is re-implemented by the author of [34]. We do not use its full version as occlusion status annotation is not available. We trained the baseline RCPR model on our 300W training set using Viola-Jones [27] face detection. 20 random initialisations are used for data augmentation at the training time. ## 5 Evaluation <figure><img src="content_image/1507.03148/x5.png"><figcaption>Figure 4: Head pose estimation result. Left, learning curve of head posenetwork, with y axis the Root Mean Square Error (RMSE) and x axis the numberof epochs; middle, absolute mean error on test set; right, example results ofhead pose estimation.</figcaption></figure> ### Head Pose Estimation We first evaluate the performance of head pose estimation. As we discussed before, it is very difficult to get the ground truth head pose for face images acquired in uncontrolled conditions. We calculate the pose based on the annotated facial landmark locations. We apply the trained deep ConvNet model on the test images of 300W and measure the performance. The result is shown in Fig. 4. The absolute mean errors of the head pose pitch, yaw, roll angles are 5.1${}^{\circ}$, 4.2${}^{\circ}$ and 2.4${}^{\circ}$, respectively. Some example results are shown on the right. Despite the work by Zhu & Ramanan [39] is conceptually similar to our work in terms of simutaneuous head pose and facial landmarks estimation, we do not compare to it here because their work can only estimate very sparse head pose yaw angles (e.g. -15${}^{\circ}$, 0${}^{\circ}$ , 15 ${}^{\circ}$ ). ### Face Alignment We first show the effectiveness of head pose based initialisation by comparing with the baseline strategy of the CPR framework [29, 5], i.e., generating random initialisations from training samples. The comparison is shown in Fig. 5. As can be seen on the left figure, by using one initialisation projected from 3D face shape, we obtain similar performance to the baseline approach with 5 initialisation shapes, and much better performance than that uses only one random initialisation shape. Similar superior performance is obtained by using nearest neighbour initialisation scheme, as shown on the right. By using more head pose based initialisations, we gain even better results, though the improvement is minor. It is worthy noting that by using our proposed initialisation scheme, we are able to decrease the number of failure cases (sample-wise average alignment error $>$ 0.1) from 130 to 69 (scheme 1) and from 130 to 72 (scheme 2), nearly 50%. Those samples are usually with large head pose variations and difficult for conventional face alignment methods. Moreover, by using one set of initialisation, the whole test procedure on one typical image takes 3.8 ms (0.3 ms for head pose estimation and 3.5 ms for cascaded face alignment). <figure><img src="content_image/1507.03148/x8.png"><figcaption>Figure 5: Our proposed head pose based initialisation scheme vs. randominitialisation scheme. Left, our 3D face shape based scheme; right, ourNearest Neighbour (NN) based scheme.</figcaption></figure> We further compare the proposed method with recent state of the art methods including 5 holistic based methods (SDM [30], IFA [2], LBF [20], CFAN [36], TCDCN [37]) and 3 local based methods (GNDPM [26], DRMF [1], CCNF [3]). SDM and DRMF are trained using the Multi-PIE [12] dataset and detect 49 and 66 facial landmarks respectively. The rest of them are with models trained on 300W datasets. When we run their model on the test images, we use the _best_ bounding boxes for a fair comparison. Best bounding box refers to Viola-Jones detection for SDM and RCPR and tight face detection provided by 300w dataset for the rest of them. The comparison is shown in Fig. 6. As can be seen, our proposed method shows competitive performance. We also compare the performance on another type of common face detection, HeadHunter, given its best performance in face detection. The result is shown on the right of Fig. 6. We observe that the performance of most methods deteriorate significantly when testing on HeadHunter face bounding boxes. Our method provides most stable result, despite the fact that the HeadHunter face bounding box is more overlapped with the face detection from 300W (both are tight boxes of facial landmarks) than with Viola-Jones face detection. We believe this robustness to face bounding box changes is partially due to our head pose based initialisation strategy. <figure><img src="content_image/1507.03148/x10.png"><figcaption>Figure 6: Comparison with recent methods. Left, results from the best facedetection of each method; right, results from the common HeadHunter facedetection. Pose-RCPR is our proposed method using only 1 initialisation from3D.</figcaption></figure> ## 6 Conclusion and Future Work In this paper we first demonstrate that most recent face alignment methods show failure cases when large head pose variation is present. Based on the fact that cascaded face alignment is initialisation dependent, we proposed supervised initialisation schemes based on explicit head pose estimation. We use deep convolutional networks for head pose estimation and produce initialisation shape by either projecting a 3D face shape to the test image or searching nearest neighbour shapes from the training set. We demonstrated that using a more reliable initialisation is able to improve the face alignment performance with around 50% failure decreasing. It also shows comparable or better performance when comparing to recent face alignment approaches. Although we have managed to decrease the failure cases to a certain degree, we have not fully solved this problem. There are several interesting directions for future research. First, using head pose based initialisation shapes in the training stage may further boost the performance. Second, we only test our method on RCPR, we believe the proposed scheme can be naturally applied to other cascaded face alignment methods. It also raises several interesting questions. Do we need to make the cascaded learning model better for face alignment or to make the initialisation more reliable? Do we need more uniformly distributed data or a better model in order to make face alignment work better in wider range of head pose variations? We are going to investigate on these problem in our future research. ## Acknowledgement The work is sponsored by Cambridge VBRAD project from Jaguar-Land-Rover. We gratefully acknowledge NVIDIA for the donation of the Tesla GPU used for this research. ## References * [Asthana et al.(2013)Asthana, Zafeiriou, Cheng, and Pantic] Akshay Asthana, Stefanos Zafeiriou, Shiyang Cheng, and Maja Pantic. Robust discriminative response map fitting with constrained local models. In _Proc. IEEE Conf. Comput. Vis. 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0801.2747	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 117075, "num_imgs": 21, "llama3_tokens_count": 33890 }	[ "content_image/0801.2747/x1.png", "content_image/0801.2747/x2.png", "content_image/0801.2747/x3.png", "content_image/0801.2747/x4.png", "content_image/0801.2747/x5.png", "content_image/0801.2747/x6.png", "content_image/0801.2747/x7.png", "content_image/0801.2747/x8.png", "content_image/0801.2747/x9.png", "content_image/0801.2747/x10.png", "content_image/0801.2747/x11.png", "content_image/0801.2747/x12.png", "content_image/0801.2747/x13.png", "content_image/0801.2747/x14.png", "content_image/0801.2747/x15.png", "content_image/0801.2747/x16.png", "content_image/0801.2747/x17.png", "content_image/0801.2747/x18.png", "content_image/0801.2747/x19.png", "content_image/0801.2747/x20.png", "content_image/0801.2747/x21.png" ]	# Energetics and Structural Properties of Trapped Two-Component Fermi Gases J. von Stecher Department of Physics and JILA, University of Colorado, Boulder, CO 80309-0440 Chris H. Greene Department of Physics and JILA, University of Colorado, Boulder, CO 80309-0440 D. Blume JILA, University of Colorado, Boulder, CO 80309-0440 Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814 February 27, 2024 ###### Abstract Using two different numerical methods, we study the behavior of two-component Fermi gases interacting through short-range $s$-wave interactions in a harmonic trap. A correlated Gaussian basis-set expansion technique is used to determine the energies and structural properties, i.e., the radial one-body densities and pair distribution functions, for small systems with either even or odd $N$, as functions of the $s$-wave scattering length and the mass ratio $\kappa$ of the two species. Particular emphasis is put on a discussion of the angular momentum of the system in the BEC-BCS crossover regime. At unitarity, the excitation spectrum of the four-particle system with total angular momentum $L=0$ is calculated as a function of the mass ratio $\kappa$. The results are analyzed from a hyperspherical perspective, which offers new insights into the problem. Additionally, fixed-node diffusion Monte Carlo calculations are performed for equal-mass Fermi gases with up to $N=30$ atoms. We focus on the odd-even oscillations of the ground state energy of the equal-mass unitary system having up to $N=30$ particles, which are related to the excitation gap of the system. Furthermore, we present a detailed analysis of the structural properties of these systems. ## I Introduction Pure Fermi systems with essentially any interaction strength can be realized experimentally with ultracold atomic gases. In most experiments to date, large samples of atomic Li or K are trapped optically in two different hyperfine states, in the following simply referred to as “spin-up” and “spin-down” states. By tuning an external magnetic field in the vicinity of a Fano-Feshbach resonance [1; 2; 3; 4], the interspecies $s$-wave scattering length can be varied from non-interacting to infinitely strongly-interacting (either attractive or repulsive). This tunability is unique to atomic systems, and it has enabled for the first time quantitative experimental studies of the crossover from the molecular BEC-regime to the atomic BCS-regime [5; 6; 7; 8; 9; 10]. Since the systems studied experimentally are in general large, many observations have been explained quite successfully by applying theoretical treatments based on the local density approximation (LDA); see, e.g., Ref. [11] and references therein. The LDA uses the equation of state of the homogeneous system as input, and, in general, accurately describes the properties of the system near the trap center, where the density changes slowly. However, it fails to accurately describe the properties of the system near the edge of the cloud, where the density varies more rapidly. In a different set of experiments, atomic Fermi gases are loaded into an optical lattice with variable barrier height [12; 13; 14]. In the regime where the tunneling of atoms between neighboring lattice sites can be neglected, each lattice site provides an approximately harmonic confining potential for the atoms at that site. Through the application of a so-called “purification scheme” [15], experimentalists are now able to realize systems with a deterministic number of atoms per site. So far, optical lattices have been prepared with one or zero atoms per site, with two or zero atoms per site, and with three or zero atoms per site. Optical-lattice experiments thus allow for the simultaneous preparation of multiple copies of identical few-particle systems. We anticipate that these experiments will be extended to larger atom samples in the future, thereby opening the possibility to study systematically how the properties of the system change as functions of the number of atoms. Transitions from few- to many-body systems have, e. g., been studied experimentally in metal and rare gas clusters [16; 17], and it is exciting that the experimental study of this transition in dilute gaseous systems is within reach. A mature body of theoretical work has also investigated the manner in which bulk electronic, magnetic and superfluid properties can be understood by studying small or modest-size clusters [18; 19]. This paper presents theoretical results for trapped two-component Fermi gases with up to $N=30$ fermions, which shed light on the few- to many-body transition from a microscopic or few-body point of view. To solve the many-body Schrödinger equation we use two different numerical methods, a correlated Gaussian (CG) basis set expansion approach and a fixed-node diffusion Monte Carlo (FN-DMC) approach. The CG approach allows for the determination of the entire energy spectrum and eigenstates with controlled accuracy (i.e., no approximations are employed and the convergence can be systematically improved). If we demand an accuracy of the order of 2% or better, our current CG implementation limits us to treating systems with up to $N=6$ atoms (and to the lowest 10 or twenty eigen states). To the best of our knowledge, no other such calculations exist for dilute fermionic few-body systems ($N=4-6$) with short-range interactions. The FN-DMC method, in contrast, can be applied to larger systems but its accuracy crucially depends on the quality of the many-body nodal surface, which is in general unknown. Moreover, the FN-DMC approach as implemented here treats only ground state properties for the chosen symmetry. Careful comparisons of the ground state energy and structural properties calculated by the FN-DMC and CG approach for different interaction strengths validate the construction of the nodal surfaces employed for $N\leq 6$. We expect, and provide some evidence, that our nodal surfaces constructed to describe the energetically lowest-lying gas-like state of larger $N$ are also quite accurate. Specifically, we calculate the energy of the energetically lowest-lying gas-like state of trapped two-species Fermi gases as a function of the number of particles $N$, the $s$-wave scattering length $a_{s}$ and the mass ratio $\kappa$. Our ground state energies for even and odd $N$ can be readily combined to determine the excitation gap, which is related to pairing physics. For small systems, we additionally determine and discuss the excitation spectrum. Furthermore, we present pair correlation functions, which provide further insights into the pair formation process, and radial density profiles for the ground state. Finally, we elaborate on the interpretation of the behaviors within a framework that uses hyperspherical coordinates. This connection has been summarized in an earlier paper [20]. Here, we present additional results and discuss in more detail how the even-odd oscillations emerge in the hyperspherical framework. Our analysis provides an alternative means, complementary to conventional many-body theory, for understanding the excitation gap at unitarity. The remainder of this paper is organized as follows. Section II introduces the Hamiltonian of the system under study, reviews the definitions of the normalized energy crossover curve and the excitation gap, and summarizes some peculiar properties of the unitary gas using hyperspherical coordinates. Section III summarizes the CG and FN-DMC approaches, and provides some implementation details specific to the problem at hand. Section IV presents our results for the ground state energies, the excitation spectrum and structural properties. Finally, Sec. V concludes. ## II Theoretical background ### Hamiltonian The main objective of this article is to obtain and interpret solutions to the many-body time-independent Schrödinger equation for a trapped two-component Fermi gas with short-range interactions. The model Hamiltonian for $N_{1}$ fermions of mass $m_{1}$ and $N_{2}$ fermions of mass $m_{2}$ reads \[H=\sum_{i=1}^{N_{1}}\left(\frac{-\hbar^{2}}{2m_{1}}\nabla_{i}^{2 }+\frac{1}{2}m_{1}\omega^{2}\vec{r}_{i}^{2}\right)+\] \[\sum_{i^{\prime}=1}^{N_{2}}\left(\frac{-\hbar^{2}}{2m_{2}}\nabla_ {i^{\prime}}^{2}+\frac{1}{2}m_{2}\omega^{2}\vec{r}_{i^{\prime}}^{2}\right)+ \sum_{i=1}^{N_{1}}\sum_{i^{\prime}=1}^{N_{2}}V_{0}(r_{ii^{\prime}}).\] (1) Here, $\vec{r}_{i}$ and $\vec{r}_{i^{\prime}}$ denote the position vector of the $i$th mass $m_{1}$ fermion and the $i^{\prime}$th mass $m_{2}$ fermion, respectively. Both atom species experience a trapping potential characterized by the same angular frequency $\omega$. For equal masses, this is indeed the case in ongoing experiments. For unequal masses, however, the two atomic species typically experience different trapping frequencies. Our restriction to equal trapping frequencies reduces the parameter space which otherwise would be impractical to explore numerically. Furthermore, our CG calculations simplify for equal trapping frequencies because the center-of-mass and relative motions decouple in this case. The studies presented here for unequal masses but equal frequencies complement our earlier study [21], which treats two-component Fermi gases with unequal masses that experience trapping frequencies $\omega_{1}$ and $\omega_{2}$ adjusted so that $m_{1}\omega_{1}=m_{2}\omega_{2}$. In Eq. (II.1), $V_{0}$ is a short-range two-body potential between each pair of mass $m_{1}$ and mass $m_{2}$ atoms. We characterize the strength of $V_{0}$ by the $s$-wave scattering length $a_{s}$, which can be varied experimentally through the application of an external magnetic field in the vicinity of a Fano-Feshbach resonance. Here, we model this situation by changing the depth of $V_{0}$; our results should be applicable to systems with a broad $s$-wave Fano-Feshbach resonance and vanishingly small $p$-wave interactions. The present study considers two-component Fermi gases with either even or odd $N$, where $N=N_{1}+N_{2}$.Because odd-even oscillations serve as one major subject of this study, we set $N_{1}=N_{2}$ for even $N$, and $N_{1}=N_{2}\pm 1$ for odd $N$. In addition to the scattering length $a_{s}$, we vary the mass ratio $\kappa$, \[\kappa=m_{1}/m_{2}.\] (2) Throughout, we take $m_{1}\geq m_{2}$ so that $\kappa\geq 1$. In most cases, we measure lengths in units of the oscillator length $a_{ho}$, $a_{ho}=\sqrt{\hbar/(2\mu\omega)}$, which is defined in terms of the reduced mass $\mu$, $\mu=m_{1}m_{2}/(m_{1}+m_{2})$. It has been shown previously [22; 23; 24; 20] that small equal-mass two-component Fermi gases, which interact through short-range two-body potentials with infinitely large $a_{s}$ that support no $s$-wave bound state, support no tightly-bound many-body states with negative energy. For unequal mass systems the situation is different [25; 26; 23; 27]. Trimers consisting of two heavy particles and one light particle that interact through short-range potentials support tightly-bound states with negative energy if the mass ratio and the scattering length are sufficiently large. Reference [21] discussed the role of non-universal trimer states for unequal-mass systems in some detail, and we return to this discussion in Sec. IV.1. Throughout this work, we restrict our analysis to gas-like states, consisting of atomic fermions, molecular bosons or both. To solve the Schrödinger equation for eigenstates of $H$, we use two different numerical methods: a correlated Gaussian (CG) basis set expansion technique and a fixed-node diffusion Monte Carlo (FN-DMC) technique. For numerical convenience, we utilize different short-range potentials $V_{0}$ in our CG and FN-DMC calculations. We adopt a purely attractive Gaussian interaction potential defined as \[V_{0}(r)=-d\exp\left(-\frac{r^{2}}{2R_{0}^{2}}\right)\] (3) in the CG calculations, and a square well interaction potential defined as \[V_{0}(r)=\left\{\begin{array}[]{cl}-d&\mbox{for }r<R_{0}\\ 0&\mbox{for }r>R_{0}\end{array}\right.\] (4) in the FN-DMC calculations. For a fixed range $R_{0}$, the potential depth $d$ is adjusted so that the $s$-wave scattering length $a_{s}$ takes the desired value. The range $R_{0}$ is selected so that $R_{0}\ll a_{ho}$. The premise is that the properties of two-component Fermi gases with short-range interactions (or at least the universal state properties) are determined by the $s$-wave scattering length $a_{s}$ alone, and independent of the details of the underlying two-body potential if the range $R_{0}$ is chosen sufficiently small. Ideally, we would consider the limit $R_{0}=0$. This is, however, impossible within the numerical frameworks employed. Thus, we perform calculations for different finite $R_{0}$, which allows us to approximately extrapolate to the $R_{0}=0$ limit and to estimate the dependence of our results on $R_{0}$, i.e., to estimate the scale of the finite-range effects. ### Energy crossover curve and excitation gap The energetically lowest-lying gas-like states of two-component Fermi gases with short-range interactions determine the normalized energy crossover curve $\Lambda_{N}^{(\kappa)}$ and the excitation gap $\Delta(N)$. To simplify the notation, the energetically lowest-lying gas-like state is referred to as the ground state in this section. The BCS and BEC limits of the crossover can be treated perturbatively. For small $\|a_{s}\|$ and $a_{s}<0$, the system behaves like a weakly-interacting atomic Fermi gas whose leading order properties beyond the non-interacting degenerate Fermi gas are determined by $a_{s}$. For attractive two-body potentials that generate small $a_{s}$ and $a_{s}>0$, in contrast, the system behaves like a weakly-interacting molecular Bose gas whose properties are to leading order determined by $a_{dd}$, where $a_{dd}$ denotes the dimer-dimer scattering length. (One can also have small, positive $a_{s}$ with purely repulsive two-body potentials that have no bound molecular states, but these systems behave quite differently and will not be considered in this paper.) In the strongly-interacting regime (large $\|a_{s}\|$), perturbation theory cannot be applied and it is not clear _a priori_ whether the system behaves more like an atomic gas or a molecular gas, or like neither of the two. The definition of the normalized energy crossover curve $\Lambda_{N_{1},N_{2}}^{(\kappa)}$ introduced in Refs. [21; 28] for even $N$ can be extended to odd $N$, \[\Lambda_{N_{1},N_{2}}^{(\kappa)}=\frac{E(N_{1},N_{2})-N_{d}E(1,1) -3N_{f}/2\hbar\omega}{E_{NI}-\frac{3}{2}N\hbar\omega}.\] (5) Here, $E(N_{1},N_{2})$ denotes the ground state energy of the trapped two-component gas consisting of $N_{1}$ fermions with mass $m_{1}$ and $N_{2}$ fermions with mass $m_{2}$. In Eq. (5), $N_{d}$ is defined by \[N_{d}=\min\{N_{1},N_{2}\},\] (6) and corresponds to the number of dimers formed on the BEC side, i.e., in the regime where $a_{s}$ is small and positive. $N_{f}$ is defined by \[N_{f}=\|N_{1}-N_{2}\|;\] (7) it represents the number of unpaired atoms on the BEC side, and takes the value 0 for even $N$ and 1 for odd $N$. In Eq. (5), $E_{NI}$ denotes the ground state energy of the non-interacting two-component Fermi gas consisting of $N$ atoms, where— as before— $N=N_{1}+N_{2}$. The $E_{NI}$ can be evaluated as the sum of the noninteracting energies of polarized Fermi gases $E^{p}_{NI}$ with $N_{1}$ and $N_{2}$ particles, $E_{NI}(N)=E^{p}_{NI}(N_{1})+E^{p}_{NI}(N_{2})$. Following Ref. [29], the $E^{p}_{NI}(N_{i})$ can be written in terms of the shell number $n_{s}$, the energy of the closed shell subsystem $E^{cs}_{NI}(n_{s})$, and the corresponding magic number $N^{cs}$, \[E^{p}_{NI}(N_{i})=E^{cs}_{NI}(n_{s})+\left(\frac{3}{2}+n_{s}\right)(N-N^{cs}) \hbar\omega.\] (8) The shell number $n_{s}$ represents the number of closed shells and is given by \[n_{s}=\mbox{Int}\left[\frac{1}{g(N_{i})}+\frac{g(N_{i})}{3}-1\right],\] (9) where \[g(N_{i})=\sqrt[3]{3\left(27N_{i}-\sqrt{3(243N_{i}^{2}-1)}\right)}\] (10) and Int[$x$] is the integer part of $x$. Finally, the energy of the closed shell subsystem $E^{cs}_{NI}(n_{s})$ and the corresponding magic number $N^{cs}$ are \[N^{cs}=\frac{n_{s}(n_{s}+1)(n_{s}+2)}{6}\,\,\,\,\,\mbox{and}\] (11) \[\frac{E^{cs}_{NI}(n_{s})}{\hbar\omega}=\frac{(n_{s}-1)n_{s}(n_{s} +1)(n_{s}+2)}{8}+\frac{3N^{cs}}{2}.\] (12) On the positive $a_{s}$ side where a high-lying two-body bound state exists, a significant fraction of the ground state energy of the $N$ fermion system is determined by the binding energy of the trapped dimer, which depends on $R_{0}$. To reduce the dependence of $\Lambda_{N_{1},N_{2}}^{(\kappa)}$ on the range $R_{0}$, the energy $E(1,1)$ of $N_{d}$ trapped dimer pairs is subtracted in Eq. (5). Thus, $\Lambda_{N_{1},N_{2}}^{(\kappa)}$ depends to a good approximation only on $a_{s}$, $\kappa$, $N_{1}$ and $N_{2}$, and not on the details of the underlying two-body potential (see also Sec. IV.1). By construction, $\Lambda_{N_{1},N_{2}}^{(\kappa)}$ changes from one on the weakly-interacting BCS side (small $\|a_{s}\|$ and $a_{s}<0$) to zero on the weakly-interacting molecular BEC side (small, positive $a_{s}$). The weakly-interacting regimes, where $\|a_{s}\|\ll a_{ho}$, can be treated perturbatively assuming zero-range interactions, i.e., a Fermi pseudopotential [30]. For small $\|a_{s}\|$ and $a_{s}<0$, the energy within first order perturbation theory becomes \[E\approx E_{NI}+\hbar\omega\,C_{N_{1},N_{2}}^{\kappa}\,\frac{a_{ s}}{a_{ho}},\] (13) where $C_{N_{1},N_{2}}^{\kappa}$ is a dimensionless quantity. In general, the evaluation of $C_{N_{1},N_{2}}^{\kappa}$ is a bit cumbersome since there is no unique ground state and degenerate perturbation theory must be applied. When both $N_{1}$ and $N_{2}$ correspond to closed shells, then $C_{N_{1},N_{2}}^{\kappa}$ can be calculated straightforwardly analytically [21], \[C_{N_{1},N_{2}}^{\kappa}=4\pi a_{ho}^{3}\int\rho_{1}^{NI}(\vec{r})\rho_{2}^{NI }(\vec{r})d\vec{r}.\] (14) Here, $\rho_{i}^{NI}(\vec{r})$ is the density of a one-component non-interacting gas with $N_{i}$ fermions of mass $m_{i}$, normalized so that $\int\rho_{i}^{NI}(\vec{r})d\vec{r}=N_{i}$. Alternatively, one can approximate the $\rho_{i}^{NI}$ by the Thomas-Fermi density profiles. This approximation should be quite accurate in the large $N$ limit. <figure><img src="content_image/0801.2747/x1.png"><figcaption>Figure 1: (Color Online) C1N1,N2 coefficients divided by ENI as a function ofN. Circles correspond to L=0 ground state, squares to L=1 ground state andtriangles to L=2 ground state. A solid line connects the odd-N values while adashed line connects the even-N values.</figcaption></figure> To obtain the $C_{N_{1},N_{2}}^{\kappa}$ for open-shell systems, we apply first-order degenerate perturbation theory. This calculation additionally allows us to obtain the angular momentum quantum number $L$ of the ground state. Figure 1 and Table 1 present the results for $N\leq 20$ and $\kappa=1$. The coefficients $C_{N_{1},N_{2}}^{1}$ increase monotonically with increasing $N$ and show a slight odd-even staggering. In general, the coefficients $C_{N_{1},N_{2}}^{1}$ for even $N$ are comparatively higher than those for odd $N$, implying a smaller energy for even $N$ than for odd $N$ and suggesting that, even in the perturbative regime, the odd-even oscillations are already present. We note that the $C_{N_{1},N_{2}}^{1}$ coefficients for even $N$ shown in Fig. 1 clearly reflect the shell-closure at $N=8$. N \| L \| C1N1,N2 \| N \| L \| C1N1,N2 ---\|---\|---\|---\|---\|--- 2 \| 0 \| 2√2π \| 12 \| 0 \| 12.2274 3 \| 1 \| 3√2π \| 13 \| 0 \| 13.1651 4 \| 0 \| 132√2π \| 14 \| 0 \| 15.2382 5 \| 1 \| 152√2π \| 15 \| 2 \| 16.1642 6 \| 0 \| 11√2π \| 16 \| 0 \| 18.2445 7 \| 1 \| 12√2π \| 17 \| 2 \| 19.1735 8 \| 0 \| 312√2π \| 18 \| 0 \| 21.2476 9 \| 0 \| 1458√2π \| 19 \| 2 \| 177932√2π 10 \| 0 \| 9.21052 \| 20 \| 0 \| 194532√2π 11 \| 0 \| 10.1980 \| \| \| Table 1: Angular momentum L and coefficient C1N1,N2 for the ground state of equal-mass two-component Fermi gases in the weakly-attractive regime. Here, we consider N2=N1 for even N and N1=N2+1 for odd N. In the weakly-interacting molecular BEC regime, the two-component Fermi system should behave like a system that consists of $N_{d}$ bosonic molecules and $N_{f}=0$ or $1$ fermions. In first order perturbation theory the ground state energy of such a system is given by \[E\approx N_{d}E(1,1)+\hbar\omega\frac{3N_{f}}{2}+\hbar\omega\, \frac{N_{d}(N_{d}-1)}{2}\sqrt{\frac{2}{\pi}}\,\frac{a_{dd}}{a_{ho}^{(dd)}}\] \[+\hbar\omega\,N_{d}N_{f}\sqrt{\frac{2}{\pi}}\,\frac{a_{ad}}{a_{ho }^{(ad)}}.\] (15) Here, $a_{dd}$ and $a_{ad}$ denote the dimer-dimer and atom-dimer scattering lengths, respectively. The oscillator lengths $a_{ho}^{(dd)}$ and $a_{ho}^{(ad)}$ for the dimer-dimer and atom-dimer systems, $a_{ho}^{(dd)}=\sqrt{\hbar/(2\mu_{dd}\omega)}$ and $a_{ho}^{(ad)}=\sqrt{\hbar/(2\mu_{ad}\omega)}$, are defined in terms of the reduced mass $\mu_{dd}$ of the dimer-dimer system and the reduced mass $\mu_{ad}$ of the atom-dimer system, respectively. The limiting behaviors of the BEC-BCS crossover curve can be used to guide the construction of the many-body nodal surface, which is a crucial ingredient for our FN-DMC calculations (see Sec. III.2). In the weakly-interacting molecular BEC regime, even $N$ systems consist of $N/2$ dimers. Each molecule is expected to be in its rotational ground state, leading to a many-body wave function with total angular momentum $L=0$. For odd $N$ systems, the extra fermion is expected to occupy the lowest $s$-wave orbital, leading, as in the even $N$ case, to a many-body wave function with $L=0$. Thus, the angular momentum of even $N$ systems is expected to be the same along the crossover while that of odd $N$ systems is expected to change (for $N=3$, this has been pointed out recently by two independent groups [31; 32]). This symmetry change introduces a kink in the normalized energy curve $\Lambda_{N_{1},N_{2}}^{(\kappa)}$ for odd $N$ and in the excitation gap $\Delta(N)$ (see below) at the scattering length where the symmetry change or inversion occurs. In addition to the energy crossover curve, we calculate the excitation gap $\Delta(N)$, which characterizes the odd-even oscillations of two-component Fermi systems, as a function of $N$. For homogeneous two-component Fermi systems with equal masses, the excitation gap $\Delta$, which equals half the energy it takes to break a pair, is quite well understood. In the weakly interacting BCS regime, the excitation gap $\Delta$ becomes exponentially small [33; 34], indicating vanishingly little pairing. In the deep BEC regime, on the other hand, the excitation gap approaches half the binding energy of the free-space dimer, indicating essentially complete pairing: By adding an extra particle to the odd $N$ system, the energy of the total system changes by approximately the binding energy of the free-space dimer. In addition to these limiting cases, the excitation gap of the equal-mass two-component Fermi system has been determined throughout the crossover regime by the FN-DMC method [35; 36; 37]. For unequal-mass systems, in contrast, the behavior of the gap is much less studied and understood [38; 39; 40; 41; 42]. To define the excitation gap $\Delta(N)$ for trapped unequal-mass systems, we set $N=2n+1$ and assume $N$ to be odd. The unequal-mass system is characterized by two chemical potentials, the chemical potential $\mu_{1}(N)$ for species one and the chemical potential $\mu_{2}(N)$ for species two (see, _e.g._, Ref. [43]), \[E(n+1,n)=E(n,n)+\mu_{1}(2n+1)+\Delta(2n+1)\] (16) and \[E(n,n+1)=E(n,n)+\mu_{2}(2n+1)+\Delta(2n+1).\] (17) Here, $\Delta(2n+1)$ denotes the excitation gap. If $\Delta(2n+1)$ vanishes— as is the case for the normal system—, then Eqs. (16) and (17) reduce to the “usual” chemical potentials. Furthermore, $\mu_{1}(N)$ and $\mu_{2}(N)$ coincide for equal-mass systems. To determine $\mu_{1}(N)$, $\mu_{2}(N)$ and $\Delta(N)$, we need an additional relationship. In condensed matter physics, one typically considers the average of the two chemical potentials, \[\frac{1}{2}\left[\mu_{1}(2n+1)+\mu_{2}(2n+1)\right]=\] \[\frac{1}{2}\left[E(n+1,n+1)-E(n,n)\right].\] (18) Since the average chemical potential is defined in terms of the energy of the next smaller and the next larger balanced systems, it is independent of the odd-even oscillations. Equations (16) through (II.2) can be solved for $\mu_{1}(2n+1)$, $\mu_{2}(2n+1)$ and $\Delta(2n+1)$, \[\mu_{1}(2n+1)=\frac{E(n+1,n+1)-E(n,n+1)}{2}\] \[+\frac{E(n+1,n)-E(n,n)}{2},\] (19) \[\mu_{2}(2n+1)=\frac{E(n+1,n+1)-E(n+1,n)}{2}\] \[+\frac{E(n,n+1)-E(n,n)}{2},\] (20) and \[\Delta(2n+1)=\frac{E(n+1,n)+E(n,n+1)}{2}\] \[-\frac{E(n,n)+E(n+1,n+1)}{2}.\] (21) Note that the energies $E(n+1,n)$ and $E(n,n+1)$ are equal for equal masses. The excitation gap $\Delta(N)$ and the chemical potentials $\mu_{1}(N)$ and $\mu_{2}(N)$ depend on $N$, $\kappa$, $\omega$ and $a_{s}$. Ultimately, one of the goals is to relate the excitation gaps calculated for the trapped and the homogeneous systems. For equal masses and equal frequencies, the densities of the two trapped species overlap fully. Hence, one might expect that the excitation gaps of the homogeneous and inhomogeneous systems can be related via the local density approximation (LDA), which predicts that $\Delta(N)$ scales with $N$ as $N^{1/3}$. Connecting the excitation gaps for the homogeneous and trapped systems in this way breaks down, however, if the extra particle sits near the edge of the gas cloud; this is the region that is poorly described by the LDA. Indeed, we present some evidence that the extra particle sits for $N\gtrsim 11$ near the cloud edge. For unequal masses, the connection between the two excitation gaps becomes even more challenging, because one now has to first determine whether the trapped system exhibits phase separation or not [40; 44]. For the trapped system, the density mismatch can be quantified by comparing the density overlap $O_{N_{1},N_{2}}^{\kappa}$, \[O_{N_{1},N_{2}}^{\kappa}=a_{ho}^{3}\int\rho_{1}(\vec{r})\rho_{2} (\vec{r})d\vec{r},\] (22) of the unequal-mass system with that of the equal-mass system for a given scattering length $a_{s}$. In Eq. (22), the one-body densities $\rho_{i}(\vec{r})$ and the oscillator length $a_{ho}$ depend on $\kappa$. In the non-interacting limit, the normalized density mismatch $O_{N_{1},N_{2}}^{\kappa}/O_{N_{1},N_{2}}^{1}$ reduces to $C_{N_{1},N_{2}}^{\kappa}/C_{N_{1},N_{2}}^{1}$. In this case, $O_{N_{1},N_{2}}^{\kappa}/O_{N_{1},N_{2}}^{1}$ equals one for all $\kappa$ if $N_{1}=N_{2}=1$ (see Table II of Ref. [21]). For larger $N$, however, $O_{N_{1},N_{2}}^{\kappa}/O_{N_{1},N_{2}}^{1}$ decreases from 1 to a finite value that is smaller than one as $\kappa$ varies from one to infinity. In particular, the Thomas Fermi approximation predicts $O_{N_{1},N_{2}}^{\kappa}/O_{N_{1},N_{2}}^{1}=315\pi/1024\sqrt{2}\approx 0.683$ for large non-interacting systems ($N_{1}=N_{2}$) with large $\kappa$. For the small unequal-mass systems considered in Sec. IV, we find that the density mismatch for finite $a_{s}$ is smaller than that for $a_{s}=0$. ### Hyperspherical formulation at unitarity The two-component Fermi gas at unitarity is characterized by a diverging scattering length, i.e., $1/a_{s}=0$. In this regime, the underlying two-body potential, for sufficiently small $R_{0}$, has no characteristic length scale, thus leaving only the size of the system itself. This elimination of the two-body length scale is the key to obtaining a number of analytical results; a particularly appealing framework for deriving these results employs hyperspherical coordinates. The hyperspherical formulation has been primarily developed in the context of few-body systems [45; 46; 47; 48; 49; 50]. More recently, some properties of Bose and Fermi gases with essentially arbitrary number of atoms have been explained successfully within this formulation [51; 52; 53]. The ability to treat both small and large systems on equal footing makes the hyperspherical formulation particularly suited for studying the transition from few- to many-body systems. We define the hyperspherical coordinates by first separating off the center-of-mass vector $\vec{R}_{CM}$, and by then dividing the remaining $3N-3$ coordinates into the hyperradius $R$ and $3N-4$ hyperangles, collectively denoted by $\Omega$. The hyperradius $R$ is defined by \[\mu_{N}R^{2}=\sum_{i=1}^{N_{1}}m_{1}r_{i}^{2}+\sum_{i^{\prime}=1} ^{N_{2}}m_{2}r_{i^{\prime}}^{2}-MR_{CM}^{2},\] (23) and can be viewed as a coordinate that measures the overall size of the system. Here, $M$ denotes the total mass of the system, $M=m_{1}N_{1}+m_{2}N_{2}$, and $\mu_{N}$ an arbitrary mass scaling factor. Usually, the value of $\mu_{N}$ is chosen so that the hyperradial potential curves $V_{s_{\nu}}(R)$, defined below, approach physically motivated asymptotic values as $R\rightarrow\infty$. In the adiabatic approximation [47], the relative wave function $\Psi^{rel}(R,\Omega)$ reduces to \[\Psi^{rel}(R,\Omega)=R^{-(3N-4)/2}F_{\nu n}(R)\Phi_{\nu}(R;\Omega).\] (24) The antisymmetric Pauli correlations are built into the channel functions $\Phi_{\nu}(R;\Omega)$ at the outset. In addition, the $\Phi_{\nu}(R;\Omega)$ account for a significant fraction of the two-body correlations of the system. Within the hyperspherical approximation, the description of the many-body system reduces to solving a one-dimensional Schrödinger equation in the hyperradial coordinate $R$, \[\left(-\frac{\hbar^{2}}{2\mu_{N}}\frac{d^{2}}{dR^{2}}+V_{s_{\nu}} (R)+\frac{1}{2}\mu_{N}\omega^{2}R^{2}\right)F_{\nu n}(R)\] \[=E_{\nu n}^{rel}F_{\nu n}(R).\] (25) The effective hyperradial potential $V_{s_{\nu}}(R)$ includes part of the kinetic energy and a contribution due to the short-range two-body interactions. Assuming zero-range interactions, the adiabatic approximation becomes exact for a subclass of universal states of the unitary two-component Fermi gas [54]. For these states, the channel functions $\Phi_{\nu}$ obey specific boundary conditions imposed by the zero-range pseudopotential and become independent of $R$. Furthermore, the functional form of the hyperradial potentials $V_{s_{\nu}}(R)$ can be derived analytically [54; 55], \[V_{s_{\nu}}(R)=\frac{\hbar^{2}s_{\nu}(s_{\nu}+1)}{2\mu_{N}R^{2}}.\] (26) The eigen energies of Eq. (25) are then given by \[E_{\nu n}^{rel}=\left(s_{\nu}+2n+\frac{3}{2}\right)\hbar\omega,\] (27) where $n$ is a non-negative integer, and the hyperradial wave functions $F_{\nu n}(R)$ (not normalized) by \[F_{\nu n}(R)=R^{s_{\nu}+1}L_{n}^{(s_{\nu}+1/2)}(R^{2}/{\cal{L}}^ {2})\exp\left(-\frac{R^{2}}{2{\cal{L}}^{2}}\right),\] (28) where ${\cal{L}}$ denotes the oscillator length associated with $\mu_{N}$, ${\cal{L}}=\sqrt{\hbar/(\mu_{N}\omega)}$, and $L_{n}^{(s_{\nu}+1/2)}$ the Laguerre polynomial. The total energy $E_{\nu n}$ is obtained from $E_{\nu n}^{rel}$ by adding the center of mass energy. The spacing between states labeled by the same $\nu$ is $2\hbar\omega$ and is thus independent of $s_{\nu}$. This implies that knowledge of the lowest eigenenergy $E_{\nu 0}^{rel}$ in each hyperradial potential curve determines the entire energy spectrum. This property of the spectrum has also been shown using the scale invariance properties of unitary systems [56]. Transitions between vibrational levels that lie within a given hyperradial potential curve $V_{s_{\nu}}(R)$ can be driven by an excitation operator that depends on $R$ only. Such a driving field results in a ladder of excitation frequencies of the form $2k\hbar\omega$, where $k$ denotes an integer. On the other hand, transitions between states living in different hyperradial potential curves (labeled by $\nu$ and $\nu^{\prime}$) require the driving field to depend on $\Omega$, or stated more generally, the excitation operator must not commute with the fixed-hyperradius Hamiltonian. The corresponding excitation frequencies are, in general, non-integer multiples of $2\hbar\omega$ and depend on the difference between $s_{\nu}$ and $s_{\nu^{\prime}}$. Thus, knowledge of the entire excitation spectrum requires determining all $s_{\nu}$. Moreover, the coefficients $s_{\nu}$ of the three-body system play a role in determining the three-body recombination rate for large and negative $a_{s}$ [57], and the lifetime of weakly bound dimers for large and positive $a_{s}$ [57]. Similarly, one may expect that the $s_{\nu}$ of larger systems play a role in determining the corresponding quantities for larger systems. Section IV.3 presents evidence of the $2\hbar\omega$ energy spacing and determines the $s_{\nu}$ coefficients for the four-particle system for various mass ratios. Equation (23) defines the hyperradius $R$ without the CM motion. Alternatively, we can define a hyperradius $R^{\prime}$, \[MR^{\prime 2}=\mu_{N}R^{2}+MR_{CM}^{2},\] (29) which includes the CM motion and represents the rms radius of the system. In the adiabatic approximation, the total wave function $\Psi(R^{\prime},\Omega^{\prime})$ can be written in terms of the new hyperradius $R^{\prime}$ as $\Psi(R^{\prime},\Omega^{\prime})=R^{\prime-(3N-1)/2}\bar{F}_{\nu n}(R^{\prime} )\bar{\Phi}(R^{\prime};\Omega^{\prime})$, where $\Omega^{\prime}$ collectively denotes the $3N-1$ hyperangles. Equations (25) and (26) remain valid if $R$, $\mu_{N}$ and $F_{\nu n}$ are replaced by $R^{\prime}$, $M$ and $\bar{F}_{\nu n}$, respectively. The eigen values of the hyperradial Schrödinger equation equal the eigenenergies $E_{\nu n}$ of the total system. Defining $x=R^{\prime}/R^{\prime}_{NI}$ and $\epsilon_{\nu n}=E_{\nu n}/E_{NI}$, the hyperradial Schrödinger equation can be rewritten as \[\left(-\frac{1}{2\mu_{eff}}\frac{d^{2}}{dx^{2}}+\frac{s_{\nu}(s_{ \nu}+1)}{2\mu_{eff}x^{2}}+\frac{1}{2}x^{2}\right)\bar{F}_{\nu n}(x)\] \[=\epsilon_{\nu n}\bar{F}_{\nu n}(x),\] (30) where $\mu_{eff}=E_{NI}^{2}/(\hbar\omega)^{2}$. Above, $R^{\prime}_{NI}$ denotes the rms radius of the non-interacting system; it can be, using the virial theorem [58; 54], expressed in terms of the energy $E_{NI}$ of the non-interacting two-component Fermi gas, \[R^{\prime}_{NI}=\sqrt{\langle R^{\prime 2}\rangle_{NI}}=\sqrt{ \frac{\hbar}{M\omega}}\sqrt{\frac{E_{NI}}{\hbar\omega}}.\] (31) The dimensionless coefficients $\bar{C}_{N}$, \[\bar{C}_{N}=\frac{s_{0}(s_{0}+1)}{\mu_{eff}}=\frac{s_{0}(s_{0}+1) \hbar^{2}\omega^{2}}{E_{NI}^{2}},\] (32) characterize the ground state of the system at unitarity. The scaled hyperradius $x$ and the scaled energies $\epsilon_{\nu n}$ remain finite in the large $N$ limit and are thus particularly well suited to discuss the large $N$ limit (see Sec. IV.2). For small systems, in contrast, some properties of the system can be highlighted more naturally using the unscaled hyperradius $R$ or $R^{\prime}$. The coefficients $s_{\nu}$ describe both the trapped and free systems, and can be related to the universal parameter $\xi$ of the homogeneous system [20]. The hyperspherical framework thus connects few- and many-body quantities and allows one to bridge the gap between atomic and condensed matter physics. ## III Numerical techniques ### Correlated Gaussian approach The CG method has proven capable of providing an accurate description of trapped few-body systems with short-range interactions [59; 21; 20]. The CG method expands the many-body wave function $\Psi$ in terms of a set of basis functions $\Phi_{\{d_{ij}\}}$, \[\Psi(\vec{r}_{1},\cdots,\vec{r}_{N})=\sum_{\{d_{ij}\}}C_{\{d_{ij}\}}\,\Phi_{\{ d_{ij}\}}(\vec{r}_{1},\cdots,\vec{r}_{N}),\] (33) where the $C_{\{d_{ij}\}}$ denote expansion coefficients and the ${\{d_{ij}\}}$ a set of widths. Each basis function has the form: \[\Phi_{\{d_{ij}\}}=\mathcal{S}\left\{\psi_{0}(\vec{R}_{CM})\exp\left(-\sum_{j>i =1}^{N}r_{ij}^{2}/(2d_{ij}^{2})\right)\right\}.\] (34) Here, $\psi_{0}$ is the ground state wavefunction associated with the center-of-mass vector $\vec{R}_{CM}$, and the operator $\mathcal{S}$ ensures that the basis functions have the proper symmetry under exchange of two fermions of the same species. Due to the simplicity of the basis functions, the elements of the Hamiltonian and overlap matrices can be calculated analytically [60; 61]. Since the basis functions depend only on the center of mass vector and the interparticle distances, i.e., Gaussians centered around $r_{ij}=0$, the resulting eigenenergies correspond to eigenstates with zero relative angular momentum $L_{rel}$ and zero total angular momentum $L$; throughout this work, we do not consider center-of-mass excitations so that $L_{rel}=L$ for all systems investigated. To determine the eigenenergies of states of the $N$-atom system with non-vanishing $L_{rel}$, we add a spectator atom and solve the Schrödinger equation for the $(N+1)$-atom system. The extra particle does not interact with the rest of the system but can have non-vanishing angular momentum. This trick allows us to describe non-zero angular momentum states of the $N$-atom system. We find, e.g., that the ground state of the equal-mass three- and five-particle systems at unitarity has $L_{rel}=1$. To illustrate how the energies calculated by the CG method converge with respect to the size of the basis set, we consider the three-body system with $L=0$ at unitarity. <figure><img src="content_image/0801.2747/x2.png"><figcaption>Figure 2: Convergence of the energetically lowest-lying energies as a functionof the size D of the basis set for N=3 (L=0) at unitarity. The range R0 isfixed at 0.01aho. Solid lines connect the CG energies (filled circles) of agiven state for ease of viewing.</figcaption></figure> We define $E_{D}$ as the eigenenergies obtained for an optimized basis set of size $D$. The optimization of the basis functions for a given size $D$ is performed using the basic ideas of the stochastic variational approach [61]. The size of the basis set is then increased and the new basis functions are optimized. Figure 2 shows an example of the convergence of the lowest few eigenenergies for $R_{0}=0.01a_{ho}$ as a function of $D$. The largest $D$ considered in this study is $700$, and the energies have been tested and are approximately converged for this $D$ value. Thus, Fig. 2 shows the normalized difference between $E_{700}$ and $E_{D}$ for the lowest few eigenenergies. Figure 2 shows that the basis set can be improved systematically. For larger number of particles, the size of the basis set needs to be increased. For $N=5$ and $6$, the size of the basis set is increased up to approximately $D=10^{4}$. The $N=6$ energies reported in Ref. [20], e.g., are calculated for $D=1.6\times 10^{4}$. Here, we analyze the convergence of these energies as a function of $1/D$. Since the energies behave approximately linearly as a function $1/D$, we can extrapolate straightforwardly to the limit $D\rightarrow\infty$. The extrapolated energies for $\nu=0$ are $E_{00}=8.48\hbar\omega$, $E_{01}=10.50\hbar\omega$ and $E_{02}=12.50\hbar\omega$. $E_{00}$ and $E_{01}$ agree with those reported in Ref. [20] for $D=1.6\times 10^{4}$ while $E_{02}$ is only $0.02\hbar\omega$ lower than the previously reported value. For $\nu=1$ and $2$, the extrapolated energies are $E_{10}=10.43\hbar\omega$ and $E_{20}=10.99\hbar\omega$; these energies are lower by $0.01\hbar\omega$ than those reported in Ref. [20]. While the extrapolated energies are most likely closer to the exact eigenenergies than the energies calculated for $D=1.6\times 10^{4}$, we note that the extrapolated energies are no longer variational, i.e., they no longer provide upper bounds to the exact eigenenergies. Our analysis of the $\nu\leq 2$ excited energies shows that the extrapolated energies follow the expected $2\hbar\omega$ spacing more closely than those calculated for the largest $D$ considered, suggesting that the extrapolation procedure is indeed justified. In general, the convergence of the energies with respect to the basis set depends on the scattering length $a_{s}$ and the number of states considered. Usually, an accurate determination of the spectrum at unitarity requires a larger basis than the determination of the spectrum on both the weakly-interacting BEC and BCS sides. For equal-mass systems, a converged basis at unitarity usually describes the spectrum in the entire crossover region accurately. Of the equal-mass systems treated, the $N=5$ ($L=1$) calculations have been the hardest to converge. For $L=1$ states, we can estimate the uncertainty of the calculations by monitoring the energy of the spare non-interacting particle, which is known analytically. For example, for the $N=5$ equal-mass calculations presented in Ref. [20], the energies of the spare non-interacting particle deviate from the exact solution by approximately $0.01\hbar\omega$, which is less than $1\%$. We find that systems with large $\kappa$ are typically harder to converge than the corresponding equal-mass systems. To analyze the effects of finite range interactions we study the eigenenergies of the three-particle system at unitarity ($L=0$) as a function of the range $R_{0}$. <figure><img src="content_image/0801.2747/x3.png"><figcaption>Figure 3: (Color Online) Three-body energies Eν0 at unitarity for L=0 [(a)ν=0, (b) ν=1 and (c) ν=2] as a function of the range R0. Symbols show the CGenergies and solid lines the linear extrapolation to the R0=0 limit.</figcaption></figure> Figures 3(a) through (c) show the energies for the lowest state in the hyperradial potential curve $V_{s_{\nu}}(R)$ with $\nu=0,1$ and 2. The energies show a linear dependence on $R_{0}$, and can thus be extrapolated straightforwardly to the zero range limit. Neglecting the basis set error, which is estimated to be smaller than the uncertainty of the extrapolation, we find $E_{00}=4.66622(1)\hbar\omega$, $E_{10}=7.62738(2)\hbar\omega$, and $E_{20}=9.61466(4)\hbar\omega$. Our three-body energies compare favorably with those calculated using the $s_{\nu}$ coefficients, $\nu=0$ and 1, determined by Ref. [57] in Eq. (27), $E_{00}=4.6662220\hbar\omega$ and $E_{10}=7.6273521\hbar\omega$. Section IV.1 reports three-particle energies for equal masses for $R_{0}=0.01a_{ho}$, which— according to Fig. 3— agree to better than $0.02\hbar\omega$ with those calculated in the zero-range limit. We additionally performed systematic studies of the dependence of the energies on the range $R_{0}$ for the three-body system with equal and unequal masses in the weakly-interacting molecular BEC regime, where two-body bound states form (see Secs. II.2 and IV.1), and for the four-body system. For the five- and six-body calculations, it is prohibitively expensive to perform calculations for different $R_{0}$. For these systems, we estimate the finite range effects based on our findings for the $N=3$ and 4 systems. In addition to the energies, Sec. IV.4 reports structural properties calculated by the CG approach. The one-body density and the pair-distribution functions are extracted from the total wave function $\Psi$ calculated by the CG approach by integrating $\Psi^{2}$ over the relevant Jacobi coordinates. ### Fixed-node diffusion Monte Carlo approach For larger systems, the CG approach in our current implementation becomes prohibitively expensive and we instead determine first-principles solutions of the time-independent Schrödinger equation using Monte Carlo techniques. In this study, we use the FN-DMC method [62; 63], a variant of the diffusion Monte Carlo (DMC) method, to determine solutions for up to $N=30$ fermions. The DMC method, which interprets the system’s wave function as a density, allows for the accurate determination of the energy of nodeless ground states but is not suited to determine the energy of excited states of bosonic systems or of fermionic systems. To treat systems whose eigenfunctions have nodes, the DMC algorithm has to be modified slightly. Here, we adopt the FN-DMC method, which obtains a solution of the Schrödinger equation that has the same symmetry as a so-called guiding function $\psi_{T}$. The FN-DMC method provides, to within statistical uncertainties, an upper bound to the exact eigen energy of the many-boson or many-fermion system, i.e., to the lowest-lying state with the same symmetry as $\psi_{T}$. If the nodal surface of $\psi_{T}$ coincides with that of the exact eigenfunction, then the FN-DMC method results in the exact eigen energy of the system. In general, however, the nodal surface of the exact eigenfunction is not known and the FN-DMC results depend crucially on the quality of the nodal surface of $\psi_{T}$. In this work, we consider three different parametrizations of the nodal surface of two-component Fermi systems. The guiding function $\psi_{T1}$ reads \[\psi_{T1}=\prod_{i=1}^{N_{1}}\Phi_{1}(\vec{r}_{i})\times\prod_{i^ {\prime}=1}^{N_{2}}\Phi_{2}(\vec{r}_{i^{\prime}})\times F_{node}^{T1}(\vec{r}_ {1},\cdots,\vec{r}_{N_{2}})\times\] \[\prod_{i<j}^{N_{1}}g_{11}(r_{ij})\times\prod_{i^{\prime}<j^{ \prime}}^{N_{2}}g_{22}(r_{i^{\prime}j^{\prime}})\times\prod_{i,i^{\prime}}^{N_ {1},N_{2}}g_{12}(r_{ii^{\prime}}).\] (35) The function $F_{node}^{T1}$ determines the nodal structure of $\psi_{T1}$ and is, for even $N$ and $N_{1}=N_{2}$, constructed by anti-symmetrizing a product of pair functions $f$ [64], \[F_{node}^{T1}={\cal{A}}(f(r_{11^{\prime}}),f(r_{22^{\prime}}), \cdots,f(r_{N_{1}N_{2}})),\] (36) where ${\cal{A}}$ is the antisymmetrization operator. The pair function $f$ is given by the free-space two-body solution [64]: $f$ coincides with the free-space two-body bound state solution for positive scattering length $a_{s}$, and with the free-space scattering solution, calculated at the scattering energy $E_{rel}$, for negative $a_{s}$. For $N=6$, we treat $E_{rel}$ as a variational parameter and find a reduction of the energy of 1 or 2% for a finite $E_{rel}$ compared to $E_{rel}=0$. For larger $N$, we simply use $E_{rel}=0$. For odd $N$, we add a single particle orbital $\phi_{nl}$ in Eq. (36) so that $F_{node}^{T1}$ becomes, for $N_{1}=N_{2}+1$ [65; 35], \[F_{node}^{T1}=\] \[{\cal{A}}(f(r_{11^{\prime}}),\cdots,f(r_{N_{1}-1,N_{2}}),\phi_{nl }(\vec{r}_{N_{1}}/a_{ho}^{(1)}))=\] \[\mbox{det}\left\|\begin{array}[]{cccc}f(r_{11^{\prime}})&\cdots&f( r_{1N_{2}})&\phi_{nl}(\vec{r}_{1}/a_{ho}^{(1)})\\ f(r_{21^{\prime}})&\cdots&f(r_{2N_{2}})&\phi_{nl}(\vec{r}_{2}/a_{ho}^{(1)})\\ \vdots&&\vdots&\vdots\\ f(r_{N_{1}1^{\prime}})&\cdots&f(r_{N_{1}N_{2}})&\phi_{nl}(\vec{r}_{N_{1}}/a_{ ho}^{(1)})\end{array}\right\|,\] (37) where $a_{ho}^{(i)}=\sqrt{\hbar/(m_{i}\omega)}$. We consider a number of different single particle orbitals $\phi_{nl}$, and determine the optimal $nl$ values by performing a series of FN-DMC calculations. For the lowest $n$ and $l$, the orbitals read $\phi_{00}(\vec{r}/a_{ho}^{(1)})=1$, $\phi_{01}(\vec{r}/a_{ho}^{(1)})=z/a_{ho}^{(1)}$, $\phi_{20}(\vec{r}/a_{ho}^{(1)})=1-2(r/a_{ho}^{(1)})^{2}/3$ and $\phi_{02}(\vec{r}/a_{ho}^{(1)})=3(z/a_{ho}^{(1)})^{2}-(r/a_{ho}^{(1)})^{2}$. In Eq. (III.2), the $\Phi_{i}$ ($i=1$ and 2) denote Gaussian single particle orbitals that depend on a width parameter $b_{i}$, $\Phi_{i}(\vec{r})=\exp(-r^{2}/(2b_{i}^{2}))$. If $b_{i}=\sqrt{\hbar/(m_{i}\omega)}$, $\Phi_{i}$ coincides with the ground state orbital of the harmonic oscillator. The parameters $b_{1}$ and $b_{2}$ are optimized variationally. For even $N$ ($N_{1}=N_{2}$) and equal masses, we require $b_{1}=b_{2}$. At unitarity, e.g., we find that the $b_{i}$ are smaller than the $a_{ho}^{(i)}$, reflecting the attractive nature of the interspecies interaction potential. If $b_{i}=a_{ho}^{(i)}$, the product $\Phi_{i}(\vec{r})\phi_{nl}(\vec{r}/a_{ho}^{(i)})$ equals the harmonic oscillator wave function $\phi_{nl0}^{(HO)}(\vec{r}/a_{ho}^{(i)})$. In Eq. (III.2), the pair functions $g_{11}$, $g_{22}$ and $g_{12}$ are introduced to improve the variational energy and to additionally ensure that the structural properties calculated at the VMC and FN-DMC levels agree at least qualitatively. The pair functions $g_{11}$ and $g_{22}$ allow for the effective repulsion between equal fermions to be accounted for, \[g_{ii}(r)=\exp(-p_{i}r^{-q_{i}})\] (38) for $i=1$ and $2$. The parameters $p_{1}$, $p_{2}$, $q_{1}$ and $q_{2}$ are optimized variationally. For even $N$ and equal masses, we require $p_{1}=p_{2}$ and $q_{1}=q_{2}$. The pair function $g_{12}$ is parametrized in terms of the three variational parameters $t$, $p_{12}$ and $q_{12}$, \[g_{12}(r)=1+t\exp(-p_{12}r^{-q_{12}}).\] (39) The parameters $t$, $p_{12}$ and $q_{12}$ are optimized under the constrained that $g_{12}\geq 0$. The guiding function $\psi_{T1}$ is expected to provide a good description of the system in the weakly-interacting molecular BEC regime, where we expect bound dimer pairs to form. Section IV.2 shows that this wave function also provides a good description of the unitary gas for sufficiently large $N$. This is in agreement with FN-DMC studies for the homogeneous system [64]. Since each pair function $f$ has vanishing relative orbital angular momentum, the total angular momentum $L$ of $\psi_{T1}$ is 0 for even $N$ and $N_{1}=N_{2}$. For odd $N$, $L$ of $\psi_{T1}$ is determined by the angular momentum of $\phi_{nl}$, i.e., $L=l$. In addition to $\psi_{T1}$, we consider the guiding function $\psi_{T2}$, \[\psi_{T2}=\prod_{i=1}^{N_{1}}\Phi_{1}(\vec{r}_{i})\times\prod_{i^ {\prime}=1}^{N_{2}}\Phi_{2}(\vec{r}_{i^{\prime}})\times\] \[\Psi_{node}^{T2}(\vec{r}_{1},\cdots,\vec{r}_{N_{2}})\times\prod_{ i,i^{\prime}}^{N_{1},N_{2}}\bar{f}(r_{ii^{\prime}}).\] (40) The nodal surface of $\psi_{T2}$ is determined by $\Psi_{node}^{T2}$, which is defined so that the product $\prod_{i=1}^{N_{1}}\Phi_{1}(\vec{r}_{i})\times\prod_{i^{\prime}=1}^{N_{2}}\Phi _{2}(\vec{r}_{i^{\prime}})\times\Psi_{node}^{T2}$ coincides for $b_{i}=a_{ho}^{(i)}$ with the wave function of $N$ trapped non-interacting fermions. Thus, the nodal surface of $\psi_{T2}$ coincides with that of the corresponding non-interacting system. The pair function $\bar{f}$ coincides with the pair function $f$ introduced above for $r\leq R_{m}$, where $R_{m}$ is a matching point determined variationally. For $r>R_{m}$, $\bar{f}$ is given by $c_{1}+c_{2}\exp(-\alpha r)$. The parameters $c_{1}$ and $c_{2}$ are determined by the condition that $\bar{f}$ and its derivative be continuous at $r=R_{m}$ while $\alpha$ is optimized variationally. The guiding function $\Psi_{T2}$ is expected to provide a good description of the system in the weakly-interacting BCS regime. In this regime, we construct the guiding function so that its angular momentum agrees with that predicted analytically (see Table 1). Section IV.2 shows that the guiding function $\Psi_{T2}$ also provides a good description of small fermionic systems at unitarity. Finally, the guiding function $\psi_{T3}$ is constructed following Eqs. (3) and (4) of Ref. [66]. We find that $\psi_{T3}$ gives the lowest energy for $N=11$. Expectation values $\langle A\rangle$ of operators $A$ that do not commute with the Hamiltonian cannot be calculated as straightforwardly by the FN-DMC method as the energy. Here, we use the mixed estimator $\langle A\rangle_{mixed}$ [67; 63], \[\langle A\rangle_{mixed}=2\langle A\rangle_{DMC}-\langle A\rangle _{VMC}.\] (41) In Eq. (41), $\langle A\rangle_{VMC}$ denotes the expectation value calculated by the VMC method and $\langle A\rangle_{DMC}$ that calculated by the FN-DMC method. We note that some algorithms for the calculation of pure estimators exist [68; 69] but we do not use them in this work. In some cases, we optimize the variational parameters, collectively denoted by $\vec{p}$, by not only minimizing the energy expectation value but by additionally ensuring that $\psi_{T}$ captures selected structural properties correctly. To this end, we compare the structural properties calculated by the VMC method for a given $\vec{p}_{0}$ with those obtained by the FN-DMC method, which uses $\psi_{T}(\vec{p}_{0})$ as a guiding function, and then choose a new parameter set $\vec{p}_{1}$ so that the VMC structural properties calculated using $\vec{p}_{1}$ agree better with the FN-DMC structural properties calculated using $\vec{p}_{0}$. This procedure is repeated till the VMC and FN-DMC structural properties and energy expectation values agree sufficiently well. For equal-mass systems with $N\leq 20$, our VMC energies are at most 15% higher than the corresponding FN-DMC energies. The optimization strategy employed here is similar in spirit to that discussed in Ref. [70] for the homogeneous system. ## IV Results ### Ground state energy in the crossover regime This section discusses the behavior of the crossover curve and the excitation gap for $N=3$ for different mass ratios $\kappa$. This odd $N$ study complements our earlier results for even $N$ [21]. Our analysis for $N=4$ showed that the crossover curve $\Lambda_{N}^{(\kappa)}$ is independent of the details of the two-body potential and allowed us to extract the dimer-dimer scattering length $a_{dd}$ and the dimer-dimer effective range $r_{dd}$ as a function of $\kappa$. The $a_{dd}$ and $r_{dd}$ results from Ref. [21] are summarized in Table 2. Furthermore, for larger even $N$ systems, we determined the validity regimes of the analytically calculated limiting behaviors in the weakly-interacting molecular BEC and atomic BCS regimes. Our even $N$ study resulted in a deeper understanding of some of the peculiarities of trapped systems and emphasized similarities and differences between the trapped and κ \| add/as (a) \| add/as (b) \| rdd/as (a) \| rdd/as (b) ---\|---\|---\|---\|--- 1 \| 0.608(2) \| 0.64(1) \| 0.13(2) \| 0.12(4) 4 \| 0.77(1) \| 0.79(1) \| 0.15(1) \| 0.23(1) 8 \| 0.96(1) \| 0.98(1) \| 0.28(1) \| 0.38(2) 12 \| 1.10(1) \| 1.08(2) \| 0.39(2) \| 0.55(2) 16 \| 1.20(1) \| 1.21(3) \| 0.55(2) \| 0.60(5) 20 \| 1.27(2) \| 1.26(5) \| 0.68(2) \| 0.74(5) Table 2: Dimer-dimer scattering length add and dimer-dimer effective range rdd obtained using (a) the CG spectrum and (b) the FN-DMC energies. The reported uncertainties reflect the uncertainties due to the fitting procedure; the potential limitations of the FN-DMC method to accurately describe the energetically lowest-lying gas-like state, e.g., are not included here (see Sec. IIIB of Ref. vonstechtbp ). homogeneous systems. The behavior of odd $N$ systems is rich and, in many cases, qualitatively different from that of even $N$ systems. One characteristic of odd $N$ systems is the possible change of the angular momentum of the ground state as the scattering length is tuned through the BEC-BCS crossover region (see Sec. II.2 and Refs. [31; 32]). <figure><img src="content_image/0801.2747/x4.png"><figcaption>Figure 4: (Color Online) Normalized energy (E−E(1,1)−3ℏω/2)/ℏω for N=3 as afunction of aho/as calculated by the CG approach (lines). E denotes the three-body energy for L=0 (solid lines) and for L=1 (dashed lines): (a) equal-massatoms [κ=1, E=E(2,1)=E(1,2)], (b) two heavy atoms and one light atom [κ=4,E=E(2,1)], and (c) two light atoms and one heavy atom [κ=4, E=E(1,2)]. Thenormalized energy crossover curve Λ(κ)3, Eq. (5), coincides with the dashedand solid lines, respectively, depending on whether the three-particle groundstate has L=1 or 0. In the CG calculations, the range R0 of the two-bodypotential is fixed at 0.01aho. For comparison, crosses and circles showselected FN-DMC energies for L=0 and L=1, respectively.</figcaption></figure> Figure 4 shows the three-particle energy $E$, with the energy $E(1,1)+3\hbar\omega/2$ subtracted, for $L=0$ (solid lines) and $L=1$ (dashed lines). The upper panel shows results for $\kappa=1$, and the two lower panels for $\kappa=4$ [panels (b) and (c) consider the three-particle system with a spare heavy and a spare light particle, respectively]. The ground state has $L=1$ for $a_{ho}/a_{s}\rightarrow-\infty$ and $L=0$ for $a_{ho}/a_{s}\rightarrow\infty$, independent of $\kappa$ and independent of whether the spare particle is heavy or light. For equal masses, the change of symmetry occurs at $a_{s}\approx a_{ho}$. For $\kappa=4$, in contrast, it occurs at $a_{s}\approx 0.3a_{ho}$ if the extra particle is a heavy atom [panel (b)] and at $a_{s}\approx 3a_{ho}$ if the extra particle is a light atom [panel (c)]. The dashed and solid lines shown in Fig. 4 coincide with the normalized crossover curve $\Lambda_{N_{1},N_{2}}^{(\kappa)}$, Eq. (5), in the region where the ground state of the three-particle system has $L=1$ and $0$, respectively. The normalized crossover curve $\Lambda_{3}^{(\kappa)}$ changes from 1 in the weakly-interacting molecular BEC regime to 0 in the weakly-interacting BCS regime. We find that the normalized $L=1$ energy curve for two heavy atoms and one light atom [Fig. 4(b)] depends notably on the range of the underlying two-body potential if the scattering length $a_{s}$ is positive. For example, the normalized energy curve changes by as much as 20% if the range $R_{0}$ of the two-body potential changes from $0.01a_{ho}$ to $0.02a_{ho}$. This comparatively large dependence on $R_{0}$ indicates that the properties of the system with two heavy atoms and one light atom are not fully determined by the $s$-wave scattering length for the ranges considered. In the $R_{0}\to 0$ limit, the $\kappa=4$ system is expected to behave universal [23; 27]. We speculate that the comparatively strong dependence of the normalized energy curve on the range for $a_{s}>0$ is related to the fact that the three-particle system supports, for sufficiently large $\kappa$, bound states with negative energy. For comparison, circles and crosses in Fig. 4 show selected three-particle energies calculated by the FN-DMC method for $L=0$ and $L=1$, respectively. The good agreement with the CG results (lines) indicates that the FN-DMC method can be used to accurately describe different symmetry states. Our CG energies for equal-mass systems interacting through short-range potentials presented in Fig. 4(a) can be compared with those of Kestner and Duan [31] obtained for zero-range interactions. Our $L=1$ energy curve agrees with that of Kestner and Duan for all scattering lengths $a_{s}$ considered. The $L=0$ energy curve, however, only agrees for $a_{s}<0$. For $a_{s}>0$, our results are noticeably lower than those of Kestner and Duan. As shown below, our $a_{s}>0$ results for $L=0$ predict the correct atom-dimer scattering length suggesting that our energies should be very close to those for $R_{0}=0$ and that the disagreement is not due to finite-range effects. We speculate that the results of Kestner and Duan might not be fully converged for $a_{s}>0$ although other possibilities cannot be excluded. Figures 5(a) and (b) present the BCS and BEC limiting behaviors for an equal mass system with $N=3$. The perturbative expression, Eq. (II.2), on the BEC side is expected to be applicable if $R_{0}\ll a_{s}\ll a_{ho}$; thus, we choose a small $R_{0}$, i.e., $R_{0}=0.005a_{ho}$, in the CG calculations. The energy is in this region determined by the atom-dimer scattering length $a_{ad}$ [see Eq. (II.2)]. The CG energies change linearly with $a_{s}$, showing that $a_{ad}$ is proportional to $a_{s}$, i.e., $a_{ad}=c_{ad}a_{s}$. A simple linear fit to the CG results predicts $c_{ad}\approx 1.21$, in good agreement with previous studies [71; 72], which found $a_{ad}\approx 1.2a_{s}$. A solid line in Fig. 5(a) shows the resulting linear expression. A more sophisticated analysis accounts for the energy-dependence of $a_{ad}$ [73; 74], which results in a more reliable determination of $c_{ad}$ and also a determination of the effective range $r_{ad}$ [21]. Considering the three lowest energy levels on the BEC side [21], we obtain $c_{ad}\approx 1.18(1)$ and $r_{ad}\approx 0.08(1)a_{s}$. It was suggested earlier [24] that the atom-dimer system is characterized by a soft-core repulsion with range of the order of $a_{s}$; our calculations support this general picture but predict a range about ten times smaller than $a_{s}$. On the BCS side, the first order correction varies also linearly with $a_{s}$. Circles in Fig. 5(b) show the CG results while the solid line shows the prediction from Eq. (13). Good agreement is observed in both limiting behaviors. <figure><img src="content_image/0801.2747/x5.png"><figcaption>Figure 5: (Color Online) Limiting behavior of the ground state energy for N=3equal mass fermions. (a) Energy correction ΔE=E(2,1)−E(1,1)−3ℏω/2 on the BECside. Circles show the CG results while the solid line shows the first ordercorrection for aad≈1.2as. (b) Energy E(2,1) on the BCS side. Circles show theCG results while the solid line shows the first order correction on the BCSside.</figcaption></figure> Our CG energies for $N=2$, $3$ and $4$ can be readily combined to determine the excitation gap $\Delta(3)$, Eq. (II.2). <figure><img src="content_image/0801.2747/x6.png"><figcaption>Figure 6: (Color Online) Excitation gap Δ(N) for N=3 as a function of aho/ascalculated by the CG approach for κ=1 (solid line) and κ=4 (dashed line).Circles present the BEC limiting behavior 3ℏω/2−E(1,1)/2 which is independentof κ. The inset shows a blow-up of the region where Δ(3) is smallest; in thisregion, the dependence of Δ(3) on κ is most pronounced. The dash-dotted lineshows the limiting behavior for κ=1 obtained by approximating the E(N) in Eq.(II.2) by their perturbative values, Eq. (13).</figcaption></figure> Figure 6 shows the excitation gap $\Delta(3)$ as a function of $a_{ho}/a_{s}$ for two different mass ratios, i.e., $\kappa=1$ and $4$. In the weakly-interacting molecular BEC regime, the excitation gap approaches $3\hbar\omega/2-E(1,1)/2$ (circles), independent of the mass ratio. In the weakly-interacting BCS regime, however, the excitation gap depends on the mass ratio (see inset of Fig. 6). For equal masses, $\Delta(3)$ is very well described by the perturbative expression for $a_{s}\lesssim-0.5a_{ho}$ (dash-dotted line in the inset). Figure 6 shows that $\Delta(3)$ is smaller for $\kappa=4$ than for $\kappa=1$. Intuitively, this might be expected since the radial densities of the two species do not fully overlap for unequal masses (recall, we consider the case where species one and two experience the same trapping frequency). Thus, the pairing mechanism is expected to be less efficient in the unequal-mass system, especially on the BCS side, than in the equal-mass system. The next section discusses the behavior of the excitation gap at unitarity in more detail. ### Ground state energy at unitarity This section explores the odd-even behavior of two-component Fermi gases at unitarity. In particular, we present the excitation gap for equal-mass systems with up to $N=30$ fermions and interpret the behaviors of these systems within the hyperspherical framework. We also discuss the excitation gap for small unequal-mass systems. Table 3 summarizes selected CG and FN-DMC energies for small equal-mass systems at unitarity. Some of these energies were already reported in Refs. [21; 20], and we include them in Table 3 for comparative purposes. N \| L \| E/(ℏω) (CG) \| E/(ℏω) (FN-DMC) ---\|---\|---\|--- 3 \| 0 \| 4.682 \| 4.67(3) 3 \| 1 \| 4.275 \| 4.281(4) 4 \| 0 \| 5.028 \| 5.051(9) 5 \| 0 \| 8.03 \| 8.10(3) 5 \| 1 \| 7.53 \| 7.61(1) 6 \| 0 \| 8.48 \| 8.64(3) 7 \| 0 \| \| 11.85(5) 7 \| 1 \| \| 11.36(2) 8 \| 0 \| \| 12.58(3) 9 \| 0 \| \| 15.84(6) 9 \| 1 \| \| 15.69(1) Table 3: CG and FN-DMC energies E at unitarity for small equal-mass systems with angular momentum L=0 and 1. The CG energies are calculated for the Gaussian interaction potential with R0=0.01aho for N=3 and 4, and with R0=0.05aho for N=5 and 6. The FN-DMC energies are calculated for the square well interaction potential with R0=0.01aho. The guiding functions ψT1 and ψT2, Eqs. (III.2) and (III.2), are used to obtain the energies of states with L=0 and 1, respectively. A comparison of the CG and FN-DMC energies for $N\leq 6$ shows that the FN-DMC energies agree to within 2% with the CG energies for both $L=0$ and $1$ states. This agreement suggests that the nodal surface used in the FN-DMC calculations is quite accurate. Thus, Table 3 shows that the FN-DMC method allows not only for an accurate description of the ground state but also of excited states. For $N=9$, the energy of the $L=1$ state is by only about $0.15\hbar\omega$ smaller than that of the $L=0$ state. The ground state energies for larger $N$ are reported in Table II of Ref. [20]. For both even and odd $N$ ($N>9$), we find that the angular momentum of the lowest energy state at unitarity is zero. Our FN-DMC energies thus suggest that the total angular momentum of the lowest energy states at unitarity has $L=1$ for small odd $N$ systems and $L=0$ for larger odd $N$ systems. We note that this conclusion depends crucially on the construction of the nodal surface entering the FN-DMC calculations. For $N=19$, e.g., the energies at unitarity for $L=2$ and 1 are less than $0.8\hbar\omega$ higher than the $L=0$ energy; thus, the definite determination of the ordering of the states at unitarity with different angular momenta remains a challenge for odd-$N$ systems with $N>9$. For homogeneous systems, the ground state energy per particle at unitarity $E_{u}$ is related to the energy per particle $E_{FG}$ of the non-interacting system by a universal proportionality constant $\xi$, $E_{u}=\xi E_{FG}$ [35; 64; 37]. Applying this result to the trapped unitary system with even $N$ through the LDA, the ground state energy $E_{00}(N)$ of the trapped system becomes directly proportional to the energy $E_{NI}$ of the non-interacting trapped system [21], \[E_{00}(N)=\sqrt{\xi}E_{NI}.\] (42) An analysis of our FN-DMC energies for $N=2-30$ suggests that the trapped unitary system shows little shell structure. This motivates us to “smooth” the non-interacting energies, i.e., we approximate $E_{NI}$ by the extended Thomas-Fermi expression [75], \[E_{NI,ETF}=\hbar\omega\frac{1}{4}(3N)^{4/3}\left(1+\frac{1}{2}(3N)^{-2/3} \right).\] (43) To determine the proportionality constant $\xi$, we fit our even $N$ energies for $N=2-30$ to the expression $\sqrt{\xi_{tr}}E_{NI,ETF}$. We find $\xi_{tr}=0.467$, and denote the resulting energies by $E_{fit}$. This value is in very good agreement with our previous result, $\xi_{tr}=0.465$, obtained by including the energies for $N=2-20$ only [21]. Circles in Fig. 7 show the residual energy $E_{00}(N)-E_{fit}$ for both even and odd $N$. For even $N$, the energy difference $E_{00}(N)-E_{fit}$ is at most $0.15\hbar\omega$ (except for $N=30$, for which the error bar is large). This suggests that the energies of the trapped unitary system are indeed quite well described by $\sqrt{\xi_{tr}}E_{NI,ETF}$; in other words, our energies show little residual shell structure. <figure><img src="content_image/0801.2747/x7.png"><figcaption>Figure 7: (Color Online) Excitation gap Δ(N) (squares) and residual energyE00(N)−Efit (circles) for equal-mass Fermi systems at unitarity as a functionof N calculated from the FN-DMC energies. Triangles show Δ(N) calculated usingdensity functional theory bulgac07 .</figcaption></figure> As expected, the odd $N$ energies are not even quantitatively described correctly by $E_{00}(N)-E_{fit}$. Instead, Fig. 7 shows that the residual energy $E_{00}(N)-E_{fit}$ for odd $N$ (circles) agrees quite well with the excitation gap $\Delta(N)$ (squares). For comparison, triangles in Fig. 7 show the excitation gap calculated using DFT [76]. The good agreement between the DFT and FN-DMC results is encouraging. The ground-state energies $E_{00}(N)$ determine the coefficients $s_{0}$ [see Eq. (27)] of the hyperradial potential $V_{s_{0}}(R)$ [see Eq. (26)]. <figure><img src="content_image/0801.2747/x8.png"><figcaption>Figure 8: Hyperradial potential curves V(R) for equal-mass two-componentFermi systems with (a) vanishing interactions and (b) infinitely stronginteractions as a function of R. The hyperradial potential curves naturallyappear ordered as N increases: Solid lines correspond, from bottom to top, toN=4−20 (N even), while dashed lines correspond, from bottom to top, to N=3−19(N odd).</figcaption></figure> Figures 8(a) and (b) show the lowest hyperradial potential curves $V(R)$ [$V(R)=V_{s_{0}}(R)+V_{trap}(R)$, where $V_{trap}(R)=\frac{1}{2}\mu_{N}\omega^{2}R^{2}$ and $\mu_{N}=m$] for $N=3-20$ in the non-interacting limit and at unitarity, respectively. The small $R$ behavior of $V(R)$ is dominated by $V_{s_{0}}(R)$ while the large $R$ behavior of $V(R)$ is dominated by $V_{trap}(R)$. Comparison of Figs. 8(a) and (b) shows that the attractive interactions lead to a lowering of the potential curves at unitarity compared to those of the non-interacting system. Furthermore, the $V(R)$ at unitarity appear “staggered”, i.e., odd-even oscillations are visible, reflecting the finite excitation gap at unitarity. In the non-interacting limit, in contrast, the excitation gap is zero and no odd-even staggering of the hyperradial potential curves is visible. To extrapolate to the large $N$ limit, Fig. 9 shows the normalized coefficients $\bar{C}_{N}$, Eq. (32) with $E_{NI}$ replaced by $E_{NI,ETF}$, as a function of $N$ (just as in our analysis of the energies $E_{00}$ we find it useful to smooth the energies $E_{NI}$). <figure><img src="content_image/0801.2747/x9.png"><figcaption>Figure 9: (Color Online) Normalized coefficients ¯CN, Eq. (32) with ENIreplaced by ENI,ETF, as a function of N; values for even N are shown bycircles and values for odd N by crosses. The dash-dotted line shows the valueξ=0.42 obtained by FN-DMC calculations for the homogeneous system astr04c ;carl05 , while a dashed curve shows the value ξ=0.508 obtained with arenormalization procedure vonstech07 . The inset shows the same quantities asa function of 1/N instead of N.</figcaption></figure> The coefficients $\bar{C}_{N}$ oscillate between two smooth curves, a curve for even $N$ (circles) and a curve for odd $N$ (crosses). As $N$ increases, the difference between the two curves decreases. In the large $N$-limit, the value of $\bar{C}_{N}$ for two-component Fermi gases at unitarity should approach the universal parameter $\xi$ [20]. This can be shown by relating the ground state energy obtained within the hyperspherical framework, Eq. (27), to the LDA prediction (see above), or by applying renormalized zero-range interactions within the hyperspherical framework [55]. The dash-dotted and dashed lines in Fig. 9 show the $\xi$ value obtained by FN-DMC calculations for the homogeneous system ($\xi=0.42$) [64; 37] and the $\xi$ value obtained with a renormalization procedure ($\xi=0.508$) [77], respectively. It is generally believed that the FN-DMC calculations provide the most reliable estimate for $\xi$ to date. For comparison, our energies for the trapped system predict $\xi_{tr}=0.467$ (see above). The circles in Fig. 9 approach this value. We attribute the fact that $\xi_{tr}$ is larger than the corresponding value of the bulk system, i.e., $\xi=0.42$, to the comparatively small system sizes ($N\leq 30$) included in our analysis. If this was true, we would expect the circles in the main part of Fig. 9 to turn around at larger $N$ values. We note that we cannot rule out that the nodal surface entering our FN-DMC calculations might not be optimal. In addition to equal-mass unitary systems, we study small systems with unequal masses at unitarity. Figure 10 shows the excitation gap $\Delta(N)$ <figure><img src="content_image/0801.2747/x10.png"><figcaption>Figure 10: (Color Online) Circles show the excitation gap Δ(N) for N=3 as afunction of the mass ratio κ at unitarity. Triangles and squares show thechemical potentials μ1(3) and μ2(3), respectively.</figcaption></figure> for $N=3$ at unitarity as a function of the mass ratio $\kappa$. $\Delta(3)$ decreases from about $0.8\hbar\omega$ for $\kappa=1$ to about $0.3\hbar\omega$ for $\kappa=8$. A decrease of the excitation gap as a function of $\kappa$ has recently also been reported for the homogeneous unequal-mass system at unitary [78]. To better understand the decrease of $\Delta(N)$ with increasing $\kappa$, triangles and squares in Fig. 10 show the chemical potentials $\mu_{1}(3)$ and $\mu_{2}(3)$ for the two species. The decrease of $\mu_{1}$ is related to the fact that trimers with negative energy form for sufficiently large $\kappa$. We additionally note that the densities of the light and heavy particles do not fully overlap. This effect is unique to the trapped system (the study of the homogeneous system with unequal masses [78] assumes equal densities of the two components and full pairing). Simple arguments lead one to conclude that a partial density overlap as opposed to a full density overlap leads to a decrease of the excitation gap. Thus, it is not clear if the decrease of $\Delta(3)$ visible in Fig. 10 with $\kappa$ is due to the same mechanisms that lead to a decrease of $\Delta$ in the homogeneous system or due to the specifics of the trapping potentials, or possibly both. ### Excitation spectrum at unitarity Excitation spectra of two-component Fermi gases are rich. For four equal-mass fermions, e.g., Ref. [59] shows how the spectrum evolves from the non-interacting limit for small $\|a_{s}\|$, $a_{s}<0$, to different families in the small $a_{s}$ region, $a_{s}>0$: One family consists of states that describe two bound dimers, another consists of states that describe a bound dimer plus two atoms, and yet another consists of states that describe a gas. Between these two limiting cases is the unitary region where the eigenspectrum is expected to be characterized by unique properties, similar to those of the non-interacting system (see Sec. II.3). In particular, in the unitary regime families of eigenenergies separated by $2\hbar\omega$ are expected to exist [54]. This prediction has recently been verified for up to six particles with equal masses to within numerical accuracy, i.e., to within 2% [20]. Here we extend our analysis to unequal-mass systems with $N=4$ and $L=0$. Circles in Fig. 11 show the zero angular-momentum energy spectrum calculated by the CG approach for four particles at unitarity as a function of $\kappa$. The range of the Gaussian potential is $R_{0}=0.01a_{ho}$. <figure><img src="content_image/0801.2747/x11.png"><figcaption>Figure 11: (Color Online) Four-body energy spectrum for L=0 at unitarity as afunction of κ. Circles correspond to the numerical results obtained by the CGapproach. Solid, dashed and dash-dotted lines show the energies Eν0+2nℏω forν=0, 1 and 2, respectively (n=0,1,⋯).</figcaption></figure> To analyze the eigenenergies, we employ the hyperspherical framework. Assuming that the separation of the wave function (see Sec. II.3) holds for the short-range interactions considered here, we expect that the energy spectrum consists of families of energy levels separated by $2\hbar\omega$. Solid lines show the energies $E_{00}+2n\hbar\omega$ ($n$ non-negative integer), where $E_{00}$ denotes the lowest positive energy of the spectrum (for sufficiently large $\kappa$, negative energy states form; these are not shown in Fig. 11). The agreement between the solid lines and the CG energies indicates that the $2\hbar\omega$ spacing, predicted for zero-range interactions, is fulfilled within our numerical accuracy. We repeat this exercise for the next family of energy levels: Dashed lines show the energy $E_{10}+2n\hbar\omega$, where $E_{10}$ corresponds to the lowest positive energy not yet assigned to a family. Similarly, dash-dotted lines connect states belonging to the third family. In addition to the just outlined assignment of quantum numbers, we checked in a few cases that the hyperradial wave functions $F_{\nu n}(R)$ corresponding to the energies $E_{\nu n}$ possess $n$ hyperradial nodes (see also Sec. IV.5). The lines in Fig. 11 show a crossing of energy levels belonging to different families at $\kappa\approx 4$. In close vicinity to this crossing, the spacing may not be exactly $2\hbar\omega$. As already pointed out in the previous section, the energies $E_{\nu 0}$ determine the coefficients $s_{\nu}$ of the hyperradial potential curves $V_{s_{\nu}}(R)$. Table 4 summarizes the three smallest coefficients for various $\kappa$. κ \| s0 \| s1 \| s2 \| κ \| s0 \| s1 \| s2 ---\|---\|---\|---\|---\|---\|---\|--- 1 \| 2.03 \| 4.46 \| 5.05 \| 8 \| 2.45 \| 3.81 \| 5.29 2 \| 2.09 \| 4.41 \| 4.88 \| 9 \| 2.45 \| 3.74 \| 5.35 3 \| 2.18 \| 4.27 \| 4.90 \| 10 \| 2.42 \| 3.68 \| 5.39 4 \| 2.27 \| 4.15 \| 4.98 \| 11 \| 2.37 \| 3.62 \| 5.39 5 \| 2.34 \| 4.04 \| 5.06 \| 12 \| 2.29 \| 3.57 \| 5.30 6 \| 2.40 \| 3.95 \| 5.15 \| 13 \| 2.17 \| 3.51 \| 5.18 7 \| 2.43 \| 3.88 \| 5.22 \| \| \| \| Table 4: Coefficients sν of the hyperradial potential curves Vsν(R), Eq. (26), for the N=4 system with L=0 for various mass ratios κ. To the best of our knowledge, these are the first calculations of the $s_{\nu}$ for four-particle systems with unequal masses. ### Structural properties along the BEC-BCS crossover In addition to the energetics, we analyze the one-body densities and pair distribution functions of two-component Fermi systems in the crossover regime for different $\kappa$. While the densities $\rho_{i}(\vec{r})$ of $L=0$ states are spherically symmetric, those of states with $L>0$ are not spherically symmetric. In the following, we determine the averaged radial densities $\rho_{i}(r)$, normalized so that $4\pi\int\rho_{i}(r)r^{2}dr=N_{i}$; $4\pi r^{2}\rho_{i}(r)/N_{i}$ tells one the probability of finding a particle with mass $m_{i}$ at a distance $r$ from the center of the trap. If $N_{1}=N_{2}$ and $m_{1}=m_{2}$, the radial one-body densities $\rho_{1}(r)$ and $\rho_{2}(r)$ coincide. If $m_{1}$ and $m_{2}$ or $N_{1}$ and $N_{2}$ differ, however, the radial one-body densities $\rho_{1}(r)$ and $\rho_{2}(r)$ are, in general, different. We also determine the averaged radial pair distribution functions $P_{ij}(r)$, normalized so that $4\pi\int P_{ij}(r)r^{2}dr=1$; $4\pi r^{2}P_{ij}(r)$ tells one the probability to find a particle of mass $m_{i}$ and a particle of mass $m_{j}$ at a distance $r$ from each other. For notational simplicity, we refer to the radial one-body densities as one-body densities and to the radial pair distribution functions as pair distribution functions in the following. Figure 12 shows the pair distribution <figure><img src="content_image/0801.2747/x12.png"><figcaption>Figure 12: (Color Online) Pair distribution functions P12(r), multiplied byr2, for equal mass two-component Fermi systems with N=3 and L=0 (dashedlines), N=3 and L=1 (dash-dotted lines), and N=4 and L=0 (solid lines)obtained by the CG approach for three different scattering lengths as: (a)as=−aho (BCS regime), (b) 1/as=0 (unitarity), and (c) as=0.1aho (BEC regime).The pair distribution function for N=4 and as=0.1aho [solid line in panel (c)]is shown in more detail in Fig. 13.</figcaption></figure> function $P_{12}(r)$ for $N=3$ (dash and dash-dotted lines correspond to $L=0$ and $1$, respectively) and $N=4$ (solid lines) along the crossover for $\kappa=1$. Panel (a) shows results for $a_{s}=-a_{ho}$, panel (b) for $1/a_{s}=0$ and panel (c) for $a_{s}=0.1a_{ho}$. Interestingly, the pair distribution functions for $N=3$ and 4 show a similar overall behavior. In the BCS regime [Fig. 12(a)], the quantity $P_{12}(r)r^{2}$ shows a minimum at small $r$ (for very small $r$, $P_{12}(r)r^{2}$ goes smoothly but steeply to zero; this rapid change of $P_{12}(r)r^{2}$ is hardly visible on the scale shown in Fig. 12). At unitarity [Fig. 12(b)], $P_{12}(r)r^{2}$ shows a maximum at small $r$ and a second peak at larger $r$. In the BEC regime [Fig. 12(c)], the two-peak structure is notably more pronounced. The peak at small $r$ indicates the formation of tightly-bound dimers (one dimer for $N=3$ and two dimers for $N=4$), while the peak between $1a_{ho}$ and $2a_{ho}$ is related to the presence of larger atom-atom distances set approximately by the atom-dimer distance for the three-body system and the dimer-dimer distance for the four-body system. This interpretation suggests that the three-particle system has one small and one large interspecies distance, and the four-particle system has two small and two large interspecies distances. Indeed, integrating $P_{12}(r)$ for $N=3$ and 4 from $0$ to the $r$ value at which $P_{12}(r)r^{2}$ exhibits the minimum, we find that the likelihood of being at small distances (forming a molecule) and being at large distances is the same. We now analyze the pair distribution function $P_{12}(r)$ for $N=4$ <figure><img src="content_image/0801.2747/x13.png"><figcaption>Figure 13: (Color Online) (a) Circles show the pair distribution functionP12(r), multiplied by r2, for as=0.1aho (BEC regime) calculated by the CGapproach for N=4 and κ=1 [note, this quantity is also shown by a solid line inFig. 12(c)]. For comparison, the dash-dotted line (blue online) shows P12(r)r2for two atoms of mass m with the same scattering length but normalized to 1/2,the dashed line (red online) shows P12(r)r2 for two trapped bosonic moleculesof mass 2m interacting through a repulsive effective potential with add=0.6as,and the dotted line (green online) shows P12(r)r2 for two trapped non-interacting bosonic molecules of mass 2m. Panel (b) shows a blow-up of thesmall r region.</figcaption></figure> more quantitatively. Dash-dotted lines in Figs. 13(a) and (b) show the pair distribution function $P_{12}(r)$, multiplied by $r^{2}$, for two trapped atoms with $a_{s}=0.1a_{ho}$ (normalized to $1/2$). This dimer curve is essentially indistinguishable from the small $r$ part of the four-particle pair distribution function (circles). To describe the large $r$ part of the four-particle pair distribution function, we consider two bosonic molecules of mass $2m$, which interact through an effective repulsive potential with dimer-dimer scattering length $a_{dd}\approx 0.6a_{s}$ [23; 21]. The dashed line in Fig. 13(a) shows the pair distribution function for this system under external confinement. This dashed curve is essentially indistinguishable from the large $r$ part of the pair distribution function for the four-particle system. For comparison, a dotted line shows the pair distribution function for two non-interacting trapped bosons of mass $2m$. Figure 13 indicates that the effective repulsive interaction between the two dimers is crucial for reproducing the structural properties of the four-body system accurately. Our analysis shows that the entire pair distribution function $P_{12}(r)$ of the four-body system in the weakly-interacting molecular BEC regime can be described quantitatively in terms of a “dimer picture”. We now return to Fig. 12 and discuss how the symmetry-inversion of the $N=3$ system along the crossover (see Sec. IV.1) is reflected in $P_{12}(r)$. In the BCS regime and at unitarity [Figs. 12(a) and (b)], $P_{12}(r)$ shows less structure for $L=1$ than for $L=0$. In the weakly-interacting molecular BEC regime [Fig. 12(c)], the pair distribution function for $L=0$ nearly coincides with that for $L=1$ at small $r$ but is more compact than that for $L=1$ at large $r$. Next, we analyze how the behaviors of the pair distribution functions $P_{12}(r)$ for $N=3$ and $4$ change along the crossover if the mass ratio is changed from $\kappa=1$ to $4$. Figure 14 shows the pair distribution functions for $\kappa=4$. For $N=3$, we consider three-particle systems with either a spare light particle or with a spare heavy particle. The pair distributions for the three-particle system with two light particles and one heavy particle are notably broader than those for the three-particle system with one light particle and two heavy particles. This behavior can be attributed to the fact that $a_{ho}^{(1)}>a_{ho}^{(2)}$. Besides this, a comparison of the pair distribution functions shown in Fig. 14 for $\kappa=4$ and those shown in Fig. 12 for $\kappa=1$ reveals that the overall behavior of the $P_{12}(r)$ is similar. <figure><img src="content_image/0801.2747/x14.png"><figcaption>Figure 14: (Color Online) Pair distribution function P12(r), multiplied by r2,for two-component Fermi gases with κ=4 for different scattering lengths as:(a) as=−aho (BCS regime), (b) 1/as=0 (unitarity), and (c) as=0.1aho (BECregime). Dashed and dash-dotted lines show P12(r)r2 for N=3 (two heavyparticles) with L=0 and 1, respectively. Circles and squares show P12(r)r2 forN=3 (two light particles) with L=0 and 1, respectively. Solid lines showP12(r)r2 for N=4 with L=0.</figcaption></figure> Figures 15(a) and (b) show the one-body densities for $\kappa=1$ and $4$, respectively. In the non-interacting limit [the solid lines show $\rho_{1}(r)$ and the circles show $\rho_{2}(r)$], the sizes of $\rho_{1}(r)$ and $\rho_{2}(r)$ are determined by $a_{ho}^{(1)}$ and $a_{ho}^{(2)}$, respectively. As is evident in Fig. 15, the density of the light particles extends to larger $r$ than the density of the heavy particles. The density mismatch for $\kappa=4$ between the two one-body densities decreases as $a_{s}$ is tuned through the strongly-interacting regime to the weakly-interacting molecular BEC side. In the weakly-interacting molecular BEC regime, two molecules consisting each of a heavy and a light particle form. In this regime, the size of the system is determined by the molecular trap length and the densities $\rho_{1}(r)$ and $\rho_{2}(r)$ [triangles and dash-dotted line in Fig. 15(b)] nearly coincide. Furthermore, the densities are to a very good approximation described by the one-body density for two bosonic molecules of mass $m_{1}+m_{2}$ interacting through an effective repulsive interaction characterized by the dimer-dimer scattering length ($a_{dd}\approx 0.77a_{s}$ for $\kappa=4$ [23; 21]). <figure><img src="content_image/0801.2747/x15.png"><figcaption>Figure 15: (Color Online) One-body densities ρ1(r) and ρ2(r) for N=4 and (a)κ=1 and (b) κ=4 for different scattering lengths as [for κ=1, ρ1(r) and ρ2(r)coincide and only ρ2(r) is shown]: Circles and solid lines show ρ1(r) andρ2(r) for as=0, squares and dashed lines show ρ1(r) and ρ2(r) for 1/as=0, andtriangles and dash-dotted lines show ρ1(r) and ρ2(r) for as=0.1aho. Note,ρ2(r) for κ=4 and as=0 [solid line in panel (b)] is multiplied by a factor ofthree to enhance the visibility.</figcaption></figure> We have also analyzed the pair distribution functions $P_{ii}(r)$ for $\kappa=1$ and $4$ (not shown). The small $r$ region of the $P_{ii}(r)$ is controlled by the Pauli exclusion principle between identical fermions. In the weakly-interacting molecular BEC regime, the pair distribution functions $P_{11}(r)$ and $P_{22}(r)$ nearly coincide even for $\kappa=4$. In this regime, the pair distribution functions $P_{ii}(r)$ are well approximated by that for two particles of mass $m_{1}+m_{2}$ interacting with a repulsive potential characterized by $a_{dd}$. ### Structural properties at unitarity This section discusses selected structural properties of two-component equal mass Fermi gases at unitarity with up to $N=20$ atoms. For small systems ($N\leq 6$), we present structural properties calculated using both the CG and the FN-DMC methods. For larger systems, however, our interpretation relies solely on the structural properties calculated by the FN-DMC method. To assess the accuracy of the nodal surfaces employed in our FN-DMC calculations as well as of the accuracy of the mixed estimator [see Eq. (41) in Sec. III.2], Figs. 16(a) and (b) compare the pair distribution functions $P_{12}(r)$ for the three-particle system with $L=1$ and the four-particle system with $L=0$, respectively, calculated by the CG and the FN-DMC methods. <figure><img src="content_image/0801.2747/x16.png"><figcaption>Figure 16: (Color Online) Pair distribution functions P12(r), multiplied byr2, at unitarity for equal mass Fermi systems with (a) N=3 (L=1) and (b) N=4(L=0) atoms calculated by the CG method (solid lines) and by the FN-DMC method(circles). The agreement is excellent.</figcaption></figure> The agreement between the pair distribution functions calculated by the CG method (solid lines) and the FN-DMC method (circles) is very good, validating the construction of the nodal surface of $\psi_{T}$. Furthermore, the good agreement suggests that the mixed estimator results, for the guiding functions employed, in structural properties very close to those one would obtain by an exact estimator. Figure 17 shows the pair distribution functions $P_{12}(r)$ calculated by the FN-DMC method for equal mass Fermi systems with $N=3-20$ at unitarity. <figure><img src="content_image/0801.2747/x17.png"><figcaption>Figure 17: (Color Online) Dashed and solid lines show the pair distributionfunctions P12(r), multiplied by r2Npair (Npair denotes the number ofinterspecies distances), for equal mass Fermi systems at unitarity with even N(N=4,6,⋯,20) and odd N (N=3,5,⋯,19), respectively, calculated by the FN-DMC.Beyond r≈aho, P12(r)r2Npair is smallest for N=3 and largest for N=20.</figcaption></figure> To simplify the comparison, Fig. 17 shows the even $N$ results as a dashed line and the odd $N$ results as a solid line. Furthermore, $P_{12}(r)r^{2}$ is multiplied by the number $N_{pair}$ of interspecies distances so that the $N=3$ distribution function has the smallest and the $N=20$ distribution function the largest amplitude for $r\gtrsim a_{ho}$. The pair distribution functions for even $N$ show a similar behavior for all $N$ considered; both the small $r$ and the large $r$ peaks grow monotonically and smoothly with increasing $N$. For odd $N$, in contrast, the small $r$ peak changes somewhat discontinuously at $N\approx 11$. This behavior can be attributed to the guiding functions employed. For even $N$, the guiding function $\psi_{T1}$, whose nodal surface is constructed from the two-body solution, gives the lowest energy for all $N$ (except for $N=4$). For odd $N$, however, $\psi_{T2}$ results in a lower energy for $N\leq 9$, $\psi_{T3}$ for $N=11$, and $\psi_{T1}$ for $N\geq 13$. Thus, the pair distribution functions clearly reveal how the structural properties depend on the nodal surface employed in the FN-DMC calculations and provide much deeper insights into the different $\psi_{T}$ employed than a mere comparison of the energies. For $N\geq 13$, Fig. 17 indicates that the amplitudes of the scaled interspecies pair distribution functions are nearly the same for neighboring systems. For example, the quantities $P_{12}(r)r^{2}N_{pair}$ for $N=18$ and 19 agree to a good approximation, suggesting that one can think of the $N=18$ system as consisting of nine pairs, and of the $N=19$ system as consisting of nine pairs plus a spare atom. Note that this interpretation hinges critically on the nodal surface employed in our FN-DMC calculations; a small change in the nodal structure of the guiding function may change the small $r$ behavior of the pair distribution functions non-negligibly. We next investigate in Fig. 18 where the spare particle is located in the odd-$N$ systems at unitarity. <figure><img src="content_image/0801.2747/x18.png"><figcaption>Figure 18: (Color Online) The one-body density ρ1(r) (solid lines) is shownfor N=3, 9 and 15 (for r>0.5aho, from bottom to top), together with the one-body density ρ2(r) (dashed line) for N=3, 9 and 15 (for r>0.5aho, from bottomto top) for equal mass two-component Fermi gases at unitarity calculated bythe FN-DMC method.</figcaption></figure> This figure shows the one-body densities $\rho_{1}(r)$ (solid lines) and $\rho_{2}(r)$ (dashed lines) for $N=3$, 9 and 15. For $N=3$, the difference between $\rho_{1}(r)$ and $\rho_{2}(r)$ is roughly constant across the trap. The behavior is similar for $N=9$. Interestingly, the densities for $N=9$ show a minimum at $r=0$, reflecting the fact that the FN-DMC calculations employ the nodal surface of the ideal Fermi gas, i.e., use $\psi_{T2}$ [Eq. (III.2)]. For $N=15$, the nodal surface employed is constructed from the two-body solution [see Eq. (III.2)], and consequently, the behavior of the density profiles differs from that for the smaller $N$. The densities $\rho_{1}(r)$ and $\rho_{2}(r)$ nearly coincide at small $r$. At large $r$, however, the density $\rho_{1}(r)$ has a larger amplitude than $\rho_{2}(r)$ (recall $N_{1}=N_{2}+1$). Our data for $N=15$ indicate that the spare particle is not distributed uniformly throughout the trap but has an increased probability to be found near the edge of the cloud. Possible consequences of this finding for the excitation gap have already been discussed in Refs. [20; 79]. To quantify the analysis of the one-body densities, we integrate $\rho_{i}(r)$ over $r$, \[\bar{N}_{i}(r)=4\pi\int_{0}^{r}\rho_{i}(r^{\prime})r^{\prime 2}dr ^{\prime}.\] (44) For a finite upper integration limit, $\bar{N}_{i}(r)$ monitors how many of the $N_{i}$ particles are located between zero and $r$. Figure 19 shows $\bar{N}_{i}(r)$ for $N=3$, 9 and 15. As in Fig. 18, the results for component one are shown by solid lines and those for component two by dashed lines; in the large $r$ limit, the $\bar{N}_{i}(r)$ equal $N_{i}$, as expected. One can now read off nicely, <figure><img src="content_image/0801.2747/x19.png"><figcaption>Figure 19: (Color Online) Solid and dashed lines show the integratedquantities ¯N1(r) and ¯N2(r) [Eq. (44)], respectively, as a function of r fora two-component Fermi gas at unitarity. At large r, the curves correspond frombottom to top to N=3, 9 and 15.</figcaption></figure> in which $r$-regions the densities of the two components agree and where they disagree. For $N=3$, e.g., the two atoms of component one and the one atom of component two are added over approximately the same $r$-region. For $N=15$, in contrast, the first five atoms of the two components are located in the region with $r\lesssim 1.5a_{ho}$; this core region can be considered “fully paired”. The last three atoms of component one and the last two atoms of component two form, loosely speaking, a “partially paired or unpaired outer shell”. We find similar behaviors for the odd-$N$ systems with $N=13$, 17 and 19. It will be interesting to see if this interpretation holds for larger $N$, and if this information can be used to shed light on the phase diagram of asymmetric Fermi gases [80; 81]. To further verify the validity of the special properties of two-component Fermi gases at unitarity as well as to further assess the accuracy of our guiding functions employed in the FN-DMC calculations, we analyze the hyperradial densities $\bar{F}_{00}^{2}(x)$ for various $N$. Symbols in Fig. 20 show the dimensionless hyperradial density $\bar{F}_{00}^{2}(x)$ calculated using the mixed Monte Carlo estimator, Eq. (41), for $N=3$ to 10. Here, $x$ is the dimensionless <figure><img src="content_image/0801.2747/x20.png"><figcaption>Figure 20: (Color Online) The hyperradial density ¯F200(x) is shown as afunction of the dimensionless hyperradius x for N=3−10, calculated using themixed Monte Carlo estimator (symbols) and the analytical expression with theFN-DMC energies (lines), respectively. The maximum of ¯F200(x) is smallest forN=3 and largest for N=10; the MC results are shown by filled circles for N=3,open circles for N=4, filled squares for N=5, open squares for N=6, filledtriangles for N=7, open triangles for N=8, filled diamonds for N=9 and opendiamonds for N=10.</figcaption></figure> hyperradius defined just above Eq. (30) and the normalization of the $\bar{F}_{00}$ is chosen so that $\int_{0}^{\infty}\bar{F}_{00}^{2}(x)dx=1$. The dimensionless hyperradius $x$ is scaled by $R^{\prime}_{NI}$, which we evaluate by approximating $E_{NI}$ in Eq. (31) by $E_{NI,EFT}$. This is similar to the “smoothing procedure” discussed in the context of Figs. 7 and 9. The hyperradial densities become more compact as $N$ increases, owing to the increase of the effective mass $\mu_{eff}$ entering into the effective hyperradial Schrödinger equation [Eq. (30)] with increasing $N$. Furthermore, the maximum of the hyperradial densities occurs at slightly larger $x$ values for odd $N$ systems than for even $N$ systems, in agreement with the odd-even staggering discussed in Sec. IV.2 in the context of the hyperradial potential curves $V(R)$. In the limit of zero-range interactions, the adiabatic approximation is expected to be exact (see Sec. II.3). In this case, the functional form of the hyperradial wave functions is known analytically [see Eq. (28)], and can be compared with the Monte Carlo results obtained for short-range potentials by sampling the total wave function and integrating over all coordinates but the hyperradius. Solid lines in Fig. 20 show the hyperradial densities $\bar{F}_{00}^{2}(x)$ for $N=3$ to $10$ predicted analytically for zero-range interactions, using the FN-DMC energies $E_{00}$ listed in Table 3 of this paper and Table II of Ref. [21]. The agreement between the analytical results and the Monte Carlo results obtained for finite range potentials is quite good. On the one hand, this agreement lends numerical support for the separability or near separability of the total wave function into a hyperradial and a hyperangular part. On the other hand, the good agreement suggests that the nodal surface employed in our MC calculations is appropriate. <figure><img src="content_image/0801.2747/x21.png"><figcaption>Figure 21: (Color Online) Hyperradial density F20n(R) for n=1,2 and 3. Here,we choose μN=m (L=aho). The solid lines show the analytical solutions whilethe circles show the numerical results obtained by integrating (Ψrel)2calculated by the CG method over all coordinates but the hyperradius R.</figcaption></figure> Finally, we analyze the hyperradial densities calculated by the CG approach for the $N=6$ system. Since the CG approach allows for the determination of excited states, this analysis allows us to verify that the $2\hbar\omega$ spacing reported in Ref. [20] and discussed in Sec. II.3 corresponds indeed to breathing-mode excitations, i.e., to excitations along the hyperradial coordinate. To extract the hyperradial densities, we integrate the square of the wavefunction $\Psi^{rel}$ over all the coordinates but the hyperradius. If the universal behavior is fulfilled, then the hyperradial densities should coincide with the square of the analytically determined $F_{\nu n}(R)$, Eq. (28), which are shown in Fig. 21 by solid lines. The integration over the hyperangular coordinates is carried out using Monte Carlo integration techniques. Symbols in Fig. 21 show the resulting hyperradial densities $F_{\nu n}^{2}(R)$ for $\nu=0$ and $n=0,1$ and $2$. The numerically determined hyperradial densities indicate that the excitations are to a good approximation located along the $R$ coordinate, supporting the interpretation of the $2\hbar\omega$ spacing within the hyperspherical framework. The agreement between the numerical and analytical results is excellent for the ground state [see Fig. 21(a)]. For the excited states with $n=1$ and $2$, the small deviations between the numerical and analytical results may be due to finite range effects or not fully converged numerical results. ## V Conclusions This paper presents a microscopic picture of the properties of ultracold two-component fermionic systems in a trap. Complementing previous studies [21; 20], we focus on the energetics of odd $N$ systems, and the structural properties of both odd and even $N$ systems. For sufficiently few particles, we solve the Schrödinger equation for equal and unequal mass systems, starting from a model Hamiltonian with short-range interspecies $s$-wave interactions using the CG approach. This basis set expansion technique allows for the determination of the eigenspectrum and eigenstates with controlled accuracy throughout the BEC-BCS crossover. We find that the spectrum and the structural properties of small trapped two-component Fermi systems change qualitatively throughout the crossover regime. An analysis of the energies of the $N=3$ systems in the weakly-interacting BEC and BCS regimes allows us to determine the validity regime of the analytically determined perturbative expressions for small $\|a_{s}\|$. Furthermore, we find that the angular momentum of the $N=3$ ground state changes from $L=1$ in the weakly-attractive BCS regime to $L=0$ in the weakly-repulsive BEC regime for all mass ratios considered. By additionally treating the $N=2$ and 4 systems, we determine the excitation gap $\Delta(3)$ throughout the crossover region: For equal frequencies, the excitation gap decreases for all scattering lengths with increasing mass ratio. For $N=4$ systems with $\kappa\leq 13$, we determine the $L=0$ excitation spectrum at unitarity. The spectrum determines the $s_{\nu}$ coefficients of the hyperradial potential curves and also verifies within our numerical accuracy the $2\hbar\omega$ spacing prediction, which was derived analytically assuming universality [54]. We verified in a number of cases that the $2\hbar\omega$ spacing corresponds indeed to breathing-mode excitations. Our analysis of the energetics is complemented by studies of the structural properties. For the four-particle system with equal and unequal masses, e.g., we show how the pair distribution functions in the $a_{s}>0$ region (small $a_{s}$) can be described by a system of two molecules interacting through an effective dimer-dimer potential with positive dimer-dimer scattering length. A similar analysis was carried out for the $N=3$ system and we verified that this system behaves as an interacting system of an atom and a dimer. Our small $N$ studies have implications for optical lattice experiments. Our results can be applied directly if each optical lattice site is approximately harmonic in the non-tunneling regime. In this context, Ref. [31] already pointed out, including the energies of the two- and three-particle system, that the occupation of optical lattice sites with three equal-mass atoms is unfavorable. To start with let us consider a system with two lattice sites and six atoms (three of each species). We imagine that the lattice sites are loaded by adiabatically turning up the barrier height between the two sites. It follows from our energies calculated for $N=2$ through $4$, that the system ends up with unequally occupied lattice sites at the end of the ramp: For both equal and unequal masses, and for all scattering lengths, the energy of the two-site lattice is minimal if two atoms occupy one site and the other four atoms occupy the other site. This is simply a consequence of the fact that $\Delta(3)$ is positive throughout the crossover (note, the energy of the unequal mass system might be further lowered if we consider the formation of pentamers and sextamers with negative energies; these states are not included in our analysis). The arguments presented here for just two lattice sites generalize readily to lattices with multiple sites. Instead of ramping up the barrier height adiabatically, we now imagine a fast non-adiabatic ramp. In this case, the likelihood of finding three atoms per lattice site at the end of the ramp is finite. Since the excitation frequencies for two-, three- and four-particle systems are different, a “purification sweep” [15] can be used to then prepare a system with either three or no particles per site. These three-particle systems could be investigated spectroscopically (see, e.g., Sec. IV.3 for a discussion of the excitation spectrum of the four-particle system). Alternatively, one might ask whether it would be possible to measure the odd-even physics by adiabatically lowering the lattice barrier and monitoring the point at which tunneling sets in. In addition to systems with equal numbers of atoms in the two species, we consider an optical lattice with twice as many heavy as light atoms. If the mass ratio is sufficiently large, trimers consisting of two heavy atoms and one light atom with negative energy can form at each lattice site, paving the way for spectroscopic studies of these delicate systems. Furthermore, by starting with a bound trimer in a deep lattice and then lowering the lattice height, a gas consisting of bound trimers can possibly be prepared. To extend the studies of the energetics and structural properties to larger systems, we employed the FN-DMC technique. This approach determines the lowest energy of a state that has the same symmetry as a so-called guiding function and thus an upper bound to the exact eigenenergy. Detailed comparisons of the energies and the structural properties calculated by the CG and FN-DMC approaches benchmark the nodal surfaces employed for systems with $N\leq 6$. In the strongly-correlated unitary regime, e.g., the FN-DMC energies for equal-mass two-component Fermi systems agree with the CG energies to within 2%. Our even $N$ energies ($N\leq 30$) for equal-mass systems at unitarity show vanishingly small shell structure. Applying the local density approximation and approximating the non-interacting energies by the corresponding extended Thomas-Fermi expression, we find $\xi_{tr}=0.467$, which is somewhat larger than the value of the homogeneous system, $\xi_{hom}=0.42$. We note that the expression $\sqrt{\xi_{tr}}E_{NI,EFT}$ describes the equal-mass energies for $N\leq 30$ at unitary very well; the small disagreemet between $\xi_{tr}$ and $\xi_{hom}$ is most likely due to the small number of particles considered in the present work. Combining the even and odd $N$ energies, we find that the excitation gap $\Delta(N)$ at unitarity increases with $N$. Also, the one-body densities and pair distribution functions at unitarity are studied for up to $N=20$. For odd $N$ with $N\gtrsim 11$, we observe that the extra “unpaired” particle is located predominantly near the edge of the cloud, in agreement with previous predictions [20; 79]. This suggests that the LDA cannot be applied to determine the excitation gap. Furthermore, we find that the hyperradial densities of the lowest gas-like state with $N\leq 10$ calculated for short-range interactions by the FN-DMC method agree with the analytically predicted ones, indicating that the lowest gas-like state does indeed behave universally. Selected hyperradial densities for larger $N$ were already presented in Ref. [20]. The energies and structural properties for the equal-mass two-component Fermi systems at unitarity presented in this paper may serve as a benchmark for other calculations. Recently, e.g., a DFT treatment determined the energies for systems with up to $N=20$ particles [76]. 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1006.1376	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 41760, "num_imgs": 2, "llama3_tokens_count": 13694 }	[ "content_image/1006.1376/x1.png", "content_image/1006.1376/x3.png" ]	# Improved Constraints on Isotropic Shift and Anisotropies of the Speed of Light using Rotating Cryogenic Sapphire Oscillators Michael A. Hohensee hohensee@berkeley.edu Department of Physics, Harvard University Department of Physics, University of California, Berkeley Paul L. Stanwix School of Physics, The University of Western Australia Harvard-Smithsonian Center for Astrophysics Michael E. Tobar School of Physics, The University of Western Australia Stephen R. Parker School of Physics, The University of Western Australia David F. Phillips Harvard-Smithsonian Center for Astrophysics Ronald L. Walsworth Department of Physics, Harvard University Harvard-Smithsonian Center for Astrophysics February 26, 2024 ###### Abstract We demonstrate that Michelson-Morley tests, which detect direction-dependent anisotropies in the speed of light, can also be used to place limits upon isotropic deviations of the vacuum speed of light from $c$, as described by the photon sector Standard Model Extension (SME) parameter $\tilde{\kappa}_{tr}$. A shift in the speed of light that is isotropic in one inertial frame implies anisotropic shifts in others. Using observer Lorentz covariance, we derive the time-dependent variations in the relative resonance frequencies of a pair of electromagnetic resonators that would be generated by such a shift in the rest frame of the Sun. A new analysis of a recent experimental test of relativity using this result constrains $\tilde{\kappa}_{tr}$ with a precision of $7.4\times 10^{-9}$. This represents the first constraint on $\tilde{\kappa}_{tr}$ by a Michelson-Morley experiment and the first analysis of a single experiment to simultaneously set limits on all nine non-birefringent terms in the photon sector of the SME. pacs: 03.30.+p, 11.30.Cp, 06.30.Ft, 12.60.-i ## I Introduction Lorentz invariance is a cornerstone of both General Relativity and the Standard Model of Particle Physics, and as such has been the subject of many experimental investigations over the past century. Much of this work has focused upon the properties and propagation of light in different reference frames, beginning with the pioneering work of Michelson-Morley [1], Kennedy-Thorndike [2], and Ives-Stilwell [3]. The purpose and interpretation of these experiments has varied with the development of physical theories throughout the century, ranging from attempts to observe the properties of a luminiferous aether, to determining whether space-time exhibits Lorentz as opposed to some other symmetry, and to more recent searches for the imprint of physics beyond the Standard Model. These most recent studies presume that physics is invariant under “passive” transformations of the observer reference frame, while leaving open the possibility that the theory is not Lorentz invariant under “active” boosts of the rest frame of the system under test. This could happen if known particles interact with fields not accounted for by the Standard Model, or indeed if Lorentz symmetry turns out to be explicitly broken. In this context, modern implementations of Michelson-Morley, Kennedy-Thorndike, and Ives-Stilwell tests [4; 5; 6; 7; 8] are used to look for evidence of such Lorentz violation in the form of modifications of the dispersion relation for light and other Standard Model particles. In particular, these tests search for deviations of the phase velocity of light in vacuum from the canonical value. These deviations can be orientation- and also polarization-dependent, and in general give the vacuum the properties of a potentially birefringent or anisotropic polarizable medium. Such effects can be parameterized by the Standard Model Extension (SME) [9; 10], which provides an effective field theory framework for determining the experimental consequences of a perturbative Lorentz violation. Observation of Lorentz violation in a physical system would provide clues about the structure of physics at experimentally inaccessible energy scales. Modern Michelson-Morley experiments usually consist of a pair of orthogonally mounted electromagnetic resonators that are rotated in order to modulate their orientation in space. The observable is the difference in their resonant frequencies; Lorentz violations will manifest as periodic variations in the signal at frequencies related to the rotation and its harmonics. Hence, such experiments are typically considered to be sensitive only to anisotropies in the speed of light. Here, we extend the analysis of [10] to explicitly derive the sensitivity of Michelson-Morley tests to deviations in the speed of light that are isotropic in a given inertial reference frame. Furthermore, using this result we report upon a new analysis of data from a recent experiment [4] that constrains all nine non-birefringent CPT-even photon-sector SME coefficients, summarized in table 1. In particular, our analysis constrains the isotropic shift parameter $\|\tilde{\kappa}_{\rm tr}\|$, the first such result from this form of experiment. Although this constraint is overshadowed by recent results based on collider physics [11; 12], it is an improvement upon results obtained from experiments intended to constrain $\tilde{\kappa}_{\rm tr}$ such as relativistic ion spectroscopy [5]. ~κXYe− \| 0.8 (0.6) \| ~κXXe−−~κYYe− \| 0.2 (1.0) \| ~κXYo+ \| -1.5 (1.2) ---\|---\|---\|---\|---\|--- ~κXZe− \| 1.5 (1.3) \| ~κZZe− \| 143 (179) \| ~κXZo+ \| 1.7 (0.7) ~κYZe− \| 1.7 (1.3) \| ~κtr \| -1.5 (0.74) \| ~κYZo+ \| 0.2 (0.7) Table 1: Fitted values and uncertainties of the non-birefringent photon-sector parameters of the SME for the results reported here. (~κe− in 10−16, ~κo+ in 10−12 and ~κtr in 10−8). ## II Michelson-Morley Tests of the SME In general, Lorentz violation in the electromagnetic sector of the SME causes vacuum birefringence and polarization-independent shifts in the phase velocity of light in vacuum ($c_{\rm ph}$) relative to the canonical velocity ($c$). Vacuum birefringence has been constrained to better than one part in $10^{37}$ by observations of linearly polarized light from distance gamma ray bursts [13], so is neglected in this analysis. The remaining polarization-independent shifts can be parameterized for a specified reference frame (e.g., the frame in which the sun is at rest) using nine degrees of freedom: one to describe the average deviation of $c_{\rm ph}$ from $c$ over all possible directions of propagation, five to describe the difference in the average speed of light moving forward and backwards along any given direction, and three more to describe the difference in $c_{\rm ph}$ for light moving in one direction relative to a counterpropagating beam. To leading order, the SME uses the scalar $\tilde{\kappa}_{\rm tr}$, the $3\times 3$ symmetric traceless $\tilde{\kappa}_{e-}^{jk}$ matrix with five degrees of freedom, and the $3\times 3$ antisymmetric $\tilde{\kappa}_{o+}^{jk}$ matrix with three degrees of freedom to parameterize these shifts. In terms of these $\tilde{\kappa}$’s, the free electromagnetic Lagrangian becomes [10] \[\!\!\!\!\!{\cal L}=\frac{1}{2}\left[(1+\tilde{\kappa}_{\rm tr}) \vec{E}^{2}-(1-\tilde{\kappa}_{\rm tr})\vec{B}^{2}\right]+\vec{E}\cdot(\tilde{ \kappa}_{o+})\cdot\vec{B}\\ +\frac{1}{2}\vec{E}\cdot(\tilde{\kappa}_{e-})\cdot\vec{E}+\frac{1 }{2}\vec{B}\cdot(\tilde{\kappa}_{e-})\cdot\vec{B}\;,\] (1) where $\vec{E}$ and $\vec{B}$ are the standard electromagnetic fields in vacuum. Although the total Lagrangian remains invariant under changes in an observer’s inertial frame, the parts proportional to the $\tilde{\kappa}$ coefficients are not term by term invariant. If, for example, the speed of light in a reference frame $S$ is $c_{\rm ph+}$ for a wave with wavevector $\vec{k}$, and $c_{\rm ph-}$ for waves traveling in the opposite direction, and $c_{\rm ph+}=c_{\rm ph-}\neq c$, then observer Lorentz invariance requires that the phase velocity of these two waves must differ from one another in any reference frame $S^{\prime}$, arrived at from $S$ via a boost along $\vec{k}$. This difference must also be reflected in the values taken by the $\tilde{\kappa}$’s when the Lagrangian is expressed in terms of the fields in $S^{\prime}$. The $\tilde{\kappa}$’s mix with one another under rotations and boosts of the observer coordinate frame. Therefore, results from a series of identical experiments performed in different inertial frames may be used to obtain constraints on all nine of the non-birefringent $\tilde{\kappa}$’s, even though any individual experiment might only be sensitive to a subset. It is convenient to select a standard inertial frame in which to compare the results of different experimental tests of Lorentz invariance, and to express the numerical values (or limits) on the SME coefficients. We adopt the Sun Centered Celestial Equatorial Frame (SCCEF), following [10; 14], which is defined with the coordinate origin at the Sun, X and Y lie in the plane of the Earth’s equatorial plane, with X pointing towards the Earth at the autumnal equinox. Let us now consider the Michelson-Morley laboratory experiment. Following on from Eq. (1), it can be shown that if any of the $\tilde{\kappa}$ parameters are nonzero the difference frequency between the electromagnetic modes of a pair of identical, orthogonally mounted resonators is given by [10; 15] \[\frac{\delta\nu}{\nu}=S_{e}\left\{\left[(\tilde{\kappa}_{e-})^{xx}_{\rm lab}-( \tilde{\kappa}_{e-})^{yy}_{\rm lab}\right]\cos{2\theta}-2(\tilde{\kappa}_{e-}) ^{xy}_{\rm lab}\sin{2\theta}\right\},\] (2) where $S_{e}$ is a sensitivity factor specific to the resonator modes and materials, $\theta$ is the angle of the resonators’ axes relative to the $x$ and $y$ coordinate axes, which are in turn defined by the system configuration when $\theta=0$. Thus, in a given inertial reference frame Michelson-Morley experiments directly constrain the value of $\tilde{\kappa}_{e-}^{jk}$ in the laboratory. In practice, however, changes in the Earth’s motion relative to the Sun during its orbit also allow us to set limits on the magnitudes of $\tilde{\kappa}_{o+}^{jk}$ [10] in the Sun’s rest frame. We show in this work that the relationship can be further extended to constrain the magnitude of $\tilde{\kappa}_{\rm tr}$, as is derived in detail in Appendix B, and outlined in the following section. ## III Sensitivity to the isotropic $\tilde{\kappa}_{\rm tr}$ ωi \| CC,ωi \| Num. Weight (×10−10) \| CS,ωi \| Num. Weight (×10−10) ---\|---\|---\|---\|--- ω⊕ \| - \| \| 12β2⊕sin2ηsin2χ~κtr \| -32.1 2ω⊕ \| −12β2⊕sin2η(1+cos2χ)~κtr \| -9.85 \| - \| ω⊕+2Ω⊕ \| - \| \| −12β2⊕sin2χ(1−cosη)sinη~κtr \| 1.42 ω⊕−2Ω⊕ \| - \| \| 12β2⊕sin2χ(1+cosη)sinη~κtr \| −33.5 2ω⊕+2Ω⊕ \| 14β2⊕(1+cos2χ)(1−cosη)2~κtr \| 0.209 \| - \| 2ω⊕−2Ω⊕ \| 14β2⊕(1+cos2χ)(1+cosη)2~κtr \| 116 \| - \| ωi \| SC,ωi \| Num. Weight (×10−10) \| SS,ωi \| Num. Weight (×10−10) ω⊕ \| −β2⊕cos2ηsinχ~κtr \| -60.9 \| - \| 2ω⊕ \| - \| \| −β2⊕sin2ηcosχ~κtr \| 8.13 ω⊕+2Ω⊕ \| β2⊕sinχsinη(1−cosη)~κtr \| 6.82 \| - \| ω⊕−2Ω⊕ \| −β2⊕sinχsinη(1+cosη)~κtr \| −161 \| - \| 2ω⊕+2Ω⊕ \| - \| \| 12β2⊕cosχ(1−cosη)2~κtr \| −0.172 2ω⊕−2Ω⊕ \| - \| \| 12β2⊕cosχ(1+cosη)2~κtr \| −95.9 Table 2: Contributions of ~κtr, as defined in the SCCEF, to the amplitude of sidereal variations in the Michelson-Morley observable normalized for the experimental sensitivity Se, in terms of the relative orientation and boost of the laboratory frame relative to the SCCEF. η is the declination of the Earth’s orbit relative to its spin, taken to be 23.27∘, and χ is the colatitude of the laboratory, 121.82∘ in Perth, Australia. The actual magnitude of each signal due to ~κtr for an experiment in Perth, Australia is indicated by the numerical weight. Although ~κtr does generate signals at the frequency ω⊕ of the sidereal day, and also at 2ω⊕, the magnitude of such contributions is strongly suppressed relative to those from ~κe− and ~κo+, which respectively are of order unity and 10−4. At all other frequencies, the signals from ~κe− and ~κo+ are suppressed relative to ~κtr. This, combined with the far more stringent bounds set upon ~κe− and ~κo+ from other experiments Mueller:2007 , allows us to ignore all but the contribution of ~κtr to signals at ω⊕±2Ω⊕ and 2ω⊕±2Ω⊕, where Ω⊕ is the frequency of the sidereal year. Although the general form of the time-dependence of the Earthbound lab-frame $\tilde{\kappa}$’s can be derived using the observer covariance of the action, the terms contributing to (2) proportional to the value of $\tilde{\kappa}_{\rm tr}$ in the Sun-Centered Celestial Equatorial Frame (SCCEF) can be obtained using simpler arguments. The resonator in a Michelson-Morley experiment is sensitive only to anisotropies that the SCCEF $\tilde{\kappa}_{\rm tr}$ generates in the laboratory frame, which in turn must depend solely upon the orientation of the lab with respect to the lab’s boost relative to the SCCEF. The maximum difference signal is generated when the axis of one resonator is most parallel to the boost from the SCCEF, while the axis of the other is as nearly perpendicular to the boost as possible. In general, this will happen twice per solar day, although the precise times that they occur will vary over the course of a year. For example, an experiment with one cavity axis aligned East to West in the lab sees a peak daily $\tilde{\kappa}_{\rm tr}$-induced shift in that cavity maximized during the summer and winter solstices, while its peak shift is minimized at the equinoxes. Since the frequency shift between two identical resonators given by Eq. (2) is the same up to a constant factor for any such pair [15], we can analyze the simple case of a pair of Fabry-Perot cavities aligned orthogonally to one another along the $x$- and $y$-axes. The resonance frequencies of each of the cavities are then $\nu=\frac{mc_{\rm ph}}{2L}$, where $L$ is the length of the cavity, $m=1,2,3,\dots$, and $c_{\rm ph}=(c_{\rm ph}^{+}+c_{\rm ph}^{-})/2$ is the average phase velocity of light moving back and forth along the cavities’ axes. Variations in the phase velocity of light along the cavity axes yields the frequency difference \[\frac{\delta\nu_{x}}{\nu_{x}}-\frac{\delta\nu_{y}}{\nu_{y}}=\frac{1}{2}(\rho_{ x+}+\rho_{x-}-\rho_{y+}-\rho_{y-}),\] (3) where $c\rho_{j\pm}=\delta c_{j\pm}$ is the shift in the vacuum phase velocity of light parallel ($+$) or anti-parallel ($-$) to the $j$-axis due to $\tilde{\kappa}$ in the laboratory frame. The problem now reduces to finding the mean speed of light along the laboratory $x$- and $y$-axes in terms of $\tilde{\kappa}_{\rm tr}$ in the SCCEF. Observer Lorentz covariance of the SME implies that the overall Lagrangian remains a Lorentz scalar, although the action may not be term-by-term Lorentz invariant [9]. This means that although the speed of light might be frame-dependent, the velocity of a particular electromagnetic wave must transform in the same manner as any other velocity under Lorentz boosts –i.e. according to the relativistic velocity addition formula. Given the boost of the laboratory relative to the SCCEF, we can then determine the anisotropic shift in the speed of light as seen in the laboratory arising from an isotropic shift (nonzero $\tilde{\kappa}_{\rm tr}$) in the SCCEF. We seek a solution that is leading order in the $\tilde{\kappa}$’s, and so the laboratory anisotropies induced by a nonzero $\tilde{\kappa}_{\rm tr}$ in the SCCEF can depend only upon $\tilde{\kappa}_{\rm tr}$. To second order in the laboratory boost $\beta=v/c$ relative to the SCCEF, we then obtain \[\frac{1}{2}(\rho_{x+}+\rho_{x-})=-\tilde{\kappa}_{\rm tr}-(\beta^{2}+\beta_{x} ^{2})\tilde{\kappa}_{\rm tr},\] (4) and similarly for the mean speed of light along the $y$-axis. More details of this derivation can be found in Appendix A. The differential signal produced by a pair of orthogonally mounted resonators must then be given by \[\frac{\delta\nu}{\nu}\propto\frac{\delta\nu_{x}}{\nu_{x}}-\frac{\delta\nu_{y}} {\nu_{y}}=(\beta_{y}^{2}-\beta_{x}^{2})\tilde{\kappa}_{\rm tr}.\] (5) For an experiment which rotates about the laboratory $z$-axis with angular frequency $\omega_{R}$, we find that the variation of the Lorentz-violating frequency shift in time is given by \[\frac{\delta\nu}{\nu}=S(T)\sin{2\omega_{R}T}+C(T)\cos{2\omega_{R}T},\] (6) where \[S(T) =(\tilde{\kappa}_{\rm tr})\times\sum_{i}\Big{[}S_{S,i}\sin(\omega _{i}T)+S_{C,i}\cos(\omega_{i}T)\Big{]},\] (7) \[C(T) =(\tilde{\kappa}_{\rm tr})\times\sum_{i}\Big{[}C_{S,i}\sin(\omega _{i}T)+C_{C,i}\cos(\omega_{i}T)\Big{]}.\] (8) The overall modulation of the signal by $2\omega_{R}$ follows from the fact that (3) is unchanged when we exchange $+x\leftrightarrow-x$ and $+y\leftrightarrow-y$. The remaining modulation frequencies $\omega_{i}$ are various harmonics of and beats between the frequencies of the sidereal day $\omega_{\oplus}$ and the sidereal year $\Omega_{\oplus}$, as derived in the Appendicies. The weights $S_{S,i}$, $S_{C,i}$, $C_{S,i}$, and $C_{C,i}$ most relevant to $\tilde{\kappa}_{\rm tr}$ are summarized in table 2. ## IV Experiment and Analysis The bounds presented here arise out of a new analysis of data from an experiment performed at the University of Western Australia [4]. This experiment searched for Lorentz violating signals by monitoring the difference frequency between two microwave cryogenic sapphire oscillators (CSOs) as a function of orientation and time. The details of this experiment and the operation of CSOs in general has been reported elsewhere [17; 18; 4; 15], so we will provide only a brief description here. Each CSO relies upon a high Q-factor ($\sim 2\times 10^{8}$) sapphire loaded cylindrical resonant cavity, excited in the $WGH_{8,0,0}$ whispering gallery mode at approximately 10 GHz by a Pound stabilized loop oscillator circuit. The two resonators are mounted one above the other with their cylindrical axes orthogonal in the horizontal plane. The experiment was continuously rotated in the laboratory with a period of 18 seconds. When resonantly excited, the sapphire crystals support standing waves with the dominant electric and magnetic fields pointing in the axial and radial directions respectively. For such whispering gallery modes, the Poynting vector is directed around the crystal circumference. The resonant frequency of each crystal is directly proportional to the integrated phase velocity of light along the closed path followed by the resonant mode, and is thus sensitive to Lorentz violation in the photon sector of the SME. Note that in this experiment a significant fraction of the mode field exists in the sapphire crystal, therefore the resonance frequency could also be perturbed by Lorentz violation in the electron sector. However, the relevant SME parameters for electrons have been constrained by other experiments [19; 16; 14] to the degree that they do not make significant contributions to these results, so we assume that electrons are fully Lorentz-symmetric. <figure><img src="content_image/1006.1376/x1.png"><figcaption></figcaption></figure> Data was collected from this experiment over a period of 400 days, with a useful duty cycle of 30$\%$. The data analysis is complicated by three main issues, each of which are addressed using specific techniques that in turn constitute the 3 steps of our analysis process. The first is the size of the data set. Processing the entire data set simultaneously is computationally intensive, so the data is initially reduced using the same technique described in [4]. The data is demodulated in quadrature at twice the frequency of the experiment rotation over an integer number of cavity rotation periods, $m$, generating a reduced demodulated data set consisting of S(T${}_{i}$) and C(T${}_{i}$) coefficients of equations (7) and (8), centered at the mean time of the demodulated data block, T${}_{i}$. This reduces the size of the data set by $12\times m$ (12 measurements during each of the $m$ rotations). Figure 1 shows a typical subset of the data acquired continuously over 6 days, demodulated in blocks of 50 periods. In addition to reducing the size of the data, demodulation also effectively filters noise. In the final analysis for the results presented here we chose to use 500 periods, which maximizes the signal to noise ratio of the data while satisfying the Nyquist sampling rate (providing more than 2 data points per half day). The second main issue is the presence of jumps in the data, which are due to non-stationary noise sources such as sudden stress release in the resonator [20]. When analyzing the data using standard regression techniques, such as Least Squares, these jumps mimic temporal signals resulting in incorrect parameter estimates. One solution to this problem is to remove short sections of data containing these jumps, identified using an unbiased method, albeit at the cost of reducing the useful duty cycle of the experiment. In this work we employed an alternative approach of taking the derivative of the data, which involves differencing successive data points [21]. Signal jumps manifest in the derivative as singular outliers (illustrated in Figure 1) to which the Least Squares analysis is less susceptible. This is preferable since no data is excluded, the signal to noise is maximised, and no bias is applied to the data. For nonzero $\tilde{\kappa}$’s, the derivative of the data will vary according to the derivative of (7) and (8): \[\frac{dS(T)}{dT}=(\tilde{\kappa}_{\rm tr})\times\sum_{i}\Big{[}\omega_{i}S_{S, i}\cos(\omega_{i}T)-\omega_{i}S_{C,i}\sin(\omega_{i}T)\Big{]},\] (9) \[\frac{dC(T)}{dT}=(\tilde{\kappa}_{\rm tr})\times\sum_{i}\Big{[}\omega_{i}C_{S, i}\cos(\omega_{i}T)-\omega_{i}C_{C,i}\sin(\omega_{i}T)\Big{]}.\] (10) The third and final step of the analysis is to fit the frequencies of interest to the data using Least Squares regression. Ordinary Least Squares (OLS) regression assumes that the Power Spectral Density (PSD) of the residuals is white. Figure 2 shows the PSD of the data following demodulation over 2 periods of rotation (Increasing the number of rotations over which the data is averaged truncates the PSD curves, acting as a low pass filter). For frequency offsets above $10^{-4}$ Hz the noise is white; near the frequencies of interest, $\omega_{\oplus}$ and $2\omega_{\oplus}$ ($\sim 10^{-5}$ Hz), however, a power law with $\alpha=0.5$ describes the power spectral density. Similarly, once differentiated, the PSD exhibits a power law with $\alpha=0.75$, which is then used in the third part of this analysis. To account for the noise color of the data we use a Weighted Least Squares (WLS) technique that whitens the noise by pre-multiplying the data and the fit model with a weighting matrix. The weighting matrix is determined using a fractional differencing technique [22] that corrects for serially correlated noise, as determined from $\alpha$. Different frequencies are used to set limits on $\tilde{\kappa}_{\rm tr}$ and the $\tilde{\kappa}_{e-}$ and $\tilde{\kappa}_{o+}$ components, allowing a simultaneously fit of all nine components using the coefficients in table 2 and the others already derived in [4]. <figure><img src="content_image/1006.1376/x3.png"><figcaption>Figure 2: Power spectral densities of residuals from the S(T) demodulated(averaged over 2 rotations) least squares data analysis (top graphs) of thenormal data (blue curves) and the derivative of the data (red curves). Powerlaws are fitted around the frequencies of interest (bottom graphs).</figcaption></figure> ## V Conclusion Using a more sophisticated analysis of data collected in [4], we have tightened the limits set by this experiment on the magnitude of all the non-birefringent $\tilde{\kappa}$ coefficients of the SME by a factor between 1.5 and 4, as summarized in table 1. We have explicitly demonstrated that Michelson-Morley experiments are sensitive to isotropic shifts in the vacuum speed of light, and thus for the first time, we report a simultaneous set of bounds on all nine of the non-birefringent $\tilde{\kappa}$ coefficients. The new limit on $\tilde{\kappa}_{\rm tr}$ is an improvement of more than a factor of 11 over limits obtained by relativistic ion spectroscopy [5], marking the first time that a low energy experiment has been able to surpass the sensitivity of such tests. ###### Acknowledgements. This work was supported by the National Science Foundation and the Australian Research Council. We thank Alan Kostelecky for encouragement and useful discussions. ## Appendix A Consider a beam of light moving with velocity $\vec{u}$ along the $x$-axis in the laboratory, which itself moves with velocity $\vec{v}=c\vec{\beta}$ relative to the SCCEF, and define \[\vec{u}_{\|\|} =\frac{\vec{v}\cdot\vec{u}}{\|v\|^{2}}\vec{v} \vec{u}_{\perp} =\vec{u}-\vec{u}_{\|\|}.\] (11) The velocity $\vec{s}$ of that beam of light as measured in the SCCEF must be \[\vec{s}/c=\frac{\vec{v}/c+\vec{u}_{\|\|}/c+\vec{u}_{\perp}/(c\gamma)}{1+\vec{v} \cdot\vec{u}/c^{2}}.\] (12) Since we are interested solely in the contribution of the SCCEF $\tilde{\kappa}_{\rm tr}$ to our experiment, and not in terms proportional to products of the $\tilde{\kappa}$’s, we may assume $\rho_{x\pm}$ and $\rho_{y\pm}$ are such that the speed of light in the SCCEF is isotropic and equal to $c(1-\tilde{\kappa}_{\rm tr})$. Taking the norm of (12) yields \[(1-\tilde{\kappa}_{\rm tr})^{2}=\frac{\beta^{2}-(1+\rho_{x\pm})\left((1+\rho_{ x\pm})(\beta^{2}-\beta_{x}^{2}-1)\mp 2\beta_{x}\right)}{\left(1\pm\beta_{x}(1+ \rho_{x\pm})\right)^{2}},\] (13) which to second order in $\beta$ and first order in $\tilde{\kappa}_{\rm tr}$, becomes \[\frac{1}{2}\left(\rho_{x+}+\rho_{x-}\right)=-\tilde{\kappa}_{\rm tr}-(\beta^{2 }+\beta_{x}^{2})\tilde{\kappa}_{\rm tr}.\] (14) Note that we have neglected terms proportional to $\tilde{\kappa}_{\rm tr}\rho_{x\pm}$, since $\rho_{x\pm}$ is of the same order as $\tilde{\kappa}_{\rm tr}$. We can repeat the above argument to obtain the mean velocity of light along the $y$-axis to find that the dependence of $\delta\nu/\nu$ on $\tilde{\kappa}_{\rm tr}$ is given by \[\frac{\delta\nu}{\nu}\simeq\frac{\delta\nu_{x}}{\nu_{x}}-\frac{\delta\nu_{y}}{ \nu_{y}}=\left(\beta_{y}^{2}-\beta_{x}^{2}\right)\tilde{\kappa}_{\rm tr}.\] (15) The detailed form of the boost $\vec{\beta}$ from the SCCEF as defined in the laboratory frame is [10] \[\vec{\beta}=\mathcal{R}\left(\begin{matrix}\beta_{\oplus}\sin\Omega_{\oplus}T \\ -\beta_{\oplus}\cos\eta\cos\Omega_{\oplus}T\\ -\beta_{\oplus}\sin\eta\cos\Omega_{\oplus}T\end{matrix}\right),\] (16) where we have neglected the contribution of the earth’s rotation $\beta_{L}\simeq 10^{-6}$ to the boost vector, $T$ is the time since the last vernal equinox, and the rotation $\mathcal{R}$ which reorients the SCCEF to align with the laboratory frame, with $\hat{z}$ pointing upwards and $\hat{x}$ pointing south, is given by \[\mathcal{R}=\left(\begin{matrix}\cos\chi\cos\omega_{\oplus}T_{\oplus}&\cos\chi \sin\omega_{\oplus}T_{\oplus}&-\sin\chi\\ -\sin\omega_{\oplus}T_{\oplus}&\cos\omega_{\oplus}T_{\oplus}&0\\ \sin\chi\cos\omega_{\oplus}T_{\oplus}&\sin\chi\sin\omega_{\oplus}T_{\oplus}& \cos\chi\end{matrix}\right).\] (17) Here $\chi$ is the colatitude of the laboratory, $\eta$ is the declination of the Earth’s orbit relative to its spin, $\omega_{\oplus}$ and $\Omega_{\oplus}$ are the Earth’s annual and sidereal frequencies, and $\beta_{\oplus}\simeq 0.994\times 10^{-4}$ is the Earth’s orbital speed. The time $T_{\oplus}$ is not the same as $T$, and represents the time as measured in the SCCEF since that frame’s $Y$-axis coincided with the laboratory $y$-axis [10]. We can account for the active rotation of the experiment [4] by redefining $\mathcal{R}$ so as to be aligned with the resonator axes: \[\mathcal{R}=\left(\begin{matrix}\cos\omega_{R}T&-\sin\omega_{R}T& 0\\ \sin\omega_{R}T&\cos\omega_{R}T&0\\ 0&0&1\end{matrix}\right)\\ \cdot\left(\begin{matrix}\cos\chi\cos\omega_{\oplus}T_{\oplus}& \cos\chi\sin\omega_{\oplus}T_{\oplus}&-\sin\chi\\ -\sin\omega_{\oplus}T_{\oplus}&\cos\omega_{\oplus}T_{\oplus}&0\\ \sin\chi\cos\omega_{\oplus}T_{\oplus}&\sin\chi\sin\omega_{\oplus}T_{\oplus}& \cos\chi\end{matrix}\right).\] (18) Insertion of $\vec{\beta}$ into (15) yields modulations with the form of Eq. (6), described in part III. ## Appendix B This appendix presents the general form of transformations of the non-birefringent $\tilde{\kappa}$ coefficients under an arbitrary boost $\vec{\beta}$ from one inertial frame ($S$) to another ($S^{\prime}$). This derivation rests upon the assumption of Lorentz invariance under observer transformations: that the Lagrangian is an overall Lorentz scalar quantity. In what follows, we neglect the contribution of SME coefficients which give rise to vacuum birefringence, as these have been constrained to be of order $10^{-37}$ or less [13], and do not contribute to the non-birefringent physics at leading order [9]. With this assumption in hand, we may begin with the Lagrangian of Eq. (1), defined in frame $S$ in terms of the fields $\vec{E}$ and $\vec{B}$ as \[\!\!\!\!\!{\cal L}=\frac{1}{2}\left[(1+\tilde{\kappa}_{\rm tr}) \vec{E}^{2}-(1-\tilde{\kappa}_{\rm tr})\vec{B}^{2}\right]+\vec{E}\cdot(\tilde{ \kappa}_{o+})\cdot\vec{B}\\ +\frac{1}{2}\vec{E}\cdot(\tilde{\kappa}_{e-})\cdot\vec{E}+\frac{1 }{2}\vec{B}\cdot(\tilde{\kappa}_{e-})\cdot\vec{B}\;.\] (19) In the boosted frame $S^{\prime}$, both the fields and the $\tilde{\kappa}$’s transform, while the total Lagrangian remains constant, yielding \[\!\!\!\!\!\mathcal{L}=\mathcal{L}^{\prime}=\frac{1}{2}\left[(1+ \tilde{\kappa}_{\rm tr}^{\prime})\vec{E^{\prime}}^{2}-(1-\tilde{\kappa}_{\rm tr }^{\prime})\vec{B^{\prime}}^{2}\right]+\vec{E^{\prime}}\cdot(\tilde{\kappa}_{o +}^{\prime})\cdot\vec{B^{\prime}}\\ +\frac{1}{2}\vec{E^{\prime}}\cdot(\tilde{\kappa}_{e-}^{\prime}) \cdot\vec{E^{\prime}}+\frac{1}{2}\vec{B^{\prime}}\cdot(\tilde{\kappa}_{e-}^{ \prime})\cdot\vec{B^{\prime}}\;.\] (20) Since the fields transform normally [10], the boosted fields $\vec{E^{\prime}}$ and $\vec{B^{\prime}}$ can be written in terms of the unprimed fields as [23] \[\vec{E^{\prime}} =\gamma(\vec{E}+\vec{\beta}\times\vec{B})-\frac{\gamma^{2}}{ \gamma+1}\vec{\beta}(\vec{\beta}\cdot\vec{E}),\] (21) \[\vec{B^{\prime}} =\gamma(\vec{B}-\vec{\beta}\times\vec{E})-\frac{\gamma^{2}}{ \gamma+1}\vec{\beta}(\vec{\beta}\cdot\vec{B}).\] (22) Substituting (21) and (22) into (20) allows us to determine the relationship between the primed $\tilde{\kappa}$’s in $S^{\prime}$ and the unprimed $\tilde{\kappa}$’s in $S$ via the term-by-term equality of all factors of $E_{j}E_{k}$, $B_{j}B_{k}$, and $E_{j}B_{k}$ which appear on both sides. This yields the following general form of the non-birefringent $\tilde{\kappa}$’s in the boosted frame: \[\begin{split}\tilde{\kappa}_{\rm tr}^{\prime}& =\left(1+\frac{\|\beta\|^{2}}{3}\right)\gamma^{2}\tilde{\kappa}_{ \rm tr}+\frac{2}{3}\gamma^{2}\left[(\beta_{x}^{2}-\beta_{y}^{2})\tilde{\kappa} _{e-}^{yy}+(\beta_{x}^{2}-\beta_{z}^{2})\tilde{\kappa}_{e-}^{zz}\right]\\ &\quad\quad-\frac{4}{3}\gamma^{2}\left[\beta_{x}\beta_{y}\tilde{ \kappa}_{e-}^{\prime xy}+\beta_{x}\beta_{z}\tilde{\kappa}_{e-}^{\prime xz}+ \beta_{y}\beta_{z}\tilde{\kappa}_{e-}^{\prime yz}-\beta_{z}\tilde{\kappa}_{o+} ^{\prime xy}+\beta_{y}\tilde{\kappa}_{o+}^{\prime xz}-\beta_{x}\tilde{\kappa}_ {o+}^{\prime yz}\right],\end{split}\] (23) \[\begin{split}\tilde{\kappa}_{e-}^{\prime yy}& =\frac{2}{3}\left[(1-3\beta_{y}^{2})\gamma^{2}-1\right]\tilde{ \kappa}_{\rm tr}+\frac{(\beta_{x}^{2}-\beta_{z}^{2})\gamma^{2}}{3(\gamma+1)^{2 }}\left[1+\gamma(2-\gamma(3\beta_{y}^{2}-1))\right]\tilde{\kappa}_{e-}^{zz}\\ &\quad+\left[\frac{1}{3}(2+(1-\beta_{z}^{2})\gamma^{2})+\frac{ \beta_{y}^{2}(\beta_{y}^{2}-\beta_{x}^{2})\gamma^{4}}{(\gamma+1)^{2}}-\frac{2 \beta_{y}^{2}\gamma^{2}(\gamma-2)}{3(\gamma+1)}\right]\tilde{\kappa}_{e-}^{yy} \\ &\quad+\frac{2\gamma^{2}(2+\gamma+(3\beta_{y}^{2}-1)\gamma^{2})}{ 3(\gamma+1)^{2}}\left[\beta_{x}\beta_{y}\tilde{\kappa}_{e-}^{xy}+\beta_{x} \beta_{z}\tilde{\kappa}_{e-}^{xz}+\beta_{y}\beta_{z}\tilde{\kappa}_{e-}^{yz} \right]-\frac{2\beta_{x}\beta_{z}\gamma^{2}}{\gamma+1}\tilde{\kappa}_{e-}^{xz} \\ &\quad+\frac{2}{3}\gamma^{2}\left(1-\frac{3\beta_{y}^{2}\gamma}{ \gamma+1}\right)\left[\beta_{z}\tilde{\kappa}_{o+}^{xy}-\beta_{y}\tilde{\kappa }_{o+}^{xz}+\beta_{x}\tilde{\kappa}_{o+}^{yz}\right]+2\beta_{y}\gamma\tilde{ \kappa}_{o+}^{xz},\end{split}\] (24) \[\begin{split}\tilde{\kappa}_{e-}^{\prime zz}& =\frac{2}{3}\left[(1-3\beta_{z}^{2})\gamma^{2}-1\right]\tilde{ \kappa}_{\rm tr}+\frac{(\beta_{x}^{2}-\beta_{y}^{2})\gamma^{2}}{3(\gamma+1)^{2 }}\left[1+\gamma(2-\gamma(3\beta_{z}^{2}-1))\right]\tilde{\kappa}_{e-}^{yy}\\ &\quad+\left[\frac{1}{3}(2+(1-\beta_{y}^{2})\gamma^{2})+\frac{ \beta_{z}^{2}(\beta_{z}^{2}-\beta_{x}^{2})\gamma^{4}}{(\gamma+1)^{2}}-\frac{2 \beta_{z}^{2}\gamma^{2}(\gamma-2)}{3(\gamma+1)}\right]\tilde{\kappa}_{e-}^{zz} \\ &\quad+\frac{2\gamma^{2}(2+\gamma+(3\beta_{z}^{2}-1)\gamma^{2})}{ 3(\gamma+1)^{2}}\left[\beta_{x}\beta_{y}\tilde{\kappa}_{e-}^{xy}+\beta_{x} \beta_{z}\tilde{\kappa}_{e-}^{xz}+\beta_{y}\beta_{z}\tilde{\kappa}_{e-}^{yz} \right]-\frac{2\beta_{x}\beta_{y}\gamma^{2}}{\gamma+1}\tilde{\kappa}_{e-}^{xy} \\ &\quad+\frac{2}{3}\gamma^{2}\left(1-\frac{3\beta_{z}^{2}\gamma}{ \gamma+1}\right)\left(\beta_{z}\tilde{\kappa}_{o+}^{xy}-\beta_{y}\tilde{\kappa }_{o+}^{xz}+\beta_{z}\tilde{\kappa}_{o+}^{yz}\right)-2\beta_{z}\gamma\tilde{ \kappa}_{o+}^{xy},\end{split}\] (25) \[\begin{split}\tilde{\kappa}_{e-}^{\prime xy}& =-2\beta_{x}\beta_{y}\gamma^{2}\tilde{\kappa}_{\rm tr}+\frac{ \beta_{x}\beta_{y}(\beta_{y}^{2}-\beta_{x}^{2})\gamma^{4}}{(\gamma+1)^{2}} \tilde{\kappa}_{e-}^{yy}\\ &\quad-\frac{\beta_{x}\beta_{y}\gamma^{2}(1+\gamma+(\beta_{x}^{2} -\beta_{z}^{2})\gamma^{2})}{(\gamma+1)^{2}}\tilde{\kappa}_{e-}^{zz}\\ &\quad+\left(1+\frac{\gamma^{2}(\beta_{x}^{2}+\beta_{y}^{2})}{ \gamma+1}+\frac{2\beta_{x}^{2}\beta_{y}^{2}\gamma^{4}}{(\gamma+1)^{2}}\right) \tilde{\kappa}_{e-}^{xy}+\frac{\beta_{y}\beta_{z}\gamma^{2}}{\gamma+1}\left(1+ \frac{2\beta_{x}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{e-}^{xz}\\ &\quad+\frac{\beta_{x}\beta_{z}\gamma^{2}}{\gamma+1}\left(1+\frac {2\beta_{y}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{e-}^{yz}-\frac{2 \beta_{x}\beta_{y}\beta_{z}\gamma^{3}}{\gamma+1}\tilde{\kappa}_{o+}^{xy}\\ &\quad+\beta_{x}\gamma\left(1+\frac{2\beta_{y}^{2}\gamma^{2}}{ \gamma+1}\right)\tilde{\kappa}_{o+}^{xz}-\beta_{y}\gamma\left(1+\frac{2\beta_{ x}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{o+}^{yz},\end{split}\] (26) \[\begin{split}\tilde{\kappa}_{e-}^{\prime xz}& =-2\beta_{x}\beta_{z}\gamma^{2}\tilde{\kappa}_{\rm tr}+\frac{ \beta_{x}\beta_{z}(\beta_{z}^{2}-\beta_{x}^{2})\gamma^{4}}{(\gamma+1)^{2}} \tilde{\kappa}_{e-}^{zz}\\ &\quad-\frac{\beta_{x}\beta_{z}\gamma^{2}(1+\gamma+(\beta_{x}^{2} -\beta_{y}^{2})\gamma^{2})}{(\gamma+1)^{2}}\tilde{\kappa}_{e-}^{yy}\\ &\quad+\frac{\beta_{y}\beta_{z}\gamma^{2}}{\gamma+1}\left(1+\frac {2\beta_{x}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{e-}^{xy}+\left(1+ \frac{\gamma^{2}(\beta_{x}^{2}+\beta_{z}^{2})}{\gamma+1}+\frac{2\beta_{x}^{2} \beta_{z}^{2}\gamma^{4}}{(\gamma+1)^{2}}\right)\tilde{\kappa}_{e-}^{xz}\\ &\quad+\frac{\beta_{x}\beta_{y}\gamma^{2}}{\gamma+1}\left(1+\frac {2\beta_{z}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{e-}^{yz}-\beta_{x} \gamma\left(1+\frac{2\beta_{z}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{ o+}^{xy}\\ &\quad+\frac{2\beta_{x}\beta_{y}\beta_{z}\gamma^{3}}{\gamma+1} \tilde{\kappa}_{o+}^{xz}-\beta_{z}\gamma\left(1+\frac{2\beta_{x}^{2}\gamma^{2} }{\gamma+1}\right)\tilde{\kappa}_{o+}^{yz},\end{split}\] (27) \[\begin{split}\tilde{\kappa}_{e-}^{\prime yz}& =-2\beta_{y}\beta_{z}\gamma^{2}\tilde{\kappa}_{\rm tr}+\frac{ \beta_{y}\beta_{z}\gamma^{2}}{\gamma+1}\left(1+\frac{(\beta_{y}^{2}-\beta_{x}^ {2})\gamma^{2}}{(\gamma+1)^{2}}\right)\tilde{\kappa}_{e-}^{yy}\\ &\quad+\frac{\beta_{y}\beta_{z}\gamma^{2}}{\gamma+1}\left(1+\frac {(\beta_{z}^{2}-\beta_{x}^{2})\gamma^{2}}{(\gamma+1)^{2}}\right)\tilde{\kappa} _{e-}^{zz}\\ &\quad+\frac{\beta_{x}\beta_{z}\gamma^{2}}{\gamma+1}\left(1+\frac {2\beta_{y}^{2}\gamma^{2}}{\gamma+1}\right)\tilde{\kappa}_{e-}^{xy}+\frac{ \beta_{x}\beta_{y}\gamma^{2}}{\gamma+1}\left(1+\frac{2\beta_{z}^{2}\gamma^{2}} {\gamma+1}\right)\tilde{\kappa}_{e-}^{xz}\\ &\quad+\left(1+\frac{\gamma^{2}(\beta_{y}^{2}+\beta_{z}^{2})}{ \gamma+1}+\frac{2\beta_{y}^{2}\beta_{z}^{2}\gamma^{4}}{(\gamma+1)^{2}}\right) \tilde{\kappa}_{e-}^{yz}-\beta_{y}\gamma\left(1+\frac{2\beta_{z}^{2}\gamma^{2} }{\gamma+1}\right)\tilde{\kappa}_{o+}^{xy}\\ &\quad+\beta_{z}\gamma\left(1+\frac{2\beta_{y}^{2}\gamma^{2}}{ \gamma+1}\right)\tilde{\kappa}_{o+}^{xz}-\frac{2\beta_{x}\beta_{y}\beta_{z} \gamma^{3}}{\gamma+1}\tilde{\kappa}_{o+}^{yz},\end{split}\] (28) \[\begin{split}\tilde{\kappa}_{o+}^{\prime xy}& =2\beta_{z}\gamma^{2}\tilde{\kappa}_{\rm tr}-\frac{(\beta_{y}^{2} -\beta_{x}^{2})\beta_{z}\gamma^{3}}{\gamma+1}\tilde{\kappa}_{e-}^{yy}-\left( \beta_{z}\gamma+\frac{(\beta_{z}^{2}-\beta_{x}^{2})\beta_{z}\gamma^{3}}{\gamma +1}\right)\tilde{\kappa}_{e-}^{zz}\\ &\quad-\frac{2\beta_{x}\beta_{y}\beta_{z}\gamma^{3}}{\gamma+1} \tilde{\kappa}_{e-}^{xy}-\gamma\left(1+\frac{2\beta_{z}^{2}\gamma^{2}}{\gamma+ 1}\right)\left[\beta_{x}\tilde{\kappa}_{e-}^{xz}+\beta_{y}\tilde{\kappa}_{e-}^ {yz}\right]\\ &\quad+\gamma\tilde{\kappa}_{o+}^{xy}+\frac{\beta_{z}\gamma^{2}(1 +2\gamma)}{\gamma+1}\left[\beta_{z}\tilde{\kappa}_{o+}^{xy}-\beta_{y}\tilde{ \kappa}_{o+}^{xz}+\beta_{x}\tilde{\kappa}_{o+}^{yz}\right],\end{split}\] (29) (30) (31) In the limit that only $\tilde{\kappa}_{\rm tr}$ has significant value in the SCCEF, the Michelson-Morley observable (2) becomes \[\frac{\delta\nu}{\nu}=S_{e}(2\gamma^{2})\left\{\left[\beta_{y}^{2}-\beta_{x}^{ 2}\right]\cos{2\theta}+2\beta_{x}\beta_{y}\sin{2\theta}\right\}(\tilde{\kappa} _{\rm tr})_{\odot},\] (32) where $(\tilde{\kappa}_{\rm tr})_{\odot}$ represents the value of $\tilde{\kappa}_{\rm tr}$ in the SCCEF, yielding the same result as reported in (5), and derived in Appendix A. Note that these transformations are only appropriate between concordant frames [24], in which the effects of the $\tilde{\kappa}$ parameters are perturbative. As can be seen from Eqs. (23) through (31), boosts between frames with large relative $\gamma$ can enhance the effects of various $\tilde{\kappa}$’s by up to $\gamma^{2}$ in one frame relative to the other. In general, $\gamma^{2}$ should be much smaller than the smallest inverse fractional shift $1/\rho$ in the speed of light that is generated by the $\tilde{\kappa}$’s. Thus the maximum boost relative to the SCCEF for which the above relations are useful in the absence of more complete knowledge of the underlying physics at high energy scale is limited by the most poorly bounded of the $\tilde{\kappa}$’s. ## References * (1) A.A. Michelson, American Journal of Science 22, 120 (1881). * (2) R.J. Kennedy and E.M. Thorndike, Phys. Rev. 42, 400 (1932). * (3) H.E. Ives and G.R. Stilwell, JOSA 28, 215 (1938). * (4) P.L. Stanwix, M.E. Tobar, P. Wolf, C.R. Locke and E.N. Ivanov, Phys. Rev. D 74, 081101 (2006). * (5) S. Reinhardt, G. Saathoff, H. Buhr, L.A. Carlson, A. Wolf, D. Schwalm, S. Karpuk, C. Novotny, G. Huber and M. Zimmermann, Nature Physics 3, 861 (2007). * (6) M.E. Tobar, E.N. Ivanov, P.L. Stanwix, J.-M.G. le Floch and J.G. Hartnett, Phys. Rev. D 80, 125024 (2009). * (7) Ch. Eisele, A.Yu. Nevsky and S. Schiller, Phys. Rev. Lett. 103, 090401 (2009). * (8) S. Herrmann, A. Senger, K. Möhle, M. Nagel, E.V. Kovalchuk and A. Peters, Phys. Rev. D 80, 105011 (2009). * (9) D. Colladay and V.A. Kostelecký, Phys. Rev. D 55, 6760 (1997); Phys. Rev. D 58, 116002 (1998). * (10) V.A. Kostelecký and M. Mewes, Phys. Rev. D 66, 056005 (2002). * (11) B. Altschul, Phys. Rev. D 80, 091901(R) (2009). * (12) M.A. Hohensee, R. Lehnert, D.F. Phillips and R.L. Walsworth, Phys. Rev. Lett. 102, 170402 (2009); Phys. Rev. D 80, 036010 (2009). * (13) V.A. Kostelecký and M. Mewes, Phys. Rev. Lett. 97, 140401 (2006). * (14) V.A. Kostelecký and N. Russell, arXiv:0801.0287v3 (2010). * (15) M.E. Tobar, P.L. Stanwix, M. Susli, P. Wolf, C.R. Locke and E.N. Ivanov, Lect. Notes Phys. 702, 416 (2006). * (16) H. Müller, P.L. Stanwix, M.E. Tobar, E.N. Ivanov, P. Wolf, S. Herrmann, A. Senger, E. Kovalchuk and A. Peters, Phys. Rev. Lett. 99, 050401 (2007). * (17) C.R. Locke, E.N. Ivanov, J.G. Hartnett, P.L. Stanwix, and M.E. Tobar, Rev. Sci. Instrum. 79, 051301 (2008). * (18) P.L. Stanwix, M.E. Tobar, P. Wolf, M. Susli, C.R. Locke, E.N. Ivanov, J. Winterflood, and F. van Kann, Phys. Rev. Lett. 95, 040404 (2005). * (19) B. Altschul, Phys. Rev. D 74, 083003 (2006). * (20) M.E. Tobar, E.N. Ivanov, C.R. Locke, P.L. Stanwix, J.G. Hartnett, A.N. Luiten, R.B. Warrington, P.T.H. Fisk, M.A. Lawn, M. Wouters, S. Bize, G. Santarelli, P. Wolf, A. Clairon and P. Guillemot, IEEE Trans. UFFC 53, 2386 (2006). * (21) M.E. Tobar, P. Wolf, S. Bize, G. Santarelli and V.V. Flambaum, Phys. Rev. D 81, 022003 (2010). * (22) L.S. Schmidt, Metrologia 40, S305 (2003). * (23) J.D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, New York, NY, (1999). * (24) V.A. Kostelecký and R. Lehnert, Phys. Rev. D 63, 065008 (2001).
1411.5861	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 25362, "num_imgs": 2, "llama3_tokens_count": 7960 }	[ "content_image/1411.5861/x1.png", "content_image/1411.5861/x2.png" ]	# A Comparison of Skewed and Orthogonal Lattices in Gaussian Wiretap Channels Alex Karrila and Camilla Hollanti, _Member, IEEE_ Department of Mathematics and Systems Analysis Aalto University, Finland Emails: firstname.lastname@aalto.fi ###### Abstract We consider lattice coset-coded transmissions over a wiretap channel with additive white Gaussian noise (AWGN). Examining a function that can be interpreted as either the legitimate receiver’s error probability or the eavesdropper’s correct decision probability, we rigorously show that, albeit offering simple bit labeling, orthogonal nested lattices are suboptimal for coset coding in terms of both the legitimate receiver’s and the eavesdropper’s probabilities. ## I Introduction We consider a wiretap set-up, in which a message is transmitted to its legitimate receiver Bob in the presence of Eve the eavesdropper. Eve is assumed to have unlimited computational power, but to experience an additional noise compared to Bob. Lattice coset coding is utilized to maximize Eve’s confusion, cf. [1, 2]. Bob’s lattice is referred to as the code lattice or dense lattice, and Eve’s lattice as the sparse or coarse lattice. The channel is assumed to exhibit additive white Gaussian noise (AWGN) but no fading. The respective channel equations for Bob and Eve are \[\mathbf{y}_{b}=\mathbf{x}+\mathbf{n}_{b},\quad\mathbf{y}_{e}=\mathbf{x}+ \mathbf{n}_{e},\] where $\mathbf{y}$ is the received vector, $\mathbf{x}$ the transmitted coset-coded vector, and $\mathbf{n}$ is AWGN with respective variances $\sigma_{b}^{2}<\sigma_{e}^{2}$. In finding optimal lattice wiretap codes, there are three main objectives: * Maximizing the data rate $R$, which is determined by the size of the codebook $\mathcal{C}$ and the decoding delay $n$ as \[R=\frac{\log_{2}\|\mathcal{C}\|}{n}\] bits per channel use (bpcu). * Minimizing the legitimate receiver’s decoding error probability. * Minimizing the eavesdropper’s probability of correct decision. Considering only the first two problems, the largest codebooks for a fixed transmission power and an upper bound for the receiver’s error probability are the solutions to the widely investigated sphere-packing problem. This results in lattices that are typically nonorthogonal (see, e.g., [4]). Orthogonal lattices have still traditionally been preferred due to an easy-to-implement bit-labeling algorithm, namely the Gray-mapping. Due to this mapping, the encoding and decoding procedures are more straightforward for orthogonal lattices than for nonorthogonal, i.e., skewed lattices. Nevertheless, computationally efficient closest-point algorithms such as the sphere decoder also exist for nonorthogonal lattices (for an explicit construction, see [3], Sec. 4). In [5, 6], it was also demonstrated how skewed lattices can be efficiently encoded and decoded by using a modified power-controlled sphere decoder or sphere decoding adjoined with minimum-mean-square-error generalized-decision-feedback-equalization (MMSE-GDFE), both resulting in optimal (maximum-likelihood) performance. Hence, skewed lattices should not be excluded when searching for optimal lattices, in particular in the light of the present paper showing that they are not only better in terms of Bob’s performance, but also in terms of confusing the eavesdropper. Similarly to the sphere-packing problem in Bob’s case, we now include the third objective in our consideration. Our approach is to fix the data rate and the transmission power and then compare skewed and orthogonal lattices from the point of view of the latter two objectives in an AWGN channel. We study an expression that has two alternative interpretations as either the receiver’s error probability (REP) for any lattice code in an AWGN channel or the eavesdropper’s correct decision probability (ECDP) for a lattice coset code in an AWGN channel. We prove the following results (notation will be defined in the subsequent section). * Skewing Bob’s orthogonal code lattice $\Lambda_{b}$ will decrease the REP of any code. * Skewing Eve’s orthogonal sparse lattice $\Lambda_{e}$ will decrease the ECDP of any lattice coset code. * Combining the previous two results, the common set-up of the dense lattice $\Lambda_{b}$ being orthogonal and the commonly used choice of an orthogonal sublattice $\Lambda_{e}=2^{k}\Lambda_{b}$ are suboptimal in terms of both the ECDP and the REP. According to whether Gray-labeling is insisted or not, this common set-up can be improved by either choosing a skewed sublattice of the same orthogonal dense lattice $\Lambda_{b}$, leaving Bob’s lattice orthogonal and the REP suboptimal, or skewing both lattices. These results suggest that skewed lattices deserve more attention in the study of the AWGN wiretap channels even though their encoding and decoding are admittedly somewhat more complicated than that of orthogonal lattices. It is also worthwhile to keep in mind that in any practical system, an outer error correcting code, e.g., a low-density parity-check (LDPC) code, is used in addition to the inner lattice code. The true decoding bottle-neck in this case is the outer code requiring soft input, not the lattice code. ## II Preliminaries In this section, we present some necessary definitions and their information-theoretic interpretations. Definition II.1: _A lattice is a discrete additive subgroup of $\mathbb{R}^{n}$._ Any point in a lattice $\Lambda\subset\mathbb{R}^{n}$ can be expressed in terms of a _generator matrix_$M\in\mathbb{R}^{n\times m}$ as follows \[\Lambda=\{\mathbf{x}\in\mathbb{R}^{n}\|\mathbf{x}=M\omega,\omega\in\mathbb{Z}^{ m}\}.\] We assume that the columns of $M$ are linearly independent over $\mathbb{Z}$ and hence, the _lattice coordinates_$\omega$ of a lattice point are unique. If $m=n$, the lattice is of _full rank_. A _sublattice_ of a lattice of dimension $m$ in $\mathbb{R}^{n}$ is an additive subgroup; it has a generator matrix $MZ$, where $Z\in\mathbb{Z}^{m\times k}$. Here $k$ is the dimension of the sublattice and for a square matrix $Z$, \[\|\Lambda_{b}/\Lambda_{e}\|=\|\det Z\|.\] The _volume_$\text{Vol}(\Lambda)$ of the lattice $\Lambda$ is the volume of the fundamental parallellotope spanned by the column vectors of $M$, given by \[\text{Vol}(\Lambda)=\|\det M\|\] for full-rank lattices. Remark II.2: _Differing from some information theory references, here vectors are identified with column matrices and the lattice generator vectors with the columns of the generator matrix $M$._ Definition II.3: _The dual lattice $\Lambda^{\star}$ of a full-rank lattice $\Lambda$ generated by $M$ is the one generated by_ \[M^{-T}:=(M^{-1})^{T}=(M^{T})^{-1}.\] Theorem II.4 (The Poisson formula for lattices): _Let $\Lambda$ be a full-rank lattice with generator $M$ and let $f:\mathbb{R}^{n}\to\mathbb{C}$ be a continuous function with $\int_{\mathbf{x}\in\mathbb{R}^{n}}\|f(\mathbf{x})\|d^{n}x<\infty$ and $\sum_{\mathbf{t}\in\Lambda^{\star}}\|\hat{f}(\mathbf{t})\|<\infty$ such that the partial sums of $\sum_{\mathbf{t}\in\Lambda}\|f(\mathbf{t}+\mathbf{u})\|$ converge uniformly whenever $\mathbf{u}$ is restricted onto a compact set. Then,_ \[\sum_{\mathbf{t}\in\Lambda}f(\mathbf{t})=\|\det M\|^{-1}\sum_{\mathbf{t}\in \Lambda^{\star}}\hat{f}(\mathbf{t})\] _where the Fourier transform is defined as_ \[\hat{f}(\mathbf{t})=\int_{\mathbf{y}\in\mathbb{R}^{n}}e^{-i2\pi\mathbf{y}\cdot \mathbf{t}}f(\mathbf{y})dy.\] Proof:: The proof is given in [7]. We point out that the condition on the continuity of $f$ is essential for the proof and is missing in the book. The function that we will optimize is the following. Definition II.5: _The psi function $\psi_{\Lambda}(x)$ of a lattice $\Lambda$ at a point $x\in\mathbb{R}_{+}$ is given by_ \[\psi_{\Lambda}(x)=\sum_{\mathbf{t}\in\Lambda}e^{-x\\|\mathbf{t}\\|^{2}}.\] This is a variant of lattice theta series restricted on the imaginary axis, $\psi_{\Lambda}(x)=\Theta_{\Lambda}(ix/\pi)$. The convergence properties of the psi series follow from those of the theta series. Interpretation II.6: _In [2], an upper approximation for the ECDP $P_{c,e}$ for a lattice coset code is derived as_ \[P_{c,e}\leq\frac{\text{Vol}(\Lambda_{b})}{(\sqrt{2\pi}\sigma_{e})^{n}}\psi_{ \Lambda_{e}}\left(\frac{1}{2\sigma_{e}^{2}}\right).\] (1) _Here $\Lambda_{b}$ is the dense and $\Lambda_{e}$ the sparse lattice, inteded for the receiver and the eavesdropper, respectively. The lattices are assumed to be of full rank and the eavesdropper’s noise is assumed to be AWGN with variance $\sigma_{e}^{2}$. The inequality (1) is tight for large $\sigma_{e}$. For small $\sigma_{e}$, the upper bound is larger than $1$ and hence useless._ _On the other hand, using the union bound technique as is done in [8, Appendix II] for Rayleigh-fading channels and setting the Rayleigh fading coefficients equal to one, the REP can be approximated from above as:_ \[P_{e,b}\leq 1/2\left(\psi_{\Lambda_{b}}\left(\frac{1}{8\sigma_{b}^{2}}\right)- 1\right).\] (2) _This formula is valid for any lattice code $\Lambda_{b}$ (not just a coset code) in an AWGN channel and the approximation is good for small receiver’s noise variances $\sigma_{b}^{2}$._ _Based on these two formulae and the fact that the variances $\sigma_{e}^{2}$ and $\sigma_{b}^{2}$ vary with the random channels, our subsequent aim will be to provide inequalities of the form $\psi_{\Lambda_{1}}(x)<\psi_{\Lambda_{2}}(x)$ for all $x\in\mathbb{R}_{+}$. When comparing different lattices sharing the same dimension, their volumes are first normalized to one. This ensures that for a relatively large fixed transmission power, the finite codebooks carved from the infinite lattices will be approximately equally large, and hence we can fairly compare the lattice codes without considering the actual data rates, as these will coincide._ Remark II.7: _Due to the obvious connection between the formulae (2) and (1) for the REP and ECDP, respectively, one would intuitively guess that a solution for the sphere-packing problem also yields an optimal ECDP. This, however, does not seem to work on the level of mathematical proofs; Eq. (2) is obtained by the union bound technique, whereas in the sphere-packing problem, the upper bound for REP is based on integrating a Gaussian function over a ball, yielding a much tighter bound for large receiver’s noise variances $\sigma_{b}$ or, equivalently, for small arguments of $\psi$. To minimize the ECDP, we want to minimize $\psi$ for small arguments. Hence, even if the sphere-packing probability bound is small, it does not provide us with immediate information as to how small the $\psi$ function is for small arguments, i.e., how small the ECDP is._ ## III Skewing an orthogonal lattice In this section, we show that skewing a lattice will always improve a code both in terms of Eve’s and Bob’s probabilities. Lemma III.1: _For any full-rank lattice $\Lambda$,_ \[\psi_{\Lambda}(x)>\sum_{\mathbf{t}\in\Lambda}e^{-x\\|\mathbf{t}+\mathbf{u}\\|^{2}}\] (3) _for any $\mathbf{u}\not\in\Lambda$._ Proof:: Denote summands of the respective sides as $g(\mathbf{t})=e^{-x\\|\mathbf{t}\\|^{2}}$ and $f(\mathbf{t})=e^{-x\\|\mathbf{t}+\mathbf{u}\\|^{2}}$, so $f(\mathbf{t})=g(\mathbf{t}+\mathbf{u})$. Then, by the elementary properties of Fourier transform, we have $\hat{f}(\mathbf{t})=\hat{g}(\mathbf{t})e^{-i2\pi\mathbf{t}\cdot\mathbf{u}}$. Hence, using the Poisson formula, \[\sum_{\mathbf{t}\in\Lambda}e^{-x\\|\mathbf{t}+\mathbf{u}\\|^{2}}\] \[= \sum_{\mathbf{t}\in\Lambda}f(\mathbf{t})\] \[= \|\det M\|^{-1}\sum_{\mathbf{t}\in\Lambda^{\star}}\hat{g}(\mathbf{t })e^{-i2\pi\mathbf{t}\cdot\mathbf{u}},\] where $M$ is the generator matrix of $\Lambda$. In continuation, we will use the knowledge that the Fourier transform of the gaussian function $g$ is another gaussian, hence a real, positive and even function. (The explicit form of $\hat{g}$ could be calculated but it is not necessary.) First, since $\hat{g}$ is even, the imaginary parts $-i\hat{g}(\pm\mathbf{t})\sin(\pm 2\pi\mathbf{t}\cdot\mathbf{u})$ of the summand for lattice $\Lambda^{\star}$ points $\pm\mathbf{t}$ cancel out, yielding \[\sum_{\mathbf{t}\in\Lambda}e^{-x\\|\mathbf{t}+\mathbf{u}\\|^{2}}\] \[= \|\det M\|^{-1}\sum_{\mathbf{t}\in\Lambda^{\star}}\hat{g}(\mathbf{t })\cos(2\pi\mathbf{t}\cdot\mathbf{u}).\] Next, we need the positivity of $\hat{g}$ to be able to approximate the cosine by $1$. First, note that we assumed $\mathbf{u}\not\in\Lambda$, equivalently, $\mathbf{u}=M\omega_{1}$ with some component of $\omega_{1}$, say the $j^{th}$ one $\omega_{1,j}$, not integer. Also note that $\Lambda^{\star}\ni\mathbf{t}=M^{-T}\omega_{2},$ where $\omega_{2}\in\mathbb{Z}^{n}$. Hence, $\mathbf{t}\cdot\mathbf{u}=\omega_{2}^{T}M^{-1}M\omega_{1}=\omega_{2}^{T}\omega _{1}$. Now, choosing the lattice point $\mathbf{t}$ such that $\omega_{2}=\mathbf{e}_{j}$, we immediately see that $\mathbf{t}\cdot\mathbf{u}=\omega_{1,j}\not\in\mathbb{Z}$ and $\cos(2\pi\mathbf{t}\cdot\mathbf{u})<1$. Hence, replacing $\cos(2\pi\mathbf{t}\cdot\mathbf{u})$ by $1$ in the preceding step, we get a strict inequality \[\sum_{\mathbf{t}\in\Lambda}e^{-x\\|\mathbf{t}+\mathbf{u}\\|^{2}}\] (4) \[< \|\det M\|^{-1}\sum_{\mathbf{t}\in\Lambda^{\star}}\hat{g}(\mathbf{t})\] (5) \[= \|\det M\|^{-1}\|\det M^{-T}\|^{-1}\sum_{\mathbf{t}\in\Lambda}\hat{ \hat{g}}(\mathbf{t})\] (6) \[= \sum_{\mathbf{t}\in\Lambda}\hat{\hat{g}}(\mathbf{t}),\] (7) where we have again applied the the Poisson formula to (5). Finally, the double Fourier transform is in general a reflection operator, so $\hat{\hat{g}}(\mathbf{t})=g(-\mathbf{t})$, and using the fact that $g(-\mathbf{t})=g(\mathbf{t})$ we obtain the result, \[\sum_{\mathbf{t}\in\Lambda}e^{-x\\|\mathbf{t}+\mathbf{u}\\|^{2}}\] \[< \sum_{\mathbf{t}\in\Lambda}g(\mathbf{t})\] \[= \psi_{\Lambda}(x).\] Definition III.2: _Let $\Lambda_{o}$ be a full-rank orthogonal lattice in $\mathbb{R}^{n}$ with generator vectors $a_{1}\mathbf{e}_{1},...,a_{n}\mathbf{e}_{n}$, $a_{i}>0$ for $1\leq i\leq n$. We call a lattice $\Lambda_{s}\neq\Lambda_{o}$ a skewing of $\Lambda_{o}$, if it has a generator matrix that is an upper triangular matrix with the diagonal elements $a_{1},...,a_{n}$._ This definition has a simple geometric interpretation, depicted in Fig. 1. <figure><img src="content_image/1411.5861/x1.png"><figcaption>Fig. 1: The fundamental parellellotopes of (a) a square lattice (b) itsskewing.</figcaption></figure> We point out that skewing can be interpreted as a matrix operation. If $M_{o}$ and $M_{s}$ are the generator matrices of $\Lambda_{o}$ and $\Lambda_{s}$, respectively, then the non-singular skewing matrix $S$ can be solved from the matrix equation \[M_{s}=SM_{o}.\] (8) This equation is non-singular, since \[\det M_{o}=\prod_{i=1}^{n}a_{i}\neq 0\] and \[\det M_{s}=\prod_{i=1}^{n}a_{i}=\det M_{o}\] by the determinant rule of upper triangular matrices. This also implies that $\det S=1$. Now we are ready to state the main theorem. After this, we will provide an illustrative interpretation of the theorem and prove it. Theorem III.3: _For a skewing $\Lambda_{s}$ of a full-rank orthogonal lattice $\Lambda_{o}$,_ \[\psi_{\Lambda_{s}}(x)<\psi_{\Lambda_{o}}(x)\] _for all $x>0$._ Interpretation III.4: _Skewings provide several easy ways to improve lattice coset codes. We point out that since skewing keeps the lattice volume constant ($\det S=1$ in matrix representation), it will not affect the size of a spherical codebook. Hence, a lattice comparison between skewings only requires considering the ECDP and the REP. With ths knowledge, the theorem has the following immediate implications._ * _Comparing a dense lattice_ $\Lambda_{b,o}$ _and its skewings_ $\Lambda_{b,s}$_, Theorem_ III.3 _applied to Eq. (_2_) shows that the REP is always smaller for the skewings_ $\Lambda_{b,s}$_. This holds for all codes, not just coset codes._ * _Consider a coset code arising from a fixed nonorthogonal lattice_ $\Lambda_{b}$_. Then, to minimize the ECDP (_1_), it seems that_ $\Lambda_{e}$ _should not be chosen orthogonal (if orthogonal sublattices exist). Note that then no skewing of the orthogonal_ $\Lambda_{e}$ _is necessarily a sublattice of_ $\Lambda_{b}$_, so this is just heuristics._ * _Consider a typical set-up of_ $\Lambda_{b,o}$ _generated by_ $M=\text{diag}(a_{1},...a_{n})$ _being orthogonal and_ $\Lambda_{e,o}=2^{k}\Lambda_{b,o}$_. In this case both the REP and the OCDP are suboptimal. There are two remedies:_ * _First, we can skew both_ $\Lambda_{e,o}$ _and_ $\Lambda_{b,o}$_. Skewing by_ $S$ _so that the skewed lattices_ $\Lambda_{b,s}$ _and_ $\Lambda_{e,s}$ _are generated by_ $SM$ _and_ $2^{k}SM$_, respectively, will yield a nonorthogonal lattices but preserve the volumes:_ $\text{Vol}(\Lambda_{b,s})=\text{Vol}(\Lambda_{b,o})$ _(since_ $\det S=1$_). Hence, applying this and Theorem_ III.3 _in Eqs. (_2_) and (_1_), we see that skewing will decrease both the REP and the ECDP. However, the skewed lattice will not allow for a simple Gray mapping, or in other words, the Gray mapping is not guaranteed to give an optimal bit-labeling._ * _Second, we can only opt for skewing the sublattice_ $\Lambda_{e}$_, while leaving_ $\Lambda_{b}$ _orthogonal. This means that the REP will remain suboptimal, but the lattice will allow for Gray labeling and maintains simpler encoding and decoding for Bob along the lines discussed in the introduction. Moreover, the ECDP is decreased. The idea is that if_ $\Lambda_{b,o}$ _is orthogonal and generated by_ $\text{diag}(a_{1},...a_{n})$_, and_ $\Lambda_{e,o}=2^{k}\Lambda_{b}$_, then any sublattice_ $\Lambda_{e,s}$ _generated by_ $MZ$_, where_ $Z$ _is an upper triangular integer matrix with diagonal etries_ $2^{k}$_, is easily proven to be a skewing of_ $\Lambda_{e,o}$ _(or equal to_ $\Lambda_{e,o}$_). Then, applying Theorem_ III.3 _to Eq. (_1_), we see that_ $\Lambda_{e,s}$ _will yield a lower ECDP._ Proof:: Let us use the notation of Definition III.2. Furthermore, denote by $\Lambda_{s,k}$ the embedding into $\mathbb{R}^{k}$, $k\leq n$, of $\Lambda_{s}\cap\mathbb{R}^{k}\times 0^{n-k}$. Equivalently, $\Lambda_{s,k}$ is the lattice in $\mathbb{R}^{k}$ generated by the $k$ first columns of the generator $M_{s}$ of $\Lambda_{s}$. Continuing to ease the notation, denote the projection of the $k^{th}$ column of $M_{s}$ onto $\mathbb{R}^{k-1}$ by $\mathbf{m}_{k}$, so the $k^{th}$ column is $a_{k}\mathbf{e}_{k}+\mathbf{m}_{k}$. Now, it is apparent from the definition of $\Lambda_{s,k}$ that \[\psi_{\Lambda_{s}}(x)=\psi_{\Lambda_{s,n}}(x),\] (9) and that \[\psi_{\Lambda_{s,1}}(x)=\sum_{\omega_{1}\in\mathbb{Z}}e^{-x\omega_{1}^{2}a_{1} ^{2}}.\] (10) On the other hand, using Lemma III.1 for the lattice $\Lambda_{s,k-1}$, $2\leq k\leq n$, which is a full-rank lattice of $\mathbb{R}^{k-1}$, we obtain \[\psi_{\Lambda_{s,k}}(x)\] (11) \[= \sum_{\mathbf{t}\in\Lambda_{s,k}}e^{-x\\|\mathbf{t}\\|^{2}}\] \[= \sum_{\omega_{k}\in\mathbb{Z}}\quad\sum_{\mathbf{t}^{(k-1)}\in \Lambda_{s,k-1}}e^{-x\omega_{k}^{2}a_{k}^{2}-x\\|\mathbf{t}^{(k-1)}+\omega_{k} \mathbf{m}_{k}\\|^{2}}\] \[= \sum_{\omega_{k}\in\mathbb{Z}}e^{-x\omega_{k}^{2}a_{k}^{2}}\sum_{ \mathbf{t}^{(k-1)}\in\Lambda_{s,k-1}}e^{-x\\|\mathbf{t}^{(k-1)}+\omega_{k} \mathbf{m}_{k}\\|^{2}}\] \[\leq \sum_{\omega_{k}\in\mathbb{Z}}e^{-x\omega_{k}^{2}a_{k}^{2}}\sum_{ \mathbf{t}^{(k-1)}\in\Lambda_{s,k-1}}e^{-x\\|\mathbf{t}^{(k-1)}\\|^{2}}\] \[= \left(\sum_{\omega_{k}\in\mathbb{Z}}e^{-x\omega_{k}^{2}a_{k}^{2}} \right)\psi_{\Lambda_{s,k-1}}(x)\] and, as stated in Lemma III.1, the equality holds if and only if $\omega_{k}\mathbf{m}_{k}\in\Lambda_{s,k-1}$ for all $\omega_{k}\in\mathbb{Z}$, equivalently, $\mathbf{m}_{k}\in\Lambda_{s,k-1}$. This is furthermore equivalent to that the $k^{th}$ column $a_{k}\mathbf{e}_{k}+\mathbf{m}_{k}$ of $M_{s}$ can be replaced by $a_{k}\mathbf{e}_{k}$ without changing the lattice $\Lambda_{s}$. Next, starting from Eq. (9), using the identity (11) inductively, and finally using Eq. (10), we obtain \[\psi_{\Lambda_{s,n}}(x) \leq \prod_{i=1}^{n}\left(\sum_{\omega_{i}\in\mathbb{Z}}e^{-x\omega_{i }^{2}a_{i}^{2}}\right)\] \[= \psi_{\Lambda_{s,o}}(x).\] The equality holds if and only if it has been possible to modify, for all $k$, the $k^{th}$ column of $M_{s}$ into $a_{k}\mathbf{e_{k}}$ without changing the lattice generated by $M_{s}$. But this is equivalent to $M_{s}$ and $(a_{1}\mathbf{e}_{1},...,a_{n}\mathbf{e}_{n})=M_{o}$ generating the same orthogonal lattice $\Lambda_{o}$. This is impossible by the definition of a skewing. Hence, for any skewing $\Lambda_{s}$ of $\Lambda_{o}$, we have a strict inequality \[\psi_{\Lambda_{s}}(x)<\psi_{\Lambda_{o}}(x)\] for all $x>0$. This completes the proof. Example III.5: _The Gosset lattice $E_{8}$ has the generator matrix $M_{s}$ given by [4]_ \[\left(\begin{array}[]{r r r r r r r r}2&-1&0&0&0&0&0&1/2\\ 0&1&-1&0&0&0&0&1/2\\ 0&0&1&-1&0&0&0&1/2\\ 0&0&0&1&-1&0&0&1/2\\ 0&0&0&0&1&-1&0&1/2\\ 0&0&0&0&0&1&-1&1/2\\ 0&0&0&0&0&0&1&1/2\\ 0&0&0&0&0&0&0&1/2\\ \end{array}\right),\] _so it is a skewing of the orthogonal lattice $\Lambda$ generated by $M_{o}=\text{diag}(2,1,...,1,1/2)$. The theta series of the Gosset lattice is expressible by the Jacobi theta functions as [4]_ \[\Theta_{E_{8}}(z)=1/2(\vartheta_{2}(q)^{8}+\vartheta_{3}(q)^{8}+\vartheta_{4}( q)^{8}),\] _where $q=e^{i\pi z}$. The theta series of the orthogonal lattice $\Lambda$ is_ \[\Theta_{\Lambda}(z)=\prod_{j=1}^{n}\vartheta_{3}(q_{j}),\] (12) _where $q_{j}=e^{i\pi a_{j}^{2}z}$ and $a_{j}$ is the $j^{th}$ diagonal element of $M_{o}$. Now, recalling that $\psi_{\Lambda}(x)=\Theta_{\Lambda}(ix/\pi)$, we can compare the psi series of these two lattices by evaluating Jacobi theta functions. The plots of the psi functions are depicted in Fig. 2. The figure shows that $\psi_{E_{8}}(x)<\psi_{\Lambda}(x)$ for all $x$, as predicted by Theorem III.3._ _In coset coding, this has the following interpretation: $E_{8}$ and $\Lambda$ are both index $2^{8}$ subgroups of $\frac{1}{2}\mathbb{Z}^{8}$. If Bob’s lattice is $\frac{1}{2}\mathbb{Z}^{8}$, then the coset lattices $E_{8}$ and $\Lambda$ will yield the same code rates, but $E_{8}$ with a better secrecy._ <figure><img src="content_image/1411.5861/x2.png"><figcaption>Fig. 2: The psi functions of an orthogonal lattice Λ=Lo and its skewing E8.</figcaption></figure> ## IV Conclusions In the construction of lattice codes for AWGN wiretap channels, skewed lattices should be taken more seriously. Namely, we have proved that orthogonal lattices are suboptimal not only in terms of the receiver’s error probability as we already know from the sphere-packing theorems, but also in terms of the eavesdropper’s correct decision probability when using lattice coset codes. Hence, the design of secure lattice codes should ideally be based on skewed lattices. However, due to implementation purposes, one may opt for only skewing the eavesdropper’s lattice, while preserving the orthogonality of the legitimate receiver’s lattice, which results in suboptimal performance but easy-to-implement algorithms for Bob, as well as improved security. ## V Acknowledgments This work was carried out during A. Karrila’s MSc thesis project. The Department of Mathematics and Systems Analysis at Aalto University is gratefully acknowledged for the financial support. C. Hollanti is financially supported by the Academy of Finland grants #276031, #282938, and #283262, and by a grant from Magnus Ehrnrooth Foundation, Finland. The support from the European Science Foundation under the ESF COST Action IC1104 is also gratefully acknowledged. ## References * [1] A. D. Wyner, “The wire-tap channel”, _Bell System Technical Journal_, Vol. 54, Oct 1975. * [2] F. Oggier, P. Solé and J.-C. Belfiore, “Lattice codes for the wiretap Gaussian channel: construction and analysis”, arXiv 1103.4086v3, 2013. * [3] E. Viterbo and F. Oggier, “Algebraic number theory and code design for Rayleigh fading channels”, _Foundations and Trends in Communications and Information Theory_, Vol. 1, No. 3, Now Publishers Inc., 2004. * [4] J. H. Conway and N. J. A. Sloane, _Sphere packings, lattices and groups_, Springer-Verlag, 1988. * [5] C. Hollanti and K. Ranto, “Maximal orders in space–time coding: construction and decoding”, _International Symposium on Information Theory and Its Applications ISITA 2008_, pp.1–5, 7-10 Dec. 2008. * [6] K. R. Kumar and G. Caire, “Space–time codes from structured lattices”, _IEEE Transactions on Information Theory_, vol. 55, no. 2, pp. 547–556, Feb. 2009. * [7] W. Ebeling, _Lattices and codes. A course partially based on lectures by Friedrich Hirzebruch_, 3rd Ed., Springer Spectrum, 2013. * [8] J. Boutros, E. Viterbo, C. Rastello and J.-C. Belfiore, “Good lattice constellations for both Rayleigh fading and Gaussian channels”, _IEEE Transactions on Information Theory_, vol. 42, no 2, March 1996.
1802.04276	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 71883, "num_imgs": 14, "llama3_tokens_count": 21523 }	[ "content_image/1802.04276/x1.png", "content_image/1802.04276/x2.png", "content_image/1802.04276/x5.png", "content_image/1802.04276/x6.png", "content_image/1802.04276/x10.png", "content_image/1802.04276/x11.png", "content_image/1802.04276/x12.png", "content_image/1802.04276/x13.png", "content_image/1802.04276/x14.png", "content_image/1802.04276/x15.png", "content_image/1802.04276/x16.png", "content_image/1802.04276/x17.png", "content_image/1802.04276/x18.png", "content_image/1802.04276/x19.png" ]	# Black-hole kicks from numerical-relativity surrogate models Davide Gerosa dgerosa@caltech.edu François Hébert fhebert@caltech.edu Leo C. Stein leostein@tapir.caltech.edu TAPIR 350-17, California Institute of Technology, 1200 E California Boulevard, Pasadena, CA 91125, USA March 2, 2024 ###### Abstract Binary black holes radiate linear momentum in gravitational waves as they merge. Recoils imparted to the black-hole remnant can reach thousands of km/s, thus ejecting black holes from their host galaxies. We exploit recent advances in gravitational waveform modeling to quickly and reliably extract recoils imparted to generic, precessing, black-hole binaries. Our procedure uses a numerical-relativity surrogate model to obtain the gravitational waveform given a set of binary parameters; then, from this waveform we directly integrate the gravitational-wave linear momentum flux. This entirely bypasses the need for fitting formulas which are typically used to model black-hole recoils in astrophysical contexts. We provide a thorough exploration of the black-hole kick phenomenology in the parameter space, summarizing and extending previous numerical results on the topic. Our extraction procedure is made publicly available as a module for the Python programming language named surrkick. Kick evaluations take $\sim 0.1$ s on a standard off-the-shelf machine, thus making our code ideal to be ported to large-scale astrophysical studies. † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction Gravitational waves (GWs) carry energy, linear momentum, and angular momentum, and are therefore responsible for the final evolutionary stages of compact binary systems. As energy and angular momentum are dissipated away, the two objects inspiral and eventually merge. The GW-driven orbital decay of two neutron stars was first observed by pulsar timing, leading to a major confirmation of Einstein’s theory of general relativity Taylor and Weisberg (1982). The first landmark detection of GWs was from a binary black hole (BH) which was brought to merger by those same GWs that ultimately reached our detectors Abbott, B. P. _et al._ (LIGO and Virgo Scientific Collaboration) (2016). Similar to how the dissipation of energy and angular momentum causes the orbit of a BH binary to shrink, the emission of linear momentum through GWs causes the binary’s center of mass to recoil Bonnor and Rotenberg (1961); Peres (1962). The key property to generate a GW recoil (or “kick”) is asymmetry. It is straightforward to show that symmetry prevents linear momentum dissipation during the inspiral and merger of equal-mass, nonspinning BHs. Conversely, a generic BH binary radiates GWs anisotropically: linear momentum is preferentially emitted in some direction, and the binary consequently recoils. BH kicks were first studied using the post-Newtonian (PN) approximation (e.g., Refs Fitchett (1983); Kidder (1995); Blanchet _et al._ (2005)) but their full astrophysical relevance was only realized after numerical relativity (NR) simulations of BH mergers became possible Pretorius (2005); Campanelli _et al._ (9); Baker _et al._ (2006). Most of the linear momentum is emitted during the last few orbits and merger, which corresponds to the highly dynamical, fully nonlinear regime that can only be captured with NR simulations. In particular, simulations showed that BHs formed following a merger may be imparted recoil velocities of up to $5000$ km/s Campanelli _et al._ (11); González _et al._ (12); Tichy and Marronetti (2007); Lousto and Zlochower (2011). The striking astrophysical consequences of these findings were quickly realized (e.g., Refs. Sesana (2007); Schnittman and Buonanno (2007); Gualandris and Merritt (2008); Shields and Bonning (2008); Holley-Bockelmann _et al._ (2008); Blecha and Loeb (2008)): BH recoils might exceed the escape speed of even the most massive galaxies in the Universe Merritt _et al._ (2004); Gerosa and Sesana (2015), thus making galactic ejections a possible outcome of binary mergers Redmount and Rees (1989). Recoiling BHs might give rise to a variety of electromagnetic signatures Komossa (2012) —notably a kinematical offset of a set of broad emission lines— which led to the identifications of a few observational candidates Komossa _et al._ (2008); Civano _et al._ (2012); Decarli _et al._ (2014); Koss _et al._ (2014); Chiaberge _et al._ (2017); Kim _et al._ (2017); Kalfountzou _et al._ (2017) (see also Refs. Lena _et al._ (2014); Raffai _et al._ (2016); Blecha _et al._ (2016) for detection strategies). As the system recoils, a Doppler shift of the emitted GWs can provide a possible direct observational signature of BH kicks within the reach of future space- and ground-based GW observatories Gerosa and Moore (2016). Since NR simulations are far too expensive to be performed in astrophysical population studies, BH kicks have mostly been modeled using fitting formulas based on PN theory and calibrated to NR simulations (e.g., Refs. Campanelli _et al._ (36); González _et al._ (37); Lousto and Zlochower (2008, 2008); Lousto _et al._ (2012); Lousto and Zlochower (2013)). These “black box” expressions return the final kick of the BH remnant given the intrinsic parameters (mass ratio and spins) of the merging binary at some initial separation. Another so far unexplored possibility to model BH kicks is to compute the flux of linear momentum in GWs using a waveform approximant that can be quickly evaluated in parameter space. Linear momentum dissipation, however, is encoded in both differences between the dominant $l=2,m=\pm 2$ modes and higher harmonics ($l>2$) Brügmann _et al._ (2008). This approach, therefore, requires an inspiral-merger-ringdown approximant able to model both higher harmonics (crucial to linear momentum flux) and misaligned spins (which are known to generate the largest kicks). In this paper we present the first attempt in this direction using the recent NR surrogate model by Blackman _et al._ Blackman _et al._ (42) — the first waveform approximant able to model generic precessing systems with higher harmonics. In contrast with the available fitting formulas, our procedure provides not only the final kick speed $v_{k}$, but also the entire velocity accumulation profile $\mathbf{v}(t)$. We present a thorough exploration of BH recoils for generic systems, which summarizes and extends various previous findings in a coherent fashion. Our numerical code, surrkick, is publicly available and allows for reliable computation of the radiated quantities (energy, linear momentum, and angular momentum) at a moderate computational cost. Our implementation is therefore ideal to be ported to larger-scale astrophysical codes which require fast estimates of BH kicks, such as galaxy merger-tree simulations, populations synthesis studies, and GW event-rate predictions. This paper is organized as follows. Section II introduces the main tools of our analysis. Section III presents results and comparisons with other methods. Section IV explores the numerical accuracy of our procedure. Section V briefly describes the implementation and usage of our public code. Section VI draws conclusions and future prospects. Unless otherwise stated, we use relativists’ units $c=G=1$. ## II Methods ### Numerical-relativity surrogate models Surrogate models interpolate a set of precomputed GW signals and make use of advanced decomposition and interpolation schemes to quickly produce waveforms for any desired point in parameter space. Surrogate models are typically optimized to accurately reproduce the complex gravitational-wave strain, here expanded in terms of spin-weighted spherical harmonics Thorne (1980) \[h(t,\theta,\phi,\boldsymbol{\lambda}) =h_{+}(t,\theta,\phi,\boldsymbol{\lambda})-ih_{\times}(t,\theta, \phi,\boldsymbol{\lambda})\] \[=\sum_{l=2}^{\infty}\sum_{m=-l}^{+l}h^{lm}(t,\boldsymbol{\lambda} )\;_{-2}Y_{lm}(\theta,\phi)\,,\] (1) where $t$ denotes time, $\theta$ and $\phi$ describe the GW propagation direction, and the symbol $\boldsymbol{\lambda}$ encodes all the binary’s intrinsic parameters. For quasicircular BH binaries, these are the mass ratio $q$ and spin vectors $\boldsymbol{\chi_{1}},\boldsymbol{\chi_{2}}$ (the total mass $M$ is a free scale). Surrogate models have been presented for both effective-one-body Field _et al._ (2014); Pürrer (2014, 2016) and NR waveforms Blackman _et al._ (47, 47, 42). In this paper we use the NR waveform surrogate model NRSur7dq2 Blackman _et al._ (42) to generate our waveforms. NRSur7dq2 is the very first model able to cover the seven-dimensional parameter space describing generic precessing systems. NRSur7dq2 is trained on 886 NR waveforms generated with the Spectral Einstein Code (SpEC) Kidder _et al._ (2000) and interpolated using the technique put forward in Ref. Field _et al._ (2014). It provides modes $h^{lm}$ up to $l\leq 4$ for binaries with mass ratios $q=m_{2}/m_{1}\in[0.5,1]$ and dimensionless spin magnitudes $\chi_{1},\chi_{2}\in[0,0.8]$; updates to extend its validity range are under active development. The model has been shown to be extremely accurate at reproducing the gravitational-wave strain $h$: it outperforms all other available waveform approximants by several orders of magnitude, reaching a level of accuracy comparable to the NR simulations used in the training process Blackman _et al._ (42). Waveforms generated with NRSur7dq2 span the time range $-4500M\leq t\leq 100M$, where $t=0$ is defined as the time that maximizes the total waveform amplitude $\mathcal{A}^{2}(t)=\sum_{l,m}\|h^{lm}(t)\|^{2}$. The initial time $t=-4500M$ corresponds to about $20$ orbits before merger and the final value $t=100M$ allows for a full dissipation of the signal. Values of $h^{lm}$ are computed at carefully selected time nodes Blackman _et al._ (42) and later interpolated in time using standard cubic univariate B-splines. More specifically, NRSur7dq2 provides the distance-independent dimensionless strain, extrapolated to $\mathcal{I}^{+}$, i.e. $\lim_{r\to\infty}rh/M$ where $r$ is the distance from the binary’s center of mass and $M$ is the total mass of the binary at the beginning of the evolution. NRSur7dq2 allows for the spin directions to be specified at a reference time $-4500M\leq t_{\rm ref}\leq-100M$, in a frame defined such that the more (less) massive BH sits on the positive (negative) x-axis and the Newtonian orbital angular momentum $\mathbf{L}$ lies along the z-axis. Unless otherwise stated, we use $t_{\rm ref}=-100M$. ### Radiated energy and momenta Multipolar expansions for the radiated energy, linear momentum and angular momentum have been worked out in detail in Ref. Ruiz _et al._ (2008) (derived from Refs. Thorne (1980); Lousto and Zlochower (2007)). We report their expressions here for completeness.¹ Whenever terms with $l<2$ or $\|m\|>l$ are present in the following summations, their coefficients are intended to be zero. In practice, one is also limited to $l\leq l_{\rm max}$ (where, e.g., $l_{\rm max}=4$ for NRSur7dq2 waveforms and $l_{\rm max}=8$ for SpEC waveforms). [FOOTNOTE:1][ENDFOOTNOTE] The energy flux emitted in GWs is provided in terms of the first time derivative of the complex strain $\dot{h}$ and reads: \[\frac{dE}{dt}=\lim_{r\rightarrow\infty}\frac{r^{2}}{16\,\pi}\sum_ {l,m}\,\left\|\dot{h}^{l,m}\right\|^{2}\;.\] (2) When integrating to obtain $E(t)$ we set the integration constant $E_{0}$ to account for the binding energy dissipated in GWs at times $t<-4500M$, before the start of our waveforms, thus enforcing $\lim_{t\to-\infty}E(t)=0$. A straightforward Newtonian calculation yields Peters (1964) \[\frac{E_{0}}{M}=\left(\frac{5}{1024}\frac{q^{3}}{(1+q)^{6}}\dot{E}_{0}\right)^ {1/5},\] (3) where $\dot{E}_{0}$ is estimated from Eq. (2) by averaging over the first $100M$ in time. We have verified that corrections up to 2PN (including spin effects Arun _et al._ (2009)) have a negligible impact on $E_{0}$. One can then define the time-dependent (Bondi) mass of the binary, \[M(t)=M-E(t)+E_{0}\,,\] (4) such that $M(t)$ at the beginning of our waveforms is equal to $M$. The mass of the post-merger BH in units of the total mass of the binary at early times is \[\frac{\lim_{t\to+\infty}M(t)}{{\lim_{t\to-\infty}M(t )}}=1-\frac{\lim_{t\to+\infty}E(t)}{M+E_{0}}.\] (5) The emitted linear momentum is also fully specified by $\dot{h}$ and crucially includes mixing between modes with different $l$ and $m$: \[\frac{dP_{x}}{dt}= \lim_{r\to\infty}\frac{r^{2}}{8\,\pi}\Re\Bigg{[}\sum_{l,m}\,\dot{ h}^{l,m}\Big{(}a_{l,m}\,\dot{\bar{h}}^{l,m+1}\] \[+b_{l,-m}\,\dot{\bar{h}}^{l-1,m+1}-b_{l+1,m+1}\,\dot{\bar{h}}^{l+ 1,m+1}\Big{)}\Bigg{]}\;,\] (6) \[\frac{dP_{y}}{dt}= \lim_{r\to\infty}\frac{r^{2}}{8\,\pi}\Im\Bigg{[}\sum_{l,m}\,\dot{ h}^{l,m}\Big{(}a_{l,m}\,\dot{\bar{h}}^{l,m+1}\] \[+b_{l,-m}\,\dot{\bar{h}}^{l-1,m+1}-b_{l+1,m+1}\,\dot{\bar{h}}^{l+ 1,m+1}\Big{)}\Bigg{]}\;,\] (7) \[\frac{dP_{z}}{dt}= \lim_{r\to\infty}\frac{r^{2}}{16\pi}\sum_{l,m}\,\dot{{h}}^{l,m} \Big{(}c_{l,m}\,\dot{\bar{h}}^{l,m}\] \[+d_{l,m}\,\dot{\bar{h}}^{l-1,m}+d_{l+1,m}\,\dot{\bar{h}}^{l+1,m} \Big{)}\;,\] (8) where the upper bar denotes complex conjugation and \[a_{l,m} = \frac{\sqrt{(l-m)\,(l+m+1)}}{l\,(l+1)}\;,\] (9) \[b_{l,m} = \frac{1}{2\,l}\,\sqrt{\frac{(l-2)\,(l+2)\,(l+m)\,(l+m-1)}{(2l-1)( 2l+1)}}\;,\] (10) \[c_{l,m} = \frac{2\,m}{l\,(l+1)}\;,\] (11) \[d_{l,m} = \frac{1}{l}\,\sqrt{\frac{(l-2)\,(l+2)\,(l-m)\,(l+m)}{(2l-1)(2l+1) }}\;.\] (12) The integration constant for the $d\mathbf{P}/dt$ integration is chosen so that the average of $\mathbf{P}$ over the first $1000M$ in time, where linear momentum emission is expected to be negligible, is zero. By conservation of linear momentum, the time profile of the kick imparted to the system is² [FOOTNOTE:2][ENDFOOTNOTE] \[\mathbf{v}(t)=-\frac{{P_{x}}(t)\mathbf{\hat{x}}+{P_{y}}(t)\mathbf{\hat{y}}+{P_ {z}}(t)\mathbf{\hat{z}}}{M(t)}\,,\] (13) and the final velocity of the post-merger remnant BH is \[\mathbf{v_{k}}=\lim_{t\to\infty}\mathbf{v}(t)\,.\] (14) One can further integrate $\mathbf{v}(t)$ in time to obtain the trajectory $\mathbf{x}(t)=\int\mathbf{v}(t)dt$. Although the binary trajectory is a coordinate-dependent notion, the time integral of the linear momentum dissipated in GWs can be interpreted as the motion of the spacetime’s center of mass seen by an observer at $\mathcal{I}^{+}$ Flanagan and Nichols (2017). The angular momentum carried by GWs involves both $h$ and $\dot{h}$: \[\frac{dJ_{x}}{dt}= \lim_{r\rightarrow\infty}\frac{r^{2}}{32\pi}\:\Im\Bigg{[}\sum_{l, m}\,h^{l,m}\Big{(}f_{l,m}\,\dot{\bar{h}}^{l,m+1}\] \[+f_{l,-m}\,\dot{\bar{h}}^{l,m-1}\Big{)}\Bigg{]}\;,\] (15) \[\frac{dJ_{y}}{dt}= -\lim_{r\rightarrow\infty}\frac{r^{2}}{32\pi}\:\Re\Bigg{[}\sum_{l ,m}\,h^{l,m}\Big{(}f_{l,m}\,\dot{\bar{h}}^{l,m+1}\] \[-f_{l,-m}\,\dot{\bar{h}}^{l,m-1}\Big{)}\Bigg{]}\;,\] (16) \[\frac{dJ_{z}}{dt}= \lim_{r\rightarrow\infty}\frac{r^{2}}{16\pi}\:\Im\Bigg{[}\sum_{l, m}\,m\,h^{l,m}\,\dot{\bar{h}}^{l,m}\Bigg{]}\;,\] (17) where \[f_{l,m}=\sqrt{l(l+1)-m(m+1)}\;.\] (18) When integrating $d\mathbf{J}/dt$, we do not adjust the integration constant to account for the angular momentum radiated before the beginning of our waveforms. Contrary to the binding energy, the Newtonian angular momentum of a binary system diverges as separation grows ($J\propto\sqrt{r}$). We perform all differentiations and integrations required to extract these radiated quantities analytically on the spline interpolants provided by NRSur7dq2, over the range $-4500M\leq t\leq 100M$. The $t\to\infty$ limits [e.g. Eqs. (5) and (14)] are approximated with values at $t=100M$. ## III Results ### Anatomy of the kick Nonspinning BH binaries do not receive any recoil for both $q=1$ (because of symmetry) and $q=0$ (which corresponds to the test-particle limit). Recoils are present in between these two limits. Figure 1 shows the kick profile $\mathbf{v}(t)$ for a series of BH mergers with $q=0.5,\dots,1$. Axisymmetry prevents linear momentum dissipation along the direction of the orbital angular momentum, i.e. $\mathbf{v}(t)\cdot\mathbf{\hat{z}}=0$ (within numerical errors; see Sec. IV.1). The binary’s center of mass oscillates in the orbital plane x-y during the inspiral, until the merger halts these oscillations and imparts the final recoil. The kick velocity grows as $q$ decreases, reaching $v_{k}\simeq 148$ km/s for $q=0.5$. The largest kick achievable for a nonspinning system is $v_{k}\simeq 175$ km/s and corresponds to $q\sim 0.36$ González _et al._ (37), which is outside the parameter space currently covered by NRSur7dq2. The trajectory of the spacetime’s center of mass $\mathbf{x}(t)$ for $q=0.5$ and $\chi_{1}=\chi_{2}=0$ is shown in the left panel of Fig. 2. One last oscillation occurs after merger, and is responsible for most of the kick. This effect is also visible in Fig. 1, where we see the system typically accelerates at $t\sim 10M$ after merger, with the final burst of linear momentum radiation lasting only for a few $M$ in time. Interestingly, the projection of the recoil profile along the final kick direction $\mathbf{v}(t)\cdot\mathbf{\hat{v}_{k}}$ is not monotonic after merger: the binary suddenly decelerates at about $t\sim 15M$, after which the imparted velocity settles down to the asymptotic value $v_{k}$. This effect has been dubbed _antikick_ Rezzolla _et al._ (2010), and turns out to be a rather generic feature of BH mergers (cf. Sec. III.2 below). <figure><img src="content_image/1802.04276/x1.png"><figcaption>Figure 1: Kick profile v(t) projected along ^x, ^y, ^z and the direction ofthe final kick ^vk for a series of non-spinning BH binaries with mass ratioranging from q=0.5 (light orange) to q=1 (black). The binary’s center of massoscillates in the orbital plane during the inspiral; the final recoil isimparted with a sudden acceleration at t∼10M after the peak-amplitude time.</figcaption></figure> <figure><img src="content_image/1802.04276/x2.png"><figcaption>Figure 2: Center-of-mass trajectory x(t)=∫v(t)dt for three binaryconfigurations as described in the legends. The circle markers on each curvecorrespond to t=0. The left panel shows a recoil due to mass asymmetry only:the center of mass oscillates in the orbital plane during the inspiral and isfinally pushed after merger. The middle panel shows a complicated interplay ofmass and spin asymmetry, with the initial oscillations being greatly distortedat merger by the superkick effect. Finally, the right panel shows the simplertrajectory of a binary receiving a very large kick of ∼3000 km/s. An animatedversion of this figure is available at[davidegerosa.com/surrkick](https://davidegerosa.com/surrkick).</figcaption></figure> <figure><img src="content_image/1802.04276/x5.png"><figcaption>Figure 3: Radiated energy E(t) for binaries with mass ratio q=0.5 and spins ofmagnitude χ1=χ2=0.8 (anti)aligned to the orbital angular momentum. Fourconfigurations are shown —up-up, down-down, up-down, down-up— where the termbefore (after) the hyphen refers to the spin of the heavier (lighter) BH beingco-/counter-aligned with the binary’s orbital angular momentum. Forcomparison, we also show E(t) for a non-spinning system with the same massratio. Because of the orbital hang-up effect, BH binaries with (anti-)alignedspins radiate more (less) energy compared to non-spinning systems with thesame mass ratio.</figcaption></figure> <figure><img src="content_image/1802.04276/x6.png"><figcaption>Figure 4: Kick profile v(t) projected along ^x, ^y, ^z and the direction ofthe final kick ^vk for binaries with mass ratio q=1 (left) and q=0.5 (right),and spins of magnitude χ1=χ2=0.8 (anti)aligned to the orbital angularmomentum. Four configurations are shown: up-up, down-down, up-down, down-up,where the term before (after) the hyphen refers to the spin of the heavier(lighter) BH being co-/counter-aligned with the binary’s orbital angularmomentum. Kicks from non-precessing systems lie in the binary’s orbital plane,with the spin kicks being more pronounced for the up-down and down-upconfigurations in accordance with PN predictions.</figcaption></figure> BH spins introduce additional sources of linear momentum dissipation. The impact of aligned spins on the radiated energy and linear momentum profile is illustrated in Figs. 3 and 4, respectively. In particular, we study BH binaries with spin magnitude $\chi_{1}=\chi_{2}=0.8$ and four different spin orientations: $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{z}}=\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{z}}=1$ (up-up), $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{z}}=\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{z}}=-1$ (down-down), $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{z}}=-\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{z}}=1$ (up-down), $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{z}}=-\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{z}}=-1$ (down-up), where $\mathbf{\hat{z}}=\mathbf{\hat{L}}$ at $t_{\rm ref}=-100M$. Although the up-down configuration is generically unstable to spin precession Gerosa _et al._ (55), the instability develops on longer timescales and can therefore be neglected in this context. The orbital hang-up effect Damour (2001); Campanelli _et al._ (57); Scheel _et al._ (2015) causes binaries with spins co- (counter-) aligned with the binary’s angular momentum to merge later (sooner) compared to non-spinning systems with the same mass ratio. Consequently, the energy emitted in GWs increases (decreases) if the total spin $\mathbf{S}=m_{1}^{2}\boldsymbol{\chi_{1}}+m_{2}^{2}\boldsymbol{\chi_{2}}$ is (anti-)aligned with $\mathbf{L}$ (c.f. Fig. 3). For $q=1$ (Fig. 4, left panel), moderately large recoils of $v_{k}\sim 350$ km/s are achieved for the up-down and down-up configurations, in agreement with the PN predictions $v_{k}\propto\|\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{L}}-\boldsymbol{\hat {\chi}_{2}}\cdot\mathbf{\hat{L}}\|$ Kidder (1995) (see Baker _et al._ (2007); Lousto and Zlochower (2008) for numerical explorations). The recoil is mostly imparted in the orbital plane, but its magnitude is somewhat smaller than the mass-asymmetry case explored above and reduces to a single burst of linear momentum emitted at $t\sim 10M$, preceded by a smaller one in the opposite direction at $t\sim-5M$. The $q=1$ up-up configuration presents some linear momentum emitted perpendicular to the orbital plane, resulting in $v_{k}\sim 50$ km/s. This is the inherent error scale in our model, as symmetry implies $v_{k}=0$ for both the up-up and down-down configuration at $q=1$ Boyle _et al._ (2008); Boyle and Kesden (2008), see Sec. IV.1. For binaries with unequal masses and aligned spins (Fig. 4, right panel), both the orbital hang-up and the mass asymmetry effect are present: the binary’s center of mass first oscillates in the orbital plane (because $q\neq 1$) and then receive a further push at $t\sim 10M$ (because $\boldsymbol{\chi_{i}}\cdot\mathbf{\hat{z}}\neq 0$). [FIGURE:S3.F5][ENDFIGURE] <figure><img src="content_image/1802.04276/x10.png"><figcaption>Figure 6: Left panel: Recoil velocities for a series of right-left binarieswith q=1 and χi=0.8 initialized at various reference times tref; the orangecircle marks the reference time used in Fig. 5. Right panel: Recoil velocitiesfor BH binaries with q=1 and χ1=−χ2=[0.8cosα,0.8sinα,0] (such that α=0corresponds to the right-left configuration) at tref=−100M. The angle αcorresponds to a rotation of both spins about the orbital angular momentum,and is degenerate with the reference time at which spins are specified. Graycrosses mark the same configuration in both panels.</figcaption></figure> The largest kicks are achieved for BHs merging with misaligned spins Campanelli _et al._ (11); González _et al._ (12); Campanelli _et al._ (36); Tichy and Marronetti (2007); Brügmann _et al._ (2008); Lousto and Zlochower (2011). Figure 5 shows kick profiles for four binary configurations with spins $\chi_{i}=0.8$ lying in the orbital plane: $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{x}}=\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{x}}=1$ (right-right), $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{x}}=\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{x}}=-1$ (left-left), $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{x}}=-\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{x}}=1$ (right-left), $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{x}}=-\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{x}}=-1$ (left-right), where $\mathbf{\hat{x}}$ is defined as the axis connecting the lighter to the heavier BH at $t_{\rm ref}$. For reasons clarified below, here we take $t_{\rm ref}=-125M$. Kicks as large as $\sim 2820$ km/s are achieved for the right-left and left-right configurations, which correspond to the _superkick_ scenario discovered in Refs. Campanelli _et al._ (11); González _et al._ (12). During the inspiral, frame dragging from the two holes acts constructively and pushes the binary’s center of mass up and down along the direction of the orbital angular momentum $\mathbf{\hat{z}}$. The final kick is imparted as the BHs merge and the last of these oscillations is abruptly interrupted. The phenomenology is rather similar to the case of aligned spins studied above, although with the key difference that in this case linear momentum is emitted along the binary’s orbital angular momentum, not orthogonal to it. It is worth noting that binaries with these large kicks present a remarkably simple accumulation profile: the acceleration $d\mathbf{P}/dt$ is well described by a Gaussian centered at $t\sim 10M$ with width $\sigma\sim 5M$ (cf. Brügmann _et al._ (2008) and Sec. III.2 below). Conversely, frame dragging from the two BHs add destructively for the right-right and left-left binaries. This cancellation is perfect (within numerical errors, cf. Sec. IV.1) if the two spins have the same magnitude $m_{1}^{2}\chi_{1}=m_{2}^{2}\chi_{2}$ (Fig. 5, left panel). For $q=0.5$ and $\chi_{i}=0.8$ (Fig. 5, right panel), the dynamics is dominated by the largest spin and the four configurations reach values between 650 and 1530 km/s. Interestingly, smaller mass ratios excite a sizable kick along the orbital plane of $\sim 300$ km/s, which exceed the recoil imparted to nonspinning systems with the same $q$ of about a factor $\sim 2$ (cf. Fig. 1). The spacetime trajectory $\int\mathbf{v}(t)dt$ for one such binary is illustrated in the middle panel of Fig. 2: the center of mass oscillates at early time, undergoes a complicated motion right before merger, after which the superkick effect becomes dominant. To the best of our knowledge, this mass-spin asymmetry mixing in the kick profile has not been reported elsewhere. <figure><img src="content_image/1802.04276/x11.png"><figcaption>Figure 7: Velocity accumulation profile v(t) projected along the direction ofthe final kick ^vk for binaries with q=1 and antiparallel spins of magnitudeχ1=χ2=0.8 lying in the orbital plane. The rotation angle α (defined ascosα=^χ1⋅^x=−^χ2⋅^x) controls the orbital phase at merger and thus sets thevelocity of the center of mass when the final kick is imparted. Curves arecolored according to α as it spans from −π (black) to π (orange).</figcaption></figure> Superkick velocities critically depend on the orbital phase at merger, as it controls the abrupt interruption of the oscillatory behavior described above. In the left panel of Fig. 6 we study a series of right-left binaries ($q=1$, $\chi_{1}=\chi_{2}=0.8$, $\boldsymbol{\hat{\chi}_{1}}\cdot\mathbf{\hat{x}}=-\boldsymbol{\hat{\chi}_{2}} \cdot\mathbf{\hat{x}}=1$) specified at various reference times $t_{\rm ref}/M\in[-250,-100]$. The final kick velocity $v_{k}$ shows a clear sinusoidal dependence, as already found in, e.g., Refs. Brügmann _et al._ (2008); Lousto _et al._ (2012); Zlochower and Lousto (2015). The peaks (e.g. at $t\sim-125M$) correspond to configurations for which the center-of-mass velocity happens to be at its maximum when the last oscillation is interrupted. The orbital phase at merger can also be controlled by an overall rotation of both spins about the orbital angular momentum. The right panel of Fig. 6 shows $v_{k}$ for binaries with spins $\boldsymbol{\hat{\chi}_{1}}=-\boldsymbol{\hat{\chi}_{2}}=[\cos\alpha,\sin \alpha,0]$ specified at $t_{\rm ref}=-100M$ (a similar series of NR simulations was reported in Ref. Brügmann _et al._ (2008)). The right-left (left-right) configuration corresponds to $\alpha=0$ ($\pi$). The two curves in Fig. 6 span the very same range, showing that the angle $\alpha$ and the reference time $t_{\rm ref}$ are indeed degenerate. In practice, this means that only binaries with a specific orbital phase at merger are subject to superkicks, thus making their occurrence very rare. Figure 7 shows the velocity accumulation profile for the same series of binaries with different values of $\alpha$: the BH merger abruptly stops the center-of-mass oscillation at different phases, thus setting the final kick velocities. As first noted in Refs. Lousto and Zlochower (2011, 2013), binaries with partially aligned spins give rise to BH kicks even larger than those imparted to binaries in the superkick configuration. Equal-mass, maximally spinning BH binaries are predicted to reach $v_{k}\sim 5000$ km/s for spins misaligned by angles $\theta_{i}=\cos^{-1}(\boldsymbol{\hat{\chi}_{i}}\cdot\mathbf{L})\sim 50^{\circ}$. These recoils were dubbed _hang-up kicks_, and are due to a combination of the BH frame-dragging addition (responsible for superkicks) and the orbital hang-up effect (which enhances the energy radiated in GWs for aligned spins). To check that our model reproduces these hang-up kicks, we generate $10^{5}$ binaries with $q=1$, $\chi_{1}=\chi_{2}=0.8$, and isotropic spin orientations. The largest kick detected is $v_{k}\sim 3300$ km/s, and is obtained for $\theta_{1}\sim\theta_{2}\sim 57^{\circ}$. For the same values of $q$, $\chi_{1}$ and $\chi_{2}$, the hang-up kick fitting formula of Refs. Lousto and Zlochower (2011, 2013) returns a largest kick of $\sim 3500$ km/s (a more careful comparison is postponed to Sec. III.2). The spacetime trajectory corresponding to one of these cases is shown in the right panel of Fig. 2, confirming our earlier claims that large kicks present rather simple accumulation profiles. <figure><img src="content_image/1802.04276/x12.png"><figcaption>Figure 8: Kick profiles for a right-left binary with q=0.5 and χ1=χ2=0.8projected along various random directions ^n. Curves are colored from black toorange according to the final projected kick limt→∞v(t)⋅^n.</figcaption></figure> Finally, Fig. 8 explores projection effects of the kick accumulation profile. For a single system with $q=0.5$ and $\chi_{1}=\chi_{2}=0.8$ in the right-left configuration, we show the projection of $\mathbf{v}(t)$ along various randomly chosen directions $\mathbf{\hat{n}}$. Although some features are solid, the kick profile appears rather different if viewed from different orientations. This behavior is important to model BHs recoiling into astrophysical environments with well-defined geometries, such as accretion disks Rossi _et al._ (2010); Corrales _et al._ (2010), and to implement the effect of the BH kick in waveform models through the induced Doppler shift Gerosa and Moore (2016). ### Statistical exploration and comparison with fitting formulas After exploring the main features of the kick profile in controlled scenarios, we now turn our attention to statistical samples. We generate a sample of $10^{6}$ binaries with mass ratio uniform in $q\in[0.5,1]$ and spins uniformly distributed in volume with magnitude $\chi_{i}\leq 0.8$. Figure 9 shows the distributions of total energy, linear momentum, and angular momentum radiated in GWs by this BH binary population. The energy and angular momentum distributions are roughly symmetric, with peaks at $E\sim 0.045M$ and $J\sim 0.45M^{2}$, respectively. The recoil distribution peaks at $v_{k}\sim 0.001c$, with a long tail extending up to $v_{k}\sim 0.01c\sim 3000$ km/s. Figure 9 also shows predictions for $v_{k}$ obtained with fitting formulas currently available in the literature. In particular, we use the expressions summarized in Ref. Gerosa and Kesden (2016), which are calibrated on various numerical simulations from Refs. Campanelli _et al._ (36); González _et al._ (37); Lousto and Zlochower (2008); Lousto _et al._ (2012); Lousto and Zlochower (2013, 2008). Although kick predictions for individual binaries might differ significantly, the two methods largely agree on the overall distribution. We note, however, that the fitting formula tends to overestimate the number of binaries receiving large recoils. In particular, the fractions of binaries with $v_{k}>2000$ km/s are $\sim 2.4$% and $\sim 3.2$% for the surrogate extraction and fitting formula, respectively. The largest kicks found in these distributions are $v_{k}\sim 3160$ km/s (surrogate) and $v_{k}\sim 3330$ km/s (fit). We speculate that this disagreement might be due to the calibration of the hang-up kick terms in the fitting formula, which was only performed with $q=1$ simulations (cf. Ref. Sperhake (2015) for a critical discussion on this point). Although some runs for unequal-mass binaries with largely misaligned spins have been presented Campanelli _et al._ (36); Baker _et al._ (2008); Lousto and Zlochower (2009); Zlochower and Lousto (2015), the effect of the mass ratio on the largest kick might not be fully captured by the expressions currently available. Figure. 9 also compares the total radiated energy extracted from the surrogate model against the final-mass fitting formula of Barausse _et al._ (2012), corrected according to Eq. (5). Agreement is found at the $\sim 2\%$ level: the median for the surrogate (fit) estimate of $E/M$ is $\sim 0.047$ ($\sim 0.046$) with standard deviations of $\sim 0.008$ ($\sim 0.009$). The authors of Ref. Jiménez-Forteza _et al._ (2017) presented a careful analysis comparing different estimates of the energy radiated following BH mergers and reported similar, if not higher, differences between various approaches. <figure><img src="content_image/1802.04276/x13.png"><figcaption>Figure 9: Distribution of radiated linear momentum vk (left panel), energy E(top right panel) and angular momentum J (bottom right panel) for adistribution of binaries with mass ratio uniformly distributed in [0.5,1] andspin of magnitude χi<0.8 uniformly distributed in volume. Our results(“Surrogate”) are compared to the model summarized inRef. Gerosa and Kesden(2016) based on Refs. Campanelli _et al._ (2007b); González _et al._(2007b); Lousto and Zlochower (2008); Lousto _et al._ (2012); Lousto andZlochower (2013, 2008) (“Fitting formula”): the two distributions largelyagree, although differences are present for large values of vk.</figcaption></figure> <figure><img src="content_image/1802.04276/x14.png"><figcaption>Figure 10: Kick profiles v(t) for a sample of BH binaries with uniform massratio and isotropic spin directions projected along random directions ^n.Curves are normalized according to the final projected kick vk⋅^n and arecolored according to the total kick magnitude vk. The dashed blue linecorresponds to a Gaussian acceleration profile of width σ=8M centered att=10M, which well approximates the largest kick in our sample. Smaller kicksrequire more complicated profiles to be modeled carefully.</figcaption></figure> In order to highlight the “shape” of the kick, Fig. 10 shows 200 velocity accumulation profiles $\mathbf{v}(t)$ from the same binary distribution projected along random directions $\mathbf{\hat{n}}$ and normalized to the value of the final kick $\mathbf{v_{k}}\cdot\mathbf{\hat{n}}$. Despite the remarkable complexity explored above, the kick accumulation profiles present very robust features. In particular, profiles are simpler for binaries receiving large recoils, for which the acceleration $d\mathbf{v}/dt\cdot\mathbf{\hat{n}}$ is well approximated by a single Gaussian with mean $t=10M$ and width $\sigma=8M$. Smaller kicks, on the other hand, present more complicated profiles which typically include an antikick Rezzolla _et al._ (2010). These findings corroborate the approach of Ref. Gerosa and Moore (2016), where $\mathbf{v}(t)\cdot\mathbf{\hat{n}}$ was modeled with a basis of damped oscillatory functions. We stress that the population explored here is far from being astrophysically relevant. Astrophysical processes (such as the Bardeen-Petterson effect in the case of disk accretion Bardeen and Petterson (1975) and tidal interactions for stellar-mass BH progenitors Hut (1981)) deeply modify the BH spin orientations, thus affecting the expected kick distribution Lodato and Gerosa (2013); Miller and Krolik (2013); Gerosa _et al._ (75). Moreover, PN effects in the long inspiral before merger have been shown to preferentially suppress or enhance recoils in specific regions of the parameter space Kesden _et al._ (2010); Gerosa _et al._ (75). ## IV Accuracy ### Exploiting symmetries Before presenting a detailed comparison with NR simulations, we first perform internal tests of our kick extraction procedure by leveraging the symmetries of the problem. For instance, equal-mass nonspinning systems are not expected to recoil ($v_{k}=0$). Our extraction procedure returns $v_{k}\sim 10^{-5}$, which has to be considered a numerical error. Following Refs. Boyle _et al._ (2008); Boyle and Kesden (2008), we further exploit this argument using other symmetries of the system. In particular: 1. $q=1$ and $\boldsymbol{\chi_{1}}=\boldsymbol{\chi_{2}}$ imply $v_{k}=0$. 2. Aligned spins ($\boldsymbol{\chi_{1}}\parallel\mathbf{\hat{L}}$ and $\boldsymbol{\chi_{2}}\parallel\mathbf{\hat{L}}$) force the recoil to be confined to the orbital plane ($\mathbf{v_{k}}\cdot\mathbf{\hat{L}}=0$); this property is independent of $q$. 3. For $q=1$ and spins with opposite orbital-plane components ($\boldsymbol{\chi_{1}}\cdot\mathbf{\hat{L}}=\boldsymbol{\chi_{2}}\cdot\mathbf{ \hat{L}}$ and $\boldsymbol{\chi_{1}}\times\mathbf{\hat{L}}=-\boldsymbol{\chi_{2}}\times \mathbf{\hat{L}}$) the kick is restricted to be orthogonal to the orbital plane ($\mathbf{v_{k}}\parallel\mathbf{\hat{L}}$). Some of the special cases encountered in Sec. III.1 belong to these classes. For instance, equal-mass nonspinning systems are a trivial example of all categories. The $q=1$ up-up, down-down, right-right and left-left cases shown in Figs. 4 and 5 are an instance of (i) and are therefore expected to have $v_{k}=0$. All up-up, down-down, up-down and down-up configurations are an instance of (ii), while right-left and left-right binaries with $q=1$ are an instance of (iii). These symmetries are investigated in the three panels of Fig. 11, respectively. For the top panel, we generate binaries with $q=1$ and random spins $\boldsymbol{\chi_{1}}=\boldsymbol{\chi_{2}}$ uniform in volume with magnitude $<0.8$. For the middle panel, we take $q$ to be uniformly distributed in $[0.5,1]$, generate $\boldsymbol{\chi_{i}}\cdot\mathbf{\hat{z}}$ uniformly in $[-0.8,0.8]$, and set all of the x and y components of the spins to zero. For the bottom panel, we fix $q=1$, generate $\boldsymbol{\chi_{1}}$ uniform in volume with magnitude $<0.8$, and set $[\chi_{2x},\chi_{2y},\chi_{2z}]=[-\chi_{1x},-\chi_{1y},\chi_{1z}]$. The values of $v_{k}$, $\|\mathbf{v_{k}}\cdot\mathbf{\hat{z}}\|$ and $\|\mathbf{v_{k}}\times\mathbf{\hat{z}}\|$ shown in Fig. 11 are expected to be zero under symmetries (i), (ii) and (iii), respectively. We see that symmetry (i) exhibits the largest violations. The absolute largest deviations are $\sim 6\times 10^{-4}c\sim 180$ km/s, which is therefore a generous upper limit of our numerical errors. The median of the errors is as small as $\sim 1.1\times 10^{-4}c$, while the 90th percentile is $\sim 2.8\times 10^{-4}c$. Symmetries (ii) and (iii) are better preserved, with a precision which is roughly an order of magnitude higher. The error medians for both are $\sim 1.5\times 10^{-5}c$. <figure><img src="content_image/1802.04276/x15.png"><figcaption>Figure 11: Test of the kick numerical extraction by exploiting some of thesymmetries of the system. All quantities shown in these plots are expected tobe zero; deviations are interpreted as numerical inaccuracies of ourextraction procedure. Top panel, symmetry (i): equal mass binaries with thesame spin vectors are expected to have zero kicks. Middle panel, symmetry(ii): binaries with generic mass ratio and aligned spins are expected to havekicks in the orbital plane. Bottom panel, symmetry (iii): equal-mass binarieswith opposite orbital-plane spin components and same aligned components areexpected to have kicks directed along the binary’s orbital angular momentum.Each panel contains a sample of 104 binaries generated as described in thetext. Dashed (dotted) lines show medians (90th percentiles) of thedistributions.</figcaption></figure> It is worth noting that the errors reported here are rather conservative, as they take into account inaccuracies accumulated throughout the entire extraction pipeline—from the NR simulations that were used to calibrate NRSur7dq2, to the surrogate waveform interpolations, and finally the numerical operations described in this paper. ### Comparison with numerical relativity simulations: SpEC <figure><img src="content_image/1802.04276/x16.png"><figcaption>Figure 12: Accuracy of the surrogate extraction of the kick velocity vkcompared to NR simulations from SpEC. Filled histograms show distributions ofvk extracted by both approaches, while the black dashed line shows residualsbetween the two methods. Solid thin lines explore some of the possible causesof the observed differences: the orange line shows a lower limit on the NRextraction accuracy, computed using the two highest resolutions available; thepurple line shows residuals between NR kicks extracted with lmax=8 (default)and lmax=4 (corresponding to the highest modes available in NRSur7dq2); thegreen line shows residuals in the surrogate extraction when the same NR runsare reproduced setting either tref=−100M or tref=−4500M.</figcaption></figure> <figure><img src="content_image/1802.04276/x17.png"><figcaption>Figure 13: Comparison between BH kicks extracted from NR SpEC simulations(horizontal) and the surrogate model NRSur7dq2 (vertical). The NR runs usedhere are the same that entered the surrogate model calibration, which was notdesigned to model large kicks specifically. 50th and 90th percentiles areshown with dashed and dotted lines, respectively. Red crosses mark the fourcases explored in Fig. 14.</figcaption></figure> <figure><img src="content_image/1802.04276/x18.png"><figcaption>Figure 14: Linear momentum profiles P(t) projected along the direction of thefinal kick ^vk for four selected NR simulations from SpEC compared topredictions obtained with the surrogate model. These same four cases aremarked with crosses in Fig. 13. While the vast majority of the kickmorphologies are faithfully represented, some outliers are present. An exampleis provided in the bottom right panel, where profiles are in good agreementbefore merger but then diverge at t∼10M.</figcaption></figure> We now estimate the accuracy of our extraction procedure by directly comparing our results to numerical relativity simulations from the SpEC code Kidder _et al._ (2000). In particular, we compare against the 744 simulations³ used to construct NRSur7dq2 Blackman _et al._ (42). These simulations constitute the majority of the waveforms available in the SpEC catalog Mroué _et al._ (2013) in the relevant parameter range, and especially so for generic spin orientations. This is not the most ideal comparison: each of these numerical simulations occupies a special point in the binary parameter space of the surrogate model. However, it is worth noting that (i) the surrogate waveforms do not reproduce the NR waveforms exactly, even at the parameter-space location of the simulations that entered the training process; and (ii) NRSur7dq2 was designed to maximize the overlap between the interpolated and the NR strain $h$, not to accurately model BH kicks. The comparison to NR simulations will therefore be sensitive to errors from the surrogate’s reproduction of the training set of gravitational waveforms, but insensitive to errors from the surrogate’s interpolation between these waveforms. [FOOTNOTE:3][ENDFOOTNOTE] Recoils are extracted from SpEC waveforms using the expressions reported in Sec. II.2, and normalized by the remnant mass computed from the BH horizon at the end of the SpEC simulation. We include modes up to $l_{\rm max}=8$ from the highest-resolution data. To compare with the surrogate kick, we must determine the correct binary parameters by first time shifting and rotating the NR waveforms consistently with NRSur7dq2 (per criteria given in Sec. II.1) and then measuring the BH spins at $t_{\rm ref}=-4500M$ as in Blackman _et al._ (42). Consequently the surrogate is evaluated with $t_{\rm ref}=-4500M$. Filled histograms in Fig. 12 show the distributions of $v_{k}$ obtained for both the NR and surrogate extractions. Differences $\Delta v_{k}$ between the two (thick dashed line) are typically $\sim 10^{-4}c$; 90% of the simulations are reproduced within $\Delta v_{k}=5.5\cdot 10^{-4}c$. In this histogram we also plot several sources of error to evaluate their importance. One of these is the difference between NR kicks extracted from different resolutions of each SpEC simulation —a solid upper limit on the accuracy of the NR kick extraction. This also presents a tail up to $\sim 2\cdot 10^{-3}c$, similar to that of $\Delta v_{k}$. The selection of the reference time $t_{\rm ref}$ in the surrogate extraction is a marginally smaller effect, with tail up to $\sim 10^{-3}c$. The contribution of higher-order modes $l>4$ to the NR kick is a subdominant effect and contributes only on the scale of $\sim 10^{-5}c$. Finally, the error from evaluating the kick at a finite time $t=100M$, instead of taking the kick’s $t\to\infty$ limit, is negligible: the NR kicks extracted at $t=100M$ and $135M$ (each simulation has a different final time in $[139M,165M]$) differ by $\sim 10^{-8}c$ only. The surrogate-to-NR comparison is also presented as a scatter plot in Fig. 13, which shows how the surrogate kick extraction faithfully reproduces the vast majority of the simulations. A few outliers with $\Delta v_{k}\sim 2\cdot 10^{-3}c$ are present in the bottom-center panel of the figure (also in Fig. 12 as the tail of the $\Delta v_{k}$ distribution), for which our surrogate extraction underestimates the value of $v_{k}$. These are cases where the surrogate model fails to correctly reproduce some cycles in the waveform’s higher harmonics around the time of merger, when the majority of the kick is being accumulated. We note that cases with large $\Delta v_{k}$ are preferentially located at the high-spin edge of the NRSur7dq2 parameter space: the three outliers mentioned above, and $\sim 2/3$ among the 5% of cases with the largest $\Delta v_{k}$, have $\chi_{1}=\chi_{2}=0.8$. This occurs because the error of the SpEC simulations, and consequently the surrogate model waveforms, increases towards this maximum-spin boundary. Restricting to the 464 NR simulations (or $\sim 2/3$ of the sample) with zero or one spin of magnitude $\chi=0.8$, we find that the surrogate reproduces 90% of the kicks within $\Delta v_{k}$ of $3.8\cdot 10^{-4}c\sim 113$ km/s. The error is about twice as large for the 280 simulations (or $\sim 1/3$ of the sample) with $\chi_{1}=\chi_{2}=0.8$, with 90% of the kicks being within $7.7\cdot 10^{-4}c\sim 232$ km/s. Finally, Fig. 14 shows comparisons for the kick accumulation profiles $\mathbf{P}(t)\cdot\mathbf{\hat{v}_{k}}$ in four selected cases. We find the the surrogate model reproduces not only the kick magnitude $v_{k}$, but also the morphology of the time accumulation profile for the vast majority of the NR simulations. The lower left panel of Fig. 14 shows one of the few outliers, which has $\Delta v_{k}\sim 3\cdot 10^{-3}c$. The NR and surrogate profiles diverge around $t\sim 10M$, when the surrogate fails to capture the merger waveform. These two curves appear similar to the kick profiles of Fig. 7, suggesting that the surrogate model fails to reconstruct the orbital phase at merger. Even if NRSur7dq2 well reproduces the strain $h$, its small errors might propagate to the phase of center-of mass-oscillation causing a relatively large error on the final kick velocity. ### Comparison with numerical relativity simulations: LazEv <figure><img src="content_image/1802.04276/x19.png"><figcaption>Figure 15: Distribution of BH kicks extracted from 132 NR LazEV simulationsLousto and Zlochower (2013); Zlochower and Lousto (2015), rescaled between theminimum and maximum kicks obtained from NRSur7dq2 [cf. Eq. (19)]. If 0≤νk≤1,there exists a suitable choice of tref for which the surrogate modelreproduces the NR value of the kick. On the other hand, the NR data cannot bereproduced if νk<0 or νk>1.</figcaption></figure> Finally, we compare our results against NR simulations performed by the RIT group with the LazEv code Zlochower _et al._ (2005). This additional comparison is noteworthy because not only were these simulations not used in the surrogate calibration, but they were performed with a completely different numerical scheme (for a detailed comparison between SpEC and LazEv see Ref. Lovelace _et al._ (2016)). We compare against several series of simulations performed by Lousto and Zlochower that vary over the relative azimuthal projection of the spin (i.e. the angle $\alpha$ defined in Sec. III.1) Lousto and Zlochower (2013); Zlochower and Lousto (2015). Of the 223 NR simulations described in these references, 132 of them lie within the parameter range covered by NRSur7dq2.⁴ We extract horizon masses, spins, and final kicks from the relevant tables in Refs. Lousto and Zlochower (2013); Zlochower and Lousto (2015); then, we use the mass ratios and spins as inputs to NRSur7dq2. Case-by-case comparisons between the RIT simulations and the surrogate model are not possible because differences in gauges preclude us from converting their initial separations to our $t_{\mathrm{ref}}$’s. We can, however, check for each case whether there exists a choice of $t_{\mathrm{ref}}$ for which the surrogate reproduces the reported value of the kick. [FOOTNOTE:4][ENDFOOTNOTE] To this end, we rescale each of the RIT kick values $v_{k}^{(\rm NR)}$ with an affine transformation determined by the minimum and maximum surrogate kicks $v_{k}^{(\rm surr)}$ as $t_{\mathrm{ref}}$ is varied over the range $t_{\rm ref}/M\in[-4500,-100]$, while holding all other parameters fixed: \[\nu_{k}=\frac{v_{k}^{(\rm NR)}-\min_{t_{\rm ref}}\,v_{k}^{(\rm surr)}}{\max_{t _{\rm ref}}\,v_{k}^{(\rm surr)}-\min_{t_{\rm ref}}\,v_{k}^{(\rm surr)}}\,.\] (19) Therefore the kicks from Refs. Lousto and Zlochower (2013); Zlochower and Lousto (2015) that can be reproduced lie in the range $0\leq\nu_{k}\leq 1$. The resulting distribution of $\nu_{k}$ is shown in Fig. 15. We find that $0\leq\nu_{k}\leq 1$ for $117/132\simeq 89\%$ of the simulations. The remaining simulations cannot be matched by our procedure; in particular, the surrogate underestimates the NR result in $15/132\simeq 11\%$ of the cases for which $\nu_{k}>1$ (no simulations are found with $\nu_{k}<0$). We stress, however, that these disagreements are very moderate, with $\nu_{k}<1.12$ over all the simulations we analyzed. The different comparisons presented in this section show that the surrogate kick extraction reaches precisions similar to those of the NR simulations that entered its calibration, well respects the symmetries of the problem, and matches kick results obtained with an independent NR code. We quote an overall average precision of $40$ km/s on the surrogate extraction of $v_{k}$. Method \| Description \| Equation \| Default inputs ---\|---\|---\|--- sur() \| Instance of the surrogate class from NRSur7dq2. \| \| q \| Binary mass ratio q∈[0.5,1]. \| \| q=1. chi1 \| Spin vector χ1 of the heavier BH at tref. \| \| χ1=[0,0,0]. chi2 \| Spin vector χ2 of the lighter BH at tref. \| \| χ2=[0,0,0]. t_ref \| Reference time tref/M∈[−4500,−100]. \| \| tref/M=−100. times \| Time nodes ti/M∈[−4500,100]. \| \| lmax \| Largest available l mode (lmax=4 in NRSur7dq2). \| \| h(l,m) \| Modes of the complex GW strain hlm. \| Eq. (1) \| hdot(l,m) \| Modes of the time derivative ˙hlm \| \| dEdt \| Energy flux dE/dt. \| Eq. (2) \| Eoft \| Radiated energy profile E(t). \| \| Erad \| Total radiated energy limt→∞E(t). \| \| Moft \| Mass profile M(t). \| Eq. (4) \| Mrad \| Mass of the remnant BH limt→∞M(t). \| \| Mfin \| Mass of the remnant BH in units of the mass at t=−∞. \| Eq. (5) \| dPdt \| Linear momentum flux dP/dt \| Eqs. (6-8) \| Poft \| Radiated linear momentum profile P(t). \| \| Prad \| Total radiated linear momentum limt→∞\|P(t)\|. \| \| voft \| Recoil velocity profile v(t). \| Eq. (13) \| kickcomp \| Kick velocity, vector vk=limt→∞v(t). \| Eq. (14) \| kick \| Kick velocity, magnitude vk. \| \| kickdir \| Kick velocity, unit vector ^vk=vk/vk. \| \| dJdt \| Angular momentum flux dJ/dt. \| Eqs. (15-17) \| Joft \| Radiated angular momentum profile J(t). \| \| Jrad \| Total radiated angular momentum limt→∞\|J(t)\|. \| \| xoft \| Center-of-mass trajectory x(t)=∫v(t)dt. \| \| Table 1: Main methods of the surrkick class. A class instance has to be initialized with e.g. sk=surrkick.surrkick(q=1,chi1=[0,0,0],chi2=[0,0,0],t_ref=-100). Methods can then be accessed with e.g. sk.voft. ## V Code distribution and usage Our numerical code, surrkick, is publicly available as a module for the Python programming language. The latest stable release is kept updated on the Python Package Index (PyPI) and can be installed via pip install surrkick Python packages numpy (van der Walt _et al._, 2011), scipy Jones _et al._ (01), matplotlib (Hunter, 2007), h5py Collette (2013), pathos McKerns _et al._ (2011), tdqm da Costa-Luis _et al._ (2017), NRSur7dq2 Blackman _et al._ (42) and precession Gerosa and Kesden (2016) are specified as dependencies and will automatically be installed if missing. The surrkick module has to be imported with import surrkick from within a Python environment. Information on all classes, methods, and functions of the code can be obtained from the code docstrings using Python’s help function. surrkick is hosted under version control on GitHub at github.com/dgerosa/surrkick, where development versions are available. Further information and code outputs can be found at davidegerosa.com/surrkick. surrkick is structured as an add-on to any waveform approximant. In particular, it will be straightforward to update it as new surrogate models become available. The code is currently compatible with Python 2; porting to Python 3 is foreseen. Results in this paper were obtained with version 1.1 of surrkick. All of the main functionalities of the code are provided as methods of a single class surrkick.surrkick. An instance of the class is created providing mass ratio $q$, spin vectors $\boldsymbol{\chi}_{i}$ and reference time $t_{\rm ref}/M$: sk=surrkick.surrkick(q=1,chi1=[0,0,0], chi2=[0,0,0],t_ref=-100) A list of the relevant methods is provided in Table 1. All quantities are returned in units of the binary’s total mass (i.e. $c=G=M=1$). Time profiles are evaluated at the time nodes sk.times. For instance, the following code snippet computes the final kick imparted to a right-left binary with $q=0.5$ and $\chi_{1}=\chi_{2}=0.8$, and plots the velocity profile $\mathbf{v}(t)$ projected along $\mathbf{\hat{x}}$, $\mathbf{\hat{y}}$, $\mathbf{\hat{z}}$ and $\mathbf{\hat{v}_{k}}$. import surrkick import matplotlib.pyplot as plt sk=surrkick.surrkick(q=0.5,chi1=[0.8,0,0], chi2=[-0.8,0,0]) print "vk/c=", sk.kick plt.plot(sk.times,sk.voft[:,0],label="x") plt.plot(sk.times,sk.voft[:,1],label="y") plt.plot(sk.times,sk.voft[:,2],label="z") plt.plot(sk.times,surrkick.project(sk.voft, sk.kickdir),label="vk") plt.xlim(-100,100) plt.legend() plt.show() The class surrkick.plots provides tools to reproduce all figures and results presented in this paper. The snippet above is implemented as surrkick.plots.minimal(). Performance of the code was evaluated on a single processor of an Intel Xeon CPU E5-2660 v3 @2.60GHz averaging over $10^{3}$ binaries with generic parameters. Computation of $v_{k}$ takes $\sim 0.1$ s, where $\sim 50$ ms are spent evaluating $h$ from NRSur7dq2 Blackman _et al._ (42) and $\sim 50$ ms are spent integrating the energy and linear momentum fluxes. These low execution times make our code ideal to be ported into large-scale computational studies. ## VI Conclusions New waveform approximants able to model precessing BH binaries with higher harmonics have been recently developed for GW detection and parameter estimation. Here we have shown, for the first time, how these tools present an interesting by-product, namely the quick and reliable estimation of energy and momenta radiated in GWs during BH inspirals and mergers. In particular, the dissipation of linear momentum is responsible for powerful BH recoils, which might even eject BHs from their host galaxies. We exploited the recent NR surrogate model NRSur7dq2 Blackman _et al._ (42) to explore the phenomenology of the recoil velocity profile $\mathbf{v}(t)$ imparted to generic binaries as they merge. Our findings are implemented in the numerical code surrkick, which is made available to the community as a module for the Python programming language. Our extraction procedure inherits both strengths and weaknesses of NRSur7dq2. The model can reproduce the GW strain with mismatches $\sim 10^{-3}$, orders of magnitude better than any other model currently available. This translates into an average accuracy $\Delta v_{k}/c\lesssim 10^{-4}$ on the recoil estimates. The model has only been calibrated on BH binaries with mass ratios $q\geq 0.5$ and spin magnitudes $\chi_{i}\leq 0.8$. Both NRSur7dq2 and surrkick can in principle be used outside this range, but those extrapolations have not been tested accurately. NRSur7dq2 provides evolutions over a time $\Delta t\sim 5000M$, corresponding to $\sim 20$ orbits before merger. While this is a severe limitation for waveform modeling (because low-mass systems spend many more cycles in the sensitivity windows of the detectors), it is irrelevant for kick estimation. Linear momentum emission is concentrated in a small time window ($2\sigma\sim 20M$) around merger which is well covered by NRSur7dq2. The tools presented here provide an alternative way to estimate BH kicks which, contrary to fitting formulas, does not require specific ansätze. Moreover, they provide information on the full $\mathbf{v}(t)$ profile, not just the final recoil velocity $v_{k}$. With executions times of $\sim 0.1$ s, our approach allows for quick and reliable implementations of BH kicks in a variety of astrophysical studies, from galaxy evolution codes to population synthesis studies of compact binaries. Future developments include building new NR surrogate models specifically designed to accurately reproduce mass, spin, and recoil of the post-merger BH. ###### Acknowledgements. We thank Jonathan Blackman, Chad Galley, Mark Scheel, Ulrich Sperhake, Saul Teukolsky, and Vijay Varma for fruitful discussions and technical help. D.G. is supported by NASA through Einstein Postdoctoral Fellowship Grant No. PF6-170152 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under Contract NAS8–03060. F.H. acknowledges the support of the Sherman Fairchild Foundation, and NSF grants PHY-1404569, PHY-1708212, and PHY-1708213 at Caltech. L.C.S. acknowledges the support of NSF grant PHY-1404569 and the Brinson Foundation. 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1704.04042	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 67926, "num_imgs": 6, "llama3_tokens_count": 20045 }	[ "content_image/1704.04042/x1.png", "content_image/1704.04042/x2.png", "content_image/1704.04042/x3.png", "content_image/1704.04042/x4.png", "content_image/1704.04042/x5.png", "content_image/1704.04042/x6.png" ]	# Higgs-mode radiance and charge-density-wave order in 2H-NbSe${}_{2}$ Romain Grasset Université Paris Diderot, Sorbonne Paris Cité, CNRS Laboratoire Matériaux et Phénomènes Quantiques, UMR 7162 75013, Paris, France Tommaso Cea IMDEA Nanoscience, C/Faraday 9, 28049 Madrid, Spain Graphene Labs, Fondazione Istituto Italiano di Tecnologia, Via Morego, 16163 Genova, Italy ISC-CNR and Department of Physics, Sapienza University of Rome, P.le A. Moro 5, 00185 Rome, Italy Yann Gallais Maximilien Cazayous Alain Sacuto Université Paris Diderot, Sorbonne Paris Cité, CNRS Laboratoire Matériaux et Phénomènes Quantiques, UMR 7162 75013, Paris, France Laurent Cario Institut des Matériaux Jean Rouxel (IMN), Université de Nantes - CNRS, 2 rue de la Houssiniére, BP 32229, 44322 Nantes Cedex 03, France. Lara Benfatto lara.benfatto@roma1.infn.it ISC-CNR and Department of Physics, Sapienza University of Rome, P.le A. Moro 5, 00185 Rome, Italy Marie-Aude Méasson marie-aude.measson@neel.cnrs.fr Université Paris Diderot, Sorbonne Paris Cité, CNRS Laboratoire Matériaux et Phénomènes Quantiques, UMR 7162 75013, Paris, France February 21, 2024 ###### Abstract Despite being usually considered two competing phenomena, charge-density-wave and superconductivity coexist in few systems, the most emblematic one being the transition metal dichalcogenide 2H-NbSe${}_{2}$. This unusual condition is responsible for specific Raman signatures across the two phase transitions in this compound. While the appearance of a soft phonon mode is a well-established fingerprint of the charge-density-wave order, the nature of the sharp sub-gap mode emerging below the superconducting temperature is still under debate. In this work we use external pressure as a knob to unveil the delicate interplay between the two orders, and consequently the nature of the superconducting mode. Thanks to an advanced extreme-conditions Raman technique we are able to follow the pressure evolution and the simultaneous collapse of the two intertwined charge-density-wave and superconducting modes. The comparison with microscopic calculations in a model system supports the Higgs-type nature of the superconducting mode and suggests that charge-density-wave and superconductivity in 2H-NbSe${}_{2}$ involve mutual electronic degrees of freedom. These findings fill the knowledge gap on the electronic mechanisms at play in transition metal dichalcogenides, a crucial step to fully exploit their properties in few-layers systems optimized for devices applications. pacs: 74.70.Ad,71.45.Lr ,74.20.-z,74.25.nd,74.62.Fj † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction The symmetry breaking across an electronic phase transition always occurs along with the emergence of new collective excitations. The charge-density-wave (CDW) electronic instability is accompanied by the softening of a phonon coupled to the electronic density at ${\bf Q}_{CDW}$ and dressed by the amplitude fluctuations of the CDW order parameters which develops below T${}_{CDW}$Lee _et al._ (1974); Grüner (1988). This new mode, also called amplitudon, is Raman activeKlein (1982); Cea and Benfatto (2014) and has been detected in several CDW dichalcogenides, including 2H-NbSe${}_{2}$Tsang _et al._ (1976); Méasson _et al._ (2014); Xi _et al._ (2015). In the superconducting (SC) state, two additional collective excitations of the superconducting order parameter are expected: a massless Nambu-Goldstone phase mode, which is pushed to the plasmon frequency in a charged superconductor, and a massive amplitude mode, also named Higgs mode for the analogy with the Higgs boson in high-energy physicsNagaosa (1999). In principle, the Higgs mode remains ‘dark’ to spectroscopy probes, since it weakly couples to the electromagnetic fieldCea _et al._ (2016) and is overdampedVolkov and Kogan (10); Kulik _et al._ (1981); Littlewood and Varma (1982); Cea and Benfatto (2014); Cea _et al._ (2015). Indeed its energy coincides with the threshold $2\Delta_{SC}$ of the quasiparticle continuum, $\Delta_{SC}$ being the superconducting gapVolkov and Kogan (10); Kulik _et al._ (1981); Littlewood and Varma (1982); Varma (2002); Cea and Benfatto (2014). Even though some recent reports investigated the possibility to detect it via optical spectroscopy in strongly-disordered superconductorsSherman _et al._ (2015) or intense THz fieldMatsunaga _et al._ (2013, 2014), its presence and observability are still under strong debatePekker and Varma (2015); Cea _et al._ (2015, 2016); Cheng _et al._ (2016); Tsuji _et al._ (2016). On the other hand, when superconductivity coexists with a CDW order the Higgs mode has the unique opportunity to become visible via its coupling to the soft CDW phonon mode. 2H-NbSe${}_{2}$ is one of the few systems where the two orders coexist, with a CDW and superconducting instabilities at $T_{CDW}=33$ K and $T_{c}=7$ K, respectively. This mechanism has been proposed long agoLittlewood and Varma (1982, 1981) to explain the dramatic changes of the Raman spectrum of 2H-NbSe${}_{2}$ below $T_{c}$, where a sharp peak develops below $2\Delta_{SC}$ by stealing spectral weight from the soft phonon peak Sooryakumar and Klein (1981); Méasson _et al._ (2014). Nonetheless, other mechanisms could give rise to sharp superconducting resonances, as observed in other multiband superconductorsBlumberg _et al._ (2007); Böhm _et al._ (2014), making the assignment of the superconducting peak to the Higgs mode problematic. Tuning the delicate interplay between the CDW and superconductivity is achievable by application of high pressure or by lowering the dimensionality of the system. Pressure experiments in 2H-NbSe${}_{2}$ showed that above a critical pressure of 4 GPa the CDW order disappears while superconductivity remains almost unaffectedBerthier _et al._ (1976); Suderow _et al._ (2005); Feng _et al._ (2012); Leroux _et al._ (2015) (Fig. 1(a)). This behavior is in striking contrast to what is found by reducing the sample dimensionality since, there, CDW order is significantly reinforced while superconducting transition temperature is halved for monoloayer systemXi _et al._ (2015); Staley _et al._ (2009); Ugeda _et al._ (2015). All these observations triggered intense theoretical effortsJohannes _et al._ (2006); Calandra _et al._ (2009); Zhu _et al._ (2015); Flicker and van Wezel (2015); Weber _et al._ (2016) to explain the origin of the CDW and superconducting transitions in bulk and few-layers 2H-NbSe${}_{2}$, by accounting for the different role of the electron-phonon coupling and Fermi-surface nesting. So far, pressure effects have been addressed via signatures of the lattice and its dynamics Feng _et al._ (2012); Leroux _et al._ (2015). Raman probe under pressure has the advantage to assess directly the evolution of the _electronic_ degrees of freedom with the pressure-driven CDW softening, without the additional complication of the interaction with the substrate, relevant in devices based on few-layers 2H-NbSe${}_{2}$. Here we use an advanced low-temperature and high-pressure technique to probe the Raman signatures of the CDW and superconducting excitations across the phase diagram of 2H-NbSe${}_{2}$. Besides, we compute the evolution of the Raman response within a microscopic model for the coexisting CDW and superconducting order. Our findings point to the assignment of the SC peak to the Higgs fluctuations as the most likely interpretation. This result not only provides a perspective for the microscopic mechanisms at play in the coexisting states of 2H-NbSe${}_{2}$, but it also shows that the Higgs-mode radiance is a direct fingerprint of charge ordering. <figure><img src="content_image/1704.04042/x1.png"><figcaption>Figure 1: Raman scattering under hydrostatic pressure and at low temperatureof 2H-NbSe2. (a) (P,T) Phase diagram of 2H-NbSe2 drawn from resistivitymeasurementsSuderow _et al._ (2005); Jérome _et al._ (1976). Theincommensurate CDW collapses at a critical pressure of ∼ 4 GPa, a pure SCstate persists up to at least 10 GPa. Large colored circles mark the (P,T)positions of the experimental spectra reported in Fig. 2(a). (b) MembraneDiamond anvil cell designed for a large numerical aperture collection (greencone), low Raman signal from the environment and access to low temperature (∼3 K). The 350 μm diameter pressure chamber containing the freshly cleaved2H-NbSe2 sample and rubies as pressure gauge is depicted below. (A) Thermallink between the metallic gasket and the cold finger of the cryostat made ofhigh conductivity Cooper wires. (c) Raman spectra of 2H-NbSe2 in the E2gsymmetry at 2 GPa and 3 K in the coexisting region of charge-density-wave andsuperconducting orders. The ⋆ designs a CDW-phonon mode. Consistently with thetheory, the two collective excitations at 14 cm−1 and 40 cm−1 at 2 GPa areassigned to the superconducting Higgs mode and the charge-density-waveamplitudon mode.</figcaption></figure> ## II Experimental details Single crystals of 2H-NbSe${}_{2}$ were synthesized at 750${}^{\circ}$C using the iodine-vapour transport method as described elsewhereFisher and Sienko (1980). The crystallographic quality of several crystals was checked by X-ray diffraction. All crystals revealed a hexagonal cell with parameters a=b=3.44$\angstrom$ and c=12.54$\angstrom$ in agreement with the 2H polytype. We have adapted an original optical experimental setup (Fig. 1(b)) to probe low-energy Raman excitations under extreme conditions of pressure and low temperature. This report constitutes the first experiment of this type, successfully reaching low-energy down to 7 cm${}^{-1}$ (0.85 meV), down to $\sim$ 3 K and up to 5 GPa. We have thus been able to track simultaneously the superconducting and the CDW modes under a broad range of hydrostatic pressure. Raman scattering measurements have been performed on freshly cleaved 2H-NbSe${}_{2}$ crystal with an incident angle of $\sim$ 30${}^{\circ}$ with respect to the sample surface normal and in a membrane diamond anvils cell designed for a large numerical aperture together with a low Raman signal from the environment of the sample as described in Buhot _et al._ (2015). The pressure cell was cooled down in a closed-cycle ${}^{4}$He cryostat with a base temperature of 3 K. As sketched in Fig. 1(b), a OFHC Cooper braid between the metallic gasket and the cold finger is used as a cryogenic leakage. This, together with a low laser incident power (typically 0.1 mW) and control of the size of the incident laser spot (about 20 $\mu$m diameter), allowed to reach low temperature, estimated between 3 and 3.5 K, low-enough to measure a Raman signature of the superconducting state of 2H-NbSe${}_{2}$ ($T_{c}$ = 7.2 K). We have used a triple-grating spectrometer Jobin Yvon T64000 equipped with a liquid-nitrogen-cooled CCD detector and the 532 nm excitation line from a solid-state laser. The polarization of the incoming and outgoing light are in the (ab) plane of the sample. In this configuration, in parallel and crossed polarizations we select the A${}_{1g}$+E${}_{2g}$ and the E${}_{2g}$ symmetries, respectively. Birefringence of the diamonds under pressure mixes the effective polarization of lights. Extraction of pure E${}_{2g}$ and $A_{1g}$ symmetries was done by scaling the E${}_{2g}$ phonon mode at about 250 cm${}^{-1}$. The fluorescence of ruby has been used as a pressure gauge. The pressure transmitting medium is ${}^{4}$He. It does not show any particular Raman features at low energy (down to 7 cm${}^{-1}$), down to 3 K and up to 10 GPa. ## III Collapse of the collective modes A typical spectrum of 2H-NbSe${}_{2}$ under high pressure (2 GPa) and at low temperature in the coexisting region of charge-density-wave (CDW) and superconducting (SC) states is displayed in Fig 1(c) in a large Raman shift range. In the E${}_{2g}$ symmetry, beyond the single phonons E${}^{2}_{2g}$ and E${}^{1}_{2g}$ (at 26 and 252 cm${}^{-1}$, respectively), a second-order phonon peak ($\sim$ 130 cm${}^{-1}$) and a CDW-phonon ($\sim$ 200 cm${}^{-1}$, marked with $\star$) are observed. The last one, also observed in the A${}_{1g}$ symmetry, is a signature of the CDW ordering, most probably a phonon mode folded to the zone center due to the CDW ordering. It remains a hard mode up to T${}_{CDW}$ while the low energy collective modes at $\sim$ 15 cm${}^{-1}$ and $\sim$ 40 cm${}^{-1}$ soften upon approaching the ordering temperatures T${}_{c}$ and T${}_{CDW}$, respectively. Both these SC and CDW collective modes are visible in the fully symmetrical A${}_{1g}$ (CDW soft mode at 39 cm${}^{-1}$ and SC mode at 19 cm${}^{-1}$ at ambient pressure) and the in-plane symmetry breaking E${}_{2g}$ (CDW soft mode at 43.5 cm${}^{-1}$ and SC mode at 14 cm${}^{-1}$ at ambient pressure) channels. The CDW soft mode has been already identified in previous literatureTsang _et al._ (1976); Méasson _et al._ (2014) with the so called amplitudon, i.e. the instability phonon dressed by amplitude fluctuations of the CDW. As we will discuss below, we assign the SC mode in the coexisting region of CDW and SC states to a signature of the amplitude fluctuations of the SC order parameter, so we will denote it as Higgs mode in what follows. As already discussed Méasson _et al._ (2014), a partial spectral weight transfer from the CDW soft mode to the SC Higgs mode exists with decreasing temperature. While the Higgs mode rises, the CDW soft mode looses spectral weight, as observed in the two different symmetries E${}_{2g}$ and A${}_{1g}$ (see also Fig. 5). <figure><img src="content_image/1704.04042/x2.png"><figcaption>Figure 2: Collapse of the superconducting Higgs mode in the puresuperconducting state of 2H-NbSe2, measurements and theoretical predictions.(a) Raman spectra in the E2g symmetry measured at various (P,T) positions asidentified Fig. 1(a): in the coexisting SC+CDW (green) and pure CDW (blue)states at ambient pressure and in the pure SC (brown) and paramagnetic (red)state at high pressure. Both the CDW amplitudon and the SC Higgs modesdisappear at high pressure. A small Cooper-pairs breaking peak remains at2ΔSC. 2ΔSC is marked by the grey band ranging from 2ΔSC measured by STMGuillamón _et al._ (2008a) at ambient pressure to the value we extrapolate athigh pressure accordingly to the increase of Tc with pressure Suderow _etal._ (2005). Inset: Raman spectra in the pure SC state of 2H-NbSe2 (above 4GPa) and non-CDW NbS2 (0 GPa) versus the Raman shift normalized to 2ΔSCGuillamón _et al._ (2008a, b). (b) Theoretical Raman responses calculated ina microscopic model (see text and Appendix A) in the four phases (SC+CDW, CDW,SC,PM) for comparison with the experimental spectra in (a). t is the hoppingterm. The parameters are: ΔSC/t=0.025, gCDW/t=0.14 and 0.12 in the SC+CDW andSC phases, respectively. The spectra are well-reproduced in all phases. (c)Raman response of 2H-NbSe2 in the pure superconducting phase above Pc in theE2g and A1g symmetries. The black line is the theoretical Raman response of aCooper-pairs breaking peak in a two-gaps (or anisotropic gap) s-wavesuperconductor in the BCS regime with an additional electronic background β.The form of β is : β(ω)=aω/√b+cω2. It barely affects the shape of the Cooper-pairs breaking peak.</figcaption></figure> By applying high pressure above 4 GPa (Fig. 1(a)) a pure superconducting state is reached and, as presented in Fig. 2(a), both the CDW amplitudon and the Higgs mode disappear. On the other hand, in the E${}_{2g}$ symmetry a weak SC signature persists, with marked differences with respect to the sharp SC Higgs mode seen at $P<P_{c}$=4 GPa. Indeed, its intensity is a factor of $\sim 8$ smaller and its energy suddenly hardens. As shown in the inset of Fig. 2(a), the superconducting Raman response of the compound 2H-NbS${}_{2}$, which lacks the CDW state at ambient pressure Méasson _et al._ (2014), matches perfectly that of 2H-NbSe${}_{2}$ above the critical pressure P${}_{c}$, in the pure superconducting state, as long as the Raman shift is scaled by $2\Delta_{SC}$ ($\Delta_{SC}$ is calculated from the value of the superconducting gap measured by STM Guillamón _et al._ (41) and its pressure dependence is scaled as $T_{c}$(P)Suderow _et al._ (2005)). Both superconducting thresholds are positioned at $2\Delta_{SC}$ as expected for a simple Cooper-pairs breaking peak. As pointed out by many recent measurementsNoat _et al._ (2015); Rahn _et al._ (2012); Fletcher _et al._ (2007); Jing _et al._ (2008); Rodrigo and Vieira (2004); Borisenko _et al._ (2009), 2H-NbSe${}_{2}$ is an s-wave superconductor with either an anisotropic gap or multiple gaps. This property affects the shape of the Raman Cooper-pairs breaking peak. As shown in Fig. 2(c), the Raman spectrum can be properly reproduced by defining an anisotropic gap which varies from a minimum value of $\Delta_{SC}^{s}$=0.92 meV to a maximum value of $\Delta_{SC}^{L}$=1.38 meV Guillamón _et al._ (39). We cannot distinguish here between the presence of multiple gaps or a single anisotropic gap and we do not exclude $k_{z}$ dependency in the real compound 2H-NbSe${}_{2}$Weber _et al._ (2016); Noat _et al._ (2015). Mainly our fit provides evidence for the nature of the SC peak above P${}_{c}$, i.e. a Cooper-pairs breaking peak, with insight into the energy scale of the superconducting gap. Consistently with this assignment, in the A${}_{1g}$ symmetry there is no signature of the pure superconducting state reached in 2H-NbSe${}_{2}$ above the critical pressure, due to Coulomb screening effect Devereaux and Einzel (1995); Devereaux and Hackl (2007); Cea and Benfatto (2016) (see Fig. 2(c)). The disappearance of the sharp SC mode below $2\Delta_{SC}$ in the pure superconducting phase demonstrates unambiguously its intimate link with the coexisting charge-density-wave order. These findings, consistently with the theory discussed below, support the Higgs type assignment of the sharp SC mode below P${}_{c}$. ## IV Comparison with a microscopic model In 2H-NbSe${}_{2}$ the phonon coupled to the CDW belongs to an acoustic branch, so the single-phonon mode is not visible as a finite-energy peak in ${\bf q}$$\sim$ 0 Raman spectroscopy above $T_{CDW}$. Below $T_{CDW}$ the intermediate electron-hole excitations which couple directly to light allow to make the phonon mode at ${\bf Q}_{CDW}$ Raman visible at ${\bf q}=0$. This gives rise to the soft phonon modes at $\sim~{}40~{}$cm${}^{-1}$. In a general approach, the Raman response below $T_{CDW}$ can be schematically written as \[\chi"(\omega)=Z_{eff}(T,\Delta_{CDW})\frac{\Gamma_{ph}}{(\omega^{2}-\Omega_{0} ^{2}(T))^{2}+\Gamma_{ph}^{2}}\] (1) where the soft mode frequency $\Omega_{0}(T)$ and damping $\Gamma_{ph}$ are both determined by the CDW amplitude fluctuations and the prefactor $Z_{eff}\sim\Delta_{CDW}^{2}$ grows proportionally to the CDW order parameterCea and Benfatto (2014). The frequency $\Omega_{0}(T)$ also scales approximately with the CDW gap, so it goes to zero at $T_{CDW}$, even though the Raman peak disappears already at $T\simeq 0.9T_{CDW}$ due to the strong suppression of $Z_{eff}$. While the assignment of the soft CDW peak to the amplitudon is well established in the literatureTsang _et al._ (1976); Littlewood and Varma (1982); Klein (1982); Méasson _et al._ (2014); Xi _et al._ (2015); Weber _et al._ (2011) the interpretation of the additional peak emerging upon entering the SC state has been somehow controversial. The first suggestionsBalseiro and Falicov (1980); Littlewood and Varma (1982) assumed that the amplitudon can be treated as an ordinary ${\bf q}=0$ Raman-active soft phonon, and considered how the proximity of $\Omega_{0}$ to the scale $2\Delta_{SC}$ can modify the phonon spectral function itself. Balseiro and FalikovBalseiro and Falicov (1980) proposed that the SC peak originates from an ordinary self-energy correction of the phonon due to the coupling to electronic excitations, whose quasiparticle spectrum changes after the gap openingZeyher and Zwicknagl (1990). This mechanism is analogous to the one proposed to interpret the changes in the lineshape of finite-momentum strongly-damped phonons measured by neutron scattering in systems like YNi${}_{2}$B${}_{2}$C or LuNi${}_{2}$B${}_{2}$CAllen _et al._ (1997); Kawano _et al._ (1996); Weber _et al._ (2008). However, as correctly pointed out by Littlewood and Varma later onLittlewood and Varma (1982), a ${\bf q}=0$ symmetric (A${}_{1g}$) phonon couples also to the long-range Coulomb interactionsZeyher and Zwicknagl (1990), which renormalize to zero the self-energy phononic corrections in the particle-hole channel. In contrast, if the soft phonon is coupled to the Higgs fluctuations, the self-energy corrections due to SC amplitude fluctuations are not affected by the Coulomb screening. Then this mechanism can lead to sharp sub-gap peaks, even in the A${}_{1g}$ symmetry. In the case of this last scenario, the SC signature is not the pure Higgs mode. Rather we observe its manifestation on the CDW amplitudon, which is split thanks to the interaction with the Higgs fluctuations. The milestone idea by Littlewood and Varma has been put later on firmer groundsBrowne and Levin (57); Cea and Benfatto (2014). Browne and LevinBrowne and Levin (57) explained that the coupling between the soft CDW phonon and the Higgs fluctuations originates microscopically from the intertwined amplitude fluctuations of the two CDW and SC order parameters. More recently, Cea and BenfattoCea and Benfatto (2014) computed explicitly the Raman response, evaluating the intermediate electron-hole processes which make the CDW phonon Raman visible, i.e. the effective charge $Z_{eff}$ in Eq. (1) above. The microscopic identification of the coupling between the amplitudon and the Higgs implies that the CDW and SC order parameters should overlap at least in part of the Fermi surface, so that their amplitude fluctuations talk to each other via a modification of the same electronic density-of-states. As it has already been shown in Ref. Cea and Benfatto (2014), the calculation of the Raman response within a microscopic model system for the coexisting state is able to reproduce successfully the main feature of the experiments at ambient pressure. In addition such a microscopic approach clarifies that accounting only for the change in the particle-hole spectrum of excitations due to the superconducting gap opening is not enough to reproduce the strong sub-gap peak (see Appendix A for further details). Indeed, in contrast to the usual case of a metallic-to-superconductor transition considered e.g. in Ref. Balseiro and Falicov (1980); Allen _et al._ (1997); Zeyher and Zwicknagl (1990), here the quasiparticle spectrum above $T_{c}$ is already gapped by the CDW gap, being then weakly affected by the opening of the superconducting one. Thus the changes in the phonon lineshape when going from the CDW to the superconducting state cannot be simply ascribed to a redistribution of the charge excitations across $2\Delta_{SC}$, as described in the previous workBalseiro and Falicov (1980); Allen _et al._ (1997); Zeyher and Zwicknagl (1990) focusing on standard phonon. Besides, even if considered as an ordinary phonon mode, the soft mode in 2H-NbSe${}_{2}$ is at $\sim 2\cdot 2\Delta_{SC}$ and its tail does not overlap with $2\Delta_{SC}$. So the mechanism of spectral-weight redistribution around $2\Delta_{SC}$Balseiro and Falicov (1980); Allen _et al._ (1997); Zeyher and Zwicknagl (1990) fails to reproduce the intense sub-gap peak, even in the E${}_{2g}$ symmetry. Here, following the approach of Ref. Cea and Benfatto (2014), we model the pressure effects by a continuous suppression of the couplings in the CDW and superconducting channels, in order to reproduce the suppression of the CDW gap while keeping $\Delta_{SC}$ almost constant (see Appendix A for further details). As a control parameter playing the role of the pressure we then use the relative change $\alpha=2(g_{CDW}^{0}-g_{CDW})/g^{0}_{CDW}$ of the CDW coupling $g_{CDW}$. The Raman intensities are then computed in the various phases (pure SC, CDW+SC, CDW such as measured) (see Fig. 2(b)). The spectra have the same absolute units, so the scaling of the intensities is respected. The theoretical Raman response is consistent with our measurements: the Higgs mode manifests as a secondary peak of the CDW soft phonon, which is the mode Raman visible. Thus it appears as a sub-gap intense peak only when it coexists with a CDW state. When the CDW disappears the Raman response in the pure superconducting state displays only a broad and weak signature at 2$\Delta_{SC}$. Besides, the hardening and damping of the amplitudon mode upon entering the SC state is a direct consequence of its coupling to the collective electronic excitations, whose density-of-states gets redistributed from below to above $2\Delta_{SC}$ in the superconducting state. It is actually observed experimentally at all pressures below 4 GPa (see Fig. 4(a,b)). <figure><img src="content_image/1704.04042/x3.png"><figcaption>Figure 3: Pressure dependence of the Raman spectra of 2H-NbSe2 in thecoexisting superconducting and charge-density-wave phases, experiments andtheory. Raman response at 8 K (b,c) in the CDW state and at 3 K (e,f) in theCDW+SC state for various pressures up to the critical pressure. The spectraare normalized to the E2g phonon and consecutively shifted up. (a,d)Theoretical Raman response computed microscopically, with frequency given inunits of the hopping parameter t, which sets the energy scale. The pressuredependence is simulated by suppressing the CDW coupling gCDW with respect toits value g0CDW at ambient pressure, with α=2(g0CDW−gCDW)/g0CDW. Theexperiments at a given P/Pc are compared to calculations at α/αc, whereαc=0.30 is the critical coupling at which CDW order disappears.</figcaption></figure> ## V Tuning the interplay between CDW and SC with pressure To further unveil the interplay between CDW and SC in 2H-NbSe${}_{2}$ we have finely tuned the pressure in the coexisting CDW+SC state, both experimentally and theoretically. Figure 3(b,c) and 3(e,f) reports the experimental results in the E${}_{2g}$ and A${}_{1g}$ symmetries above and below T${}_{c}$, respectively, from ambient pressure up to 3.67 GPa, corresponding to $P/P_{c}=0.92$. The intensities are normalized on the high energy E${}_{2g}$ phonon mode. As one can see in the upper panels, the CDW amplitudons gradually soften, enlarge and loose intensity with increasing pressure. At $P=3.54$ GPa ($P/P_{c}=0.89$), the amplitudon is barely visible above $T_{c}$, even though the critical pressure has not been reached yet. In contrast, as shown Fig. 3(e,f), the SC Higgs peaks are visible up to $P_{c}$, leading to the remarkable effect that the radiance of the SC Higgs signature guarantees that a residual CDW order is present. At the same time, as the system is cooled below T${}_{c}$, the amplitudon shifts to higher energy and gets enlarged, demonstrating a clear coupling between the two peaks. All these features are well reproduced by our calculations shown in Fig. 3(a,d). As we mentioned before, the CDW instability is progressively suppressed as the relative CDW coupling $\alpha$ increases, up to the critical value $\alpha_{c}=0.3$ where it disappears. We then compare the experimental results for $P/P_{c}$ to our calculations at the corresponding $\alpha/\alpha_{c}$. In the model, the softening of the energy of the amplitudon for increasing $\alpha$ is due to the suppression of the CDW gap, since the CDW amplitude fluctuations are peaked at $2\Delta_{CDW}$. Simultaneously the Raman intensity $Z_{eff}\sim\Delta^{2}_{CDW}$ is rapidly suppressed, making the Raman signature of the CDW amplitudon above $T_{c}$ barely visible already at $\alpha/\alpha_{c}=0.8$, in agreement with the experiments. On the other hand, since the Higgs mode is much sharper than the amplitudon at any pressure, even in this regime near $\alpha_{c}$, it is clearly visible, giving a clear fingerprint of the existence of a CDW order. <figure><img src="content_image/1704.04042/x4.png"><figcaption>Figure 4: (a) Pressure evolution of the energy of the amplitudon in the E2gsymmetry above and below Tc compared to the evolution of the amplitudon in themicroscopic model. Inset: Evolution of the energy of the folded CDW phonon,denoted with * (190 cm−1) in the spectra of Fig 1(b), measured at 8 K. (b)Pressure evolution of the width of the amplitudon in the E2g symmetry aboveand below Tc compared to the evolution of the amplitudon in the microscopicmodel. (c) Pressure dependence of the energy of the Higgs mode normalized toits value at zero pressure in the two A1g and E2g channels, and in themicroscopic model. From 0 to Pc=4 GPa, the A1g mode softens, qualitativelyfollowing the behavior of the Higgs mode in the microscopic model, whereas theE2g one is constant, showing strong symmetry-dependent behavior. Inset:Pressure dependence of the Higgs mode in both symmetries compared to the pair-breaking threshold 2ΔSC, with ΔSC extrapolated from STM measurementsGuillamón_et al._ (2008a) at ambient pressure scaled with the pressure evolution ofTcSuderow _et al._ (2005).</figcaption></figure> The detailed pressure dependence of the energy and width of both the amplitudon and the Higgs mode is reported in Fig. 4. The CDW amplitudons gradually soften with increasing pressure and harden upon entering the SC state at all pressures. This tendency is well reproduced by our microscopic model (see Fig. 4(a)). For the sake of completeness we also show in the inset of Fig. 4(a) the pressure dependence of the folded CDW-phonon mode at $\sim~{}190~{}cm^{-1}$ (marked with $\star$ in Fig. 1(c)). As one can see, in contrast to the amplitudons, its energy hardens linearly with pressure in the same way the regular A${}_{1g}$ and E${}_{2g}$ phonons do. More precisely, the rate of increase of the energy of the A${}_{1g}$, E${}^{1}_{2g}$ and the folded CDW-phonon mode is similar at about 1$\%$ per GPa. In panel 4(b) we compare the pressure evolution of the width of the amplitudon above and below $T_{c}$ with the theoretical calculations. The experimental trends are very well captured by the model, with an overall broadening of the amplitudon upon entering the SC state at a given pressure or as the pressure increases. The larger variations in temperature found theoretically can be ascribed to the presence of a sharper phonon peak in the calculations above $T_{c}$ (see also Fig. 3(a)). Since the broadening of the phonon peak is provided by residual quasiparticle scattering events from ungapped regions of the Fermi surface, it crucially depends on details of the band structure in the CDW state. In panel 4(c) we summarize the evolution of the SC Higgs peaks energy in the two channels. As highlighted in the inset, the Higgs mode always lies below $2\Delta_{SC}$, where the usual Cooper-pairs breaking peak would be instead expected. In addition, the A${}_{1g}$ peak softens by 30 $\%$ and the energy of the E${}_{2g}$ peak stays constant whereas the critical temperature $T_{c}$ rises with pressure. The absence of scaling between the SC peaks energy and T${}_{c}$ in the SC+CDW coexisting phase indicates that, in both symmetries, the SC mode below P${}_{c}$ is not simple a Cooper-pairs breaking peak. Moreover, this is an additional evidence that the SC mode, even in the E${}_{2g}$ symmetry where Coulomb screening is not effective, is not a SC peak originates from an ordinaryBalseiro and Falicov (1980); Allen _et al._ (1997); Zeyher and Zwicknagl (1990) self-energy correction of the phonon, since this mechanism would predict that the peak position follows the pressure evolution of $2\Delta_{SC}$. While in the A${}_{1g}$ channel the Higgs mode has the same pressure trend than our calculations, in the E${}_{2g}$ symmetry the Higgs mode energy shows a relatively different behavior. To understand such a discrepancy one should notice that in our simplified model the CDW instability occurs at a single ${\bf Q}_{CDW}$ vector equivalent to half of the reciprocal-lattice wavevector. In this situation, the CDW phonon is only visible in the fully symmetrical A${}_{1g}$ channel. On the other hand, in 2H-NbSe${}_{2}$ the CDW instability can occur at three equivalent ${\bf Q}^{i}_{CDW}$ wavevectors connected by a $2\pi/3$ rotation. This guarantees that the amplitudon has a finite projection in both A${}_{1g}$ and E${}_{2g}$ symmetriesKlein (1982). When the SC state forms the Higgs fluctuations renormalize the frequencies of the phonon modes corresponding to lattice displacements at the three ${\bf Q}^{i}_{CDW}$ wavevector. If the electron-phonon coupling has a non-trivial momentum dependence, as it has been emphasized recentlyCalandra _et al._ (2009); Flicker and van Wezel (2015), one cannot exclude a splitting of the Higgs signatures, which reflects in a different trend of the subgap peak observed in the A${}_{1g}$ and E${}_{2g}$ channels under pressure. A full understanding of this issue requires a calculation within a microscopic model for 2H-NbSe${}_{2}$, which is beyond the scope of the present paper. <figure><img src="content_image/1704.04042/x5.png"><figcaption>Figure 5: Spectral weight transfer between the superconducting mode and thecharge-density-wave mode. (a) Raman spectra of 2H-NbSe2 at 3.2 GPa above andbelow Tc in the E2g symmetry. The spectral weight transfer is depicted ingreen. (b,c) Difference (in %) of the total spectral weight of both modesbetween 3 K and 8 K (below/above Tc) as a function of pressure in E2g (b) andA1g (c) symmetries. At every pressure and in both symmetries, the totalspectral weight of the Higgs mode and the amplitudon is relatively conserved(∼20%) within the large error bars.</figcaption></figure> Finally, in Fig. 5 we show in details the transfer of spectral weight between the amplitudon and the Higgs mode below $P_{c}$. Even though in Raman spectroscopy the total spectral weigh is not constrained by a sum rule, as it happens for optical spectroscopy, previous work at ambient pressure has shown Méasson _et al._ (2014) that upon entering the superconducting state the rise of the Higgs mode happens at the expense of the CDW amplitudon. This finding is also observed for increasing pressure up to the collapse of both modes, as shown in Fig. 5(b) and (c) for the E${}_{2g}$ and A${}_{1g}$ symmetries, respectively. The total spectral weight of the two modes is approximately conserved (at $\pm$ 20$\%$) when the system is cooled below $T_{c}$, even though error bars here are larger as compared to experiments at ambient pressure Méasson _et al._ (2014). This approximate conservation of spectral weight further demonstrates the existence of a direct coupling between the SC Higgs mode and the CDW amplitudon at every pressure below $P_{c}$. ## VI Discussion The comparison between the theoretical calculations and the experiments points to the assignment of the SC mode in the CDW and SC coexisting region to a signature of the Higgs mode carried out by the CDW amplitudon as the most likely. In particular, the sudden disappearance of the SC signature as $\Delta_{CDW}=0$ agrees with the general prediction of Eq. (1) that its Raman visibility is only guaranteed by the presence of a soft, Raman-active CDW amplitudon, allowing for $Z_{eff}\neq 0$. Indeed, other possible interpretations based on the multiband structure of 2H-NbSe${}_{2}$, like a Leggett mode or a Bardasis-Schrieffer mode, cannot be easily reconciled with this behavior, since they are intimately related to the properties of the SC state, which barely changes as a function of pressure. For example the Leggett mode, due to the relative fluctuations of the SC phases in two bands, becomes Raman visible thanks to the hole/electron character of the various bandsBlumberg _et al._ (2007); Cea and Benfatto (2016), which are not expected to change with pressure. The Bardasis-Schrieffer mode can manifest as a sharp mode below $2\Delta_{SC}$Monien and Zawadowski (1990). It originates from subleading pairing fluctuations, so it should be observed in a Raman channel orthogonal to the one where the driving SC instability occursMonien and Zawadowski (1990). Thus, since 2H-NbSe${}_{2}$ is a $s$-wave superconductor the Bardasis-Schrieffer mode should _not_ be visible in the $A_{1g}$ channelMonien and Zawadowski (1990), in contrast with our experimental results. As we mentioned above, we computed the Raman intensity in the coexisting CDW and SC state with a model Hamiltonian (see Appendix A) that is not intended to reproduce realistically 2H-NbSe${}_{2}$. This implies for example that in our approach the CDW originates from a Fermi-surface nesting instability, while in 2H-NbSe${}_{2}$ it has been clearly shown that the electronic susceptibility gets strongly enhanced at the ordering wavevector ${\bf Q}_{CDW}$ only when the momentum dependence of the electron-phonon coupling is taken into accountCalandra _et al._ (2009); Flicker and van Wezel (2015). However, once this effect is included the description of the CDW state does not differ conceptually from a standard Peierls mechanismFlicker and van Wezel (2015), making our approach suitable to include the microscopic ingredients specific of 2H-NbSe${}_{2}$. ## VII Conclusions In summary, we report the pressure dependence of the A${}_{1g}$ and E${}_{2g}$ Raman active modes related to the charge-density-wave and superconducting orders in the transition metal dichalcogenide 2H-NbSe${}_{2}$ up to $\sim$ 5GPa. We showed that the soft CDW modes, the so-called amplitudons, and the sub-gap SC peaks, the Higgs mode, collapse in the pure SC state above 4 GPa while only the expected Cooper-pairs breaking peak at $2\Delta_{SC}$ persists in the E${}_{2g}$ symmetry. In the coexisting CDW and SC state, the CDW amplitudon modes soften, enlarge and loose intensity while the intensity of the SC Higgs mode, in both symmetries, decreases but remains sizable even when the amplitudons are almost invisible. These results reveal that the radiance of the SC Higgs mode guarantees that a residual CDW order is present. At all pressure up to 4 GPa and in both A${}_{1g}$ and E${}_{2g}$ symmetries, we observed a shift to higher energy of the CDW amplitudons and their enlargement upon entering the SC state as well as a transfer of spectral weight from the CDW amplitudons to the SC Higgs peaks. The pressure trends of the two intertwined CDW and SC modes are well reproduced by our exact calculation of the Raman response within a microscopic model system for the CDW and SC coexisting states. This implies that CDW and SC order parameters must overlap at least in part of the Fermi surface. Thus our experimental and theoretical findings support the Higgs type assignment of the superconducting mode in the coexisting CDW+SC region and in both A${}_{1g}$ and E${}_{2g}$ channels. Interestingly, from 0 to 4 GPa, the A${}_{1g}$ superconducting Higgs peak softens by 30$\%$ whereas the E${}_{2g}$ one is constant. An explanation of this strong symmetry-dependent behavior might requires a calculation within a microscopic model for 2H-NbSe${}_{2}$. Our findings also point out that Raman spectroscopy represents the best suited probe to investigate interplay and competition between charge-density-wave and superconducting coexisting orders, notably when varying the dimensionality, in few-layers systems. ## Appendix A Theoretical model The general derivation of the Raman response in the coexisting CDW+SC state has been recently provided in Ref.Cea and Benfatto (2014). Its explicit form depends on the band structure and on the electron-phonon coupling. In order to simplify the derivation we adopt here the same model system used in Ref. Cea and Benfatto (2014). Even thought it does not provide a complete microscopic description of 2H-NbSe${}_{2}$, it contains the main ingredient needed to describe the interplay of CDW and SC in this system, i.e a momentum-dependent CDW, which leaves part of the Fermi surface ungapped below $T_{CDW}$. This condition makes energetically possible a SC gap opening, and it also allows for a coexisting SC and CDW state in part of the Fermi surface, leading to a coupling between the amplitude fluctuations of the two order parameters. We then start from a single-band model on the square lattice (lattice spacing $a=1$) with band dispersion $\xi_{\bf k}\equiv\epsilon_{\bf k}-\mu=-2t(\cos k_{x}+\cos k_{y})-\mu$, where $t=1$ is the hopping and $\mu$ is the chemical potential. The CDW instability is driven by the microscopic coupling $g_{CDW}$ of the electrons to a phonon of energy $\omega_{0}$ \[H_{CDW}=g_{CDW}\sum_{{\bf k}\sigma}\gamma_{\bf k}c^{\dagger}_{{\bf k}+{\bf Q} \sigma}c_{{\bf k}\sigma}(b^{+}_{\bf Q}+b_{-{\bf Q}}),\] (2) where the $\gamma_{\bf k}=\|\cos k_{x}-\cos k_{y}\|$ factor modulates the CDW in the momentum space. Near half filling ($\mu=0$) the nesting of the Fermi surface at the CDW vector ${\bf Q}=(\pi,\pi)$ allows for a CDW instability to occur, with new bands $\xi_{\pm}=-\mu\mp\sqrt{\epsilon_{\bf k}^{2}+\Delta_{CDW}^{2}\gamma_{\bf k}^{2}}$ and a CDW order parameter $\Delta_{CDW}=(4g^{2}/\omega_{0})\sum_{{\bf k}\sigma}\langle\gamma_{\bf k}c^{ \dagger}_{{\bf k}\sigma}c_{{\bf k}+{\bf Q}\sigma}\rangle$. The superconductivity originates from a BCS-like interaction term \[H_{SC}=-(U/N)\sum_{q}\Phi_{\Delta}^{\dagger}({\bf q})\Phi_{\Delta}({\bf q}),\] (3) where $\Phi_{\Delta}({\bf q})\equiv\sum_{\bf k}c_{-{\bf k}+{\bf q}/2\downarrow}c_{{ \bf k}+{\bf q}/2\uparrow}$ is the pairing operator and $N$ is the number of lattice sites. When treated at mean-field level it leads to the following Green’s function $G_{0}^{-1}(\mathbf{k},i\omega_{n})$, defined on the basis of a generalized 4-components Nambu spinor $\Psi^{\dagger}_{\bf k}(i\omega_{n})\equiv(c^{\dagger}_{\mathbf{k}\uparrow}(i \omega_{n}),c^{\dagger}_{\mathbf{k}+\mathbf{Q}\uparrow}(i\omega_{n}),c_{- \mathbf{k}\downarrow}(-i\omega_{n}),c_{-\mathbf{k}-\mathbf{Q}\downarrow}(-i \omega_{n}))$ that accounts for the CDW band folding: \[G_{0}^{-1}(\mathbf{k},i\omega_{n})\equiv i\omega_{n}-\begin{pmatrix}\hat{h}&&- \Delta_{SC}\sigma_{0}\\ -\Delta_{SC}\sigma_{0}&&-\hat{h}\end{pmatrix},\] (4) where $\omega_{n}=(2n+1)\pi T$ is the fermionic Matsubara frequency, $\Delta_{SC}$ is the superconducting gap, $\sigma_{i}$ denotes the Pauli matrices and $\hat{h}$ is a $2\times 2$ matrix: \[\hat{h}=\begin{pmatrix}\epsilon_{\bf k}-\mu&&-\Delta_{CDW}\gamma_{\bf k}\\ -\Delta_{CDW}\gamma_{\bf k}&&-\epsilon_{\bf k}-\mu\end{pmatrix}.\] (5) The eigenvalues of the matrix $\hat{h}$ represent the two CDW bands $\xi_{\pm}$, while in the superconducting state the full Green’s function (4) has four possible poles, corresponding to the energies $\pm E_{\pm}({\bf k})$, with $E_{\pm}=\sqrt{\xi_{\pm}^{2}+\Delta_{SC}^{2}}$. For the sake of simplicity we will consider in the following the half-filled case, which allows for an easier treatment of the fluctuations in both superconducting and CDW sectors, without loss of generality of the main conclusions (see Appendix B of ref. Cea and Benfatto (2014) for further details on the case of general filling). The Raman response of the previous model has been derived in Ref. Cea and Benfatto (2014). Its general structure can be written as \[\chi_{RR}(i\Omega_{n})=\chi^{0}_{RR}-g_{CDW}^{2}\chi^{2}_{R,CDW}(i\Omega_{n})D _{ph}(i\Omega_{n})\] (6) where $\Omega_{n}=2\pi nT$ is the bosonic Matsubara frequency, $\chi^{0}_{RR}=\langle\rho_{R}\rho_{R}\rangle$ is the bare electronic Raman response, $D_{ph}(i\Omega_{n})$ is the Green’s function of the ${\bf Q}$ phonon and $\chi_{R,CDW}=\langle\rho_{R}\delta\Delta_{CDW}\rangle$ is the response function coupling the electronic Raman density $\rho_{R}$ to the amplitude fluctuations $\delta\Delta_{CDW}$ of the CDW order parameter. Eq. (6) establishes that when the system enters the CDW state the phonon coupled via Eq. (2) to the electronic charge fluctuations at the ${\bf Q}_{CDW}$ ordering vector becomes Raman active. In addition, the spectral function itself of the phonon changes dramatically due to its coupling to the electronic charge fluctuations at ${\bf Q}_{CDW}$. As usual, the coupling of the phonon to the particle-hole excitations is described by self-energy corrections, which renormalize in general its bare frequency $\omega_{0}$ and introduce a finite broadeningAllen _et al._ (1997); Zeyher and Zwicknagl (1990); Balseiro and Falicov (1980): \[D_{ph}(i\Omega_{n})=-\frac{2\omega_{0}}{(i\Omega_{n})^{2}-\omega_{0}^{2}- \Sigma(i\Omega_{n})}.\] (7) However, in contrast to the case considered in Allen _et al._ (1997); Zeyher and Zwicknagl (1990) of ordinary phonons in metal, the ordering of the electronic charge at ${\bf Q}_{CDW}$ below $T_{CDW}$ implies that the CDW phonon is directly coupled to the amplitude fluctuations of the CDW order parameterLee _et al._ (1974); Browne and Levin (57); Cea and Benfatto (2014), justifying its denomination as ”amplitudon”. More explicitly one then has \[\Sigma(i\Omega_{n})=2g^{2}_{CDW}\omega_{0}\chi_{CDW}(i\Omega_{n}),\quad\chi_{ CDW}=\langle\delta\Delta_{CDW}\delta\Delta_{CDW}\rangle.\] (8) After analytical continuation to real frequencies the renormalized phonon frequency $\Omega_{0}$ below $T_{CDW}$ is then defined, after Eq. (7), as a solution of the equation \[\Omega_{0}^{2}\equiv\omega_{0}^{2}+\Sigma^{\prime}(\Omega_{0})=\omega_{0}^{2}[ 1+(2g_{CDW}^{2}/\omega_{0})\chi^{\prime}_{CDW}(\Omega_{0})]\] (9) which leads to a temperature dependent $\Omega_{0}(T)$ scaling approximately as the CDW order parameter. In particular $\Omega_{0}\to 0$ as $T\to T_{CDW}$ and $\Omega_{0}(T=0)$ is much smaller than the bare frequency $\omega_{0}$Lee _et al._ (1974); Cea and Benfatto (2014). As a consequence, near $\Omega_{0}$ Eq. (6) assumes the form of Eq. (1), with $\Gamma_{ph}\simeq-\Sigma^{\prime\prime}(\Omega_{0})$ and $Z_{eff}\simeq 2g_{CDW}^{2}\omega_{0}\chi^{\prime 2}_{R,CDW}(\Omega_{0})$. When entering the superconducting state the phonon self-energy is modified in two ways. First, the two CDW bands $\xi_{\pm}=\mp\sqrt{\epsilon_{\bf k}^{2}+\Delta_{CDW}^{2}\gamma_{\bf k}^{2}}$ are further gapped by the superconducting gap $\Delta_{SC}$, so that the dispersion becomes $E_{\pm}=\sqrt{\epsilon_{\bf k}^{2}+\Delta_{CDW}^{2}\gamma_{\bf k}^{2}+\Delta_{ SC}^{2}}$. This reflects in general in a change of the self-energy (8). However, in contrast to the standard case of the metal-to-superconductor transitionAllen _et al._ (1997); Zeyher and Zwicknagl (1990), the change of quasiparticle dispersion has weak effects on the phonon lineshape, due to the fact that the electronic excitations are already gapped by the CDW gap at $T>T_{c}$. This is explicitly shown in Fig. 6, where we report the change in the phonon spectral function due only to the modifications of the self-energy (8) below $T_{c}$. The phonon slightly softens, but no new peak develops below $2\Delta_{SC}$. However, in the mixed state the phonon self-energy acquires a new term which represents in diagrammatic language a vertex corrections in the particle-particle channelLittlewood and Varma (1982); Browne and Levin (57); Cea and Benfatto (2014). The full self-energy is then computed as: \[\Sigma(i\Omega_{n})=2g_{CDW}^{2}\omega_{0}\chi_{CDW}(i\Omega_{n})-2g_{CDW}^{2} \omega_{0}\chi^{2}_{SC,CDW}/X_{SC}\] (10) Here $\chi_{SC,CDW}=\langle\delta\Delta_{SC}\delta\Delta_{CDW}\rangle$ is the response function measuring the change in the superconducting amplitude induced by a fluctuation of the CDW gap, and vice versa. As we mentioned in the main text, it provides a direct coupling between the phonon mode and the Higgs mode, whose fluctuations are described by the function $X_{SC}(i\Omega_{n})=2/U+\chi_{SC}(i\Omega_{n})$, where $\chi_{SC}=\langle\delta\Delta_{SC}\delta\Delta_{SC}\rangle$. More specifically, the Higgs resonance occurs when $X^{\prime}_{SC}(\omega=2\Delta_{SC})=0$. As a consequence, the equation (9), which defines the poles of the phonon propagator probed by the Raman response (6), acquires an additional solution at $\omega<2\Delta_{SC}$, see Fig. 6. In addition the phonon signature at $\Omega_{0}$ changes considerably, with a broadening and hardening analogous to the experimental observations. We also note that, as discussed in Ref. Cea and Benfatto (2014), the broadening of the Higgs resonance due to its decay in particle-hole excitations is less pronounced in the coexisting CDW+SC state, since the quasiparticle spectrum above $2\Delta_{SC}$ remains partly gapped by the CDW gap. This explains why the Higgs resonance coupled to the phonon mode appears so sharp in 2H-NbSe${}_{2}$. The above Eq.s (6), (8) and (10) are generic to any band structure, and explain the general mechanism of generation of the amplitudon below $T_{CDW}$ and of its Higgs signature in the mixed CDW+SC state. To evaluate the pressure dependence of the Raman spectra we computed explicitly their evolution in our toy model. The various susceptibilities listed above are then easily derived using the definitions of the various operators and of the Green’s function (4) above. We then obtain for the fermionic susceptibility the general structure: \[\chi_{A}=\sum_{\bf k}\frac{R_{A}({\bf k})}{E_{\bf k}((i\Omega_{n})^{2}-4E_{\bf k }^{2})}\tanh(\beta E_{\bf k}/2)\] (11) where $\beta=1/T$, $E_{\bf k}=\sqrt{\epsilon_{\bf k}^{2}+\Delta_{CDW}^{2}\gamma^{2}_{\bf k}+\Delta _{SC}^{2}}$ and the form factor $R_{A}(\bf k)$ depends on the susceptibility under consideration: \[R_{R,CDW} = 8\Gamma({\bf k})\Delta_{CDW}\gamma^{2}_{\bf k}\epsilon_{\bf k},\] (12) \[R_{CDW}=R_{SC} = 4\epsilon_{\bf k}^{2},\] (13) \[R_{SC,CDW} = -8\Delta_{SC}\Delta_{CDW}\gamma^{2}_{\bf k}\] (14) <figure><img src="content_image/1704.04042/x6.png"><figcaption>Figure 6: Change in the phonon spectral function in the transition from theCDW (solid red line) to the superconducting state with (solid blue line) andwithout (dashed blue line) coupling to the Higgs. The bare phonon energy istaken at ω0=0.16t=3.2(2ΔSC), so the softening of the phonon frequency from ω0to a value Ω0 of the order of 2ΔSC is due to the coupling to the CDW amplitudefluctuations, encoded in self-energy (8). Below Tc the bare self-energy (8) isweakly affected (dashed blue line) by the superconducting gap opening, due tothe fact that the electronic excitations were already gapped by the CDW gap.However, the coupling to the Higgs encoded in the full self-energy (10) leadsto a sharp additional sub-gap peak, and to a hardening and broadening of thephonon spectral function.</figcaption></figure> In the present model the Raman response is found different from zero only in the $A_{1g}$ channel where $\Gamma({\bf k})=\cos k_{x}+\cos k_{y}\propto\epsilon_{\bf k}$, leading to a term proportional to $\epsilon_{\bf k}^{2}$ in Eq. (12) that survives under momentum integration. For the same reason, we find that the phonon does not couple to the total charge density, as obtained by setting $\Gamma({\bf k})=1$ in Eq. (12). This also means that the phonon response is not screened by the long-range Coulomb interactions, mediated by density fluctuations. Even though the present model does not capture the microscopic band structure of 2H-NbSe${}_{2}$, we expect that a similar mechanism is at play in this system as well, explaining the lack of Coulomb screening of the A${}_{1g}$ CDW phonon observed experimentally. The CDW and superconducting order parameters are computed by solving self-consistently the two equations: \[\Delta_{CDW} = \Delta_{CDW}\frac{2g^{2}}{\omega_{0}N}\sum_{\bf k}\frac{\gamma_{ \bf k}^{2}}{E_{\bf k}}\tanh(\beta E_{\bf k}/2)\] (15) \[\Delta_{SC} = \Delta_{SC}\frac{U}{2N}\sum_{\bf k}\frac{1}{E_{\bf k}}\tanh(\beta E _{\bf k}/2)\] (16) Here we performed the calculations for the following choice of parameters at $P=0$: $\omega_{0}=0.16t$, $g_{0}=0.14t$, $U_{0}=0.97t$. To simulate the effect of pressure we suppressed progressively the CDW effective coupling $g^{2}/\omega_{0}$ (see Eq. (15 above) up to the value $g=0.117t$, where CDW order disappears. As a consequence $(g_{0}^{2}-g^{2})/g_{0}^{2}\simeq 2(g-g_{0})/g_{0}=\alpha$ with the definition of the $\alpha=2(g_{0}-g)/g_{0}$ given in the text. Simultaneously, we slightly suppressed the SC coupling down to $U=0.79t$, in order to keep $T_{c}$ almost constant as in the experiments. Notice that in principle in the present model at half-filling the direct coupling between the Higgs mode and the Raman density, $\chi_{R,SC}=\langle\rho_{R}\delta\Delta_{SC}\rangle$ is not zero. This is a quite peculiar effect of the band structure considered, that is not expected to hold in 2H-NbSe${}_{2}$ where the bands are approximately parabolic. In this situation indeed the Raman density scales as the total density and the direct coupling of the Higgs mode to the e.m. field is vanishingly small, as in ordinary superconductorsCea and Benfatto (2014); Cea _et al._ (2015, 2016). For this reason we did not include explicitly this coupling in the above calculations, and we refer the reader to Ref. 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0903.3939	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 64494, "num_imgs": 9, "llama3_tokens_count": 19770 }	[ "content_image/0903.3939/x1.png", "content_image/0903.3939/x3.png", "content_image/0903.3939/x4.png", "content_image/0903.3939/x6.png", "content_image/0903.3939/x7.png", "content_image/0903.3939/x9.png", "content_image/0903.3939/x11.png", "content_image/0903.3939/x12.png", "content_image/0903.3939/x13.png" ]	# Characterizing multipartite symmetric Dicke states under the effects of noise S. Campbell M. S. Tame M. Paternostro School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom February 20, 2024 ###### Abstract We study genuine multipartite entanglement (GME) in a system of $n$ qubits prepared in symmetric Dicke states and subjected to the influences of noise. We provide general, setup-independent expressions for experimentally favorable tools such as fidelity- and collective spin-based entanglement witnesses, as well as entangled-class discriminators and multi-point correlation functions. Besides highlighting the effects of the environment on large qubit registers, we also discuss strategies for the robust detection of GME. Our work provides techniques and results for the experimental communities interested in investigating and characterizing multipartite entangled states by introducing realistic milestones for setup design and associated predictions. pacs: 03.67.-a, 03.67.Mn, 42.50.Dv, 03.67.Lx The ability to classify entangled states [1; 2; 3] and quantify their degree of correlation [4] is advancing together with the capacity to experimentally produce interesting and useful forms of quantum correlated many-body systems. The range of experimentally available multipartite entangled states is witnessing a steady growth, stimulated by remarkable achievements such as the generation of eight-qubit W states [1; 5], ten-qubit GHZ-like states [6; 7] and various sizes of cluster states [8; 9]. Very successful linear optics experiments have been conducted exploring the many entangled classes of up to four-photon quantum correlated states [11; 12; 10] and such possibilities are foreseeable in other physical systems as well [13]. Very recently, this range of interesting states has been enriched with the experimental generation of six-qubit symmetric Dicke states and their evaluation in multiparty quantum networking protocols [14; 15; 16], together with the exploration of a substantial part of the Dicke class of entangled states. Dicke states of $n$ qubits are defined as the eigenstates of both the total spin operator $\hat{J}^{2}=\sum_{k=x,y,z}\hat{J}^{2}_{k}$ and its $z$-component $\hat{J}_{z}$, where $\hat{J}_{k}=(1/2)\sum^{n}_{j=1}\hat{\sigma}^{j}_{k}$ and $\{\hat{\sigma}^{j}_{x},\hat{\sigma}^{j}_{y},\hat{\sigma}^{j}_{z}\}$ is the set of Pauli matrices of qubit $j$. The model from which this class of states arise has been the focus of extensive investigations regarding super-radiance, quantum phase transition, correlation and entropic properties [17]. A Dicke state of $n$ qubits and $k$ excitations $\|{D^{(k)}_{n}}\rangle$ is given by \[\|D^{(k)}_{n}\rangle=\frac{1}{\sqrt{C^{k}_{n}}}\sum_{l}\hat{P}_{l}\|0_{0}\dots 0 _{k}1_{k+1}\dots 1_{n}\rangle,\] (1) where $C^{k}_{n}$ is the binomial coefficient and $\hat{P}_{l}$ is the set of all distinct permutations of $0$’s and $1$’s. Symmetric Dicke states having $k=n/2$ are particularly interesting in virtue of the fact that they are associated with the largest eigenvalues of the set of observables $\{\hat{J}^{2},\hat{J}_{z}\}$. Important theoretical studies [18] and experimental progress in characterizing these states have been accompanied by the development and optimization of special tools explicitly designed in order to “detect” the presence of genuine multipartite entanglement (GME) in a state. We define GME as states in which all subsystems are entangled with each other. In this context, a remarkable contribution has come from the introduction and use of _entanglement witness_ operators (see [4; 19] and references within), which with often only modest experimental effort allow for the discrimination between separable, biseparable and fully GME states without requiring the complete knowledge of the state at hand. In fact, it is usually the case that the complete detection and quantification of entanglement in a state requires knowledge of the full density matrix of the system. As this may not always be possible, entanglement witnesses provide us with viable ways to detect entanglement through only partial information. Various forms of witnesses have been formulated recently, the most prominent being based on the use of state fidelity [20] and collective spin-qubit operators [21; 22]. It is very important to study the resilience of such characterization tools to the influences of unavoidable interactions between the constituents of a given system and their surrounding environment. These give rise to phenomena of dissipation and decoherence which have a negative impact on the entanglement content of a state. Knowing beforehand how a chosen method for GME-characterization is able to cope with such spoiling mechanisms is not only interesting, but also pragmatically useful. It allows one to make predictions about the performance of a setup and to determine in which direction technological progress should be made, in order to circumvent noise and reliably reveal quantum effects for fundamental studies and applications in quantum information tasks. Our work is performed precisely in this important direction. We concentrate on the class of symmetric Dicke states of $n$ qubits, which exhibit interesting quantum properties and are usable resources for quantum networking tasks such as quantum secret sharing, telecloning and open-destination teleportation [10; 15]. We study the behavior of a variety of methods for revealing Dicke-class GME, including fidelity- and collective spin-based entanglement witnesses, as well as less-explored but valuable tools. We provide many setup-independent results that can be adapted to those experimental situations where local measurement settings can be reliably and easily arranged (as in Refs. [11; 13; 18]). This is a crucial requirement of the detection schemes addressed throughout this work. We also discuss feasible techniques for the improvement of the resilience of GME-detection to the influences of noise. The remainder of the paper is organized as follows. Section I briefly introduces the types of noisy channels considered in this study. Section II studies the behavior, under noisy mechanisms of collective-spin based entanglement witnesses of the form experimentally implemented in Refs. [15; 16] and proposes an original way to gain robustness. In Section III we extend our investigation to more common fidelity-based entanglement witnesses. These are subjected to filtering operations in a way that stretches their tolerance to environmental effects and allows for a faithful detection of GME. We apply our techniques also to the class of $W$ states and highlight some unexpected differences among the channels. In Section IV, the GME properties of symmetric Dicke states are highlighted by exploiting the interesting network of entanglement shared by reduced two-qubit states obtained by tracing out $n-2$ elements. Section V approaches the problem of reliably discerning the entanglement class of a given noisy state. In Section VI we explore ways to reveal the behavior of multi-qubit quantum coherence under the influence of noise, while Section VII briefly summarizes our findings. Finally, some technical details are presented in two appendices. ## I Decoherence Models In our study, we address a selection of physically relevant multi-qubit noisy channels affecting the class of states in question. Our choice encompasses a wide range of possible mechanisms that are likely to affect a given experimental setting designed to achieve multipartite entanglement. We use an effective picture for the action of a completely positive trace-preserving map given by the operator-sum representation [20]. Within this formalism, a single-qubit noisy process is described by a set of Kraus operators $\{\hat{K}_{\mu}\}$, satisfying the completeness property $\sum_{\mu}\hat{K}^{\dagger}_{\mu}\hat{K}_{\mu}=\hat{\openone}$, such that, calling $\rho_{0}$ the initial state of a qubit, its evolution is given by $\rho_{ch}=\$_{ch}(\rho_{0})=\sum_{\mu}\hat{K}_{\mu}\rho_{0}\hat{K}_{\mu}^{\dagger}$. In what follows, for the sake of convenience, $\gamma$ indicates the characteristic channel’s influence rate, regardless of its specific nature and $ch$ is a label for the channel. We start by considering the Kraus decomposition of a zero-temperature _amplitude damping_ (AD) mechanism (2) This physically corresponds to an energy dissipation process: the system undergoing AD has a finite probability $e^{-\gamma}$ to lose an excitation (here, $\gamma$ is an effective dimensionless rate characterizing the whole process). The second process we consider is represented by pure _phase damping_ (PD) (or dephasing), which is a phase-scrambling and energy-preserving mechanism described by the two operators \[\hat{K}^{pd}_{0}=\sqrt{\frac{1+e^{-\gamma}}{2}}\hat{\openone},~{}~{}~{}~{}~{} \hat{K}^{pd}_{1}=\sqrt{\frac{1-e^{-\gamma}}{2}}\hat{\sigma}_{z}.\] (3) It is easy to see that the action of PD on a single-qubit density matrix is to exponentially decrease the off-diagonal terms (at an effective rate $\gamma$), leaving the populations unaffected. Finally, we consider a _depolarizing_ channel (DP), which (with probability $\gamma$) mixes a given _one-qubit_ state with the maximally mixed state $\hat{\openone}/2$. Its action is given by the four-operator Kraus representation \[\hat{K}^{dp}_{0}=\sqrt{1-\frac{3\gamma}{4}}\hat{\openone},~{}~{}\hat{K}^{dp}_{ k}=\sqrt{\frac{\gamma}{4}}\hat{\sigma}_{k}~{}~{}~{}~{}(k=1,2,3),\] (4) which gives $\rho_{dp}=(1-\gamma)\rho_{0}+\gamma\hat{\openone}/2$. Therefore, the effect of a DP channel is to effectively add white noise to a given single-qubit state. This correspondence will become useful in our study. The above situation is different from that of a _collective_ DP mechanism, which would add white noise to a multipartite state of $n$ qubits $\rho_{mp}$, leading to $(1-\gamma)\rho_{mp}+\gamma\hat{\openone}/2^{n}$. When considering $n$-partite registers, the effects of equal noisy channels, each affecting an individual qubit of the system, can be accounted for by considering $q^{n}$$n$-party tensor products of Kraus operators $\hat{\cal K}^{ch}_{j}$’s, where $q$ is the number of channel operators of a single-qubit channel. That is \[\rho_{ch}=\sum^{q^{n}}_{j=1}\hat{\cal K}^{ch}_{j}\rho_{mp}\hat{\cal K}^{ch{ \dagger}}_{j},\] (5) Our task is now to provide an analysis as general as possible of the effects of such environmental channels on a variety of experimentally viable tools for multipartite entanglement. ## II Collective-spin based entanglement witness Collective-spin operators are useful tools for the investigation of GME. Spin-squeezing inequalities fall into this class [21] and have been extensively studied as well as experimentally implemented. More recently, collective-spin operators that are not directly related to spin squeezing have been formulated and shown to be particularly effective when symmetric, permutation invariant states are studied. One can construct the witness operator [25] \[\hat{\cal W}^{s}_{n}=b_{bs}\hat{\openone}-(\hat{J}^{2}_{x}+\hat{J}^{2}_{y}),\] (6) where $b_{bs}$ is the maximum expectation value of $\hat{S}_{n}=\hat{J}^{2}_{x}+\hat{J}^{2}_{y}$ over the class of biseparable states of $n$ qubits. Finding $\langle\hat{\cal W}^{s}_{n}\rangle\!<\!0$ for a given state implies GME. The biseparable bound, $b_{bs}$ can be numerically calculated (see Appendix A and Ref. [25]). The witness can be implemented with only two local measurement settings making it experimentally appealing and realizable in many physical settings (linear optics, circuit or cavity quantum-electodynamics). In particular, this tool has been used in [11; 16] for the case of four and six-qubit states. Quite often, Eq. (6) fails to detect GME in non-ideal symmetric Dicke states which have been affected by noise at their generation stage. Through a suitable modification, as experimentally demonstrated in [15], we can provide such a witness with greater flexibility in detecting GME in noise affected symmetric Dicke states. Let us introduce the generalized collective-spin witness \[\hat{\cal W}^{s}_{n}(\alpha)=b_{bs}(\alpha)-\hat{S}_{n}(\alpha)~{}~{}(\alpha \in\mathbb{R}),\] (7) where $\hat{S}_{n}(\alpha)=\hat{J}^{2}_{x}+\hat{J}^{2}_{y}+\alpha\hat{J}^{2}_{z}$. Here, we shall discuss how Eq. (7) offers more robustness to the noisy channels introduced in Sec. I than the standard Eq. (6). We notice that the bi-separability bound is now a function of new parameter $\alpha$. Using numerics it is seen that in general $b_{bs}(\alpha)\!<\!b_{bs}(0)$ for $\alpha<0$, which implies that at $\alpha\neq{0}$ the threshold for detection of GME is lowered. Consequently, we restrict our study to the case of negative $\alpha$. Let us start with an AD channel and its effect on the state $\|D^{(n/2)}_{n}\rangle$. We call $\rho_{ad}$ the channel-affected version of this state, $\rho_{ad}=\$_{ad}(\|D^{(n/2)}_{n}\rangle\langle D^{(n/2)}_{n}\|)$, calculated as described in Sec. I. One finds \[\text{Tr}[\hat{S}_{n}(\alpha)\rho_{ad}] =\!\frac{n}{2}+\frac{n\alpha}{4}(n-1)(1-e^{-\gamma})^{2}\] (8) \[+\frac{n^{2}}{4}e^{-\gamma}+\frac{n\alpha}{4}(1-e^{-2\gamma}).\] <figure><img src="content_image/0903.3939/x1.png"><figcaption>Figure 1: (Color Online) (a) Collective-spin based entanglement witness⟨^Sn(α)⟩ for an AD-affected \|D(2)4⟩ state. We show the lines corresponding toγ∈[0,0.3] and highlight the cases where the modified entanglement witnessturns out to be advantageous: each full line is such that ⟨^Ws4(0)⟩>0 with⟨^Ws4(α<0)⟩<0 in a region of values of α. The behavior of the biseparabilitybound is also shown. (b) Same as panel (a) but for an AD-affected \|D(3)6⟩state. We show the lines corresponding to γ∈[0,0.16]. In both panels, theshading highlights the biseparability region.</figcaption></figure> This formula is easily proven by noticing that, under a given noisy channel, the set of Pauli matrices of qubit $j$ changes as $\hat{\sigma}^{j}_{k,ch}\!=\!\sum_{\mu}\hat{K}^{{\dagger}}_{\mu}\hat{\sigma}^{j }_{k}\hat{K}_{\mu}$. For AD, this leads to [27] \[\hat{\sigma}^{j}_{k,ad}=e^{-\gamma/2}\hat{\sigma}^{j}_{k},~{}~{}\hat{\sigma}^{ j}_{z,ad}=(1-e^{-\gamma})\hat{\openone}+e^{-\gamma}\hat{\sigma}^{j}_{z},\] (9) where $k=x,y$. From these, it is easy to check that $\text{Tr}[\hat{\sigma}^{\otimes 2}_{x,y}\otimes\hat{\openone}^{\otimes{(n-2)}} \rho_{ad}]\!=\!ne^{-\gamma}/(2n-2)$ while $\text{Tr}[\hat{\sigma}^{\otimes 2}_{z}\otimes\hat{\openone}^{\otimes{(n-2)}} \rho_{ad}]\!=\!(1-e^{-\gamma})^{2}-e^{-\gamma}/(n-1)$. By using the explicit decomposition of $\hat{J}^{2}_{k}$ given in Appendix A it is straightforward to see that \[\text{Tr}[(\hat{J}^{2}_{x}+\hat{J}^{2}_{y})\rho_{ad}]\!=\!\frac{n }{4}(2+ne^{-\gamma}),\] (10) \[\text{Tr}[\hat{J}^{2}_{z}\rho_{ad}]\!=\!\frac{1}{4}[n(n-1)(1-e^{- \gamma})^{2}+{n}(1-e^{-2\gamma})],\] which leads directly to Eq. (8). We can infer two consequences. From the first identity of Eq. (10) we see that for $\gamma\geq\ln[n^{2}/(4b_{bs}-2n)]$, we have $b_{bs}(0)\geq\text{Tr}[(\hat{J}^{2}_{x}+\hat{J}^{2}_{y})\rho_{ad}]$. This signals the failure of Eq. (6) and quantifies the amount of AD noise a given experimental set-up can tolerate. The second identity of Eq. (10) shows that, different to a pure symmetric Dicke state, $\langle\hat{J}^{2}_{z}\rangle\neq{0}$ for $\rho_{ad}$ and strongly depends on $\gamma$. From Eq. (8) one finds that the dependence of the expectation value of $\hat{S}_{n}(\alpha)$ on $\alpha$ is linear, with a gradient coefficient determined by $\langle\hat{J}^{2}_{z}\rangle$, justifying the inclusion of such a term in the witness. For a decohered state such that $b_{bs}(0)\geq\text{Tr}[(\hat{J}^{2}_{x}+\hat{J}^{2}_{y})\rho_{ad}]$, the inclusion of the $z$-dependent term may be able to _pull_ the expectation value of $\hat{\cal W}^{s}_{n}(\alpha)$ below zero, for a set value of $\alpha$. This is possible if $\langle\hat{J}^{2}_{z}\rangle$ is smaller than the gradient coefficient of the tangent to $b_{bs}(\alpha)$ at that point. In Fig. 1 we show two instances of such a possibility for the cases of $n=4$ and $6$, for which $b_{bs}(0)\sim 5.23$ and $11.018$ respectively (see Appendix A and Refs. [15; 16]). Let us discuss the $n=6$ case: At $\gamma\geq 0.116$ the witness in Eq. (6) cannot detect GME in any AD-affected $\|D^{(3)}_{6}\rangle$. However, we find instances of decohered states that can be detected as GME via $\hat{\cal W}_{6}^{s}(\alpha<0)$, although $\gamma>0.116$. Such a possibility goes beyond the example given here. In fact, we have strong numerical evidence that for $\alpha=0$, GME cannot be revealed for $\gamma\geq{0.076}$ for an AD-affected $\|D^{(4)}_{8}\rangle$. However, for $\alpha\in[-5,-0.1]$ such states are detected as GME via our modified witness $\hat{\cal W}_{6}^{s}(\alpha)$ despite the fact that they correspond to $\gamma\in[0.18,0.20]$. Although a proof for any $n$ is difficult (due to the computational problem of quantifying $b_{bs}(\alpha)$ for large registers of qubits), this shows robustness of $\hat{\cal W}^{s}_{n}(\alpha)$ under the influences of AD channels. <figure><img src="content_image/0903.3939/x3.png"><figcaption>Figure 2: (Color Online) (a) ⟨^Sn(α)⟩ for a DP-affected \|D(2)4⟩ state. We showthe lines corresponding to γ∈[0,0.16] and highlight the cases where themodified entanglement witness turns out to be advantageous: each full line issuch that ⟨^Ws4(0)⟩>0 with ⟨^Ws4(α<0)⟩<0 in a region of values of α. Thebehavior of the biseparability bound is also shown. (b) Same as in (a) but forn=6. In both panels, the shading highlights the biseparability region.</figcaption></figure> A similar analysis conducted with respect to DP noise leads to the relation \[\hat{\sigma}^{j}_{k,dp}=(1-\gamma)\hat{\sigma}^{j}_{k}~{}~{}~{}(k=x,y,z),\] (11) which tells us that the $\langle\hat{\sigma}^{\otimes{2}}_{x,y}\otimes\hat{\openone}^{\otimes(n-2)}\rangle$ are given by the expressions valid for a pure $\|D^{(n/2)}_{n}\rangle$ state, multiplied by $(1-\gamma)^{2}$. Following the same procedure as in the AD case leads us to \[\text{Tr}[(\hat{J}^{2}_{x}+\hat{J}^{2}_{y})\rho_{dp}]\!=\!\frac{n }{2}+\frac{n^{2}}{4}(1-\gamma)^{2}\] (12) \[\text{Tr}[\hat{J}^{2}_{z}\rho_{dp}]\!=\!\frac{n}{4}\gamma(2- \gamma),\] from which the expectation value of the collective-spin based entanglement witness can be determined. In this case too, the effectiveness of using $\hat{\cal W}_{n}^{s}(\alpha)$ is clearly revealed by the existence of situations where GME at non-zero $\alpha$ is detected, as shown in Fig. 2, while this is not the case if the standard witness is used. Finally, we address the PD channel, for which we have [27] \[\hat{\sigma}^{j}_{k,pd}=e^{-\gamma(1-\delta_{kz})}\hat{\sigma}^{j}_{k}~{}~{}~{ }(k=x,y,z),\] (13) where we have introduced the Kronecker-delta function $\delta_{kz}$ which is equal to $1$ for $k=z$ and $0$ otherwise. This clearly implies that a symmetric Dicke state affected by PD noise will keep the pure-state property $\text{Tr}[\hat{J}^{2}_{z}\rho_{pd}]=0$, for any $n$ and $\gamma$, therefore making the strategy highlighted so far ineffective. The expectation value of the witness operator reads, regardless of $\alpha$, as \[\text{Tr}[\hat{S}_{n}(\alpha)\rho_{pd}]\!=\!\frac{n}{2}+\frac{n^{2}}{4}e^{-2 \gamma},\] (14) which is unable to detect GME as soon as $\gamma\geq\frac{1}{2}\ln[n^{2}/(4b_{bs}-2n)]$. It remains to be checked whether, for instance, an approach such as the one used in Ref. [16; 26] could be used in order to build up a resilient entanglement witness in this case as well. ## III Fidelity-based entanglement witness and filtering ### Symmetric Dicke States We now investigate fidelity-based witnesses [4; 19]. Despite generally requiring more local measurement settings than collective-spin based witnesses, they offer a more specific state characterization. Assuming that the state to study remains close to the Dicke class, the fidelity-based entanglement witness with the following general form can be used to detect GME \[\hat{\mathcal{W}}_{n}=c_{n}\hat{\openone}-\|D^{(n/2)}_{n}\rangle\langle D^{(n/2 )}_{n}\|,\] (15) where $c_{n}$ is the maximum overlap between $\|D^{(n/2)}_{n}\rangle$ and any possible biseparable state of $n$ qubits. Quantitatively, $c_{n}=n/(2n-2)$ for the class of symmetric Dicke states [19; 25]. When the AD-affected symmetric Dicke state is studied, based on our previous considerations, it is straightforward to find that \[\text{Tr}[{\hat{\cal W}_{n}\rho_{ad}}]=\frac{n}{2n-2}-e^{-\frac{n\gamma}{2}},\] (16) regardless of $n$. <figure><img src="content_image/0903.3939/x4.png"><figcaption>Figure 3: (Color Online) (a) Expectation value of an entanglement witness fora symmetric six-qubit Dicke state undergoing AD at a dimensionless rate γ. Thesolid curve corresponds to the unfiltered witness while the dashed curve isfor the filtered one. By filtering one can increase the range of γ where thewitness is still able to detect GME. (b) Expectation value of the entanglementwitness for a symmetric six-qubit Dicke state undergoing PD type of noise. Thesolid curve is for the unfiltered witness while the dashed curve correspondsto the filtered case. No advantage is achieved, in this case, upon filtering.</figcaption></figure> For GME detection we require $\text{Tr}[{\hat{\cal W}_{n}\rho_{ad}}]<0$, which is guaranteed for $\gamma<(2/n)\ln[(1/c_{n})]$. For $n=4$ ($n=6$) the witness is implementable with $9$ ($21$) local measurement settings (see Appendix B and Refs. [15; 16]). The limiting amount of sustainable AD noise for $n=4$ is $\gamma=0.203$ and decreases as larger Dicke states are examined (we have $\gamma=0.170$ for $n=6$, see Fig. 3(a)). Fig. 4(a) shows this “pulling-back” effect for $n=4,..,50$. As previously done, we seek to devise experimentally-friendly techniques to detect GME for larger amounts of noise. To achieve this task, we apply _filtering operations_ [23; 19] to the fidelity-based entanglement witness. The local nature of the filters cannot alter the amount of entanglement in a state. However, they may allow for an increase in the noise allowed before a witness starts failing to detect GME. For $n$-qubit states, we thus construct filters of the form \[\hat{\cal F}=\bigotimes^{n}_{j=1}\hat{\cal F}_{j}\] (17) where $\hat{\cal F}_{j}$’s are local invertible operators. This technique has already been investigated for a variety of states in Refs. [1; 6; 8; 24] and here we extend and generalize its use to arbitrarily-sized symmetric Dicke states. We must be careful to ensure that the correct normalization is taken. To this end we impose the constraint Tr[$\hat{\mathcal{W}}_{n}$]=Tr[$\hat{\mathcal{W}}_{n}^{\mathcal{F}}$] with $\hat{\mathcal{W}}_{n}^{\mathcal{F}}$ the filtered witness operator. This is sufficient to ensure that the expectation value arising from the new witness is comparable to the expectation value from the unfiltered one. Therefore \[\hat{\mathcal{W}}^{\mathcal{F}}_{n}=\frac{\textrm{Tr}[\hat{\mathcal{W}}_{n}] \hat{\mathcal{F}}\hat{\mathcal{W}}_{n}\hat{\mathcal{F}}^{\dagger}}{\textrm{Tr} [\hat{\mathcal{F}}\hat{\mathcal{W}}_{n}\hat{\mathcal{F}}^{\dagger}]}.\] (18) Filtering may lead to an increase in the number of required local measurement settings. However, there are indications that the class of filters given by [24] \[\hat{\mathcal{F}}_{j}=\left(\begin{array}[]{ll}1&0\\ 0&y_{j}\\ \end{array}\right)\] (19) with $y_{j}$ a positive real number, guarantees for the smallest possible number of necessary measurement settings. <figure><img src="content_image/0903.3939/x6.png"><figcaption>Figure 4: (Color Online) (a) Tr[^Wnρad] plotted against the dimensionless ADrate γ for n=4→50. Solid (Dashed) lines are for unfiltered (filtered)witnesses. As n grows, the filtering-induced gain is reduced. (b) Values of γat which the filtered witness surpasses the threshold value −10−3 and theunfiltered one starts to become non-negative with n. The lower (upper) curveis for the unfiltered (filtered) case. The advantage achieved upon filteringis clear as it slowly decays as n increases.</figcaption></figure> A direct calculation, performed using (for convenience) $y_{j}=y~{}(\forall{j=1,..,n})$, leads to \[\begin{split}\text{Tr}[{\hat{\mathcal{W}}^{\mathcal{F}}_{n}\rho_{ ad}}]&\!=\!\frac{(2^{n}n-2n+2)e^{-\frac{n\gamma}{2}}}{n\left(y^{2 }+1\right)^{n}-(2n-2)y^{n}}\\ &\times[\frac{n}{2n-2}\left(y^{2}+e^{\gamma}-1\right)^{n/2}-y^{n} ].\end{split}\] (20) This is then minimized with respect to the filtering parameter $y$ for any set value of $\gamma$, ensuring that the expectation value of the witness would not depend on $y$ at all [24]. The effect of this procedure can be clearly seen in Fig. 3(a): the local filtering increases the amount of decoherence tolerated while the witness is still able to detect GME in a symmetric Dicke state [28]. The actual detection of GME via negativity of a witness operator clearly depends on the errors associated with the experimentally determined value of $\text{Tr}[{\hat{\mathcal{W}}^{\mathcal{F}}_{n}\rho_{ad}}]$: a small (albeit negative) value is likely to be covered up by the corresponding error bar. Although the quantification of any acceptable lower bound for significant GME detection is a setup-dependent issue, based on current linear optics implementation, it is reasonable to expect that, upon collection of a sufficiently large sample of data, values of $\text{Tr}[{\hat{\mathcal{W}}^{\mathcal{F}}_{n}\rho_{ad}}]\sim-10^{-3}$ can still be discerned from zero. We thus fix a threshold of this order of magnitude and seek the smallest value of $\gamma$ at which the filtered witness (for a given $n$) surpasses it. This provides a practical lower limit to the mathematically rigorous performance of such tool. Fig. 4(b) reveals the advantage acquired upon filtering for $n=4\rightarrow{20}$: the gap with respect to the unfiltered case is quite considerable and marks the success of this strategy. Quantitatively, the upper points in Fig. 4(b) are well fitted by the function $\ln[(\frac{n-b_{1}}{n})^{\frac{2}{n}}{c_{1}}^{\frac{a_{1}}{n}}]$ with $a_{1}=1.54$, $b_{1}=-1.54$ and $c_{1}=4.73$. We now study the detection of GME via a fidelity-based witness in symmetric Dicke states affected by PD type of noise and find \[\text{Tr}[{\hat{\mathcal{W}}_{n}\rho_{pd}}]=\frac{n}{2n-2}-\frac{1}{C^{n/2}_{n }}\sum_{k}^{{n}/{2}}(C^{k}_{n/2})^{2}e^{-\gamma(n-2k)}.\] (21) One can easily find that the general qualitative features highlighted for the case of the AD-related study of an unfiltered witness hold in this case as well: as $n$ grows, the values of $\gamma$ at which the expectation value of Eq. (21) become positive are quickly pushed towards zero. However, an important remark is due: different to the case of AD noise, for PD the filtering technique employed above does not increase the range of tolerated noise. In fact, one can easily see that the net effect of the application of filtering operators is that $\text{Tr}[{\hat{\mathcal{W}}^{\cal F}_{n}\rho_{pd}}]=y^{n}\text{Tr}[{\hat{ \mathcal{W}}_{n}\rho_{pd}}]$. Therefore one cannot shift the value of $\gamma$ at which GME is unambiguously revealed. This is shown by the dashed line in Fig. 3(b), where we see that the expectation value of the filtered operator tracks its unfiltered version in the negative semi-space [29]. The reason behind such a behavior is clearly understood by the noticing that, \[\text{Tr}[{\hat{\mathcal{W}}^{\cal F}_{n}\rho_{pd}}]=\text{Tr}[\hat{\cal W}_{n }\sum^{2^{n}}_{j=0}\hat{\cal F}^{\dagger}\hat{\cal K}^{pd}_{j}\|D^{(n/2)}_{n} \rangle\langle D^{(n/2)}_{n}\|\hat{\cal K}^{pd{\dagger}}_{j}\hat{\cal F}],\] (22) where, as defined in Sec. I, $\hat{\cal K}^{pd}_{j}$’s are the $n$-qubit Kraus operators for the whole PD-affected register. We can thus interpret this equation as the expectation value of the unfiltered witness over a symmetric Dicke state affected by new Kraus operators, each given by $\hat{\cal F}^{\dagger}\hat{\cal K}^{pd}_{j}$. It is matter of direct calculation to see that the resulting density matrix $\tilde{\rho}_{pd}$ is simply $y^{n}\rho_{pd}$, thus demonstrating our claim. This is obviously not the case for an AD channel, in virtue of the different form of single-qubit Kraus operators required in that case. In order to exclude any limitations induced by the choice of the specific form of filtering operator, we repeated our analysis using general invertible operators, still finding no improvement upon filtering. A very similar situation is encountered for the case of DP channels acting on the qubits of the register. In this case, the number of Kraus operators involved in the evolution of a given state grows as $4^{n}$. This makes any approach to the problem intractable for large $n$. Therefore, we have not been able to produce a general formula for the fidelity-based entanglement witness nor a simple argument to explain why, in this case as well, filtering is non-effective in providing noise-robust GME detection. In Figs. 5(a) and (b) we give evidence of this for $n=4$ and $6$, comparing the unfiltered witness with a few instances of filtering. Moreover, panel (c) shows that the filtered witness has an absolute minimum at $y=1$ in its region of negativity, demonstrating the ineffectiveness of filtering. <figure><img src="content_image/0903.3939/x7.png"><figcaption>Figure 5: (Color online) (a) Expectation values of filtered and unfilteredfidelity-based entanglement witnesses for DP-affected \|D(2)4⟩ plotted againstthe characteristic DP rate γ (dimensionless). Each dashed line is for afiltered witness with y∈[0,3] at steps of 0.1. (b) Same as in panel (a) butfor \|D(3)6⟩. (c) Filtered entanglement witness for DP affected \|D(3)6⟩ againstboth y and γ. In its negativity region, the witness is minimized at y=1.</figcaption></figure> ### W-States With minimal changes to the analysis performed with respect to the symmetric Dicke states, we can also study $\|D^{(1)}_{n}\rangle$, which are commonly referred to as $n$-qubit W states [1], thus showing the versatility of the techniques employed in this paper. We consider the fidelity-based witness [19] \[\hat{\mathcal{W}}_{w}=\frac{n-1}{n}\openone-\|D^{(1)}_{n}\rangle\langle D^{(1)} _{n}\|,\] (23) where in order to distinguish this case for the previously treated one, $\omega_{ch}$ is used to indicate the density matrix resulting from the application of channel $ch$ to a pure $n$-qubit W state. Under AD, the expectation values of Eq. (23) for the unfiltered and filtered cases are given by \[\text{Tr}[\hat{\cal W}_{n}\omega_{ad}] =\frac{n-1}{n}\hat{\openone}-e^{-\gamma},\] (24) \[\text{Tr}[\hat{\cal W}^{\mathcal{F}}_{n}\omega_{ad}] =\frac{[2^{n}(n-1)-n][(n-1)(1-e^{-\gamma})-y^{2}{e}^{-\gamma}]}{n [(n-1)\left(y^{2}+1\right)^{n}-ny^{2}]}\] The effect of filtering once again increases the amount of tolerated noise for the witness to still be able to identify GME. Considering PD, we see that the addition of filtering has no benefit for the witnesses (for the same reasons explained in the discussion put forward in Sec. III.1). We find the expectation value of the witness to be \[\text{Tr}[\hat{\cal W}_{n}\omega_{pd}]=\frac{n-2}{n}-\frac{(n-1)e^{-2\gamma}}{ n}.\] (25) Under appropriate conditions, these results are in agreement with the study reported for $n=4$ in [24]. As for the DP channel, our investigation reveals that, at least for $n=4$ and $6$, a small advantage is gained by filtering the fidelity-based witness, although we do not have a closed analytical form for any $n$. ## IV Reduced-State Based Entanglement Witness We now consider a different manifestation of GME in symmetric Dicke states based on the observation of entanglement residing in the two-qubit reduced states of $\|D_{n}^{(n/2)}\rangle$. Tracing out $n\!-\!2$ qubits, we find \[\varrho=\alpha_{n}\|\psi^{+}\rangle\langle\psi^{+}\|+\frac{(1-\alpha_{n})}{2}[\|0 0\rangle\langle 00\|+\|11\rangle\langle 11\|]\] (26) with $\alpha_{n}=n/[2(n-1)]$ for $n\geq 4$ [21] and $\|\psi^{\pm}\rangle=(\|01\rangle\pm\|10\rangle)/\sqrt{2}$. Here, $\varrho$ has fidelity [20]$\langle\hat{F}_{\psi^{+}}\rangle={\rm Tr}[\|\psi^{+}\rangle\langle\psi^{+}\| \varrho]\equiv\alpha_{n}$ with respect to $\|\psi^{+}\rangle$. Using the fidelity-based entanglement witness [19] \[\hat{\cal W}_{r}=\frac{1}{2}\openone-\|\psi^{+}\rangle\langle\psi^{+}\|,\] (27) we have $\langle\hat{\cal W}_{r}\rangle=1/2-\langle\hat{F}_{\psi^{+}}\rangle$. Thus, a fidelity $\langle\hat{F}_{\psi^{+}}\rangle>1/2$ detects the presence of entanglement in the two-qubit reduced state $\varrho$. This is always possible for all choices of pairs of qubits in $\|D_{n}^{(n/2)}\rangle$, as $\alpha_{n}>1/2~{}\forall~{}n$. However, clearly, this may not be true for other states, including $\|D_{n}^{(n/2)}\rangle$ subjected to the noise channels outlined in the Section I. In general, we define two qubits of a multipartite state $\|\phi\rangle$ as _connected_ if their reduced density matrix is such that $\langle\hat{F}_{\psi^{+}}\rangle>1/2$. In this sense, a symmetric Dicke state gives rise to a connected set of reduced states. For any given $n$, one can construct a graph having qubits $j=1,..,n$ at its vertices. Two vertices are joined by an edge if and only if they are connected in the sense explained above. According to this definition, symmetric Dicke states give rise to fully connected (complete) graphs, as shown for $n=4$ and $6$ in Fig. 6 (a). In order to relate the witness $\hat{\cal W}_{r}$ to observables in an experiment, we decompose the fidelity $\langle\hat{F}_{\psi^{+}}\rangle$ into expectation values of Pauli operators as follows \[\langle\hat{F}_{\psi^{+}}\rangle\!=\!\frac{1}{4}(1+{\rm Tr}[\hat{\sigma}_{x} \otimes\hat{\sigma}_{x}\varrho]+{\rm Tr}[\hat{\sigma}_{y}\otimes\hat{\sigma}_{ y}\varrho]-{\rm Tr}[\hat{\sigma}_{z}\otimes\hat{\sigma}_{z}\varrho]),\] (28) showing that only the local measurement settings $\sigma_{x}^{\otimes 6}$, $\sigma_{y}^{\otimes 6}$ and $\sigma_{z}^{\otimes 6}$ are required for an experimental implementation. In fact, any two-qubit correlation ${\rm Tr}[\sigma_{k}^{i}\otimes\sigma_{k}^{j}\varrho]$ can be obtained from the corresponding data. This puts the method described here on an equal footing with the collective-spin based witness described in Sec. II, in terms of required experimental effort. We now investigate how this method copes with noise affecting the class of symmetric Dicke states. By using Eqs. (9), (11) and (13), together with Eq. (28), it is straightforward to show that \[\langle\hat{F}_{\psi^{+}}\rangle^{ad} =\frac{1}{2}e^{-2\gamma}(\alpha_{n}+e^{\gamma}(1+\alpha_{n})-1),\] (29) \[\langle\hat{F}_{\psi^{+}}\rangle^{dp} =\alpha_{n}(\gamma-1)^{2}-\frac{1}{4}(\gamma-2)\gamma,\] \[\langle\hat{F}_{\psi^{+}}\rangle^{pd} =\frac{1}{2}(1+e^{-2\gamma})\alpha_{n}.\] In Fig. 6(b) we show the behavior of $\langle\hat{F}_{\psi^{+}}\rangle^{ch}$ for the three channels. As soon as $\gamma\geq\text{ln}[(1+\alpha_{n}+(\alpha_{n}^{2}+6\alpha_{n}-3)^{1/2})/2]$, $\gamma\geq 1-(4\alpha_{n}-1)^{-1/2}$ and $\gamma\geq-\frac{1}{2}\text{ln}[(1-\alpha_{n})/\alpha_{n}]$ respectively, one finds disconnected sets in an AD-, DP- and PD-affected $\|D^{(n/2)}_{n}\rangle$ state using $\langle\hat{\cal W}_{r}\rangle$. These thresholds are larger than those corresponding to the use of a collective-spin based entanglement witness for any value of $n$ that we could quantitatively consider (see discussion in Sec. II), as a result of the smaller dimension of the states being tested. <figure><img src="content_image/0903.3939/x9.png"><figcaption>Figure 6: (a) Fully connected graphs for \|D(2)4⟩ and \|D(3)6⟩. The verticesrepresent qubits and the edges represent the presence of entanglement withintheir reduced-state, as detected by using ⟨^Wr⟩. (b) Effects of noise on thedetection of entanglement in the reduced two-qubit states of \|D(n/2)n⟩ using⟨^Wr⟩. Here, the solid (red), dashed (blue) and dotted (green) linescorrespond to ⟨^Fψ+⟩ for the AD, DP and PD channels respectively. The shadedgrey area corresponds to the region in which ⟨^Wr⟩ fails to detectentanglement and the corresponding graph in (a) becomes completelydisconnected.</figcaption></figure> ## V State discrimination via characteristic operators The variety of ways multipartite entanglement can be shared by an $n$-qubit register requires ways to determine if an experimental state belongs to one of the known classes of entanglement. This can be achieved by relying on the formalism of _state discriminators_ [11], which have been experimentally implemented and used in order to assess the classes of four-qubit entangled states [11]. Here, we study the reliability of such methods for symmetric Dicke states suffering effects of noisy channels. Consider multi-qubit operators having $\|D^{(n/2)}_{n}\rangle$ as a non-degenerate eigenstate associated with the largest possible eigenvalue of the operator’s spectrum. We call such operators characteristic of the state $\|D^{(n/2)}_{n}\rangle$. At least one operator having these features exists, _i.e._ the fidelity operator $\|D^{(n/2)}_{n}\rangle\langle D^{(n/2)}_{n}\|$. However, this is not the only option and other characteristic operators can be designed, requiring far less local measurement settings than the fidelity one. A systematic approach would require the decomposition of the fidelity operator into tensor products of single-qubit Pauli operators (see Appendix B). Out of them, only the genuine $n$-qubit correlators should be selected to be combined together in a way so as to construct a proper characteristic operator. For instance, for $n=6$ qubits prepared in $\|D^{(3)}_{6}\rangle$ we can build up the Bell-Mermin operator [31]$\hat{\cal B}_{D^{(3)}_{6}}\!=\!\sum_{k=x,y}\hat{\cal O}_{k}/20$ with $\hat{\cal O}_{k}\!=\!\hat{\sigma}^{1}_{k}\otimes[\hat{\sigma}^{\otimes{5}}_{k} -\sum_{l}\hat{P}_{l}(\hat{\sigma}^{\otimes{3}}_{k}\otimes\hat{\sigma}^{\otimes {2}}_{z}-\hat{\sigma}_{k}\otimes\hat{\sigma}^{\otimes{4}}_{z})]$. This is characteristic for $\|D^{(3)}_{6}\rangle$, which is an eigenstate with associated eigenvalue $1$ (the maximum within $\hat{\cal B}_{D^{(3)}_{6}}$’s spectrum). However, if we simply take the negative terms $\hat{\cal D}_{D^{(3)}_{6}}\!\propto\!-\sum_{k=x,y}\hat{\sigma}^{1}_{k}\otimes \sum_{l}\hat{P}_{l}(\hat{\sigma}_{k}^{\otimes 3}\otimes\hat{\sigma}^{\otimes{2 }}_{z})$ in this expression, we find that $\|D^{(3)}_{6}\rangle$ is still an eigenstate of maximum eigenvalue (enforced to be $1$ upon renormalization). Representatives of any other class of entanglement will achieve expectation values smaller than $1$. This can be used for effective entanglement-class discrimination. For instance, six-qubit GHZ states transformed by local unitary operations (LU) or stochastic local operations supported by classical communication (SLOCC) [20] yield expectation values no-larger than $0.833$. Thus, if an experimental state $\rho_{exp}$ of $n=6$, thought to be close to the symmetric Dicke family, gives $\langle\hat{\mathcal{D}}_{\rho_{exp}}\rangle>0.833$, one can exclude any GHZ-like character. This can be adapted to any other class of GME states. Here, without affecting the generality of our study, we concentrate on the discrimination between noise-affected symmetric Dicke states and the $n$-qubit GHZ class. We thus construct the streamlined characteristic operators \[\begin{split}\hat{\mathcal{D}}_{D^{(n/2)}_{n}}\!=&-{ \cal N}\sum_{k=x,y}\hat{\sigma}^{1}_{k}\otimes\sum_{l}\hat{P}_{l}(\hat{\sigma} _{k}^{\otimes n-3}\otimes\hat{\sigma}^{\otimes{2}}_{z}),\end{split}\] (30) where ${\cal N}$ is a normalisation factor taken so that the eigenvalue corresponding to $\|{D^{(n/2)}_{n}}\rangle$ is $1$ and found by noticing that $-\langle D^{(n/2)}_{n}\|{\hat{\sigma}^{\otimes{n-2}}_{k}\otimes\hat{\sigma}^{ \otimes{2}}_{z}}\|D^{(n/2)}_{n}\rangle=n/(2n-2)$ for $k=x,y$ and that the number of possible permutations involved in Eq. (30) is $C^{n-3}_{n-1}$, so that ${\cal N}={2}/[n(n-2)]$. We thus see the effects that a channel has on the expectation value of discrimination operators. For AD, PD and DP noise, respectively, are \[\text{Tr}[{\hat{\mathcal{D}}_{D^{(n/2)}_{n}}\rho_{ad}}] \!=\!-e^{-\frac{n\gamma}{2}}(e^{\gamma}-2),\] (31) \[\text{Tr}[{\hat{\mathcal{D}}_{{D^{(n/2)}_{n}}}\rho_{pd}}] \!=\!e^{-(n-2)\gamma},\] \[\text{Tr}[{\hat{\mathcal{D}}_{{D^{(n/2)}_{n}}}\rho_{dp}}] \!=\!(1-\gamma)^{n}.\] The first two equations can be understood by using arguments analogous to those valid for collective-spin witness operators. The third identity of Eq. (31) is clarified considering the analogy between depolarizing channel and single-qubit white noise. <figure><img src="content_image/0903.3939/x11.png"><figcaption>Figure 7: (Color online) State discrimination via characteristic operators forsymmetric Dicke states with n=4,6,8,10. The continuous lines show the behaviorof AD-affected \|D(n/2)n⟩, the dashed lines are for a PD channel, while thelowest dotted lines are for a DP mechanism. Each channel is assumed to becharacterized by a rate γ. The horizontal lines show the maximum expectationvalues of the characteristic operators ^DD(n/2)n over SLOCC-equivalent GHZstates of n qubits. The shading highlights the regions where discriminationbetween symmetric Dicke states and GHZ class is no longer possible.</figcaption></figure> In Fig. 7 we show the behavior of Eqs. (31) for $n=4,6,8$ and $10$ as the amount of the respective noise-influence increases. We also show the bound associated with the $n$-qubit GHZ class. Thus, the shading in Fig. 7 represents the region where discrimination between one of the channel-affected Dicke states and the GHZ class is not possible. After a certain noise strength, we can no longer determine if the state being studied is a noise-affected Dicke state or a state from the GHZ class. The decay increases more sharply with larger $n$ and the DP channel has the worst effects. We are working on the design of a strategy based on filtering operations which might allow one to gain robustness of this discrimination procedure against noise. However, the potential of this tool is already seen at the level of practical implementability. For instance, the $n=6$ version of Eq. (30) can be decomposed as $\hat{\mathcal{D}}_{D^{(3)}_{6}}\!=\!\sum_{k=x,y}\!\hat{\sigma}^{1}_{k}\! \otimes\![\frac{1}{6}\bigotimes^{6}_{j=2}(\hat{\sigma}^{j}_{z}\!+\!\hat{\sigma }^{j}_{k})\!-\!\frac{1}{6}\bigotimes^{6}_{j=2}(\hat{\sigma}^{j}_{z}\!-\!\hat{ \sigma}^{j}_{k})-\!\frac{1}{12}\bigotimes^{6}_{j=2}(\hat{\sigma}^{j}_{z}+2\hat {\sigma}^{j}_{k})\!+\!\frac{1}{12}\bigotimes^{6}_{j=2}(\hat{\sigma}^{j}_{z}-2 \hat{\sigma}^{j}_{k})+{5}{}\bigotimes^{6}_{j=2}\hat{\sigma}^{j}_{k}]$, which only requires 10 local measurement settings for its implementation. Finding out the explicit decomposition of discrimination operators for general $n$ is, however, a daunting problem. ## VI Correlation Function In this Section we shift the focus of our discussion from GME to the quantum-coherence properties of the class of states under investigation. Our approach here considers the expectation value of the $n$-qubit correlation operator \[\hat{\cal C}(\vartheta)\!=\!(\cos\vartheta\hat{\sigma}_{k}+\sin\vartheta\hat{ \sigma}_{j})^{\otimes{n}}\] (32) with $k\neq{j}=x,y,z$. Through this, one probes the coherence of each element of the register along a direction (in the single-qubit Bloch sphere) lying in the plane formed by the unit vectors $\bf{k}$ and $\bf{j}$. The use of such a multi-qubit correlator is common in the assessment of the properties of GHZ states for quantum metrology purposes [32]: as a result of $n$-qubit coherence, when $k=x$ and $j=y$ the expectation value of Eq. (32) oscillates with $\vartheta$ at a frequency that depends on $n$. Here, we shall investigate collective coherence in an $n$-qubit symmetric Dicke state by means of Eq. (32). This requires the implementation of a single measurement setting per value of $\vartheta$ and is thus an experimentally favorable tool for multipartite state characterization. For the sake of definiteness, here we concentrate on $k=x$ and $j=z$, although any other choice is equally suitable. For a pure $\|D^{(n/2)}_{n}\rangle$ state, the expectation value of the multi-point correlator is easily found using basic combinatorial arguments and the symmetries in the class of states at hand. We have \[\langle D^{(n/2)}_{n}\|\hat{\cal C}(\vartheta)\|D^{(n/2)}_{n}\rangle\!=\!\sum^{n /2}_{k=0}(-1)^{k}(C^{k}_{n/2})^{2}(\cos\vartheta)^{n-2k}(\sin\vartheta)^{2k}.\] (33) In general, Eq. (33) exhibits an $n$-dependent oscillatory behavior whose features strongly depend also on the parity of $n/2$, as shown in Fig. 8. The expression corresponding to $n=6$ has been used in Ref. [15] in order to contribute to the characterization of $\|D^{(3)}_{6}\rangle$. <figure><img src="content_image/0903.3939/x12.png"><figcaption>Figure 8: (Color online) Behavior of the expectation value of the n-qubitcorrelation function calculated over symmetric Dicke states \|D(n/2)n⟩ withn=4,6,8 and 10. A parity-dependent effect related to the number n/2 ofexcitations in the state being studied is seen: states with an even (odd)number of excitations give rise to a positive (negative) expectation value atθ=π/2. Moreover, the position of secondary maxima and minima is n-dependent.The beating is an effect of quantum coherence in the state.</figcaption></figure> Here, we shall study the behavior of $\langle\hat{\cal C}(\vartheta)\rangle$ when a symmetric Dicke state is subjected to noise effects. By using again the expressions of the single-qubit Pauli operators transformed upon the action of a given channel (see Sec. II), one can prove that Eq. (33) remains almost invariant under environmental action. The only modification is at the level of oscillation amplitudes, which are changed by a $\gamma$-dependent factor. Explicitly, we have found the following universal form \[\text{Tr}[\hat{\cal C}(\vartheta)\rho_{ch}]\!=\!\sum^{n/2}_{k=0}(C^{k}_{n/2})^ {2}\Gamma^{ch}_{n,k}(\gamma)(\cos\vartheta)^{n-2k}(\sin\vartheta)^{2k}\] (34) with $ch=\{ad,pd,dp\}$ and $\Gamma^{ad}_{n,k}(\gamma)\!=\!e^{-(\frac{n}{2}-k)\gamma}(1-2e^{-\gamma})^{k}$, $\Gamma^{pd}_{n,k}(\gamma)\!=\!(-1)^{k}e^{-({n}-2k)\gamma}$ and $\Gamma^{dp}_{n,k}(\gamma)\!=\!(-1)^{k}(1-\gamma)^{n}$ being the channel-specific factors responsible for the loss of coherence in the state. Our claim is that the beating effect responsible for the rich oscillatory structures shown in Fig. 8 arises only in virtue of the quantum coherences within $\|D^{(n/2)}_{n}\rangle$. In fact, while $\Gamma^{dp}_{n,k}(\gamma)\rightarrow{0}$ as $\gamma$ grows, so as to progressively kill any oscillations, $\Gamma^{ad}_{n,n/2}(\gamma)\rightarrow{1}$ and $\Gamma^{pd}_{n,n/2}(\gamma)\rightarrow{(-1)^{n/2}}$, with $\Gamma^{ad,pd}_{n,k}(\gamma)\to 0~{}\forall{k<n/2}$. This implies that, as the coherences in the $n$-qubit state disappear under an AD or PD channel, the modulus of corresponding correlation functions simply becomes $\|(\sin\vartheta)^{n}\|$. Thus, although an oscillatory behavior is still kept in these cases, no beating is found. A difference can be found between the trends corresponding to AD and PD channels: a PD channel would progressively destroy the off-diagonal elements of a density matrix without affecting its populations. Asymptotically, this results in a diagonal state whose only non-zero entries are those corresponding to states having $n/2$$\|0\rangle$’s. If we now take $\vartheta=\pi/2$ in the $n$-qubit correlation operator, we easily understand that its expectation value over the PD-affected state can only be $\pm{1}$, depending on the parity of $n/2$. On the other hand, such a parity-effect is absent in the AD case. Indeed, asymptotically, this channel would reduce any symmetric Dicke state to its collective ground state, which can only give $\langle\hat{\cal C}(\pi/2)\rangle=1,~{}\forall{n}$. These features are well illustrated in Figs. 9(a), (b) and (c) for the case of a symmetric six-qubit Dicke state. <figure><img src="content_image/0903.3939/x13.png"><figcaption>Figure 9: (Color online) We show Tr[^C(ϑ)ρad] [panel (a)], Tr[^C(ϑ)ρpd] [panel(b)] and Tr[^C(ϑ)ρdp] [panel (c)] against ϑ and the respective γ for a noiseaffected \|D(3)6⟩ state. Parity effects associated with the various channelsconsidered are shown. Panels (d) and (e) show the AD and PD case,respectively, when the classical asymptotic behavior is subtracted.</figcaption></figure> Such an asymptotic classical behavior tends to mask the trend that the beating follows in the changes undergone by the state. We have thus stripped the correlation functions from such contributions by subtracting their Fourier-series expansions from the analogous one of $\text{Tr}[\hat{\cal C}(\vartheta)\rho_{ad,pd}]$. The results, which highlight the sole decrease in visibility of the fringe of beating induced by quantum correlations, are shown in Figs. 9(d) and (e), which show an evident exponential decay against $\gamma$. Such a procedure is not necessary for the DP, which smoothly flattens the correlation function to zero. The identification of a general trend against the number of qubits in a state is a task made difficult by the $\vartheta$-dependence of such figures of merit: states affected by different channels give rise to maxima and minima of $\text{Tr}[\hat{\cal C}(\vartheta)\rho_{ch}]$ located at different values of the angle $\vartheta$. One can extract useful indicative information by looking, for instance, at the correlations corresponding to $\vartheta=0$, for a set channel and increasing number of qubits, to find that the exponential decay of correlations induced by a growing $\gamma$ becomes faster for larger $n$. Overall, the analytic expressions provided here for this set of relevant noise channels embody a valuable tool for the experimental characterization of symmetric Dicke states. The theoretical curves can indeed be used to fit the points acquired, experimentally, by tuning the parameters of a setup in a way so as to properly select the direction $\vartheta$ along which one would like to probe quantum coherence [15]. ## VII Conclusions We have studied both GME-detecting and state-characterizing tools for the class of symmetric Dicke states of an arbitrary number of qubits, under the influences of noise. Our investigation starts from standard instruments such as fidelity-based entanglement witnesses and collective-spin operators and unveils some experimentally-friendly ways to improve their resilience to noise. Besides its pragmatic features, which make our study explicitly devoted to the prediction and evaluation of the performances of an experimental setup, we have revealed interesting characteristics of noisy channels and their influence on GME witnesses. For instance, we have clearly shown that resilience to PD channels cannot be achieved by means of simple filtering operations performed on fidelity-based entanglement witnesses. In light of the steady technological progress in a variety of experimental settings, ranging from linear optics to solid-state systems, we expect our results to be useful for the purposes of a complete characterization of multipartite entangled states generated or evolving in the presence of environmental effects. Our results should also help in the context of reliably determining the quality of experimental symmetric Dickes states for use in quantum networking tasks such as quantum secret sharing and open-destination teleportation. ###### Acknowledgements. We thank G. Cronenberg, C. Di Franco, M. S. Kim and R. Prevedel for invaluable discussions and encouragement. We acknowledge support from DEL and the UK EPSRC through QIPIRC. MP thanks EPSRC for financial support (EP/G004579/1). ## Appendix A In this Appendix, we outline the steps involved in determining the biseparability bound used in the entanglement witness based on collective-spin operators. We start by considering the collective operator \[\hat{S}_{n}=\hat{J}^{2}_{x}+\hat{J}^{2}_{y}+\alpha\hat{J}^{2}_{z}\] (A-1) with $\alpha\in\mathbb{R}$. Upon expansion of each collective-spin operator and using the “one-vs-$(n-1)$” qubit bipartition, we get the expression \[\hat{S}_{n}=(\frac{n}{2}+\frac{n\alpha}{4})\openone+\frac{1}{2}(\hat{\sigma}^{ 1}_{x}\hat{Q}_{x}+\hat{\sigma}^{1}_{y}\hat{Q}_{y}+\alpha\hat{\sigma}^{1}_{z} \hat{Q}_{z}+\hat{R}_{\alpha}),\] (A-2) where $\hat{Q}_{k}\!=\!\sum^{n}_{j=2}\hat{\sigma}^{j}_{k}$ and $\hat{R}_{\alpha}\!=\!\hat{R}_{x}+\hat{R}_{y}+\alpha\hat{R}_{z}$ with $\hat{R}_{k}\!=\!\sum^{n-1}_{j=2}\hat{\sigma}^{j}_{k}\sum^{n}_{l=j+1}\hat{ \sigma}^{l}_{k}~{}(k=x,y,z)$. We take the expectation value of $\hat{S}_{n}$ over the biseparable state of qubit $1$ and system $2\rightarrow{n}$, so that \[\langle\hat{S}_{n}\rangle\!=\!\frac{n}{2}\!+\!\frac{n\alpha}{4}\! +\!\frac{1}{2}[\sum_{k=x,y}\langle\hat{\sigma}^{1}_{k}\rangle\langle\hat{Q}_{k }\rangle\!+\!\alpha\langle\hat{\sigma}^{1}_{z}\rangle\langle\hat{Q}_{z}\rangle +\langle\hat{R}_{\alpha}\rangle]\] (A-3) \[={\frac{n}{2}\!+\!\frac{n\alpha}{4}}\!+\!\frac{1}{2}\langle{\hat{ \Omega}}_{\alpha}\rangle\leq{\frac{n}{2}+\frac{n\alpha}{4}}+\frac{1}{2}\max_{\| r\|^{2}\leq{1}}(\lambda_{\alpha})=b_{bs}(\alpha)\] with $r=(x,y,z)$, $\{\lambda_{\alpha}\}$ the set of eigenvalues of $\hat{\Omega}_{\alpha}$, $x=\langle\hat{\sigma}^{1}_{x}\rangle$ and analogous expressions for $y$ and $z$. The diagonalization of $\hat{\Omega}_{\alpha}$ can be performed numerically for a few relevant cases, therefore allowing the quantification of the upper bound for biseparability associated with this splitting. For the specific cases of $n=4,6$ and $8$, the “one-vs-$(n-1)$” splitting gives the largest bound among all other possible bipartitions, thus providing the threshold $b_{bs}(\alpha)$ for the detection of GME (see Sec. II). Quantitatively, we have $b_{bs}(0)=5.23,11.018$ and $18.83$ for $n=4,6$ and $8$ respectively. ## Appendix B We discuss the local Pauli decomposition of the fidelity operator $\|D_{n}^{(n/2)}\rangle\langle D_{n}^{(n/2)}\|$ and provide a useful method for reducing the number of local measurement settings experimentally required to measure it. To decompose a given $n$-qubit projector $\|\phi\rangle\langle\phi\|$ into local Pauli form, we use the relation \[\|\phi\rangle\langle\phi\|=\frac{1}{2^{n}}\sum_{i_{1},i_{2}...i_{n}}C_{i_{1},i_{ 2}..i_{n}}(\hat{\sigma}_{i_{1}}\otimes\hat{\sigma}_{i_{2}}\otimes..\otimes\hat {\sigma}_{i_{n}}),\] (B-1) where we have introduced the correlation tensor $C_{i_{1},i_{2}...i_{n}}=\left\langle\phi\right\|\hat{\sigma}_{i_{1}}\otimes\hat {\sigma}_{i_{2}}\otimes..\otimes\hat{\sigma}_{i_{n}}\|\phi\rangle$ and $i_{n}\in\{0,x,y,z\}$, with $\hat{\sigma}_{0}=\hat{\openone}$. Using Eq. (B-1), the correlation tensor corresponding to $\|D^{(2)}_{4}\rangle\langle{D}^{(2)}_{4}\|$ has 40 non-zero elements \[\|D^{(2)}_{4}\rangle\langle D^{(2)}_{4}\|\!=\!\frac{1}{16}(\openone ^{\otimes 4}\!+\!\sum_{k=x,y,z}\hat{\sigma}_{k}^{\otimes 4}+\frac{1}{3}\sum_{ \pi}[\sigma_{x}^{\otimes 2}\sigma_{y}^{\otimes 2}\] (B-2) \[+2(\openone^{\otimes 2}\hat{\sigma}_{x}^{\otimes 2}\!+\!\openone^ {\otimes 2}\sigma_{y}^{\otimes 2}\!+\!\openone^{\otimes 2}\sigma_{z}^{\otimes 2 }\!-\!\sigma_{x}^{\otimes 2}\sigma_{z}^{\otimes 2}\!-\!\sigma_{y}^{\otimes 2} \sigma_{z}^{\otimes 2})]),\] where $\sum_{\pi}$ indicates all distinct permutations of the operators. However, such a 40-element decomposition can be significantly streamlined by using the relation (valid for $i,j=x,y,z)$ \[\sum_{\pi}\sigma_{i}^{\otimes 2}\sigma_{j}^{\otimes 2}\!=\!\frac{1}{2}[(\sigma _{i}+\sigma_{j})^{\otimes 4}+(\sigma_{i}-\sigma_{j})^{\otimes 4}]-\sigma_{i}^{ \otimes 4}-\sigma_{j}^{\otimes 4},\] (B-3) through which one can rewrite the $\sum_{\pi}$ term of Eq. (B-2) in terms of the 9 local measurement settings:$\sigma_{x,y,z}^{\otimes 4}$, $[(\sigma_{x}\!\pm\!\sigma_{y,z})/\sqrt{2}]^{\otimes 4}$ and $[(\sigma_{y}\!\pm\!\sigma_{z})/\sqrt{2}]^{\otimes 4}$. Thus, the symmetries present in the state’s decomposition have been exploited in a way so as to reduce the number of measurements. Using the compacting techniques outlined here for $\|D_{4}^{(2)}\rangle$, one should be able to apply them to arbitrary sized symmetric Dicke states $\|D_{n}^{(n/2)}\rangle$ to obtain significant reductions in the number of local measurement settings. For example, using Eq. (B-1), one finds that the correlation tensor corresponding to $\|D^{(3)}_{6}\rangle\langle{D}^{(3)}_{6}\|$ has 544 non-zero elements \[\|D^{(3)}_{6}\rangle\langle D^{(3)}_{6}\|\!=\!\sum_{\pi}(\frac{1}{3 20}[\sigma_{x}^{\otimes 2}\sigma_{y}^{\otimes 4}+\sigma_{x}^{\otimes 4}\sigma_ {y}^{\otimes 2}-\sigma_{x}^{\otimes 2}\sigma_{y}^{\otimes 2}\sigma_{z}^{ \otimes 2}\] (B-4) \[+\openone^{\otimes 2}\sigma_{x}^{\otimes 2}\sigma_{y}^{\otimes 2} +\openone^{\otimes 2}\sigma_{z}^{\otimes 4}-\openone^{\otimes 4}\sigma_{z}^{ \otimes 2}-\openone^{\otimes 2}\sigma_{x}^{\otimes 2}\sigma_{z}^{\otimes 2}\] \[-\openone^{\otimes 2}\sigma_{y}^{\otimes 2}\sigma_{z}^{\otimes 2} ]+\frac{3}{320}[\sigma_{x}^{\otimes 2}\sigma_{z}^{\otimes 4}-\sigma_{y}^{ \otimes 2}\sigma_{z}^{\otimes 4}-\sigma_{x}^{\otimes 4}\sigma_{z}^{\otimes 2}\] \[-\sigma_{y}^{\otimes 4}\sigma_{z}^{\otimes 2}+\openone^{\otimes 4 }\sigma_{x}^{\otimes 2}+\openone^{\otimes 2}\sigma_{x}^{\otimes 4}+\openone^{ \otimes 4}\sigma_{y}^{\otimes 2}+\openone^{\otimes 2}\sigma_{y}^{\otimes 4}])\] \[+\frac{1}{64}[\sigma_{x}^{\otimes 6}+\sigma_{y}^{\otimes 6}+ \openone^{\otimes 6}-\sigma_{z}^{\otimes 6}].\] Using Eq. (B-3), together with the relations \[\sum_{\pi}\sigma_{x}^{\otimes 2}\sigma_{y}^{\otimes 2}\sigma_{z}^ {\otimes 2}=\frac{1}{4}[(\sigma_{x}+\sigma_{y}+\sigma_{z})^{\otimes 6}+(\sigma _{x}+\sigma_{y}-\sigma_{z})^{\otimes 6}\] (B-5) \[\!+\!(\sigma_{x}\!-\!\sigma_{y}\!+\!\sigma_{z})^{\otimes 6}\!+\!( \sigma_{x}\!-\!\sigma_{y}\!-\!\sigma_{z})^{\otimes 6}]\!-\!\sum_{\pi}(\sigma_{ x}^{\otimes 2}\sigma_{y}^{\otimes 4}\!+\!\sigma_{x}^{\otimes 4}\sigma_{y}^{ \otimes 2})\] \[\!-\!\sum_{\pi}\sum_{k=x,y}(\sigma_{k}^{\otimes 2}\sigma_{z}^{ \otimes 4}+\sigma_{k}^{\otimes 4}\sigma_{z}^{\otimes 2})\!-\!\sum_{k=x,y,z} \sigma_{k}^{\otimes 6},\] \[\sum_{\pi}(\sigma_{i}^{\otimes 2}\sigma_{j}^{\otimes 4}+\sigma_{i }^{\otimes 4}\sigma_{j}^{\otimes 2})\!=\!\frac{1}{2}[(\sigma_{i}\!+\!\sigma_{j })^{\otimes 6}\!+\!(\sigma_{i}-\sigma_{j})^{\otimes 6}]\] \[-\sigma_{i}^{\otimes 6}-\sigma_{j}^{\otimes 6},\] \[\sum_{\pi}(\sigma_{i}^{\otimes 2}\sigma_{j}^{\otimes 4}-\sigma_{i }^{\otimes 4}\sigma_{j}^{\otimes 2})\!=\!\frac{1}{24}\{(\sigma_{i}+2\sigma_{j} )^{\otimes 6}+(\sigma_{i}-2\sigma_{j})^{\otimes 6}\] \[-4[(\sigma_{i}+\sigma_{j})^{\otimes 6}+(\sigma_{i}-\sigma_{j})^{ \otimes 6}]-10\sigma_{i}^{\otimes 6}-136\sigma_{j}^{\otimes 6}\},\] one can rewrite the permutations of Eq. 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0805.2798	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 24206, "num_imgs": 2, "llama3_tokens_count": 7554 }	[ "content_image/0805.2798/x1.png", "content_image/0805.2798/x2.png" ]	# Cooper pair turbulence in atomic Fermi gases M. Dzero${}^{1,2}$, E. A. Yuzbashyan${}^{2}$, B. L. Altshuler${}^{1}$ ${}^{1}$ Department of Physics, Columbia University, New York, NY 10027, USA ${}^{2}$Center for Materials Theory, Rutgers University, Piscataway, NJ 08854, USA ###### Abstract We investigate the stability of spatially uniform solutions for the collisionless dynamics of a fermionic superfluid. We demonstrate that, if the system size is larger than the superfluid coherence length, the solution characterized by a periodic in time order parameter is unstable with respect to spatial fluctuations. The instability is due to the parametric excitations of pairing modes with opposite momenta. The growth of spatial modulations is suppressed by nonlinear effects resulting in a state characterized by a random superposition of wave packets of the superfluid order parameter. We suggest that this state can be probed by spectroscopic noise measurements. pacs: 05.30.Fk, 32.80.-t, 74.25.Gz How does a homogeneous interacting many-body system develop spatial modulations as a result of a uniform quench? In general, this can happen due to an interplay between the extra energy introduced by the quench and an intrinsic coupling of the degrees of freedom at various length scales [1; 2; 3; 4]. By the very nature of the problem, the energy distribution of these degrees of freedom is initially far from a state of thermodynamic equilibrium. This situation is common in a nonlinear medium and can be described by the term ”wave turbulence” [5]. In the case when one of the parameters of the medium or an applied field is periodically varied in time, wave turbulence due to a parametric instability can develop. Examples of such a phenomenon include the decay of high-frequency electric field into Lagmuir and ion-sound waves in plasma [7] and the spin-wave instability in an rf-magnetic field in dielectric ferromagnets [8]. A natural question is then whether a quench can lead to a parametric excitation of spatial modes. Oscillating homogeneous fields can be generated by a uniform quench without an external drive. One of the recent examples where this takes place is a fermionic superfluid quenched by a sudden change of the pairing strength [9; 10]. There are two types of asymptotic states the system can reach depending on the strength of the initial perturbation [11; 12]. The first type has a constant value $\Delta(t)=\Delta_{s}$, while the second one is characterized by periodic $\Delta(t)$. This analysis does not take into account the pair breaking processes and is thus valid only for $t<\tau_{\varepsilon}$, where $\tau_{\varepsilon}$ is the quasiparticle relaxation time. Moreover, the emerging asymptotic states are spatially uniform: order parameter evolution was obtained as a solution of an effectively zero dimensional problem. Results of Refs. [11; 12] are thus valid when the system size is smaller than the coherence length, $L<\xi$. The description of the order parameter evolution for $L>\xi$ requires an additional investigation. <figure><img src="content_image/0805.2798/x1.png"><figcaption>Figure 1: Asymptotic state with the periodic order parameterΔ(t)=Δs[1+qcos(2Δst)] (top left) is unstable with respect to the parametricexcitation of two modes with opposite momenta (right panel). Frequency of eachmode is half the oscillation frequency of the homogeneous order parameter.Initial exponential growth of the parametric instability is followed by atransient regime after which the condensate reaches a spatially inhomogeneouspost-threshold state (bottom left). The value of the order parameter increasesfrom white to black.</figcaption></figure> In this paper, we perform a stability analysis of the solutions for the wave function and order parameter obtained in Refs. [11; 12] with respect to spatial fluctuations. We find that the asymptotic states with constant order parameter remain stable, while the state with periodic $\Delta(t)$ does not. The physical origin of the instability lies in the possibility of parametric excitation of the spatially modulated pairing modes, Fig. 1. In a homogeneous medium, the periodic (in time) order parameter can be considered as an energy pump allowing a coupling between two spatial modes with opposite momenta, and, at the same time, providing enough energy to overcome the damping due to the scattering of Cooper pairs. Subsequent scattering effects of the resonant modes limit the initial exponential growth and result in a state with a _spatially inhomogeneous_ order parameter. We demonstrate that as the process of the parametric instability develops, the energy of the homogeneous state is transferred into that of the pairing wave packets with the typical size of the order of the coherence length, $\xi$, signaling the onset of the Cooper pair turbulence. Since the amplitudes of the wave packets are essentially random, we suggest that this state can be experimentally probed by the noise measurements. Consider the state with a periodically varying order parameter. Analytically, $\Delta(t)$ is described by the Jacobi elliptic function dn [9]. Here we assume that the amplitude of the oscillations is small. This allows us to keep only the first two terms in the Fourier series for dn: \[\Delta(t)=\Delta_{s}[1+q\cos(2\Delta_{s}t)],\quad q\ll 1.\] (1) We note that the non-dissipative dynamics within the BCS model is described by the Bogoliubov-de Gennes equations, which can be cast into the form of equations of motion for classical vector variables ${\vec{s}}_{\mathbf{p}}$: $\dot{\vec{s}}_{\mathbf{p}}={\vec{b}}_{\mathbf{p}}\times{\vec{s}}_{\mathbf{p}}$. Here ${\vec{b}}_{\mathbf{p}}=2(-\Delta(t),0,\varepsilon_{\mathbf{p}})$ plays the role of an external magnetic field. The periodic external field makes it possible for the parametric instabilities to develop. Given that the asymptotic state (1) is robust against homogeneous perturbations [9; 10], it seems natural to look for an instability with respect to spatial fluctuations. In particular, we will investigate the conditions of the parametric decay of the homogeneous pairing mode (1) with an energy $2\Delta_{s}$ into two pairing modes with energies and momenta $(\omega_{1},\mathbf{k}_{1})$ and $(\omega_{2},\mathbf{k}_{2})$, Fig. 1. In a continuous medium the energy and momentum have to be conserved: $\mathbf{k}_{1}+\mathbf{k}_{2}=0$ and $2\Delta_{s}=\omega_{1}+\omega_{2}$. These type of instabilities appear in various physical systems, see e.g. Ref. [5] for an extensive review. To analyze the stability of the asymptotic state (1), we employ the Bogoliubov-de Gennes (BdG) equations[13]: \[\begin{split}& i\dot{u}_{\mathbf{p}}(\mathbf{r},t)=\hat{\xi}u_{ \mathbf{p}}(\mathbf{r},t)+\Delta(\mathbf{r},t)v_{\mathbf{p}}(\mathbf{r},t),\\ & i\dot{v}_{\mathbf{p}}(\mathbf{r},t)=-\hat{\xi}v_{\mathbf{p}}( \mathbf{r},t)+{\bar{\Delta}}(\mathbf{r},t)u_{\mathbf{p}}(\mathbf{r},t).\end{split}\] (2) Here $\hat{\xi}=-{\vec{\nabla}}^{2}/{2m}-\mu$, $\mu$ is the chemical potential and the order parameter $\Delta(\mathbf{r},t)$ is \[\Delta(\mathbf{r},t)=g\sum\limits_{\mathbf{p}}u_{\mathbf{p}}(\mathbf{r},t){ \bar{v}}_{\mathbf{p}}(\mathbf{r},t),\] (3) where $g$ is the BCS coupling constant. We linearize Eqs. (2) with respect to the deviations $\phi_{\mathbf{p}}(\mathbf{r},t)$ and $\psi_{\mathbf{p}}(\mathbf{r},t)$ from the homogeneous solution $U_{\mathbf{p}}(t)$ and $V_{\mathbf{p}}(t)$: (4) The order parameter (3) also contains a small inhomogeneous part $\Delta(\mathbf{r},t)=\Delta(t)+\delta\Delta(\mathbf{r},t)$. Plugging (4) into (2) and Fourier transforming with respect to the spatial coordinates, we find (5) The time dependance of $U_{\mathbf{p}}(t)$ and $V_{\mathbf{p}}(t)$ (see Ref. [14]) suggests that for linear corrections (4) we write $\phi_{\mathbf{p}}(\mathbf{k},t)=a_{\mathbf{p}}(\mathbf{k},t)e^{i\xi_{\mathbf{p }}t}+b_{\mathbf{p}}(\mathbf{k},t)e^{-i\xi_{\mathbf{p}}t}$ and $\psi_{\mathbf{p}}(\mathbf{k},t)=\tilde{a}_{\mathbf{p}}(\mathbf{k},t)e^{i\xi_{ \mathbf{p}}t}+\tilde{b}_{\mathbf{p}}(\mathbf{k},t)e^{-i\xi_{\mathbf{p}}t}$ with $\xi_{\mathbf{p}}=\sqrt{\varepsilon_{\mathbf{p}}^{2}+\Delta_{s}^{2}}+O(q)$ and $\varepsilon_{\mathbf{p}}=\frac{\mathbf{p}^{2}}{2m}-\mu$. We also write \[\delta\Delta(\mathbf{k},t)=C_{\mathbf{k}}(t)e^{i\Delta_{s}t}+\widetilde{C}_{ \mathbf{k}}(t)e^{-i\Delta_{s}t},\] (6) where $C_{\mathbf{k}}(t)=c_{\mathbf{k}}e^{\nu(t-t_{0})}$, $\widetilde{C}_{\mathbf{k}}(t)=\tilde{c}_{\mathbf{k}}e^{\nu(t-t_{0})}$, $\nu$ determines the growth rate, and $t_{0}$ is a time scale when (1) is reached (in what follows we set $t_{0}=0$). In the expression (6) we have neglected the higher harmonics $e^{\pm i3\Delta_{s}t},e^{\pm i5\Delta_{s}t}$ etc. This is justified for $q\ll 1$, since their inclusion yields higher order in $q$ corrections to the growth rate $\nu$ and to the order parameter amplitudes [6]. In the linear corrections to the Bogoliubov amplitudes (see above) we also keep only the lowest harmonics $\omega=\pm\Delta_{s}$, i.e. $a_{\mathbf{p}}(\mathbf{k},t)\to(a_{1,\mathbf{p}}(\mathbf{k})e^{i\Delta_{s}t}+a _{-1,\mathbf{p}}(\mathbf{k})e^{-i\Delta_{s}t})e^{\nu t}$, $b_{\mathbf{p}}(\mathbf{k},t)\to(b_{1,\mathbf{p}}(\mathbf{k})e^{i\Delta_{s}t}+b _{-1,\mathbf{p}}(\mathbf{k})e^{-i\Delta_{s}t})e^{\nu t}$ etc. Next, we express the Bogoliubov amplitudes in terms of $c_{\mathbf{k}}$ and $\tilde{c}_{\mathbf{k}}$ by equating the coefficients in front of $e^{\pm i\Delta_{s}t}$ in Eq. (5). The resulting amplitudes are substituted into the self-consistency equation (3). One obtains a linear system for the variables $c_{\mathbf{k}},\tilde{c}_{-\mathbf{k}}^{},c_{-\mathbf{k}}^{}$ and $\tilde{c}_{\mathbf{k}}$. Expressing $\tilde{c}_{\mathbf{k}},\tilde{c}_{-\mathbf{k}}^{}$ in terms of $c_{\mathbf{k}}$ and $c_{-\mathbf{k}}^{}$, we derive \[(\omega_{\mathbf{k}}+i\gamma_{\mathbf{k}})c_{\mathbf{k}}+h_{\mathbf{k}}c_{- \mathbf{k}}^{}=0,~{}~{}(\omega_{\mathbf{k}}-i\gamma_{\mathbf{k}})c_{-\mathbf{ k}}^{}+h_{\mathbf{k}}c_{\mathbf{k}}=0,\] (7) where $\omega_{\mathbf{k}}$, $\gamma_{\mathbf{k}}$, and $h_{\mathbf{k}}$ are nonlinear functions of the growth rate $\nu$. Eqs. (7) can be interpreted as the equations of motion for a classical field $c_{\mathbf{k}}$[5]. Then, $\omega_{\mathbf{k}}$ has the meaning of the excitation spectrum of this field and $h_{\mathbf{k}}\sim O(q)$ stands for the pumping amplitude, which gives rise to a parametric instability. Finally, $\gamma_{\mathbf{k}}$ describes the damping of the parametric modes due to the intrinsic relaxation processes. Nonzero solutions of Eqs. (7) for $\nu(\mathbf{k})$ exist provided $\omega_{\mathbf{k}}^{2}(\nu)=h_{\mathbf{k}}^{2}(\nu)-\gamma_{\mathbf{k}}^{2}(\nu)$. Thus, the stability analysis is reduced to the solution of the nonlinear equation for $\nu(\mathbf{k})$. We have analyzed this equation numerically and present the results on Fig. 2. We find that the instability region is centered around $k_{m}\approx 1.6k_{\xi}$ and has a width $\delta k\approx 1.2\sqrt{q}k_{\xi}$, where $k_{\xi}=\Delta_{s}/v_{F}=1/\xi$ is the coherence wave vector. From (7) it follows that for a fixed $q$ the parametric growth will be suppressed as soon as the energy pumped into the system fully goes into dissipation. This condition determines the maximum growth rate $\nu_{m}$, i.e. $\gamma_{\mathbf{k}}(\nu_{m})=h_{\mathbf{k}}(\nu_{m})$. Our estimate yields $\nu_{m}\approx 2q\Delta_{s}$. Lastly, we have also verified that asymptotic states with constant order parameter remain stable with respect to the spatial fluctuations of the above type. <figure><img src="content_image/0805.2798/x2.png"><figcaption>Figure 2: Region of the parametric instability of the homogeneous Δ(t) (1)with respect to generation of the pairing modes with opposite momenta (k,−k).Instability growth rate is plotted for q=0.05 in the units of Δs (see Eq. (1))and momentum is in the units of kξ=Δs/vF. The maximum rate is reached atνm≈2qΔs. For small q the shape of the instability curve isν(k)≈νm−2Δs(k−1.6kξ)2/k2ξ.</figcaption></figure> The initial growth of the parametric instability (11) will be limited by nonlinear effects which lead to the transient behavior with subsequent transition into a post-threshold state. The latter is defined as a state in which Fourier components of the order parameter (6) are time independent, $C_{\mathbf{k}}(t)=c_{\mathbf{k}}$ and $\widetilde{C}_{\mathbf{k}}(t)=\tilde{c}_{\mathbf{k}}$. Below we focus on finding the resulting post-threshold state of the condensate. From the linear analysis we have seen that the fastest growing modes are the ones with a certain magnitude of the momentum. Thus, in the BdG equations (2) among the nonlinear in powers of $c_{\mathbf{k}},\tilde{c}_{\mathbf{k}}$ terms we keep the resonant ones with frequencies $\omega=\pm\Delta_{s}$ and momenta $\|\mathbf{k}\|=\|\mathbf{k}^{\prime}\|=k_{s}$, where $k_{s}$ is a new post-threshold state momentum to be determined below. The resulting set of nonlinear in $c_{\mathbf{k}}$ equations for the order parameter amplitudes can be written as Eq. (7) with renormalized coefficients \[\begin{split}&\omega_{\mathbf{k}}\to\Omega_{\mathbf{k}}=\omega_{ \mathbf{k}}(0)+\sum\limits_{\|\mathbf{k}^{\prime}\|=k_{s}}T_{\mathbf{k}\mathbf{k }^{\prime}}\|c_{\mathbf{k}^{\prime}}\|^{2},\\ & h_{\mathbf{k}}\to P_{\mathbf{k}}=h_{\mathbf{k}}(0)+\sum\limits_ {\|\mathbf{k}^{\prime}\|=k_{s}}S_{\mathbf{k}\mathbf{k}^{\prime}}c_{\mathbf{k}^{ \prime}}c_{-\mathbf{k}^{\prime}},\end{split}\] (8) where $T_{\mathbf{k}\mathbf{k}^{\prime}}$ and ${S}_{\mathbf{k}\mathbf{k}^{\prime}}$ are the scattering matrix elements. They vary slowly on a scale of $k_{\xi}$ and are almost independent of the angle between $\mathbf{k}$ and $\mathbf{k}^{\prime}$. In what follows we neglect the $k$-dependence in the scattering matrix elements, $T_{\mathbf{k}\mathbf{k}^{\prime}}=T$ and $S_{\mathbf{k}\mathbf{k}^{\prime}}=S$. Note that each contribution in (8) is either phase independent or depends on a sum of the two phases of $c_{\mathbf{k}}$ and $c_{-\mathbf{k}}^{}$. This can be interpreted as follows. There are two physical processes which limit the parametric excitations: one has to do with the reduction of the absolute value of the amplitudes, while the other is related to the phase decoherence of the two pairing modes with opposite momenta. In its spirit approximation (8) is similar to the mean-field BCS model where only diagonal in momentum terms are kept in the interaction. The inclusion of off-diagonal terms in Eq. (8) is expected to cause a broadening of the post-threshold state momentum $\delta k\sim\sqrt{q}k_{\xi}$, see Fig. 2. To determine the parameters of our post-threshold state, we insert $c_{\mathbf{k}}=\|c_{\mathbf{k}}\|e^{i\alpha_{\mathbf{k}}}$, and Eqs. (8) for $\Omega_{\mathbf{k}}$ and $P_{\mathbf{k}}$ into (7). The post-threshold state momentum $k_{s}$ is determined by the condition that the magnitude of the pumping field $\|P_{\mathbf{k}}\|$ does not exceed the damping $\gamma_{\mathbf{k}}$ for any $k$. As a result we have $\Omega_{\mathbf{k}_{s}}=0$. Phase $\Psi_{s}=\alpha_{\mathbf{k}}+\alpha_{-\mathbf{k}}$ and amplitude $\|c_{k_{s}}\|$ are given by $\sin\Psi_{s}=\gamma_{k_{s}}/h_{k_{s}}$ and $\|c_{k_{s}}\|^{2}=h_{k_{s}}\cos\Psi_{s}/\|S\|$ (the corresponding expressions for $\|\tilde{c}_{k_{s}}\|$ and $\widetilde{\Psi}_{s}=\tilde{\alpha}_{\mathbf{k}}+\tilde{\alpha}_{-{\mathbf{k}}}$ can be derived similarly). We obtain (9) where $k_{s}\approx 1.73k_{\xi}$, $c_{s}\approx 0.77$ and $w_{s}\approx 0.95$ for $\Delta_{s}=0.1\mu$. Note that the post-threshold momentum $k_{s}>k_{m}\approx 1.6k_{\xi}$, i.e. the energy cascades to smaller length scales, as expected of turbulent behavior. The individual phases $\alpha_{\mathbf{k}}$ and $\tilde{\alpha}_{\mathbf{k}}$ cannot be determined within the diagonal approximation (8). In a continuous medium, one can treat them as random variables. For the correlators we take $\langle e^{i\alpha_{\mathbf{k}}}\rangle=0$, $\langle e^{i\alpha_{\mathbf{k}_{1}}}e^{i\tilde{\alpha}_{\mathbf{k}_{2}}} \rangle=0$ and $\langle e^{i\alpha_{{\mathbf{k}}_{1}}}e^{i\alpha_{{\mathbf{k}}_{2}}}\rangle= \delta_{{\mathbf{k}}_{1},-{\mathbf{k}}_{2}}^{(3)}e^{i\Psi_{s}}$, where $\langle...\rangle$ stands for averaging over the phase distribution. To get further insight into the nature of the post-threshold state, consider the following choice for the phases $\alpha_{\mathbf{k}}=\Psi_{s}/2-\mathbf{k}\cdot\mathbf{r}_{0}$ and $\tilde{\alpha}_{\mathbf{k}}=\widetilde{\Psi}_{s}/2-{\mathbf{k}}\cdot\mathbf{r} _{0}$. It leads to a spherically symmetric wave packet, a ”bubble”, with a periodic amplitude $A(t)$ \[\begin{split}&\Delta({\vec{r}},t)=\Delta_{s}+\frac{\sqrt{q}\Delta _{s}c_{s}\sin(k_{s}\|{\vec{r}}-{\vec{r}}_{0}\|)}{k_{s}\|{\vec{r}}-{\vec{r}}_{0}\|} A(t),\\ & A(t)=e^{i(\frac{\Psi_{s}}{2}+\Delta_{s}t)}+w_{s}e^{i(\frac{ \widetilde{\Psi}_{s}}{2}-\Delta_{s}t)}.\end{split}\] (10) In general we obtain a linear combination of these bubbles (10) by writing $e^{i\alpha_{\mathbf{k}}}=\int f(\mathbf{r}_{0})e^{-i\mathbf{k}\cdot\mathbf{r}_ {0}}d^{3}\mathbf{r}_{0}$ and similarly for $e^{i\tilde{\alpha}_{\mathbf{k}}}$. Then, Eq. (9) can be viewed as a superposition of wave packets of the form (10) centered at different ${\vec{r}}_{0}$ with random amplitudes $A({\vec{r}}_{0},t)$. This suggests that the parametric instability results in a random distribution of the wave packets. It is instructive to compare (10) with $\delta\Delta({\vec{r}},t)$ at the linear stage of the parametric instability, which can be derived using our result for $\nu(k)$ (see Fig. 2) and Eqs. (6,7). Taking $c_{\mathbf{k}}=Ce^{-i(\mathbf{k}\cdot\mathbf{r}_{0}+\Delta_{s}\tau_{\mathbf{k} })}$ and $\tilde{c}_{\mathbf{k}}=c_{-\mathbf{k}}^{}$, we obtain \[\delta\Delta({\vec{r}},t)\approx\frac{Ce^{\nu_{m}t}\cos[\Delta_{s}(t-\tau)]}{ \sqrt{\Delta_{s}t}}\frac{\sin(k_{m}R)e^{-R^{2}/l^{2}(t)}}{k_{m}R},\] (11) where $l(t)\approx\xi\sqrt{\Delta_{s}t}$, $R=\|{\vec{r}}-{\vec{r}}_{0}\|$, $C$ is a constant, and $e^{i\Delta_{s}\tau_{\mathbf{k}}}=(i\gamma_{k}-\omega_{k})/h_{k}$. In deriving Eq. (11), we also assumed $k_{m}R\gg\Delta_{s}t$ and replaced a slowly varying function $\tau_{\mathbf{k}}\to\tau$. Expression (11) describes the initial formation of a wave packet (10). Note that on a time scale $(q\Delta_{s})^{-1}$ at which the order parameter deviation is of the order $\sqrt{q}\Delta_{s}$, the width of the packet is $l_{p}\approx\xi/\sqrt{q}$. The observations above help to identify features of the post-threshold state (9) relevant for the experimental verification of our theory. To be more specific, let us compute the correlator ${\cal K}({\vec{r}}_{1}-{\vec{r}}_{2},t_{1}-t_{2})=\langle\delta\Delta({\vec{r} }_{1},t_{1})\delta\Delta({\vec{r}}_{2},t_{2})\rangle$, where $t_{1}$ and $t_{2}$ are taken in the post-threshold stage. ${\cal K}({\vec{r}},t)$ characterizes the spatial and time distribution of the parametric _noise_ in the system. Using the correlators for the random functions $e^{i\alpha_{\mathbf{k}}}$ (see above), we obtain from Eq. (9) \[{\cal K}({\vec{r}},t)\propto{q\Delta_{s}^{2}}\frac{\sin(k_{s}r)\cos\Delta_{s}( t_{1}-t_{2})}{k_{s}r},\] (12) where $r=\|{\vec{r}}_{1}-{\vec{r}}_{2}\|$. This means that the spatial noise spectrum $\propto\int{\cal K}({\vec{k}},\omega)d\omega$ has a peak at the wave vector $k=k_{s}$ and decays as $1/k^{2}$ for $k\gg k_{s}$. Formation of an isolated wave packet (10) induces an oscillating supercurrent ${\vec{j}}_{s}\propto{\vec{\nabla}}\Phi({\vec{r}},t)$, where $\Phi({\vec{r}},t)$ is the phase of the order parameter. Setting $r_{0}=0$ we find that only the radial component of the current is nonzero. To the lowest order in $q$, ${\vec{j}}_{s}({\vec{r}},t)\propto\hat{e}_{r}\sqrt{q}\cos(\Delta_{s}t)[k_{s}r \cos(k_{s}r)-\sin(k_{s}r)]/(k_{s}r)^{2}$. This implies a spatial re-distribution of Cooper pairs similar to the Friedel oscillations in the density of a degenerate Fermi gas induced by a weak scattering potential. In our discussion so far we treated the pairing mode (1) giving rise to the parametric instability as an independent external field. Inclusion of the feedback on this mode as weak turbulence ($q\ll 1$) develops may modify the post-threshold state (9). We leave a detailed analysis of possible feedback effects for future studies. In the post-threshold state (9) the Fourier components of the order parameter are $c_{\mathbf{k},\omega}\sim\delta(k-k_{s})\delta(\omega-\Delta_{s})$. Inelastic scattering or thermal effects generally leads to a broadening in the momentum and frequency distributions of $c_{\mathbf{k},\omega}$[8]. The latter might cause a damping of the temporal oscillations in Eq. (9). On a time scale $t>\tau_{\varepsilon}$ dissipation due to quasi-particle scattering processes ultimately forces the system to reach an equilibrium state. Finally, we comment that in the transient regime leading to an asymptotic state with constant $\Delta(t)=\Delta_{s}$ the order parameter is $(\Delta(t)-\Delta_{s})\propto\cos(2\Delta_{s}t)/\sqrt{t}$ (cf. (1)) [10]. Oscillatory behavior suggests that this asymptotic state might never be attained owing to the development of the parametric instability of the type considered above. In conclusion, we have investigated the stability of the nonequilibrium asymptotic states of a fermionic superfluid, which can be generated e.g. by a uniform quench of the pairing strength. We have demonstrated that in a system of size $L$ larger than the coherence length $\xi$ the asymptotic state (1) with periodic in time order parameter is unstable with respect to spatial fluctuations. The instability is due to the parametric excitation of two pairing modes with opposite momenta. The initial exponential growth of deviations from the homogeneous state (11) is suppressed by nonlinear effects eventually leading to a spatially nonuniform post-threshold state described by Eq. (9). This state can be interpreted as a superposition of bubbles of the superfluid order parameter (10) with random amplitudes. The parametric instability of the uniform oscillations can be experimentally probed by spectroscopic noise measurements. M. D.’s research was financially supported by the Department of Energy, grant DE-FE02-00ER45790. E.A.Y. acknowledges the financial support by a David and Lucille Packard Foundation Fellowship for Science and Engineering, NSF award NSF-DMR-0547769, and Alfred P. Sloan Research Fellowship. B.L.A. thanks The US-Israel Binational Science Foundation for the financial support. ## References * (1) V. M. H. Ruutu _et al._, Nature (London) 382, 334 (1995). * (2) G. L. Warner and A. J. Leggett, Phys. Rev. B 71, 134514 (2005). * (3) S. Casado, W. González-Vinas, and H. Mancini, Phys. Rev. E 74, 047101 (2006). * (4) L. E. Sadler _et al._, Nature (London) 443, 312 (2006). * (5) V. E. Zakharov, V. S. L’vov and S. S. Starobinets, Sov. Phys. - Uspekhi 17, 896 (1975). * (6) L. D. Landau and E. M. Lifshitz, _Classical Mechanics_, (Pergamon Press, London,1975). * (7) V. S. L’vov and A. M. Rubenchik, Sov. Phys. - JETP 37, 263 (1973). * (8) V. S. L’vov, Sov. Phys. - JETP 42, 1057 (1976). * (9) R. A. Barankov, L. S. Levitov and B. Z. Spivak, Phys. Rev. Lett. 93, 160401 (2004). * (10) E. A. Yuzbashyan, O. Tsyplyatyev and B. L. Altshuler, Phys. Rev. Lett. 96, 097005 (2006). * (11) R. A. Barankov and L. S. Levitov, Phys. Rev. Lett. 96, 230403 (2006). * (12) E. A. Yuzbashyan and M. Dzero, Phys. Rev. Lett. 96, 230404 (2006). * (13) In this paper we consider only the case of a clean superfluid. * (14) M. Dzero, E. A. Yuzbashyan, B. L. Altshuler and P. Coleman, Phys. Rev. Lett. 99, 160402 (2007). * (15) V. E. Zakharov, V. S. L’vov and S. S. Starobinets, Sov. Phys. - JETP 32, 656 (1971).
1707.03398	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 26968, "num_imgs": 6, "llama3_tokens_count": 7701 }	[ "content_image/1707.03398/x1.png", "content_image/1707.03398/x2.png", "content_image/1707.03398/x3.png", "content_image/1707.03398/x5.png", "content_image/1707.03398/x7.png", "content_image/1707.03398/x11.png" ]	# Unidentified emission features in the R Coronae Borealis star V854 Centauri L. C. Oostrum Unidentified emission features in the R Coronae Borealis star V854 CentauriUnidentified emission features in the R Coronae Borealis star V854 CentauriUnidentified emission features in the R Coronae Borealis star V854 CentauriUnidentified emission features in the R Coronae Borealis star V854 Centauri B. B. Ochsendorf Unidentified emission features in the R Coronae Borealis star V854 CentauriUnidentified emission features in the R Coronae Borealis star V854 Centauri L. Kaper Unidentified emission features in the R Coronae Borealis star V854 CentauriUnidentified emission features in the R Coronae Borealis star V854 Centauri A. G. G. M. Tielens Unidentified emission features in the R Coronae Borealis star V854 CentauriUnidentified emission features in the R Coronae Borealis star V854 Centauri Received date / Accepted date Key Words.:circumstellar matter – stars: individual: V854 Cen¹ [FOOTNOTE:1][ENDFOOTNOTE] [ENDFOOTNOTE] ² [FOOTNOTE:2][ENDFOOTNOTE] ³ [FOOTNOTE:3][ENDFOOTNOTE] ⁴ [FOOTNOTE:4][ENDFOOTNOTE] During its 2012 decline the R Coronae Borealis star (RCB) V854 Cen was spectroscopically monitored with X-shooter on the ESO _Very Large Telescope_. The obscured optical and near-infrared spectrum exhibits many narrow and several broad emission features, as previously observed. The envelope is spatially resolved along the slit and allows for a detailed study of the circumstellar material. In this _Letter_ we report on the properties of a number of unidentified emission features (UFs), including the detection of a new one at $8692\,\mathrm{\AA}$. These UFs have been observed in the Red Rectangle, but their chemical and physical nature is still a mystery. The previously known UFs behave similarly in the Red Rectangle and V854 Cen, but are not detected in six other observed RCBs. Possibly the presence of some hydrogen is required for the formation of their carrier(s). The $\lambda$8692 UF is present in all RCBs. Its carrier is likely of a carbonaceous molecular nature, presumably different from that of the other UFs. ## 1Introduction R Coronae Borealis stars (RCBs) are rare, hydrogen deficient supergiants that exhibit strong declines in their brightness (Clayton, 2012). Only about a hundred RCBs are known in the Galaxy. Their rarity is an indication of a short evolutionary phase and/or points to a peculiar mode of stellar evolution: (i) a merger between a CO and He white dwarf or (ii) a final helium-shell flash leading to the expansion to supergiant size (Iben et al., 1996; Saio & Jeffery, 2002). The decline is thought to be due to the formation of clouds of carbon dust along the line of sight, obscuring the stellar photosphere so that the circumstellar envelope becomes observable in emission. Such a natural coronograph provides a unique opportunity to study the chemical and physical nature of the circumstellar envelope of these peculiar objects. Due to the irregularity of these events, limited spectra are available of RCBs in decline. V854 Cen is an RCB that is of particular interest because it is one of the few RCBs that include hydrogen lines in their spectra; It is the most hydrogen-rich RCB after DY Cen (Asplund et al., 1998; Jeffery & Heber, 1993). V854 Cen and DY Cen are also the only RCBs in which polycyclic aromatic hydrocarbons (PAHs) have been detected (García-Hernández et al., 7). The spectra include the weak 18.9$\,\mathrm{\mu m}$ band that is now generally attributed to C${}_{60}$****(Cami et al., 2010; Sellgren et al., 2010). Besides that, V854 Cen shows some unidentified visual emission features (UFs) in its decline spectrum (Rao & Lambert, 15). These features have only been detected in the Red Rectangle proto-planetary nebula (Schmidt & Witt, 1991). In that object, the features change in shape, intensity and peak position as a function of position in the nebula (Van Winckel et al., 2002; Wehres et al., 2011). Additional impetus for a study of the visual emission features is provided by the potential link between the visual emission bands in the Red Rectangle (RR) and the diffuse interstellar band (DIB) absorption features in the interstellar medium (Sarre et al., 1995). In this _Letter_ we show, for the first time, the spatial structure of the emission features in V854 Cen during its 2012 decline. We compare the characteristics of these features to those detected in the RR. In addition, we search for new emission features (300–2500$\,$nm) in V854 Cen, as well as in six other RCBs. ## 2Observations and data reduction Time on the ESO VLT was granted for observing V854 Cen within a window of four months in 2012. The object was monitored by the American Association of Variable Star Observers (AAVSO). When the star’s visual magnitude dropped below $m_{\mathrm{v}}=8$ (maximum-light $m_{\mathrm{v}}=7.1$), multiple spectra were obtained with VLT/X-shooter****(Vernet et al., 2011) during the decline that lasted around 2.5 months. Additionally, in 2013 X-shooter spectra were taken of V854 Cen and a few other RCBs, known to be in decline as determined from their AAVSO light curves, with the aim to search for the presence of unidentified features. V854 Cen was then at maximum light. For all observations, the highest resolution mode was used, where $R\approx 10000$, 18000, and 11500 for the UVB, VIS, and NIR arms, respectively. A log of observations is given in Table 1. The spectra were reduced using the X-shooter pipeline version 2.2.0 (Modigliani et al., 2010) and flux calibrated using spectrophotometric standard stars. Telluric correction of the 1D NIR spectra was done with Spextool (Vacca et al., 2003) using telluric standard spectra obtained at similar airmass and close in time to the targets. The resulting spectra were shifted to the rest-frame of the observed source. Target \| d \| Phase \| Obs. date \| mv \| mv \| S/N ---\|---\|---\|---\|---\|---\|--- \| (kpc) \| \| \| (max) \| (obs) \| V854 Cen \| 2.4 \| Max \| 2012-05-04 \| 7.1(1) \| 7.2 \| 90 \| \| Min \| 2012-06-14 \| \| 13.3 \| 70 \| \| Min \| 2012-06-18 \| \| 12.8 \| 130 \| \| Rise \| 2012-07-05 \| \| 9.9 \| 170 \| \| Rise \| 2012-07-05 \| \| 9.9 \| 150 \| \| Rise \| 2012-07-05 \| \| 9.9 \| 170 \| \| Max \| 2013-07-15 \| \| 7.2 \| 85 NSV 8092 \| 17.5 \| Min \| 2013-07-15 \| <11.7(2) \| 14.0 \| 30 R CrB \| 1.3 \| Min \| 2013-07-15 \| 5.7(3) \| 13.7 \| 35 RT Nor \| 9.9 \| Min \| 2013-07-15 \| 10.6(3) \| 15.2 \| 20 RZ Nor \| 10.5 \| Min \| 2013-07-15 \| 10.6(3) \| 16.2 \| 15 S Aps \| 7.1 \| Min \| 2013-07-15 \| 9.6(3) \| 14.3 \| 10 V CrA \| 6.6 \| Min \| 2013-07-15 \| 9.4(3) \| 17.5 \| 10 \tablebib (1) Samus’ et al. (2003); (2) Tisserand et al. (2013); (3) Ducati (2002). Table 1: Log of VLT/X-shooter observations. During the rise of V854 Cen, three different position angles were used. The magnitudes were measured from the acquisition images, except for V854 Cen during maximum light, for which they are saturated. For these, values from the AAVSO were used. V854 Cen distance is based on the (V−I) \- MV relation for RCBs in the LMC by Tisserand et al. (2009). For the others, a typical value of Mv=−5 (Clayton, 2012) is assumed. The signal-to-noise ratio was measured in the continuum around 6200 Å. ## 3The unidentified emission features ### Detected bands <figure><img src="content_image/1707.03398/x1.png"><figcaption>Figure 1: V854 Cen decline spectra at different offsets from the centralobject (_see inset_) and at maximum light (_bottom spectrum_). During adecline, the absorption spectrum converts into an emission spectrum, thestrongest emission lines being the Na i D doublet. Additionally, a sequence ofunidentified emission features (UFs) is clearly detected between 5800 and5860\AA, at 6617\AA, and at 8692\AA. The narrow Y ii λ6614 emission line hasbeen removed from the 1.6\arcsec and 2.0\arcsec offset spectra for plottingpurposes. The absorption lines near 5800\AA and 5854\AA are due to C i and Baii, respectively.</figcaption></figure> A section of the X-shooter spectrum of V854 Cen is shown in Fig. 1. The broad features at 5800, 5827, 5854 and 6617 Å were first recognized in V854 Cen by Rao & Lambert(15) during a deep decline ($m_{V}\sim 15$). The similarity between these features and the emission features in the RR was already noted. We expand upon this by considering the spatial and dynamical (i.e. radial velocity) structure of these features in V854 Cen. None of these bands are detected in any of the other observed RCBs. The spectra reveal the presence of a broad emission feature at 8692 Å that has not been seen before in any RCB, nor in the RR. Of the seven bands, only the $\lambda\lambda$5800, 5827, 5854, and 6617 bands are detected in the 2012 decline of V854 Cen. The absence of three of the features (at 5772, 6774 and 6997$\mathrm{\,\AA}$) may be attributed to the depth of the decline, as this decline is 2 magnitudes – or a factor ${\sim}6$ in flux – shallower than the 1992 decline, and the non-detected features are the weakest ones. The four detected bands are present in the two spectra taken during the deepest part of the decline ($m_{V}\sim 13$), just beyond the stellar continuum along the slit. The features are strongest in the spectrum taken just after the minimum, on 2012-06-14. Only in that spectrum the S/N in the features is high enough to allow for an analysis of their spatial distribution. The features are detected at different distances from the central object as shown in Fig. 1, which also shows the maximum light spectrum for comparison. The features are only very marginally detected at the Western side of the star (negative offsets), hence we focus on the Eastern side (positive offsets). None of the bands are detected in the other RCBs. The whole wavelength range of X-shooter (300-2500$\,\mathrm{nm}$) has been searched for the presence of emission features. One new feature was detected, present in _all_ V854 Cen spectra, at ${\sim}8692\,\mathrm{\AA}$. It is also detected in five of the six other observed RCBs in decline (Fig. 2). Similar to many features associated with the circumstellar material, this feature is only detected off-source. The feature is blended with two photospheric lines: Sr ii**$\lambda 8689$ and Sc i$\lambda 8694$. In order to disentangle the nebular features from the stellar ones, the off-source spectrum is divided by the on-source spectrum and the resulting spectrum renormalised. The presence of the feature during maximum-light is especially noteworthy. <figure><img src="content_image/1707.03398/x2.png"><figcaption>Figure 2: The λ8692 UF in all observed RCBs. The two dashed lines indicate theposition of the Sr ii λ8689 and Sc i λ8694 photospheric lines. The UF isdetected in all except V CrA. The position and intensity of the featureclearly vary between objects. In V854 Cen the feature is also detected duringmaximum light.</figcaption></figure> ### Spatial and dynamical structure of the bands As the circumstellar envelope of V854 is spatially resolved along the spectrograph slit, position-velocity (PV) diagrams can be produced to study the kinematic structure of the emission lines as a function of the distance from the star. In Fig. 3 we show the dynamical structure as revealed by the Na i D resonance lines, and UF8692. The other unidentified features are too weak to construct a PV diagram. Both doublet components in Na i D show a structure suggesting that the emission is produced in roughly a shell with a radius of about 3.5″ and expanding with a velocity of $250\mathrm{\,km\,s^{-1}}$. The size of the shell is based on the furthest position where the flux is more than 3$\sigma$ above the noise. PV diagrams of more extended features, confirming the size and velocity of the shell, are shown in Appendix A. The shape of the UF8692 extended emission is different from the shell traced by Na i D. It shows a change in wavelength and width for different distances from the central object. For V854 Cen, it shifts toward the red at the Eastern side of the star (i.e. positive offsets), and to the blue at the Western side of the star. Additionally, it narrows with increasing distance to the star on the Eastern side. It is also worth noting that the equivalent width (W${}_{\mathrm{eq}}$) of the feature – here defined such that an emission feature has a positive W${}_{\mathrm{eq}}$ – increases with distance to the star. This confirms that this newly found feature originates from the circumstellar material. The behaviour of the $\lambda 8692$ feature is different for each RCB. In RT Nor and RZ Nor, the width of the feature does not change significantly, while a narrowing with increasing distance is observed in the others. The position of the feature shifts in all objects. There is no clear pattern to this, in some objects only blueshift is observed, in others also redshift and/or no shift on one side of the star. As the RCBs have different distances, different physical scales are probed, but we do not find a correlation between the behaviour of this feature and the respective distances to the RCBs. <figure><img src="content_image/1707.03398/x3.png"><figcaption>Figure 3: Position along the spectrograph slit (vertical direction) againstradial velocity for the Ca II K resonance line (top) and UF at 8692 Å (bottom)in V854 Cen. The continuum flux has been subtracted and all flux within 0.7″of the central object has been set to zero to enhance the visibility of theseextended features. The thick blue line shows the expected maximum radialvelocity at each position for a spherical shell of radius 3.5″ and expandingwith a velocity of 250kms−1. One sees that the unidentified feature at 8692\AAis spatially extended as well.</figcaption></figure> The shape of the bands, when integrated over an interval along the slit, is well described by either one or two Gaussians. All features in the $\lambda 5825$ complex are measured together, as those bands are too close in wavelength to be considered separately. For the band at $5800\,\mathrm{\AA}$, two Gaussians are used to account for the asymmetric band shape, which significantly improves the fits. For the other bands a single component is used. All sharp emission lines, originating from regions close to the star (*${\sim}2R_{}$; Clayton, 1996) instead of the large-scale circumstellar material, are removed from the spectra prior to the fitting procedure. For all five detected bands, the measured peak position as function of distance to the star is shown in Fig. 4 (black squares). All features (at 1.6$\arcsec$ and 2.8$\arcsec$) are shown in Appendix B. The narrow component in the $5800\,\mathrm{\AA}$ feature (Fig. 1) is used for the band position, as this component is stronger and dominates the peak position. The same bands in the RR are known to show a blueshift with increasing distance to the central binary system (Van Winckel et al., 2002; Wehres et al., 2011). The same analysis method as for V854 Cen is applied to the RR spectra, except for the number of components: The higher S/N RR spectra require two components for the $\lambda 5854$ and $\lambda 6617$ bands in addition to $\lambda 5800$, whereas one component is sufficient for those bands in V854 Cen. The only RR spectrum covering the $\lambda 8692$ feature is from the ESPaDOnS spectrograph mounted on the CFHT, and unfortunately has insufficient signal in that wavelength region (N.L.J. Cox, priv. comm.). The measured RR band positions agree well with those provided by Wehres et al.(2011). With the exception of $\lambda 8692$, all V854 Cen and RR bands show a shift towards shorter wavelengths with increasing distance to the central object. The RR bands are clearly shifted towards the blue with respect to V854 Cen, although they are known to be redder than the V854 Cen bands close to the central binary in the RR (Van Winckel et al., 2002). The $\lambda 8692$ feature is the only feature detected on both sides of the star, and shows a redshift where the other features show a blueshift. The RR bands show a correlation between their position and width: Some bands become narrower as they shift to shorter wavelengths (Schmidt & Witt, 1991; Sarre et al., 1995; Van Winckel et al., 2002; Wehres et al., 2011). We are unable to confirm such a correlation in the V854 Cen features. It should be noted that, in the RR, the features change most rapidly close to the central binary. The distance to V854 Cen is roughly three times that of the RR ($710\mathrm{\,pc}$; Men’shchikov et al., 2002). We thus cannot exclude the presence of a width-wavelength correlation in the inner regions of the circumstellar material. <figure><img src="content_image/1707.03398/x5.png"><figcaption>Figure 4: Band positions of the unidentified features as function of distanceto the central object for V854 Cen (black squares) and the Red Rectangle (redcircles). There is no measurement of the λ6617 feature at 6\arcsec. Error barsreflect 1σ errors. All RR bands are blueshifted with respect to thecorresponding bands in V854 Cen. The λ8692 band is only covered in V854 Cenand is detected on both sides of the star.</figcaption></figure> ## 4Discussion We detect several emission features in the nebula surrounding V854 Cen, up to $3.5\,\arcsec$ or ${\sim}8500\,\mathrm{AU}$ from the central object. This is the furthest detection of the nebula so far. The broad emission lines from Na i D, H$\alpha$, and [C i] $\lambda 9850$ show a roughly spherical outflow with velocities up to $250\,\mathrm{km\,s^{-1}}$. In addition to these atomic lines, we detect six unidentified emission features, including a new one at $8692\,\mathrm{\AA}$. The width of the emission features is comparable to that of other broad emission lines. The UFs that are also detected in the RR have a similar width, even though the velocities in the RR are much lower (${\sim}7\,\mathrm{km\,s^{-1}}$ in CO; Jura et al., 1995) than in V854 Cen. This indicates that the features are intrinsically broad and hence that their width in V854 Cen is not due to Doppler broadening. The observed shift of ${\sim}1\,\mathrm{\AA}$ in the features at $5800-5854\,\mathrm{\AA}$ corresponds to ${\sim}50\,\mathrm{km\,s^{-1}}$. This shift may be due to rotational cooling, as in the RR (Wehres et al., 2011), however assuming no cooling it is an upper limit on the radial velocity of the carrier. This velociy is significantly lower than that of the Na i D-traced shell, hence the carriers are not located in that shell. There is, however, evidence for low-velocity dust in RCBs, assuming that the minimum-light optical spectrum is dominated by scattered light from the central object (García-Hernández et al., 8). This low-velocity material is the most likely environment for the carrier of the UFs. The absence of the features in other RCBs may indicate that the presence of some hydrogen is required for their formation. The $\lambda$8692 feature may provide insight in the geometry of this dust in V854 Cen, as it is the only UF that is detected clearly on both sides of the star. If we interpret the shift in this feature as a Doppler shift, it is consistent with a bipolar outflow which is being accelerated to ${\sim}50\,\mathrm{km\,s^{-1}}$. A bipolar geometry has been suggested in literature (e.g. Rao & Lambert, 14; Chesneau et al., 2014). If the other UFs are part of the same outflow, we would expect them to show a shift of ${\sim}0.3\,\mathrm{\AA}$ between their closest and furthest detection. We are, however, unable to confirm this, as this putative shift is smaller than the detected shifts in both V854 Cen and the RR. By the same argument as for the other features, the carrier of $\lambda 8692$ is not located in the high-velocity shell. For the observed bands, the shifts in the RR are consistent with a change in excitation temperature of a molecule (Wehres et al., 2011). For V854 Cen too, one would expect such a shift as the temperature in the outflow decreases. Care has to be taken when comparing UF8692 to the other features, as it is the only feature present in all spectra and the only one to show up in other RCBs. It is thus unlikely that this feature has the same carrier as one of the other emission bands. The presence of the feature in more hydrogen-deficient RCBs does indicate that the carrier is hydrogen-poor. A carbonaceous molecular nature seems likely, given the high carbon abundances of RCBs. The different behaviour of the feature in different RCBs is quite puzzling. It may be an indication of varying physical and chemical conditions in the complex circumstellar envelopes of RCBs. It has been suggested that the UFs in the RR are the emission equivalent of the DIBs (e.g. Scarrott et al., 1992). The inverted DIB spectrum does show correlations with some UFs. We are unable to identify a possible DIB counterpart for the $\lambda 8692$ feature. The closest known DIB is located at $8648.3$ Å (online catalogue¹ and N.L.J. Cox, priv. comm.), which is too far off to be the likely absorption equivalent of the $\lambda 8692$ feature. [FOOTNOTE:1][ENDFOOTNOTE] ###### Acknowledgements. The authors thank Nadine Wehres and Hans van Winckel for providing us the Red Rectangle spectra. We acknowledge the variable star observations from the AAVSO International Database contributed by observers worldwide, and used in this research. LCO would like to thank Nick Cox for discussion about DIBs and the Red Rectangle. LCO acknowledges funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 617199. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programmes 089.D-0937(A) and 091.C-0934(B). ## References * Asplund et al. (1998) Asplund, M., Gustafsson, B., Rao, N. K., & Lambert, D. 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Note that thenarrow emission line in the CH+ diagram at ∼50kms−1 is likely due to Na ii. Hαand [C i] show similar, though less pronounced behaviour.</figcaption></figure> ## Appendix BUnidentified feature spectra <figure><img src="content_image/1707.03398/x11.png"><figcaption>Figure 6: Spectra at 1.6\arcsec and 2.4\arcsec from the central object for allunidentified features. Dashed lines indicate the best-fit peak position foreach feature. Top left: λλ5800, 5827, and 5854. Top right: λ6617\. Bottomleft: λ8692. In all features except λ8692, a blueshift with increasingdistance from the central object is observed. The λ8692 feature shifts towardlonger wavelengths in the spectra shown here, but it shifts toward shorterwavelengths on the other side of the star. The 1.6\arcsec spectrum of λ8692has been stretched by a factor 8 for displaying purposes.</figcaption></figure>
0906.3493	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 62921, "num_imgs": 13, "llama3_tokens_count": 19464 }	[ "content_image/0906.3493/x1.png", "content_image/0906.3493/x2.png", "content_image/0906.3493/x3.png", "content_image/0906.3493/x4.png", "content_image/0906.3493/x5.png", "content_image/0906.3493/x6.png", "content_image/0906.3493/x7.png", "content_image/0906.3493/x8.png", "content_image/0906.3493/x9.png", "content_image/0906.3493/x10.png", "content_image/0906.3493/x11.png", "content_image/0906.3493/x12.png", "content_image/0906.3493/x13.png" ]	# The initial conditions of stellar protocluster formation I. A catalogue of Spitzer dark clouds N. Peretto 1Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, UK 1Nicolas.Peretto@manchester.ac.uk G. A. Fuller 1Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, UK 1Nicolas.Peretto@manchester.ac.uk Received; accepted Key Words.:Catalogs; Stars: formation; ISM: clouds † [FOOTNOTE:†][ENDFOOTNOTE] ###### Abstract Context:The majority of stars form in clusters. Therefore a comprehensive view of star formation requires understanding the initial conditions for cluster formation. Aims:The goal of our study is to shed light on the physical properties of infrared dark clouds (IRDCs) and the role they play in the formation of stellar clusters. This article, the first of a series dedicated to the study of IRDCs, describes techniques developed to establish a complete catalogue of Spitzer IRDCs in the Galaxy. Methods: We have analysed Spitzer GLIMPSE and MIPSGAL data to identify a complete sample of IRDCs in the region of Galactic longitude and latitude $10^{\circ}<\|l\|<65^{\circ}$ and $\|b\|<1^{\circ}$. From the 8$\mu$m observations we have constructed opacity maps and used a newly developed extraction algorithm to identify structures above a column density of N${}_{\rm{H_{2}}}\ga 1\times 10^{22}$ cm${}^{-2}$. The 24$\mu$m data are then used to characterize the star formation activity of each extracted cloud. Results: A total of 11303 clouds have been extracted. A comparison with the existing MSX based catalogue of IRDCs shows that 80$\%$ of these Spitzer dark clouds were previously unknown. The algorithm also extracts $\sim$ 20000 to 50000 fragments within these clouds, depending on detection threshold used. A first look at the MIPSGAL data indicates that between 20% and 68% of these IRDCs show 24$\mu$m point-like association.This new database provides an important resource for future studies aiming to understand the initial conditions of star formation in the Galaxy. Conclusions: ## 1Introduction <figure><img src="content_image/0906.3493/x1.png"><figcaption>Figure 1: These images show the GLIMPSE Spitzer 8μm emission of 3 randomIRDCs from our sample. These illustrate the diversity in shape and size ofIRDCs.</figcaption></figure> The majority of stars form in groups from few tens to few hundreds of objects (e.g. Lada & Lada, 2003). So, understanding cluster formation is key to understanding the formation of stars. Clusters form from the gas located in the densest parts of molecular clouds, within structures called clumps (Blitz, 1993). These clumps fragment into an assembly of protostellar cores which collapse to produce stars, forming ‘protoclusters’. By definition, protoclusters are active star forming regions, with jets, flows and heating sources (e.g. Bally et al., 2006) which rapidly start to shape their surroundings. From the study of these protoclusters, it is therefore difficult to back track to the initial conditions of their formation. On the other hand, clumps which are on the verge of forming protostars, but which have not formed any yet, are structures unpolluted by star formation activity and must still reflect the initial conditions of the formation of protoclusters. Looking for, and studying such ‘pre-protoclusters’ is crucial for our understanding of star formation processes. Only a tiny percentage of the material in any molecular cloud forms stars. These star-forming regions are traced by various signposts of star formation activity such as the presence of strong infrared sources, outflows, jets, methanol and water masers and compact HII regions. The problem with identifying pre-protoclusters is that by definition these signposts are not yet present. Other means are thus necessary to find such objects. The two infrared satellites ISO and MSX have been important tools for this purpose. Indeed, the large infrared surveys these satellites carried out identified infrared dark structures, seen in absorption from 7 to 25 $\mu$m against the background emission (Perault et al., 1996; Hennebelle et al., 2001; Egan et al., 1998; Simon et al., 27) . Millimeter molecular lines (e.g. Carey et al., 1998; Teyssier et al., 2002; Pillai et al., 2006) and dust continuum observations (e.g. Teyssier et al., 2002; Rathborne et al., 2006) have clearly demonstrated that these infrared dark clouds are dense, cold structures, possibly being the progenitors of protoclusters (Simon et al., 28). Rathborne et al.(2006) even suggested that the dust continuum “cores” observed in these IRDCs are the direct progenitors of massive stars. However, the wide range of mass and size of these IRDCs clearly suggests that they cannot all be evolving along the same evolutionary path and they must lead to the formation of a large range of different stellar contents. So far, the study of the earliest stages of the formation of protoclusters have mostly focussed on the closest objects such as $\rho$-Oph(e.g. Motte et al., 1998; André et al., 2007), Perseus (Hatchell et al., 2005; Enoch et al., 2006), NGC2264 (e.g. Peretto et al., 2006; Teixeira et al., 2006). The results of these studies set important constraints on models of star formation, but may not be representative of the formation of stars throughout the Galaxy. The only way to define such a representative view is through studies of large unbiased samples of the precursors of stellar clusters. In this paper we identify and characterise the IRDCs detected using the Spitzer GLIMPSE and MIPSGAL archive data. The high angular resolution of the Spitzer data provides a detailed probe of the structure of these sources while the high sensitivity of IRAC and MIPS allows us to detect previously unseen deeply embedded protostars/protoclusters. Section 2 of this paper presents the Spitzer archive data used for this study. Section 3 will discuss the construction of 8$\mu$m opacity maps for IRDCs, while Section 4 will focus on the conversion from 8$\mu$m opacity to H${}_{2}$ column density. The extraction of structures within these maps will be discussed in Section 5. A comparison with the MSX catalogue of IRDCs is in Section 6 while Section 7 summarizes our initial study. The nature of these dark clouds and their star formation actively are discussed in more detail in subsequent papers (Peretto & Fuller, in preparation). ## 2A large survey of infrared dark clouds: Spitzer archive data IRDCs are seen in silhouette against the infrared background emission (see Fig. 1) and as a sample are likely to contain protoclusters and pre-protoclusters. Even when large scale (sub)millimetre surveys of the Galactic plane become available and these objects can be detected through their dust emission, IRDCs and studies of the absorption towards these sources will remain important. Not only can the IRDCs be studied at high angular resolution at infrared wavelengths, but unlike the (sub)millimetre emission, their column density can be measured from the absorption independent of the dust temperature. The first large survey of IRDCs was undertaken by Simon et al.(27) using the mid-infrared data of the MSX satellite. In total, Simon et al. detected more than 10000 IRDCs, with sizes larger than (36${}^{\prime\prime}$)${}^{2}$ and flux density more than 2 MJy/sr ($>2$ times the rms noise of the MSX images) below the mid-infrared radiation field. Within these IRDCs they extracted more than 12000 IRDC “cores”. Simon et al.(28) performed a follow up of a sub-sample of few hundreds sources for which they were able to determine distances. They found that these IRDCs are very similar to CO molecular clumps (e.g. Blitz, 1993). In the GLIMPSE and MIPSGAL surveys the Spitzer satellite has resurveyed a large fraction of the Galactic plane at infrared wavelengths ($10\hbox{${}^{\circ}$}<\|l\|<65\hbox{${}^{\circ}$},0<\|b\|<1\hbox{${}^{\circ}$}$). These data have both better angular resolution (2${}^{\prime\prime}$ vs 20${}^{\prime\prime}$ at 8$\mu$m ) and sensitivity (0.3 MJy/sr vs 1.2 MJy/sr at 8$\mu$m ) than the MSX data, as well as wider wavelength coverage.The IRAC (3.6, 4.5, 5.8, 8 $\mu$m) GLIMPSE and MIPS (24, 70, 160 $\mu$m) MIPSGAL observations provide a unique opportunity to shed light on the role of IRDCs during the earliest stages of star formation. Despite a smaller coverage of the Galactic plane by Spitzer, an initial comparison of the MSX IRDC catalogues with the Spitzer observations indicated that the Spitzer data contained IRDCs undetected by MSX in the same region of the Galaxy. Therefore an unbiased search of the Spitzer GLIMPSE data has been undertaken to identify IRDCs. Many IRDCs can been seen in silhouette up to at least 24 $\mu$m, providing a wide wavelength range over which they can be studied in absorption. However several factors affect the choice of the optimal wavelength at which to identify and study the overall cloud properties. These include the strength and uniformity of the background emission and the number of foreground and background stars and in principle, the wavelength dependence of the dust extinction law, although recent work suggests that from 4.5 to 8$\mu$m, the three last bands observed by Spitzer/IRAC, the extinction is a relatively flat function of wavelength (Lutz et al., 1996; Indebetouw et al., 2005; Román-Zúñiga et al., 2007). The angular resolution of the observations is highest at the shortest wavelengths, but in these bands a very high density of stars is detected and high degree of structure in the relatively weak background emission makes analysis of the images at these wavelengths complex. Overall, inspection of the Spitzer data shows that the strength and relative smoothness of the background emission together with the relatively low density of stars make the IRAC 8 $\mu$m band the most suitable for this initial study of a large sample of objects. The GLIMPSE and MIPSGAL data have been reduced and calibrated automatically to produce the so called post-Basic Calibrated Data (post BCD). The typical flux uncertainty for point-like sources is $\sim 2\%$ at 8$\mu$m (Reach et al., 2005) while the position uncertainty is less than 0.3${}^{\prime\prime}$(IRAC manual V8.0: http://ssc.spitzer.caltech.edu/documents/SOM/). However, because we are not looking at point-like sources but extended objects, a calibration factor has to be applied on the PBCD 8$\mu$m images (Reach et al., 2005). This calibration factor, CF, is a function of the aperture radius, R${}_{a}$, for the source under investigation (http://ssc.spitzer.caltech.edu/irac/calib/extcal/). The relation between CF and R${}_{a}$ in arcseconds, at 8$\mu$m is $CF=1.37\times\exp(-R_{a}^{0.33})+0.74$. Because the typical size of the structure we analyse is about one arcminute, in the analysis which follows we applied a calibration factor of 0.8 to the PBCD 8$\mu$m images. A different calibration factor would not change the opacities of the IRDCs we calculated, but would imply different related intensities (Table 1). <figure><img src="content_image/0906.3493/x2.png"><figcaption>Figure 2: Schematic view a typical IRDC flux density profile. The variablemeanings used in the rest of the text are illustrated on this figure. In thisfigure, Ifore has been set to a particular value, e.g., 38 MJy/sr, but inpractice, it can be anywhere between Izl and Imin.</figcaption></figure> ## 3Opacity distribution of IRDCs ### Principle Infrared dark clouds are structures seen in absorption against the background emission. The strength of the absorption is directly related to the opacity along the line of sight. Following the notation of Bacmann et al.(2000), the relation between the opacity $\tau_{\lambda}$ and the intensity at wavelength at $\lambda$, emerging from the cloud $I_{\lambda}$, is given by \[I_{8\mu\rm{m}}=I_{\rm{bg}-8\mu\rm{m}}\times\exp(-\tau_{8\mu\rm{m}})+I_{\rm{ fore}-8\mu\rm{m}}\] (1) where $I_{\rm{bg}-8\mu\rm{m}}$ is the intensity of the background emission at 8$\mu$m, and $I_{\rm{fore}-8\mu\rm{m}}$ is the foreground emission. In the following for simplicity we drop the 8$\mu$m label on the variable names, except on the opacity. If we know the foreground and background intensities we can invert Eq. (1) and infer the spatial distribution of the opacity within an infrared dark cloud, \[\tau_{8\mu\rm{m}}=-\ln\left(\frac{I-I_{\rm{fore}}}{I_{\rm{bg}}}\right)\] (2) $I_{\rm{fore}}$ and $I_{\rm{bg}}$ are related to each other by $I_{\rm{MIR}}=I_{\rm{bg}}+I_{\rm{fore}}$ where $I_{\rm{MIR}}$ is the observed mid-infrared radiation field and can be estimated directly from the $8\mu$m images (see Fig. 2). A lower limit on $I_{\rm{fore}}$ is given by the intensity of the zodiacal light, $I_{\rm{zl}}$, in the direction of the cloud, while an upper limit is given by the minimum intensity within the cloud, $I_{\rm{min}}$. However, with the extinction data only, it is impossible to find the exact value of $I_{\rm{fore}}$ for a given cloud. The determination of $I_{\rm{fore}}$ is crucial to infer the spatial opacity distribution of a given IRDC. To illustrate this point, we computed the opacity of the cloud profile shown in Fig. 2 for three different values of $I_{\rm{fore}}$ (Fig. 3). On this figure we see that, with increasing $I_{\rm{fore}}$, the opacity increases significantly everywhere in the cloud, and even more sharply at the peak. These opacity variations are even more drastic for shallower clouds. It is therefore important to constrain $I_{\rm{fore}}$ when calculating the opacity distribution of an IRDC. <figure><img src="content_image/0906.3493/x3.png"><figcaption>Figure 3: Calculated opacity profiles of the IRDC plotted in Fig.2corresponding to 3 different assumptions on the foreground intensity. Thesolid line shows Ifore=Izl (i.e Ifore=0.25×Imin), the dotted lineIfore=0.7×Imin and the dashed line Ifore=0.9×Imin .</figcaption></figure> Of course it is also possible that at least some the IRDCs are saturated and their intensity profiles become flattened. In such cases, it becomes impossible to recover the central structure of the clouds through the extinction maps. Moreover, such flattening could lead to an incorrect interpretation of the final opacity profiles of IRDCs. \begin{tabular}{c c c c c c c c c c c c c c c c c c} \hline \hline Number & Name & \multicolumn{2}{c}{Coordinates} & $I_{\rm{min}}$ & $I_{\rm{MIR}}$ & $\delta I_{\rm{MIR}}$ & $\Delta X$ & $\Delta Y$ & $\alpha$ & R${}_{\rm eq}$ & $\tau_{\rm{peak}}$ & $\tau_{\rm{av}}$ & $\tau_{\rm{sat}}$ & frag & \multicolumn{2}{c}{star} & $\sigma_{\rm{star}}$ \\ & & RA(J2000) & Dec(J2000) & (MJy/sr) & (MJy/sr) & & (${}^{\prime\prime}$) & (${}^{\prime\prime}$) & (${}^{\circ}$) & (${}^{\prime\prime}$) & & & & & Field & IRDC & (arcmin${}^{-2}$) \\ \hline 1 & SDC9.22+0.169 & 18:05:30.40 & -20:53:16.0 & 37.9 & 55.7 & 0.07 & 42.9 & 29.8 & 76 & 32.3 & 1.16 & 0.50 & 4.45 & 2 & y & y & 0.92 \\ 2 & SDC9.25+0.144 & 18:05:39.69 & -20:52:25.6 & 40.8 & 56.9 & 0.04 & 26.9 & 20.5 & -66 & 23.9 & 0.95 & 0.48 & 4.47 & 1 & y & n & 0.95 \\ 3 & SDC9.256+0.133 & 18:05:42.92 & -20:52:27.8 & 41.7 & 56.7 & 0.01 & 13.5 & 10.8 & -42 & 13.1 & 0.85 & 0.48 & 4.47 & 1 & n & n & 1.33 \\ 4 & SDC9.301+0.126 & 18:05:50.01 & -20:50:16.8 & 43.0 & 56.7 & 0.00 & 17.9 & 7.2 & -89 & 12.6 & 0.75 & 0.45 & 4.47 & 1 & n & n & 1.34 \\ 5 & SDC9.328+0.031 & 18:06:14.76 & -20:51:39.7 & 46.2 & 60.8 & 0.02 & 11.4 & 10.0 & 87 & 11.6 & 0.74 & 0.47 & 4.53 & 1 & y & n & 1.14 \\ 6 & SDC9.403+0.26 & 18:05:33.02 & -20:41:00.7 & 33.7 & 44.5 & 0.02 & 11.2 & 7.3 & -87 & 10.2 & 0.75 & 0.44 & 4.22 & 1 & y & n & 0.93 \\ 7 & SDC9.432+0.163 & 18:05:58.31 & -20:42:21.8 & 38.7 & 51.4 & 0.03 & 13.3 & 8.4 & -50 & 11.8 & 0.76 & 0.44 & 4.37 & 1 & n & n & 1.02 \\ 8 & SDC9.461+0.138 & 18:06:07.50 & -20:41:32.8 & 40.0 & 53.0 & 0.03 & 17.8 & 6.9 & 22 & 12.0 & 0.77 & 0.42 & 4.40 & 1 & n & n & 1.08 \\ 9 & SDC9.624+0.187 & 18:06:16.80 & -20:31:35.6 & 37.7 & 53.5 & 1.58 & 321.6 & 203.8 & -75 & 188.7 & 4.30 & 0.61 & 4.41 & 6 & y & y & 0.78 \\ 10 & SDC9.629-0.061 & 18:07:13.25 & -20:38:38.7 & 38.0 & 51.7 & 0.03 & 23.0 & 16.1 & 30 & 18.0 & 0.85 & 0.45 & 4.37 & 1 & y & n & 0.94 \\ 11 & SDC9.635+0.296 & 18:05:53.87 & -20:27:49.8 & 24.1 & 39.7 & 0.08 & 29.6 & 15.1 & -60 & 21.5 & 1.96 & 0.58 & 4.11 & 1 & y & n & 0.76 \\ 12 & SDC9.689+0.000 & 18:07:06.92 & -20:33:41.5 & 39.6 & 52.9 & 0.01 & 24.6 & 11.8 & 57 & 15.9 & 0.79 & 0.46 & 4.40 & 1 & n & n & 1.01 \\ 13 & SDC9.692-0.55 & 18:09:10.79 & -20:49:34.6 & 27.2 & 35.3 & 0.02 & 18.6 & 6.8 & -69 & 12.7 & 0.69 & 0.41 & 3.99 & 1 & n & n & 0.94 \\ 14 & SDC9.737-0.239 & 18:08:06.57 & -20:38:08.1 & 37.9 & 52.1 & 0.05 & 30.4 & 12.4 & 4 & 20.3 & 0.91 & 0.48 & 4.38 & 1 & n & n & 0.97 \\ 15 & SDC9.762-0.567 & 18:09:23.29 & -20:46:20.9 & 26.6 & 37.3 & 0.01 & 12.1 & 7.5 & -48 & 10.7 & 0.98 & 0.51 & 4.05 & 1 & y & y & 0.79 \\ 16 & SDC9.787-0.156 & 18:07:54.17 & -20:33:06.8 & 40.1 & 61.2 & 0.06 & 58.7 & 26.3 & -26 & 37.1 & 1.38 & 0.57 & 4.54 & 2 & y & n & 1.13 \\ 17 & SDC9.796-0.028 & 18:07:26.89 & -20:28:53.3 & 43.5 & 59.8 & 0.04 & 36.6 & 17.7 & -1 & 25.7 & 0.92 & 0.50 & 4.52 & 1 & n & n & 1.25 \\ 18 & SDC9.798-0.707 & 18:09:59.42 & -20:48:32.9 & 29.1 & 40.5 & 0.36 & 63.7 & 45.8 & 2 & 47.5 & 0.90 & 0.47 & 4.13 & 1 & y & n & 0.72 \\ 19 & SDC9.819-0.141 & 18:07:54.93 & -20:31:00.7 & 44.8 & 62.2 & 0.02 & 39.0 & 12.5 & 37 & 20.1 & 0.95 & 0.44 & 4.56 & 1 & y & n & 1.25 \\ 20 & SDC9.825-0.03 & 18:07:30.72 & -20:27:26.0 & 41.0 & 58.3 & 0.08 & 78.9 & 19.9 & 78 & 32.7 & 1.02 & 0.47 & 4.49 & 2 & y & n & 1.16 \\ 21 & SDC9.844+0.752 & 18:04:38.31 & -20:03:31.4 & 23.1 & 30.8 & 0.05 & 24.1 & 13.6 & -52 & 16.8 & 0.76 & 0.43 & 3.86 & 1 & y & n & 0.56 \\ 22 & SDC9.845-0.138 & 18:07:57.47 & -20:29:34.3 & 37.0 & 63.5 & 0.05 & 67.9 & 35.1 & 0 & 37.5 & 2.28 & 0.51 & 4.58 & 2 & y & n & 1.16 \\ 23 & SDC9.852-0.034 & 18:07:35.07 & -20:26:07.8 & 30.5 & 59.3 & 0.17 & 119.4 & 55.6 & -73 & 76.5 & 4.95 & 0.87 & 4.51 & 12 & y & y & 1.19 \\ 24 & SDC9.859-0.746 & 18:10:15.75 & -20:46:27.5 & 33.1 & 61.3 & 1.94 & 294.5 & 150.3 & -7 & 162.8 & 3.08 & 0.79 & 4.54 & 4 & y & y & 0.63 \\ 25 & SDC9.864-0.102 & 18:07:51.78 & -20:27:30.2 & 49.1 & 64.5 & 0.02 & 35.2 & 17.3 & 84 & 23.4 & 0.74 & 0.44 & 4.59 & 1 & y & n & 1.18 \\ 26 & SDC9.872-0.767 & 18:10:22.02 & -20:46:23.9 & 42.5 & 69.3 & 1.51 & 180.5 & 113.6 & -66 & 126.3 & 2.29 & 0.79 & 4.67 & 2 & y & y & 0.55 \\ 27 & SDC9.878-0.11 & 18:07:55.37 & -20:26:58.8 & 35.5 & 65.7 & 0.07 & 59.2 & 40.9 & 26 & 49.3 & 4.98 & 0.83 & 4.61 & 4 & y & y & 1.18 \\ 28 & SDC9.889-0.747 & 18:10:19.58 & -20:44:56.8 & 61.7 & 99.2 & 0.20 & 26.5 & 23.2 & -16 & 24.8 & 2.01 & 0.92 & 5.02 & 1 & y & n & 0.59 \\ 29 & SDC9.895-0.749 & 18:10:20.74 & -20:44:41.4 & 73.5 & 105.0 & 0.04 & 12.6 & 6.4 & 18 & 9.9 & 1.07 & 0.54 & 5.08 & 1 & n & n & 0.57 \\ 30 & SDC9.904-0.699 & 18:10:10.70 & -20:42:43.9 & 58.7 & 78.9 & 0.38 & 39.5 & 19.2 & 43 & 26.9 & 0.73 & 0.47 & 4.80 & 1 & y & n & 0.49 \\ \hline \end{tabular} Table 1: SDC properties for the first 30 out of 11303 in the catalogue. The full catalogue is available online. The columns give a running number (1), the name of the source based in its Galactic coordinates (2), the right ascension and declination (in J2000) of the cloud peak (3,4), the minimum 8$\mu$m emission towards the cloud ($I_{\rm{min}}$) (5), the background 8$\mu$m emission ($I_{\rm{MIR}}$) (6), the maximum $I_{\rm{MIR}}$ variation within the IRDC ($\delta I_{\rm{MIR}}$, Sec. 3.3) (7), the size of the cloud along its major and minor axes in arcseconds (8,9), the position angle of the major axis of the cloud in degrees East of North (10), the equivalent radius ($R_{\rm{eq}}$; Sec. 5.1) of the cloud (11), the peak and average optical depth of the cloud at 8$\mu$m (12,13), the optical depth at 8$\mu$m at which the absorption would be saturated (14), the number of fragments in the cloud identified with $\tau_{\rm{step}}=0.35$ (15; Sec. 5.2), whether there are 24$\mu$m stars in the field (16) and in the cloud (17; Sec. 5.3); and the density of stars around the cloud (18). ### Constraining $I_{\rm{fore}}$ Comparison of the infrared extinction and millimeter emission can be used to constrain the infrared foreground emission towards an IRDC by requiring that both techniques give the same column density towards the source. For this purpose we have used the 38 IRDC 1.2mm dust continuum images Rathborne et al.(2006) obtained with the IRAM 30m telescope at 11${}^{\prime\prime}$angular resolution. The 1.2mm emission can be translated into an 8$\mu$m opacity, $\tau_{\rm{em}}$, using the equation \[\tau_{\rm{em}}=\frac{S_{\rm{peak}}\times R_{\kappa}}{B_{1.2}(T_{d})\times \Omega_{\rm{30m}}}\] (3) where S${}_{\rm{peak}}$ is the 1.2mm dust continuum emission peak of the source, $R_{\kappa}$ is the specific dust opacity ratio between 8$\mu$m and 1.2mm, $B_{1.2}(T_{d})$ is the Planck function at 1.2mm for the dust temperature $T_{d}$, and $\Omega_{30m}$ is the solid angle at 1.2mm of the IRAM 30m telescope beam. The value for $R_{\kappa}$ is not well constrained: different models of dusts provide different values of $R_{\kappa}$. Given the chemical composition of the emitting/absorbing dust the value of $R_{\kappa}$ can be as large as 2000 for interstellar dust in diffuse clouds (e.g. Draine, 2003), decreasing to 750 for dense clouds (e.g. Ossenkopf & Henning, 1994; Johnstone et al., 2003). Given the dense and cold nature of IRDCs, we adopted the value $R_{\kappa}=750$, and a dust temperature of 15 K, which gives \[\tau_{\rm{em}}=0.02\times S_{\rm{peak}}\] (4) with $S_{\rm{peak}}$ in mJy/beam. After smoothing the Spitzer 8$\mu$m images of the 38 IRDCs observed by Rathborne et al.(2006) to the same resolution as the the IRAM 30m 1.2mm images, we have constructed their 8$\mu$m opacity maps assuming $I_{\rm{fore}}=I_{\rm{zl}}$ (i.e. the lower limit on the foreground emission). A direct comparison between these opacity maps and the ones calculated from the 1.2mm dust continuum images becomes then possible. However the observations of the 8$\mu$m absorption and 1.2mm emission are not equally sensitive to all of the dust along the line of sight. Regions of low column density are more easily detected in absorption than in emission. For this reason, we selected only clear corresponding peaks in both type of images, ending up with 57 “cores” (emission peaks and absorption minima) which have been used for the comparison. Amongst these cores 11 show 24$\mu$m point-like emission. Figure 4 shows the resulting comparison for these 57 cores, the “starless” ones (those without associated 24$\mu$m emission) are marked with open triangles while the “protostellar” ones are marked with red stars. Also shown are the three lines: $\tau_{\rm{abs}}=\tau_{\rm{em}}$ (solid line), $\tau_{\rm{abs}}=2\times\tau_{\rm{em}}$, and $\tau_{\rm{abs}}=0.5\times\tau_{\rm{em}}$ (dashed lines). In the figure there is a clear separation between the starless sources and those objects associated with a 24$\mu$m point-like source. For the sources associated with 24$\mu$m point-like emission, the values of $\tau_{\rm{em}}$ are on average higher than for the starless sources. The $\tau_{\rm{em}}/\tau_{\rm{abs}}$ ratio is on average $\sim 2.9$ for the starless sources with a dispersion of 1.1, while it is $\sim 7.2$ for the sources with stars with a dispersion of 3.8. This reflects that the latter group of sources have stronger 1.2mm emission (a factor of $\sim 2.5$ ), which translates to higher opacities for the same assumed dust temperature. This clearly shows these sources are in fact either warmer with average dust temperature greater than 15 K, or else have different dust properties. On the other hand for the starless objects, the average ratio $<\tau_{\rm{em}}/\tau_{\rm{abs}}>=2.9$ is closer, but still rather far from, unity. This suggests that the value of $I_{\rm{fore}}$ is underestimated and the assumption $I_{\rm{fore}}=I_{\rm{zl}}$ is incorrect. <figure><img src="content_image/0906.3493/x4.png"><figcaption>Figure 4: Plot of the 8μm opacity estimated from the 8μm Spitzer maps (τabs)and from the 1.2mm dust continuum emission (τem). The starless sources aremarked with open triangles while those associated with 24μm point-likeemission are marked with red open star symbols. τabs has been calculatedassuming Ifore=Izl. The solid line marks the relationship τabs=τem, while thetwo dashed lines indicates τabs=0.5×τem and τabs=2×τem</figcaption></figure> Assuming that for starless cores the true 8$\mu$m opacity is given by $\tau_{\rm{em}}$, we can invert Eq. (2) to estimate the value of I${}_{\rm{fore}}$ in terms of $I_{\rm{MIR}}$. We did such a calculation for every starless core and plotted the results in Fig. 5, I${}_{\rm{MIR}}$ being measured at the position of the core on the large scale emission map (Sec. 3.3). A strong correlation is seen between I${}_{\rm{fore}}$ vs I${}_{\rm{MIR}}$. The best linear fit to this correlation is given by \[I_{\rm{fore}}=0.54\times I_{\rm{MIR}}\] (5) with a standard deviation of 0.08, minimum and maximum values of 0.4 and 0.75, respectively. This relationship allows us to compute an average foreground emission just by estimating the mid-infrared radiation for any IRDCs. Figure 6 shows $\tau_{\rm{em}}$ versus $\tau_{\rm{abs}}$ calculated using Eq. (5), but only for the starless cores this time. Here $<\tau_{\rm{em}}/\tau_{\rm{abs}}>=1.1$ with a dispersion of only 0.5. The relation in Eq. (5) gives us the maximum opacity (and equivalent column density) we can probe before reaching saturation. Indeed, the rms noise level of the 8$\mu$m images ($\sigma_{\rm{noise}}\sim 0.3$ MJy/sr) defines the minimum flux we can detect above the foreground emission. Below this value, the dust in the cloud is basically absorbing all the background emission and we cannot recover the true peak column density. This saturation opacity, $\tau_{\rm{sat}}$, is given by $\tau_{\rm{sat}}=-\ln(\sigma_{\rm{noise}}/I_{\rm{bg}})$, with $I_{\rm{bg}}=0.46\times I_{\rm{MIR}}$. The saturation opacity is calculated for every IRDC and given in Table 1. We also note that we have $I_{\rm{fore}}\simeq I_{\rm{bg}}$ as also observed by Johnstone et al.(2003) and this suggests that most of the foreground emission originates from the same place as the background emission and is local to the IRDC, and therefore the foreground emission is independent of distance to the IRDC. <figure><img src="content_image/0906.3493/x5.png"><figcaption>Figure 5: Plot of the 8μm foreground intensity calculated for 57 positions(see text) of the Rathborne et al. (2006) sample in function of the 8μm mid-infrared radiation field estimated around them. The best linear fit is shownas a red solid line.</figcaption></figure> <figure><img src="content_image/0906.3493/x6.png"><figcaption>Figure 6: Same as Fig. 4 but only for starless sources and with a 8μm opacitycalculated with Ifore=0.54×Imin. The solid line marks the relationshipτabs=τem, while the two dashed lines indicate τabs=0.5×τem and τabs=2×τem.</figcaption></figure> ### Construction of the opacity maps To construct opacity maps of IRDCs all over the Galactic plane we mosaiced the GLIMPSE 8$\mu$m and MIPSGAL 24$\mu$m images in blocks of $1^{\circ}$ in longitude by $2^{\circ}$ in latitude using the Montage software (http://montage.ipac.caltech.edu/). To allow the identification of IRDCs which cross the edges of these blocks and to allow the extraction of regions large enough for our analysis around clouds near the edges of these blocks, each consecutive block overlaps adjacent blocks by 0.5${}^{\circ}$. In principle this means our extraction could miss IRDCs larger than about $0.5^{\circ}$ in size. However the largest cloud identified by Simon et al.(27) is 27${}^{\prime}$ long. The sensitivity of the Spitzer images is such that significant numbers of stars and galaxies appear in them, even at 8$\mu$m. These need to be removed in order to produce clean mid-infrared images and opacity maps of the clouds. This has been done in two steps. First identifying the central position of stars in the field using the IDL FIND task from the Astronomy library. Second, the values in the pixels containing the star were replaced with values calculated from an average gradient plane fit to the values of the pixels surrounding the star we want to remove. While this allowed the recovery of some part of the structure of a cloud, it can also produce artifacts. Once the 8$\mu$m stars were removed, we calculated the mid-infrared radiation field $I_{\rm{MIR}}$ by smoothing each 8$\mu$m block by a normalised Gaussian of FWHM=308${}^{\prime\prime}$¹. This size is a compromise between several parameters: the typical size of an IRDC, the typical spatial scale of the 8$\mu$m emission of the Galactic plane and the computation time. Visual inspection of Spitzer images suggests that most of the clouds are filamentary with a minor axis which is not larger than a few arcminutes. The smoothing we have used is well matched to such clouds and our method will recover their exact structure. For clouds which are larger than the smoothing length, but which are centrally condensed, we will detect them but somewhat underestimate their opacity. On the other hand shallow large clouds will be missed (Section 5 and 6). Using a larger smoothing length would allow us to better detect these large clouds, but at the cost of additional processing time and more significantly, the introduction of spurious artificial clouds, especially where the background emission is weak. In any case, distinguishing between a feature due to a smooth lack of background emission or the presence of a large and low column density cloud requires observations of tracers in addition to the inferred mid-infrared extinction. We preferred to convolve the images with a Gaussian rather than using a median filter in order to better recover potential clouds adjacent to strong 8$\mu$m emitting structures. [FOOTNOTE:1][ENDFOOTNOTE] Having calculated $I_{\rm{MIR}}$ we are able to compute both I${}_{\rm{fore}}$ and I${}_{\rm{bg}}$ images (Section 3.2). Then using Eq. (2) we can construct the 8$\mu$m opacity image, but before doing so, we smoothed the 8$\mu$m images with a 4${}^{\prime\prime}$ Gaussian in order to suppress high frequency noise. A series of artifacts, and spurious clouds may arise from our method. The first one comes from potentially interpreting every decrease in the 8$\mu$m emission on spatial scale smaller than $\sim 5\hbox{${}^{\prime}$}$ as being a potential cloud. This effect is especially important at high latitudes where the mid-infrared radiation field is low. In these regions a small decrease in the intensity will be interpreted as a stronger increase in the opacity than for a similar intensity drop in a high mid-infrared radiation field environment. Identifying such spurious clouds is difficult, and only follow-ups in other tracers in emission will give a definitive answer on the nature of these sources. However, we have attempted to minimise such objects by selecting a relatively high opacity detection threshold. Another artifact can arise in regions with strong intensity gradients in the initial 8$\mu$m block where the smoothing may artifially produce features identified as clouds, although real clouds also exist in these environments (Deharveng et al., 2009). To help identify possible spurious objects in regions of large 8$\mu$m intensity variations, our catalogue (Table 1)² lists $\delta\rm{I_{MIR}}$, the normalised maximum variation of $I_{\rm{MIR}}$ within the IRDC and defined as $\delta I_{\rm{MIR}}=(I_{\rm{MIR}}^{\rm{max}}-I_{\rm{MIR}}^{\rm{min}})/I_{\rm{ MIR}}^{\rm{min}}$. Our experience suggests that clouds with $\delta I_{\rm{MIR}}>0.5$ have to be treated with caution. These clouds represent 14% of the total number of IRDCs included in our sample. Overall, after a visual inspection of every IRDC and the removal of obviously spurious IRDCs, we believe that more than 90% of the catalogued objects are true IRDCs. [FOOTNOTE:2][ENDFOOTNOTE] The tools to automatically construct the maps were mainly constructed using IDL packages. <figure><img src="content_image/0906.3493/x7.png"><figcaption>Figure 7: 8μm opacity maps for the 3 IRDCs showed in Fig. 1. The contours gofrom 0.4 to 0.8 in steps of 0.2 for the figures on the right and left, whilefor the middle figure the contours go from 0.4 to to 1.9 in steps of 0.3.</figcaption></figure> ## 4From 8$\mu$m opacities to column densities The images resulting from the analysis described above provide the spatial 8$\mu$m opacity distribution towards IRDCs. However a more useful quantity is the H${}_{2}$ column density distribution of these clouds. To convert 8$\mu$m opacities to H${}_{2}$ column densities requires a knowledge of the properties of the absorbing dust. Depending on the line of sight and on the structures observed e.g. diffuse material or dense material, the dust chemical composition and thus, the dust properties, are different. In dense clouds like IRDCs, it is believed that dust grains are larger than in the diffuse interstellar medium due to coagulation and presence of icy mantles on the grains. This is supported by ISO (Lutz et al., 1996), and more recently Spitzer (Indebetouw et al., 2005; Román-Zúñiga et al., 2007), observations which have shown that towards dense clouds, the extinction cannot be fitted by a single power-law from the near-IR up to the mid-IR (Draine & Lee, 1984). The recent work has shown that in dense clouds the extinction decreases from the near infrared to $\sim 5\mu$m and then reaches a plateau up to the silicate absorption band around 9$\mu$m. This behavior can be reproduced with dust models having $R_{v}\simeq 5$****(Weingartner & Draine, 2001), implying larger dust grains (compared to the commonly used value $R_{v}\simeq 3$ for diffuse interstellar medium). For the IRDCs we therefore adopt a value of $A_{8\mu\rm{m}}/A_{v}=0.045$****(Indebetouw et al., 2005; Román-Zúñiga et al., 2007). To convert to the molecular hydrogen column density, $N_{\rm{H_{2}}}$ we adopt \[A_{v}=10^{-21}\times N_{\rm{H_{2}}}\] (6) from Bohlin et al.(1978), although the more recent work by Draine(2003), based on the observations of Rachford et al.(2002), suggests a 50% larger column density per magnitude of extinction. To account for this, and other uncertainties, the column densities in this (and subsequent papers), have been calculated from the 8$\mu$m optical depth adopting the relation \[N_{\rm{H_{2}}}=\tau_{8\mu\rm{m}}\times 3[\pm 1]\times 10^{22}\rm{cm}^{-2}\] (7) ## 5Identification of sources Once the opacity maps have been constructed, we need to extract the information on the structures lying within them. For this purpose, we have developed a new code, largely inspired by the CLUMPFIND source extraction code of Williams et al.(1994). The operation of the code is described in Appendix A. The main differences compared to CLUMPFIND are how a source is defined and its properties determined. This new method does not assume that every pixel belongs to a source, but we define the boundaries of an object by the local minimum between closest neighbours. Then to estimate the size of the source we calculate the first and second order moments of the absorption distribution, and then we diagonalise the second order moment matrix (Appendix A). ### IRDCs In our maps, the IRDCs have been defined as connected structures lying above an opacity, $\tau_{8\mu\rm{m}}$, of 0.35 with a peak above 0.7 and a diameter greater than 4${}^{\prime\prime}$. Therefore, using Eq. (7), these detection thresholds correspond to $1\times 10^{22}$ cm${}^{-2}$ and $2\times 10^{22}$ cm${}^{-2}$, respectively. With these parameters, we have identified 11303 IRDCs (see Fig. 7). Table 1 lists the first 30 IRDCs, giving their name, coordinates, I${}_{\rm{min}}$ in MJy/sr, I${}_{\rm{MIR}}$ in MJy/sr, $\delta{\rm I_{MIR}}$ (see Sec. 3.3), $\Delta X$ the major axis size in arcseconds, $\Delta Y$ the minor axis size in arcseconds, $\alpha$ the position angle in degrees ( see Appendix A for an exact definition of these parameters), R${}_{\rm{eq}}$ the equivalent radius which corresponds to the radius of a disc having the same area as the IRDC in arcseconds, $\tau_{\rm{peak}}$ the 8$\mu$m peak opacity, $\tau_{\rm{av}}$ the $8\mu$m opacity averaged over the cloud, $\tau_{\rm{sat}}$ the saturation opacity as described in Section 3.2, the number of fragments within the IRDC (Sec. 5.2), whether there is a 24$\mu$m star in the field/IRDC or not (Sec. 5.3), and $\sigma_{\rm{star}}$ the 24$\mu$m stellar density around the IRDC in number of stars per arcminute squared. ### IRDC fragments Substructures are seen in almost every IRDC map (Fig. 7). Since column density peaks likely pinpoint the sites of the formation of the next generation of stars, identifying these peaks is crucial in identifying the initial conditions of star formation in IRDCs. We call these substructures identified within the IRDCs _fragments_. We prefer this name, rather than for example, cores, as they have been called in other papers (e.g. Rathborne et al., 2006). The term core has often been used to identify a substructure which forms one star or a small group of stars and we do not at this stage wish to imply any physical interpretation of these structures in IRDCs. Especially since we do not know the distance of the bulk of the IRDCs, we cannot infer any physical parameters such as the sizes and masses of the fragments/IRDCs. To extract the IRDC fragments, we apply the same extraction code used to identify the IRDCs (Appendix A). We applied different values of $\tau_{\rm{step}}$ in order to get a comprehensive picture of the fragmentation in these IRDCs. In total we identified 20000 to 50000 fragments depending on $\tau_{\rm{step}}$ (from 0.1 to 0.35). For each of these fragments we have measured their positions, sizes, peak and average opacity, and their 24$\mu$m star association. As an indication of the degree of fragmentation Table 1 includes the number of fragments extracted in each IRDC with $\tau_{\rm{step}}=0.35$. The nature of these fragments is discussed in detail in Peretto & Fuller (2009, in preparation). Structures \| Number of \| Req \| Aspect ratio \| τav \| τpeak \| Star association ---\|---\|---\|---\|---\|---\|--- \| Objects \| Average \| Range \| Average \| Range \| Average \| Range \| Average \| Range \| \| \| (arcsec) \| (arcsec) \| \| \| \| \| \| \| % IRDCs \| 11303 \| 31 \| 4–374 \| 2.2 \| 1.0–11.6 \| 0.15 \| 0.01–2.35 \| 1.15 \| 0.70 – 8.36 \| 20-68 Fragments \| 19838 \| 19 \| 1–205 \| 2.0 \| 1.0–11.6 \| 0.75 \| 0.01–7.88 \| 1.63 \| 0.70 – 8.36 \| 6 Table 2: Average properties of IRDCs and fragments (extracted with τstep=0.35). ### 24$\mu$m point-like sources association In order to check for star formation activity associated with the IRDCs and fragments, we analysed the 24$\mu$m MIPSGAL data, looking for point-like sources. For this purpose we used the IDL FIND task of the IDL Astronomy Library. As an initial indication of the the star formation activity of these IRDCs, we have identified all the 24$\mu$m stars lying within a box (described as _Field_ in Table 1 col. 16) of twice the calculated extent along the coordinate axes of each IRDC. Doing so, we find that 32$\%$ of the IRDCs do not have any 24$\mu$m point-like sources in such a box³. On the other hand, 20$\%$ of the IRDCs have a 24$\mu$m source lying within their boundaries (Table 1 col. 17). Therefore, the percentage of active star forming IRDCs is likely to be between 20 and 68$\%$. A more detailed analysis of the stellar content of IRDCs will be presented in a following paper. [FOOTNOTE:3][ENDFOOTNOTE] Concerning the fragments, between 1% and 6% have stars lying within their boundaries, depending on the parameters used to extract the fragments (Peretto & Fuller 2009, in preparation). We have also calculated the 24$\mu$m stellar surface density around each IRDC extracted (Table 2 col. 18). This number provides an idea of the crowding in the area around the IRDC. ### Uncertainties on the opacity estimates The main source of uncertainty in the opacity maps arises from the estimate of the foreground intensity $I_{\rm{fore}}$. As explained in Section 3, we used the relation $I_{\rm{fore}}=0.54\times I_{\rm{MIR}}$ to calculate this quantity for every cloud. However, as can be seen in Fig. 5 a dispersion of $\sim 0.1$ exists on this relation with a maximum variation of $\pm 0.25$. To assess the impact of such variations on the calculated peak opacities of the clouds we have computed for every cloud the ratio, $K$, of the peak opacity inferred assuming $I_{\rm{fore}}=C_{f}\times I_{\rm{MIR}}$ where $0.25<C_{f}<0.75$ to the peak opacity calculated with the fiducial $I_{\rm{fore}}$ (Eq. 5; $C_{f}=0.54$). Figure 8 shows the median value of this ratio as a function of $C_{f}$. For each value of $C_{f}$ we also calculated the dispersion in $K$ across the entire sample of clouds. These dispersions were all $<0.1$, except for the case $C_{f}=0.75$ where the dispersion in $K$ reached 0.3. The range in $K$ shown on Fig. 8 provides an estimate of the peak opacity uncertainty related to the choice/variation of $I_{\rm{fore}}$. In most cases this uncertainty is less than a factor of 2, but can be as large as 10 for extreme cases. On the same figure we also plot the fraction of saturated clouds in function of the adopted $I_{\rm{fore}}$. Naturally, the higher $I_{\rm{fore}}$, the higher the number of saturated clouds, reaching 80$\%$ in the most extreme case, but being less than 10$\%$ for $I_{\rm{fore}}<0.6\rm{I}_{\rm{MIR}}$. In the case of $C_{f}=0.54$, the percentage of saturated cloud is $3\%$. This is consistent with a visual inspection of the 8$\mu$m intensity profiles of a sample of clouds which indicates that less than 10% of the objects show a flattening in their inner regions, a signature of possible saturation. Another source of uncertainty is the variation of the foreground intensity relative to the background emission. Since we have shown that on average the background emission is equal to the foreground emission (Sec. 3.2), we assumed that the variations of both quantities in front and behind a cloud have the same origin, and so, the same variations. However, this assumption could be wrong. For instance one could be constant over the extent of the cloud, more likely the foreground, with the other one containing all the variations observed in the mid-infrared radiation field. The impact of such effects on the opacity estimate is similar to the one described above. Clouds with small variations in their mid-infrared radiation fields are thus better constrained than the ones with high $\delta I_{\rm{MIR}}$. As mentioned in the previous section large clouds ($>5\hbox{${}^{\prime}$}$) have opacities which are likely to be underestimated, however this effect is minor compared with those mentioned above. Overall, considering all the factors which contribute to the uncertainty in opacity, we estimate the values derived from the Spitzer data are uncertain by a factor of no more than two. This result is consistent with the observations of a subset of clouds in the 1.2mm continuum emission from the dust (Fig. 6). <figure><img src="content_image/0906.3493/x8.png"><figcaption>Figure 8: (top): Correction factor to apply to peak opacities in order tocorrect for different foreground intensities than the one we used in thisstudy. (bottom): Fraction of saturated clouds as a function of the assumptionmade on the foreground intensity.</figcaption></figure> ## 6Comparison with the MSX IRDC catalogue Simon et al.(27) undertook a systematic survey of IRDCs using MSX data. Their survey covers a larger area of the Galactic plane than ours due to the smaller coverage of GLIMPSE survey. In total, Simon et al.(27) have extracted 6721 clouds between $10\hbox{${}^{\circ}$}<\|l\|<65\hbox{${}^{\circ}$}$ and $-1\hbox{${}^{\circ}$}<b<1\hbox{${}^{\circ}$}$. For the same coverage we extracted 11303 Spitzer dark clouds, which is roughly twice as many. However, the detection limits, peak and boundary, in the two surveys are different, the simple comparison of the numbers of clouds provides only an incomplete comparison and so a more complete comparison has been performed. As illustrated by Fig. 9, it appears that a minority of IRDCs are common to both MSX and Spitzer catalogues. Actually, only 20$\%$ of the Spitzer dark clouds appear in the MSX catalogue (corresponding to 25$\%$ of MSX clouds being associated with a Spitzer dark cloud). Based on this comparison we define 3 categories of clouds: _Spitzer only_, which are clouds appearing only in our catalogue; _MSX only_, which are clouds appearing only in Simon et al. catalogue; and _both_, which are clouds appearing in both catalogues. Figure 10 shows an example of an IRDC in each of these categories. Of the _Spitzer only_ clouds, 51% do not meet the size criteria, R${}_{eq}>20\hbox{${}^{\prime\prime}$}$, imposed by Simon et al.(27) to identify the MSX IRDCs, explaining why they are not in the MSX catalogue. The remaining $\sim 30\%$ of _Spitzer only_ IRDCs are the result from the difference in the method used to estimate the background. Using a median filter of 30${}^{\prime}$ diameter, Simon et al.(27) underestimated the background almost everywhere in the inner $0<\|b\|<0.25\hbox{${}^{\circ}$}$ of the Galactic plane. As a consequence, the inferred background reaches a similar value as the IRDC itself, and therefore, an IRDC is not detected. This artifact can be seen when ploting the source fraction as a function of the Galactic latitude (Fig. 11). We see a significant difference between the distributions of MSX and Spitzer IRDCs. The MSX IRDCs have a rather flat distribution in a central 1${}^{\circ}$ region whereas the Spitzer IRDC distribution has a clear central peak decreasing sharply on both sides of it. We believe than this difference arises from the difference in the background construction. On the other hand the _MSX only_ clouds have very low contrast (opacity peaks) and are particularly large. The detection of such clouds in the MSX data has been possible due to the large background smoothing length, and the low contrast threshold used by Simon et al.(27). In order to investigate this effect and see whether our method could recover these clouds when using a larger Gaussian, we smoothed the block shown in Fig 9 to 20${}^{\prime}$, and performed the extraction of IRDCs on the resulting opacity map. Doing so, we find twice as many clouds (40%) which are in both catalogues, but in parallel 35% of Spitzer clouds which were initially detected using a smaller Gaussian are lost. The remaining _MSX only_ clouds are just too shallow to be identified given the opacity threshold we used, 0.7. In addition, looking at their 8$\mu$m emission it is not clear whether many of these clouds are real, or just a decrease in the background of the Galactic plane. Overall, we can say that 80% of our catalogue comprises IRDCs which were previously unknown and constitutes the most complete catalogue available of such objects with column density peaks above $1\times 10^{22}$ cm${}^{-2}$. <figure><img src="content_image/0906.3493/x9.png"><figcaption>Figure 9: In grey scale is the Spitzer 8μm emission of one of the blocks weconstructed around l≃30∘. The black circles indicate the position and size ofthe Spitzer IRDCs identified in this study, while the red square symbols codethe position and size of the MSX IRDCs. We see on this image that the SpitzerIRDCs are more numerous where the background is stronger, while, quitesurprisingly, this is not the case for the MSX IRDCs. The MSX clouds detectedat \|b\|>0.5∘, are on average the larger clouds in the Simon et al. (2006a)sample. For most of them, we do not detect any Spitzer IRDCs at thesepositions in our standard processing (using a 5′ Gaussian) but some aredetected when using a larger smoothing function (see text).</figcaption></figure> <figure><img src="content_image/0906.3493/x10.png"><figcaption>Figure 10: Comparison of three IRDCs seen with Spitzer at 8μm illustrating the3 categories of IRDC based on their MSX and Spitzer detection. Note that thecloud detected only in the MSX catalogue (left panel) exhibits much lowerextinction than the other two objects.</figcaption></figure> <figure><img src="content_image/0906.3493/x11.png"><figcaption>Figure 11: Comparison of the latitude distribution of Spitzer and MSX darkclouds</figcaption></figure> ## 7Summary This paper, the first of a series dedicated to the study of infrared dark clouds, describes the techniques developed to establish a complete catalogue of Spitzer dark clouds. We analysed the full data set of the 8$\mu$m GLIMPSE Galactic plane to look for IRDCs. We extracted 11303 of these clouds, obtaining column density maps for each of them, and characterizing their physical properties. We also identify the substructures lying within these clouds, extracting up to $\sim 50000$ of these. Table 2 presents a summary of the average and range of properties of both the clouds and these substructures (fragments). The full table of the properties of the clouds and fragments plus images and opacity maps are available from an online database⁴. In subsequent papers we will exploit the tremendous quantity of information concerning the initial conditions for the formation of stars in the Galaxy contained within this set of IRDC column density maps. [FOOTNOTE:4][ENDFOOTNOTE] ###### Acknowledgements. This work was supported in part by the PPARC and STFC grants. We thank Hannah Stacey for her work in the early stages of identifying some of the IRDCs. We also thank Jim Jackson, Robert Simon, and Jill Rathborne for providing us with the IRAM 30m dust continuum images published in Rathborne et al.(2006)****This research made use of Montage, funded by the National Aeronautics and Space Administration’s Earth Science Technology Office, Computation Technologies Project, under Cooperative Agreement Number NCC5-626 between NASA and the California Institute of Technology. Montage is maintained by the NASA/IPAC Infrared Science Archive. ## Appendix AMethod for extracting sources <figure><img src="content_image/0906.3493/x12.png"><figcaption>Figure 12: Illustration of our extraction method. This figure shows theopacity profile of a typical IRDC. The bottom dashed line shows the opacitythreshold beneath which structures are ignored. The dotted lines show thedifferent slices through the cloud, every slice being separated by τstep. Theupper dashed line shows the opacity corresponding to the local minimum , τlev,between the two local peaks shown on that plot. In such a cloud, our methodwould extract one IRDC (colored area) and two fragments (colored area +dashed-dotted lines) within it.</figcaption></figure> <figure><img src="content_image/0906.3493/x13.png"><figcaption>Figure 13: 8μm opacity map of the middle IRDC shown in Fig. 1. Our extractionmethod detected 7 fragments within this IRDCs when τstep=0.1. The blackcontours mark the τlev value (boundary contour) for each fragment. The sizesand position angle are also given in between brackets. We can see that thesevalues give a reasonable description of the shape of the fragments (and IRDC)</figcaption></figure> We developed a new code to extract sources from our opacity maps. The first part of our algorithm is mainly based on the same principle as the one developed by Williams et al.(1994) for CLUMPFIND. We set two main parameters which are the lowest contour level under which we do not consider any structure, $\tau_{\rm{thres}}$, and a step in unit of the map, $\tau_{\rm{step}}$. Then we look at every local peak between two consecutive levels, up to the maximum of our image. The number of local peaks gives us the number of fragments we will extract from the image, unless the final estimated size is lower than the final angular resolution or that the amplitude between the peak of the fragment and its external boundary is less than $\tau_{\rm{step}}$. Then we have to determine the pixels we associate to each local peak. For this, for every peak, we go down, level by level, and check if the local peak we are looking at is the only one in this contour. If yes, we look at the following contour and do the same job. If there is more than one local peak within the contour we look for the local minimum between these two peaks, $\tau_{\rm{lev}}$, and the pixels lying above $\tau_{\rm{lev}}$ and associated with the considered peak define the extent of the fragment. In order to measure the size of the clouds and fragments, we did not want to assume any particular shape for the source. So, once we have identify all the pixels associated with a given peak, we estimate first the center of gravity of the cores, $(X_{CG},Y_{CG})$, using \[X_{CG}=\frac{\sum_{i=1}^{N}V_{i}\times x_{i}}{\sum_{ i=1}^{N}V_{i}}\\ Y_{CG}=\frac{\sum_{i=1}^{N}V_{i}\times y_{i}}{\sum_{ i=1}^{N}V_{i}}\] (8) where $V_{i}$ is the value of the $i$th pixel, $x_{i}$ and $y_{i}$ its coordinates, and N is the number of pixels. Then, we calculate the matrix of moment of inertia, I: \[I=\left[\begin{array}[]{cc}I_{xx}&I_{xy}\\ I_{yx}&I_{yy}\\ \end{array}\right]\] (9) with \[I_{xx}=\sum_{i=1}^{N}V_{i}(y_{i}-Y_{CG})^{2}\\\] (10) \[I_{yy}=-\sum_{i=1}^{N}V_{i}(x_{i}-X_{CG})^{2}\\\] (11) \[I_{xy}=I_{yx}=\sum_{i=1}^{N}V_{i}(x_{i}-X_{CG})(y_{i}-Y_{CG})\] (12) Finally, we diagonalize I in order to obtain its two eigenvalues and eigenvectors. From this we can easily calculate the position angle $\alpha$ of the major axis (given by the vector associated by the smallest eigenvalue). 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1307.2606	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 36869, "num_imgs": 8, "llama3_tokens_count": 10940 }	[ "content_image/1307.2606/x1.png", "content_image/1307.2606/x2.png", "content_image/1307.2606/x3.png", "content_image/1307.2606/x4.png", "content_image/1307.2606/x5.png", "content_image/1307.2606/x6.png", "content_image/1307.2606/x7.png", "content_image/1307.2606/x9.png" ]	# Top Quark as a Dark Portal and Neutrino Mass Generation John N. Ng Alejandro de la Puente Theory Group, TRIUMF, Vancouver BC V6T 2A3, Canada ###### Abstract We present a new model for radiatively generating Majorana active neutrino masses while incorporating a viable dark matter candidate. This is possible by extending the Standard Model with a single Majorana neutrino endowed with a dark parity, a colour electroweak singlet scalar, as well as a colour electroweak triplet scalar. Within this framework, the $up$-type quarks play a special role, serving as a portal for dark matter, and a messenger for neutrino mass generation. We consider three benchmark scenarios where the abundance of dark matter can match the latest experimental results, while generating neutrino masses in the milli-electronvolt range. We show how constraints from lepton flavour violation, in particular the branching fraction of $\mu\to e\gamma$, can place lower bounds on the coupling between our dark matter candidate and top quarks. Furthermore, we show that this coupling can also be constrained using collider data from the Tevatron and the LHC. † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction We now have compelling evidence for the existence of three active neutrino species [1]. Radiochemical experiments such as Homestake, Gallex/GNO and SAGE [2; 3; 4] together with the SuperK and SNO experiments [6; 7] have narrowed down the mass patterns to three possibilities: A normal or an inverted hierarchy or almost degenerate masses. Moreover, the absolute scale of neutrino masses remains unknown. Additionally, the last mixing angle, $\theta_{13}$, has been measured by several reactor experiments [8; 9; 10] and the T2K accelerator experiment [11]. This is a breakthrough for the standard picture of neutrino oscillations since it now paves the way towards measuring CP violation in the lepton sector. Furthermore, the evidence for neutrino masses represents one clear motivation for new physics beyond the Standard Model (SM). Within the SM, neutrinos are massless; they can be accommodated in a variety of ways such as incorporating new degrees of freedom and/or new effective interactions. Extending the SM model in this way allows us to be sensitive to new high energy scales. Take for example the Type I seesaw mechanism [12], where the SM is extended with a singlet Majorana fermion that couples to left-handed leptons through the Higgs, as with the charged leptons. This class of models generates viable neutrino masses with a Majorana mass scale $\gtrsim 10^{14}$ GeV and Yukawa interaction of order one. Such a high seesaw scale can arise from Grand Unified models such as SO(10) [13]. However, such a high scale for new physics makes the mechanism impossible to test. A TeV scale Majorana mass is also possible in models such as left-right symmetric models, (for a recent discussion see [14]). Models with flavour symmetries are also used to explain the neutrino masses (see [15; 16] for recent reviews). Models where neutrino masses are radiatively generated have also been studied. In particular, the simplest model where neutrino masses are induced as one-loop radiative corrections was first introduced in [17]. In this class of models a charged scalar singlet under the SM gauge group couples to left-handed lepton doublets and one is able to generate active neutrino masses of the right order with a charged scalar mass scale as low as a TeV. Neutrino masses may also arise as two loop radiative corrections in extensions of the SM with an additional singlet charged scalar and a doubly charged scalar [18; 19; 20]. The main motivation for this class of models is that they employ new physics at the TeV scale and hence can be probed at the LHC. The nature of the neutrino mass matrix can be accessed through data on neutrino oscillations. In the gauge basis the mass matrix can be parametrized in the following way: \[m_{\alpha\beta}=\sum_{i}m_{i}U_{\alpha i}U^{}_{\beta i},\] (1) where $\alpha,\beta=e,\mu,\tau$, $i=1,2,3$ and $U_{\alpha,i}$ are the neutrino mixing matrix elements. In general, if neutrinos are Majorana fermions then two new independent degrees of freedom, the Majorana phases, exist and are usually assigned to the $m_{i}$’s. The experimental status on the neutrino oscillation parameters is summarized in [21] \[\Delta m^{2}_{21} = 7.59^{+0.20}_{-0.18}\times 10^{-5}~{}\text{eV}^{2}\] \[\sin^{2}\theta_{12} = 0.312^{+0.017}_{-0.015}\] \[\|\Delta m^{2}_{31}\| = \left\{\begin{array}[]{rl}2.45\pm 0.09\times 10^{-3}~{}\text{eV}^ {2}~{}\text{Normal~{}Hierarchy}\\ 2.34^{+0.10}_{-0.09}\times 10^{-3}~{}\text{eV}^{2}~{}~{}~{}\text{Inverted~{} Hierarchy}\\ \end{array}\right.\] \[\sin^{2}\theta_{23} = 0.51\pm 0.06.\] (2) Another strong indicator of physics beyond the SM is the ample evidence pointing towards the existence of dark matter [22; 23]. Velocity dispersion and rotation curves of galaxies suggest the existence of non-luminous matter that is not composed by any of the known SM particles. Furthermore, the most recent data from Plank estimates a cold dark matter cosmological parameter $\Omega_{DM}h^{2}=0.1199\pm 0.0027$ [1] or roughly $26.8\%$ of the universe’s total energy. Unfortunately, all experimental evidence for dark matter is due to its gravitational properties and its identity remains unknown to date. One candidate explanation for dark matter is the existence of a weakly interacting massive particle (WIMP). Supersymmetric models are known to provide a natural WIMP candidate, usually the lightest superpartner. The abundance of these particles in the universe is determined by their self-annihilation rate in relation to the expansion of the universe. When the expansion rate dominates over the rate of annihilation, interaction among dark matter particles becomes less efficient and their density becomes a constant or “freezes out”. There is, however, some possible signal regions for WIMP scattering with nuclei in direct detection experiments, most notably the DAMA/LIBRA result [24] and CRESST [25]. Experimental upper limits on the WIMP-nucleon cross section have also been found by various experiments [26; 27; 28]. In this work we present a new model for radiatively generating Majorana active neutrino masses while incorporating a viable dark matter candidate. This is possible by extending the SM with a single electroweak singlet Majorana neutrino, $N_{R}$, to which we also assign an odd parity, referred to as dark parity (DP). We also add a colour electroweak singlet scalar, $\psi$, and a colour electroweak triplet scalar, $\chi$. These are, respectively, odd and even under DP. Furthermore, all SM fields have even DP. Of the two particles that are odd under DP, we assume $N_{R}$ to be the lightest. Such an assignment makes $N_{R}$ a good dark matter candidate since it will be stable, as we engineer DP to be unbroken. It also forbids the usual coupling of $N_{R}$ to the SM lepton doublet and the Higgs doublet and hence no Dirac mass term is generated. The new colour scalars couple to quark fields, in particular the $up$-type quarks. In our framework, the $up$-type quarks play two roles: the first one, serving as a messenger for neutrino mass generation. This is possible given the rich structure of the Lagrangian which is used to radiatively generate masses for the left-handed neutrinos via the exchange of the exotic colour scalars at three loops. The second role is as a portal for dark matter, where the relic abundance of dark matter is reproduced through renormalizable interactions between the Majorana neutrino and $up$-type quarks via $\psi$. We study three benchmark scenarios where the abundance of dark matter can match the latest experimental results, while generating neutrino masses in the milli-electronvolt range. Our model is consistent with constraints from lepton flavour violation and collider data from the Tevatron and the LHC. The idea of using a discrete symmetry such as a $Z_{2}$ parity to forbid a Dirac mass term for the neutrinos and identify $N_{R}$ as a dark matter candidate was first proposed in [29]. Radiative neutrino masses are generated by the use of Higgs triplets or inert doublets. Here we explore a new avenue by making use of colour scalars which allow neutrino masses to be generated at the 3-loop level. Furthermore, the phenomenology at the LHC is richer by virtue that it is very efficient in producing new colour degrees of freedom. ## 2 Model The model we consider in this study is an extension to the SM that incorporates a dark matter candidate and generates Majorana masses for the active left-handed neutrinos, radiatively and at the three loop level. Within this framework, $N_{R}$ couples to right handed $up$-type quarks through a colour electroweak singlet scalar, $\psi$. Furthermore, we incorporate a coupling between the electroweak lepton doublets and the $up$-type quark doublets through a colour electroweak triplet, $\chi$. The new physics can be parametrized in the following way: \[{\cal L}_{BSM} = \sum_{i}y_{\psi}^{i}\overline{u^{i}}P_{L}N^{c}\psi+\sum_{\ell,i} \left\{\lambda_{\ell}^{i}\left[\overline{u^{i}}P_{R}\left(\chi_{1}\nu^{c}_{ \ell}+\chi_{2}\ell^{c}\right)\right.\right.\] (3) \[+ \left.\left.\overline{d^{i}}P_{R}\left(\chi_{3}\ell^{c}-\chi_{2} \nu^{c}_{\ell}\right)\right]\right\}+\text{hc},\] where $l=e,\mu,\tau$ and $i=1,2,3$ is the quark family index. The coupling $y^{i}_{\psi}$ denotes the strength of the interaction between $N_{R}$ and $u^{i}_{R}$ via $\psi$, while $\lambda^{i}_{l}$ the strength between the quark doublets $(u^{i},d^{i})_{L}$ and $(\nu,l)_{L}$ via $\chi$. Throughout this work, we make use of $P_{R/L}=\frac{1\pm\gamma_{5}}{2}$. Furthermore, unless otherwise stated, we work in the charged fermion mass basis. Under the SM gauge group $SU(3)_{c}\times SU(2)_{W}\times U(1)_{Y}$, $\psi$ transforms as a ${\bf(3,1,2/3)}$ and we write the field $\chi$ as \[\chi=\begin{pmatrix}\chi_{2}/\sqrt{2}&\chi_{1}\\ \chi_{3}&-\chi_{2}/\sqrt{2}\end{pmatrix}\] (4) which transforms as a ${\bf(3,3,-1/3)}$. These assignments yield electric charges of $Q=2/3,-1/3,-4/3$ for $\chi_{1},\chi_{2}$ and $\chi_{3}$ respectively. The gauge covariant derivatives for the scalars are given by \[{\cal L}_{kin}=(D_{\mu}\psi)^{\dagger}(D^{\mu}\psi)+\mathrm{Tr}(D_{\mu}\chi)^{ \dagger}(D^{\mu}\chi),\] (5) where \[D_{\mu}\psi = \left(\partial_{\mu}-ig_{s}G^{a}_{\mu}\lambda^{a}-ig^{\prime}( \frac{2}{3})B_{\mu}\right)\psi\] \[D_{\mu}\chi = \partial_{\mu}-ig_{s}G^{a}_{\mu}\lambda^{a}\chi-\frac{ig}{2}[W^{i }_{\mu}\sigma^{i},\chi]-ig^{\prime}(\frac{-1}{3})B_{\mu}\chi.\] The implicit sums are over the generators $\lambda^{a}$ of SU(3), $a=1,...8$, and the generators $\sigma^{i}$ of SU(2), $i=1,2,3$. Within this framework a $N_{R}$ Majorana mass term, $\frac{1}{2}\,M_{N_{R}}\,\bar{N^{c}_{R}}\,N_{R}$, can be added. This term is even under DP, and we treat $M_{N_{R}}$ as a free parameter. We further assume that $M_{N_{R}}<m_{\psi}$ which makes $N_{R}$ a suitable dark matter candidate¹. [FOOTNOTE:1][ENDFOOTNOTE] The gauge and $Z_{2}$ invariant potential is given by \[V(H,\psi,\chi)=-\mu^{2}H^{\dagger}H+\frac{\lambda}{4!}(H^{ \dagger}H)^{2}+m_{\chi}^{2}\mathrm{Tr}(\chi^{\dagger}\chi)\] \[+\lambda_{\chi}(\mathrm{Tr}(\chi^{\dagger}\chi))^{2}+m_{\psi}^{2} \psi^{\dagger}\psi+\lambda_{\psi}(\psi^{\dagger}\psi)^{2}+\kappa_{1}H^{\dagger }H\mathrm{Tr}(\chi^{\dagger}\chi)\] \[+\kappa_{2}H^{\dagger}\chi^{\dagger}\chi H+\kappa_{3}H^{\dagger}H \psi^{\dagger}\psi+\rho\mathrm{Tr}(\chi^{\dagger}\chi)\psi^{\dagger}\psi\] (7) where $H$ is the SM Higgs field. In order not to have a colour breaking vacuum we take $m_{\chi}^{2},m_{\psi}^{2}$ to be positive. Since all the new physics terms we have incorporated are of dimension four, the theory remains renormalizable. Of particular interest to us is the last term of Equation (2), since this coupling would play a role in the radiative generation of the active neutrino masses. In addition, the DP remains exact even after electroweak symmetry breaking. ## 3 Dark Matter As mentioned in the previous section, the unbroken $Z_{2}$ symmetry stabilizes $N_{R}$. Due to the interaction introduced in Equation (1), the mechanism that leads to a reduction in the relic abundance of $N_{R}$ is via $t$-channel annihilation into right-handed top and charm quarks through the exchange of the colour electroweak singlet scalar, $\psi$. In this work we consider three benchmark points which depict three important regions of parameter space: $M_{N_{R}}=80,150,450$ GeV. The evolution of the comoving particle density is given by the Boltzmann equation \[\frac{\dot{n}}{n_{eq}}=\Gamma\cdot\left(\frac{n^{2}}{n^{2}_{eq}}-1\right)-3H \frac{n}{n_{eq}}\] (8) where $n$ is the particle density at time $t$ and $n_{eq}$ is the density at equilibrium, $H$ is the Hubble expansion rate and $\Gamma$ parametrizes the interaction rate, $\Gamma=\left<\sigma v\right>n_{eq}$, where $\left<\sigma v\right>$ denotes the thermally average annihilation cross section. By solving numerically the above equation one can find the temperature at which particles depart from equilibrium and freeze out. This temperature is given by \[x_{FO}\equiv\frac{m}{T_{FO}}\approx\log\left(0.038g\frac{mM_{Pl}\left<\sigma v \right>}{g^{1/2}_{}x^{1/2}_{FO}}\right),\] (9) where $g$ denotes the number of degrees of freedom of the particle under consideration and $g_{}$ the number of relativistic degrees of freedom at the freeze out temperature. The present day relic abundance is then given by \[\Omega_{DM}h^{2}\approx\frac{1.07\times 10^{9}~{}\text{GeV}^{-1}}{Jg^{1/2}_{} M_{Pl}},\] (10) where \[J\equiv\int^{\infty}_{x_{FO}}\frac{\left<\sigma v\right>}{x^{2}}dx.\] (11) The thermalized cross section at temperature $T$ can be calculated from the annihilation cross section of our dark matter candidate, $N_{R}$. The thermalized cross section is given by \[\left<\sigma_{N_{R}N_{R}}v\right>=\int^{\infty}_{4M^{2}_{N_{R}}}ds\frac{(s-4M^ {2}_{N_{R}})s^{1/2}K_{1}\left(s^{1/2}/T\right)}{8M_{N_{R}}TK_{2}^{2}(M_{N_{R}} /T)}\sigma(s),\] (12) where $\sigma(s)$ is the annihilation cross section as a function of the center of mass energy squared of the interaction, and $K_{1}(z)$, $K_{2}(z)$ are Modified Bessel function of the first and second kind respectively. We calculated the relic abundance using the latest version of MicOMEGAs [30] and the model files were generated with the latest version of FeynRules [31]. We carried out a scan over three parameters, $y_{\psi}^{t,c}$, and $m_{\psi}$, for the three benchmark points. The results are shown in Figures 1 and 2. <figure><img src="content_image/1307.2606/x1.png"><figcaption>Figure 1: The normalized relic abundance in the ytψ−mψ plane. The grey regioncorresponds to the region of parameter space consistent with a Majorananeutrino with mass MNR=150 GeV contributing 75−100% of the dark matter relicabundance. The region in maroon corresponds to MNR=450 GeV.</figcaption></figure> <figure><img src="content_image/1307.2606/x2.png"><figcaption>Figure 2: The normalized relic abundance in the ycψ−mψ plane. The black regioncorresponds to the region of parameter space consistent with a Majorananeutrino with mass MNR=80 GeV contributing 75−100% of the dark matter relicabundance. The region in grey and maroon correspond to MNR=150,450 GeVrespectively.</figcaption></figure> The dependence of the relic abundance on $y^{t}_{\psi}$ and $m_{\psi}$ is shown in Figure 1. The grey region denotes the parameter space consistent with a relic with mass $M_{N_{R}}=150$ GeV contributing $75-100\%$ of the dark matter relic abundance, while the maroon region is for a relic with mass $M_{N_{R}}=450$ GeV. In Figure 2 we show the dependence of the relic abundance as a function of $y^{c}_{\psi}$ and $m_{\psi}$. The grey and maroon regions correspond to a relic with mass of $150$ and $450$ GeV respectively. The scattered behaviour of the grey and maroon regions in Figures 1 and 2 is due to the fact that a combination of annihilation channels are open: $c\bar{c}$ and $t\bar{c}/c\bar{t}$ for a $150$ GeV Majorana neutrino and $c\bar{c}$, $t\bar{t}$ and $t\bar{c}/c\bar{t}$ for a $450$ GeV Majorana neutrino. This is not the case for a relic with $M_{N_{R}}=80$ GeV, where the $c\bar{c}$ annihilation channel is the only one open. Here one finds that the relic abundance depends only on $y^{c}_{\psi}$ and $m_{\psi}$. For an $80$ GeV Majorana neutrino, the region consistent with $75-100\%$ of the relic abundance is depicted by the black region in Figure 2. ## 4 Radiative Neutrino Mass generation The conserved DP allows us to identify $N_{R}$ as a candidate for dark matter and it also forbids Dirac neutrino mass terms for the active neutrinos, $\nu_{i}$. Therefore, the usual seesaw mechanism is not operative in this model. However, the Lagrangian of Equation (3) has enough structure to radiatively generate masses for $\nu_{i}$ via the exchange of the exotic colour scalars. In particular, it has the novel feature of using the $t_{R}$ and $c_{R}$ quarks as a portal to communicate with the dark sector and as messengers for the neutrinos. Within this framework, the lowest order diagram for neutrino mass generation is at 3-loops. The diagram is due to exchanges of both $\psi$ and $\chi$ fields. This is depicted in Figure 3 which gives the $\ell,\ell^{\prime}$ element of the active neutrino mass matrix $M_{\nu}$. <figure><img src="content_image/1307.2606/x3.png"><figcaption>Figure 3: 3-loop generation of a Majorana mass for active neutrinos from thet-quark. The crosses on the fermion lines indicate mass insertions. Similardiagrams from the c-quark will also play a role although it gives smallercontribution.</figcaption></figure> This mechanism yields finite contributions to all the elements of $M_{\nu}$ and it is best seen using the mass insertion technique. The $\ell\ell^{\prime}$ element of the active neutrino mass matrix is given by \[(M_{\nu})_{\ell\ell^{\prime}}=\sum_{i,j}K^{ij}\lambda_{\ell}^{i}\lambda_{\ell^ {\prime}}^{j}\] (13) where $i,j=u,c,t$ and $K^{ij}$ which controls the scale of neutrino masses is given by \[K^{ij} = \frac{y_{\psi}^{i}y^{j}_{\psi}\rho}{(16\pi^{2})^{3}}\frac{m_{i}m_ {j}\,M_{N_{R}}}{(m_{\chi}^{2}-m_{i}^{2})(m^{2}_{\chi}-m_{j}^{2})}I(m_{\chi}^{2 },m_{\psi}^{2}),\] \[I = \int_{0}^{\infty}du\,\frac{u}{u+M_{N_{R}}^{2}}\] (14) \[\cdot \left[\int_{0}^{1}dx\ln\left(\frac{m_{\chi}^{2}(1-x)+m_{\psi}^{2} x+ux(1-x)}{m_{i}^{2}(1-x)+m_{\psi}^{2}x+ux(1-x)}\right)\right]^{2}.\] From the above equation we see that the $u$-quark yields a negligible contribution to the neutrino masses and we can concentrate on the $t$ and $c$ quarks. Furthermore, if only one type of quark is involved in the neutrino mass generation, then Equation (13) gives rise to two massless active neutrinos, excluded by experimental data. Therefore, at least two quark families must come into play. To simplify the model we assume that the top quark gives the main contribution and also demand that $\lambda_{e,\mu}^{c}<<\lambda^{c}_{\tau}$, such that the $c$-quark contribution only modifies the $3,3$ element of $M_{\nu}$. These requirements are sufficient to lift the degeneracy of two massless neutrinos. <figure><img src="content_image/1307.2606/x4.png"><figcaption>Figure 4: Kt,t factor as a function of the NR−tR coupling, ytψ. The region inblack corresponds to a Majorana neutrino with MNR=80 GeV while the grey andmaroon regions correspond to Majorana neutrino masses of 150 and 450 GeVrespectively.</figcaption></figure> Using this framework for neutrino mass generation we analyzed the parameter space consistent with 75-100$\%$ of the dark matter relic abundance, and calculated the $K^{ij}$ factors. In Figure 4 we show the $K^{t,t}$ factor as a function of $y^{t}_{\psi}$. The black region corresponds to a Majorana neutrino with $M_{N_{R}}=80$ GeV and the grey and maroon regions correspond to Majorana neutrino masses of $150$ and $450$ GeV respectively. We use a colour electroweak triplet with mass $m_{\chi}=1$ TeV and a scalar potential coupling between $\chi$ and $\psi$ of $\rho=0.1$. The bulk of the neutrino mass is due to $K^{t,t}$ since $K^{t,t}\gg K^{c,c}$. The $K^{t,t}$ parameter ranges from $\sim$one meV to 100 eV for parameter points responsible for 75-100$\%$ of the dark matter relic abundance. This range of $K^{t,t}$ values can naturally provide this model with a milli-electronvolt active neutrino mass. It is easy to see why the neutrino masses are naturally small. Let us consider the t-quark contribution. The 3-loop suppression yields a factor of $10^{-7}$. Since the LHC has not seen any new colour states we can assume that $m_{\chi}>1~{}\mathrm{TeV}$. A further suppression comes from $(\frac{m_{t}}{m_{\chi}})^{2}\sim 10^{-2}$. For $M_{N_{R}}=100$ GeV, the factor $(\frac{M_{N_{R}}}{m_{\chi}})^{2}$ gives another $10^{-2}$ suppression. Therefore, sub-eV active neutrinos are natural in this model and no fine tuning of $y_{\psi}^{t,c}$ or $\rho$ is required. ## 5 $\mu\to e\gamma$ From Equation 3 one can see that the colour electroweak triplet scalar states will give rise to lepton flavour violating decays. In particular, the decay $\mu\to e\gamma$ can be used to place a lower bound on the $y^{t}_{\psi}$ coupling. In our framework, the branching fraction of $\mu\to e\gamma$ is given by \[Br(\mu\to e\gamma)=1.8\left(\frac{\mathrm{TeV}}{m_{\chi}}\right)^{4}\times 10^ {-6}\|\lambda_{\mu}^{t}\lambda_{e}^{t}+\lambda_{\mu}^{c}\lambda_{e}^{c}\|^{2}.\] (15) Given that $K^{t,t}\gg K^{c,c}$, we see that we have no sensitivity to $\lambda^{c}_{\mu}\lambda^{c}_{e}$ in the definition of $M_{\nu}$. In this work we have maximized the contribution to the branching fraction in the limit where $\lambda^{c}_{\mu}\lambda^{c}_{e}\sim\lambda^{t}_{\mu}\lambda^{t}_{e}$. We then extract the value of $\lambda^{t}_{\mu}\lambda^{t}_{e}$ using the results from Figure 4 together with the latest values of $m_{e\mu}$ [32] and the current experimental upper bound on $Br(\mu\to e\gamma)\leq 2.4\times 10^{-12}$ [33]. In the analysis, we have used the best fit range for $m_{e\mu}$ assuming a normal hierarchy of active neutrino masses, $\|m_{e\mu}\|=1.5-8.8$ meV [32]. We have also fixed the colour electroweak triplet mass to $m_{\chi}=1$ TeV. The branching fraction can then be written in the following way: \[Br(\mu\to e\gamma)=7.2\times 10^{-6}\left(\frac{m_{e\mu}}{K^{t,t}}\right)^{2}\] (16) Our results are shown in Figure 5, where we plot the normalized branching fraction, $\xi(\mu\to e\gamma)=Br(\mu\to e\gamma)/Br(\mu\to e\gamma)_{exp}$, as a function of $y^{t}_{\psi}$ using the lower limit on $m_{e\mu}$; and in Figure 6 using the upper limit on $m_{e\mu}$. The black region corresponds to $M_{N_{R}}=80$ GeV while the grey and maroon regions to $M_{N_{R}}=150,450$ GeV respectively. We see that the lower bound on $y^{t}_{\psi}$ increases with decreasing $M_{N_{R}}$. This behaviour is due to the fact that the branching fraction is proportional to $M^{2}_{N_{R}}$ while it is inversely proportional to $(y^{t}_{\psi})^{4}$. In particular, we find an upper bound of $y^{t}_{\psi}\lesssim 0.3-0.4$ for $M_{N_{R}}=80$ GeV and $y^{t}_{\psi}\lesssim 0.2-0.3,0.18-0.2$ for $M_{N_{R}}=150,450$ GeV. <figure><img src="content_image/1307.2606/x5.png"><figcaption>Figure 5: Lower limit on the branching fraction normalized to the experimentalupper bound as a function of ytψ using the best fit values for meμ using anelectroweak triplet scalar mass, mψ=1 TeV. The black region corresponds toMNR=80 GeV while the grey and maroon regions correspond to MNR=150,450 GeVrespectively.</figcaption></figure> <figure><img src="content_image/1307.2606/x6.png"><figcaption>Figure 6: Upper limit on the branching fraction normalized to the experimentalupper bound as a function of ytψ using the best fit values for meμ using anelectroweak triplet scalar mass, mψ=1 TeV. The black region corresponds toMNR=80 GeV while the grey and maroon regions correspond to MNR=150,450 GeVrespectively.</figcaption></figure> An important fact to note is that the constraints placed on $y^{t}_{\psi}$ using the current experimental bound on $Br(\mu\to e\gamma)$ are not at all sensitive to the mass of the colour electroweak singlet scalar. This scalar plays an important role in mediating the annihilation of the Majorana neutrinos. As we will see below, bounds on the mass of this scalar as well as upper bounds on the $y^{t}_{\psi}$ can be obtained using collider data. ## 6 Collider constraints This model is also highly constrained by data from high energy colliders such as the Tevatron and the LHC. In particular, our model yields two very distinct signatures for which very stringent bounds exist. We used Madgraph 5 [34] to calculate the parton-level signal prediction and implemented the initial and final state radiation using Pythia [35]. Our signal acceptances were are calculated with the PGS detector simulation implementing the cuts in the corresponding LHC and Tevatron analyses. <figure><img src="content_image/1307.2606/x7.png"><figcaption></figcaption></figure> One constraint is due to dijet plus missing energy (MET) searches at the Tevatron. The latest bounds on this process were carried out by the CDF collaboration using $p\bar{p}$ collisions at a center of mass energy of $\sqrt{s}=1.96$ TeV and $2.$ fb${}^{-1}$ of integrated luminosity [36]. Within our framework, two channels can lead to this final state. The first one is $t\bar{t}$ production followed by a three body decay of the top quark into two Majorana neutrinos and a charm quark, $t\to N_{R}N_{R}c$. This channel is open as long as $N_{R}$ has a mass below $\sim 86$ GeV. The second channel is through pair production of two colour electroweak singlets, followed by the decay $\psi\to N_{R}c$. These two channels are sensitive to $y_{\psi}^{t,c}$ and $m_{\psi}$. In order to generate exclusions on all three parameters of our model we implemented the experimental sample with tight kinematic thresholds of MET $>100$ GeV and $H_{T}>225$ GeV, where $H_{T}$ denotes the scalar sum of the two jet transverse energies: \[H_{T}=E_{T}(\text{jet}_{1})+E_{T}(\text{jet}_{2})\] (17) The second constraint is due to top squark pair production in $pp$ collisions with a center of mass energy of $\sqrt{s}=8$ TeV and $19.5$ fb${}^{-1}$ of integrated luminosity. We used the results obtained with the Compact Muon Solenoid (CMS) detector at the LHC. This search looks for decays of a stop squark into a top quark and a neutralino [37]. Top squarks are the scalar partners of the top quark in supersymmetric extensions of the SM such as the Minimal Supersymmetric Standard Model (MSSM), and the neutralino is a linear combination of the fermionic partners of the neutral gauge bosons and the two neutral Higgs bosons. Within our framework, the colour electroweak singlet, $\psi$, has the same gauge quantum numbers as the top squark but additional decay modes, in particular $\psi\to N_{R}c$. We apply the CMS constraint using their cut based analysis for three different MET cuts: $>150,200,300$ GeV. In Figure 7, we show the parameter regions excluded for an 80 GeV Majorana neutrino from the four experimental observables mentioned at the beginning of this section. On the top, we plot the excluded region in the $y^{t}_{\psi}-m_{\psi}$ plane for $y^{c}_{\psi}=0.25$. The region labeled 1 corresponds to regions of parameter space excluded by the CMS observable with MET $>200$ GeV, while the regions labeled 2 and 3 correspond to MET $>150$ and $>300$ GeV respectively. The plot at the bottom corresponds to a value of $y^{c}_{\psi}=0.5$. For this value of $y^{c}_{\psi}$ the excluded region is smaller since the branching fraction of $\psi\to N_{R}t$ is reduced, and thus, the CMS analysis is less sensitive to our model. From the plots in Figure 7 we also see that no region is excluded by the CDF experiment for $m_{\psi}>300$ GeV. This is not true for masses below $300$ GeV, where the CDF experiment rules out the entire model for $M_{N_{R}}=80$ GeV. <figure><img src="content_image/1307.2606/x9.png"><figcaption></figcaption></figure> In Figure 8, we show the regions of parameter space excluded for a 150 GeV Majorana neutrino. The plot on the top corresponds to $y^{c}_{\psi}=0.25$ while the plot on the bottom to $y^{c}_{\psi}=0.5$. For this benchmark point, the CDF observable is sensitive to regions where $m_{\psi}$ lies above 200 GeV but it is not able to exclude any of that region of parameter space. Therefore, the only relevant observable is the CMS analysis, which is able to exclude a region of parameter space where $320\lesssim m_{\psi}\lesssim 550$ for $y^{t}_{\psi}\gtrsim 0.4$ and $y^{c}_{\psi}=0.25$. Again, the excluded region is significantly smaller for larger values of $y^{c}_{\psi}$, since the branching fraction of $\psi\to N_{R}t$ is suppressed. The above collider constraints were also applied to a Majorana neutrino with $M_{N_{R}}=450$ GeV. We found that these constraints were not strong enough to rule out any of the parameter space consistent with $75-100\%$ of the dark matter relic abundance. Furthermore, we found that for Majorana neutrinos with masses below $20$ GeV, the CDF data on dijet+MET was enough to exclude it as a viable dark matter candidate. ## 7 Discussion In this study, we have investigated the possibility of extending the Standard Model with an electroweak singlet Majorana neutrino, stabilized by a new $Z_{2}$ symmetry, to explain the abundance of the dark matter in the universe. In this model, we coupled the dark matter candidate to $up$-type quarks via a new colour electroweak singlet scalar. Throughout the study we considered three benchmark scenarios: $M_{N_{R}}=80,150,450$ GeV. We found that the main annihilation channels were into right-handed top and charm quarks depending on the Majorana neutrino mass. Furthermore, we found that when all annihilation channels were open, we were able to generate $75-100\%$ of the dark matter relic abundance with a wide range of couplings, $y^{t,c}_{\psi}$, and scalar masses, $m_{\psi}$. This however was not the case for $M_{N_{R}}=80$ GeV, where the only available annihilation channel was into charm quarks. In this case we found a very clear dependence of the coupling $y^{c}_{\psi}$ on $m_{\psi}$. We have also investigated the possibility of radiatively generating Majorana masses for the active neutrinos of the Standard Model by incorporating a colour electroweak triplet scalar in addition to the colour electroweak singlet scalar. This setup allowed us to generate active neutrino masses at three loops. We found that the neutrino mass was mostly sensitive to the $y^{t}_{\psi}$ coupling, and that for points consistent with $75-100\%$ of the dark matter relic abundance, neutrino masses in the meV to 100 eV range are natural, with data favouring the lower values. We have considered two types of constraints. The first one arising from the lepton flavour violating decay, $\mu\to e\gamma$. We found that the current experimental bound on the branching fraction placed lower bounds on the coupling $y^{t}_{\psi}$ independent on the colour electroweak singlet mass, $m_{\psi}$. This lower bound was also higher for lighter Majorana neutrinos. The second constraint was due to two different collider searches. We found that these constraints place upper bounds on the coupling $y^{t}_{\psi}$. These constraints were also dependent on $m_{\psi}$ and $y^{c}_{\psi}$; the latter responsible for the size of the excluded region, since this coupling modifies the branching fraction of $\psi\to N_{R}t$. Our framework offers an attractive avenue that naturally generates small active neutrino masses while providing a large range of masses for a viable dark matter candidate. The model we presented here is a minimal one as only couplings to $t$ and $c$ quarks are employed. The model also predicts new colour degrees of freedom which lie below the TeV scale, and are now being probed at the LHC. 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1304.0841	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 211624, "num_imgs": 42, "llama3_tokens_count": 58136 }	[ "content_image/1304.0841/x1.png", "content_image/1304.0841/x2.png", "content_image/1304.0841/x3.png", "content_image/1304.0841/x4.png", "content_image/1304.0841/x5.png", "content_image/1304.0841/x7.png", "content_image/1304.0841/x9.png", "content_image/1304.0841/x11.png", "content_image/1304.0841/x12.png", "content_image/1304.0841/x13.png", "content_image/1304.0841/x14.png", "content_image/1304.0841/x15.png", "content_image/1304.0841/x18.png", "content_image/1304.0841/x19.png", "content_image/1304.0841/x21.png", "content_image/1304.0841/x27.png", "content_image/1304.0841/x28.png", "content_image/1304.0841/x30.png", "content_image/1304.0841/x31.png", "content_image/1304.0841/x33.png", "content_image/1304.0841/x37.png", "content_image/1304.0841/x38.png", "content_image/1304.0841/x39.png", "content_image/1304.0841/x40.png", "content_image/1304.0841/x41.png", "content_image/1304.0841/x42.png", "content_image/1304.0841/x43.png", "content_image/1304.0841/x45.png", "content_image/1304.0841/x46.png", "content_image/1304.0841/x48.png", "content_image/1304.0841/x49.png", "content_image/1304.0841/x50.png", "content_image/1304.0841/x51.png", "content_image/1304.0841/x52.png", "content_image/1304.0841/x53.png", "content_image/1304.0841/x56.png", "content_image/1304.0841/x58.png", "content_image/1304.0841/x59.png", "content_image/1304.0841/x60.png", "content_image/1304.0841/x61.png", "content_image/1304.0841/x62.png", "content_image/1304.0841/x63.png" ]	# Evidence of Electron Neutrino Appearance in a Muon Neutrino Beam K. 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Autiero Université de Lyon, Université Claude Bernard Lyon 1, IPN Lyon (IN2P3), Villeurbanne, France M. Barbi University of Regina, Department of Physics, Regina, Saskatchewan, Canada G.J. Barker University of Warwick, Department of Physics, Coventry, United Kingdom G. Barr Oxford University, Department of Physics, Oxford, United Kingdom M. Bass Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. M. Batkiewicz H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland F. Bay ETH Zurich, Institute for Particle Physics, Zurich, Switzerland S.W. Bentham Lancaster University, Physics Department, Lancaster, United Kingdom V. Berardi INFN Sezione di Bari and Università e Politecnico di Bari, Dipartimento Interuniversitario di Fisica, Bari, Italy B.E. Berger Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. S. Berkman University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada I. Bertram Lancaster University, Physics Department, Lancaster, United Kingdom D. Beznosko State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. S. Bhadra York University, Department of Physics and Astronomy, Toronto, Ontario, Canada F.d.M. Blaszczyk Louisiana State University, Department of Physics and Astronomy, Baton Rouge, Louisiana, U.S.A. A. Blondel University of Geneva, Section de Physique, DPNC, Geneva, Switzerland C. Bojechko University of Victoria, Department of Physics and Astronomy, Victoria, British Columbia, Canada S. Boyd University of Warwick, Department of Physics, Coventry, United Kingdom D. Brailsford Imperial College London, Department of Physics, London, United Kingdom A. Bravar University of Geneva, Section de Physique, DPNC, Geneva, Switzerland C. Bronner Kyoto University, Department of Physics, Kyoto, Japan D.G. Brook-Roberge University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada N. Buchanan Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. R.G. Calland University of Liverpool, Department of Physics, Liverpool, United Kingdom J. Caravaca Rodríguez Institut de Fisica d’Altes Energies (IFAE), Bellaterra (Barcelona), Spain S.L. Cartwright University of Sheffield, Department of Physics and Astronomy, Sheffield, United Kingdom R. Castillo Institut de Fisica d’Altes Energies (IFAE), Bellaterra (Barcelona), Spain M.G. Catanesi INFN Sezione di Bari and Università e Politecnico di Bari, Dipartimento Interuniversitario di Fisica, Bari, Italy A. Cervera IFIC (CSIC & University of Valencia), Valencia, Spain D. Cherdack Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. G. Christodoulou University of Liverpool, Department of Physics, Liverpool, United Kingdom A. Clifton Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. J. Coleman University of Liverpool, Department of Physics, Liverpool, United Kingdom S.J. Coleman University of Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A. G. Collazuol INFN Sezione di Padova and Università di Padova, Dipartimento di Fisica, Padova, Italy K. Connolly University of Washington, Department of Physics, Seattle, Washington, U.S.A. L. Cremonesi Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom A. Curioni ETH Zurich, Institute for Particle Physics, Zurich, Switzerland A. Dabrowska H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland I. Danko University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania, U.S.A. R. Das Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. S. Davis University of Washington, Department of Physics, Seattle, Washington, U.S.A. M. Day University of Rochester, Department of Physics and Astronomy, Rochester, New York, U.S.A. J.P.A.M. de André Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France P. de Perio University of Toronto, Department of Physics, Toronto, Ontario, Canada G. De Rosa INFN Sezione di Napoli and Università di Napoli, Dipartimento di Fisica, Napoli, Italy T. Dealtry STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom Oxford University, Department of Physics, Oxford, United Kingdom S. Dennis University of Warwick, Department of Physics, Coventry, United Kingdom STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom C. Densham STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom F. Di Lodovico Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom S. Di Luise ETH Zurich, Institute for Particle Physics, Zurich, Switzerland J. Dobson Imperial College London, Department of Physics, London, United Kingdom O. Drapier Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France T. Duboyski Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom F. Dufour University of Geneva, Section de Physique, DPNC, Geneva, Switzerland J. Dumarchez UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France S. Dytman University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania, U.S.A. M. Dziewiecki Warsaw University of Technology, Institute of Radioelectronics, Warsaw, Poland M. Dziomba University of Washington, Department of Physics, Seattle, Washington, U.S.A. S. Emery IRFU, CEA Saclay, Gif-sur-Yvette, France A. Ereditato University of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High Energy Physics (LHEP), Bern, Switzerland L. Escudero IFIC (CSIC & University of Valencia), Valencia, Spain A.J. Finch Lancaster University, Physics Department, Lancaster, United Kingdom E. Frank University of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High Energy Physics (LHEP), Bern, Switzerland M. Friend High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan Y. Fujii High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan Y. Fukuda Miyagi University of Education, Department of Physics, Sendai, Japan A. Furmanski University of Warwick, Department of Physics, Coventry, United Kingdom V. Galymov IRFU, CEA Saclay, Gif-sur-Yvette, France A. Gaudin University of Victoria, Department of Physics and Astronomy, Victoria, British Columbia, Canada S. Giffin University of Regina, Department of Physics, Regina, Saskatchewan, Canada C. Giganti UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France K. Gilje State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. T. Golan Wroclaw University, Faculty of Physics and Astronomy, Wroclaw, Poland J.J. Gomez-Cadenas IFIC (CSIC & University of Valencia), Valencia, Spain M. Gonin Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France N. Grant Lancaster University, Physics Department, Lancaster, United Kingdom D. Gudin Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia D.R. Hadley University of Warwick, Department of Physics, Coventry, United Kingdom A. Haesler University of Geneva, Section de Physique, DPNC, Geneva, Switzerland M.D. Haigh Oxford University, Department of Physics, Oxford, United Kingdom P. 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Jamieson University of Winnipeg, Department of Physics, Winnipeg, Manitoba, Canada R.A. Johnson University of Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A. J.H. Jo State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. P. Jonsson Imperial College London, Department of Physics, London, United Kingdom K.K. Joo Chonnam National University, Institute for Universe & Elementary Particles, Gwangju, Korea C.K. Jung State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. A. Kaboth Imperial College London, Department of Physics, London, United Kingdom H. Kaji University of Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan T. Kajita University of Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan H. Kakuno University of Tokyo, Department of Physics, Tokyo, Japan J. 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Nagasaki Kyoto University, Department of Physics, Kyoto, Japan T. Nakadaira High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan M. Nakahata University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan T. Nakai Osaka City University, Department of Physics, Osaka, Japan K. Nakajima Osaka City University, Department of Physics, Osaka, Japan K. Nakamura High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan S. Nakayama University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan T. Nakaya Kyoto University, Department of Physics, Kyoto, Japan K. Nakayoshi High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan D. Naples University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania, U.S.A. T.C. Nicholls STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom C. Nielsen University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada M. Nirkko University of Bern, Albert Einstein Center for Fundamental Physics, Laboratory for High Energy Physics (LHEP), Bern, Switzerland K. Nishikawa High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan Y. Nishimura University of Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan H.M. O’Keeffe Oxford University, Department of Physics, Oxford, United Kingdom Y. Obayashi University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan R. Ohta High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan K. Okumura University of Tokyo, Institute for Cosmic Ray Research, Research Center for Cosmic Neutrinos, Kashiwa, Japan T. Okusawa Osaka City University, Department of Physics, Osaka, Japan W. Oryszczak University of Warsaw, Faculty of Physics, Warsaw, Poland S.M. Oser University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada M. Otani Kyoto University, Department of Physics, Kyoto, Japan R.A. Owen Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom Y. Oyama High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan M.Y. Pac Dongshin University, Department of Physics, Naju, Korea V. Palladino INFN Sezione di Napoli and Università di Napoli, Dipartimento di Fisica, Napoli, Italy V. Paolone University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, Pennsylvania, U.S.A. D. Payne University of Liverpool, Department of Physics, Liverpool, United Kingdom G.F. Pearce STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom O. Perevozchikov Louisiana State University, Department of Physics and Astronomy, Baton Rouge, Louisiana, U.S.A. J.D. Perkin University of Sheffield, Department of Physics and Astronomy, Sheffield, United Kingdom Y. Petrov University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada E.S. Pinzon Guerra York University, Department of Physics and Astronomy, Toronto, Ontario, Canada P. Plonski Warsaw University of Technology, Institute of Radioelectronics, Warsaw, Poland E. Poplawska Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom B. Popov UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France M. Posiadala University of Warsaw, Faculty of Physics, Warsaw, Poland J.-M. Poutissou TRIUMF, Vancouver, British Columbia, Canada R. Poutissou TRIUMF, Vancouver, British Columbia, Canada P. Przewlocki National Centre for Nuclear Research, Warsaw, Poland B. Quilain Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-Ringuet, Palaiseau, France E. Radicioni INFN Sezione di Bari and Università e Politecnico di Bari, Dipartimento Interuniversitario di Fisica, Bari, Italy P.N. Ratoff Lancaster University, Physics Department, Lancaster, United Kingdom M. Ravonel University of Geneva, Section de Physique, DPNC, Geneva, Switzerland M.A.M. Rayner University of Geneva, Section de Physique, DPNC, Geneva, Switzerland M. Reeves Lancaster University, Physics Department, Lancaster, United Kingdom E. Reinherz-Aronis Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. F. Retiere TRIUMF, Vancouver, British Columbia, Canada A. Robert UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France P.A. Rodrigues University of Rochester, Department of Physics and Astronomy, Rochester, New York, U.S.A. E. Rondio National Centre for Nuclear Research, Warsaw, Poland S. Roth RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany A. Rubbia ETH Zurich, Institute for Particle Physics, Zurich, Switzerland D. Ruterbories Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. R. Sacco Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom K. Sakashita High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan F. Sánchez Institut de Fisica d’Altes Energies (IFAE), Bellaterra (Barcelona), Spain E. Scantamburlo University of Geneva, Section de Physique, DPNC, Geneva, Switzerland K. Scholberg Duke University, Department of Physics, Durham, North Carolina, U.S.A. J. Schwehr Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. M. Scott Imperial College London, Department of Physics, London, United Kingdom D.I. Scully University of Warwick, Department of Physics, Coventry, United Kingdom Y. Seiya Osaka City University, Department of Physics, Osaka, Japan T. Sekiguchi High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan H. Sekiya University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan D. Sgalaberna ETH Zurich, Institute for Particle Physics, Zurich, Switzerland M. Shibata High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan M. Shiozawa University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan S. Short Imperial College London, Department of Physics, London, United Kingdom Y. Shustrov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia P. Sinclair Imperial College London, Department of Physics, London, United Kingdom B. Smith Imperial College London, Department of Physics, London, United Kingdom R.J. Smith Oxford University, Department of Physics, Oxford, United Kingdom M. Smy University of California, Irvine, Department of Physics and Astronomy, Irvine, California, U.S.A. J.T. Sobczyk Wroclaw University, Faculty of Physics and Astronomy, Wroclaw, Poland H. Sobel University of California, Irvine, Department of Physics and Astronomy, Irvine, California, U.S.A. M. Sorel IFIC (CSIC & University of Valencia), Valencia, Spain L. Southwell Lancaster University, Physics Department, Lancaster, United Kingdom P. Stamoulis IFIC (CSIC & University of Valencia), Valencia, Spain J. Steinmann RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany B. Still Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom A. Suzuki Kobe University, Kobe, Japan K. Suzuki Kyoto University, Department of Physics, Kyoto, Japan S.Y. Suzuki High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan Y. Suzuki University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan T. Szeglowski University of Silesia, Institute of Physics, Katowice, Poland M. Szeptycka National Centre for Nuclear Research, Warsaw, Poland R. Tacik University of Regina, Department of Physics, Regina, Saskatchewan, Canada TRIUMF, Vancouver, British Columbia, Canada M. Tada High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan S. Takahashi Kyoto University, Department of Physics, Kyoto, Japan A. Takeda University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan Y. Takeuchi Kobe University, Kobe, Japan H.A. Tanaka University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada M.M. Tanaka High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan M. Tanaka High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan I.J. Taylor State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. D. Terhorst RWTH Aachen University, III. Physikalisches Institut, Aachen, Germany R. Terri Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom L.F. Thompson University of Sheffield, Department of Physics and Astronomy, Sheffield, United Kingdom A. Thorley University of Liverpool, Department of Physics, Liverpool, United Kingdom S. Tobayama University of British Columbia, Department of Physics and Astronomy, Vancouver, British Columbia, Canada W. Toki Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. T. Tomura University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan Y. Totsuka C. Touramanis University of Liverpool, Department of Physics, Liverpool, United Kingdom T. Tsukamoto High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan M. Tzanov Louisiana State University, Department of Physics and Astronomy, Baton Rouge, Louisiana, U.S.A. Y. Uchida Imperial College London, Department of Physics, London, United Kingdom K. Ueno University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan A. Vacheret Oxford University, Department of Physics, Oxford, United Kingdom M. Vagins University of California, Irvine, Department of Physics and Astronomy, Irvine, California, U.S.A. G. Vasseur IRFU, CEA Saclay, Gif-sur-Yvette, France T. Wachala Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. A.V. Waldron Oxford University, Department of Physics, Oxford, United Kingdom C.W. Walter Duke University, Department of Physics, Durham, North Carolina, U.S.A. D. Wark STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom Imperial College London, Department of Physics, London, United Kingdom M.O. Wascko Imperial College London, Department of Physics, London, United Kingdom A. Weber STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory, Warrington, United Kingdom Oxford University, Department of Physics, Oxford, United Kingdom R. Wendell University of Tokyo, Institute for Cosmic Ray Research, Kamioka Observatory, Kamioka, Japan R.J. Wilkes University of Washington, Department of Physics, Seattle, Washington, U.S.A. M.J. Wilking TRIUMF, Vancouver, British Columbia, Canada C. Wilkinson University of Sheffield, Department of Physics and Astronomy, Sheffield, United Kingdom Z. Williamson Oxford University, Department of Physics, Oxford, United Kingdom J.R. Wilson Queen Mary University of London, School of Physics and Astronomy, London, United Kingdom R.J. Wilson Colorado State University, Department of Physics, Fort Collins, Colorado, U.S.A. T. Wongjirad Duke University, Department of Physics, Durham, North Carolina, U.S.A. Y. Yamada High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, Japan K. Yamamoto Osaka City University, Department of Physics, Osaka, Japan C. Yanagisawa State University of New York at Stony Brook, Department of Physics and Astronomy, Stony Brook, New York, U.S.A. S. Yen TRIUMF, Vancouver, British Columbia, Canada N. Yershov Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia M. Yokoyama University of Tokyo, Department of Physics, Tokyo, Japan T. Yuan University of Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A. A. Zalewska H. Niewodniczanski Institute of Nuclear Physics PAN, Cracow, Poland L. Zambelli UPMC, Université Paris Diderot, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Paris, France K. Zaremba Warsaw University of Technology, Institute of Radioelectronics, Warsaw, Poland M. Ziembicki Warsaw University of Technology, Institute of Radioelectronics, Warsaw, Poland E.D. Zimmerman University of Colorado at Boulder, Department of Physics, Boulder, Colorado, U.S.A. M. Zito IRFU, CEA Saclay, Gif-sur-Yvette, France J. Żmuda Wroclaw University, Faculty of Physics and Astronomy, Wroclaw, Poland February 20, 2024 ###### Abstract The T2K collaboration reports evidence for electron neutrino appearance at the atmospheric mass splitting, $\|$$\Delta m^{2}_{32}$$\|$$\approx 2.4\times 10^{-3}$$\rm\,eV^{2}$. An excess of electron neutrino interactions over background is observed from a muon neutrino beam with a peak energy of $0.6$$\mathrm{\,Ge\kern-1.0ptV}$ at the Super-Kamiokande (SK) detector 295 km from the beam’s origin. Signal and background predictions are constrained by data from near detectors located 280 m from the neutrino production target. We observe 11 electron neutrino candidate events at the SK detector when a background of $3.3\pm 0.4$(syst.) events is expected. The background-only hypothesis is rejected with a $p$-value of 0.0009 (3.1$\sigma$), and a fit assuming $\nu_{\mu}\rightarrow\nu_{e}$ oscillations with $\textrm{sin}^{2}2\theta_{23}$=1, $\delta_{CP}$=0 and $\|\Delta m^{2}_{32}\|=2.4\times 10^{-3}$ $\rm\,eV^{2}$ yields $\textrm{sin}^{2}2\theta_{13}$=$0.088{}^{+0.049}_{-0.039}$(stat.+syst.). pacs: 14.60.Pq,14.60.Lm,12.27.-a,29.40.ka † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] The T2K Collaboration ## I Introduction The phenomena of neutrino oscillations through the mixing of massive neutrinos have been well established by experiments observing neutrino interaction rates from solar Cleveland _et al._ (1998); Hirata _et al._ (1989); Abdurashitov _et al._ (1994); Anselmann _et al._ (1994); Fukuda _et al._ (2001); Ahmad _et al._ (2002); Arpesella _et al._ (2008), atmospheric Fukuda _et al._ (1998); Hirata _et al._ (1988); Becker-Szendy _et al._ (1992); Ahlen _et al._ (1995); Allison _et al._ (1997); Wendell _et al._ (2010), reactor Abe _et al._ (2008) and accelerator Ahn _et al._ (2006); Adamson _et al._ (2012, 17); Abe _et al._ (18) sources. With few exceptions, such as the results from the LSND Aguilar-Arevalo _et al._ (2001) and MiniBooNE collaborations Aguilar-Arevalo _et al._ (20), the observations are consistent with the mixing of three neutrinos, governed by three mixing angles: $\theta_{12}~{}\approx 34^{\circ}$, $\theta_{23}\approx 45^{\circ}$ and $\theta_{13}$; and an as-yet-undetermined CP-violating phase, $\delta_{CP}$. Neutrino mixing also depends on three mass states, $m_{i}$, and therefore two independent mass splittings, $\|\Delta m^{2}_{32}\|\approx 2.4\times 10^{-3}$ $\rm\,eV^{2}$ (atmospheric) and $\Delta m^{2}_{21}\approx 7.6\times 10^{-5}$ $\rm\,eV^{2}$ (solar), where $\Delta m^{2}_{ij}={m_{i}}^{2}-{m_{j}}^{2}$. Additional understanding of neutrino mixing can be gained by observing the appearance of one flavor of neutrino interactions in a beam of another flavor through charged current interactions. Recently, T2K Abe _et al._ (21) has reported on the appearance of electron neutrinos in a beam of muon neutrinos, and the OPERA Agafonova _et al._ (2010) and Super-Kamiokande Abe _et al._ (23) collaborations have reported on the appearance of tau neutrinos from accelerator-based and atmospheric muon neutrino sources, respectively. The oscillations of $\nu_{\mu}\rightarrow\nu_{e}$ that T2K searches for are of particular interest since the observation of this mode at a baseline over energy ratio ($L/E$) of $\sim 1$ GeV$/500$ km implies a non-zero value for the mixing angle $\theta_{13}$. Until recently, the mixing angle $\theta_{13}$ had only been constrained to be less than 11${}^{\circ}$ by reactor Apollonio _et al._ (2003) and accelerator Adamson _et al._ (25, 2013) neutrino experiments. With data collected through 2011, the T2K experiment found the first indication of non-zero $\theta_{13}$ in the oscillation of muon neutrinos to electron neutrinos Abe _et al._ (21). Since then, a non-zero value of $\theta_{13}=9.1^{\circ}\pm 0.6^{\circ}$ Beringer _et al._ (2012) has been confirmed from the disappearance of reactor electron anti-neutrinos observed by the Daya Bay An _et al._ (2012), RENO Ahn _et al._ (2012) and Double Chooz Abe _et al._ (30) experiments. In this paper, T2K updates its measurement of electron neutrino appearance using additional data collected through 2012 and improved analysis methods. The probability for electron neutrino appearance in a muon neutrino beam with energy $E_{\nu}$ of $\mathcal{O}(1)$ GeV propagating over a baseline $L$ of $\mathcal{O}(100)$ km is dominated by the term (in units of $c,\hbar=1$): \[P_{\nu_{\mu}\rightarrow\nu_{e}}\approx\textrm{sin}^{2}\theta_{23}~{}\textrm{ sin}^{2}2\theta_{13}~{}\textrm{sin}^{2}\frac{\Delta m^{2}_{32}L}{4E_{\nu}}.\] (1) This leading term is identical for neutrino and antineutrino oscillations. Since the probability depends on $\textrm{sin}^{2}\theta_{23}$, a precise determination of $\theta_{13}$ requires measurements of $\theta_{23}$. The dependence on $\textrm{sin}^{2}\theta_{23}$ can lift the degeneracy of solutions with $\theta_{23}>\pi/4$ and $\theta_{23}<\pi/4$ that are present when $\theta_{23}$ is measured from muon neutrino survival, which depends $\textrm{sin}^{2}2\theta_{23}$. The electron neutrino appearance probability also includes sub-leading terms which depend on $\delta_{CP}$ and terms that describe matter interactions Freund (2001): \[P_{\nu_{\mu}\rightarrow\nu_{e}}= \frac{1}{(A-1)^{2}}~{}\textrm{sin}^{2}2\theta_{13}~{}\textrm{sin} ^{2}\theta_{23}~{}\textrm{sin}^{2}[(A-1)\Delta]\] \[-(+) \frac{\alpha}{A(1-A)}~{}\textrm{cos}\theta_{13}~{}\textrm{sin}2 \theta_{12}~{}\textrm{sin}2\theta_{23}~{}\textrm{sin}2\theta_{13}\times\] \[\textrm{sin}\delta_{CP}~{}\textrm{sin}\Delta~{}\textrm{sin}A \Delta~{}\textrm{sin}[(1-A)\Delta]\] \[+ \frac{\alpha}{A(1-A)}~{}\textrm{cos}\theta_{13}~{}\textrm{sin}2 \theta_{12}~{}\textrm{sin}2\theta_{23}~{}\textrm{sin}2\theta_{13}\times\] \[\textrm{cos}\delta_{CP}~{}\textrm{cos}\Delta~{}\textrm{sin}A \Delta~{}\textrm{sin}[(1-A)\Delta]\] \[+ \frac{\alpha^{2}}{A^{2}}~{}\textrm{cos}^{2}\theta_{23}~{}\textrm{ sin}^{2}2\theta_{12}~{}\textrm{sin}^{2}A\Delta\] Here $\alpha=\frac{\Delta m^{2}_{21}}{\Delta m^{2}_{32}}<<1$, $\Delta=\frac{\Delta m^{2}_{32}L}{4E_{\nu}}$ and $A=2\sqrt{2}G_{F}N_{e}\frac{E_{\nu}}{\Delta m^{2}_{32}}$, where $N_{e}$ is the electron density of the Earth’s crust. In the three-neutrino paradigm CP violation can only occur when all three mixing angles, including $\theta_{13}$, have non-zero values. The second term has a negative sign for neutrinos and a positive sign for antineutrinos and violates CP, which suggests the possibility of observing CP violation by measuring the difference in the electron neutrino appearance probabilities for neutrinos and antineutrinos. Since the CP-violating term can only appear in an appearance probability, a measurement of $\nu_{e}$ appearance, such as the one described in this paper, is an important milestone towards future searches for CP violation. The $A$ dependence in the oscillation probability arises from matter effects and introduces a dependence on the sign of $\Delta m^{2}_{32}$. We refer to $\Delta m^{2}_{32}>0$ as the normal mass hierarchy and $\Delta m^{2}_{32}<0$ as the inverted mass hierarchy. This paper is organized as follows. Section II is a brief overview of the T2K experiment and the data-taking periods. Section III summarizes the analysis method and components, including the flux (Section IV), neutrino interaction model (Section V) and near detector and far detector data samples (Section VI and Section VIII respectively). The fit to near detector data, described in Section VII, is used to constrain the far detector rate and associated uncertainties. Finally, Section IX describes how the far detector $\nu_{e}$ sample is used to estimate $\textrm{sin}^{2}2\theta_{13}$. ## II Experimental Overview and Data Collection The T2K experiment Abe _et al._ (32) is optimized to observe electron neutrino appearance in a muon neutrino beam. We sample a beam of muon neutrinos generated at the J-PARC accelerator facility in Tokai-mura, Japan, at baselines of 280 m and 295 km from the neutrino production target. The T2K neutrino beam line accepts a 31 GeV/$c$ proton beam from the J-PARC accelerator complex. The proton beam is delivered in 5 $\mu$s long spills with a period that has been decreased from 3.64 s to 2.56 s over the data-taking periods described in this paper. Each spill consists of 8 equally spaced bunches (a significant subset of the data was collected with 6 bunches per spill) that are $\sim 15$ ns wide. The protons strike a 91.4 cm long graphite target, producing hadrons including pions and kaons, and positively charged particles are focused by a series of three magnetic horns operating at 250 kA. The pions, kaons and some muons decay in a 96 m long volume to produce a predominantly muon neutrino beam. The remaining protons and particles which have not decayed are stopped in a beam dump. A muon monitor situated downstream of the beam dump measures the profile of muons from hadron decay and monitors the beam direction and intensity. We detect neutrinos at both near (280 m from the target) and far (295 km from the target) detectors. The far detector is the Super-Kamiokande (SK) water Cherenkov detector. The beam is aimed 2.5${}^{\circ}$ (44 mrad) away from the target-to-SK axis to optimize the neutrino energy spectrum for the oscillation measurements. The off-axis configuration Beavis _et al._ (1995); 34; 35 takes advantage of the kinematics of pion decays to produce a narrow band beam. The angle is chosen so that the spectrum peaks at the first oscillation maximum, as shown in Fig. 1, maximizing the signal in the oscillation region and minimizing feed-down backgrounds from high energy neutrino interactions. This optimization is possible because the value of $\|$$\Delta m^{2}_{32}$$\|$ is already relatively well known. <figure><img src="content_image/1304.0841/x1.png"><figcaption>Figure 1: The muon neutrino survival probability (top) and electron neutrinoappearance probabilities (middle) at 295 km, and the unoscillated neutrinofluxes for different values of the off-axis angle (OA) (bottom). Theappearance probability is shown for two values of the phase δCP, and fornormal (NH) and inverted (IH) mass hierarchies.</figcaption></figure> The near detectors measure the properties of the beam at a baseline where oscillation effects are negligible. The on-axis INGRID detector Otani _et al._ (2010); Abe _et al._ (37) consists of 16 modules of interleaved scintillator/iron layers in a cross configuration centered on the nominal neutrino beam axis, covering $\pm 5$ m transverse to the beam direction along the horizontal and vertical axes. The INGRID detector monitors the neutrino event rate stability at each module, and the neutrino beam direction using the profile of event rates across the modules. The off-axis ND280 detector is a magnetized multi-purpose detector that is situated along the same direction as SK. It measures the neutrino beam composition and energy spectrum prior to oscillations and is used to study neutrino interactions. The ND280 detector utilizes a 0.2 T magnetic field generated by the refurbished UA1/NOMAD magnet and consists of a number of sub-detectors: side muon range detectors (SMRDs Aoki _et al._ (2013)), electromagnetic calorimeters (ECALs), a $\pi^{0}$ detector (P0D Assylbekov _et al._ (2012)) and a tracking detector. The tracking detector is composed of two fine-grained scintillator bar detectors (FGDs Amaudruz _et al._ (2012)) sandwiched between three gaseous time projection chambers (TPCs Abgrall _et al._ (41)). The first FGD primarily consists of polystyrene scintillator and acts as the target for most of the near detector neutrino interactions that are treated in this paper. Hence, neutrino interactions in the first FGD are predominantly on carbon nuclei. The ND280 detector is illustrated in Fig. 2, where the coordinate convention is also indicated. The $x$ and $z$ axes are in the horizontal plane, and the $y$ axis is vertical. The origin is at the center of the magnet, and the magnetic field is along the $x$ direction. The $z$ axis is the direction to the far detector projected to the horizontal plane. <figure><img src="content_image/1304.0841/x2.png"><figcaption>Figure 2: An exploded illustration of the ND280 detector. The description ofthe component detectors can be found in the text.</figcaption></figure> The SK far detector Fukuda _et al._ (2003), as illustrated in Fig. 3, is a 50 kt water Cherenkov detector located in the Kamioka Observatory. The cylindrically-shaped water tank is optically separated to make two concentric detectors : an inner detector (ID) viewed by 11129 inward-looking 20 inch photomultipliers, and an outer detector (OD) with 1885 outward-facing 8 inch photomultipliers. The fiducial volume is defined to be a cylinder whose surface is 2 m away from the ID wall, providing a fiducial mass of 22.5 kt. Cherenkov photons from charged particles produced in neutrino interactions form ring-shaped patterns on the detector walls, and are detected by the photomultipliers. The ring topology can be used to identify the type of particle and, for charged current interactions, the flavor of the neutrino that interacted. For example, electrons from electron neutrino interactions undergo large multiple scattering and induce electromagnetic showers, resulting in fuzzy ring patterns. In contrast, the heavier muons from muon neutrino interactions produce Cherenkov rings with sharp edges. <figure><img src="content_image/1304.0841/x3.png"><figcaption>Figure 3: An illustration of the SK detector.</figcaption></figure> The T2K experiment uses a special software trigger to associate neutrino interactions in SK to neutrinos produced in the T2K beam. The T2K trigger records all the photomultiplier hits within $\pm 500$ $\mu$s of the beam arrival time at SK. Beam timing information is measured spill-by-spill at J-PARC and immediately passed to the online computing system at SK. The time synchronization between the two sites is done using the Global Positioning System (GPS) with $<150$ ns precision and is monitored with the Common-View method Allan and Weiss (1980). Spill events recorded by the T2K triggers are processed offline to apply the usual SK software triggers used to search for neutrino events, and any candidate events found are extracted for further T2K data analysis. Spills used for the far detector data analysis are selected by beam and SK quality cuts. The primary reason spills are rejected at SK is due to the requirement that there are no events in the $100$ $\mu$s before the beam window, which is necessary to reject decay electrons from cosmic-ray muons. In this paper we present neutrino data collected during the three run periods listed in Table 1. The total SK data set corresponds to $3.01\times 10^{20}$ protons on target (POT) or $4\%$ of the T2K design exposure. About 50% of the data, the Run 3 data, were collected after T2K and J-PARC recovered from the 2011 Tohoku earthquake. A subset of data corresponding to $0.21\times 10^{20}$ POT from Run 3 was collected with the magnetic horns operating at 205 kA instead of the nominal value of 250 kA. The size of the total data set is approximately two times that of T2K’s previously published electron neutrino appearance result Abe _et al._ (21). Run Period \| Dates \| Integrated POT by SK ---\|---\|--- Run 1 \| Jan. 2010-Jun. 2010 \| 0.32×1020 Run 2 \| Nov. 2010-Mar. 2011 \| 1.11×1020 Run 3 \| Mar. 2012-Jun. 2012 \| 1.58×1020 Table 1: T2K data-taking periods and the integrated protons on target (POT) for SK data collected in those periods. We monitor the rate and direction of the neutrino beam over the full data-taking period with the INGRID detector. As illustrated in Fig. 4, the POT-normalized neutrino event rate is stable to within 1%, and the beam direction is controlled well within the design requirement of 1 mrad, which corresponds to a 2% shift in the peak energy of the neutrino spectrum. <figure><img src="content_image/1304.0841/x4.png"><figcaption>Figure 4: The time dependence of the POT-normalized reconstructed neutrinoevent rate (a) and the beam direction (b) measured by INGRID. The error barsshow the statistical uncertainty only. The points shown for the directionmeasurement include sequential data grouped in periods of stable beamconditions.</figcaption></figure> ## III Analysis Overview We search for $\nu_{\mu}\rightarrow\nu_{e}$ oscillations via charged current quasi-elastic (CCQE) interactions of $\nu_{e}$ at SK. Since the recoil proton from the target nucleus is typically below Cherenkov threshold, these events are characterized by a single electron-like ring and no other activity. The most significant background sources are $\nu_{e}$ from muon and kaon decays that are intrinsic to the neutrino beam, and neutral current $\pi^{0}$ (NC$\pi^{0}$) events where the detector response to the photons from the $\pi^{0}$ decay is consistent with a single electron-like ring. The selection of $\nu_{e}$ candidates is described in Section VIII. We estimate the oscillation parameters and produce confidence intervals using a model that describes the probabilities to observe $\nu_{e}$ candidate events at SK in bins of electron momentum (magnitude and direction), as described in Section IX. The probabilities depend on the values of the oscillation parameters as well as many nuisance parameters that arise from uncertainties in neutrino fluxes, neutrino interactions, and detector response. The point where the likelihood is maximum for the observed data sample gives the oscillation parameter estimates, and the likelihood ratio at other points is used to construct confidence intervals on the parameters. We model the neutrino flux with a data-driven simulation that takes as inputs measurements of the proton beam, hadron interactions and the horn fields Abe _et al._ (44). The uncertainties on the flux model parameters arise largely from the uncertainties on these measurements. The flux model and its uncertainties are described in Section IV. We model the interactions of neutrinos in the detectors assuming interactions on a quasi-free nucleon using a dipole parametrization for vector and axial form factors. The nuclei are treated as a relativistic Fermi gas, and outgoing hadrons are subject to interactions in the nucleus, so-called “final state interactions”. We validate the neutrino interaction model with comparisons to independent neutrino cross section measurements at $\mathcal{O}(1)$ GeV and pion scattering data. We set the uncertainties on the interaction model with comparisons of the model to data and alternate models. The neutrino interaction model and its uncertainties are described in Section V. We further constrain the flux and interaction model parameters with a fit to samples of neutrino interaction candidates in the ND280 detector. Selections containing a negative muon-like particle provide high purity samples of $\nu_{\mu}$ interactions, which constrain both the $\nu_{\mu}$ flux that determines signal and NC$\pi^{0}$ backgrounds at SK, and the intrinsic $\nu_{e}$ flux. In the energy range of interest, the intrinsic $\nu_{e}$ are predominantly produced from the decay chain $\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$, $\mu^{+}\to e^{+}+\nu_{e}+\bar{\nu}_{\mu}$, and to a lesser extent by three-body kaon decays. Hence, the $\nu_{e}$ flux is correlated with the $\nu_{\mu}$ flux through the production of pions and kaons in the T2K beam line. The charged current interactions that make up most of the ND280 samples constrain the charged current interaction model. While $\nu_{e}$ interactions are indirectly constrained by $\nu_{\mu}$ interactions, we also include uncertainties which account for differences between the $\nu_{\mu}$ and $\nu_{e}$ cross section model. The ND280 neutrino interaction sample selection is described in Section VI, and the fit of the neutrino flux and interaction models to this data is described in Section VII. ## IV Neutrino Flux Model We simulate the T2K beam line to calculate the neutrino flux at the near and far detectors in the absence of neutrino oscillations, and this flux model is used as an input to predict neutrino interaction event rates at the detectors. The flux simulation begins with the primary proton beam upstream of the collimator that sits in front of the T2K target. The interactions of particles in the target, beam line components, decay volume walls and beam dump, and their decays, are simulated. The simulation and its associated uncertainties are driven by measurements of the primary proton beam profile, measurements of the magnetic fields of the T2K horns, and hadron production data, including NA61/SHINE measurements Abgrall _et al._ (45, 2012). First, we model the interactions of the primary beam protons and subsequently produced particles in the graphite target with a FLUKA 2008 47; Battistoni _et al._ (2007) simulation. We pass any particles that exit the target into a GEANT3 Brun _et al._ (1994) simulation that tracks particles through the magnetic horns and decay region, and decays hadrons and muons to neutrinos. The hadron interactions in the GEANT3 simulation are modeled with GCALOR Zeitnitz and Gabriel (1993). To improve agreement between selected hadron interaction measurements and the simulation, we weight simulated events based on the stored information of the true initial and final state hadron kinematics for hadron interactions in events producing neutrinos. The predicted flux at the SK and ND280 detectors, including systematic errors, is shown in Fig. 5. Here we describe the methods for weighting the flux and evaluating uncertainties based on proton beam measurements, hadron interaction data, alignment measurements, horn current and field measurements, and the beam direction measurement from the INGRID detector. More details of the flux calculation are described in Ref. Abe _et al._ (44). <figure><img src="content_image/1304.0841/x5.png"><figcaption>Figure 5: The T2K flux prediction at SK (a) and ND280 (b) for neutrinos andantineutrinos with systematic error bars. The flux above Eν=10 GeV is notshown; the flux is simulated up to Eν=30 GeV.</figcaption></figure> ### Weighting and systematic error evaluation methods To tune the flux model and study its uncertainties, adjustments are made by weighting events based on kinematics of the hadron interactions or the primary proton. The sensitivities to nuisance parameters that arise from such uncertainties as the hadron production model, proton beam profile, or horn currents, are evaluated by their effect on the predicted neutrino spectrum. We use one of two approaches for each uncertainty source, depending on whether the uncertainty source has correlations that need to be treated. For error sources described by a number of correlated underlying parameters, we use weighting methods when possible. The nuisance parameters are sampled according to their covariance and the corresponding flux predictions for the $k$ samples, $\phi^{k}$, are calculated. A large number of parameters sets, N (typically 500 or more), are used to calculate the fractional covariance using: \[v_{ij}=\frac{1}{N}\sum_{k=1}^{N}\frac{(\phi_{i}^{nom}-\phi_{i}^{k})(\phi^{nom} _{j}-\phi_{j}^{k})}{\phi_{i}^{nom}\phi_{j}^{nom}}.\] (3) Here $\phi_{i}^{nom}$ is the nominal flux prediction and $i$ specifies a neutrino energy bin, flavor and detector at which the flux is evaluated. We evaluate hadron interaction and proton beam profile uncertainties with this method. For systematic variations that cannot be treated by weighting simulated events, such as misalignment of beam line elements or changes to the horn currents, we produce new simulated event samples with $\pm 1\sigma$ variations of the nuisance parameters and calculate the fractional covariance matrix: \[v_{ij} =\frac{1}{2}\frac{(\phi_{i}^{nom}-\phi_{i}^{+})(\phi^{nom}_{j}- \phi_{j}^{+})}{\phi_{i}^{nom}\phi_{j}^{nom}}\] \[+\frac{1}{2}\frac{(\phi_{i}^{nom}-\phi_{i}^{-})(\phi^{nom}_{j}- \phi_{j}^{-})}{\phi_{i}^{nom}\phi_{j}^{nom}}.\] (4) $\phi_{i}^{+}$ and $\phi_{i}^{-}$ are the flux prediction for $+1\sigma$ and $-1\sigma$ variations of the nuisance parameter. We evaluate horn and target alignment and horn current and field uncertainties with this method. The total fractional flux covariance matrix is the sum of fractional flux covariance matrices calculated for each source of uncertainty. For the fits to data described in Sections VII and IX, variations of the flux prediction are modeled with parameters $b_{i}$ that scale the normalization of the flux in bins of neutrino energy and flavor at a given detector. The covariance matrix of the $b_{i}$, $(V_{b})_{ij}$, is simply the total fractional flux covariance matrix described here. Since the $b_{i}$ are separated for the near and far detectors, their covariances account for the correlations between the flux predictions at the two detectors. The covariances can therefore be used directly in simultaneous fits of near and far detector data or to calculate the uncertainty on the ratio of flux spectra at the two detectors. The following sections describe each source of flux systematic uncertainty. ### Proton beam monitoring and simulation We simulate the proton beam according to the proton orbit and optics parameters measured by the proton beam position and profile monitors, and the number of protons measured by the intensity monitors. These monitors are described elsewhere Abe _et al._ (32); Bhadra _et al._ (2013). We measure proton beam properties for each run period by reconstructing the beam profile at the upstream end of the collimator that sits before the T2K target for each beam spill. The sum of profiles for each beam spill, weighted by the number of protons, gives the proton beam profile that we input to the flux simulation. Table 2 summarizes the measured mean position, angle, emittance, Twiss $\alpha$ parameter McDonald and Russell (1989) and width of the proton beam at the collimator, and their uncertainties for a typical run period. The largest contributions to the flux uncertainty from the proton beam simulation arise from the alignment uncertainties of the beam monitors. The effect of the proton beam profile uncertainty on the flux is studied by varying the parameters in Table 2 within their uncertainties while accounting for the parameter correlations. The uncertainties on $Y$ and $Y^{\prime}$ are dominant and are studied on a simulated “wide beam” flux sample that has a profile in the $y-y^{\prime}$ (proton vertical position and angle) plane that covers the measured uncertainties. The wide beam sample is weighted for variations of $Y$ and $Y^{\prime}$ and the effect on the flux is studied. The variations correspond to shifts in the off-axis angle of $\sim 0.35$ mrad, or shifts in the off-axis spectrum peak of $\sim 10$ MeV. \| X Profile \| Y Profile ---\|---\|--- Parameter \| Central Value \| Error \| Central Value \| Error X,Y (mm) \| 0.00 \| 0.35 \| -0.37 \| 0.38 X′,Y′ (mrad) \| 0.03 \| 0.07 \| 0.07 \| 0.28 σ (mm) \| 4.03 \| 0.14 \| 4.22 \| 0.12 ϵ (π mm mrad) \| 4.94 \| 0.54 \| 6.02 \| 3.42 α \| 0.33 \| 0.08 \| 0.34 \| 0.41 Table 2: Summary of measured proton beam profile parameters and uncertainties at the collimator for a typical run period : mean position (X,Y) and angle (X′,Y′), width (σ), emittance (ϵ), and Twiss parameter (α). ### Hadron production data, weighting and uncertainties The pion and kaon differential production measurements we use to weight the T2K flux predictions are summarized in Table 3. Experiment \| Beam Mom. \| Target \| Particles ---\|---\|---\|--- NA61/SHINE Abgrall _et al._ (2011b, 2012) \| 31 GeV/c \| C \| π±, K+ Eichten et al. Eichten _et al._ (1972) \| 24 GeV/c \| Be, Al, … \| p, π±, K± Allaby et al. Allaby _et al._ (1970) \| 19.2 GeV/c \| Be, Al, … \| p, π±, K± BNL-E910 Chemakin _et al._ (2008) \| 6.4-17.5 GeV/c \| Be \| π± Table 3: Differential hadron production data relevant for the T2K neutrino flux predictions. We weight charged meson differential production multiplicities to the NA61/SHINE $\pi^{+}/\pi^{-}$ Abgrall _et al._ (45) and $K^{+}$ Abgrall _et al._ (2012) thin target production data, which covers most of phase space relevant for the off-axis flux. We use additional kaon differential production data from Eichten _et al._ Eichten _et al._ (1972) and Allaby _et al._ Allaby _et al._ (1970) to weight $K^{+}$ multiplicities in the phase space not covered by the NA61/SHINE measurements, and for $K^{-}$ multiplicities. To estimate the uncertainty of pion production by secondary protons, we use differential pion production data from the BNL-E910 experiment Chemakin _et al._ (2008) that were collected in interactions with proton beam energies less than the T2K primary proton beam energy. We use measurements of the inelastic cross sections for proton, pion, and kaon beams with carbon and aluminum targets Abrams _et al._ (1970); Allaby _et al._ (1969); Allardyce _et al._ (1973); Bellettini _et al._ (1966); Bobchenko _et al._ (1979); Carroll _et al._ (1979); Cronin _et al._ (1957); Chen _et al._ (1955); Denisov _et al._ (1973); Longo and Moyer (1962); Vlasov _et al._ (1978) to weight based on particle interaction and absorption rates in the flux prediction. In particular, NA61/SHINE measures the inclusive “production” cross section of 31 GeV/$c$ protons on carbon: $\sigma_{prod}=229.3\pm 9.2$ mb Abgrall _et al._ (45). The production cross section is defined as: \[\sigma_{prod}=\sigma_{inel}-\sigma_{qe}.\] (5) Here, $\sigma_{qe}$ is the quasi-elastic scattering cross section, _i.e._ scattering off of individual bound nucleons that breaks up or excites the nucleus, but does not produce additional hadrons. The inclusive production cross section is used in the weighting of the flux prediction, and the quasi-elastic cross section is subtracted from measurements where necessary. We apply hadron interaction-based weights to simulated events in two steps. The multiplicity of pions and kaons produced in interactions of nucleons on the target nuclei is defined as: \[\frac{dn}{dp}(p,\theta)=\frac{1}{\sigma_{prod}}\frac{d\sigma}{dp}(p,\theta).\] (6) Here $p$ and $\theta$ are the momentum and angle relative to the incident particle of the produced particle in the lab frame. We apply multiplicity weights that are the ratio of the measured and simulated differential multiplicities: \[W(p,\theta)=\frac{[\frac{dn}{dp}(p,\theta)]_{data}}{{[\frac{dn}{dp}(p,\theta)] _{MC}}}.\] (7) We adjust the interaction rates of protons, charged pions and charged kaons as well, with weights that account for attenuation in the target: \[W = \frac{\sigma_{prod}^{\prime}}{\sigma_{prod}}e^{-x(\sigma_{prod}^{ \prime}-\sigma_{prod})\rho}.\] (8) Here $\rho$ is the number density of nuclear targets in the material, $\sigma_{prod}$ is the original inclusive production cross section in the simulation, $\sigma_{prod}^{\prime}$ is the inclusive production cross section to which the simulation is being weighted, and $x$ is the distance traversed by the particle through the material. The total weight is the product of weights from all materials through which the particle propagates. For pion and kaon production in secondary nucleon interactions, or in the phase space covered by the alternative kaon production data sets, we converted weights to an $x_{F}-p_{T}$ dependence, where $p_{T}$ is the transverse momentum of the produced particle and $x_{F}$ is the Feynman $x$ Feynman (1969) defined as: \[x_{F}=p_{L}/p_{max}.\] (9) Here $p_{L}$ is the longitudinal momentum of the produced particle in the center of mass frame, and $p_{max}$ is the maximum momentum the produced particle can have. We apply the $x_{F}-p_{T}$ dependent weights after converting simulated hadron interactions to the $x_{F}-p_{T}$ basis. This method assumes that the pion and kaon multiplicities expressed in the $x_{F}-p_{T}$ basis are independent of the collision center of mass energy. The effect of the hadron interaction weighting on the SK $\nu_{\mu}$ and $\nu_{e}$ flux are shown as the ratios of weighted to nominal flux in Fig. 6. The weighting of pion multiplicities is a $10\%$ effect at low energy, while the weighting of kaon multiplicities affects the flux by as much as $40\%$ in the high energy tail. The large weighting effect for kaons is due to the underestimation of kaon production above kaon momenta of 3 GeV/$c$ in the simulation. The effect of the inclusive production cross section weighting on the flux prediction is less than 4% for all energies. <figure><img src="content_image/1304.0841/x7.png"><figcaption>Figure 6: Ratio of the hadron interaction weighted flux to the nominal fluxfor νμ (a), νe (b) flux predictions at SK. The effects of the pion production,kaon production and inclusive production cross section weighting are shownseparately and in total.</figcaption></figure> The uncertainties on the hadron multiplicity measurements contribute to the total uncertainty on the flux. Typical NA61/SHINE $\pi^{\pm}$ data points have $\sim 7\%$ systematic error, corresponding to a maximum uncertainty of 6$\%$ on the flux. In addition, we evaluate uncertainties on the $x_{F}$ scaling assumption (less than 3%), and regions of the pion phase space not covered by data (less than 2%). The dominant source of uncertainty on the kaon production is the statistical uncertainty on the NA61/SHINE measurements. The uncertainties on the inclusive production cross section measurements reflect the discrepancies that are seen between different measurements at similar incident particle energies. These discrepancies are similar in size to $\sigma_{qe}$ and may arise from ambiguities in the actual quantity being measured by each experiment. We apply an uncertainty equal to the $\sigma_{qe}$ component to the inclusive production cross section measurements (typically larger than the individual measurement errors), and the uncertainty propagated to the flux is less than 8% for all energies. We apply an additional uncertainty to the production of secondary nucleons, for which no adjustments are made in the current flux prediction The uncertainty is based on the discrepancy between the FLUKA modeling of secondary nucleon production and measurements by Eichten et. al. Eichten _et al._ (1972) and Allaby et. al. Allaby _et al._ (1970). The uncertainty propagated to the flux is less than 10% for all energies. The neutrino energy-dependent hadron interaction uncertainties on the SK $\nu_{\mu}$ and $\nu_{e}$ flux predictions are summarized in Fig. 7, and represent the dominant source of uncertainty on the flux prediction. <figure><img src="content_image/1304.0841/x9.png"><figcaption>Figure 7: The fractional hadron interaction errors on νμ (a), νe (b) fluxpredictions at SK.</figcaption></figure> ### Horn and target alignment and uncertainties The horns are aligned relative to the primary beam line with uncertainties of 0.3 mm in the transverse x direction and 1.0 mm in the transverse y direction and beam direction. The precision of the horn angular alignment is 0.2 mrad. After installation in the first horn, both ends of the target were surveyed, and the target was found to be tilted from its intended orientation by 1.3 mrad. We have not included this misalignment in the nominal flux calculation, but the effect is simulated and included as an uncertainty. We also simulate linear and angular displacements of the horns within their alignment uncertainties and evaluate the effect on the flux. The total alignment uncertainty on the flux is less than 3% near the flux peak. ### Horn current, field and uncertainties We assume a $1/r$ dependence of the magnetic field in the flux simulation. The validity of this assumption is confirmed by measuring the horn field using a Hall probe. The maximum deviation from the calculated values is 2% for the first horn and less than 1% for the second and third horns. Inside the inner conductor of a spare first horn, we observe an anomalous field transverse to the horn axis with a maximum strength of 0.065 T. Flux simulations including the anomalous field show deviations from the nominal flux of up to 4%, but only for energies greater than $1$ GeV. The absolute horn current measurement uncertainty is 2% and arises from the uncertainty in the horn current monitoring. We simulate the flux with $\pm$5 kA variations of the horn current, and the effect on the flux is 2% near the peak. ### Off-axis angle constraint from INGRID The muon monitor indirectly measures the neutrino beam direction by detecting the muons from meson decays, while the INGRID on-axis neutrino detector directly measures the neutrino beam direction. The dominant source of uncertainty on the beam direction constraint is the systematic uncertainty on the INGRID beam profile measurement, corresponding to a $0.35$ mrad uncertainty. We evaluate the effect on the flux when the SK or ND280 off-axis detectors are shifted in the simulation by 0.35 mrad. ### Summary of flux model and uncertainties The T2K flux predictions at the ND280 and SK detectors have been described and are shown in Fig. 5. We use the flux predictions as inputs to calculate event rates at both the ND280 and SK detectors. To evaluate the flux related uncertainties on the event rate predictions, we evaluate the fractional uncertainties on the flux prediction in bins of energy for each neutrino flavor. The bin edges are: * $\nu_{\mu}$: 0.0, 0.4, 0.5, 0.6, 0.7, 1.0, 1.5, 2.5, 3.5, 5.0, 7.0, 30.0 GeV * $\bar{\nu}_{\mu}$: 0.0, 1.5, 30.0 GeV * $\nu_{e}$: 0.0, 0.5, 0.7, 0.8, 1.5, 2.5, 4.0, 30.0 GeV * $\bar{\nu}_{e}$: 0.0, 2.5, 30.0 GeV We choose coarse binning for the antineutrino fluxes since they make a negligible contribution for the event samples described in this paper. The neutrino flux has finer bins around the oscillation maximum and coarser bins where the flux prediction uncertainties are strongly correlated. The uncertainties on the ND280 $\nu_{\mu}$, SK $\nu_{\mu}$ and SK $\nu_{e}$ flux predictions are shown in Fig. 8 and the correlations are shown in Fig. 9. The correlations shown are evaluated for the binning described above. The ND280 $\nu_{\mu}$ and SK $\nu_{\mu}$ flux predictions have large correlations, indicating the $\nu_{\mu}$ interaction rate at the near detector can constrain the unoscillated $\nu_{\mu}$ interaction rate at the far detector. The SK $\nu_{e}$ flux is also correlated with the ND280 $\nu_{\mu}$ flux, since the $\nu_{\mu}$ and $\nu_{e}$ both originate from the $\pi\rightarrow\mu+\nu_{\mu}$ decay chain or kaon decays. This correlation also allows us to constrain the expected intrinsic $\nu_{e}$ rate at the far detector by measuring $\nu_{\mu}$ interactions at the near detector. <figure><img src="content_image/1304.0841/x11.png"><figcaption>Figure 8: The fractional uncertainties on the ND280 νμ, SK νμ and SK νe fluxevaluated for the binning used in this analysis. This binning is coarser thanthe binning shown in Fig. 7 and includes the correlations between merged bins.</figcaption></figure> <figure><img src="content_image/1304.0841/x12.png"><figcaption>Figure 9: The correlations of the flux uncertainties in the bi bins for theND280 νμ and SK νμ and νe fluxes. The axes are the bins in neutrino energy foreach flavor/detector combination and are proportional to the neutrino energyup to 10 GeV.</figcaption></figure> ## V Neutrino Interaction Model We input the predicted neutrino flux at the ND280 and SK detectors to the NEUT Hayato (2009) neutrino interaction generator to simulate neutrino interactions in the detectors. Fig. 10 illustrates the neutrino-nucleon scattering processes modeled by NEUT at the T2K beam energies. The dominant interaction at the T2K beam peak energy is charged current quasi-elastic scattering (CCQE): \[\nu_{\ell}+N\rightarrow\ell+N^{\prime},\] (10) where $\ell$ is the corresponding charged lepton associated with the neutrino’s flavor (electron or muon), and $N$ and $N^{\prime}$ are the initial and final state nucleons. Above the pion production threshold, single pion production contributes to charged current interactions (CC1$\pi$): \[\nu_{\ell}+N\rightarrow\ell+N^{\prime}+\pi,\] (11) and neutral current interactions (NC1$\pi$): \[\nu+N\rightarrow\nu+N^{\prime}+\pi.\] (12) In the high energy tail of the T2K flux, multi-pion and deep inelastic scattering (DIS) processes become dominant. <figure><img src="content_image/1304.0841/x13.png"><figcaption>Figure 10: The NEUT νμ interaction cross section per nucleon on 16O with abreakdown by interaction process. The “NC Other” curve includes neutralcurrent coherent pion production, resonant charged pion production, multi-pionproduction and deep inelastic scattering. The predicted νμ flux spectrum at SKwith no oscillations is shown for comparison.</figcaption></figure> ### NEUT simulation models CCQE interactions in NEUT are simulated using the model of Llewellyn Smith Llewellyn Smith (1972), with nuclear effects described by the relativistic Fermi gas model of Smith and Moniz Smith and Moniz (1972, 1975). Dipole forms for the vector and axial-vector form factors in the Llewellyn Smith model are used, with characteristic masses $M_{V}=0.84\,\mathrm{GeV}$ and $M_{A}=1.21\,\mathrm{GeV}$ respectively in the default simulation. The Fermi momentum $p_{F}$ is set to $217\,\mathrm{MeV/c}$ for carbon and $225\,\mathrm{MeV/c}$ for oxygen, and the binding energy is set to 25 MeV for carbon and 27 MeV for oxygen. NEUT simulates the production of pions via the excitation of hadronic resonances using the model of Rein and Sehgal Rein and Sehgal (1981). The simulation includes 18 resonances below $2\,\mathrm{GeV}$, along with interference terms. In the energy range relevant for T2K, resonance production is dominated by the $\Delta(1232)$. For 20% of the $\Delta$s produced within a nucleus, NEUT also simulates pion-less $\Delta$ decay, in which the $\Delta$ de-excites in the nuclear medium without the emission of pions. NEUT includes the production of pions in coherent scattering of the neutrino on the target nucleus based on the Rein and Sehgal model. Multi-pion and DIS interactions in NEUT are simulated using the GRV98 parton distribution functions Gluck _et al._ (1998). Where the invariant mass of the outgoing hadronic system ($W$) is in the range $1.3<W<2.0$ GeV/$c^{2}$, a custom program is used Nakahata _et al._ (1986), and only pion multiplicities of greater than one are considered to avoid double counting with the Rein and Sehgal model. For $W>2.0$ GeV/$c^{2}$ PYTHIA/JETSET Sjostrand (1994) is used. Corrections to the small $Q^{2}$ region developed by Bodek and Yang are applied Bodek and Yang (2003). NEUT uses a cascade model to simulate the interactions of hadrons as they propagate through the nucleus. For pions with momentum below $500\;\mathrm{MeV}/c$, the method of Salcedo _et al._ Salcedo _et al._ (1988) is used. Above pion momentum of $500\;\mathrm{MeV}/c$ the scattering cross sections are modeled using measurements of $\pi^{\pm}$ scattering on free protons de Perio (2011). Additional details on the NEUT simulation can be found elsewhere Abe _et al._ (32). ### Methods for varying the NEUT model Uncertainties in modeling neutrino interactions are a significant contribution to the overall systematic uncertainty in the $\nu_{e}$ appearance analysis reported in this paper. In the rest of this section, we describe these uncertainties with nuisance parameters that vary the NEUT interaction models. The parameters, listed in Table 4, are chosen and their central values and uncertainties are set to cover the systematic uncertainties on the interaction models derived from comparisons of NEUT to external data or alternative models. They are a combination of free parameters in the NEUT model and ad-hoc empirical parameters. The parameter values and uncertainties are further constrained by the fit to neutrino data from the T2K ND280 detector, as described in Section VII. To tune the NEUT model parameters and evaluate the effect of neutrino interaction uncertainties, adjustments are carried out by applying weights to simulated NEUT event samples from T2K or external experiments, such as MiniBooNE. CCQE Cross Section --- MQEA \| The mass parameter in the axial dipole form factor for quasi-elastic interactions xQE1 \| The normalization of the quasi-elastic cross section for Eν<1.5 GeV xQE2 \| The normalization of the quasi-elastic cross section for 1.5<Eν<3.5 GeV xQE3 \| The normalization of the quasi-elastic cross section for Eν>3.5 GeV Nuclear Model for CCQE Interactions (separate parameters for interactions on O and C) xSF \| Smoothly changes from a relativistic Fermi gas nuclear model to a spectral function model pF \| The Fermi surface momentum in the relativistic Fermi gas model Resonant Pion Production Cross Section MRESA \| The mass parameter in the axial dipole form factor for resonant pion production interactions xCC1π1 \| The normalization of the CC resonant pion production cross section for Eν<2.5 GeV xCC1π2 \| The normalization of the CC resonant pion production cross section for Eν>2.5 GeV xNC1π0 \| The normalization of the NC1π0 cross section x1πEν \| Varies the energy dependence of the 1π cross section for better agreement with MiniBooNE data Weff \| Varies the distribution of Nπ invariant mass in resonant production xπ−less \| Varies the fraction of Δ resonances that decay or are absorbed without producing a pion Other xCCcoh. \| The normalization of CC coherent pion production xNCcoh. \| The normalization of NC coherent pion production xNCother \| The normalization of NC interactions other than NC1π0 production xCCother \| Varies the CC multi-π cross section normalization, with a larger effect at lower energy →xFSI \| Parameters that vary the microscopic pion scattering cross sections used in the FSI model xνe/νμ \| Varies the ratio of the CC νe and νμ cross sections Table 4: The parameters used to vary the NEUT cross section model and a brief description of each parameter. ### NEUT model comparisons to external data and tuning A detailed description of the NEUT model tuning using external data comparisons can be found in Appendix A. Here we provide a brief summary. #### v.3.1 FSI model tuning and uncertainty The NEUT FSI model includes parameters which alter the microscopic pion interaction probabilities in the nuclear medium. The central values of these parameters and their uncertainties are determined from fits to pion scattering data Ashery _et al._ (1981); Jones _et al._ (1993); Giannelli _et al._ (2000). We consider variations of the FSI parameters within the uncertainties from the fit of the pion scattering data, and evaluate the uncertainties on the predicted event rates for ND280 and SK selections. #### v.3.2 CCQE model uncertainty The most detailed measurement of CCQE scattering on light nuclei in the region of $1\,\mathrm{GeV}$ neutrino energy has been made by MiniBooNE, which has produced double-differential cross sections in the muon kinetic energy and angle, $(T_{\mu},\cos\theta_{\mu})$ Aguilar-Arevalo _et al._ (82). We compare the agreement of NEUT to the MiniBooNE CCQE data in addition to our own near detector measurement of CCQE events (Section VI) since the MiniBooNE detector has 4$\pi$ acceptance, providing a kinematic acceptance of the leptons that more closely matches the SK acceptance for the selection described in Section VIII. This is illustrated in Fig. 11, which compares the predicted true $Q^{2}$ distributions for CCQE events in the ND280 CCQE selection, the MiniBooNE CCQE selection, and the SK selection for $\nu_{e}$ appearance candidates. <figure><img src="content_image/1304.0841/x14.png"><figcaption>Figure 11: The predicted Q2 distributions for CCQE interactions in the ND280CCQE selection, the MiniBooNE CCQE selection, and the SK νe appearanceselection.</figcaption></figure> In order to allow the ND280 data to constrain the CCQE model, we use the difference of the NEUT nominal value and the best-fit value from fit to MiniBooNE data to set the uncertainty on $M^{QE}_{A}$, $\sigma_{M_{A}^{QE}}=0.43\,\mathrm{GeV}$. We also set the uncertainty on the low energy CCQE normalization, $x_{1}^{QE}$, to the size of the MiniBooNE flux uncertainty, 11%. The results of the MiniBooNE fit are discussed in more detail in Appendix A. To allow for the discrepancy in CCQE cross section at $\mathcal{O}(1)$ GeV measured by MiniBooNE and at $\mathcal{O}(10)$ GeV measured by NOMAD Lyubushkin _et al._ (2009), we employ independent CCQE normalization factors for $(1.5<E_{\nu}<3.5)$ GeV ($x_{2}^{QE}$) and $E_{\nu}>3.5$ GeV ($x_{3}^{QE}$), each with a prior uncertainty of 30% and a nominal value of unity. Alternate explanations have been proposed to reconcile the MiniBooNE data with a $M^{QE}_{A}\approx 1.0$ GeV derived from electron scattering and NOMAD data Martini _et al._ (2009, 2010); Amaro _et al._ (2011); Bodek _et al._ (2011); Nieves _et al._ (2012). These models typically modify the cross section either by enhancing the transverse component of the cross section, or by adding an additional multi-nucleon process to the existing cross section, where the neutrino interacts on a correlated pair of nucleons. Future improvements to the NEUT generator may include a full implementation of alternate CCQE models. However, these models would also require modifications to the kinematics of the exiting nucleons, but no consensus has been reached yet in the field as to how the nucleons should be treated. We consider two possible effects of alternate CCQE models on the $\nu_{e}$ appearance analysis. First, the effect in $Q^{2}$ for these models is often similar to increasing $M^{QE}_{A}$ and Nieves _et al._ (2012) shows that other improvements to the CCQE cross section can be represented by an experiment-specific $M^{QE}_{A}$(effective), so the increase to the overall cross section from these models is approximately covered by the uncertainty on $M^{QE}_{A}$. Second, a multi-nucleon process would appear as a CCQE-like interaction in the SK detector, but the relationship between the neutrino energy and the lepton kinematics is different than for quasi-elastic scatters, which may affect the determination of oscillation parameters Meloni and Martini (2012); Lalakulich and Mosel (2012). Other processes also appear CCQE-like and have a different relationship between lepton kinematics and neutrino energy, such as non-QE events with no pions in the final state (pion-less $\Delta$ decay). The uncertainty on these events indirectly accounts for the effect of multi-nucleon models as these events affect the extracted oscillation parameters in a way similar to how multi-nucleon models would. #### v.3.3 Single pion production model tuning and uncertainty Measurements of single pion production cross sections on light nuclei in the T2K energy range have been made by MiniBooNE Aguilar-Arevalo _et al._ (91, 92, 93), and K2K, which used a 1000 ton water Cherenkov detector Nakayama _et al._ (2005). We perform a joint fit to the MiniBooNE measurements of charged current single $\pi^{+}$ production (CC1$\pi^{+}$), charged current single $\pi^{0}$ production (CC1$\pi^{0}$) and neutral current single $\pi^{0}$ production (NC1$\pi^{0}$). As shown in Appendix A, we compare the NEUT best-fit derived from the MiniBooNE single pion data with the K2K measurement, which is of particular interest since it is the same nuclear target as SK. \| Nominal value \| Penalty \| best-fit \| Error ---\|---\|---\|---\|--- MRESA (GeV) \| 1.21 \| \| 1.16 \| 0.10 Weff \| 1 \| \| 0.48 \| 0.14 xCCother \| 0 \| 0.40 \| 0.36 \| 0.39 Normalizations: \| \| \| \| xCCcoh \| 1 \| \| 0.66 \| 0.70 xCC1π1 \| 1 \| \| 1.63 \| 0.32 xNCcoh \| 1 \| 0.30 \| 0.96 \| 0.30 xNC1π0 \| 1 \| \| 1.19 \| 0.36 NC 1π± \| 1 \| 0.30 \| 0.98 \| 0.30 NC multi-pion/DIS \| 1 \| 0.30 \| 0.99 \| 0.30 Table 5: Parameters used in the single pion fits, and their best-fit values and uncertainties. The 1σ value of the penalty term is shown for parameters which are penalized in the fit. Where parameters are defined in a manner consistent with the T2K data fits, the same parameter name is used. <figure><img src="content_image/1304.0841/x15.png"><figcaption>Figure 12: Differential cross sections for CC1π+ Q2 (top), CC1π0 Q2 (middle)and NC1π0 pπ0 (bottom) used in the single-pion fits to MiniBooNE data, and theNEUT nominal and best-fit predictions. The MiniBooNE data point errors arestatistical+systematic.</figcaption></figure> The parameters listed in Table 5 are varied in the fit to the MiniBooNE single pion data and their best-fit values and uncertainties are listed. The parameters include $M_{A}^{RES}$, the axial mass in the Rein and Sehgal model, the empirical parameter, $W_{\textrm{eff}}$, discussed in the next paragraph, and parameters that vary the normalization of various interaction modes. Contributions to the samples from CC multi-pion/DIS ($x_{CCother}$) interactions, NC coherent interactions, NC1$\pi^{\pm}$ interactions and NC multi-pion/DIS interactions are relatively small, so the MiniBooNE samples have little power to constrain the associated parameters which are discussed in Section V.3.4. Penalty terms for these parameters are applied using the prior uncertainties listed in Table 5. The $W_{\textrm{eff}}$ parameter alters the single pion differential cross section as a function of pion-nucleon invariant mass $W$, providing a means to change the shape of the NEUT prediction for NC1$\pi^{0}$$\mathrm{d}\sigma/\mathrm{d}p_{\pi^{0}}$ differential cross section. Uncertainties in the NC1$\pi^{0}$ pion momentum distribution enter into the $\nu_{e}$ appearance analysis, as the momentum and angular distributions of $\nu_{e}$ candidates from NC1$\pi^{0}$ interactions depend on the kinematic distribution of the $\pi^{0}$. The NEUT predicted $p_{\pi^{0}}$ spectrum, shown in the bottom plot of Fig. 12 is broader than the observed MiniBooNE data. A decrease to the $W_{\textrm{eff}}$ parameter results in a more sharply-peaked $p_{\pi^{0}}$ spectrum, and achieves agreement between the NEUT prediction and the measured cross section; $W_{\textrm{eff}}$ does not alter the total cross section. Future changes to the NEUT model that may eliminate the need for $W_{\textrm{eff}}$ include refinements of the treatment of formation time effects, which have been shown to affect the pion momentum distribution Golan _et al._ (2012), or modifications to the contribution of higher order resonances relative to $\Delta(1232)$. The fitted data and NEUT model are shown in Fig. 12. We propagate the fitted parameter values for $M_{A}^{RES}$, $x^{CC1\pi}_{1}$ and $x^{NC1\pi^{0}}$ and their correlated uncertainties to the fits of ND280 and SK data. The remaining parameters from the fit to MiniBooNE data are marginalized. We evaluate additional uncertainties on these parameters by re-running the fit to MiniBooNE data with variations of the FSI model and pion-less $\Delta$ decay turned off. The deviations of the fitted parameter values due to these FSI or pion-less $\Delta$ decay variations are applied as parameter uncertainties, increasing the uncertainties on $M_{A}^{RES}$, $x^{CC1\pi}_{1}$ and $x^{NC1\pi^{0}}$ to 0.11 GeV, 0.43 and 0.43 respectively. The fitted $W_{\textrm{eff}}$ parameter value is not applied to the T2K predictions, but the difference between the nominal value of $W_{\textrm{eff}}$ and the best-fit value from the MiniBooNE data fit is treated as an uncertainty. An additional uncertainty in the energy-dependent pion production cross section is considered since we observe a discrepancy between the fitted NEUT model and the MiniBooNE CC1$\pi^{+}$ data, as shown in Fig. 13. We introduce a parameter $x_{1\pi E_{\nu}}$ that represents the energy-dependent tuning which brings the NEUT prediction into agreement with the MiniBooNE data. Uncertainties on the ND280 and SK predictions include the difference between the resonant pion production with and without this energy-dependent tuning. <figure><img src="content_image/1304.0841/x18.png"><figcaption>Figure 13: The CC1π+ cross section as a function of energy as measured byMiniBooNE, with the NEUT nominal and best-fit models. The treatment in theanalysis of the disagreement between the best-fit NEUT and data is discussedin the text.</figcaption></figure> The fits to MiniBooNE data constrain the normalization of CC$1\pi$ resonant production below 2.5 GeV. Above 2.5 GeV, we apply a separate normalization uncertainty of 40% on the parameter $x_{2}^{CC1\pi}$. This uncertainty covers the maximum difference between MiniBooNE CC1$\pi^{+}$ data and NEUT at $E_{\nu}\approx 2$ GeV and is conservative given the CC inclusive cross section measurements Wu _et al._ (2008) made at higher energies. #### v.3.4 Other interaction channels We evaluate the uncertainty on CC coherent pion production based on measurements by K2K Hasegawa _et al._ (2005) and SciBooNE Hiraide _et al._ (2008) which place upper limits on the CC coherent production that are significantly less than the Rein and Sehgal model prediction. Since no clear CC coherent signal has been observed at $\mathcal{O}(1)$ GeV , we apply a 100% normalization uncertainty to the NEUT CC coherent pion production ($x^{CCcoh}$). SciBooNE’s measurement of the NC coherent pion cross section at $\mathcal{O}(1)$ GeV Kurimoto _et al._ (2010) is in good agreement with the Rein and Sehgal model prediction; the uncertainty on this channel is set to 30% based on the SciBooNE measurement and is represented by a normalization parameter, $x^{NCcoh}$. We define a single parameter $x^{NCother}$ that varies the normalization of the NC resonant $\pi^{\pm}$, NC elastic and NC multi-pion/DIS/other resonant modes. The uncertainty on this normalization parameter is set to 30%. As there is little NC resonant $\pi^{\pm}$ data, the uncertainty on the NC resonant $\pi^{\pm}$ processes is set to be the same size as the agreement shown in Section V.3.3 for the NC resonant 1$\pi^{0}$ cross section (30%). The NC multi-pion and DIS model was tuned to agree with the CC/NC data using the NEUT predicted CC DIS cross section; the uncertainties on this phenomenological model are set to cover the size of the uncertainties of the CC/NC data Musset and Vialle (1978); Kim _et al._ (1981) (30%). The CC multi-pion/DIS interactions contribute to the ND280 samples discussed in Section VI. At energies greater than 4 GeV, these modes dominate the inclusive cross section and are constrained by measurements of the inclusive cross section Adamson _et al._ (2010) with $\approx$10% uncertainties. At lower energies the constraint from the inclusive cross section measurements is weaker since other interactions modes are significant. Hence, we apply an uncertainty that is $10\%$ at high energies and increases to $40\%$ near the threshold for multi-pion production. The model is adjusted by applying a weight: \[w=1+\frac{x_{CCother}}{E_{\nu}(\mathrm{GeV})}.\] (13) The parameter $x_{CCother}$ is allowed to vary around a nominal value of 0 with a prior uncertainty of 0.4 GeV. ### Nuclear model uncertainties NEUT models nuclei with a relativistic Fermi gas model (RFG) using a Fermi momentum $p_{F}$ from electron scattering data Moniz _et al._ (1971). We evaluate the uncertainty on the CCQE cross section for variations of $p_{F}$ within its uncertainty of $30$ MeV/$c$. This uncertainty covers the uncertainty from the electron scattering data and has been inflated to cover possible discrepancies in the CCQE cross section at low $Q^{2}$. The uncertainty is applied independently for interactions on carbon and oxygen targets. We also consider alternatives to the RFG model of the nuclei by making comparisons to a spectral function nuclear model implemented in the NuWro neutrino interaction generator Ankowski and Sobczyk (2006). The discrepancy in CCQE interactions models with the RFG and spectral function are assigned as uncertainty and represented by the parameter $x_{SF}$ which smoothly varies the predicted lepton kinematics between the RFG ($x_{SF}=0$) and spectral function ($x_{SF}=1$) models. We apply the uncertainties for the nuclear model independently for carbon and oxygen cross sections. ### $\nu_{e}$ cross section uncertainty Differences between $\nu_{\mu}$ and $\nu_{e}$ in the cross section are also considered, as the CC $\nu_{\mu}$ sample at ND280 is used to infer the CC $\nu_{e}$ rate at the far detector. The spectral function uncertainty is calculated separately for $\nu_{\mu}$ and $\nu_{e}$ as well as target material. In addition, an overall 3% uncertainty on the ratio of $\nu_{\mu}$ and $\nu_{e}$ CC neutrino-nucleon cross sections ($x_{\nu_{e}/\nu_{\mu}}$) is included, based on calculations Day and McFarland (2012) over T2K’s energy range. ### Summary of the neutrino cross section model, tuning and uncertainties The cross section model parameters values and uncertainties are listed in Table 6. These priors are used as inputs to fits to the T2K ND280 and SK data sets, and include the results of the MiniBooNE single pion model fit. For parameters related to the nuclear modeling, such as $x_{SF}$, $p_{F}(^{12}$C) and $p_{F}(^{16}$O), we apply separate uncorrelated parameters for the modeling of interactions on ${}^{12}$C and ${}^{16}$O. Hence, the fit to ND280 data does not constrain the nuclear modeling parameters used when modeling interactions at SK. Of the remaining parameters, we treat them as correlated for ND280 and SK if they are strongly constrained by ND280 data. These parameters include the CCQE cross section parameters, $M_{A}^{QE}$, $x^{QE}_{1}$, and the CC1$\pi$ cross section parameters, $M_{A}^{RES}$, $x^{CC1\pi}_{1}$. To preserve the correlations between NC and CC parameters from the fit to MiniBooNE single pion data, $x^{NC1\pi^{0}}$ is also propagated. All other parameters are not well constrained by the ND280 data and are applied separately for ND280 and SK interaction modeling. Parameter \| Input Value \| Uncertainty ---\|---\|--- MQEA (GeV) \| 1.21 \| 0.43 xQE1 \| 1.00 \| 0.11 xQE2 \| 1.00 \| 0.30 xQE3 \| 1.00 \| 0.30 xSF \| 0.0 \| 1.0 pF(12C) (MeV/c) \| 217 \| 30 pF(16O) (MeV/c) \| 225 \| 30 MRESA (GeV) \| 1.16 \| 0.11 xCC1π1 \| 1.63 \| 0.43 xCC1π2 \| 1.00 \| 0.40 xNC1π0 \| 1.19 \| 0.43 x1πEν \| off \| on Weff \| 1.0 \| 0.51 xπ−less \| 0.2 \| 0.2 xCCcoh. \| 1.0 \| 1.0 xNCcoh. \| 1.0 \| 0.3 xNCother \| 1.0 \| 0.3 xCCother (GeV) \| 0.0 \| 0.4 xνe/νμ \| 1.0 \| 0.03 Table 6: The parameters used to vary the NEUT cross section model along with the values used in the ND280 fit (input value) and uncertainties prior to the ND280 and SK data fits. ## VI ND280 Neutrino Data We select samples of CC $\nu_{\mu}$ interactions in the ND280 detector, which are fitted to constrain the flux and cross section models, as described in Section VII. CC $\nu_{\mu}$ interaction candidates are divided into two selections, one enhanced in CCQE-like events, and the second consisting of all other CC interactions, which we refer to as the CCnonQE-like selection. While the $\nu_{e}$ flux and interaction models are constrained by the CC $\nu_{\mu}$ data, we also select a sample enhanced in CC $\nu_{e}$ interactions to directly verify the modeling of the intrinsic $\nu_{e}$ rate. ### ND280 simulation The ND280 detector response is modeled with a GEANT4-based Agostinelli _et al._ (2003); Allison _et al._ (107) Monte Carlo (MC) simulation, using the neutrino flux described in Section IV and the NEUT simulation. The MC predictions presented in this section are not calculated with the cross section parameter tuning described in Table 5. Neutrino interactions are generated up to 30 GeV for all flavors from the unoscillated flux prediction, with a time distribution matching the beam bunch structure. The ND280 subdetectors and magnet are represented with a detailed geometrical model. To properly represent the neutrino flux across a wider range of off-axis angles, a separate simulation is run to model neutrino interactions in the concrete and sand which surround ND280. The scintillator detectors, including the FGD, use custom models of the scintillator photon yield, photon propagation including reflections and attenuation, and electronics response and noise. The gaseous TPC detector simulation includes the gas ionization, transverse and longitudinal diffusion of the electrons, propagation of the electrons to the readout plane through the magnetic and electric field, and a parametrization of the electronics response. Further details of the simulation of the individual detectors of ND280 can be found in Refs Abe _et al._ (32); Amaudruz _et al._ (2012). ### $\nu_{\mu}$ candidate selection We select CC $\nu_{\mu}$ interactions by identifying the muons from $\nu_{\mu}N\to\mu^{-}X$ interactions, which may be accompanied by hadronic activity $X$ from the same vertex. Of all negatively charged tracks, we identify the highest momentum track in each event that originates in the upstream FGD (FGD1) and enters the middle TPC (TPC2) as the $\mu^{-}$ candidate. The negatively charged track is identified using curvature and must start inside the FGD1 fiducial volume (FV) that begins 48 mm inward from the edges of FGD1 in $x$ and $y$ and 21 mm inward from the upstream FGD1 edge in $z$. In this analysis we use only selected tracks with a vertex in FGD1, since it provides a homogeneous target for neutrino interactions. To reduce the contribution from neutrino interactions upstream of the FGD1 FV, any tracks which pass through both the upstream TPC (TPC1) and FGD1 are rejected. This also has the consequence of vetoing backward-going particles from the CC interaction vertex, so the resulting selection is predominantly forward-going $\mu^{-}$. The $\mu^{-}$ candidate track energy loss is required to be consistent with a muon. The identification of particles (PID) is based on a truncated mean of measurements of energy loss in the TPC gas, from which a discriminator function is calculated for different particle hypotheses. We apply the discriminator to select muon candidates and reject electron and proton tracks. The TPC PID and TPC performance are described in more detail elsewhere Abgrall _et al._ (41). Events passing the previously described cuts comprise the CC-inclusive sample, and the number of selected events and the MC predictions are listed in Table 7. These data correspond to $2.66\times 10^{20}$ POT. The predictions include a correction for the event pile-up that is not directly modeled by the Monte Carlo simulation of the detector. The pile-up correction takes into account the presence of neutrino interactions in the same beam bunch originating in the sand and material surrounding the detector. The size of this correction ranges between 0.5% and 1% for the different run periods. Of CC $\nu_{\mu}$ interactions in the FGD1 FV, 47.6% are accepted by the CC-inclusive selection, and the resulting selection is 88.1% pure. The largest inefficiency of the CC-inclusive selection is from high angle particles which do not traverse a sufficient distance through the TPC to pass the selection criteria. We divide the CC-inclusive $\nu_{\mu}$ events into two mutually exclusive samples sensitive to different neutrino interaction types: CCQE-like and CCnonQE-like. As the CCQE neutrino interaction component typically has one muon and no pions in the final state, we separate the two samples by requiring the following for the CCQE-like events: * Only one muon-like track in the final state * No additional tracks which pass through both FGD1 and TPC2. * No electrons from muon decay at rest in FGD1 (Michel electron) A Michel electron will typically correspond to a stopped or low energy pion that decays to a muon which stops in FGD1, and is identified by looking for a time-delayed series of hits in FGD1. The Michel electron tagging efficiency is 59%. Events in the CC-inclusive selection which do not pass the CCQE-like selection comprise the CCnonQE-like sample. Example event displays for ND280 events are shown in Fig. 14. <figure><img src="content_image/1304.0841/x19.png"><figcaption>Figure 14: Event displays of example ND280 CCQE-like (a) and CCnonQE-like (b)selected events.</figcaption></figure> The numbers of selected events in the data and nominal prediction for the CCQE-like and CCnonQE-like selections are shown in Table 8. Table 9 shows the composition of the CC, CCQE-like and CCnonQE-like selections according to the generated neutrino interaction categories in the Monte Carlo. The CCQE-like sample contains 40.0% of all CCQE interactions in the FGD1 FV, and CCQE interactions comprise 69.5% of the CCQE-like sample. Fig. 15 shows the distributions of events binned in the muon momentum ($p_{\mu}$) and cosine of the angle between the muon direction and the $z$-axis ($\cos\theta_{\mu}$) for both data and the prediction. In addition, we check the stability of the neutrino interaction rate with a Kolmogorov-Smirnov (KS) test of the accumulated data and find $p$-values of 0.20, 0.12, and 0.79 for the CC-inclusive, CCQE-like and CCnonQE-like samples, respectively. Both CCQE-like and CCnonQE-like samples provide useful constraints on the neutrino flux and neutrino interaction models. The CCQE-like sample includes the dominant neutrino interaction process at the T2K beam peak energy (CCQE) and the CCnonQE-like sample is sensitive to the high energy tail of the neutrino flux, where relatively few CCQE interactions occur. The fit of the flux and cross section models to these data, further described in Section VII, uses two-dimensional $p_{\mu}$ and $\cos\theta_{\mu}$ distributions for the CCQE-like and CCnonQE-like samples. We use a total of 20 bins per each sample, where $p_{\mu}$ is split into 5 bins and $\cos\theta_{\mu}$ is split into 4 bins. The data and the expected number of events for this binning are shown in Table 10. \| Data \| MC ---\|---\|--- Good negative track in FV \| 21503 \| 21939 Upstream TPC veto \| 21479 \| 21906 μ PID \| 11055 \| 11498 Table 7: Number of data and predicted events for the ND280 CC-inclusive selection criteria. \| CCQE-like \| CCnonQE-like ---\|---\|--- \| Data \| MC \| Data \| MC TPC-FGD track \| 6238 \| 6685 \| 4817 \| 4813 Michel electron \| 5841 \| 6244 \| 5214 \| 5254 Table 8: Number of data and predicted events for the ND280 CCQE-like and CCnonQE-like selection criteria, after the CC-inclusive selection has been applied. Event type \| CC-inclusive \| CCQE-like \| CCnonQE-like ---\|---\|---\|--- CCQE \| 44.4 \| 69.5 \| 14.7 CC resonant 1π \| 21.4 \| 14.5 \| 29.6 CC coherent π \| 2.8 \| 1.7 \| 4.0 All other CC \| 18.8 \| 3.7 \| 36.8 NC \| 3.0 \| 1.3 \| 5.1 ¯¯¯νμ \| 0.7 \| 0.2 \| 1.2 out of FV \| 7.8 \| 7.6 \| 8.0 sand interactions \| 1.1 \| 1.6 \| 0.5 Table 9: Breakdown of the three ND280 CC samples by true interaction type as predicted by the MC simulation. <figure><img src="content_image/1304.0841/x21.png"><figcaption>Figure 15: Muon momentum for the CC-inclusive (a), CCQE-like (c), and CCnonQE-like (e) samples. Cosine of the muon angle for the CC-inclusive (b), CCQE-like(d), and CCnonQE-like (f) samples. The errors on the data points are thestatistical errors.</figcaption></figure> CCQE-like sample --- \| pμ (MeV/c) \| 0-400 \| 400-500 \| 500-700 \| 700-900 \| >900 −1<cosθμ≤0.84 \| 854 (807.7) \| 620 (655.6) \| 768 (821.2) \| 222 (255.0) \| 222 (233.0) 0.84<cosθμ≤0.90 \| 110 (107.2) \| 110 (116.3) \| 235 (270.6) \| 133 (153.5) \| 159 (194.7) 0.90<cosθμ≤0.94 \| 62 (69.1) \| 67 (74.0) \| 142 (179.0) \| 90 (121.4) \| 228 (274.6) 0.94<cosθμ≤1.0 \| 92 (95.4) \| 73 (85.4) \| 184 (216.5) \| 160 (174.8) \| 1310 (1339.0) CCnonQE-like sample \| pμ (MeV/c) \| 0-400 \| 400-500 \| 500-700 \| 700-900 \| >900 −1<cosθμ≤0.84 \| 560 (517.9) \| 262 (272.2) \| 418 (400.3) \| 256 (237.8) \| 475 (515.0) 0.84<cosθμ≤0.90 \| 83 (80.3) \| 42 (35.8) \| 83 (80.2) \| 86 (74.8) \| 365 (389.8) 0.90<cosθμ≤0.94 \| 46 (58.6) \| 37 (33.8) \| 60 (63.1) \| 39 (56.4) \| 462 (442.6) 0.94<cosθμ≤1.00 \| 75 (76.6) \| 33 (43.2) \| 91 (93.4) \| 85 (87.2) \| 1656 (1694.7) Table 10: Data (MC) pμ and cosθμ events split in bins as used by the fit described in Section VII at ND280. ### Detector Response Modeling Uncertainties We consider systematic uncertainties on the modeling of the detection efficiency and reconstruction of events which affect: * the overall efficiency for selecting CC interactions * the reconstructed track properties ($p_{\mu}$, $\cos\theta_{\mu}$) * the sample (either CCQE-like or CCnonQE-like) in which the event is placed We estimate uncertainties from each category with a variety of control samples that include beam data, cosmic events and simulated events. The uncertainty on the efficiency for selecting CC $\nu_{\mu}$ interactions is propagated from uncertainties on: the data quality criteria applied to the tracks, track reconstruction and matching efficiencies, PID, and determination of the track curvature. We also consider the uncertainty on the detector mass. The systematic uncertainty on the track momentum determination is from uncertainties on the magnetic field absolute value and field non-uniformity. Small imperfections in the magnetic and electric fields can affect the path of the drift electrons, causing a distorted image of the track and a possible bias in the reconstructed momentum. The size of these distortions is constrained from laser calibration data and MC simulations using magnetic field measurements made prior to detector installation. The overall momentum scale is determined from the magnitude of the magnetic field component transverse to the beam direction, $B_{x}$, which is inferred from the measured magnetic coil current. The momentum resolution is determined in data from studies of tracks which traverse multiple TPCs; the individual momentum calculated for a single TPC can be compared to the momentum determined by nearby TPCs to infer the momentum resolution in data and MC simulation. The primary causes of event migration between the CCQE-like and CCnonQE-like samples are external backgrounds or interactions of pions. External backgrounds in the samples are due to three sources: cosmic rays, neutrino interactions upstream in the surrounding sand and concrete, and neutrino interactions in the ND280 detector outside the FV (out of FV). Interactions from the sand or concrete contribute to the number of tracks in the selected event, which can change a CCQE-like event to a CCnonQE-like event. Interactions that occur outside of the FGD1 FV are about 7.6% of the total selected CC-inclusive sample. Sources include neutrino interactions in FGD1 outside of the FV, or particles produced in interactions downstream of FGD1 that travel backwards to stop in the FGD1 FV. Pion absorption and charge exchange interactions in the FGD material can also reduce the probability that a charged pion produces a track in TPC2, affecting the identification of an event as CCQE-like or CCnonQE-like. The uncertainty on the GEANT4 modeling of pion inelastic scattering is evaluated by comparing the GEANT4 model to pion scattering data. For each source of systematic uncertainty, we generate a $40\times 40$ covariance matrix with entries for each pair of ($p_{\mu}$,$\cos\theta_{\mu}$) bins. These matrices represent the fractional uncertainty on the predicted numbers of events in each ($p_{\mu}$,$\cos\theta_{\mu}$) bin for each error source. The binning used is the same as shown in Table 10, where the first 20 bins correspond to the CCQE-like sample and the second 20 correspond to the CCnonQE-like sample. The total covariance matrix $V_{d}$ is generated by linearly summing the covariance matrices for each of the systematic uncertainties. Fig. 16 shows the bin-to-bin correlations from the covariance matrix, which displays the feature of anti-correlations between bins in the CCQE-like and CCnonQE-like samples arising from systematic error sources, such as the pion absorption uncertainty, that migrate simulated events between samples. Table 11 summarizes the range of uncertainties across the ($p_{\mu}$,$\cos\theta_{\mu}$) bins and the uncertainty on the total number of events. <figure><img src="content_image/1304.0841/x27.png"><figcaption>Figure 16: The bin-to-bin correlation matrix from the systematic covariancematrix for the νμ selected sample at ND280. The bins are ordered by increasingcosθμ in groups of increasing muon momentum (p0 to p4) for the two selections.</figcaption></figure> Systematic error \| Error Size (%) ---\|--- \| Minimum and \| Total fractional \| maximum fractional \| error \| error \| B-Field Distortions \| 0.3 - 6.9 \| 0.3 Momentum Scale \| 0.1 - 2.1 \| 0.1 Out of FV \| 0 - 8.9 \| 1.6 Pion Interactions \| 0.5 - 4.7 \| 0.5 All Others \| 1.2 - 3.4 \| 0.4 Total \| 2.1 - 9.7 \| 2.5 Table 11: Minimum and maximum fractional errors among all the (pμ,cosθμ) bins, including the largest error sources. The last column shows the fractional error on the total number of events, taking into account the correlations between the (pμ,cosθμ) bins. ### Intrinsic $\nu_{e}$ candidate selection We also select a sample of CC $\nu_{e}$ interactions to check the consistency of the predicted and measured intrinsic $\nu_{e}$ rates. The CC $\nu_{\mu}$ selections described earlier provide the strongest constraint on the expected intrinsic $\nu_{e}$ rate, through the significant correlation of the $\nu_{\mu}$ flux to the $\nu_{e}$ flux. However, a CC $\nu_{e}$ selection at the near detector provides a direct and independent measurement of the intrinsic $\nu_{e}$ rate. We select CC $\nu_{e}$ interactions by applying the same criteria as described in Section VI.2, except that the energy loss for the highest momentum negatively charged particle is required to be consistent with an electron instead of a muon, and interactions in FGD2 are used to increase the sample size. For electrons of momenta relevant to T2K, the energy loss is 30–40% larger than for muons at the same momenta, and so electrons and muons are well separated since the TPC energy loss resolution is less than $8$% Abgrall _et al._ (41). In addition, for tracks which reach the downstream ECAL, we use the information from the ECAL to remove events in which the lepton candidate is consistent with a muon. A muon that crosses the ECAL produces a narrow track while an electron releases a large part of its energy, producing an electromagnetic shower. We developed a neural network to distinguish between track-like and shower-like events. For this analysis we select only shower-like events. The total number of selected events in the electron candidate sample is 927. The signal efficiency for selecting CC $\nu_{e}$ interactions in the FGD1 and FGD2 FV is 31.9% with an overall 23.7% purity. For higher momenta the relative purity of the selection increases (42.1% for $p_{e}>300$ ${\mathrm{\,Me\kern-1.0ptV\!/}c}$). The majority of selected $\nu_{e}$ are from kaon decay (80%). The dominant background events ($78\%$ of the total background) are low energy electrons produced by photon conversion in the FGDs, called the $\gamma$ background. The photons come from $\pi^{0}$ decays, where the $\pi^{0}$s are generated in $\nu_{\mu}$ interactions either in the FGD or in the material which surrounds the FGD. A total of $7$% of the remaining background events are misidentified muons coming from $\nu_{\mu}$ interactions. The probability for a muon to be misidentified as an electron is estimated to be less than 1% across most of the relevant momentum range. This probability is determined using a highly pure ($>$99%) sample of muons from neutrino-sand interactions. Finally, background not belonging to the two previous categories is mainly due to protons and pions produced in NC and CC $\nu_{\mu}$ interactions in the FGD. Fig. 17 (a) shows the momentum distribution of the highest momentum track with negative charge for each event in the selected electron candidate sample. We estimate the uncertainties on the detector response modeling for the electron candidate sample in the same manner as described in Section VI.3, with additional uncertainties considered for the FGD2 interactions in the selection, and for electron-PID selection. The total detector response systematic uncertainty on the electron candidate sample is 5.7%, with the TPC PID (3.8%) uncertainty as the largest. The rate of intrinsic $\nu_{e}$ interactions is determined with a likelihood fit to reconstructed momenta of electron candidate events. To constrain the large background from photons, a control sample of positron (positive charge, electron PID tracks) candidates is used. Fig. 17 (b) shows the momentum distribution of candidate positrons. The sample is composed of positrons at lower energies and protons at higher energies. We simultaneously fit the electron and positron candidate samples to determine the photon background and $\nu_{e}$ signal rate normalizations. The misidentified muon background component is fixed according to the estimate from the pure muon control sample, and other smaller background sources are fixed according to the nominal predictions. Neutrino flux, neutrino cross section, and detector response uncertainties are included in the likelihood fit. The inferred rate of CC $\nu_{e}$ events in data from the likelihood fit normalized by the prediction is $0.88\pm 0.10(stat.)\pm 0.15(syst.)$. The measured $\nu_{e}$ rate at the near detector is consistent with the prediction within systematic uncertainties. The neutrino flux and cross section systematic uncertainties are the dominant contributions to the total systematic error on the $\nu_{e}$ rate. In Section VII we show the $\nu_{e}$ rate after the flux and cross section parameters are tuned by the fit to the CC $\nu_{\mu}$ data. <figure><img src="content_image/1304.0841/x28.png"><figcaption>Figure 17: (a) Momentum distribution of the highest momentum track withnegative charge for each event in the electron candidate sample at ND280. Theinset shows the region with momentum ≥300 MeV/c. (b) Momentum distribution ofthe highest momentum track with positive charge for each event of the positroncandidate control sample. The “Other Backgrounds” component is mainly due toprotons and pions from NC and CC νμ interactions in the FGD. The energy lossof positrons and protons (pions) is similar at p≈1000 MeV/c (200 MeV/c),resulting in the presence of these particles in the positron candidate sample.</figcaption></figure> ## VII ND280 Constraint on the Neutrino Flux and Cross Section Models The rate of neutrino interactions measured at the ND280 detector has power to constrain the neutrino flux and interaction models used to predict the $\nu_{e}$ candidate event rate at the SK detector. The predicted SK $\nu_{e}$ signal and neutral current background both depend directly on the unoscillated $\nu_{\mu}$ flux, while the intrinsic $\nu_{e}$ background depends on the $\nu_{e}$ flux. As shown in Fig. 9, both the SK $\nu_{\mu}$ and $\nu_{e}$ flux predictions are correlated to the ND280 $\nu_{\mu}$ flux prediction through the underlying data and assumptions applied in the flux calculation. Both the SK $\nu_{e}$ signal and intrinsic $\nu_{e}$ background also depend on the charged current interaction model. Hence, a fit to the CC-inclusive events from ND280 can constrain flux and cross section nuisance parameters relevant to the SK prediction. We fit the near detector CCQE-like and CCnonQE-like $\nu_{\mu}$ data to determine tuned values of the $\nu_{\mu}$ and $\nu_{e}$ flux parameters and cross section model parameters, described in Sections IV and V respectively. The fit includes the marginalization of nuisance parameters describing uncertainties in the simulation of the detector response and parameters describing parts of the neutrino interaction model that are not correlated for ND280 and SK selections. The tuned parameters are then applied to predict the $\nu_{e}$ signal and background interactions at SK. The fit also incorporates constraints on the flux and cross section models determined independently from the ND280 data constraint to properly propagate all constraints to the SK event rate predictions. ### ND280 likelihood The fit maximizes a likelihood that includes the binned likelihood of the ND280 data and the prior constraints on the flux model, the interaction model, and the detector response model: \[\begin{split}&\mathcal{L}_{ND}(\vec{b},\vec{x},\vec{d}\|N^{d}_{i}) =\pi_{flux}(\vec{b})\pi_{xsec}(\vec{x})\pi_{det}(\vec{d})\times\\ &\prod_{i=1}^{N_{bins}}\frac{[N^{p}_{i}(\vec{b},\vec{x},\vec{d})] ^{N^{d}_{i}}e^{-N^{p}_{i}(\vec{b},\vec{x},\vec{d})}}{N^{d}_{i}!}.\end{split}\] (14) $\pi_{flux}(\vec{b})$, $\pi_{xsec}(\vec{x})$, $\pi_{det}(\vec{d})$ are multivariate normal distributions that are functions of the flux ($\vec{b}$), neutrino cross section ($\vec{x}$) and detector response ($\vec{d}$) nuisance parameters. These functions encode the prior constraints on the nuisance parameters and depend on the nominal parameter values and the parameter errors or covariance matrices described in previous sections. The likelihood includes the product of the Poisson probabilities for the $N_{bins}=40$ bins of the CCQE-like and CCnonQE-like selections. For each bin the predicted number of events, $N^{p}_{i}(\vec{b},\vec{x},\vec{d})$, is evaluated based on the values of the nuisance parameters, and compared to the measurement, $N^{d}_{i}$. To obtain fit results that more closely follow a $\chi^{2}$ distribution Baker and Cousins (1984), we define the likelihood ratio: \[\mathcal{L}_{ratio}=\frac{\mathcal{L}_{ND}(\vec{b},\vec{x},\vec{d}\|N^{d}_{i})} {\mathcal{L}_{ND}(\vec{b}^{0},\vec{x}^{0},\vec{d}^{0},N^{p}_{i}=N^{d}_{i}\|N^{d }_{i})}\] (15) Here the denominator is the likelihood evaluated with $N^{p}_{i}$ set equal to $N^{d}_{i}$ and the nuisance parameters set to their nominal values: $\vec{b}^{0}=1$, $\vec{x}^{0}$, $\vec{d}^{0}=1$; both $\vec{b}$ and $\vec{d}$ have nominal values of 1. The quantity that is minimized is $-2\textrm{ln}(\mathcal{L}_{ratio})$: \[\begin{split}&-2\textrm{ln}(\mathcal{L}_{ratio})=\\ & 2\sum_{i=1}^{N_{bins}}N^{p}_{i}(\vec{b},\vec{x},\vec{d})-N^{d}_ {i}+N^{d}_{i}\textrm{ln}[N^{d}_{i}/N^{p}_{i}(\vec{b},\vec{x},\vec{d})]\\ &+\sum_{i=1}^{N_{b}}\sum_{j=1}^{N_{b}}(1_{i}-b_{i})(V_{b}^{-1})_{ i,j}(1-b_{j})\\ &+\sum_{i=1}^{N_{x}}\sum_{j=1}^{N_{x}}(x^{0}_{i}-x_{i})(V^{-1}_{x })_{i,j}(x^{0}_{j}-x_{j})\\ &+\sum_{i=1}^{N_{bins}}\sum_{j=1}^{N_{bins}}(1-d_{i})(V_{d}(\vec{ b},\vec{x})^{-1})_{i,j}(1-d_{j})\\ &+\textrm{ln}\left(\frac{\|V_{d}(\vec{b},\vec{x})\|}{\|V_{d}(\vec{b} ^{0},\vec{x}^{0})\|}\right).\end{split}\] (16) The predicted number of events in each observable bin, $N^{p}(\vec{b},\vec{x},\vec{d})$ depends on the value of the $\vec{b}$, $\vec{x}=(\vec{x}^{norm},\vec{x}^{resp})$, and $\vec{d}$ nuisance parameters: \[N^{p}_{i}=d_{i}\sum_{j}^{E_{\nu}bins}\sum_{k}^{Int.modes}{b_{j}x^{norm}_{k}(E_ {j})w_{i,j,k}(\vec{x}^{resp})T^{p}_{i,j,k}}.\] (17) The $T^{p}_{i,j,k}$ are the nominal Monte Carlo templates that predict the event rate for bins in the observables, $i$, true neutrino energy, $j$, and neutrino interaction modes, $k$. The $\vec{b}$ parameters multiply the flux prediction in bins of true neutrino energy. The detector response parameters, $\vec{d}$, multiply the expected number of events in each observable ($p_{\mu}$,$\cos\theta_{\mu}$) bin. The $\vec{x}$ are included in the prediction in one of two ways. The $x_{k}^{norm}$ are cross section parameters that multiply the neutrino cross section normalization for a given true neutrino energy bin and one of the $k$ interaction modes. We model the effect of the remaining cross section parameters, $\vec{x}^{resp}$, with pre-calculated response functions, $w_{i,j,k}(\vec{x}^{resp})$, that have a value of 1 for the nominal parameter settings and can have a non-linear dependence on the cross section parameters. The remaining terms in Eq. 16 correspond to the prior constraints on the flux, cross section and detector response models discussed in earlier sections. $V_{b}$ is the prior fractional covariance matrix, corresponding to Figures 8 and 9. The covariances of flux predictions at ND280 and SK are included so that the fit to ND280 data can constrain the SK flux parameters. The prior covariance matrix for the neutrino interaction parameters, $V_{x}$, is diagonal for most parameters with entries corresponding to the errors listed in Table 6. Correlations are included for the parameters constrained by the fit to MiniBooNE single pion data. The $V_{d}$ fractional covariance matrix, with correlations shown in Fig. 16, incorporates the simulated detector efficiency and reconstruction uncertainties, final state interaction errors and Monte Carlo statistical errors. The final term in the likelihood is present since the Monte Carlo statistical errors included in $V_{d}$ depend on the $\vec{b}$ and $\vec{x}$ parameters through the weights applied to the simulated events. Since $V_{d}$ is not constant, the determinant from the multivariate normal distribution, $\pi_{det}(\vec{d})$, cannot be dropped from the $-2\textrm{ln}(\mathcal{L}_{ratio})$. ### Parameter propagation and marginalization This fitting method extrapolates the ND280 constraint on the neutrino flux and interaction model to the far detector prediction through the simultaneous variation of ND280 and SK flux parameters, and the constraint on the common interaction model parameters. After the $-2\textrm{ln}(\mathcal{L}_{ratio})$ is minimized, we apply a subset of the fitted parameter values to the calculation of the expected $\nu_{e}$ candidate rate at SK. The subset of parameters which are substantially constrained by the ND280 data sets and are also relevant to the event rate prediction at SK are listed in Table 12. Since they are not used to calculate the predicted event rates at SK, the flux parameters for ND280, nuclear model-dependent cross section parameters, and detector response systematic parameters are marginalized by integrating out their dependence in $-2\textrm{ln}(\mathcal{L}_{ratio})$ under the assumption of a quadratic dependence near the minimum. The remaining cross section parameters do not affect the SK event prediction substantially and these are also marginalized. ### ND280 fit results The resulting ($p_{\mu}$,$\cos\theta_{\mu}$) distributions from the fit to the ND280 samples are shown in Fig. 18. We evaluate the post-fit agreement between model and data by generating 2000 pseudo-experiments with statistical and systematic variations, and fitting them to obtain the minimum $-2\textrm{ln}(\mathcal{L}_{ratio})$ value for each pseudo-experiment. The distribution of these values resembles a $\chi^{2}$ distribution of 41 degrees of freedom. Thus the value $[-2\textrm{ln}(\mathcal{L}_{ratio})]_{min}=29.7$ from the fit to data indicates that the data are consistent with the prediction within the prior uncertainties assigned for the neutrino flux model, neutrino interaction model, and detector response model. <figure><img src="content_image/1304.0841/x30.png"><figcaption>Figure 18: The fitted pμ,cosθμ bins from the ND280 CCQE-like (left) andCCnonQE-like (right) samples. All values in the plot are divided by the shownbin width. The pμ>900 MeV/c bin additionally contains the overflow bin, and isnormalized by a bin width of 1100 MeV/c. The prediction prior to the fit usesthe modifications to the NEUT model parameters derived from fits of theMiniBooNE single pion data.</figcaption></figure> The propagated neutrino flux and cross section parameter values prior to and after the fit are listed in Table 12. The fit decreases the flux prediction near the spectrum peak to improve agreement with the data. In addition to modifying the parameter central values and uncertainties, the fit also sets the correlations between parameters. Prior to the fit, the flux and cross section model parameters have no correlation, but the fit introduces anti-correlations, as shown in Fig. 19. The anti-correlations arise because the event rate depends on the product of the neutrino flux and the neutrino interaction cross section. Parameter \| Prior Value \| Fitted Value ---\|---\|--- νμ 0.0-0.4 GeV \| 1.00±0.12 \| 0.98±0.09 νμ 0.4-0.5 GeV \| 1.00±0.13 \| 0.99±0.10 νμ 0.5-0.6 GeV \| 1.00±0.12 \| 0.98±0.09 νμ 0.6-0.7 GeV \| 1.00±0.13 \| 0.93±0.08 νμ 0.7-1.0 GeV \| 1.00±0.14 \| 0.84±0.08 νμ 1.0-1.5 GeV \| 1.00±0.12 \| 0.86±0.08 νμ 1.5-2.5 GeV \| 1.00±0.10 \| 0.91±0.08 νμ 2.5-3.5 GeV \| 1.00±0.09 \| 0.95±0.07 νμ 3.5-5.0 GeV \| 1.00±0.11 \| 0.98±0.08 νμ 5.0-7.0 GeV \| 1.00±0.15 \| 0.99±0.11 νμ >7.0 GeV \| 1.00±0.19 \| 1.01±0.15 ¯νμ 0.0-1.5 GeV \| 1.00±0.12 \| 0.95±0.10 ¯νμ >1.5 GeV \| 1.00±0.11 \| 0.95±0.10 νe 0.0-0.5 GeV \| 1.00±0.13 \| 0.96±0.10 νe 0.5-0.7 GeV \| 1.00±0.12 \| 0.96±0.10 νe 0.7-0.8 GeV \| 1.00±0.14 \| 0.96±0.11 νe 0.8-1.5 GeV \| 1.00±0.10 \| 0.94±0.08 νe 1.5-2.5 GeV \| 1.00±0.10 \| 0.97±0.08 νe 1.5-4.0 GeV \| 1.00±0.12 \| 0.99±0.09 νe >4.0 GeV \| 1.00±0.17 \| 1.01±0.13 ¯νe 0.0-2.5 GeV \| 1.00±0.19 \| 0.97±0.18 ¯νe >2.5 GeV \| 1.00±0.14 \| 1.02±0.11 MQEA (GeV) \| 1.21±0.45 \| 1.33±0.20 MRESA (GeV) \| 1.16±0.11 \| 1.15±0.10 xQE1 \| 1.00±0.11 \| 0.96±0.09 xCC1π1 \| 1.63±0.43 \| 1.61±0.29 xNC1π01 \| 1.19±0.43 \| 1.19±0.40 Table 12: Prior and fitted values and uncertainties of the propagated neutrino flux and cross section model parameters. <figure><img src="content_image/1304.0841/x31.png"><figcaption>Figure 19: The neutrino flux and cross section parameter correlations before(a) and after (b) the fit to the ND280 data. The flux parameters are orderedby increasing energy with the binning listed in Table 12 (the correlations ofthe antineutrino flux parameters are not shown in this figure). The crosssection parameter ordering is: MQEA, MRESA, CCQE low energy normalization,CC1π low energy normalization and NC1π0 normalization.</figcaption></figure> ### Consistency checks with ND280 data We perform a consistency check of the fit results by applying the fitted parameters to the ND280 MC simulation and investigating the data and predicted rates in more finely binned kinematic distributions. Fig. 20 shows the level of agreement in the muon momentum and angle distributions of the CCQE and CCnonQE-like samples before and after the fit constraint to the flux and cross section models are applied. The fitted flux and cross section models show improved agreement with the data. <figure><img src="content_image/1304.0841/x33.png"><figcaption>Figure 20: Comparisons of the pμ (left column) and cosθμ (right column)distributions for CCQE-like νμ selected events in (a) and (b) and CCnonQE-likeνμ selected events in (c) and (d). The solid line represents the NEUT nominalprediction and the hatched region represents the post-fit MC prediction. Thedots are the data events. Below each graph, the data/MC ratio is shown forboth the NEUT nominal prediction (empty triangle) and post-fit MC prediction(full triangle). The error on the points is the statistical error on the data.</figcaption></figure> We also apply the fitted flux and cross section parameters to the ND280 CC $\nu_{e}$ simulation. Adopting the same analysis as in Section VI.4 while using the fitted cross section and flux parameters, we measure the ratio of inferred to predicted CC $\nu_{e}$ rate to be $0.91\pm{}0.10\mathrm{(stat.)}\pm{}0.10\mathrm{(syst.)}$. The CC $\nu_{e}$ rate remains consistent within the reduced systematic uncertainties after tuning. To check the modeling of NC$\pi^{0}$ production, we measure the rate of single $\pi^{0}$ with the P0D detector using a data set corresponding to $8.55\times 10^{19}$ POT. The ratio of the measured to the predicted rate is found to be $0.84\pm{}0.16\mathrm{(stat.)}\pm{}0.18\mathrm{(syst.)}$. When normalized to the corresponding ratio from the ND280 CC $\nu_{\mu}$ selection, we measure a ratio of $0.81\pm{}0.15\mathrm{(stat.)}\pm{}0.14\mathrm{(syst.)}$, indicating that the predicted rate is consistent with the measured rate within errors. ## VIII SK Electron Neutrino Selection For a non-zero value of $\theta_{13}$, we expect an oscillated $\nu_{\mu}\rightarrow\nu_{e}$ flux with a peak oscillation probability near $600$ MeV at the SK detector. To detect the oscillated $\nu_{e}$, we select SK events with a single electron-like Cherenkov light ring, providing a sample that is enhanced in CCQE $\nu_{e}$ interactions. Additional cuts are applied to reduce the backgrounds from intrinsic $\nu_{e}$ contamination of the beam and $\pi^{0}$ background. The selection is described here. ### The SK detector simulation We simulate the predicted event distributions at the far detector with the neutrino flux prediction up to 30 GeV, the NEUT cross section model, and a GEANT3-based detector simulation. The $\nu_{e}$ signal events from $\nu_{\mu}\to\nu_{e}$ oscillation are produced using the predicted $\nu_{\mu}$ spectrum without oscillations, and the $\nu_{e}$ cross section; oscillations probabilities are applied after the simulation. Additionally, the intrinsic $\nu_{\mu}$, $\overline{\nu}_{\mu}$, $\nu_{e}$ and $\overline{\nu}_{e}$ components of the beam are generated from the intrinsic flux predictions without oscillations. SKDETSIM, a GEANT3-based simulation of the SK detector, simulates the propagation of particles produced in the neutrino interactions in the SK detector. We use the GCALOR physics package to simulate hadronic interactions in water since it successfully reproduces pion interaction data around 1 GeV. For pions with momentum below 500 MeV, however, we use custom routines based on the cascade model used by NEUT to simulate interactions of final state hadrons. SKDETSIM models the propagation of light in water, considering absorption, Rayleigh scattering, and Mie scattering as possible interactions. The parameters employed in the models of these processes have been tuned using a number of laser calibration sources Fukuda _et al._ (2003). Example event displays for simulated SK events are shown in Fig. 21. <figure><img src="content_image/1304.0841/x37.png"><figcaption>Figure 21: Example event displays for the SK simulation of a) νμ CCQE (singlewell-defined ring from the muon), b) νe CCQE (single diffuse ring from theelectron) and c) νμ NC1π0 interactions (two diffuse rings from the π0 →γγdecay). The images show the detected light pattern at the ID wall, with thecylindrical SK detector shown as a flat projection. The color indicates theamount of charge detected by the PMT, with purple dots corresponding to theleast amount of charge, and red the most.</figcaption></figure> As a final step, we scale the predicted events according to the constrained flux and cross section models from the fit to the ND280 $\nu_{\mu}$ CC-inclusive data, and according to the oscillation probability. The three-neutrino oscillation probability, including matter effects, is calculated for each event with the parameter values shown in Table 13, unless otherwise noted. Parameter \| Value ---\|--- Δm221 \| 7.6×10−5 eV2 \|Δm232\| \| 2.4×10−3 eV2 sin2θ12 \| 0.32 sin22θ23 \| 1.0 δCP \| 0 Mass hierarchy \| Normal ν travel length \| 295 km Earth matter density \| 2.6 g/cm3 Table 13: Default neutrino oscillation parameters and earth matter density used for the MC prediction. ### Neutrino event selection We select fully contained (FC) events, which deposit all of their Cherenkov light inside the SK inner detector (ID), by applying the following selection criteria. First, any photomultiplier tubes (PMTs) which register sufficient charge, a “PMT hit”, in the outer detector (OD) are associated with other nearby PMT hits to form clusters. Events with greater than $15$ hits in the highest charge OD cluster are rejected. Second, most of the low energy (LE) events are removed by requiring that the total charge from ID PMT hits in a 300 ns time window must be above 200 photoelectrons (p.e.), corresponding to visible energy, $E_{vis}$, above 20 MeV. Visible energy is defined as the energy of an electromagnetic shower that produces the observed amount of Cherenkov light. In order to remove events caused by radioactivity very close to the PMT, a third cut removes events in which a single ID PMT hit has more than half of the total charge in a 300 ns time window. The final FC selection cut rejects events with ID photomultipliers which produced light because of a discharge around the dynode, called “flasher” events. The cut identifies flasher events from their timing distribution, which is much broader than neutrino events, and from a repeating pattern of light in the detector. However, neutrino events are sometimes misidentified as flasher events when the neutrino interaction vertex is close to the ID wall. There have been a total of 8 events that have been rejected by the flasher cut during all run periods. From event time information and visual inspections, it is clear that all eight events are induced by beam neutrino interactions. The predicted number of rejected beam events from this cut is 3.71 events; the probability to observe 8 or more events when 3.71 are expected is 3.6%. All eight events have vertices close to the ID wall, and would be rejected by the fiducial cut. We define the quantity $\Delta T_{0}$, which is the timing of the event relative to the leading edge of the spill, accounting for the travel time of the neutrino from production to detection. Fig. 22 shows the $\Delta T_{0}$ distribution of all FC, OD and LE events within $\pm 500\,\mu$s of the beam arrival time; the spill duration is about $5\,\mu$s. A clear peak at $\Delta T_{0}=0$ is seen for the FC sample. We observe five FC events outside of the $5\,\mu$s spill window. The expected number of such out-of-time FC events, mainly low energy events and atmospheric neutrino events, is estimated to be 3.3 from data collected when the beam is not present. Fig. 23 shows the $\Delta T_{0}$ distribution of FC events within the spill window. We correct the $\Delta T_{0}$ of each event to account for the position of the neutrino interaction vertex and the photon propagation time from the interaction vertex to the PMTs. The far detector event timing clearly exhibits the eight bunch beam timing structure. The eight dotted vertical lines in the figure represent the 8 bunch centers at intervals of 581 ns from a fit to the observed FC event timing. The RMS value of the residual time distribution between each FC event and the closest of the fitted bunch center times is about 25 ns. <figure><img src="content_image/1304.0841/x38.png"><figcaption>Figure 22: ΔT0 distribution of all FC, OD and LE events observed in the±500μs T2K windows. The OD histogram is stacked on the FC histogram, and theLE histogram is stacked on the OD and FC histograms.</figcaption></figure> <figure><img src="content_image/1304.0841/x39.png"><figcaption>Figure 23: ΔT0 distribution of FC events zoomed in on the spill time observedduring T2K Run 1+2 and Run 3. The eight dotted vertical lines represent the581 ns interval bunch center position fitted to the observed FC event times.</figcaption></figure> We require the $\Delta T_{0}$ for selected FC events to be between $-0.2$ $\mu$s to $10$ $\mu$s. We observe 240 such in-time fully contained events. We extract a fully contained sample within the fiducial volume (FCFV) by further requiring $E_{\rm vis}$ to be above 30 MeV and the reconstructed vertex be 2 m away from the ID wall. We observe 174 such FCFV events, while the expected accidental contamination from events unrelated to the beam, mostly atmospheric neutrino interactions, is calculated to be 0.005 events. CC $\nu_{e}$ interactions ($\nu_{e}N\to e^{-}X$) are identified in SK by detecting a single, electron-like ring; at the energy of the T2K neutrino beam, most of the produced particles other than the electron are below Cherenkov threshold or do not exit the nucleus. The main backgrounds are intrinsic $\nu_{e}$ contamination in the beam and NC interactions with a misidentified $\pi^{0}$. The analysis relies on the well-established reconstruction techniques developed for other data samples in SK Ashie _et al._ (2005). The single, electron-like ring selection criteria are unchanged from our previous measurement of electron neutrino appearance Abe _et al._ (21), and were determined from MC studies before data-taking commenced. We select CC $\nu_{e}$ candidate events which satisfy the following criteria: 1. The event is fully contained in the ID and the reconstructed vertex is within the fiducial volume (FCFV) 2. There is only one reconstructed ring 3. The ring is electron-like 4. The visible energy, $E_{\rm vis}$, is greater than 100 MeV 5. There is no Michel electron 6. The event’s invariant mass is not consistent with a $\pi^{0}$ mass 7. The reconstructed neutrino energy, $E_{\nu}^{\rm rec}$, is less than 1250 MeV The $E_{\rm vis}$ cut removes low energy NC interactions and electrons from the decay of unseen muons and pions, such as cosmic muons outside the beam time window or muons below Cherenkov threshold. A Michel electron is an electron from muon decay which is identified by looking for a time-delayed ID-PMT hit peak after the primary neutrino interaction. In order to reduce NC $\pi^{0}$ events, we utilize a special fitter which reconstructs each event with a two photon ring hypothesis. It searches for the direction and energy of the second ring which maximizes the likelihood based on the light pattern of the event Barszczak (2005). Fig. 24 shows the invariant mass $M_{\rm inv}$ distribution of the two photon rings for the data and simulation. As shown in the figure, the NC background component peaks around the $\pi^{0}$ invariant mass, hence events with $M_{\rm inv}>105$ MeV/$c^{2}$ are cut. Finally, the energy of the parent neutrino is computed assuming CCQE kinematics and neglecting Fermi motion as follows: \[E_{\nu}^{\rm rec}=\frac{m_{p}^{2}-(m_{n}-E_{b})^{2}-m_{e}^{2}+2(m_{n}-E_{b})E_ {e}}{2(m_{n}-E_{b}-E_{e}+p_{e}\cos\theta_{e})},\] (18) where $m_{p}$ is the proton mass, $m_{n}$ the neutron mass, and $E_{b}=27$ MeV is the binding energy of a nucleon inside a ${}^{16}$O nucleus. $E_{e}$, $p_{e}$, and $\theta_{e}$ are the reconstructed electron energy, momentum, and angle with respect to the beam direction, respectively. We select $E_{\nu}^{\rm rec}<1250$ MeV since the signal at high energy is expected to be small for the atmospheric mass splitting, and the intrinsic $\nu_{e}$ background is dominant in this region, as shown in Fig. 25. <figure><img src="content_image/1304.0841/x40.png"><figcaption>Figure 24: Distribution of invariant mass Minv when each event is forced tobe reconstructed as two photon rings. The data are shown as points with errorbars (statistical only) and the MC predictions are in shaded histograms. Thelast bin shows overflow entries. The arrow shows the selection criterionMinv<105 MeV/c2.</figcaption></figure> <figure><img src="content_image/1304.0841/x41.png"><figcaption>Figure 25: Distribution of the reconstructed neutrino energy spectrum of theevents which pass all νe appearance signal selection criteria with theexception of the energy cut. The data are shown as points with error bars(statistical only) and the MC predictions are in shaded histograms. The arrowshows the selection criterion Erecν<1250 MeV.</figcaption></figure> \| Data \| MC total \| CC νμ \| CC νe \| NC \| CC νμ→νe ---\|---\|---\|---\|---\|---\|--- (0) interaction in FV \| n/a \| 311.4 \| 158.3 \| 8.3 \| 131.6 \| 13.2 (1) fully contained in FV \| 174 \| 180.5 \| 119.6 \| 8.0 \| 40.2 \| 12.7 (2) single ring \| 88 \| 95.7 \| 68.4 \| 5.1 \| 11.4 \| 10.8 (3) e-like \| 22 \| 26.4 \| 2.7 \| 5.0 \| 8.0 \| 10.7 (4) Evis>100 MeV \| 21 \| 24.1 \| 1.8 \| 5.0 \| 6.9 \| 10.4 (5) no delayed electron \| 16 \| 19.3 \| 0.3 \| 4.0 \| 5.9 \| 9.1 (6) not π0-like \| 11 \| 13.0 \| 0.09 \| 2.8 \| 1.6 \| 8.5 (7) Erecν<1250 MeV \| 11 \| 11.2 \| 0.06 \| 1.7 \| 1.2 \| 8.2 Table 14: Event reduction for the νe appearance search at the far detector. After each selection criterion is applied, the numbers of observed and MC expected events of CC νμ, intrinsic CC νe, NC, and the CC νe signal, are given. All MC samples include three-neutrino oscillations for sin22θ13=0.1, δCP=0, and normal mass hierarchy. \| Data \| MC total \| CC νμ \| CC νe \| NC \| CC νμ→νe ---\|---\|---\|---\|---\|---\|--- (0) interaction in FV \| n/a \| 299.0 \| 158.5 \| 8.6 \| 131.6 \| 0.3 (1) fully contained in FV \| 174 \| 168.5 \| 119.8 \| 8.2 \| 40.2 \| 0.3 (2) single ring \| 88 \| 85.4 \| 68.5 \| 5.3 \| 11.4 \| 0.2 (3) e-like \| 22 \| 16.1 \| 2.7 \| 5.2 \| 8.0 \| 0.2 (4) Evis>100 MeV \| 21 \| 14.1 \| 1.8 \| 5.2 \| 6.9 \| 0.2 (5) no delayed electron \| 16 \| 10.6 \| 0.3 \| 4.2 \| 5.9 \| 0.2 (6) not π0-like \| 11 \| 4.8 \| 0.09 \| 2.9 \| 1.6 \| 0.2 (7) Erecν<1250 MeV \| 11 \| 3.3 \| 0.06 \| 1.8 \| 1.2 \| 0.2 Table 15: Same as Table 14 but with MC prediction for sin22θ13=0. The numbers of observed events after each selection criterion, and the MC predictions for $\sin^{2}2\theta_{13}=0.1$ and $\sin^{2}2\theta_{13}=0$, are shown in Tables 14 and 15, respectively. Eleven events remain in the data after all $\nu_{e}$ appearance signal selection criteria are applied. Using the MC simulation, we estimate the $\nu_{e}$ appearance signal efficiency in the SK FV to be 62%, while the rejection rates for CC $\nu_{\mu}$ +$\overline{\nu}_{\mu}$, intrinsic CC $\nu_{e}$ +$\overline{\nu}_{e}$, and NC are $>99.9$%, 80%, and 99%, respectively. More than half of the remaining background is due to intrinsic CC $\nu_{e}$ interactions (57% for $\sin^{2}2\theta_{13}=0.1$). The fraction of CCQE events in the CC $\nu_{e}$ signal and background are 80% and 65%, respectively. NC interactions constitute 41% of the total surviving background, 80% of which are due to $\pi^{0}$ mesons and 6% of which originate from NC single photon ($\Delta\to N\gamma$) production. Additional checks of the eleven data events are performed. From visual inspection, it appears that all events have only a single, electron-like Cherenkov ring. A KS test of the observed number of $\nu_{e}$ candidate events as a function of accumulated POT is compatible with the normalized event rate being constant ($p$-value = 0.48) as shown in Fig. 26. Fig. 27 shows the $(x,y)$ and $(r^{2},z)$ distributions of the reconstructed vertices of observed $\nu_{e}$ candidate events. As we previously reported, the first 6 candidate events were clustered near the edge of the FV in the upstream beam direction. We observe no such clustering in the newly observed 5 events (pink points in the figure). All event vertices are $x<0$ in the SK coordinate system which is not related to the beam direction. Other T2K neutrino selections with larger event samples, such as the CC $\nu_{\mu}$ selection, populate the entire $x$ and $y$ region. Figure 28 shows the distribution of distance from the ID wall to the vertex along the beam direction for events passing all $\nu_{e}$ selection cuts except the FV cut. A KS test to this distribution yields a $p$-value of 0.06. In addition, a dedicated selection of penetrating particles produced in upstream, out-of-FV neutrino interactions shows no indication of an excess. <figure><img src="content_image/1304.0841/x42.png"><figcaption>Figure 26: The cumulative number of observed νe candidate events as afunction of accumulated POT. The vertical dashed lines separate the threerunning periods, and the dotted line indicates the horn current change duringRun 3. The solid line indicates a hypothesis of constant event rate.</figcaption></figure> <figure><img src="content_image/1304.0841/x43.png"><figcaption>Figure 27: a) Two-dimensional (x,y) distribution of the reconstructed vertexpositions of observed νe candidate events. b) Two-dimensional (r2=x2+y2,z)distribution of the reconstructed vertex positions of the observed νecandidate events. The arrow indicates the neutrino beam direction and thedashed line indicates the fiducial volume boundary. Black markers are eventsobserved during Run 1+2, and pink markers are events from Run 3. Open squaresrepresent events which passed all the νe selection cuts except for thefiducial volume cut.</figcaption></figure> <figure><img src="content_image/1304.0841/x45.png"><figcaption>Figure 28: Distribution of events which pass νe selection cuts except for theFV cut as a function of the distance from the ID wall to the vertex,calculated along the beam direction. The solid line indicates the expecteddistribution for signal (sin22θ13=0.1) and background, and the backgroundprediction is shown with the dashed line. The hatch-filled histogram shows thesubset of background whose true vertex is outside the ID.</figcaption></figure> ### SK efficiency and reconstruction uncertainties We have studied the systematic uncertainties on the simulation of the SK event selection efficiency and reconstruction using comparisons of data and MC control samples. The error on the FC event selection is estimated to be 1%, with a dominant contribution from the flasher event rejection. We evaluate the flasher rejection uncertainty from the difference in the cut efficiency between the atmospheric neutrino data and MC simulation. We estimate the uncertainty on the fiducial volume definition to be 1% by comparing the reconstructed vertex distributions of observed and simulated cosmic-ray muons which have been independently determined to have stopped inside the ID. We estimate an energy scale uncertainty of 2.3% from comparisons of distributions between cosmic-ray data and simulated samples. These samples include the reconstructed momentum spectrum of electrons from the decay of cosmic ray muons, cosmic-ray muons which stop in SK and have similar energies to the T2K neutrino events, and the reconstructed mass of neutral pions from atmospheric neutrino interactions. The error on the number of $\nu_{e}$ candidate events due to the uncertainty on the delayed, decay-electron tagging efficiency is 0.2%. We evaluate this uncertainty from a comparison of the tagging efficiency between cosmic-ray stopped muon data and MC samples. The remaining uncertainties on the detection efficiency are evaluated in categories corresponding to the particles exiting the target nucleus. The “CC $\nu_{e}$ single electron” category is comprised of interactions where a single electron is emitted and is the only detectable particle in the final state. The “CC $\nu_{e}$ other” category includes all other CC $\nu_{e}$ interactions not in the CC $\nu_{e}$ single electron category. NC events are also classified based on the particle type which exits the nucleus. The “NC single $\pi^{0}$” category includes events with only one $\pi^{0}$ in the detector. The topological light pattern of the rings provides the information needed to construct quantities used in the selection: the number of rings (cut 2), particle identification (cut 3) and the invariant mass (cut 6). We evaluate the systematic error on the efficiency of each of the three topological cuts on the selection with a fit to SK atmospheric neutrino data using MC simulation-based templates. We create two control samples in the SK atmospheric neutrino data set which are sensitive to CC $\nu_{e}$ single electron and CC $\nu_{e}$ other event types. The $\nu_{e}$ enriched control samples pass the FCFV, $E_{\rm vis}>100$ MeV criteria; however the number of decay electrons in the event is used to separate QE-like (single ring) from nonQE-like (multiple rings) instead of the ring-counting algorithm. Each control sample is divided into one “core” sub-sample, which passes the three topological cuts, and three “tail” sub-samples, where events have failed one of the three topological cuts. The sub-samples are further divided into 17 bins (labeled with index $i$) in $p_{e}$ and $\theta_{e}$, the reconstructed electron momentum and angle with respect to the beam direction, so that we can evaluate the dependence of the systematic errors on these kinematic variables. The expected number of events in all sub-samples depends on the efficiency of each topological cut, $\vec{\epsilon}=\{\epsilon_{1ring},\epsilon_{PID},\epsilon_{inv.mass}\}$, and parameters which represent systematic uncertainties on the event rate, $\vec{\alpha}$. The $\vec{\alpha}$ parameters include uncertainties on the atmospheric neutrino flux normalization, the absolute cross section of CC non-QE and NC interactions, the $\nu_{e}$/$\nu_{\mu}$ relative cross section, and the energy dependence of the CCQE cross section. We perform a $\chi^{2}$ fit to the atmospheric control samples, allowing the $\vec{\epsilon}$ and $\vec{\alpha}$ parameters to vary. We extract the uncertainties on the CC $\nu_{e}$ single electron and CC $\nu_{e}$ other event categories based on the effect of the selection cuts on the efficiency $\vec{\epsilon}$ within the fit to the control samples. We estimate the bias as the difference between the fitted value and the nominal value of the event rate for two categories (CC $\nu_{e}$ single electron and CC $\nu_{e}$ other) over 17 reconstructed $(p_{e},\theta_{e})$ bins. The correlations between bins are considered. We also include uncertainties on the event categories determined from the fit; the fit uncertainties are treated as uncorrelated between bins. For the CC $\nu_{e}$ single electron category, the bias is estimated to be 1-9% across all bins, while the fit uncertainty is 4-8% across all bins. The bias and fit uncertainty for the CC $\nu_{e}$ other category are 27% and 14%, respectively; this component is a small contribution to the signal and background prediction, and so the momentum and angular dependence of the uncertainty is ignored. As described later, we use these errors and their correlations as inputs for deriving the total SK systematic error on the T2K $\nu_{e}$ appearance candidate events. NC interactions producing a single exclusive photon via radiative decays of $\Delta$ resonances (NC1$\gamma$) are a background to the $\nu_{e}$ appearance signal, as the photon ring is very similar to an electron ring. We evaluated the difference in the selection efficiency between the single photon MC sample and the single electron MC sample to estimate the uncertainty on the selection efficiency of NC1$\gamma$ events. The difference in relative efficiencies is no larger than 1%, so we assign an additional 1% uncertainty, added in quadrature to the uncertainty on single electron rings estimated from the CC $\nu_{e}$ single electron sample efficiency, as the uncertainty on the selection efficiency for NC1$\gamma$ background events. We evaluate the systematic uncertainty for events where the muon decays in flight with a MC study. The Cherenkov ring of the electron from a muon which decays in flight tends to be in the same direction as the parent muon, and therefore these events look similar to CC $\nu_{e}$ interactions. We estimate the uncertainty on the expected number of of muon-decay-in-flight background events to be 16%, with the largest contribution from the uncertainty on the muon polarization. The fraction of muons which decay in flight in the selected $\nu_{e}$ candidate event sample is estimated to be smaller than 1%, and so this uncertainty does not contribute substantially to the total uncertainty on the $\nu_{e}$ candidates. The efficiency of NC$1\pi^{0}$ events for the $\nu_{e}$ selection criteria is determined to be 6% from the MC simulation. To evaluate the systematic uncertainty for events with a $\pi^{0}$ in the final state, we construct “hybrid-$\pi^{0}$” control samples. The “hybrid-$\pi^{0}$” samples contain events where a $\pi^{0}$ is constructed using one simulated photon ring and a second electron-like ring from the SK atmospheric or cosmic-ray samples. The simulated photon ring kinematics are chosen such that the two rings follow the decay kinematics of a $\pi^{0}$. The hybrid samples are constructed with electron rings from data (hybrid-$\pi^{0}$ data) and the simulation (hybrid-$\pi^{0}$ MC), and the comparison of the two is used to evaluate the systematic uncertainties. We investigate the systematic error coming from the higher-energy ring and the lower-energy ring separately. The “primary” sample uses electron rings from the SK atmospheric samples, with the electron ring having higher energy than the simulated photon ring. In the “secondary” sample the electron ring has a lower energy than the photon ring. Below 60 MeV, electrons from cosmic-ray muons are used; otherwise the electrons from the SK atmospheric samples are used. We compare the efficiency of the $\nu_{e}$ selection criteria on $\pi^{0}$ events in the hybrid-$\pi^{0}$ data and hybrid-$\pi^{0}$ MC samples in each of the 17 ($p_{e}$,$\theta_{e}$) bins. We apply the efficiency differences as correlated systematic errors among bins, while the statistical errors on the efficiency differences are applied as uncorrelated systematic errors. For the NC single $\pi^{0}$ component, we estimate correlated errors in each ($p_{e}$,$\theta_{e}$) bin to be between 2-60%, and uncorrelated errors are between 15-50%. The assigned errors are larger in the lower momentum bins, where the $\pi^{0}$ selection efficiency is lower. We evaluate the systematic uncertainties on events with one or more charged particles above Cherenkov threshold and a $\pi^{0}$ by using hybrid-$\pi^{0}$ control samples with additional simulated rings for the extra particles. Finally, we combine all systematic uncertainties on the $\nu_{e}$ appearance signal selection at SK into a single covariance matrix. The covariance matrix has bins in the observable kinematic variables, ($p_{e}$,$\theta_{e}$) or $E_{\nu}^{rec}$, for the four event categories: signal CC $\nu_{e}$, background CC $\nu_{\mu}$, CC $\nu_{e}$, and NC. We use this covariance matrix to model the systematic uncertainties on the simulated detector efficiency and reconstruction in the oscillation fits described in Section IX. The fractional errors as a function of the both the electron momentum and angle are shown in Fig. 29. <figure><img src="content_image/1304.0841/x46.png"><figcaption>Figure 29: The fractional errors on SK νe signal (top) and NC background(bottom) predictions from the SK detector response uncertainty as a functionof the electron candidate momentum and angle.</figcaption></figure> ## IX Oscillation Fit Method and Results The $\nu_{e}$ appearance oscillation signal is an excess of $\nu_{e}$ candidates over background. Table 14 and Table 15 show the predicted number of $\nu_{e}$ candidate events after we apply the tuned neutrino flux and cross section parameters discussed in Sec. VII. If $\textrm{sin}^{2}2\theta_{13}$=0.1, we expect 11.2 events, and if $\textrm{sin}^{2}2\theta_{13}$=0, we expect 3.3. We evaluate the systematic uncertainties on the expected signal and background event rates due to the uncertainties on the flux model, neutrino interaction cross section model and SK reconstruction efficiencies, as summarized in Section IX.1. The probability to observe 11 or more events based on the predicted background of $3.3\pm 0.4$ (syst.) events is $9\times 10^{-4}$, equivalent to an exclusion significance of $3.1\sigma$. This rate-only hypothesis test makes no assumptions about the energy spectrum of the candidate events or their consistency with the neutrino oscillation hypothesis; it is a statement that we observe an excess of electron-like events over background. The background model includes expected $\nu_{\mu}$$\rightarrow$$\nu_{e}$ oscillation through the solar term shown in Eq. I, which corresponds to 0.2 events. The reported $p$-value corresponds to the probability to observe 11 or more events from background sources and oscillations that depend on the $\theta_{12}$ mixing angle. If instead we consider the probability to observe 11 or more events from background sources only, the $p$-value is $6\times 10^{-4}$. We fit the $\nu_{e}$ candidate sample in the three-neutrino mixing paradigm to estimate $\textrm{sin}^{2}2\theta_{13}$. The dominant effect of a non-zero $\textrm{sin}^{2}2\theta_{13}$ is to increase the overall rate of $\nu_{e}$ events. However, spectral information, _e.g._ electron momentum and angle with respect to the T2K beam direction, $(p_{e},\theta_{e})$, or reconstructed neutrino energy, $E_{\nu}^{rec}$, can be used to further separate the signal from background. Fig. 30 shows the area-normalized $(p_{e},\theta_{e})$ distribution for the $\nu_{e}$ candidate events predicted by the SK simulation. The signal CC $\nu_{e}$ are predominantly CCQE, and peaked at $E_{\nu}\approx 0.6$ GeV, near the first oscillation maximum and neutrino flux peak. This results in a clear kinematic correlation across the $(p_{e},\theta_{e})$ distribution for signal events. This peak is also visible in the $E_{\nu}^{rec}$ distribution for signal events, shown in Fig. 31. Conversely, the backgrounds to the $\nu_{e}$ signal populate a wider range of kinematic space. The NC backgrounds are predominantly photons misidentified as electron neutrino candidates, when one photon from $\pi^{0}$ decay is not reconstructed, or when the two photons are co-linear. This background predominantly populates the low momentum and forward angle region as well as the signal region. The intrinsic beam $\nu_{e}$ ($\overline{\nu}_{e}$) backgrounds have a larger contribution of events at higher energy than the oscillated $\nu_{e}$, and so more often produce electrons with high momentum in the forward direction. <figure><img src="content_image/1304.0841/x48.png"><figcaption>Figure 30: (pe,θe) distribution for νe signal (top left), νμ background (topright), νe background (middle left), ¯¯¯νμ background (middle right) and ¯¯¯νebackground (bottom left). Each distribution is normalized to unit area.</figcaption></figure> <figure><img src="content_image/1304.0841/x49.png"><figcaption>Figure 31: The MC reconstructed neutrino energy distributions of νμ→νe CCsignal, intrinsic νe CC background and NC background components in the νecandidate event sample. The histograms are normalized to the same area.</figcaption></figure> We find that based on studies of the $E_{\nu}^{rec}$ and $(p_{e},\theta_{e})$ kinematic distributions that the $(p_{e},\theta_{e})$ distribution has the best power to discriminate signal and background with the minimal cross section model dependence, hence we perform a two-dimensional extended maximum likelihood fit to the $(p_{e},\theta_{e})$ data distribution. Section IX.2 describes the $(p_{e},\theta_{e})$ likelihood fit to estimate $\textrm{sin}^{2}2\theta_{13}$, and Section IX.4 describes two additional fits using $E_{\nu}^{rec}$ and rate-only information for comparison to the $(p_{e},\theta_{e})$ fit. ### $\nu_{e}$ predicted event rate and systematic uncertainties The predicted number of $\nu_{e}$ candidates and the event shape distribution depend upon the flux, cross section parameters, oscillation probability, and the efficiency and resolution of the SK detector. We calculate the predicted number of events in a given momentum and angular bin ($i$) as \[N^{p}_{i}(\vec{o},\vec{f})\qquad\] (19) \[= \sum_{j}^{flux~{}type}\bigg{[}\sum_{k=1}^{E_{\nu}bins}\;b_{j,k} \cdot\Big{\{}\sum_{l=1}^{Int.modes}\;P_{j,k,l}^{osc}(\vec{o})\] \[\times x^{norm}_{k,l}w_{i,j,k,l}(\vec{x})\cdot d_{i,j,k}\cdot T_{ i,j,k,l}^{p}\Big{\}}\bigg{]}.\] Here, $T_{i,j,k,l}^{p}$ are the nominal Monte Carlo templates that predict the event rate as a function of: * momentum/angular bins ($i$). The momentum bins are 100 MeV/$c$ wide from 0 MeV/$c$ to 1500 MeV/$c$ (15 in total), and the angular bins are 10${}^{\circ}$ wide from 0${}^{\circ}$ to 140${}^{\circ}$ with one bin for $\theta_{e}>140^{\circ}$ (15 in total). The bins are ordered by increasing $\theta_{e}$ in groups of increasing momentum. * flux type ($j$) with categories for $\nu_{e}$ signal, $\nu_{\mu}$ background, $\nu_{e}$ background, $\overline{\nu}_{\mu}$ background and $\overline{\nu}_{e}$ background. * true neutrino energy ($k$) with 200 bins (50 MeV wide) from 0 GeV to 10 GeV and one bin from 10 GeV to 30 GeV. * interaction mode ($l$) with categories for CCQE, CC1$\pi$, CC coherent, CC other, NC1$\pi^{0}$, NC coherent and NC other. The systematic parameters are $\vec{f}=(b_{j,k},x_{k,l}^{norm},\vec{x},d_{i,j,k},f^{s})$. The $b_{j,k}$ vary the flux normalization, and the $x_{k,l}^{norm}$ are cross section normalization parameters. The $\vec{x}$ are cross section parameters such as $M_{A}^{QE}$ and $p_{F}$ where the effect on the prediction is modeled with response functions, $w_{i,j,k,l}$, evaluated for each combination of observable bin, flux type, neutrino energy bin and interaction mode. The $d_{i,j,k}$ are systematic parameters that vary the normalization of the prediction for each combination of observable bin, flux type and interaction mode. These parameters are used to model variations due to final state interactions (FSI) and SK efficiency uncertainties. The momentum scale variation according to the parameter $f^{s}$ is not shown in Eq. 19. The parameter $f^{s}$ scales the momentum range of the bins and the bin contents are recalculated assuming a flat momentum dependence in each bin. We compute three-neutrino oscillation probabilities, $P_{k,l,m}^{osc}(\vec{o})$, which include matter effects, according to the numerical technique defined in Ref Barger _et al._ (1980), for a given set of the oscillation parameters, $\vec{o}$. The $\delta_{CP}$ dependence is evaluated by scanning the value of $\delta_{CP}$ and fitting for $\textrm{sin}^{2}2\theta_{13}$ with $\delta_{CP}$ fixed at each scan point. The remaining oscillation parameters are always held fixed to the values listed in Table 13. Based on Eq. 19, we predict both the total number of events and the normalized $(p_{e},\theta_{e})$ shape distribution (probability density function, PDF). The predicted number of events and the predicted $(p_{e},\theta_{e})$ distribution are used in the likelihood function of the oscillation fit. The effect of the systematic uncertainties on the predicted number of events and $(p_{e},\theta_{e})$ PDF are studied by recalculating the rate and PDF under variations of the systematic parameters according to the prior probability distribution of the parameters. Table 16 summarizes the uncertainty on the predicted number of events for each systematic error source assuming $\textrm{sin}^{2}2\theta_{13}$=0 and $\textrm{sin}^{2}2\theta_{13}$=0.1. Uncertainties related to the nuclear model are applied independently for the SK prediction and are not constrained by the fit to ND280 data since the primary target nuclei are different in the ND280 (${}^{12}C$) and SK (${}^{16}O$) detectors. These uncertainties include: the nuclear model uncertainty ($x_{SF}$), the uncertainty on the Fermi momentum in the relativistic Fermi gas model ($p_{F}$), the uncertainty on the $N\pi$ invariant mass for resonant production in the nuclear medium ($W_{\textrm{eff}}$), the uncertainty on the rate of non-pionic decays of $\Delta$ resonances in the nuclear medium ($x_{\pi-less}$), and uncertainties on the final state interactions of pions in the nucleus. The nuclear model related uncertainties contribute errors on the event rate prediction of 4.8% for $\textrm{sin}^{2}2\theta_{13}$=0 and 7.0% for $\textrm{sin}^{2}2\theta_{13}$=0.1. The uncertainty on background only predicted number of events ($\textrm{sin}^{2}2\theta_{13}$=0) is larger than that of signal+background due to the larger uncertainties on the NC backgrounds (32%); the uncertainty on CC background events (14%) is comparable to that of the CC signal events. The inclusion of the ND280 measurements reduces the uncertainty on the total predicted event rate due to the flux and CCQE, CC1$\pi^{+}$ cross section model from 18.3% to 8.5% (22.6% to 5.0%), assuming $\textrm{sin}^{2}2\theta_{13}$=0. ($\textrm{sin}^{2}2\theta_{13}$=0.1). The far detector efficiency uncertainty has been reduced from 14.7% (9.4%) in the previous analysis Abe _et al._ (21) to 6.8% (3.0%) assuming $\textrm{sin}^{2}2\theta_{13}$=0.0 ($\textrm{sin}^{2}2\theta_{13}$=0.1) due to new CC $\nu_{e}$ and $\pi^{0}$ SK atmospheric control samples; the FSI uncertainty has also been reduced from 10.1% (5.4%) in the previous results to 2.9% (2.3%) in this analysis, as correlations between reconstructed bins are now taken into account (Sec. V.3.1). \| sin22θ13= ---\|--- Error source \| 0 \| 0.1 Beam flux & ν int. (ND280 meas.) \| 8.5 \| 5.0 ν int. (from other exp.) \| \| MxCCother \| 0.2 \| 0.1 MxSF \| 3.3 \| 5.7 MpF \| 0.3 \| 0.0 MxCCcoh \| 0.2 \| 0.2 MxNCcoh \| 2.0 \| 0.6 MxNCother \| 2.6 \| 0.8 Mxνe/νμ \| 1.8 \| 2.6 MWeff \| 1.9 \| 0.8 Mxπ−less \| 0.5 \| 3.2 Mx1πEν \| 2.4 \| 2.0 Final state interactions \| 2.9 \| 2.3 Far detector \| 6.8 \| 3.0 Total \| 13.0 \| 9.9 Table 16: Summary of the contributions to the total uncertainty on the predicted number of events, assuming sin22θ13=0 and sin22θ13=0.1, separated by sources of systematic uncertainty. Each error is given in units of percent. We also consider the effect on the $(p_{e},\theta_{e})$ PDF, or “shape” of $(p_{e},\theta_{e})$, as the systematic parameters are changed. Fig. 32 (Fig 33) shows the variation of the one-dimensional angular slices of the total signal+background as a function of momentum for $\textrm{sin}^{2}2\theta_{13}$=0.1 ($\textrm{sin}^{2}2\theta_{13}$=0). The main contributions to the shape systematic uncertainties for $\textrm{sin}^{2}2\theta_{13}$=0 are the SK detector efficiency and $W_{\textrm{eff}}$ parameters in the neutrino interaction models which introduce uncertainties on the $(p_{e},\theta_{e})$ distribution of $\nu_{\mu}$ (NC) background. For $\textrm{sin}^{2}2\theta_{13}$=0.1, the dominant contributions to the shape systematic uncertainties are the $\nu_{\mu}$ flux, CCQE and CC1$\pi$ cross section parameters, $x_{SF}$, and the SK detector uncertainties. <figure><img src="content_image/1304.0841/x50.png"><figcaption>Figure 32: The PDF as a function of momentum for different angular bins (10 of15 (pe,θe) bins are shown) and sin22θ13=0.1. The shaded areas represent onesigma deviations that are evaluated by fluctuating all of the systematicparameters according to a multivariate normal distribution using their priorvalues and covariance matrix.</figcaption></figure> <figure><img src="content_image/1304.0841/x51.png"><figcaption>Figure 33: The PDF as a function of momentum for for the same bins as in Fig.32 and sin22θ13=0. The shaded areas represent one sigma deviations that areevaluated by fluctuating all of the systematic parameters according to amultivariate normal distribution using their prior values and covariancematrix.</figcaption></figure> ### $\nu_{e}$ likelihood We define an extended likelihood as the product of the likelihoods for the observed number of $\nu_{e}$ candidate events (${\cal L}_{norm}$), the shape of $(p_{e},\theta_{e})$ distribution of those events (${\cal L}_{shape}$) and the constraint term for the nuisance parameters (${\cal L}_{syst}$). The normalization term, ${\cal L}_{norm}$, is defined by the Poisson probability to observe the number of $\nu_{e}$ candidate events, $N_{obs}$, given a predicted number of events, $n=\sum_{i,j}^{N_{p\theta}}N^{p}_{i,j}(\vec{o},\vec{f})$: \[{\cal L}_{norm}(\vec{o},\vec{f})=\frac{(n^{N_{obs}})e^{-n}}{N_{obs}!}\] (20) The shape term, ${\cal L}_{shape}$ is defined by the product of the probabilities that each event has a particular value of the momentum and angle $(p_{e},\theta_{e})$. We use a Bayesian marginalization technique in order to incorporate the systematic uncertainties, by integrating over all systematic parameters. Then, the only free parameter in the marginalized likelihood is $\textrm{sin}^{2}2\theta_{13}$: \[{\cal L}^{\prime}(\vec{o})=\int{\cal L}_{norm}(\vec{o},\vec{f})\times{\cal L}_ {shape}(\vec{o},\vec{f})\times{\cal L}_{syst}(\vec{f})d\vec{f}\;.\] (21) Here we assume ${\cal L}_{syst}$ is a multivariate normal distribution of the systematic parameters defined by the parameters’ prior values and covariance matrix. The oscillation parameters are obtained by maximizing the marginalized likelihood. We have studied the increase in sensitivity of the analysis from the use of kinematic $(p_{e},\theta_{e})$ information and from the ND280 fit. The difference of the log likelihood at the best-fit and at another value of $\textrm{sin}^{2}2\theta_{13}$ is calculated as: \[-2\Delta\ln{\cal L}= -2[\ln{\cal L}^{\prime}(\sin^{2}2\theta_{13})\] (22) \[-\ln{\cal L}^{\prime}(\sin^{2}2\theta_{13}^{best})]\] The likelihood in $-2\Delta\ln{\cal L}$ can include just the normalization term, or the normalization and shape term, and the systematic term in the likelihood can include the ND280 measurements or not. Fig. 34 shows the average $-2\Delta\ln{\cal L}$ curves for these three cases, for toy MC data generated at $\textrm{sin}^{2}2\theta_{13}$=0.1. We obtained a 20% improvement to $-2\Delta\ln{\cal L}$ at $\textrm{sin}^{2}2\theta_{13}$=0 when kinematic information is included; this is equivalent to a 20% increased beam exposure. Similar studies show a comparable increase of 19% for the use of ND280 information in the likelihood to reduce the systematic errors. <figure><img src="content_image/1304.0841/x52.png"><figcaption>Figure 34: The −2ΔlnL average sensitivity curve for toy MC data generated atsin22θ13=0.1 with δCP=0, normal hierarchy and 3.01×1020 POT. The likelihood isshown for three cases: where rate, shape and ND280 information is used, whereonly rate and ND280 information is used, and where rate and shape informationis used without ND280 information.</figcaption></figure> ### Results for $\textrm{sin}^{2}2\theta_{13}$ We performed the fit to the observed 11 $\nu_{e}$ candidate events by allowing $\textrm{sin}^{2}2\theta_{13}$ to vary and scanning the value of $\delta_{CP}$. Fig. 35 compares the $(p_{e},\theta_{e})$ kinematic distributions observed in data with the prediction at the best-fit value of $\textrm{sin}^{2}2\theta_{13}$. Because of the potential bias in the determination of $\textrm{sin}^{2}2\theta_{13}$ near the physical boundary of $\textrm{sin}^{2}2\theta_{13}$=0, we calculate the confidence intervals following the Feldman-Cousins (FC) method Feldman and Cousins (1998). The 68% and 90% confidence intervals calculated using the FC method and constant $-2\Delta\ln{\cal L}$ method are found to be equivalent. Assuming $\delta_{CP}$=0, the best-fit values of $\sin^{2}2\theta_{13}$ with the 68% confidence intervals are: \[\textrm{sin}^{2}2\theta_{13}=0.088{}^{+0.049}_{-0.039}\] (normal hierarchy) \[\textrm{sin}^{2}2\theta_{13}=0.108{}^{+0.059}_{-0.046}\] (inverted hierarchy) The 90% confidence intervals are: \[0.030<\sin^{2}2\theta_{13}<0.175\] (normal hierarchy) \[0.038<\sin^{2}2\theta_{13}<0.212\] (inverted hierarchy). Fig. 36 shows the 68% and 90% confidence intervals for $\sin^{2}2\theta_{13}$ and the best-fit $\textrm{sin}^{2}2\theta_{13}$ for each value of $\delta_{CP}$. <figure><img src="content_image/1304.0841/x53.png"><figcaption>Figure 35: The (pe,θe) distribution of the νe events (dots) (top) overlaidwith the prediction. The prediction includes the rate tuning determined fromthe fit to near detector information and a signal assuming the best-fit valueof sin22θ13=0.088. The angular distribution (middle) of the νe events in dataoverlaid with prediction, and the momentum distribution (bottom) with the sameconvention as above.</figcaption></figure> <figure><img src="content_image/1304.0841/x56.png"><figcaption>Figure 36: The 68% and 90% confidence intervals for sin22θ13 scanned overvalues of δCP assuming normal hierarchy (top, b) and inverted hierarchy(bottom, d) with all other oscillation parameters fixed at the values in Table13. The best-fit value of sin22θ13 for each value of δCP is also shown forthe (pe,θe) analysis. The −2ΔlnL curve for normal hierarchy (top, a) andinverted hierarchy (bottom, c) at δCP=0 are also shown vs. sin22θ13.</figcaption></figure> To compare the data with the best-fit $(p_{e},\theta_{e})$ distribution, assuming normal hierarchy and $\delta_{CP}$=0, we perform the Kolmogorov-Smirnov (KS) test. We reorder the 2D $(p_{e},\theta_{e})$ distribution into a 1D histogram, and generate 4000 toy MC experiments with the input value of $\sin^{2}2\theta_{13}=0.088$ (best-fit value) and where the observed number of events is 11. We then calculate the maximum distance for each toy experiment and determine the fraction of toy experiments for which the maximum distance is equal to or more than $0.22$, the value obtained for a KS test done on data. The $p$-value is 0.54 and therefore the $(p_{e},\theta_{e})$ distribution of data is consistent with the best-fit distribution. Fig. 36 shows the $-2\Delta\ln{\cal L}$ curve as a function of $\textrm{sin}^{2}2\theta_{13}$, for $\delta_{CP}$=0. We consider an alternate test of the background hypothesis using the value of $-2\Delta\ln{\cal L}$ at $\textrm{sin}^{2}2\theta_{13}$=0. The probability of obtaining a $-2\Delta\ln{\cal L}$ at $\textrm{sin}^{2}2\theta_{13}$=0 equal to or greater than the value observed in data, 8.8, is calculated using the distribution of $-2\Delta\ln{\cal L}$ from pseudo-experiments generated with $\textrm{sin}^{2}2\theta_{13}$=0, $\delta_{CP}$=0, normal hierarchy and fitted with the signal+background model. This test makes use of the different $(p_{e},\theta_{e})$ distributions of signal compared to background, assuming three active neutrino mixing, and yields a similar probability of $1\times 10^{-3}$ to the rate-only test presented earlier. ### Alternate analysis methods In addition to the $(p_{e},\theta_{e})$ analysis, we performed an analysis using the reconstructed neutrino energy spectrum, and a rate-only analysis. Since $E_{\nu}^{rec}$ is closely correlated to the true neutrino energy for QE interactions, it provides the simplest projection for observing the energy dependence of the oscillation probability. This analysis also provides a consistency check of the use of spectral information in the fit. We also provide an update to the previous $\nu_{e}$ appearance analysis Abe _et al._ (21), where only rate information was used. The likelihood including neutrino energy spectrum information is defined as: \[{\cal L}(\vec{o},\vec{f})={\cal L}_{norm}(\vec{o},\vec{f})\times{ \cal L}_{shape}(\vec{o},\vec{f})\times{\cal L}_{syst}(\vec{f})\] (23) In this analysis, we perform a one dimensional scan of $\textrm{sin}^{2}2\theta_{13}$ for each value of $\delta_{CP}$ while the other oscillation parameters are fixed. At each $\textrm{sin}^{2}2\theta_{13}$ point, the negative log likelihood $-2\ln{\cal L}(\vec{o},\vec{f})$ is minimized by allowing the nuisance parameters, $\vec{f}$, to vary. The best-fit value of $\sin^{2}2\theta_{13}$ is the point where $-2\ln{\cal L}(\vec{o})$ is minimized and $-2\Delta\ln{\cal L}$ is used to constructing a confidence interval for $\sin^{2}2\theta_{13}$ according to the FC method. Fig. 37 shows the observed $E_{\nu}^{rec}$ distribution for the $\nu_{e}$ events with the best-fit of the $E_{\nu}^{rec}$ analysis applied. The observed spectrum agrees with the best-fit expectation, confirmed by a KS test with a $p$-value of 0.7. The best-fit values of $\sin^{2}2\theta_{13}$, assuming $\delta_{CP}=0$, are: \[\textrm{sin}^{2}2\theta_{13}=0.092^{\,+0.049}_{\,-0.039}\] (normal hierarchy) \[\textrm{sin}^{2}2\theta_{13}=0.112^{\,+0.058}_{\,-0.047}\] (inverted hierarchy) The 90% confidence intervals are: \[0.033<\sin^{2}2\theta_{13}<0.179\] (normal hierarchy) \[0.040<\sin^{2}2\theta_{13}<0.215\] (inverted hierarchy). <figure><img src="content_image/1304.0841/x58.png"><figcaption>Figure 37: The observed Erecν distribution and prediction, assumingsin22θ13=0.092, δCP=0, and normal hierarchy. The background component is alsoshown.</figcaption></figure> The rate-only measurement only uses the number of $\nu_{e}$ events at SK to determine $\textrm{sin}^{2}2\theta_{13}$. This analysis uses the normalization likelihood ratio: \[\Delta\chi^{2}=-2\log\frac{{\cal L}_{norm}(\vec{o},\vec{f})}{{{\cal L}^{ \mathrm{best}}_{norm}}(\vec{o},\vec{f})}\] (24) where ${\cal L}_{norm}$ is defined in Eq. 20. The value of $\Delta\chi^{2}$ is calculated for the 11 observed $\nu_{e}$ candidates, in a one dimensional scan of $\textrm{sin}^{2}2\theta_{13}$ for each point of $\delta_{CP}$ with all other oscillation parameters fixed. The confidence intervals are determined using the FC method. The best-fit values of $\sin^{2}2\theta_{13}$, assuming $\delta_{CP}=0$, are: \[\textrm{sin}^{2}2\theta_{13}=0.097_{-0.041}^{+0.053}\] (normal hierarchy) \[\textrm{sin}^{2}2\theta_{13}=0.123_{-0.051}^{+0.065}\] (inverted hierarchy). The 90% confidence intervals are: \[0.034<\sin^{2}2\theta_{13}<0.190\] (normal hierarchy) \[0.044<\sin^{2}2\theta_{13}<0.236\] (inverted hierarchy). Fig. 38 shows the three analyses are consistent with each other. The rate-only analysis has a higher best-fit value of $\textrm{sin}^{2}2\theta_{13}$ than the $E_{\nu}^{rec}$, $(p_{e},\theta_{e})$ analyses. This results from the additional discriminatory power of the kinematic information to identify events as slightly more similar to the background distribution than the predicted oscillation signal. In addition, the difference between the best-fit and the 90% upper confidence interval for the rate-only analysis is larger than the other two analyses. This is due to a slight (2%) over-coverage of the rate-only analysis. <figure><img src="content_image/1304.0841/x59.png"><figcaption>Figure 38: The 68% and 90% confidence interval regions for sin22θ13 scannedover values of δCP assuming normal hierarchy (a) and inverted hierarchy (b)with the best-fit value of sin22θ13 shown for the (pe,θe) analysis. The 90%confidence interval region for the Erecν analysis and rate-only analysis areoverlaid. The best-fit values of sin22θ13 for the Erecν analysis and therate-only analysis are also shown. All other oscillation parameters are fixedat the values in Table 13.</figcaption></figure> ## X Conclusion In summary, we have reported the first evidence of electron neutrino appearance in a muon neutrino beam with a baseline and neutrino spectrum optimized for the atmospheric mass splitting. We observed 11 candidate $\nu_{e}$ events at the SK detector when $3.3\pm 0.4$(syst.) background events are expected, and rejected the background-only hypothesis with a $p$-value of 0.0009, equivalent to a 3.1$\sigma$ significance. We have employed a fit to the ND280 near detector data that constrains the parametrized neutrino flux and interaction models used to predict the event rates at SK. The ND280 constraint on the $\nu_{e}$ candidates reduced the overall systematic uncertainty to 10–13% depending on the value of $\textrm{sin}^{2}2\theta_{13}$, an important step towards precision measurements of $\nu_{e}$ appearance. The excess of events at SK corresponds to a best-fit value of $\textrm{sin}^{2}2\theta_{13}=0.088{}^{+0.049}_{-0.039}$ at 68% C.L., assuming $\delta_{CP}$=0, $\textrm{sin}^{2}2\theta_{23}$=1.0 and normal hierarchy. This result represents an important step towards constraining the unknown parameters in the three-neutrino oscillation model. The evidence of electron neutrino appearance opens the door for a rich program of experimental physics in this oscillation channel. T2K measurements of this channel will be an important input to global fits which also combine muon neutrino disappearance measurements and reactor-based measurements of $\theta_{13}$ via $\bar{\nu}_{e}$ disappearance to begin to constrain $\delta_{CP}$ and the octant of $\theta_{23}$. Future measurements of the appearance probability for antineutrinos will provide a further constraint on $\delta_{CP}$ and the mass hierarchy. ###### Acknowledgements. We thank the J-PARC accelerator team for the superb accelerator performance and CERN NA61 colleagues for providing essential particle production data and for their fruitful collaboration. We acknowledge the support of MEXT, Japan; NSERC, NRC and CFI, Canada; CEA and CNRS/IN2P3, France; DFG, Germany; INFN, Italy; Ministry of Science and Higher Education, Poland; RAS, RFBR and the Ministry of Education and Science of the Russian Federation; MEST and NRF, South Korea; MICINN and CPAN, Spain; SNSF and SER, Switzerland; STFC, U.K.; NSF and DOE, U.S.A. We also thank CERN for their donation of the UA1/NOMAD magnet and DESY for the HERA-B magnet mover system. 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These fits constrain the NEUT FSI, CCQE and resonant pion production models. The details of these fits are described here. #### a.0.1 FSI model The NEUT FSI model includes parameters which alter the pion interaction probabilities for absorption, charge exchange, and quasi-elastic scattering de Perio (2011). The values of these parameters and their uncertainties are determined from fits to pion scattering data. Fig. 39 shows the tuned cascade model compared to macroscopic measurements of the pion absorption cross section and the maximum variation of the model parameters chosen to cover the uncertainties on the data. <figure><img src="content_image/1304.0841/x60.png"><figcaption>Figure 39: Pion absorption cross section as a function of pion momentumoverlaid with π+-12C scattering data, Ashery et al. Ashery _et al._ (1981),Jones et al. Jones _et al._ (1993), and Giannelli et al. Giannelli _et al._(2000).</figcaption></figure> In total, we consider 16 variations of the FSI model parameters to cover the uncertainties on macroscopic pion scattering data. For each of the modified FSI parameter sets and the nominal NEUT model, we evaluate with weights the effect on ND280, SK or external predicted observables by calculating the covariance matrix of the predicted observables. FSI covariance matrices are generated for MiniBooNE, ND280 and SK predictions. The external data covariance matrices use observable bins from external data, such as reconstructed $(T_{\mu},\cos\theta_{\mu})$ bins for MiniBooNE. The ND280 covariance matrix corresponds to the two ND280 selections’ reconstructed ($p_{\mu}$,$\cos\theta_{\mu}$) bins and the SK covariance matrices correspond to the $\nu_{e}$ selection with either reconstructed $(p_{e},\theta_{e})$ or $E_{\nu}^{rec}$ bins: \[V_{ij}=\frac{1}{16}\sum_{k=1}^{k=16}(p_{i}^{\mathrm{nom}}-p_{i}^{k})(p_{j}^{ \mathrm{nom}}-p_{j}^{k}),\] (25) where $p_{i}^{k}$ is the expected event rate in the $i$th observable bin assuming the $k$th FSI parameter set, and $p_{i}^{\mathrm{nom}}$ is the expected event rate in the same bin assuming the nominal FSI parameter set. For the oscillation analysis, we add these FSI covariance matrices to the detector efficiency and reconstruction covariance matrices evaluated for ND280 (Section VI.3) and SK (Section VIII) selections. #### a.0.2 CCQE model As discussed in Section V.3.2, we fit the MiniBooNE measurement of the CCQE double-differential cross sections in bins of muon kinetic energy and angle, $(T_{\mu},\cos\theta_{\mu})$ Aguilar-Arevalo _et al._ (82) with the NEUT model. While the CCQE model can be directly constrained with T2K ND280 data, we also fit the MiniBooNE measurement since the MiniBooNE detector’s 4$\pi$ acceptance provides coverage for backwards produced muons that are currently excluded in the ND280 selection. To compare the NEUT model of CCQE interactions with MiniBooNE data, we use the MiniBooNE flux prediction Aguilar-Arevalo _et al._ (2009) to generate CCQE interactions. We fit the MiniBooNE double-differential cross section data with the NEUT prediction, allowing $M^{QE}_{A}$ and the overall cross section normalization to vary, by minimizing the $\chi^{2}$ defined as: \[\chi^{2}(M^{QE}_{A},\lambda)=\sum_{i=0}^{N}\left(\frac{D_{i}-\lambda M_{i}(M^{ QE}_{A})}{\sigma_{i}}\right)^{2}+\left(\frac{\lambda-1}{\sigma_{\lambda}} \right)^{2}.\] (26) Here, the sum runs over the $N$ bins in the $(T_{\mu},\cos\theta_{\mu})$ differential cross section, $D_{i}$ is the cross section measured by MiniBooNE in the $i$th bin, $M_{i}$ is the NEUT prediction in that bin and $\sigma_{i}$ is the reported shape-only component of the error on the measured cross section. The second term adds a penalty to the normalization parameter $\lambda$, which is constrained within the MiniBooNE flux uncertainty, $\sigma_{\lambda}=10.7\%$. The best-fit parameter values are $M^{QE}_{A}=1.64\pm 0.04\,\mathrm{GeV}$ and $\lambda=0.88\pm 0.02$. Fig 40 shows the measured MiniBooNE cross section as a function of $Q^{2}$ for the nominal and best-fit value of $M^{QE}_{A}$, which is well reproduced except at lowest values of $Q^{2}$. However, this value of $M^{QE}_{A}$ is significantly larger than the value of $M^{QE}_{A}=1.35\pm 0.17\,\mathrm{GeV}$ obtained by the MiniBooNE collaboration in a fit to the single-differential $\mathrm{d}\sigma/\mathrm{d}Q^{2}$ spectrum, with an uncertainty that is smaller by a factor of 4. We postulate that the difference in central values is due to deficiencies in the nuclear model at low $Q^{2}$, which MiniBooNE addressed by adding an empirical parameter $\kappa$ to modify Pauli blocking, and the lack of full correlations between the measured $(T_{\mu},\cos\theta_{\mu})$ bins which are not included in the provided uncertainties. We assume the lack of bin correlations also causes the discrepancy in the fitted uncertainty, and this is supported by the relatively small $\chi^{2}=26.9$ that is observed for 137 degrees of freedom. Furthermore, the fitted prediction for the total CCQE cross section as a function of energy is poor, as illustrated in Fig. 41. The fitted model is systematically higher than the MiniBooNE data above 1 GeV, although agreement is improved near the T2K peak energy of 600 MeV. <figure><img src="content_image/1304.0841/x61.png"><figcaption>Figure 40: The CCQE cross section as a function of Q2 (top) as measured byMiniBooNE (points), with the NEUT nominal and NEUT at the best-fit of theMiniBooNE CCQE (Tμ,cosθμ) spectrum. Ratio of data to NEUT (bottom) for nominal(dashed) and best-fit (solid).</figcaption></figure> <figure><img src="content_image/1304.0841/x62.png"><figcaption>Figure 41: The CCQE cross section as a function of neutrino energy (top) asmeasured by MiniBooNE (points), with the NEUT nominal and NEUT at the best-fitof the MiniBooNE CCQE (Tμ,cosθμ) spectrum. Ratio of data to NEUT (bottom) fornominal (dashed) and best-fit (solid).</figcaption></figure> As is discussed in Section VI, a CCQE-like selection of interactions in ND280 has power to constrain the CCQE cross section model. Since the fit to MiniBooNE data poorly reproduces the energy dependent cross section and lacks the full correlation of data points, we do not directly tune the NEUT model with the fitted value for $M^{QE}_{A}$. Instead, we set large prior uncertainties on the CCQE model parameters and allow the ND280 data to constrain the model. We set $M^{QE}_{A}$ to the NEUT nominal value (1.21 GeV), with the prior uncertainty set to the difference between the nominal value and best-fit value from the MiniBooNE fit, _viz._ $(1.64-1.21=0.43)\,\mathrm{GeV}$. We set the uncertainty on the low energy CCQE normalization, $x_{1}^{QE}$, to the size of the MiniBooNE flux uncertainty (11%). #### a.0.3 Single pion production model As discussed in Section V.3.3 we consider measurements of single pion production cross sections on light nuclei in the T2K energy range by MiniBooNE Aguilar-Arevalo _et al._ (91, 92, 93), and K2K Nakayama _et al._ (2005). We perform a joint fit to the MiniBooNE measurements of charged current single $\pi^{+}$ production (CC1$\pi^{+}$), charged current single $\pi^{0}$ production (CC1$\pi^{0}$) and neutral current single $\pi^{0}$ production (NC1$\pi^{0}$), and we check the fit results with the K2K measurement. An important feature of the MiniBooNE single pion measurements is that they are defined by the particles exiting the target nucleus, not the particles produced at the neutrino interaction vertex. The measurements do not include corrections for FSI, but do include uncertainties of interactions of the pions in the detector. To derive the NEUT predictions for these selections, we generate interactions according to the MiniBooNE flux as was done for the CCQE fits. Instead of selecting generated events based on the true neutrino interaction mode, such as CC1$\pi^{+}$, we select the events based on the presence of a single pion exiting the nucleus. Hence, multiple interaction types are present in the prediction for each of the MiniBooNE measurements. For example, CC1$\pi^{+}$ interactions chiefly result in a single charged pion exiting the nucleus, but these events may instead pass the CC1$\pi^{0}$ selection if $\pi^{+}$ undergoes single charge exchange within the nucleus. This interdependence within the MiniBooNE selections, as well as the fact that all three are predicted by the same model in NEUT, justifies the use of a joint fit to the three measurements. We fit to the measured $\mathrm{d}\sigma/\mathrm{d}Q^{2}$ spectra from CC1$\pi^{+}$ and CC1$\pi^{0}$ samples and the $\mathrm{d}\sigma/\mathrm{d}p_{\pi^{0}}$ spectrum from the NC1$\pi^{0}$ samples. MiniBooNE provides uncertainties for each of the measurement. In the case of the CC1$\pi^{0}$ and NC1$\pi^{0}$ measurements, covariance matrices account for correlations between the measured points in the spectra arising from the MiniBooNE flux model and detector response. MiniBooNE only provides diagonal errors for the CC1$\pi^{+}$ measurement. We construct a covariance matrix for the CC1$\pi^{+}$ by assuming a 10% flux uncertainty correlated across all bins and by adding an additional uncorrelated uncertainty to the diagonal terms to recover the diagonal errors provided by MiniBooNE. While the flux is shared for the three measurements, at this time no correlation between the three measurements was considered. For each of the three measured distributions ($k$) we construct the $\chi^{2}$ based on the data and NEUT prediction: \[\chi^{2}_{k}=\sum_{i}\sum_{j}[D^{k}_{i}-M^{k}_{i}(\vec{x})](C^{k}_{ij})^{-1}[D ^{k}_{j}-M^{k}_{j}(\vec{x})].\] (27) Here, $i$ and $j$ sum over the bins in the $k$th measurement, $D^{k}_{i}$ are the measured differential cross sections, $C^{k}_{ij}$ is the covariance matrix describing the uncertainty on the measurement and $M^{k}_{i}(\vec{x})$ are the NEUT predictions for each measurement. The cross section parameters that are allowed to vary in the fit, $\vec{x}$, along with their prior values and prior uncertainties for penalty terms are listed in Table 5. Contributions to the predictions from CC multi-pion/DIS ($x_{CCother}$) interactions, NC coherent interactions, NC1$\pi^{\pm}$ interactions and NC multi-pion/DIS interactions are relatively small, so penalty terms are used for the associated parameters according to the prior uncertainties. We minimize the total $\chi^{2}$ that includes the $\chi^{2}$ for each of the measurements and the penalty terms: \[\chi^{2}_{total}=\chi^{2}_{CC1\pi^{+}}+\chi^{2}_{CC1\pi^{0}}+\chi^{2}_{NC1\pi^ {0}}+\sum_{k}\frac{(s_{k}-s_{k}^{nom})^{2}}{\sigma_{k}^{2}},\] (28) where, for each penalized parameter $k$, $s_{k}$ is the value of the parameter, $s_{k}^{nom}$ is the nominal value, and $\sigma_{k}$ is the prior uncertainty assigned to the penalty parameter. In practice, the inclusion of the NC1$\pi^{0}$ covariance matrix in the fit results in a best-fit which lies outside the range of the data points. This behavior results from strongly-correlated measurements combined with a model which does not correctly describe the data D’Agostini (1994). To achieve a fit that better reproduces the central values of the data points, we only use the diagonal terms of the NC1$\pi^{0}$ covariance matrix in our fit. The missing correlations also result in uncertainties on the fit parameters which do not cover the uncertainties in the data points. To remedy this, we multiply the fit parameter uncertainties by a scale factor of 2 (2.5) for CC (NC) parameters, while keeping their correlations the same. These scale factor ensure that the flux-integrated cross section uncertainty matches that given by MiniBooNE (16% for each measurement). The results of the fit are discussed in Section V.3.3. We propagate the fitted values and uncertainties for $M_{A}^{RES}$, $x^{CC1\pi}_{1}$ and $x^{NC1\pi^{0}}$ to model the cross section in the fit to ND280 data described in Section VII. In addition, we keep parameter $W_{\textrm{eff}}$ at its nominal value, but apply an uncertainty equal to the amount it is pulled in the fit to the MiniBooNE data. We compare the results of the fitted NEUT pion production model to the NC K2K measurement. The $\mathrm{d}\sigma/\mathrm{d}p_{\pi^{0}}$ distribution measured by K2K in the 1000 ton water Cherenkov detector is shown with the nominal and tuned NEUT model in Fig. 42. As with the MiniBooNE data, the data prefer a peak at higher momentum and fewer events in the high momentum tail compared to the nominal NEUT prediction. The use of NEUT assuming the best-fit parameters from the MiniBooNE single pion production fits does not significantly improve the agreement between NEUT and the K2K data. However, the discrepancy is covered by the uncertainties on the single pion production and FSI model. <figure><img src="content_image/1304.0841/x63.png"><figcaption>Figure 42: Differential dσ/dpπ0 cross section measured by K2K and the nominaland best-fit from the MiniBooNE single pion fits NEUT predictions, with errorband showing the uncertainties after the fit to MiniBooNE data.</figcaption></figure>
1110.6880	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 36008, "num_imgs": 12, "llama3_tokens_count": 9197 }	[ "content_image/1110.6880/x1.png", "content_image/1110.6880/x2.png", "content_image/1110.6880/x3.png", "content_image/1110.6880/x4.png", "content_image/1110.6880/x5.png", "content_image/1110.6880/x6.png", "content_image/1110.6880/x7.png", "content_image/1110.6880/x11.png", "content_image/1110.6880/x16.png", "content_image/1110.6880/x21.png", "content_image/1110.6880/x26.png", "content_image/1110.6880/x28.png" ]	Testing Lattice Quantum Gravity in 2+1 Dimensions Michael K. Sachs${}^{1}$ _University of California, Davis_ _One Shields Avenue_ _Davis, CA 95616_ Borrowing techniques from cosmology, I compute the power spectrum of quantum fluctuations in (2+1)-dimensional causal dynamical triangulations, a promising discrete path integral approach to quantum gravity. The results agree with those of canonical quantization to a high degree of precision, providing strong evidence for the equivalence of the two approaches and for the validity of the discrete method. ¹ [FOOTNOTE:1][ENDFOOTNOTE] ## 1 Introduction Few quantum theories start their lives as quantum theories. Most are born in the familiar classical realm and quantized. There are many ways of performing this quantization, but they mostly fall into two broad categories. The first, path integral quantization, is based on the principal of least action in classical mechanics. An integral is constructed over all field configurations between some initial and final boundary configurations with the action contributing a phase factor to each configuration. In the second method, canonical quantization, variables in the classical theory are promoted to operators on a Hilbert space. The structure of the theory is then defined through commutators of these variables. When applying these techniques to general relativity many technical and conceptual difficulties arise. These problems are so severe that no one has yet been able to overcome them except in certain simplified cases. And the situation gets even worse; it is unclear if the results from the different quantization procedures, were we able to carry them out in full, would agree with each other [1]. So the question arises: if there are several theories that yield the same classical behavior but describe very different quantum mechanics, which is the correct quantum theory? In this paper I will examine the results from two different approaches to quantizing gravity that can be carried out in full. The first is the reduced phase space quantization of (2+1)-dimensional gravity with spherical topology. The reduced dimensionality and simple topology yield a trivial classical spacetime. This simplicity is preserved when the theory is quantized, resulting in a quantum theory with no degrees of freedom. The second approach is causal dynamical triangulations, a lattice regularization of the gravitational path integral. Smooth spacetime is approximated by a triangulated mesh and the path integral is approximated by a sum over inequivalent triangulations. The sum is computed using Monte Carlo techniques, and the resulting spacetime is analyzed using techniques borrowed from the analysis of cosmic microwave background temperature fluctuations. I find that the resulting measurements reveal an extremely spherical spacetime, indicating that the two quantization techniques yield very similar results. ## 2 (2+1)-dimensional gravity and canonical quantization The action for general relativity is the Einstein-Hilbert action \[S_{EH}=\frac{1}{16\pi G}\int_{M}d^{4}x\sqrt{-g}\left(R-2\Lambda\right),\] (1) and the Euler-Lagrange equations give us the Einstein field equations \[R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=-8\pi GT_{\mu\nu}.\] (2) As one would expect, working in 2+1 dimensions greatly simplifies the equations of motion. In $d$ dimensions (2) allows $d(d-3)$ local phase space degrees of freedom [2, 3]. In $d=4$ this gives us the 4 degrees of freedom that result in gravitational waves. In $d=3$, however, this results in zero local degrees of freedom. Additionally, in 2+1 dimensions the curvature tensor $R_{\mu\nu\rho\sigma}$ depends linearly on the Ricci tensor $R_{\mu\nu}$. This means that vacuum solutions to (2) are flat if $\Lambda=0$ and of constant curvature if $\Lambda\neq 0$[2, 3]. So it seems that choosing to work in 2+1 dimensions has given us a fairly boring universe to work with. It turns out that exactly how boring depends on the topology of spacetime. In general our manifolds $M$ will decompose into one-dimensional timelike intervals $I$ and two-dimensional spacelike manifolds $\Sigma$; more compactly $I\times\Sigma$. If we choose $\Sigma$ to be any manifold containing non-contractible curves (e.g., a torus) the resulting holonomies will yield a finite number of global degrees of freedom [3, 4]. For our purposes, we want the simplest possible scenario to work with, so we will work with a spacelike manifold in which all curves are contractible. If we choose our constant curvature to be positive, this leaves us with a two-sphere $S^{2}$. The simplest possible 2+1 dimensional topology we can look at is then $I\times S^{2}$. Quantizing even this simple spacetime is far from straightforward. A detailed description of how one goes about doing this can be found in [3]. The result is that if one chooses to apply the classical constraints _first_ and then quantize the system, one ends up with a quantized spacetime with no local or global degrees of freedom. The Hilbert space is one-dimensional; the 2-sphere that we started with remains a 2-sphere, around which there are no quantum fluctuations. ## 3 Lattice quantum gravity and covariant quantization The basic idea behind quantizing a classical theory using the path integral approach is to form an integral over all possible field configurations between some fixed initial and final configurations. The field configurations we will integrate over are essentially all of the unique geometries characterized by spacetime metrics interpolating between some starting spatial geometry and some ending spatial geometry. Along with the Einstein-Hilbert action (1), we now have all the ingredients we need to write down a formal path integral for quantum gravity: \[Z=\int\mathcal{D}[g]e^{iS_{EH}}.\] (3) In practice, expressions like (3) are usually calculated using perturbation theory. This approach fails with gravity, however, because the perturbation series is non-renormalizable [5]. In order to proceed, we need to introduce some kind of regularization scheme. We will do this by approximating the smooth manifolds $M$ by a collection of flat simplices glued together at their edges, very much like a smooth sphere is approximated by a geodesic dome. In two dimensions, the simplex used is a triangle. Since the triangle is flat, any curvature in the surface created by glued together triangles is concentrated at vertices $v$ where $i$ triangles come together, and is characterized by the deficit angle: \[\delta_{v}=2\pi-\sum_{i}\theta_{vi}.\] (4) In $n$ dimensions, triangles are replaced with $n$-dimensional simplices connected to other simplices by $(n-2)$-dimensional vertices or “bones.” The curvature at each bone is still described by (4), where the dihedral angle $\theta$ is now measured around an $(n-2)$-dimensional bone. The sum over deficit angles, weighted by the “volume” $V_{b}$ of each bone, is proportional to the integral of the scalar curvature over the surface: \[\sum_{b}V_{b}\delta_{b}=\frac{1}{2}\int_{S}R\sqrt{g}d^{n}x.\] (5) We are now ready to write down a discrete version of (1), called the Regge action [6, 7]: \[S_{EH}\to S_{R}=\frac{1}{8\pi G}\sum_{\mathrm{bones}}V_{b}\delta_ {b}-\frac{\Lambda}{8\pi G}\sum_{\mathrm{n-simplices}}V_{\mathrm{ simplex}}.\] (6) The path integral (3) over all unique field configurations is then taken to be a sum over all unique triangulations $T$ weighted by $S_{R}$: \[Z=\int\mathcal{D}[g]e^{iS_{EH}}\to Z=\sum_{T}\frac{1}{C(T)}e^{iS_ {R}},\] (7) where $\frac{1}{C(T)}$ is a symmetry factor with $C(T)$ being the order of the automorphism group for triangulation $T$[8, 9]. ### Causal dynamical triangulations Historically, discrete approaches to gravity were constructed using Euclidian building blocks. This was mostly done for technical reasons; the weight factors in (7) become real, which is usually necessary for the sums to converge. However, these approaches were unable to reproduce classical spacetimes [8]. Causal dynamical triangulations (CDT) [8] attempts to address this problem by taking the Lorentzian structure of spacetime seriously from the outset. The building blocks are Lorentzian simplices with some (n-1)-dimensional faces being spacelike and some timelike. The action (6) is constructed using these building blocks and then rotated into the Euclidian sector to aid in computation. The resulting construction exhibits three distinct phases. Two of these phases resemble the pathological results from Euclidian dynamical triangulations, a crumpled phase and a branched polymer phase, neither of which seem able to reproduce classical spacetimes. The third phase, however, is characterized by an extended geometry with small fluctuations, as one would hope for from a quantum theory of gravity. This phase is the result of building the causal structure into the theory from the outset, and is the main success of the CDT program. This paper will only focus on the (2+1)-dimensional construction, although the (3+1)-dimensional approach is very similar. Details of both cases can be found in [8]. In (2+1)-dimensional CDT we use three types of three-dimensional blocks to build our spacetime. Each block is constructed from points that lie in the constant time surface $t=t_{a}$ and the next constant time surface $t=t_{a}+1$. We will label our blocks by “(number of points at $t_{a}$, number of points at $t_{a}+1$).” With this convention our three blocks are the (3,1) simplex, the (1,3) simplex (a reflection of the (3,1) simplex) and the (2,2) simplex (Figure 1). <figure><img src="content_image/1110.6880/x1.png"><figcaption>Figure 1: The three simplices of 2+1 CDT.</figcaption></figure> We define the length of edges that lie in constant time surfaces to be $l_{\mathrm{space}}^{2}=a^{2}$ and the length of edges that connect two neighboring timelike surfaces as $l_{\mathrm{time}}^{2}=-\alpha a^{2}$; $\alpha$ is an asymmetry parameter between timelike lengths and spacelike lengths. With these definitions the Regge action (6) in three dimensions becomes [8]: \[S^{(3)}_{R}= \frac{1}{8\pi G}\left(\sum_{\begin{subarray}{c} \mathrm{spacelike}\\ \mathrm{bones\ }b\end{subarray}}\frac{1}{i}V_{b}\delta_{b}+\sum_{ \begin{subarray}{c}\mathrm{timelike}\\ \mathrm{bones\ }b\end{subarray}}V_{b}\delta_{b}\right)\] \[-\frac{\Lambda}{8\pi G}\left(\sum_{\begin{subarray}{ c}\mathrm{simplices\ }s\end{subarray}}V_{s}\right).\] (8) Following the work in [8], Lorentzian volumes $V_{i}$ and dihedrial angles $\theta_{i}$ for geometrical elements $i$ are applied to (3.1). Then a series of relationships between all of the bulk variables in the manifold are used, along with the simplifying assumption that $\alpha=1$² and a choice of units $a=1$. Lastly the entire action is Wick rotated into the Euclidean sector. The result is [8, 10]: [FOOTNOTE:2][ENDFOOTNOTE] \[S^{(3)}_{eucl}=-k_{0}N_{0}+k_{3}N_{3}\] (9) with \[k_{0}=\frac{1}{4G},\] \[k_{3}=\frac{1}{4\pi G}\left(3\arccos\frac{1}{3}-\pi\right)+\frac {\Lambda}{48\pi\sqrt{2}}\] (10) where $N_{0}$ is the number of vertices in the manifold, $N_{3}$ is the number of three-simplices, $G$ is Newton’s constant, and $\Lambda$ is the cosmological constant. Finally, the discrete path integral (7) becomes: \[Z=\sum_{T}\frac{1}{C(T)}e^{iS^{(3)}_{R}}\to Z=\sum_{ T}\frac{1}{C(T)}e^{-S^{(3)}_{eucl}},\] (11) ### Monte Carlo Moves In order to numerically implement the partition function (11), one needs to create a procedure that will explore different allowed triangulations of a given three-dimensional spacetime. This is accomplished by using the Metropolis Monte Carlo algorithm [11] with a series of so-called “moves,” each move being a simple re-triangulation of a set of initial simplices. The constraints on these moves are that they should: (i) preserve the existing time-slicing of the spacetime, (ii) preserve the topology of the spacelike slices and, (iii) be ergodic, that is, any one allowed triangulation can be transformed to any other allowed triangulation by repeated application of these moves. In three dimensions there are five of these moves [8], four of which are related by inversion symmetry. They are shown in Figure 2. Each move is attempted a roughly equal number of times and is accepted or rejected with probabilities depending on how it changes the action (9) and how it effects the local geometry. The unit of simulation time is called a sweep and is defined as $N_{3}$ attempted moves (i.e one attempted move for every simplex in the spacetime). The specific CDT implementation that was used for this work is described in [12]. <figure><img src="content_image/1110.6880/x2.png"><figcaption>Figure 2: The three Monte Carlo moves that, along with their inverses, make upthe five move set (the (4,4) move is it’s own inverse).</figcaption></figure> ## 4 Process The goal of this work is to measure the sphericity of our CDT quantized spacetime, specifically the ensemble average of the sphericity of individual spacelike slices across many sweeps of the simulation. There are four steps required to do this: (i) choose a reference spacelike slice in each sweep to be analyzed, (ii) embed this two-dimensional manifold in three dimensions, (iii) measure the distance from the three-dimensional center of geometry to the embedded manifold and, (iv) analyze the distances across many sweeps using spherical harmonics. ### Step 1: Choose reference slices Although this is the least technically challenging step of the process, it does present some conceptual difficulties. Our (2+1)-dimensional spacetime has the topology $S^{2}\times I$. The timelike interval $I$ runs from $t=1$ to $t=T$ for $T$ time-slices. For computational simplicity the first time slice is identified with the last, changing the topology to $S^{2}\times S^{1}$. When the CDT program is implemented as described in Section 3, the results are a bulge of extended spacetime and stalks with the minimal number of simplices required to maintain the $S^{2}$ input topology (Figure 3). The extended bulge performs a random walk along the time axis as we execute sweeps. Since the time axis is a circle, there is no unique choice of time coordinates for the bulge, so it makes little sense to compare a fixed time-slice across sweeps. The solution to this dilemma is to use the maximum volume slice (measured by counting the number of spacelike triangles in the slice) in each sweep as a reference. This ensures that there is at least one common bulk characteristic between any two given slices. See [9] for a further discussion related to this choice and the issues surrounding it. <figure><img src="content_image/1110.6880/x3.png"><figcaption>Figure 3: Plots of the volume per time-slice for simulation 3. The top issweep 500, before the simulation has thermalized. The bottom is sweep 76100where the volume bulge is clearly visible.</figcaption></figure> ### Step 2: Embed the slices in 3D The property of spacetimes generated by the simulation that is of interest is the the intrinsic curvature. This is defined by the deficit angle (4), which is easily measured for individual spacetimes. But because the number and location of vertices will change across sweeps, it is unclear how to measure an expectation value using this quantity. To overcome this problem it is useful to embed the individual spacetimes in three dimensions. The resulting coordinate system can then be used to compare measurements across sweeps. Of course once the spacetime is embedded, it is extrinsic not intrinsic curvature that is being measured. However, Gauss’s _Theorema Egregium_ states that the Gaussian curvature of a two-dimensional surface embedded in a flat three-manifold is intrinsic to the surface, and is completely determined by the intrinsic curvature [13]. So in 2+1 dimensions, measuring the extrinsic curvature is equivalent to measuring the intrinsic curvature. #### 4.2.1 The embedding algorithm . The points that define vertices in CDT are by necessity coordinate-free. The only information that the simulation generates is the linking between them. In order to embed a two-dimensional spacelike slice in three dimensions for analysis, we have to choose a coordinate for each point based on its relationship to its neighbors. This is accomplished by feeding the logical links between the vertices into a force-directed graphing algorithm, specifically the spring electrical embedding algorithm [14] as implemented by Mathematica 6. In this graphing algorithm, neighboring vertices are attracted to each other by a force that is proportional to the Euclidean distance between them, i.e., a spring force. Additionally, every vertex repels every other vertex with a force that falls off with the inverse square of the distance, i.e., an electrical force. The total energy of the system is given by: \[E=\sum_{i=1}^{V}\biggl{(} -C_{e}\sum_{j\neq i}\frac{x_{j}-x_{i}}{d_{ij}^{2}}\] \[+C_{s}\sum_{\begin{subarray}{c}\mathrm{neighbors}\\ \mathrm{of\ }i\end{subarray}}d_{ij}(x_{j}-x_{i})\biggr{)}^{2},\] (12) where $C_{e}$ and $C_{s}$ are constants that control the relative strength of the electrical and spring forces, respectively, and $d_{ij}$ is the Euclidean distance between vertex $i$ and $j$. This energy function is then minimized by iteratively moving each vertex in the direction of the spring force. The configuration of points that minimizes (4.2.1) is used as the final embedding of our spacelike slice. ### Step 3: Measure distances Given an embedding of our surface in flat three-dimensional space, we can, in principle, determine the extrinsic curvature from the positions $\mathbb{r}(t)$ of the points. In particular, for a spherical surface, $\mathbb{r}(t)$ will be constant, and deviations from a constant value will give a measure of the variation of the curvature. Because of this, a measurement of $\mathbb{r}(t)$ for each of the spacetimes generated by the simulation is therefore a measurement of curvature. Because of this I will focus on the measurement of $\mathbb{r}(t)$ for the remainder of this paper. To perform this measurement, rays emanating from the center of geometry of a particular slice are traced and the point at which they intersect with the manifold is calculated. The rays used are not arbitrary, but rather picked so they will point to specific locations on a 2-sphere which correspond to different pixels in the HEALPix discretization scheme (more on HEALPix in Sec. 4.4). So what is actually measured is the distance from the center of geometry to the two-dimensional surface as a function of HEALPix pixel number or $\mathbb{r}(n_{pix})$. In order to avoid correlations that are an artifact of the embedding algorithm, a random rotation is performed about the center of geometry for each set of distance measurements. ### Step 4: Spherical harmonic analysis <figure><img src="content_image/1110.6880/x4.png"><figcaption>Figure 4: Examples of HEALPix pixelations at different values of Nside (imagetaken from [15]). The value Nside is the number of divisions of the base pixelalong one side. Starting at the top left the first sphere is pixelated withthe base resolution pixels. The top right sphere is at Nside=2. The bottomright sphere is at Nside=4. The bottom left sphere is at Nside=8.</figcaption></figure> The measurements in Section 4.3 are taken with a specific method of spherical harmonic analysis in mind. HEALPix (Hierarchical Equal Area iso-Latitude Pixelisation) is an efficient method of pixelating a 2-sphere that lends itself to spherical harmonic transformations. It is used widely in cosmic microwave background analysis, including data from the WMAP and Planck experiment. Details of the HEALPix implementation can be found in [15, 16]. The basic idea is that any function $f$ on a sphere can be decomposed into a linear combination of spherical harmonics $Y_{lm}$: \[f(\theta,\phi)=\sum_{l=0}^{l_{max}}\sum_{m}a_{lm}Y_{ lm}(\theta,\phi).\] (13) The $a_{lm}$ can then be found by using the orthogonality of the $Y_{lm}$ to invert (13): \[a_{lm}=\int_{\Omega}d\Omega f(\theta,\phi)Y^{}_{lm}(\theta,\phi).\] (14) In the case of a discrete function $\hat{f}$, evaluated at pixel locations $(\theta_{p},\phi_{p})$, with $N_{pix}$ pixels, (14) can be approximated by a sum. The approximation used by HEALPix is [16] \[\hat{a}_{lm}=\frac{4\pi}{N_{pix}}\sum_{p=0}^{N_{pix}-1}Y^{}_{lm} (\theta_{p},\phi_{p})\hat{f}(\theta_{p},\phi_{p}).\] (15) The variance of the $\hat{a}_{lm}$ is the angular power spectrum, $\hat{C}_{l}$: \[\hat{C}_{l}=\frac{1}{2l+1}\sum_{m}\|\hat{a}_{lm}\|^{2}.\] (16) The resolution of HEALPix maps is characterized by the quantity $N_{side}$ which represents the number of divisions of a base-resolution pixel [15]. Examples of pixelations with different values of $N_{side}$ are shown in Figure 4. ### Errors in the embedding Although the embedding algorithm does a very good job, it is not perfect. The requirement of Section 3.1 that all link lengths be equal is very difficult for the algorithm to achieve; most links are the same length, but there are always some that are either bigger or smaller than the average. When the link lengths are different sizes, the geometry of our spacelike slices becomes distorted. This means that there is some difference between the actual geometry of our manifold and the geometry we measure in step 3 of our process. To understand how this will effect our measurements, we look at how each triangle that is intersected by a ray is distorted and apply an extra amount $\Delta r$ to our measurements. We estimate $\Delta r$ as follows: the change in the area of a triangle with changes in link length $l$ is roughly $\Delta A\sim l\Delta l$. The change in the area of a spherical spacetime with changes in the radius $r$ is roughly $A\sim r\Delta r$. Since the spherical spacetimes are constructed of triangles and the radius $r$ roughly corresponds to our ray-trace measurements, we can equate these two quantities to get an estimate of $\Delta r$: $\Delta r\sim\frac{l}{r}\Delta l$, where $l$ is the average link length for a given spacetime, $\Delta l$ is is the difference of the average of a specific triangle’s link lengths and the average link length for the spacetime, and $r$ is the measured distance to the triangle in question. This quanity $\Delta r$ is calculated for each pixel and applied to the original measurement. Then the new distances are are analyzed using the methods described in Section 4.4. The results of this additional analysis are included in Figures 8 and 9, which are discussed in Section 5. ## 5 Results In order to test the sphericity of the output from CDT, five simulations were run. The values of the bare coupling constants for each of these simulations are listed in Table 1. A “snapshot” of the spacetime was exported to a file every 100 sweeps. The largest volume spacelike slice was then chosen from each of these spacetimes and used for analysis. Examples of several of these slices, embedded in three-dimensions, are shown in Figure 5. Sim \| Vinit \| T \| k0 \| k3 \| ϵ \| Total sweeps \| Themalized sweep \| Total samples(# of subsamples) \| δS2 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 1 \| 81921 \| 64 \| 1. \| 0 \| 0. \| 78 \| 0. \| 02 \| 100000 \| 46300 \| 538 \| 0. \| 055 2 \| 81921 \| 64 \| 1. \| 5 \| 0. \| 86 \| 0. \| 02 \| 100000 \| 70100 \| 300 \| 0. \| 06 3 \| 81921 \| 64 \| 2. \| 0 \| 0. \| 94 \| 0. \| 02 \| 400100 \| 24800 \| 3745; 1873(50); 1249(50); 937(50); 749(50); 625(50) \| 0. \| 061 4 \| 81921 \| 64 \| 2. \| 5 \| 1. \| 03 \| 0. \| 02 \| 100000 \| 17800 \| 822 \| 0. \| 069 5 \| 81921 \| 64 \| 3. \| 0 \| 1. \| 12 \| 0. \| 02 \| 100000 \| 19400 \| 807 \| 0. \| 06 \| \| \| \| \| \| \| \| \| \| \| \| \| Table 1: Various parameters and results for each simulation. Vinit is the total initialization volume (i.e. the number of initial simplices), T is the number of time-slices, k0 and k3 are the bare coupling constants discussed in Section 3.1. The numbers in parentheses next to the total samples for simulation 3 are the number of random subsamples of the full set of sweeps that were made. These results were then averaged to avoid sample bias. δS2 is the deviation from being perfectly spherical discussed in Section 5.1. All of the HEALPix analysis was performed at $N_{side}=$ 32. This resolution effectively limits the power spectrum analysis to the first 100 $l$’s. At higher values than this, we start sampling the map at scales smaller than the map resolution, which results in noise in the spectrum. The power spectrum, up to $l=100$, and HEALPix map for each of the slices in Figure 5, are shown in Figure 6 and Figure 7 respectively. <figure><img src="content_image/1110.6880/x5.png"><figcaption>Figure 5: Various max volume spacelike slices from simulation 1 embedded usingthe algorithm described in section 4.2. Top left is sweep 55500, top right issweep 71800, bottom left is sweep 92700 and bottom right sweep 99900.</figcaption></figure> <figure><img src="content_image/1110.6880/x6.png"><figcaption>Figure 6: The HEALPix maps for the slices in Figure 5. Top left is sweep55500, top right is sweep 71800, bottom left is sweep 92700 and bottom rightsweep 99900.</figcaption></figure> <figure><img src="content_image/1110.6880/x7.png"><figcaption>(a)</figcaption></figure> ### The shape of spacetime As can be seen in Figure 7, individual slices vary from being spherical by non-trivial amounts. However, individual slices come from particular paths in the path integral (3), and physically, only the ensemble averages are meaningful. It is important to choose carefully when picking the quantities from which to calculate these averages. Because all notion of absolute location is washed out from sweep to sweep, averaging the distance measurements is an unreliable measure of the shape of the ensemble system. Each pixel will explore roughly the same range of allowed values, resulting in the same average distance everywhere – the measurements will “sum to zero.” If there were a non-spherical shape preferred by the system, it would be washed out in this average. We need a quantity that is rotationally invariant, i.e., some measurement that will give us the same answer for a given shape regardless of how that shape is oriented relative to a coordinate system. Luckily, the power spectrum is just such a quantity. Averaging each $l$ of the power spectrum across the entire ensemble will give us a measure of the physical shape of our spacetime. To test this assumption, simulation 3 was run for four times as many sweeps as the other simulations. This pool of sweeps was then sampled at different sizes. As sample size increases, measurements that are subject to the “sum to zero” effect will converge to average values everywhere. Measurements that are free from this defect will remain unaffected. If the power spectrum ensemble average is one of these quantities, then it should be almost the same at any reasonably large sample size. Figure (a)a confirms this. The averaged power spectra are almost identical for every $l$ at all sample sizes. <figure><img src="content_image/1110.6880/x11.png"><figcaption>(a)</figcaption></figure> Now that we have some confidence in our averaging procedure, we can examine the results for all of the simulations in Table 1. The results are shown in Figure 8. All of the simulations performed here seem to be very spherical. In each case, the $l=1$ mode is only a few percent of the the spherical or $l=0$ mode, and higher modes fall off rapidly. Also, the results are very similar for all chosen values of $k_{0}$ and $k_{3}$, implying that the shape of the spacetime in the extended phase does not depend on the values of the coupling constants. Finally, in order to give each simulation a single score to rank its sphericity, the following quantity is calculated: \[\delta S^{2}=\frac{\sum_{l=1}^{100}C_{l}}{C_{0}}.\] (17) $\delta S^{2}$ is a measure of how much power is not in the spherical $l=0$ mode. The results for each spacetime are shown in Table 1. ### The fluctuations The spacetimes produced by CDT seem quite spherical, matching the results from canonical quantization. The other prediction of the canonical procedure is that the spacetime should have no fluctuations. To test whether the CDT spacetimes exhibit this property I will look at the variance of the distance measurements: \[\sigma_{pix}^{2}=\langle r(n_{pix})^{2}\rangle-\langle r(n_{pix})\rangle^{2}.\] (18) As with the measurement in Section 5.1, it is important to note that (18) is not affected by the “sum to zero” effect, as can be seen in Figure (b)b. <figure><img src="content_image/1110.6880/x16.png"><figcaption>(a)</figcaption></figure> <figure><img src="content_image/1110.6880/x21.png"><figcaption>(a) k0=1.0</figcaption></figure> <figure><img src="content_image/1110.6880/x26.png"><figcaption>(a) Shape power spectra at different sample sizes</figcaption></figure> Equation (18) is calculated for every pixel in the HEALPix map, creating a map of the fluctuations of the spacetime (Figure 10). A power spectrum is then calculated for this map. The result for each simulation can be seen in Figure 9. Clearly non-spherical (non $l=0$) modes of fluctuations are very small, a result that fits with the predictions of the canonical quantization procedure. There is however, power in the the $l=0$ mode that does not match the canonical predictions. This zero mode power can be attributed to lattice effects in the CDT model. Because the slices are discrete, when the largest volume time-slice is selected (see Section 4.1) it will typically be slightly smaller than the maximum volume slice in the limit where the number of time-slices gets very large. This means that the volume I sample will fluctuate around its “maximum” value. Because the maximum volume is fluctuating and our ensemble spacetimes are very spherical, the radius of the spacetime will fluctuate uniformly, resulting in a large zero mode in the fluctuation power spectrum. To calculate an order of magnitude estimate for this fluctuation, the points marking the volumes of the time-slices around the maximum volume time slice were fit to a curve using a cubic curve-fitting algorithm. The maximum of this curve was then interpreted to be the “actual” maximum volume in the limit where the number of time-slices becomes large. The average difference between the “actual” value and the selected value was calculated. In order to compare this average $\Delta r$ to the zero mode of the power spectrum, one must relate the $C_{l}$ to the total power in a function $f$ via Parseval’s theorem: \[\frac{1}{4\pi}\int_{\Omega}f(\Omega)^{2}d\Omega=\sum_{l=0}^{ \infty}\sum_{m=-l}^{l}a_{lm}^{2}.\] (19) Combining this with (16) gives: \[\frac{1}{4\pi}\int_{\Omega}f(\Omega)^{2}d\Omega=\sum_{l=0}^{ \infty}(2l+1)C_{l}.\] (20) The function we are considering is the variance (18), which we will assume is constant over the sphere, and because it is so much bigger, the only $C_{l}$ that we need to look at is $C_{0}$, so: \[\sigma_{r}^{4}\sim C_{0}.\] (21) Additionally, since we are assuming that the spacetime is mostly spherical $r\sim\sqrt{V}$ and $\Delta r\sim\sqrt{V_{max}}-\sqrt{V_{measured}}$ (we are working in two dimensions, so the volume in this case is actually an area). So we have \[\Delta r\sim\sqrt{V_{max}}-\sqrt{V_{measured}}\sim C_{0}^{1/4}.\] (22) Results of this calculation are shown in Figure 12. Without any modification, the values for $\Delta r$ that we measured are smaller than the $C_{0}$s but they show similar behavior. The fit can be made better by multiplying and adding a constant to $\Delta r$. These constants arise from the fact that the assumptions we made in deriving (22) – constant variance and perfect sphericity – were not completely correct using our data set (there are small but non zero modes $>$ 0 in our simulations). The results in Figure 12 are an indication that the zero mode fluctuations are indeed induced by the CDT lattice. <figure><img src="content_image/1110.6880/x28.png"><figcaption>Figure 12: The zero mode fluctuation (C1/40) compared to the difference in thesampled radius from the actual radius due to lattice effects (Δr).</figcaption></figure> ### Small $C_{l}$ for $l>0$ The power in $l$’s greater than zero is small in all cases, but it is still non-zero. If the spacetimes were truly spherical, with no fluctuations, all of these values would be zero. Is there a small but interesting dynamic in the theory itself that causes these modes to be active? Within the context of this work it is unclear where these modes come from. It is plausible that they arise from lattice effects. If so, then as the lattice spacing decreases, theses modes would also decrease. Decreasing lattice spacing requires an increase in the initialization volume, though, and generating enough sweeps to calculate good statistics is problematic at high volume using available hardware. Studying these small, high $l$ modes is therefore left to future work. ## 6 Conclusions As Figures 8 and 9 and the analysis in Section 5 suggest, the geometry that the CDT simulations produce are very close to being perfectly spherical with no fluctuations. Because reduced phase space quantization leads to a quantum spacetime with no degrees of freedom, these results imply that, at least in (2+1) dimensions, the path integral quantization implemented by CDT is, at least very nearly, equivalent to the canonical approach. It remains to be seen if this result can be extended to other 2+1 dimensional topologies. In CDT we have, for the first time, a tool to numerically explore these fascinating theories and perhaps cast some light on the (3+1)-dimensional case. ### Acknowledgments This work was supported in part by the Department of Energy under grant DE-FG02-91ER40674. Thanks to Steve Carlip and Rajesh Kommu for all of their helpful insights. ## References * [1] S. Carlip, Rept. Prog. Phys. 64, 885 (2001), gr-qc/0108040. * [2] S. Carlip, Living Reviews in Relativity 8 (2005), gr-qc/0409039v2. * [3] S. Carlip, _Quantum Gravity in 2+1 Dimensions_ (Cambridge University Press, New York, NY, 1998). * [4] S. Carlip, Six ways to quantize (2+1)-dimensional gravity, in _Procs. of the Fith Canadian Conf. on General Relativity and Relativistic Astrophysics_, edited by R. B. Mann and R. G. McLenaghan, Singapore, 1994, World Scientific, gr-qc/9305020. * [5] M. H. Goroff and A. Sagnotti, Nuclear Physics B 266, 709 (1986). * [6] T. Regge, Il Nuovo Cimento 19, 558 (1961). * [7] J. Cheeger, W. Müller, and R. Schrader, Commun. Math. Phys. 92, 405 (1984). * [8] J. Ambjorn, J. Jurkiewicz, and R. Loll, Nucl. Phys. B610, 347 (2001), hep-th/0105267. * [9] J. Ambjorn, A. Gorlich, J. Jurkiewicz, and R. Loll, Phys. Rev. D78, 063544 (2008), arXiv:0807.4481. * [10] J. Ambjorn, J. Jurkiewicz, and R. Loll, Phys. Rev. D64, 044011 (2001), hep-th/0011276. * [11] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953). * [12] R. Kommu, abs/1110.6875. * [13] E. Weisstein, Gauss’s theorema egregium, From MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/ GausssTheoremaEgregium.html. * [14] G. D. Battista, P. Eades, R. Tamassia, and I. G. Tollis, _Graph Drawing: Algorithms for the Visualization of Graphs_ (Prentice Hall, Upper Saddle River, N.J., 1999). * [15] K. M. Gorski, B. D. Wandelt, F. K. Hansen, E. Hivon, and A. J. Banday, (1999), astro-ph/9905275. * [16] K. M. Gorski, B. D. Wandelt, F. K. Hansen, E. Hivon, and A. J. Banday, Healpix documentation, http://healpix.jpl.nasa.gov/ healpixSoftwareDocumentation.shtml.
1610.09816	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 70545, "num_imgs": 14, "llama3_tokens_count": 17656 }	[ "content_image/1610.09816/x1.png", "content_image/1610.09816/x2.png", "content_image/1610.09816/x3.png", "content_image/1610.09816/x5.png", "content_image/1610.09816/x9.png", "content_image/1610.09816/x10.png", "content_image/1610.09816/x11.png", "content_image/1610.09816/x12.png", "content_image/1610.09816/x13.png", "content_image/1610.09816/x14.png", "content_image/1610.09816/x16.png", "content_image/1610.09816/x20.png", "content_image/1610.09816/x23.png", "content_image/1610.09816/x24.png" ]	# Robust Gait Recognition by Integrating Inertial and RGBD Sensors Qin Zou, Lihao Ni, Qian Wang, Qingquan Li, and Song Wang Q. Zou, L. Ni, Q. Wang are with School of Computer Science, Wuhan University, Wuhan 430072, P.R. China (E-mails: {qzou, lhni, qianwang}@whu.edu.cn).Q. Li is with Shenzhen Key Laboratory of Spatial Smart Sensing and Service, Shenzhen University, Guangdong 518060, P.R. China (E-mail: liqq@szu.edu.cn). S. Wang is with Department of Computer Science and Engineering, University of South Carolina, Columbia, SC 29200, USA (E-mail: songwang@cec.sc.edu). ###### Abstract Gait has been considered as a promising and unique biometric for person identification. Traditionally, gait data are collected using either color sensors, such as a CCD camera, depth sensors, such as a Microsoft Kinect, or inertial sensors, such as an accelerometer. However, a single type of sensors may only capture part of the dynamic gait features and make the gait recognition sensitive to complex covariate conditions, leading to fragile gait-based person identification systems. In this paper, we propose to combine all three types of sensors for gait data collection and gait recognition, which can be used for important identification applications, such as identity recognition to access a restricted building or area. We propose two new algorithms, namely EigenGait and TrajGait, to extract gait features from the inertial data and the RGBD (color and depth) data, respectively. Specifically, EigenGait extracts general gait dynamics from the accelerometer readings in the eigenspace and TrajGait extracts more detailed sub-dynamics by analyzing 3D dense trajectories. Finally, both extracted features are fed into a supervised classifier for gait recognition and person identification. Experiments on 50 subjects, with comparisons to several other state-of-the-art gait-recognition approaches, show that the proposed approach can achieve higher recognition accuracy and robustness. Gait recognition, multi-sensor integration, person identification, dense trajectory, accelerometer. ## I Introduction Using gait, or the manner of walking, for person identification has been drawing more and more attention in recent years [1, 2, 3], due to its capability to recognize a person at a longer distance than the traditional biometrics based on face, fingerprint and iris recognition. However, in practice gait biometrics usually suffer from two issues. First, the data collected by a single type of sensors, _e.g._, a CCD camera, may only capture part of the gait features and this may limit the gait recognition accuracy. Second, gait biometrics are usually sensitive to hard-covariate conditions, _e.g._, walking with hands in pocket or with loadings. In this paper, we propose to combine gait data collected by different types of sensors to promote the gait recognition accuracy and the robustness. In the previous research, three types of sensors have been used for gait data collection and gait recognition – color sensors, depth sensors and inertial sensors. Using color sensors, _e.g._, CCD cameras, a walking person can be captured into a video, in which each frame is a 2D RGB (color) image of the person and the surrounding environment. Gait recognition on such a video is usually achieved by segmenting, tracking, and analyzing the silhouette of the walking person on each frame [4, 5, 6, 7, 8, 9, 10, 11]. The silhouette segmentation and tracking can be difficult when the color of the person is similar to the color of the surrounding environment in the video. In addition, color sensors generally capture the dynamic gait features in a 2D space. Using depth sensors, such as the line-structure light devices, it is usually easier to segment a walking person from the surrounding environment, when there is no other moving objects around. In addition, from the depth data, 3D dynamic gait features can be derived for gait recognition [12, 13, 14]. However, in practice depth data may contain noise and errors, especially at the spots with strong reflectiveness, _e.g._, on a reflective clothing, where the depth value is totally invalid. Such errors may lead to incorrect gait features and gait recognition results. Different from color and depth sensors, which are installed to capture the walking person at a distance to collect gait data, inertial sensors such as accelerometers and gyroscopes collect gait data by attaching to and moving with the person [15, 16, 17, 18, 19, 20, 21]. The inertial-sensor based gait recognition mainly benefits from the extensive use of smart phones – people always carry their smart phones and almost all the smart phones have integrated inertial sensors of accelerometers and gyroscopes. Considering the usability, the smart phone must be allowed to be placed in any pockets with different orientations when we use its inertial sensors for gait recognition. Such different placements and orientations of the sensors may vary the inertial data and affect the gait recognition accuracy [18]. In general, each type of the above-mentioned sensors can capture part of the gait features with different kinds of errors and incompleteness. For example, depth and inertial sensors capture 3D gait features and color sensors capture 2D gait features. Meanwhile, the inertial data, such as the accelerometer readings, portrait the motion pattern of the whole body and provide a general description to the gait dynamics, while the color and depth data can be used to infer the motion of many body parts and provide more detailed sub-dynamics of the gait. It is natural to assume that the gait features derived from different sensors can complement each other. This motivates the proposed approach to integrate the color, depth and inertial sensors for more accurate gait recognition. Sensitivity to complex covariate conditions is another main issue in gait biometrics [6]. For example, gait data from a sensor may look different when the same person walks with hands in pocket or with loadings. Such a difference increases the variance of a person’s gait features and reduces the gait recognition accuracy. In this paper, through carefully designed experiments, we show that the proposed approach of integrating different sensors can also improve the robustness of gait recognition under complex covariate conditions. As a practical application scenario, the proposed approach of integrating different sensors for gait recognition can be used for person identification to access a restricted area or building. As illustrated in Fig. 1, at the entrance of a restricted area, a user simply walks on a force platform to get his identity verified. During his walk, a pre-installed client application in his smart phone sends real-time inertial-sensor readings to the server by wireless communication. At the same time, color and depth sensors, mounted over the ceiling and facing the platform, collect the RGBD (color and depth) data and send them to the server. In the server, the proposed approach can integrate all the data and perform gait recognition to identify whether he is an authorized user or not. Other than a higher gait recognition accuracy, such an identification system also has good security – even if the smart phone is hacked to send forged inertial data to the server, it is difficult to forge the RGBD data since color and depth sensors are not controlled by the user. Following the scheme of identification illustrated in Fig. 1, in this paper we use accelerometer in the smart phone to collect inertial data and Microsoft Kinect to collect the RGBD (color and depth) data. We develop a new EigenGait algorithm to capture the general gait dynamics by analyzing the inertial data in the eigenspace and a new TrajGait algorithm to capture more detailed gait sub-dynamics based on the 3D trajectories extracted from the RGBD video. The extracted features on general dynamics and sub-dynamics of gait are then integrated and fed into a supervised classifier for gait recognition and person identification. In the experiments, we collect three sets of inertial and RGBD data from 50 subjects and evaluate the proposed approach under various covariate conditions. Comparison results with other approaches confirm that the gait recognition accuracy and robustness can be improved by integrating different types of sensors. The main contributions of this paper lie in four-fold. 1. First, a multi-sensor integration method is proposed for gait recognition, in which inertial sensor, color sensor and depth sensor are integrated to capture gait dynamics. The multi-sensor data fusion leads to more robust gait-recognition performance. 2. Second, an EigenGait algorithm is developed to describe the general gait dynamics by analyzing the time-series acceleration data in the eigenspace. The extracted features are more effective than that produced by Fast Fourier Transforms (FFT) or Wavelet Transforms. 3. Third, a TrajGait algorithm is proposed to describe the detailed sub-dynamics of gait by analyzing the RGBD videos. In TrajGait, 3D dense trajectories are derived from the RGBD videos and used for representing the gait features. We found that such gait features are more discriminative than the depth- or skeleton- based features in gait recognition. 4. Finally, three new datasets, with both RGBD and accelerometer data, are collected on 50 subjects. They can be used to quantitatively evaluate and compare the performance of different gait recognition methods. The remainder of this paper is organized as follows. Section II reviews the related work. Section III introduces the proposed approach, including sensor setting, data collection, gait feature extraction, and integrated gait recognition. Section IV reports the experiments and results. Section V concludes our work and briefly discuss the possible future work. ## II Related Work The ideas and experiments of gait recognition can be traced back to Cutting and Kozlowski’s work [22], in which the manner of walking, _i.e._, the gait, was found to be possible to identify a person. Since then, gait-based person identification has attracted extensive attention in both academia and industry [23], and a number of gait recognition methods have been proposed. In these methods, three types of sensors are mainly used for gait data collection, namely the color sensor, the depth sensor, and the inertial sensor. Hence, the gait recognition methods can be classified into the color-based, the depth-based, and the inertia-based. In this section, we briefly overview them, as well as a brief overview to other action-based biometrics. <figure><img src="content_image/1610.09816/x1.png"><figcaption>Fig. 1: The application scenario of the proposed approach: a gait-based personidentification system for accessing a restricted area.</figcaption></figure> Color-based methods. The color-based methods had a rapid development in the early days [4, 24, 25, 5, 26, 6, 27, 28, 29, 30, 31]. These methods can be classified into the model-free methods and the model-based methods. In the model-free methods, gait features are often extracted by analyzing the shapes, or contours, of the silhouettes in successive frames. In addition, features on the velocity, texture and color are also examined. One important work among them is the GEI (gait energy image) method [5], which represents gait dynamics by an aligned and normalized silhouettes over a gait cycle. The GEI provides a compact representation of the spatial occupancy of a person over a gait cycle. However, partitioning gait cycles from a color video is rarely easy. In [25, 32], silhouettes were produced by background subtraction, and gait features were extracted by principal component analysis. In [33] and [34], statistic methods were employed to analyze the gait characteristics on a sequence of binary silhouettes images. Motion has been exploited for gait representation [35, 36, 37]. In [35], motion is described by local binary patterns, and HMM (Hidden Markov Model) is then applied to distinguish the gait dynamics of different persons. In [38], gait motions were encoded based on a set of spatio-temporal interest points from a raw gait video. These interest points were detected by using Harris corner detector from the regions with significant movements of human body in local video volumes. In [36], motions were computed based on a sequence of silhouette images. In [37], motions were computed on multi-view color videos, and the trajectories were encoded by Fisher vectors for gait representation. The model-based approaches commonly use a priori model to match the data extracted from a video [39, 40], and parameters of the model are then used for gait recognition. For example, in [40], a pendulum model is used to describe the leg movement of the body. Similar to [37], in this paper, we also extract gait features from trajectories. However we develop a new algorithm that is totally different from [37], with availability of other sensors and a goal to extract more accurate gait dynamics. First, we segment the walking person from the background by using a depth sensor. This way, we can more accurately and reliably extract the human silhouette than many human detection algorithms [41], which only generate rectangular bounding boxes around the person. Second, we compute dense trajectories other than sparse interest points and the use of dense trajectories can encode more detailed gait dynamics. Depth-based methods. With the development of depth sensors, _e.g._, Microsoft Kinect, it is easier to segment human body from the background and many depth-based gait recognition methods have been proposed recently [12, 42, 43, 44, 14]. Under the assumption that body movements can be described by the trajectories of body joints, Munsell et al [42] proposed a full-body motion-based method for person identification. It examines the motion of skeletons, _i.e._, a number of joints tracked by the Kinect, and constructs a position matrix based on the location of the joints. All the position matrices are then dealt with by an SVD (singular value decomposition) operation for feature extraction. Following the idea of GEI, Sivapalan et al [12] proposed the use of GEV (gait energy volume) to represent gait dynamics with a sequence of gait energy images, in which reasonably good recognition accuracy can be achieved based only on the frontal depth information of gait. However, these depth-based methods characterize the gait dynamics only using the depth information and neglect more detailed gait dynamics implied in the human appearance. In [14], PDV (pose depth volume) was used to improve GEV by extracting accurate human silhouettes, in which color information is used to improve the segmentation of human mask from the depth video. But PDV does not use color information for gait representation. In [45], depth features on body joints were obtained from Kinect depth camera, and the GEI features were extracted from color images. The combined RGBD features were then used for frontal gait recognition. Different from [45], the proposed method uses color images to compute the 2D dense trajectories, which are then combined to the depth data to build dense 3D trajectories for extracting more detailed gait sub-dynamics. Inertia-based methods. Early researches on inertia-based gait recognition can be found in [15] and [16]. In [15], a portable tri-axial accelerometer device is used, and the gait is represented by the correlation of acceleration curves and the distribution of acceleration signals in the frequency domain. In [16], a template matching strategy is used for gait based person identification, in which the acceleration signals are divided by gait cycles, and then dynamic time warping is applied to check the similarity of two gait curves. In [46] and [47], gait cycles were detected and cycle matching were performed to improve the accuracy of gait recognition in the context of authentication or identification. In recent years, smart phones equipped with accelerometer and gyroscope have been widely used, which makes it easier and cheaper to conduct an inertia-based gait recognition [17, 48, 18, 21]. In [18], a Mexican-Hat wavelet transform is applied to the acceleration data to analyze the gait patterns, and most discriminative features are selected based on a Fisher-ratio value. In [49], large-scale data were collected for gait recognition, in which the accelerometer is fixed on the human body. In [50], to avoid the complications in gait-cycle detection, signature-meaningful points (SPs) on the acceleration curve were detected, and gait features extracted on SPs were used for gait recognition. In [21], the gyroscope is used to rectify the orientation of the accelerometer. The acceleration signals with orientations are calculated with autocorrelation, and converted into the frequency domain using FFT. However, the gyroscope commonly has a cumulative-error problem, which may lead to an unreliable rectification and the difficulty in determining the similarity of two gait curves. Another limitation is that the detection accuracy of previous approaches highly relies on the very accurate placement of the accelerometer sensor on the human body. This strict requirement would greatly affect the usability and flexibility of the identification system. <figure><img src="content_image/1610.09816/x2.png"><figcaption>Fig. 2: Flowchart of the proposed gait-based person identification system.</figcaption></figure> Other action-based biometrics. Also related to our work is the action- or activity- based person identification [51, 52, 53, 54, 55, 56, 57, 58, 59]. Besides gait, many other actions such as jump, run and skip are also found to be capable of identifying a person. Kobayashi and Otsu [51] proposed to identify persons from a sequence of motion images using an auto-correlation-based method. By incorporating more types of human actions, Gkalelis et al [52] presented a multi-modal method for person identification, and enhanced it by using a multi-camera setup to capture the human body from different viewing angles [53]. Recently, sparse-coding-based methods were developed for human identification based on the activities captured by videos [55, 56, 57]. In [55], a metric learning procedure was performed on the sparse-coded features to get discriminative features. In [56, 57], the discriminative power was further improved by performing a discriminative sparse projection and learning a low-dimensional subspace for feature quantization. In [58], multiple Kinects were found to improve the performance of gesture-based authentication. In [59], a generative model was presented to describe the action instance creation process and an MAP-based classifier was used for identity inference on 3D skeletal datasets captured by Kinect. ## III Proposed Method ### _System Overview_ Following the application scenario of person identification shown in Fig. 1, we let the user walk straight along a corridor for gait feature collection. The inertial sensors are with the user while the color and depth sensors are placed at the end of the corridor. In this paper, we use accelerometer in the smart phone as inertial sensors and Microsoft Kinect as color and depth sensors. This way, we collect the accelerometer readings and RGBD data for gait feature extraction and gait recognition. Note that, the color (RGB) data and depth data collected by Kinect are temporally synchronized. The flowchart of the proposed system is illustrated in Fig. 2. After data pre-processing, gait features are then extracted from the inertial data and RGBD data by using the proposed EigenGait and TrajGait algorithms, respectively. Finally, the gait features are combined as an input to the machine learning component for person identification. The proposed system can be installed at the entrance of any restricted area for person identification, such as banks, financial tower, and military base etc. The proposed gait recognition combining multiple sensors is not fully non-invasive. The inertial sensors move with the user and send the accelerometer data to the server. Therefore, the user should be notified priorly and may need to show certain level of cooperation in data collection. But from the application perspective, most, if not all, existing person person-identification systems for accessing a restricted area cannot be fully non-invasive – many of them work as a verification system where the user needs to provide his identity to the server for verification at the entrance. For such a person-identification system, the goal is to achieve good usability instead of full non-invasiveness. For better usability, a person-identification system should require as fewer human interactions and less strict cooperations as possible. For the proposed system, with appropriate settings and client applications in each user’s smart phone, the data collection, including sending the inertial and RGBD data, and possibly the user’s identity, to the server, and the whole process are fully automatic without additional human interactions. In addition, as shown in the later experiments, by combining multiple sensors, the proposed system shows higher robustness against covariate conditions. This also improves the usability by requiring less strict cooperations from the user. In the following, we first introduce the data collection and data pre-processing, and then elaborate on the EigenGait algorithm for inertia-based gait representation and the TrajGait algorithms for color- and depth-based gait representation. ### _Data Collection and Pre-processing_ In this paper, we use accelerometer to collect inertial data and Kinect to collect RGBD data. #### Iii-B1 Acceleration data We utilize a tri-axial accelerometer sensor in the smart phone to collect the acceleration data of a walking person. First, we build an application on the Android platform. Given the APIs provided by the Android SDK, we use the _android.harware.SensorManager_ package and attached event listeners to the _Sensor.Type_Accelerometer_ to collect acceleration data. The sensor is registered to the _SensorManager.Sensor_Delay_Game_ and is set a sampling rate of 50Hz on each axis. <figure><img src="content_image/1610.09816/x3.png"><figcaption></figcaption></figure> Considering the usability, in data collect we simply ask the user to put the smart phone, installed with our application, in his/her pocket with any orientation. Each user is required to walk in his/her normal pace and fast pace. Since the accelerometer is placed in the pocket with a random orientation, which varies over time during the walking, the acceleration values on each axis are collected in a time-varying direction. Therefore, the acceleration values along each axis are actually not comparable from time to time. To address this issue, we fuse the acceleration values on all three axes into one compound one. Let $Acc_{x}$, $Acc_{y}$ and $Acc_{z}$ be the acceleration values on the X, Y, and Z axes, respectively, we compute the compound acceleration value $Acc_{c}$ by $Acc_{c}$=$\sqrt{Acc_{x}^{2}+Acc_{y}^{2}+Acc_{z}^{2}}$, which is more robust against the pose change of the accelerometer over time. Figure 3 shows an acceleration data sample on the X, Y and Z axes collected by a smart phone – the periodical property of the acceleration data reflects the walking pace of the user. Figure 3 shows the compound acceleration curve, which has been partitioned at local maximum. Specifically, we sequentially consider a point as the partitioning point if it satisfies three conditions: 1) it is a local maximum (peak) along the curve, 2) its distance to the previous partitioning point is no less than 700ms, and 3) its value is greater than 4$m$/$s^{2}$. Each segment of the partitioning acceleration curve corresponds to one step in the walking. Note that, in our study, one step denotes a full step cycle consisting of a left-foot move and a right-foot move. Figures 4 (a) and (b) show 100 one-step acceleration-curve segments of an user, under normal pace and fast pace, respectively, and Figures 4 (c) and (d) show those of another user. We can see that, although the acceleration curves vary a lot between different users, the acceleration curves of the same user share similar shapes, even under different paces. <figure><img src="content_image/1610.09816/x5.png"><figcaption></figcaption></figure> #### Iii-B2 Color and depth data <figure><img src="content_image/1610.09816/x9.png"><figcaption>Fig. 5: Color and depth data collected by Kinect. Top row: a sequence of RGBimages show a person walking towards the sensors. Bottom row: thecorresponding depth images. Note that, to give a better display, we crop theimages by only showing the region around the person.</figcaption></figure> <figure><img src="content_image/1610.09816/x10.png"><figcaption>Fig. 6: Data collection under eight different hard-covariate conditions.</figcaption></figure> A Kinect 2.0 assisted with Kinect SDK v2.0¹ is applied for color and depth data collection. The Kinect is placed about 0.5m up from the ground. The RGB video stream is in 24-bit true color format with a resolution of 1280$\times$1024 pixels. The depth video stream is in VGA resolution of 640$\times$480 pixels, with 13-bit depth value. The depth sensor has a practical ranging limit of 1.2-3.5m distance when using the Kinect SDK. The sampling rate is 15 fps. Figure 5 shows a sequence of color images and depth images collected by the Kinect. The depth images shown in Fig. 5 have been normalized since a single VGA channel has only 8 bits to represent a pixel. For the computation in all the experiments, the original 13-bit depth value is used, which provides a high precision to describe the motion in the depth channel. [FOOTNOTE:1][ENDFOOTNOTE] #### Iii-B3 Three datasets Using the sensor settings as described above, we collect three datasets consisting of both RGBD data and accelerometer readings. We use these data for evaluating the performance of the proposed method, as well as the comparison methods, in the later experiments. Dataset name \| Number of subjects \| \| Acceleration --- data RGBD data \| Sub-datasets \| Walking pace \| Other information Dataset #1 \| \| 10 --- Male/Female: 7/3 \| 1,000 groups --- Normal/Fast: 1:1 \| 1,000 groups --- Normal/Fast: 1:1 \| 1-step: 5,000 samples, --- 2-steps: 5,000 samples, 3-steps: 5,000 samples, 4-steps: 5,000 samples, 5-steps: 5,000 samples, for acceleration data. Normal, Fast \| \| Acceleration data --- and RGBD data are collected independently. Dataset #2 \| \| 50 --- Male/Female: 39/11 500 samples \| 500 samples \| 2-steps: 500 samples \| Normal \| \| Acceleration data --- and RGBD data are collected at the same time. Dataset #3 \| \| 50 --- Male/Female: 39/11 \| 2,400 samples --- Normal/Fast: 1:1 \| 2,400 samples --- Normal/Fast: 1:1 2-steps: 2,400 samples \| Normal, Fast \| \| Acceleration data and --- RGBD data are collected at the same time, under 8 covariate conditions. TABLE I: Description of the three collected datasets. 1. Dataset #1. This dataset is collected on 10 subjects, containing 1,000 groups of acceleration data and 1000 groups of RGBD data – 100 groups of acceleration data and 100 groups of RGBD data are collected for each subject, with half in normal pace, and half in fast pace. The acceleration data and RGBD data are collected separately. In collecting acceleration data, each subject is required to walk along a hallway, with a length of about 60 feet. _A group of acceleration data_ is defined as the sequence of acceleration values resulting from the entire walk from one end of the hallway to the other end. We partition the acceleration data into steps as illustrated in Fig. 3(b). For all the one-step acceleration data, we temporally interpolate them into a data sequence of length 50. Based on the temporally partitioning, we create 5 sub-datasets, containing one-, two-, three-, four- and five-step long data samples, respectively. In RGBD data collection, each subject is required to walk towards the Kinect 100 times, from about 5m away to 1m away to the Kinect. The sequences of frontal color and depth images of the subjects are captured. _A group of RGBD data_ is defined as the sequence of RGBD images resulting from one full walk toward the Kinect. 2. Dataset #2. This dataset contains 500 data samples of 50 subjects, with 10 data samples for each subject. Each _data sample_ consists of a sequence of acceleration data and a sequence of RGBD data, which are collected simultaneously for one full walk of a user. For each RGBD video, a frame is preserved only if the present person is recognized with all the body joints by the Kinect SDK. Each acceleration data covers about 2 steps or more. We uniformly partition each acceleration data and generate a two-step data sample. 3. Dataset #3. This dataset contains 2,400 data samples of 50 subjects, with 48 data samples for each subject. These data are collected under different covariate conditions. In particular, in collecting Dataset #3, each subject is required to walk under eight different conditions, i.e., natural walking, left hand in pocket, right hand in pocket, both hands in pocket, left hand holding a book, right hand holding a book, left hand with loadings, and right hand with loadings, as shown in Fig. 6. For each subject, 6 data samples are collected under each condition, with 3 in fast pace and 3 in normal pace. Acceleration data and RGBD data are collected simultaneously in each data sample. The information of the above three datasets is summarized in Table I. <figure><img src="content_image/1610.09816/x11.png"><figcaption>Fig. 7: An example for EigenGaits computation. Top row: the left six figuresshow the average gait curve (^Si) of six subjects in Dataset #1, the last onegives an overlay view of the total ten curves. Middle row: the left six showthe gait-curve differences (Oi) of the six subjects in the top row,respectively, and the last one gives an overlay view of ten gait-curvedifferences. Bottom row: from left to right, the top seven eigenvectors(EigenGaits, U) computed on Dataset #1.</figcaption></figure> ### _EigenGait: eigenspace feature extraction for gait representation_ A sequence of (compound) acceleration values resulting from a walk can be plotted into a 2D curve, as illustrated in Fig. 4 and we call it a _gait curve_ in this paper. Inspired by the Eigenface algorithm [60] used for image-based face recognition, we propose an EigenGait algorithm for gait recognition based on gait curves. Let $\mathcal{A}$ = $\{\mathcal{S}_{i}\|i$ = $1,2,...,N\}$ be a set of gait curves of $N$ subjects, $\mathcal{S}_{i}$ denotes the gait curves collected for the $i$th subject. Treating a gait curve as a vector, we can compute an average gait curve for the $i$th subject as \[\hat{\mathcal{S}_{i}}=\frac{1}{M_{i}}\sum_{j=1}^{M_{i}}\mathcal{S}_{i}^{(j)},\] (1) where $M_{i}$ is the total number of gait curves collected for the $i$th subject, and $\mathcal{S}_{i}^{(j)}$ is the $j$th gait curve of the $i$th subject. Further, the overall average gait curve over all the $N$ subjects can be calculated by \[\hat{\mathcal{S}}=\frac{1}{N}\sum_{i=1}^{N}\hat{\mathcal{S}_{i}}.\] (2) Then, a gait-curve difference can be calculated by \[{\mathcal{O}_{i}}=\hat{\mathcal{S}_{i}}-\hat{\mathcal{S}}.\] (3) To better illustrate the meaning of $\mathcal{O}_{i}$, we compute them on real data. Without loss of generality, let us consider the 2-step acceleration data collected in Dataset #1. In Fig. 7, the last figure in the middle row shows the gait-curve differences of ten subjects in Dataset #1. It can be seen from Fig. 7 that, the gait-curve differences also preserve the periodic property of the original gait curve, as shown in the top row of Fig. 7, and different subjects have different gait-curve differences. Then the covariance matrix can be calculated by \[{\mathcal{C}}=\frac{1}{N}\sum_{i=1}^{N}{\mathcal{O}_{i}}{\mathcal{O}_{i}^{\top }}.\] (4) We can perform an eigen-decomposition as \[{(\mathcal{\lambda},\mathcal{U})}=\mathbf{Eigen}({\mathcal{C}}),\] (5) where $\mathcal{\lambda}$ denotes the eigenvalues, and $\mathcal{U}$ denotes the corresponding eigenvectors. Suppose the eigenvalues in $\mathcal{\lambda}$ have been sorted in descending order, we select the first $r$ elements that fulfill $\sum_{i=1}^{r}\lambda_{i}\geq 0.85\cdot\sum\mathcal{\lambda}$, and hence get $r$ corresponding eigenvectors $\{u_{1},u_{2},...,u_{r}\}$. In the bottom row of Fig. 7, the seven curves show the top seven eigenvectors of the two-step sub-dataset in Dataset #1. It can be seen from Fig. 7 that, more distinctiveness can be observed in the gait-curve differences than in the original gait curves. We can also see that, these eigenvectors preserve the shape appearance of some of the original gait curves, as shown in the top row of Fig. 7 and we call them _EigenGaits_ in this paper. When a new gait curve $s$ comes, we can project it into the eigenspace defined by the $r$ eigenvectors as \[{\mathcal{\omega}_{i}}={u_{i}^{\top}}(s-\hat{\mathcal{S}}),\ i=1,2,...,r,\] (6) and obtain an EigenGait feature vector $(\mathcal{\omega}_{1},\mathcal{\omega}_{2},...,\mathcal{\omega}_{r})$. As the acceleration data reflects the whole body motion in the walking, the extracted EigenGait features can capture the general gait dynamics. ### _TrajGait: dense 3D trajectories based gait representation_ The gait data captured by color sensor and depth sensor can be represented by a sequence of color images and depth images, respectively. These images provide useful information to describe the details of body movements, e.g., the movement of each body part. We combine the color and depth data and develop a _TrajGait_ algorithm for extracting 3D dense trajectories and describing more detailed gait sub-dynamics. <figure><img src="content_image/1610.09816/x12.png"><figcaption>Fig. 8: An example of dense trajectory points extraction. (a) A color image,(b) the corresponding depth image, (c) the segmented mask image, and (d) 2Ddense trajectories within the mask, where the red dots indicate the pointpositions in the current frame. Note that, the image mask has been fine-tunedwith image operations, including hole filling, noise removal, morphologicaloperation.</figcaption></figure> <figure><img src="content_image/1610.09816/x13.png"><figcaption>Fig. 9: Illustration of the trajectories in 3D space.</figcaption></figure> TrajGait algorithm is summarized in Algorithm 1, which contains the following four key operations: 1. computMotion One each RGB color frame, we compute the dense optical flow by the algorithm proposed by F$\ddot{a}$rneback [61]². This algorithm makes a good compromise between accuracy and speed. [FOOTNOTE:2][ENDFOOTNOTE] 2. segmentMask To focus on the walking person, we segment the person from the background, and take it as a mask in later operations. Since the Kinect SDK has provided functions for efficient human detection and joints tracking [62], we apply these functions to extract a raw human mask in each frame, and then apply some image processing techniques, including hole filling and noise removal, to get the final mask. Figure 8(c) displays a human mask segmented from the depth image in Fig. 8(b). Note that, while segmenting persons from a confusing background can be very challenging on RGBD data, it is not a serious issue in the proposed application scenario of person identification – the environment is highly controlled (e.g., a hallway) and the sensors are well set, without any other moving objects around. In this paper, the following steps are taken to obtain the human mask: (i) produce human-oriented depth image using the body-segmentation function provided by Kinect SDK (i.e., IBodyIndexFrame::AccessUnderlyingBuffer), (ii) resize the depth image to the size of the color image and interpolate the resized image using bi-cubic interpolation, (iii) binarize the depth image with a threshold $t$=113, and (iv) fill the holes and remove segments that are smaller than 1,000 pixels. ``` 1:procedure TrajGait 2: input: 3: $V_{1},V_{2},...,V_{N}$: RGB data collected for $N$ subjects, 4: $D_{1},D_{2},...,D_{N}$: the corresponding depth data, 5: $X_{1},X_{2},...,X_{N}$: the number of data samples in each set, 6: $\mathcal{K}$: the number of centers in the K-means clustering, 7: $L$: the number of frames in a trajectory, 8: output: 9: $\{H_{i}\|i=1,2,...,\mathcal{X}\}$: feature histograms for all RGBD 10: videos, where $\mathcal{X}=\sum_{1}^{N}X_{i}$. 11: 12: % Calculate the trajectories of all RGBD data: 13: for ($i$=1 to $N$) do 14: for ($j$=1 to $X_{i}$) do 15: % Compute the motion on the color video $V_{i}^{(j)}$: 16: $\mathcal{M}_{i}^{(j)}\leftarrow\textbf{computMotion}(V_{i}^{(j)})$; 17: % Segment foreground (human) from the depth video: 18: Mask${}_{i}^{(j)}\leftarrow\textbf{segmentMask}(D_{i}^{(j)})$; 19: % Calculate 3D trajectories in the RGBD channel: 20: $\mathcal{T}_{i}^{(j)}\leftarrow\textbf{calcTrajectories}(\mathcal{M}_{i}^{(j)} ,D_{i}^{(j)},\$Mask${}_{i}^{(j)},L)$; 21: $\mathcal{T}_{i}\leftarrow\textbf{putInto}(\mathcal{T}_{i}^{(j)})$; 22: end for 23: end for 24: 25: % Put all trajectories together: 26: $\mathcal{T}=\{\mathcal{T}_{i}\|i=1,2,...,N\}$; 27: % Compute a number of $\mathcal{K}$ centers using Clustering: 28: $\mathcal{Y}\leftarrow\textbf{kMeans}(\mathcal{T},\mathcal{K})$; 29: 30: % Compute trajectory histogram for each RGBD data sequence: 31: for ($i$=1 to $N$) do 32: for ($j$=1 to $X_{i}$) do 33: $H_{i}^{(j)}\leftarrow\textbf{histTrajectory}(\mathcal{T}_{i}^{(j)},\mathcal{Y})$; 34: end for 35: end for 36:end procedure ``` Algorithm 1 TrajGait algorithm 3. calcTrajectories Suppose $(x,y)$ is the coordinate of a point at a frame of the collected color data, $(z)$ is the depth value of that point in the depth video, then we can locate that point with a coordinate $(x_{t},y_{t},z_{t})$ in the RGBD space. In this way, we can treat each point in the RGBD data as a 3D point. Figure 9 illustrates the trajectories in the 3D space. The shape of a trajectory encodes the local motion patterns, which we use for gait representation. Based on the 2D dense trajectories extracted by [63] in RGB channels, we can compute the corresponding 3D trajectories. Let’s further suppose point $P_{t}=(x_{t},y_{t},z_{t})$ at frame $t$ is tracked to frame $t$+1 at the point $P_{t+1}$, then, with a given trajectory length $L$, we can describe its shape by a displacement vectors, \[\mathcal{F}=(\Delta P_{t},\Delta P_{t+1},...,\Delta P_{t+L-1}),\] (7) where $\Delta P_{t}=(P_{t+1}-P_{t})=(x_{t+1}-x_{t},y_{t+1}-y_{t},z_{t+1}-z_{t})$, and $L$ is empirically set as 15. Since the gait may be collected in various walking speed [64], the resulting vector has to be normalized to reduce deviations. As the metric in the color image is different from that in the depth image, we separately normalize them by their sums of the magnitudes of the displacement vectors. We take a normalized displacement vector as a 3D trajectory descriptor. An example of 3D trajectory descriptors derived from an RGBD data sequence is shown in Fig. 10. <figure><img src="content_image/1610.09816/x14.png"><figcaption></figcaption></figure> 4. histTrajectory We apply a bag-of-words strategy to encode the 3D trajectory descriptors. Specifically, we generate a codebook with a number of $K$ codes using a clustering technique. The standard K-means algorithm is employed here for clustering. To reduce the complexity, we cluster a subset of 1,000,000 randomly selected training samples. To increase the precision, we run K-means 10 times and keep the result with the lowest K-means clustering cost. For each RGBD sequence, the extracted 3D trajectory descriptors are quantized into a histogram by hard assignment. The resulting trajectory histograms are then used for gait representation. ### _Gait Recognition_ We achieve gait recognition using a supervised classifier. We combine the gait features extracted by EigenGait and TrajGait and feed them into a machine learning component for training and testing. The trained model can then be used to recognize new unseen data samples for gait recognition and person identification. For feature combination, we simply concatenate the EigenGait features and the TrajGait features into one single feature vector. In the machine learning component, a multiclass Support Vector Machine (SVM) classifier implemented by libSVM³ is used for both training and testing [65]. A one-vs-all classification strategy is applied. To investigate the potential relation between classification accuracy and computation efficiency, we try both the linear and non-linear SVMs. For the soft-margin constant $C$ in SVM, we consistently set it 1,000 through all the experiments. [FOOTNOTE:3][ENDFOOTNOTE] ## IV Experiments and Results <figure><img src="content_image/1610.09816/x16.png"><figcaption></figcaption></figure> In this section, we use three datasets to evaluate the performance of the proposed method, as well as the comparison methods. First, we examine the effectiveness of the proposed EigenGait algorithm and the TrajGait algorithm using Dataset #1, separately. Then, we evaluate the performance of the proposed method, i.e., the one fusing EigenGait and TrajGait, on Dataset #2 by comparing its accuracy with several state-of-the-art gait recognition methods. Finally, we test the robustness of the proposed method on Dataset #3. In particular, we try to answer the following questions: 1. How effective are the EigenGait algorithm and the TrajGait algorithm for gait recognition? How do the parameters influence their performances? 2. What is the overall performance of the proposed method? Does it work better than the state-of-the-art color-based methods, depth-based methods, and inertia-based methods? 3. How robust is the proposed method in handling gait data collected under hard-covariate conditions? In the experiments, we mainly evaluate gait recognition to address a classification problem. At the end of the section, we will also evaluate the proposed method to solve an identification problem. As a classification problem, we use the classification accuracy as a metric for performance evaluation. The classification accuracy is defined as \[Accuracy=\frac{\#\ Correctly\ Classified\ Samples}{\#\ Total\ Testing\ Samples}.\] (8) In the classification, each testing sample is classified by the pre-trained SVM and receives a score vector containing $n$ score values, where $n$ is the number of subjects in training the SVM. A score value in the score vector indicates the likelihood of this sample to be from a specific subject. The sample will be recognized as being from subject $i$ if the $i$th element is the maximum in the score vector. Compared with the ground-truth subject for the test sample, we can decide whether it is correctly classified and compute the accuracy. Walking pace \| Kernel \| 1 step \| 2 steps \| 3 steps \| 4 steps \| 5 steps \| 6 steps \| 7 steps \| 8 steps ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Normal \| KL1 \| 0.9616 \| 0.9616 \| 0.9608 \| 0.9659 \| 0.9634 \| 0.9626 \| 0.9525 \| 0.9606 KCHI2 \| 0.9522 \| 0.9510 \| 0.9471 \| 0.9520 \| 0.9449 \| 0.9454 \| 0.9393 \| 0.9354 Fast \| KL1 \| 0.8948 \| 0.9308 \| 0.9515 \| 0.9398 \| 0.9467 \| 0.9437 \| 0.9387 \| 0.9250 KCHI2 \| 0.8894 \| 0.9208 \| 0.9433 \| 0.9292 \| 0.9356 \| 0.9244 \| 0.9200 \| 0.9018 Normal&Fast \| KL1 \| 0.8790 \| 0.9048 \| 0.9242 \| 0.9183 \| 0.9213 \| 0.9247 \| 0.9094 \| 0.8977 KCHI2 \| 0.8785 \| 0.8992 \| 0.9156 \| 0.9082 \| 0.9219 \| 0.9054 \| 0.8913 \| 0.8900 TABLE II: Performance (Accuracy) of EigenGait using linear and non-linear SVM. ### _Effectiveness_ We use Dataset #1 to evaluate the performance of the EigenGait algorithm and the TrajGait algorithm. Dataset #1 is collected for 10 subjects, including five sub-datasets of the acceleration data, and one sub-dataset of the RGBD data. #### Iv-A1 EigenGait There are five acceleration sub-datasets, i.e., the one-, two-, three-, four- and five-step sub-datasets. Each sub-dataset contains 5,000 acceleration data sequences, with half in normal pace and half in fast pace. The resulting EigenGait features are of dimension 43, 85, 128, 170 and 213 for the one-, two-, three-, four- and five-step data, respectively. Note that, data in the same sub-dataset have no overlaps with each other. The EigenGait algorithm is evaluated under the normal pace, fast pace and two paces mixed, i.e., normal+fast, and the results are shown in Fig. 11(a)-(c), respectively. From Fig. 11(a)-(c), we can see that, EigenGait obtains good classification accuracy in all three cases, e.g., over 0.95 in normal pace, 0.92 in fast pace, and 0.90 in normal+fast, using 30% data for training. Moreover, EigenGait shows higher accuracy under normal pace than under fast pace. This is because, a large speed variation would occur when a person walks in a fast pace, which would increase the complexity of the gait data. Decreased performances of EigenGait can be observed in Fig. 11(c), because the mixed-pace data further increase the data complexity. As can be seen from Fig. 11(b) and (c), on a dataset with large speed variations, e.g., in fast pace, or in normal+fast, EigenGait holds lower performances on the 1-step dataset than on two or more step dataset. This is because, a 1-step data is less capable of representing the gait than a 2 or more step data. Surprisingly, the EigenGait obtains comparable performances when varying the data length from 2 to 5 steps. Considering that a 2-step data can be easily captured and efficiently computed as comparing to longer data, in our later experiments, we always choose a length of 2 steps for EigenGait features, including the experiments on Dataset #2 and Dataset #3. Further, we evaluate EigenGait’s performance using linear and non-linear SVMs. Typically, the ‘KL1’ and ‘KCHI2’ kernels are employed, respectively. Table II lists the classification accuracy under different walking paces and varied data lengths, where 50% data are used for training. It can be seen that, EigenGait generally shows higher performances using a linear SVM than using a non-linear one. This is because, the EigenGait extracts gait features in the eigenspace, which makes the feature more linearly classifiable. <figure><img src="content_image/1610.09816/x20.png"><figcaption></figcaption></figure> #### Iv-A2 TrajGait We use RGBD data in Dataset #1 to evaluate the TrajGait algorithm. Specifically, we evaluate the TrajGait under different $\mathcal{K}$ for K-means clustering, and linear and non-linear SVMs. In K-means clustering, 1,000 trajectories are randomly selected for each training sample. In feature quantization, all trajectories of each data sample are used, which may span from about 8,000 to 15,000 in our experiments. Figure 11(d) shows the TrajGait accuracy when $\mathcal{K}$=256, 512 and 1024., respectively. We can see that the TrajGait achieves classification accuracies higher than 0.98 when using over 20% data for training. A higher performance can be achieved with a larger $\mathcal{K}$, i.e., the size of the codebook. It can also be observed that, under the same $\mathcal{K}$, a non-linear SVM produces a little higher accuracies than the linear one. Considering that linear SVM performs better in EigenGait and has lower computation cost, we choose the linear SVM in the proposed gait recognition by combining the EigenGait and TrajGait features. ### _Accuracy_ We evaluate the overall performance of the proposed method, i.e., EigenGait+TrajGait, by comparing it with several other inertia-based methods, color and depth based methods. Specifically, the following methods are included in the comparison, 1. Acc_Fourier [21]: An autocorrelation operation is first applied to the acceleration data, which is then converted into the frequency domain using FFT. The top half of the coefficients are selected as the gait features. 2. Acc_Wavelet [18]: The Mexican Hat Wavelet transform is used to analyze the gait patterns from the acceleration data. 3. Acc_EigenGait: The proposed EigenGait algorithm handles the acceleration data. 4. D_Skeleton [42]: The position matrix on 20 joints are decomposed by SVD, and the resulting 220-dimensional vectors are used for gait representation. 5. D_GEV [12]: The GEV is computed on the human masks extracted from depth data. The principal component analysis is then performed the same way as in our EigenGait for gait features. 6. D_TrajGait: The displacement of a trajectory is calculated only on the depth channel, with a codebook size $\mathcal{K}$=1024. 7. RGB_TrajGait: The displacement of a trajectory is calculated on the RGB channels, with $\mathcal{K}$=1024. 8. RGBD_TrajGait: The full TrajGait algorithm, i.e., trajectories extracted from the RGBD channels, with $\mathcal{K}$=1024. 9. Acc_EigenGait+RGBD_TrajGait: The full version of the proposed method by combining EigenGait and TrajGait features. We normalize the EigenGait feature and TrajGait feature independently before concatenating them together. Afterwards, we normalize the concatenated feature as an input data for SVM. The normalization is performed using an L1-norm measure. For clarity, we use Figs. 12(a) and (b) to show the results of the acceleration-based methods and the RGBD-based methods, respectively. In Fig. 12(a), the proposed EigenGait is observed with a clear higher performance than the wavelet-based or FFT-based methods, in handling acceleration data. In Fig. 12(b) we can see that, RGBD_TrajGait obtains an accuracy over 0.90 when using 30% data for training, which is much higher than that of D_Skeleton and D_GEV. The TrajGait has a higher performance on the RGB channels than on the depth channel, which indicates that the color is more effective than the depth in representing gait sub-dynamics. Meanwhile, RGBD_TrajGait outperforms RGB_TrajGait and D_TrajGait, which simply demonstrates that the color information and depth information can complement each other in characterizing the gait. It can also be seen from Fig. 12(b) that, a boosted performance can be achieved by fusing EigenGait (handling acceleration data) and TrajGait (handling RGBD data) features, i.e., EigenGait+TrajGait, which validates the effectiveness of the proposed multi-sensor data fusion strategy. ### _Robustness_ We evaluate the robustness of the proposed method with Dataset #3, which contains 2,400 data samples of 50 subjects, under 8 hard-covariate conditions, as introduced in Section III-B3. Figure 12(c) shows the results of the proposed method and the comparison methods. We can see that, the TrajGait+EigenGait, the TrajGait, and the EigenGait achieves the top three performances among all the methods. The proposed method, i.e., TrajGait+EigenGait, stably hold an classification accuracy over 0.90 when varying the amount of training data from 10% to 90%, which indicates the proposed method can better handle these hard covariates. Moreover, we investigate the detailed performance of the proposed method by figuring out the classification accuracies on each kind of hard covariate. As shown in Fig. 13, for EigenGait, the hard covariate ‘both hands in pocket’ leads to the lowest accuracy. It is because that, the acceleration would heavily vary from normal when a person walks with both hands in the pockets. While for TrajGait, ‘a hand with loadings’ will increase the difficulty for gait recognition. This is because the loadings may bring unexpected motions in the color space, as well as in the depth space, e.g., a bag is used to carry the loadings in our case. For the skeleton-based and wavelet-based methods, the average classification accuracy is about 30% and 10% lower than the proposed method, respectively. Comparing with the turbulent performances of the comparison methods on different hard covariates, the proposed method performs rather stably. <figure><img src="content_image/1610.09816/x23.png"><figcaption>Fig. 13: Classification accuracies under 8 hard-covariate conditions using 20%of the data samples for training.</figcaption></figure> ### _Person-Identification Performance_ Finally, we evaluate the proposed method in the application scenario of person identification, as shown in Fig. 1. Half of the data samples in Dataset #3 are used for training, and the remaining half are used for querying and identification. The average ROC curve [66, 67] is employed for performance evaluation. For each subject, an ROC curve is computed on the results of a one-vs-all binary classification. The ROC curve is created by plotting the true positive rate (TPR) against the false positive rate (FPR) at varying threshold settings. The TPR and FPR are defined by \[TPR=\frac{True\ Positive}{True\ Positive+\ False\ Negative},\] (9) \[FPR=\frac{False\ Positive}{False\ Positive+\ True\ Negative}.\] (10) Then the average ROC curve is computed based on all ROC curves of 50 subjects. The larger the area under the ROC curve, the better the person-identification performance. The average ROC curves for the proposed method and the comparison methods are plotted in Fig. 14. We can see that, the proposed method by combining EigenGait and TrajGait achieves the best performance. In addition, the EigenGait and the Wavelet-based method produce competing performance, but the former achieves higher TPR than the later when the FPR is below 0.05. Thus, the EigenGait would outperform the Wavelet-based method since a lower FPR is often required in a strict identification system. It can also be observed from Fig. 14 that, the TrajGait uniformly outperforms the EigenGait which may simply indicate that the TrajGait features are more discriminative by describing the detailed gait sub-dynamics. ## V Conclusion In this paper, the inertia, color and depth sensors were integrated for accurate gait recognition and robust person identification. Specifically, the accelerometer of smart phone and the RGBD sensor of Kinect were employed for data collection. An EigenGait algorithm was proposed to process the acceleration data from inertial sensor in the eigenspace, and capture the general dynamics of the gait. A TrajGait algorithm was proposed to extract gait features on the dense 3D trajectories from the RGBD data, and capture the more detailed sub-dynamics. The extracted general dynamics and detailed sub-dynamics were fused and fed into a linear SVM for training and testing. Datasets collected from 50 subjects were used for experiments and the results showed the effectiveness of the proposed method against several existing state-of-the-art gait recognition methods. <figure><img src="content_image/1610.09816/x24.png"><figcaption>Fig. 14: ROC curves of person identification on Dataset #3 using 50% datasamples for training.</figcaption></figure> In the experiments, there are several other interesting findings. First, for the acceleration-based gait recognition, the walking pace has a potential influence on accuracy of the system. Uniform walking pace under a normal speed produces better gait recognition than mixed walking paces. Second, for the RGBD-based gait recognition, motion can be better captured by wearing textured clothes, with which we can more accurately infer the detailed gait sub-dynamics for gait recognition. 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1707.08843	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 81061, "num_imgs": 6, "llama3_tokens_count": 27225 }	[ "content_image/1707.08843/x1.png", "content_image/1707.08843/x2.png", "content_image/1707.08843/x3.png", "content_image/1707.08843/x4.png", "content_image/1707.08843/x5.png", "content_image/1707.08843/x6.png" ]	# Phase-matching-free parametric oscillators based on two dimensional semiconductors A. Ciattoni${}^{1}$ alessandro.ciattoni@spin.cnr.it A. Marini${}^{2}$ andrea.marini@icfo.es C. Rizza${}^{3}$ C. Conti${}^{4,5}$ ${}^{1}$Consiglio Nazionale delle Ricerche, CNR-SPIN, Via Vetoio 10, 67100 L’Aquila, Italy ${}^{2}$ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain ${}^{3}$Department of Industrial and Information Engineering and Economics, Via G. Gronchi 18, University of L’Aquila, I-67100 L’Aquila, Italy ${}^{4}$Institute for Complex Systems (ISC-CNR), Via dei Taurini 19, 00185, Rome, Italy ${}^{5}$Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185, Rome, Italy February 23, 2024 ###### Abstract Optical parametric oscillators are widely-used pulsed and continuous-wave tunable sources for innumerable applications, as in quantum technologies, imaging and biophysics. A key drawback is material dispersion imposing the phase-matching condition that generally entails a complex setup design, thus hindering tunability and miniaturization. Here we show that the burden of phase-matching is surprisingly absent in parametric micro-resonators adopting monolayer transition-metal dichalcogenides as quadratic nonlinear materials. By the exact solution of nonlinear Maxwell equations and first-principle calculation of the semiconductor nonlinear response, we devise a novel kind of phase-matching-free miniaturized parametric oscillator operating at conventional pump intensities. We find that different two-dimensional semiconductors yield degenerate and non-degenerate emission at various spectral regions thanks to doubly-resonant mode excitation, which can be tuned through the incidence angle of the external pump laser. In addition we show that high-frequency electrical modulation can be achieved by doping through electrical gating that efficiently shifts the parametric oscillation threshold. Our results pave the way for new ultra-fast tunable micron-sized sources of entangled photons, a key device underpinning any quantum protocol. Highly-miniaturized optical parametric oscillators may also be employed in lab-on-chip technologies for biophysics, environmental pollution detection and security. KEYWORDS: two-dimensional materials, nonlinear response, parametric oscillators, sensors, quantum sources, entangled photons, micro-resonators. ## I Introduction Optical nonlinearity in photonic materials enables an enormous amount of applications such as frequency conversion [1; 2], all-optical signal processing [3; 4], and non-classical sources [5; 6]. Parametric down-conversion (PDC) furnishes tunable sources of coherent radiation [7; 8; 9; 10; 11; 12; 13; 14] and generators of entangled photons and squeezed states of light [15; 16]. In traditional configurations, a nonlinear crystal with broken centrosymmetry and second-order nonlinearity sustains PDC [7; 8; 9; 10; 11; 12]; more recently, effective PDC was reported in centrosymmetric crystals with third-order nonlinearity [13; 14] and semiconductor microcavities [17; 18; 19]. <figure><img src="content_image/1707.08843/x1.png"><figcaption>Figure 1: Phase-matching free micron-sized parametric oscillators. a.Schematic illustration of conventional three-wave parametric coupling in bulknonlinear crystals. The effective quadratic susceptibility χ(2)eff is heavilyaffected by the mismatch Δk among the wavevectors km=nmωm/c of the pump (3),signal (1) and idler (2) waves whose destructive interference Δk≠0 hindersparametric coupling. b. Sketch of the ML-TMD based parametric oscillator. Thecavity is assembled by two Bragg mirrors separated by a dielectric layer andthe ML-TMD is placed on the left mirror. The incident (i) pump field producesboth reflected (r) and transmitted (t) pump, signal and idler fields. Thethree-waves have negligible mutual dephasing inside the nonlinear ML-TMD (withquadratic surface conductivity σnm≠0) as ℓ≃0; this enables phase-matching freeparametric coupling. c. Sketch of the geometry of MX2 ML-TMDs. Fast modulationis enabled by extrinsic doping through a gate voltage with gold contactsapplied between the ML-TMD and the Bragg mirror.</figcaption></figure> <figure><img src="content_image/1707.08843/x2.png"><figcaption>Figure 2: Electronic and optical properties of MX2. a,b. Valence EV(k) andconduction EC(k) energy bands of MoS2, where k is the electron wave-vector anda=3.19\rA is the lattice parameter. c,d. Dependence of the linear surfaceconductivities of ML-TMDs on (c) the vacuum wavelength λ (for intrinsic dopingEF=0) and on (d) the Fermi level EF ensuing from extrinsic doping (at λ=1.6μm). e. PDC mixing surface conductivities of MoS2 at λ3=800 nm as a functionof the angular frequency mismatch of down-converted signal and idler wavesΔω=ω1−ω2 rescaled to the pump angular frequency ω3. f. Dependence of the realand imaginary parts of the PDC mixing conductivity σ1,3 of MX2 ML-TMDs on theFermi level EF for λ1=λ2=1.6 μm, and λ3=0.8 μm.</figcaption></figure> <figure><img src="content_image/1707.08843/x3.png"><figcaption>Figure 3: Parametric Oscillations. Analysis of the doubly resonant parametricoscillations (DRPOs) of a cavity (with PMMA as cavity dielectric) illuminatedby a λ3=780 nm pump with micron-sized Bragg Mirrors whose stop bad is centeredat 1560 nm. In subfigure a the cavity length is used as tuning parameter fornormal incidence θ=0 whereas in b and c the incidence angle θ is the tuningparameter for two assigned cavity lengths. a1. Identification of DRPOs asintersection among the parametric oscillation (PO) curve and the signalresonance (SR) and idler resonance (IR) curves in the (L/λ3,Δω/ω3) plane. a2Intensity analysis of the degenerate (Δω=0) DRPO located at L≃0.645λ3comprising the plots of (a2.1) the intensity threshold I(i)3Th versus L/λ3 andΔω/ω3 and (a2.2-4) the intensities I(t)1,I(t)2,I(t)3 of the transmittedsignal, idler and pump fields as functions of the scaled cavity length L/λ3and the incident pump intensity I(i)3. a3 Intensity analysis of the non-degenerate (Δω≠0) DRPO located at L≃3λ3 (the panels are the analogous of thoseof a2). b1. DRPOs analysis on the (θ,Δω/ω3) plane for L as in a2. Note thatthe degenerate DRPO occurs at a small angle θ. b2. Signal intensity analysisof the DRPO showing a feasible θ range. c1. DRPOs analysis on the (θ,Δω/ω3)plane for L as in a3 revealing a variety of DRPOs ad different angles θ. c2,c3signal intensity analysis of two DRPOs identified in c1.</figcaption></figure> Since three-wave parametric coupling is intrinsically weak, one can achieve low oscillation thresholds only by doubly or triply resonant optical cavities. In addition, parametric effects are severely hampered by the destructive interference among the three waves propagating with different wavenumbers $k_{1,2,3}$ in the dispersive nonlinear medium because the momentum mismatch $\Delta k=k_{3}-k_{2}-k_{1}$ does not generally vanish (see Fig.1a). To avoid this highly detrimental effect, the use of phase-matching (PM) strategies is imperative. The commonly-adopted birefringence-PM method [20] is critically sensible to the nonlinear medium orientation. Quasi-PM [21; 22] exploits the momentum due to a manufactured long-scale periodic reversal of the sign of the nonlinear susceptibility and cannot be easily applied in miniaturized system. In semiconductors, PM is achieved by the S-shaped energy-momentum polariton dispersion in the strong coupling of excitons and photons [23; 24], only accessible at low temperatures and large pump angles. Cavity PM [25], also denoted “relaxed” PM [26], occurs in Fabry-Perot microcavities with cavity length $\ell$ shorter than the coherence length $\pi/\Delta k$; this technique drastically reduces the effective quadratic susceptibility $\chi_{\rm eff}^{(2)}$ (see Fig.1a). Any of the above mentioned PM techniques entails a non-trivial setup design that is further constrained by the need of resonance operation. In this manuscript, we show that emerging two-dimensional (2D) materials with high quadratic nonlinearity open unprecedented possibilities for tunable parametric micro-sources. Very remarkably, when illuminated with different visible and infrared waves, these novel 2D materials provide a negligible dispersive dephasing owing to their atomic-scale thickness (i.e $\ell\simeq 0$, see Fig.1b). Due to the lack of destructive interference, 2D materials support PDC without any need of satisfying a PM condition. Furthermore, these “phase-matching-free” devices turn out to be very versatile and compact, with the additional tunability offered by electrical gating of 2D materials, which provides ultrafast electrical-modulation functionality. The most famous 2D material, graphene, is not the best candidate for PDC owing to the centrosymmetric structure. In principle, a static external field may break centrosymmetry and induce a $\chi_{\rm eff}^{(2)}$, but the spectrally-flat absorption of graphene is severely detrimental for PDC. Recent years have witnessed the rise of transition metal dichalcogenides (TMDs) as promising photonic 2D materials. TMDs possess several unusual optical properties dependent on the number of layers. Bulk TMDs are semiconductors with an indirect bandgap, but the optical properties of their monolayer (ML) counterpart are characterized by a direct bandgap ranging from $\sim$ 1.55 eV to $\sim$ 1.9 eV [27; 28; 29] that is beneficial for several optoelectronic applications [30]. In addition, ML-TMDs have broken centrosymmetry and thus undergo second-order nonlinear processes [31; 32; 33; 34; 35]. Here we study PDC in micro-cavities embedding ML-TMDs; we find that the cavity design is extremely flexible if compared to standard parametric oscillators thanks to their phase-matching-free operation (see Figs.1a,1b). We demonstrate that, at conventional infrared pump intensity, parametric oscillation occurs in wavelength-sized micro-cavities with ML-TMDs. We show that the output signal and idler frequencies can be engineers thanks to the mode selectivity of doubly-resonant cavities; these frequencies are tuned by the pump incidence angle and modulated electrically by an external gate voltage. ## II Results Two hexagonal lattices of chalcogen atoms embedding a plane of metal atoms arranged at trigonal prismatic sites between the chalcogen neighbors form the structure of ML-TDMs. [29] Figure 1c shows the lattice structure of MX${}_{2}$ ML-TMDs (M $=$ Mo, W, and X $=$ S, Se), and Figs.2a and 2b report the valence and conduction bands of MoS${}_{2}$ obtained from tight-binding calculations [40; 37]. The electronic band structure of other MX${}_{2}$ materials considered is qualitatively similar. The direct bandgap is about $1.5$ eV and implies transparency for infrared radiation; the linear surface conductivity has very small real part (corresponding to absorption) and higher imaginary part at infrared wavelengths. Figure 2c shows the wavelength dependence of the linear surface conductivities of MX${}_{2}$. In the presence of an external pump field with angular frequency $\omega_{3}$, the ML-TMD second-order nonlinear processes lead to down-converted signal and idler waves with angular frequencies $\omega_{1}$ and $\omega_{2}$, such that $\omega_{3}=\omega_{1}+\omega_{2}$. Figure 2e illustrates the PDC mixing surface conductivities for MoS${}_{2}$. Both linear and nonlinear conductivities are calculated by a perturbative expansion of the tight-binding Hamiltonian of MX${}_{2}$ [see Methods and Supplementary Information (SI)]. For infrared photons with energy smaller than the bandgap, extrinsic doping by an externally applied gate voltage (see Fig.1c) modifies the optical properties and leads to an increase of absorption due to free-carrier collisions and to smaller PDC mixing conductivities. Figures 2d and 2f show the dependence of linear and nonlinear surface conductivities on the Fermi level $E_{\rm F}$. As detailed below, extrinsic doping generally leads to a decrease of PDC efficiency. Figure 1b shows the parametric oscillator design with ML-TMDs. The cavity consists of a dielectric slab (thickness $L$) surrounded by two Bragg grating mirrors (BGs); the ML-TMD is placed on the left BG inside the cavity. The cavity is illuminated from the left by an incident (i) pump field (frequency $\omega_{3}$) and the oscillator produces both reflected (r) and transmitted (t) signal and idler fields with frequencies $\omega_{1}=(\omega_{3}+\Delta\omega)/2$ and $\omega_{2}=(\omega_{3}-\Delta\omega)/2$, where $\Delta\omega$ is the beat-note frequency of the parametric oscillation (PO). As detailed in Methods, the cavity equations for the fields do not contain the momentum mismatch $\Delta k$. Indeed, due to their atomic thickness, ML-TMDs are not optically characterized by a refractive index but rather by a surface conductivity. Hence, the parametric coupling produced by the quadratic surface current of ML-TMDs is not hampered by dispersion and no PM condition is required accordingly. In order to observe signal and idler generation, only the PO condition is required along with the signal resonance (SR) and idler resonance (IR) conditions leading to a dramatic reduction of the intensity threshold (see Methods). Since there is no PM requirement, such requirements can be met by adjusting either the cavity length $L$ or the pump incidence angle $\theta$ as tuning parameters. For SR and IR, one needs highly reflective mirrors for both signal and idler (see Methods), as obtained by locating stop band of the micron-sized BGs at the half of the pump frequency $\omega_{3}/2$[37]. Figure 3 shows the PO analysis for a cavity composed of two BGs with polymethyl methacrylate (PMMA) and MoS${}_{2}$ deposited on the left mirror. The infrared pump has wavelength $\lambda_{3}=780$ nm in the spectral region where the nonlinear properties of MoS${}_{2}$ are very pronounced (see Fig.2e). The BGs are tuned with their stop bands centered at $1560$ nm ($=2\lambda_{3}$) [37]. In Fig.3a we consider the case of normal incidence $\theta=0$ and we plot the PO (black), SR (red), and IR (green) curves in the ($L/\lambda_{3},\Delta\omega/\omega_{3}$) plane. Doubly resonant POs (DRPOs) corresponding to the intersection points of these three curves [37] are labeled by dashed circles. Therefore, at pump normal incidence, degenerate ($\Delta\omega=0$) and non-degenerate ($\Delta\omega\neq 0$) DRPOs exist at specific cavity lengths. Note that such oscillations also occur for sub-wavelength cavity lengths ($L<\lambda_{3}$). Each oscillation starts when the incident pump intensity $I_{3}^{(i)}$ is larger than a threshold $I_{3Th}^{(i)}$ (see Methods) [37]. Figures 3a2 and 3a3 show the threshold for two specific degenerate and non-denegenerate DRPOs.Panels a2.1 and a3.1 of Fig.3 report the thresholds (black curves on the shadowed vertical planes) corresponding to the PO (black) curves; one can observe that the minimum thresholds occur at SR and IR (identified by the intersection between red and green curves). The minimum intensity thresholds are of the order of GW$/$cm${}^{2}$ and the non-degenerate DRPO threshold is greater than the degenerate DRPO one because the reflectivity of the Bragg mirror is maximum at $\Delta\omega=0$ (i.e. at half the pump frequency, as discussed above). In panels a2.2, a2.3 and a2.4 of Fig. 3 (and, seemingly, panels a3.2, a3.3 and a3.4) we report the basic DRPO features by plotting the intensities $I_{1}^{(t)},I_{2}^{(t)},I_{3}^{(t)}$ of the transmitted signal, idler and pump fields as functions of the scaled cavity length $L/\lambda_{3}$ and the incident pump intensity. Note that, in the considered example, the range of $L/\lambda_{3}$ where the oscillation actually occurs is rather narrow owing to the adopted BGs high reflectivity. We emphasize that tuning of the PO may be realized by the pump incidence angle $\theta$, which negligibly affects the oscillation thresholds. In Fig.3b and 3c, we analyze the DRPOs by using $\theta$ as tuning parameter for a given cavity length. In particular, in Fig.3b we consider a cavity with fixed length as in Fig.3a2. The PO, SR and IR curves of Fig.3b1 intersect at a degenerate DRPO point at $\theta\simeq 6$ deg. In Fig.3b2 we plot the transmitted signal intensity $I_{1}^{(t)}$ as a function of the pump incidence angle and intensity $I_{3}^{(i)}$; one can observe that the intensity threshold is comparable to the case in Fig.3a2 and the range of angles $\theta$ where PO occurs is of the order of a hundredth of degree and experimentally feasible. We show similar results in Figs.3c1 and 3c2, where the non-degenerate DRPO of Fig.3a3 is investigated in a cavity with slightly different length, and achieved at a finite incident angle with unchanged note-beat frequency $\Delta\omega$. A more accurate analysis of Fig.3c1 also reveals that, for a given $L$, the cavity sustains multiple DRPOs (both degenerate and non-degenerate) at different incidence angles $\theta$. In Fig.3c3 we plot the transmitted intensity of a degenerate DRPO that grows with the pump intensity above the ignition threshold. The novel PO with ML-TMDs as nonlinear media are PM-free because of the atomic size of ML-TMDs. The reported several examples of POs with MoS${}_{2}$ can be also designed by other families of ML-TMDs leading to qualitatively similar results. In the Supplementary Material, we compare the calculated dependence of the pump intensity threshold as function of wavelength $\lambda_{3}$ for parametric oscillators embedding MoS${}_{2}$, WS${}_{2}$ and MoSe${}_{2}$, WSe${}_{2}$; we find that the chosen material affects the minimal threshold intensity in a given spectral range. One can optimize the choice of the material for a desired spectral content and threshold level. A further degree of freedom offered by ML-TMDs lies in the electrical tunability through an external gate voltage, as depicted in Fig.1c. The gate voltage increases the Fermi level, and hence affects nonlinearity and absorption because of the electron-electron collision in the conduction band (see Figs.2d,f). Although electrical tunability of MX${}_{2}$ has not been hitherto experimentally demonstrated, to the best of our knowledge, we emphasize that such a further degree of freedom is absent in traditional parametric oscillators. In the Supplementary Material, we report the pump intensity threshold as a function of the Fermi level of MoS${}_{2}$, and we show that the threshold may increase by one order of magnitude. The external gate voltage can switch-off PO at fixed optical pump, and fast electrical modulation of the output signal and idler fields can be achieved. ## III Conclusions POs can be excited in micron-sized cavities embedding ML-TMDs as nonlinear media at conventional pump intensities in a PM-free regime. The cavity design remains inherently free of the complexity imposed by the need for PM and may result into doubly resonant PDC of signal and idler waves. The flexibility offered by such a novel oscillator design enables the engineering of selective degenerate or non-degenerate down-converted excitations by simply modifying the incident angle of the pump field. Furthermore, the electrical tunability of ML-TMDs can modulate fast output signal and idler waves by bringing POs below threshold. Based on our calculations, we envisage that novel parametric oscillators embedding ML-TMDs are a new technology for all the applications in which highly-miniaturized tunable source are relevant, including enviromental detection, security, biophyics, imaging and spectroscopy. PM-free ML-TMD microresonators may also potentially boost the realization of micrometric sources of entangled photons when pumped slightly below threshold, thus paving the way for the development of integrated quantum processors. ###### Acknowledgements. AM acknowledges useful discussions with F. Javier García de Abajo, Emanuele Distante and Ugo Marzolino. AC and CR thank the U.S. Army International Technology Center Atlantic for financial support (Grant No. W911NF-14-1-0315). CC acknowledges funding from the Templeton foundation (Grant number 58277 ) and PRIN NEMO (reference 2015KEZNYM). Author Contributions A.C. and A.M. conceived the idea and worked out the theory. All the authors discussed the results and wrote the paper. ## IV Methods Parametric down-conversion of MX${}_{2}$. We calculate the linear and PDC mixing surface conductivities of MX${}_{2}$ starting from the tight-binding (TB) Hamiltonian of the electronic band structure [40]. Since the properties of infrared photons with energies smaller than the bandgap are determined by small electron momenta around the K and K’ valleys, we approximate the full TB Hamiltonian as a sum of ${\bf k}\cdot{\bf p}$ Hamiltonians of first and second order $H_{0}({\bf k},\tau,s)$[37], where ${\bf k}$ is the electron wavenumber and $\tau$ and $s$ are the valley and spin indexes, respectively. We then derive the light-driven electron dynamics through the minimal coupling prescription leading to the time-dependent Hamiltonian $H_{0}\left[{\bf k}+(e/\hbar){\bf A}(t),\tau,s\right]$, where $-e$ is the electron charge, $\hbar$ is the reduced Planck constant, and ${\bf A}(t)$ is the radiation potential vector, and we obtain Bloch equations for the interband coherence and the population inversion. Finally, we solve perturbatively the Bloch equations of ML-TMDs in the weak excitation limit, obtaining the surface current density ${\bf K}(t)$ after integration over the reciprocal space \[{\bf K}(t) = {\rm Re}\left\{\sum_{j=1}^{3}\left[\hat{\sigma}^{\rm L}(\omega_{j }){\bf E}_{j}{\rm e}^{-i\omega_{j}t}\right]+\hat{\sigma}^{(1,2)}{\bf E}_{1}{ \bf E}_{2}{\rm e}^{-i\omega_{3}t}+\right.\] (1) \[\left.+\hat{\sigma}^{(1,3)}{\bf E}_{1}^{}{\bf E}_{3}{\rm e}^{-i \omega_{2}t}+\hat{\sigma}^{(2,3)}{\bf E}_{2}^{}{\bf E}_{3}{\rm e}^{-i\omega_{ 1}t}\right\},\] where $\hat{\sigma}^{\rm L}(\omega_{j})$ ($j=1,2,3$) and $\hat{\sigma}^{(l,m)}$ ($l,m=1,2,3$) are the linear and PDC surface conductivity tensors, respectively. Note that our approach is based on the independent-electron approximation and is fully justified only for infrared photons far from exciton resonances occurring at photon energies higher than $1.5$ eV [38; 39]. Parametric oscillations. The signal, idler and pump fields, labelled with subscripts $1,2,3$ respectively, have frequencies $\omega_{n}$ satisfying $\omega_{1}+\omega_{2}=\omega_{3}$. By the Transfer Matrix approach, the full electromagnetic analysis of the cavity (see Supplementary Material) yields the equations \[\Delta_{1}Q_{1}+\tilde{\sigma}_{23}Q_{2}^{}Q_{3} = 0,\] \[\Delta_{2}Q_{2}+\tilde{\sigma}_{13}Q_{1}^{}Q_{3} = 0,\] \[\Delta_{3}Q_{3}+\tilde{\sigma}_{12}Q_{1}Q_{2} = P_{3},\] (2) where $Q_{1},Q_{2},Q_{3}$ are complex amplitudes proportional to the output fields produced by the pump field which is proportional to the amplitude $P_{3}$. Here $\tilde{\sigma}_{nm}$ are scaled quadratic conductivities of the MX${}_{2}$ ML-TMD and \[\Delta_{n} = \tilde{\sigma}_{n}-\frac{c}{\omega_{n}}q_{n}\left({\frac{{r_{n}^{ \left(R\right)}-1}}{{r_{n}^{\left(R\right)}+1}}+\frac{{r_{n}^{\left(R\right)}e ^{iq_{n}L}-e^{-iq_{n}L}}}{{r_{n}^{\left(R\right)}e^{iq_{n}L}+e^{-iq_{n}L}}}} \right).\] (3) are parameters characterizing the linear cavity where $\tilde{\sigma}_{n}$ are scaled linear surface conductivities, $q_{n}=\frac{\omega_{n}}{c}\sqrt{\varepsilon\left({\omega_{n}}\right)-\sin^{2}\theta}$ are the longitudinal wavenumbers inside the dielectric slab, $\epsilon(\omega)$ is the relative permittivity of the dielectric slab, $\theta$ is the pump incidence angle whereas $r_{n}^{\left(R\right)}$ are the complex reflectivities for right illumination of the left Bragg mirror (with vacuum and the dielectric slab on its left and right sides, respectively). It is worth stressing that the phase-mismatch $\Delta k=k_{3}-k_{1}-k_{2}$ does not appear in the basic cavity equations (IV). Hence parametric coupling is here not affected by the fields destructive interference and the phase-matching constraint is strictly avoided. Paramateric oscillations (POs) are solutions of Eqs.(IV) with $Q_{1}\neq 0$ and $Q_{2}\neq 0$ and in this case the compatibility of the first two equations yields (see Supplementary Material) \[\left\|{P_{3}}\right\|^{2}\geq\frac{{\Delta_{1}\Delta_{2}^{}}}{{\tilde{\sigma}_ {23}\tilde{\sigma}_{13}^{}}}\left\|{\Delta_{3}}\right\|^{2},\] (4) which is the leading PO condition. As the right hand side of Eq. (35) is generally a complex number, for the PO we have the condition \[{\rm arg}\left(\frac{\Delta_{1}}{\tilde{\sigma}_{23}}\right)={\rm arg}\left( \frac{\Delta_{2}}{\tilde{\sigma}_{13}}\right).\] (5) Eq. (5) can be physically interpreted as a locking of the phase difference ${\arg Q_{1}}-{\arg Q_{2}^{}}$ allowing the signal and idler to oscillate. Once Eq.(5) is satisfied, Eq.(35) provides the pump threshold for the onset of PO. Due to the absolute smallness of the nonlinear surface conductivities, in order to have a feasible threshold, the cavity parameters $\|\Delta_{n}\|$ must be minimized. This can be obtained by choosing the doubly resonant condition for signal and idler corresponding to the minima of $\|\Delta_{1}\|$ and $\|\Delta_{2}\|$, respectively. In order for this minima to be very small, we need that $\|r_{1}^{\left(R\right)}\|$ and $\|r_{2}^{\left(R\right)}\|$ are very close to one. One can satisfy such a constraint by a suitable Bragg mirror design to have the stop-band centered at the half of the pump frequency $\omega_{3}/2$ since, in this case, signal and idler fields experience large mirror reflectance. Here we provide additional information on technical aspects of the theoretical methods used to model parametric down-conversion and the resulting phase-matching free resonant oscillations within the micro-cavities described in the main paper. ## Appendix A Parametric down-conversion of MX${}_{2}$ We calculate the linear and parametric down-conversion (PDC) mixing surface conductivities of monolayer (ML) transition metal dichalcogenides (TMDs) MX${}_{2}$ (M $=$ Mo, S, and X $=$ S, Se) starting from a tight-binding (TB) description of the electronic band structure of these materials [40] and studying the light-driven electron dynamics by means of Bloch equations for the valence and conduction bands. Note that our approach is based on the independent-electron approximation and is fully justified at the frequencies considered in the main paper since they are far from exciton resonances happening at photon energies of $\simeq 1.5$ eV or higher [41; 42]. In addition, for the infrared photon energies considered, the relevant valence and conduction band regions affecting the infrared response are the ones closer to the band gap around the K and K’ band edges, for which the full TB Hamiltonian can be approximated by a two-band ${\bf k}\cdot{\bf p}$ Hamiltonian ${\cal H}_{0}({\bf k},\tau,s)={\cal H}_{1}({\bf k},\tau,s)+{\cal H}_{2}({\bf k} ,\tau)$[40], where \[{\cal H}_{1}({\bf k},\tau,s)=\left[\begin{array}[]{cc}\Delta/2&t_ {0}a(\tau k_{x}-ik_{y})\\ t_{0}a(\tau k_{x}+ik_{y})&\tau s\Lambda-\Delta/2\end{array}\right],\] (6) \[{\cal H}_{2}({\bf k},\tau,s)=\left[\begin{array}[]{cc}\gamma_{1}a ^{2}k^{2}&\gamma_{3}a^{2}(\tau k_{x}+ik_{y})^{2}\\ \gamma_{3}a^{2}(\tau k_{x}-ik_{y})^{2}&\gamma_{2}a^{2}k^{2}\end{array}\right],\] (7) and $\tau,s=\pm 1$ label non-degenerate valleys and spins, while ${\bf k}=(k_{x}~{}k_{y})$ indicates the electron wave-vector. Constants \| MoS2 \| MoSe2 \| WS2 \| WSe2 ---\|---\|---\|---\|--- a (Å) \| 3.190 \| 3.326 \| 3.191 \| 3.325 Δ (eV) \| 1.658 \| 1.429 \| 1.806 \| 1.541 t0 (eV) \| 0.933 \| 0.768 \| 1.196 \| 1.016 Λ (eV) \| 0.073 \| 0.091 \| 0.211 \| 0.228 γ1 (eV) \| 0.351 \| 0.291 \| 0.443 \| 0.404 γ2 (eV) \| −0.198 \| −0.191 \| −0.211 \| −0.216 γ3 (eV) \| −0.143 \| −0.099 \| −0.199 \| −0.150 Table S1: Fitted constants of the the two-band k⋅p Hamiltonian H0(k,τ,s). The physical parameters of ${\cal H}_{0}({\bf k},\tau,s)$ are obtained by fitting the ${\bf k}\cdot{\bf p}$ valence and conduction energy bands with the ones obtained from first-principles GW simulations [40] accounting for both non-degenerate valleys and spin-orbit coupling, and are listed in Table S1. We calculate the linear and PDC mixing conductivities of ML-TMDs by introducing the time-dependent Hamiltonian ${\cal H}_{0}(\mbox{\boldmath${\kappa}$}(t),\tau,s)$, where we have replaced the electron wave-vector with the minimum coupling prescription for the electron quasi-momentum $\hbar\mbox{\boldmath${\kappa}$}(t)=\hbar{\bf k}+e{\bf A}(t)$, where $e$ is the electron charge and ${\bf A}(t)$ is the electromagnetic potential vector accounting for pump, signal, and idler waves. With this prescription, we define unperturbed and interacting Hamiltonians ${\cal H}_{0}({\bf k},\tau,s)$ and ${\cal H}_{\rm I}({\bf k},\tau,s,t)$, respectively, and write the total Hamiltonian as ${\cal H}_{\rm T}({\bf k},\tau,s,t)={\cal H}_{0}[\mbox{\boldmath${\kappa}$}(t), \tau,s]={\cal H}_{0}({\bf k},\tau,s)+{\cal H}_{\rm I}({\bf k},\tau,s,t)$, where \[{\cal H}_{\rm I}({\bf k},\tau,s,t) = \frac{e}{\hbar}[D_{x}A_{x}(t)+D_{y}A_{y}(t)]+\frac{e^{2}}{\hbar^{ 2}}[D_{xx}A_{x}^{2}(t)+D_{xy}A_{x}(t)A_{y}(t)+D_{yy}A_{y}^{2}(t)],\] (9) and the interaction operators are explicitly given by \[D_{x}=t_{0}a\tau\left[\|\psi_{\rm V}\rangle\langle\psi_{\rm C}\|+\| \psi_{\rm C}\rangle\langle\psi_{\rm V}\|\right],\] \[D_{y}=it_{0}a\left[\|\psi_{\rm V}\rangle\langle\psi_{\rm C}\|-\| \psi_{\rm C}\rangle\langle\psi_{\rm V}\|\right],\] \[D_{xx}=\gamma_{1}a^{2}\|\psi_{\rm C}\rangle\langle\psi_{\rm C}\|+ \gamma_{2}a^{2}\|\psi_{\rm V}\rangle\langle\psi_{\rm V}\|+\gamma_{3}a^{2}\left[\| \psi_{\rm V}\rangle\langle\psi_{\rm C}\|+\|\psi_{\rm C}\rangle\langle\psi_{\rm V }\|\right],\] \[D_{yy}=\gamma_{1}a^{2}\|\psi_{\rm C}\rangle\langle\psi_{\rm C}\|+ \gamma_{2}a^{2}\|\psi_{\rm V}\rangle\langle\psi_{\rm V}\|-\gamma_{3}a^{2}\left[\| \psi_{\rm V}\rangle\langle\psi_{\rm C}\|+\|\psi_{\rm C}\rangle\langle\psi_{\rm V }\|\right],\] \[D_{xy}=2i\gamma_{3}a^{2}\tau\left[\|\psi_{\rm C}\rangle\langle \psi_{\rm V}\|-\|\psi_{\rm V}\rangle\langle\psi_{\rm C}\|\right].\] In the expressions above we use the Dirac notation for the conduction $\|\psi_{\rm C}\rangle$ and valence $\|\psi_{\rm V}\rangle$ band eigenstates, and we approximate the matrix elements by their values at the band edges (${\bf k}=0$). Inserting the Ansatz $\|\psi\rangle=c_{-}\|\psi_{\rm V}\rangle+c_{+}\|\psi_{\rm C}\rangle$ in the time-dependent Schrödinger equation $i\hbar\partial_{t}\|\psi\rangle={\cal H}_{\rm T}\|\psi\rangle$, and defining the inversion population $n_{\bf k}=\|c_{+}\|^{2}-\|c_{-}\|^{2}$ and the interband coherence $\rho_{\bf k}=c_{+}c_{-}^{}$, one gets \[\dot{\rho}_{\bf k} = -\frac{i}{\hbar}(E_{\rm C}-E_{\rm V})\rho_{\bf k}-\gamma\rho_{\bf k }+\frac{ie}{\hbar^{2}}n_{\bf k}\left\{D_{x}^{\rm CV}A_{x}(t)+D_{y}^{\rm CV}A_{ y}(t)+\frac{e}{\hbar}[D_{xx}^{\rm CV}A_{x}^{2}(t)+D_{xy}^{\rm CV}A_{x}(t)A_{y} (t)+D_{yy}^{\rm CV}A_{y}^{2}(t)]\right\}+\] \[+\frac{ie^{2}}{\hbar^{3}}\left[(D_{xx}^{\rm VV}-D_{xx}^{\rm CC})A _{x}^{2}(t)+(D_{yy}^{\rm VV}-D_{yy}^{\rm CC})A_{y}^{2}(t)\right]\rho_{\bf k},\] \[\dot{n}_{\bf k} = -\frac{4e}{\hbar^{2}}{\rm Im}\left\{\rho_{\bf k}\left[D_{x}^{\rm VC }A_{x}(t)+D_{y}^{\rm VC}A_{y}(t)+\frac{e}{\hbar}[D_{xx}^{\rm VC}A_{x}^{2}(t)+D _{xy}^{\rm VC}A_{x}(t)A_{y}(t)+D_{yy}^{\rm VC}A_{y}^{2}(t)]\right]\right\},\] (11) where $E_{\rm C}({\bf k})$ and $E_{\rm V}({\bf k})$ are the conduction and valence energy bands of the unperturbed Hamiltonian ${\cal H}_{0}$, $D^{\rm CV}_{j}=\langle\psi_{\rm C}\|D_{j}\|\psi_{\rm V}\rangle$ are the interaction matrix elements, and we have introduced a phenomenological relaxation rate $\gamma=10$ ps${}^{-1}$ accounting for coherence dephasing [43]. In order to obtain the PDC surface conductivities, we consider a coherent superposition of three monochromatic fields ${\bf E}(t)={\rm Re}\left\{{\bf E}_{1}{\rm e}^{-i\omega_{1}t}+{\bf E}_{2}{\rm e }^{-i\omega_{2}t}+{\bf E}_{3}{\rm e}^{-i\omega_{3}t}\right\}$ with amplitudes ${\bf E}_{1}$, ${\bf E}_{2}$, and ${\bf E}_{3}$, and with angular frequencies $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$, respectively. In our notation, the field ${\bf E}_{3}$ indicates the external pump field, while ${\bf E}_{1}$, ${\bf E}_{2}$ label the down-converted signal and idler fields, respectively. The down-converted angular frequencies $\omega_{1}$ and $\omega_{2}$ are not independent, but are such that $\omega_{1}+\omega_{2}=\omega_{3}$ owing to energy conservation. The electromagnetic potential vector related to the coherent superposition of pump, signal, and idler waves is thus given by \[{\bf A}(t)={\rm Re}\left\{({\bf E}_{1}/i\omega_{1}){\rm e}^{-i\omega_{1}t}+({ \bf E}_{2}/i\omega_{2}){\rm e}^{-i\omega_{2}t}+({\bf E}_{3}/i\omega_{3}){\rm e }^{-i\omega_{3}t}\right\}.\] (12) We then solve perturbatively the equations above in the vanishing temperature $T\to 0$ and weak excitation limits such that $n_{\bf k}\approx-\Theta\left[E_{\rm C}({\bf k})-E_{\rm F}\right]$, where $\Theta(x)$ indicates the Heaviside step function and $E_{\rm F}$ is the Fermi energy. Taking the Ansatz $\rho_{\bf k}=\sum_{j=\pm 1,\pm 2,\pm 3}\rho_{\|j\|}^{(j/\|j\|)}{\rm e}^{i(j/\|j\|) \omega_{\|j\|}t}$ and disregarding generation of higher harmonics we obtain analytical expressions for the coefficients $\rho_{\|j\|}^{(j/\|j\|)}$, finding that the macroscopic surface current density given by \[{\bf K}(t) = -\frac{e}{4\pi^{2}\hbar}\sum_{\tau,s=-1,1}\int_{-\infty}^{+\infty }dk_{x}\int_{-\infty}^{+\infty}dk_{y}\left[\langle\psi(t)\|\nabla_{\bf k}{\cal H }_{\rm T}(t)\|\psi(t)\rangle-\langle\psi_{\rm V}\|\nabla_{\bf k}{\cal H}_{\rm T} (t)\|\psi_{\rm V}\rangle\right]=\] \[= -\frac{e}{2\pi^{2}\hbar}\sum_{\tau,s=-1,1}\int_{-\infty}^{+\infty }dk_{x}\int_{-\infty}^{+\infty}dk_{y}{\rm Re}\left\{\rho_{\bf k}(t)\left[ \nabla_{\bf k}{\cal H}_{0}^{\rm VC}+\nabla_{\bf k}D_{x}^{\rm VC}\frac{e}{\hbar }A_{x}(t)+\nabla_{\bf k}D_{y}^{\rm VC}\frac{e}{\hbar}A_{y}(t)+\right.\right.\] \[\left.\left.+\nabla_{\bf k}D_{xx}^{\rm VC}\frac{e^{2}}{\hbar^{2}} A_{x}^{2}(t)+\nabla_{\bf k}D_{xy}^{\rm VC}\frac{e^{2}}{\hbar^{2}}A_{x}(t)A_{y} (t)+\nabla_{\bf k}D_{yy}^{\rm VC}\frac{e^{2}}{\hbar^{2}}A_{y}^{2}(t)\right]+\right.\] can be recast into \[{\bf K}(t)={\rm Re}\left\{\sum_{j=1}^{3}\left[\hat{\sigma}^{\rm L}(\omega_{j}) {\bf E}_{j}{\rm e}^{-i\omega_{j}t}\right]+\hat{\sigma}^{(1,2)}{\bf E}_{1}{\bf E }_{2}{\rm e}^{-i\omega_{3}t}+\hat{\sigma}^{(1,3)}{\bf E}_{1}^{}{\bf E}_{3}{ \rm e}^{-i\omega_{2}t}+\hat{\sigma}^{(2,3)}{\bf E}_{2}^{}{\bf E}_{3}{\rm e}^{ -i\omega_{1}t}\right\},\] (14) where $\hat{\sigma}^{\rm L}(\omega_{j})$ ($j=1,2,3$) and $\hat{\sigma}^{(l,m)}$ ($l,m=1,2,3$) are the linear and PDC surface conductivity tensors, respectively, and we have neglected again generation of higher harmonics. Since centrosymmetry is broken along the $y$-direction in the notation used, the relevant components of the surface conductivity tensors for PDC are the ones such that pump, signal, and idler fields are polarized along the $y$-direction, for which \[\sigma^{\rm L}_{yy}(\omega)=\frac{ie^{2}D_{yy}^{\rm CC}}{2\pi^{2} \hbar^{2}(\omega+i\gamma)}\sum_{\tau,s=\pm 1}\int_{-\infty}^{+\infty}dk_{x} \int_{-\infty}^{+\infty}dk_{y}\Theta\left[E_{\rm F}-E_{\rm C}({\bf k})\right]+\] (15) \[+\frac{e^{2}}{4i\pi^{2}\hbar^{2}\omega}\sum_{\tau,s=\pm 1}\int_{- \infty}^{+\infty}dk_{x}\int_{-\infty}^{+\infty}dk_{y}\Theta\left[E_{\rm C}({ \bf k})-E_{\rm F}\right]\left\{\frac{\|D_{y}^{\rm CV}\|^{2}}{[(E_{\rm C}-E_{\rm V })-\hbar(\omega+i\gamma)]}+\frac{\|D_{y}^{\rm CV}\|^{2}}{[(E_{\rm C}-E_{\rm V})+ \hbar(\omega+i\gamma)]}\right\},\] \[\sigma^{\rm(1,2)}_{yyy}(\omega_{1},\omega_{2},\omega_{3})=\frac{- e^{3}}{4\pi^{2}\hbar^{3}\omega_{1}\omega_{2}}\sum_{\tau,s=\pm 1}\sum_{j=1}^{3} \int_{-\infty}^{+\infty}dk_{x}\int_{-\infty}^{+\infty}dk_{y}\Theta\left[E_{\rm C }({\bf k})-E_{\rm F}\right]\left\{\frac{D_{y}^{\rm CV}D_{yy}^{\rm VC}}{[(E_{ \rm C}-E_{\rm V})-\hbar(\omega_{j}+i\gamma)]}+\right.\] \[\left.+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[(E_{\rm C}-E_{\rm V} )+\hbar(\omega_{j}+i\gamma)]}\right\},\] \[\sigma^{\rm(1,3)}_{yyy}(\omega_{1},\omega_{2},\omega_{3})=\frac{e ^{3}}{4\pi^{2}\hbar^{3}\omega_{1}\omega_{3}}\sum_{\tau,s=\pm 1}\int_{-\infty}^ {+\infty}dk_{x}\int_{-\infty}^{+\infty}dk_{y}\Theta\left[E_{\rm C}({\bf k})-E_ {\rm F}\right]\left\{\frac{D_{y}^{\rm CV}D_{yy}^{\rm VC}}{[(E_{\rm C}-E_{\rm V })+\hbar(\omega_{1}-i\gamma)]}+\right.\] (16) \[\left.+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[(E_{\rm C}-E_{\rm V} )-\hbar(\omega_{1}-i\gamma)]}+\frac{D_{y}^{\rm CV}D_{yy}^{\rm VC}}{[(E_{\rm C} -E_{\rm V})+\hbar(\omega_{2}+i\gamma)]}+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[ (E_{\rm C}-E_{\rm V})-\hbar(\omega_{2}+i\gamma)]}+\right.\] \[\left.+\frac{D_{y}^{\rm CV}D_{yy}^{\rm VC}}{[(E_{\rm C}-E_{\rm V} )-\hbar(\omega_{3}+i\gamma)]}+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[(E_{\rm C} -E_{\rm V})+\hbar(\omega_{3}+i\gamma)]}\right\},\] \[\sigma^{\rm(2,3)}_{yyy}(\omega_{1},\omega_{2},\omega_{3})=\frac{e ^{3}}{4\pi^{2}\hbar^{3}\omega_{2}\omega_{3}}\sum_{\tau,s=\pm 1}\int_{-\infty}^ {+\infty}dk_{x}\int_{-\infty}^{+\infty}dk_{y}\Theta\left[E_{\rm C}({\bf k})-E_ {\rm F}\right]\left\{\frac{D_{y}^{\rm CV}D_{yy}^{\rm VC}}{[(E_{\rm C}-E_{\rm V })+\hbar(\omega_{1}+i\gamma)]}+\right.\] (17) \[\left.+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[(E_{\rm C}-E_{\rm V} )-\hbar(\omega_{1}+i\gamma)]}+\frac{D_{y}^{\rm CV}D_{yy}^{\rm VC}}{[(E_{\rm C} -E_{\rm V})+\hbar(\omega_{2}-i\gamma)]}+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[ (E_{\rm C}-E_{\rm V})-\hbar(\omega_{2}-i\gamma)]}+\frac{D_{y}^{\rm CV}D_{yy}^{ \rm VC}}{[(E_{\rm C}-E_{\rm V})-\hbar(\omega_{3}+i\gamma)]}+\right.\] \[\left.+\frac{D_{y}^{\rm VC}D_{yy}^{\rm CV}}{[(E_{\rm C}-E_{\rm V} )+\hbar(\omega_{3}+i\gamma)]}\right\}.\] Data reported in the main paper are obtained through the expressions above. In what follows, for convenience we will assume the simplified notation $\sigma_{n}=\sigma^{\rm L}_{yy}(\omega_{n})$ and $\sigma_{nm}=\sigma^{(n,m)}_{yyy}(\omega_{1},\omega_{2},\omega_{3})$ since the pump, signal, and idler electric fields are polarized in the $y$-direction for maximizing PDC within the micro-cavity. ## Appendix B Equations for the output fields In Fig.4 we sketch the geometry of the parametric oscillator (PO) considered in our calculations. A dielectric (PMMA) slab of thickness $L$ with a MX${}_{2}$ monolayer lying on its left side (at $z=0$) is placed between two Bragg mirrors of thickness $d$ (for convenience we choose the right mirror to be the reflected $z\rightarrow-z$ copy of the left one). The left side of the cavity is illuminated with an incident $(i)$ pump field which is a monochromatic Transverse Electric (TE) plane wave of frequency $\omega_{3}$ with incidence angle $\theta$. In addition to the reflected $(r)$ and transmitted $(t)$ pump fields, due to PDC, the cavity also produces $(r)$ and $(t)$ TE plane waves at the frequencies $\omega_{1}$ (signal) and $\omega_{2}$ (idler) such that $\omega_{3}=\omega_{1}+\omega_{2}$. It is convenient to set \[\omega_{1} = \frac{1}{2}\left(\omega_{3}+\Delta\omega\right),\] \[\omega_{2} = \frac{1}{2}\left(\omega_{3}-\Delta\omega\right),\] (18) <figure><img src="content_image/1707.08843/x4.png"><figcaption>Figure 4: Parametric oscillator geometry.</figcaption></figure> since the note-beat frequency $\Delta\omega=\omega_{1}-\omega_{2}$ is sufficient to label the signal and idler frequencies produced by a pump field of frequency $\omega_{3}$. Conservation of transverse momentum of the three fields implies that the their complex amplitudes ($\sim e^{-i\omega_{n}t}$, $n=1,2,3$) are \[{\bf{E}}_{n} = e^{i\frac{\omega_{n}}{c}x\sin\theta}\left[{A_{ny}\left(z\right){ \bf{\hat{e}}}_{y}}\right],\] \[{\bf{H}}_{n} = e^{i\frac{\omega_{n}}{c}x\sin\theta}\sqrt{\frac{{\varepsilon_{0} }}{{\mu_{0}}}}\left[{A_{nx}\left(z\right){\bf{\hat{e}}}_{x}+A_{ny}\left(z \right)\sin\theta{\bf{\hat{e}}}_{z}}\right].\] (19) Accordingly the three two-component column vectors $\left(A_{nx}(z)\:A_{ny}(z)\right)^{T}$ fully describe the field and in vacuum, i.e. outside the cavity, they are \[\begin{cases}\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}=E_{n}^{\left(i\right)}\begin{pmatrix}{-\cos\theta}\\ 1\\ \end{pmatrix}e^{i\frac{\omega_{n}}{c}\left({z+d}\right)\cos\theta}+E_{n}^{ \left(r\right)}\begin{pmatrix}{\cos\theta}\\ 1\\ \end{pmatrix}e^{-i\frac{\omega_{n}}{c}\left({z+d}\right)\cos\theta},&z<-d,\\ \begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}=E_{n}^{\left(t\right)}\begin{pmatrix}{-\cos\theta}\\ 1\\ \end{pmatrix}e^{i\frac{\omega_{n}}{c}\left({z-L-d}\right)\cos\theta},&z>L+d, \end{cases}\] (20) where $E_{n}^{\left(i\right)}$, $E_{n}^{\left(r\right)}$ and $E_{n}^{\left(t\right)}$ are field amplitudes with $E_{1}^{\left(i\right)}=E_{2}^{\left(i\right)}=0$. Resorting to the transfer matrix approach, the fields at the left ($z=0^{-}$) and right ($z=0^{+}$) sides of the MX${}_{2}$ monolayer are \[\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}_{z=0^{-}} = F_{n}\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}_{z=-d},\] \[\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}_{z=0^{+}} = B_{n}B^{\prime}_{n}\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}_{z=L+d},\] (21) where $F_{n}$, $B_{n}$ and $B^{\prime}_{n}$ are the transfer matrix describing the forward, backward and backward propagations through the left Bragg mirror, the dielectric slab and the right Bragg mirror, respectively. The transfer matrix $F_{n}$ of the left Bragg mirror is the (ordered) product of the transfer matrices of the slabs composing the mirror. For later convenience it is useful to represent this matrix as [44] \[F_{n} = \begin{pmatrix}{\frac{{cq_{n}}}{{\omega_{n}\cos\theta}}\left[{ \frac{{t_{n}^{\left(L\right)}t_{n}^{\left(R\right)}+\left({1+r_{n}^{\left(L \right)}}\right)\left({1-r_{n}^{\left(R\right)}}\right)}}{{2t_{n}^{\left(R \right)}}}}\right]}&{\frac{{cq_{n}}}{{\omega_{n}}}\left[{\frac{{-t_{n}^{\left( L\right)}t_{n}^{\left(R\right)}+\left({1-r_{n}^{\left(L\right)}}\right)\left({ 1-r_{n}^{\left(R\right)}}\right)}}{{2t_{n}^{\left(R\right)}}}}\right]}\\ {\frac{1}{{\cos\theta}}\left[{\frac{{-t_{n}^{\left(R\right)}t_{n}^{\left(L \right)}+\left({1+r_{n}^{\left(L\right)}}\right)\left({1+r_{n}^{\left(R\right) }}\right)}}{{2t_{n}^{\left(R\right)}}}}\right]}&{\left[{\frac{{t_{n}^{\left(L \right)}t_{n}^{\left(R\right)}+\left({1-r_{n}^{\left(R\right)}}\right)\left({1 +r_{n}^{\left(R\right)}}\right)}}{{2t_{n}^{\left(R\right)}}}}\right]}\\ \end{pmatrix},\] where $q_{n}=\frac{\omega_{n}}{c}\sqrt{\varepsilon\left({\omega_{n}}\right)-\sin^{2}\theta}$ are the longitudinal wavenumbers inside the dielectric slab, $\epsilon(\omega)$ is the relative permittivity of the dielectric slab whereas $r_{n}^{\left(L\right)},t_{n}^{\left(L\right)},r_{n}^{\left(R\right)},t_{n}^{ \left(R\right)}$ are the complex reflectivities $r$ and transmittivities $t$ for left $(L)$ and right $(R)$ illumination of the left Bragg mirror (with vacuum and the dielectric on its left and right sides, respectively). The other relevant transfer matrices are [44] \[B_{n} = \begin{pmatrix}{\cos\left({q_{n}L}\right)}&{i\frac{{cq_{n}}}{{ \omega_{n}}}\sin\left({q_{n}L}\right)}\\ {i{{\frac{\omega_{n}}{cq_{n}}}}\sin\left({q_{n}L}\right)}&{\cos\left({q_{n}L} \right)}\\ \end{pmatrix},\] \[B^{\prime}_{n} = \begin{pmatrix}1&0\\ 0&{-1}\\ \end{pmatrix}F_{n}\begin{pmatrix}1&0\\ 0&{-1}\\ \end{pmatrix},\] (23) where the last of Eqs. (B) is a consequence of the fact that the right Bragg mirror is the reflected image of the left one. Using Eqs. (20), Eqs. (B) yield \[\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}_{z=0^{-}} = \begin{pmatrix}{V_{nx}^{\left(i\right)}}\\ {V_{ny}^{\left(i\right)}}\\ \end{pmatrix}E_{n}^{\left(i\right)}+\begin{pmatrix}{V_{nx}^{\left(r\right)}}\\ {V_{ny}^{\left(r\right)}}\\ \end{pmatrix}E_{n}^{\left(r\right)},\] \[\begin{pmatrix}{A_{nx}}\\ {A_{ny}}\\ \end{pmatrix}_{z=0^{+}} = \begin{pmatrix}{V_{nx}^{\left(t\right)}}\\ {V_{ny}^{\left(t\right)}}\\ \end{pmatrix}E_{n}^{\left(t\right)},\] (24) where \[\begin{pmatrix}{V_{nx}^{\left(i\right)}}\\ {V_{ny}^{\left(i\right)}}\\ \end{pmatrix} = \frac{1}{{t_{n}^{\left(R\right)}}}\begin{pmatrix}{\frac{{cq_{n}}} {\omega_{n}}\left[{-t_{n}^{\left(L\right)}t_{n}^{\left(R\right)}+r_{n}^{\left( L\right)}\left({r_{n}^{\left(R\right)}-1}\right)}\right]}\\ {t_{n}^{\left(L\right)}t_{n}^{\left(R\right)}-r_{n}^{\left(L\right)}\left({r_{ n}^{\left(R\right)}+1}\right)}\\ \end{pmatrix},\] \[\begin{pmatrix}{V_{nx}^{\left(r\right)}}\\ {V_{ny}^{\left(r\right)}}\\ \end{pmatrix} = \frac{1}{{t_{n}^{\left(R\right)}}}\begin{pmatrix}{-\frac{{cq_{n}} }{{\omega_{n}}}\left({r_{n}^{\left(R\right)}-1}\right)}\\ {r_{n}^{\left(R\right)}+1}\\ \end{pmatrix},\] \[\begin{pmatrix}{V_{nx}^{\left(t\right)}}\\ {V_{ny}^{\left(t\right)}}\\ \end{pmatrix} = \frac{1}{{t_{n}^{\left(R\right)}}}\begin{pmatrix}{\frac{{cq_{n}}} {\omega_{n}}\left({r_{n}^{\left(R\right)}e^{iq_{n}L}-e^{-iq_{n}L}}\right)}\\ {\left({r_{n}^{\left(R\right)}e^{iq_{n}L}+e^{-iq_{n}L}}\right)}\end{pmatrix}.\] (25) The monolayer of MX${}_{2}$ in the presence of the above TE electromagnetic field hosts a surface current whose harmonic complex amplitudes are ${\bf K}_{n}=K_{n}\hat{\bf e}_{y}$ where \[K_{1} = \left[\sigma_{1}E_{1y}+\sigma_{23}E_{2y}^{}E_{3y}\right]_{z=0},\] \[K_{2} = \left[\sigma_{2}E_{2y}+\sigma_{13}E_{1y}^{}E_{3y}\right]_{z=0},\] \[K_{3} = \left[\sigma_{3}E_{3y}+\sigma_{12}E_{1y}E_{2y}\right]_{z=0},\] (26) showing both a linear and a quadratic response to the electric field. The effect of such surface current on the field is provided by the electromagnetic boundary conditions at $z=0$, namely ${\bf{\hat{e}}}_{z}\times\left\{{\left[{{\bf{E}}_{n}}\right]_{z=0^{+}}-\left[{{ \bf{E}}_{n}}\right]_{z=0^{-}}}\right\}={\bf{0}}$ and ${\bf{\hat{e}}}_{z}\times\left\{{\left[{{\bf{H}}_{n}}\right]_{z=0^{+}}-\left[{{ \bf{H}}_{n}}\right]_{z=0^{-}}}\right\}={\bf{K}}_{n}$ which, using Eqs. (B), can be casted within the two-component column vector description as \[\begin{pmatrix}{A_{1x}}\\ {A_{1y}}\\ \end{pmatrix}_{z=0^{+}}-\begin{pmatrix}{A_{1x}}\\ {A_{1y}}\\ \end{pmatrix}_{z=0^{-}}=\begin{pmatrix}{\tilde{\sigma}_{1}A_{1y}+\tilde{\sigma }_{23}A_{2y}^{}A_{3y}}\\ 0\\ \end{pmatrix}_{z=0^{+}},\] \[\begin{pmatrix}{A_{2x}}\\ {A_{2y}}\\ \end{pmatrix}_{z=0^{+}}-\begin{pmatrix}{A_{2x}}\\ {A_{2y}}\\ \end{pmatrix}_{z=0^{-}}=\begin{pmatrix}{\tilde{\sigma}_{2}A_{2y}+\tilde{\sigma }_{13}A_{1y}^{}A_{3y}}\\ 0\\ \end{pmatrix}_{z=0^{+}},\] \[\begin{pmatrix}{A_{3x}}\\ {A_{3y}}\\ \end{pmatrix}_{z=0^{+}}-\begin{pmatrix}{A_{3x}}\\ {A_{3y}}\\ \end{pmatrix}_{z=0^{-}}=\begin{pmatrix}{\tilde{\sigma}_{3}A_{3y}+\tilde{\sigma }_{12}A_{1y}A_{2y}}\\ 0\\ \end{pmatrix}_{z=0^{+}},\] (27) where, for each component, we have set $\tilde{\sigma}=\sqrt{\mu_{0}/\varepsilon_{0}}\sigma$, where $\varepsilon_{0}$ and $\mu_{0}$ indicate the dielectric permittivity and magnetic permeability of vacuum, respectively. After inserting Eqs. (B) along with $E_{1}^{\left(i\right)}=E_{2}^{\left(i\right)}=0$ into Eqs. (B) we obtain \[V_{1x}^{\left(t\right)}E_{1}^{\left(t\right)}-V_{1x}^{\left(r \right)}E_{1}^{\left(r\right)} =\] \[V_{1y}^{\left(t\right)}E_{1}^{\left(t\right)}-V_{1y}^{\left(r \right)}E_{1}^{\left(r\right)} = 0,\] \[V_{2x}^{\left(t\right)}E_{2}^{\left(t\right)}-V_{2x}^{\left(r \right)}E_{2}^{\left(r\right)} =\] \[V_{2y}^{\left(t\right)}E_{2}^{\left(t\right)}-V_{2y}^{\left(r \right)}E_{2}^{\left(r\right)} = 0,\] \[V_{3x}^{\left(t\right)}E_{3}^{\left(t\right)}-V_{3x}^{\left(i \right)}E_{3}^{\left(i\right)}-V_{3x}^{\left(r\right)}E_{3}^{\left(r\right)} = \tilde{\sigma}_{3}V_{3y}^{\left(t\right)}E_{3}^{\left(t\right)}+ \tilde{\sigma}_{12}V_{1y}^{\left(t\right)}V_{2y}^{\left(t\right)}E_{1}^{\left( t\right)}E_{2}^{\left(t\right)},\] \[V_{3y}^{\left(t\right)}E_{3}^{\left(t\right)}-V_{3y}^{\left(i \right)}E_{3}^{\left(i\right)}-V_{3y}^{\left(r\right)}E_{3}^{\left(r\right)} = 0,\] (28) which are six equations for the six unknown amplitudes $E_{n}^{\left(t\right)}$, $E_{n}^{\left(r\right)}$ of the signal, idler and pump fields transmitted and reflected by the cavity illuminated by incident pump field of amplitude $E_{3}^{\left(i\right)}$. The second, fourth and sixth of Eqs. (B) can be written as \[E_{1}^{\left(r\right)} = \frac{{V_{1y}^{\left(t\right)}}}{{V_{1y}^{\left(r\right)}}}E_{1}^ {\left(t\right)},\] \[E_{2}^{\left(r\right)} = \frac{{V_{2y}^{\left(t\right)}}}{{V_{2y}^{\left(r\right)}}}E_{2}^ {\left(t\right)},\] \[E_{3}^{\left(r\right)} = \frac{{V_{3y}^{\left(t\right)}}}{{V_{3y}^{\left(r\right)}}}E_{3}^ {\left(t\right)}-\frac{{V_{3y}^{\left(i\right)}}}{{V_{3y}^{\left(r\right)}}}E_ {3}^{\left(i\right)},\] (29) showing that the reflected fields can be evaluated once the transmitted fields are known. Substituting the reflected fields from Eqs. (B) into Eqs. (B) and using Eqs. (B) we eventually get \[\Delta_{1}Q_{1}+\tilde{\sigma}_{23}Q_{2}^{}Q_{3} = 0,\] \[\Delta_{2}Q_{2}+\tilde{\sigma}_{13}Q_{1}^{}Q_{3} = 0,\] \[\Delta_{3}Q_{3}+\tilde{\sigma}_{12}Q_{1}Q_{2} = P_{3},\] (30) where \[Q_{n} = \left({\frac{{r_{n}^{\left(R\right)}e^{iq_{n}L}+e^{-iq_{n}L}}}{{t _{n}^{\left(R\right)}}}}\right)E_{n}^{\left(t\right)},\] \[P_{3} = \left({\frac{{2t_{3}^{\left(R\right)}\cos\theta}}{{r_{3}^{\left(R \right)}+1}}}\right)E_{3}^{\left(i\right)},\] \[\Delta_{n} = \tilde{\sigma}_{n}-\frac{cq_{n}}{\omega_{n}}\left({\frac{{r_{n}^{ \left(R\right)}-1}}{{r_{n}^{\left(R\right)}+1}}+\frac{{r_{n}^{\left(R\right)}e ^{iq_{n}L}-e^{-iq_{n}L}}}{{r_{n}^{\left(R\right)}e^{iq_{n}L}+e^{-iq_{n}L}}}} \right).\] (31) Equations (B) are the basic equations for the output fields. Note that Eqs. (B) have been derived without resorting to any electromagnetic approximation commonly used in cavity nonlinear optics (e.g. slowly varying amplitude approximation, decoupled approximation for counter-propagating waves, etc.) and this is a consequence of the fact the overall nonlinear response of the MX${}_{2}$ monolayer is confined to a single plane. ## Appendix C Doubly Resonant Parametric Oscillation conditions Equations (B) provide the amplitudes $Q_{n}$ (proportional to the amplitudes of the transmitted signal, idler and pump fields) for a given amplitude $P_{3}$ (proportional to the amplitude of the incident pump field). Note that they always admit the solution \[Q_{1}=Q_{2}=0,\quad Q_{3}=\frac{P_{3}}{\Delta_{3}},\] (32) which describes the linear response of the cavity to the pump field without parametric oscillations (POs) in turn characterized by $Q_{1}\neq 0$ and $Q_{2}\neq 0$. On the other hand, the first and the complex conjugate of the second of Eqs. (B) are a linear system for $Q_{1}$ and $Q_{2}^{}$ and it admits nontrivial solutions only if its determinant vanishes (see Section IV below) or \[\left\|{Q_{3}}\right\|^{2}=\frac{{\Delta_{1}\Delta_{2}^{}}}{{\tilde{\sigma}_{23 }\tilde{\sigma}_{13}^{}}}.\] (33) This condition entails the occurrence of POs since if it is fulfilled, Eqs. (B) have solutions with $Q_{1}\neq 0$ and $Q_{2}\neq 0$ which are stable whereas the linear one ($Q_{1}=Q_{2}=0$) becomes unstable. Since the right hand side of Eq. (33) is generally a complex number, it is evident that POs can occur _only if_ such complex number is real and positive or \[\frac{{\Delta_{1}\Delta_{2}^{}}}{{\tilde{\sigma}_{23}\tilde{\sigma}_{13}^{}} }=\left\|\frac{{\Delta_{1}\Delta_{2}^{}}}{{\tilde{\sigma}_{23}\tilde{\sigma}_{ 13}^{}}}\right\|,\] (34) which, due to Eqs. (B) and (B), is a constraint joining the cavity length $L$, the note-beat frequency $\Delta\omega$, and the incident angle $\theta$. Geometrically, Eq. (34) represents a surface $\Sigma$ of the three-dimensional cavity state space $(L,\Delta\omega,\theta)$, and its typical slices ($\theta=0$ or $L=L_{0}$) are illustrated in Figs.3a1, 3b1 and 3c1 of the main paper (green curves). At each point of the surface $\Sigma$, PO ignites if $\|Q_{3}\|$ is sufficiently large to fulfill Eq. (33). Therefore, considering an experiment where the incident pump $\|P_{3}\|$ is gradually increased starting from the linear regime where $Q_{1}=Q_{2}=0$, the PO threshold is obtained by inserting the linear solution $Q_{3}=\frac{P_{3}}{\Delta_{3}}$ into Eq. (33), thus obtaining \[\left(\left\|{P_{3}}\right\|^{2}\right)_{th}=\frac{{\Delta_{1}\Delta_{2}^{}}}{{ \tilde{\sigma}_{23}\tilde{\sigma}_{13}^{}}}\left\|{\Delta_{3}}\right\|^{2},\] (35) which, through the second of Eqs. (B) and the relation $I_{3}^{(i)}=\frac{1}{2}\sqrt{\frac{\epsilon_{0}}{\mu 0}}\left\|E_{3}^{\left(i \right)}\right\|^{2}$, entails the intensity threshold for the incident pump. Note that the denominator of the right hand side of Eq. (35) contains the nonlinear conductivities $\tilde{\sigma}_{23}$ and $\tilde{\sigma}_{13}$ whose moduli are so small to generally yield exceedingly large and unfeasible intensity thresholds. A viable way for observing POs thus necessitates the identification of the points $(L,\Delta\omega,\theta)$ of the surface $\Sigma$ for which $\|\Delta_{1}\|$ and $\|\Delta_{2}\|$ are very close to zero. An inspection of the third of Eqs. (B) reveals that $\|\Delta_{n}\|$ can not be small if $\left\|r_{n}^{\left(R\right)}\right\|$ is not close to $1$. Therefore, by choosing Bragg mirrors with high reflectivities, the third of Eqs. (B) can be expanded up to the first order in the parameter $1-\left\|{r_{n}^{\left(R\right)}}\right\|\ll 1$ thus getting \[\Delta_{n}=\left[{\tilde{\sigma}_{n}-i\left({\frac{{2cq_{n}}}{{\omega_{n}}}} \right)\frac{{\sin\Gamma_{n}}}{{\cos\Gamma_{n}+\cos\left({q_{n}L}\right)}}} \right]+\left\{{\left({\frac{{2cq_{n}}}{{\omega_{n}}}}\right)\frac{{1+\left({ \cos\Gamma_{n}+2i\sin\Gamma_{n}}\right)\cos\left({q_{n}L}\right)}}{{\left[{ \cos\Gamma_{n}+\cos\left({q_{n}L}\right)}\right]^{2}}}}\right\}\left({1-\left\| {r_{n}^{\left(R\right)}}\right\|}\right),\] (36) where $\Gamma_{n}=q_{n}L+\arg{r_{n}^{\left(R\right)}}$. The minima of $\|\Delta_{n}\|$ are easily seen to occur for $\Gamma_{n}=m\pi$ (where $m$ is any integer) which is exactly the cavity resonance condition for the frequency $\omega_{n}$[44] and where, up to the first order of $1-\left\|{r_{n}^{\left(R\right)}}\right\|$, \[\Delta_{n}=\tilde{\sigma}_{n}+\left[{\left({\frac{{2cq_{n}}}{{\omega_{n}}}} \right)\frac{1}{{1+\cos\left({\arg r_{n}^{\left(R\right)}}\right)}}}\right] \left({1-\left\|{r_{n}^{\left(R\right)}}\right\|}\right).\] (37) Therefore, as for standard POs based on bulk nonlinear media, the pump intensity threshold is here minimum when one or more of the three fields meet the cavity resonant condition. Designing a Bragg mirror with high reflectivity for both signal and idler fields is relatively simple (see the Section III below) and therefore in this paper we consider only doubly resonant (DR) states where the signal resonance (SR) and idler resonance (IR) conditions \[q_{1}L+\arg r_{1}^{\left(R\right)} = m_{1}\pi,\] \[q_{2}L+\arg r_{2}^{\left(R\right)} = m_{2}\pi,\] (38) are both achieved whereas the pump is non-resonant. Such two equations represents two surfaces $\Sigma_{1}$ and $\Sigma_{2}$ of the space $(L,\Delta\omega,\theta)$ whose typical slices are reported in Figs.3a1, 3b1 and 3c1 of the main text (black and red lines respectively) and whose intersection describes the DR cavity states where $\|\Delta_{1}\Delta_{2}\|$ is minimum. The (nontrivial) intersection among the three surfaces $\Sigma$, $\Sigma_{1}$ and $\Sigma_{2}$ is the set of the DRPO cavity states with feasible intensity threshold. Note that if $\Sigma_{1}$ and $\Sigma_{2}$ intersect at a specific $(L,0,\theta)$ point this point also belongs to the surface $\Sigma$ since for $\Delta\omega=0$ Eqs. (34) is trivially satisfied since evidently $\omega_{1}=\omega_{2}$, $\Delta_{1}=\Delta_{2}$ and $\sigma_{23}=\sigma_{13}$. In other words a degenerate ($\omega_{1}=\omega_{2}$) DR state always supports a PO which we refer to as a degenerate DRPO. In addition, note that if $\|{\rm Re}\:\tilde{\sigma}_{n}\|\ll\|{\rm Im}\:\tilde{\sigma}_{n}\|$ and $\|{\rm Re}\:\tilde{\sigma}_{nm}\|\ll\|{\rm Im}\:\tilde{\sigma}_{nm}\|$, if both Eqs. (C) are satisfied with $\Delta\omega\neq 0$, Eq. (37) implies that $\left\|{\rm Im}\left(\frac{{\Delta_{1}\Delta_{2}^{}}}{{\tilde{\sigma}_{23} \tilde{\sigma}_{13}^{}}}\right)\right\|\ll\left\|{\rm Re}\left(\frac{{\Delta_{1 }\Delta_{2}^{}}}{{\tilde{\sigma}_{23}\tilde{\sigma}_{13}^{}}}\right)\right\|$ so that, remarkably, if both linear and nonlinear absorption are small the non-degenerate DR states are always very close to PO states. As a consequence the non-degenerate DRPOs with feasible intensity threshold are associated to those points of the surface $\Sigma$ which are as close as possible to points of the intersection between the surface $\Sigma_{1}$ and $\Sigma_{2}$. Both degenerate and non-degenerate DRPO states are labelled with a dashed disk in Figs.3a1, 3b1 and 3c1 of the main text. <figure><img src="content_image/1707.08843/x5.png"><figcaption>Figure 5: Absolute value (a) and argument (b) of the Bragg Mirror complexreflectivity r(R) for normal incidence θ=0</figcaption></figure> ## Appendix D Bragg mirror design The Bragg mirror is a periodic structure composed of $N$ bi-layers whose dielectric materials have refractive indexes $n^{(a)}$ and $n^{(b)}$ and thicknesses $a$ and $b$. If the layers’ thicknesses are chosen to satisfy the Bragg interference condition \[an^{(a)}=bn^{(b)}=\frac{\pi c}{2\bar{\omega}},\] (39) the mirror has (for normal incidence $\theta=0$), a spectral stop-band centered at $\bar{\omega}$ whose width is proportional to the refractive index contrast $\|n^{(a)}-n^{(b)}\|$[44]. Within the stop-band the mirror reflectivity is very large, the larger $N$ the closer $\|r^{(R)}(\omega)\|$ to 1. As explained above in Section II, in order for one of the three fields (pump, signal and idler) to be resonant, it is necessary a very large reflectivity of the Bragg mirror at the field angular frequency, or in other words the frequency $\omega_{n}$ has to lie within the mirror stop-band. As noted above, the degenerate DR states with $\Delta\omega=0$ (where, from Eqs. (B), $\omega_{1}=\omega_{2}=\omega_{3}/2$) rigorously supports POs so that it is convenient to set the center of the mirror stop-bad at $\bar{\omega}=\omega_{3}/2$. Due to the refractive index contrast $\|n^{(a)}-n^{(b)}\|$, this condition assures that both signal and idler fields experience very large mirror reflectivity in a range of $\Delta\omega$ and can accordingly be resonant at the same time. On the other hand the pump frequency $\omega_{3}$ is twice the central mirror frequency $\bar{\omega}$ and requiring also the pump to resonate would require very large refractive contrast. To avoid this difficulty we have chosen to leave the pump out of resonance. In the analysis reported in Fig.3 of the main text, we have set as pump wavelength $\lambda_{3}=780\:{\rm nm}$. For the Bragg Mirror we have chosen the refractive indexes $n_{a}=1.2$ and $n_{b}=2.5$ so that, in order to have the center of the stop-band at $\bar{\omega}=\omega_{3}/2$ we have chosen the thicknesses $a=\lambda_{3}/(2n^{(a)})=325\:{\rm nm}$ and $b=\lambda_{3}/(2n^{(b)})=156\:{\rm nm}$. We have also set $N=8$ for dealing with an efficient, feasible and compact Bragg mirror of length $d=N(a+b)=3848\:{\rm nm}$. Using the transfer matrix approach, the complex reflectivity $r^{(R)}$ (for normal incidence $\theta=0$) of the Bragg mirror which has vacuum and the dielectric at its left and right sides, respectively, is easily evaluated and we plot its absolute value and argument in panel (a) and (b), respectively, of Fig.5. Accordingly, the Bragg mirror stop-band is centered at $\omega_{3}/2$ and its spectral width is $\simeq 0.22$$\omega_{3}$. As a consequence, if $\omega_{1}$ and $\omega_{2}$ lie within this stop-band, signal and idler waves can resonate simultaneously since $r_{1}^{(B)}=r^{(B)}(\omega_{1})$ and $r_{2}^{(B)}=r^{(B)}(\omega_{2})$ have moduli very close to $1$. The mirror stop-band width therefore yields the note-beat frequency range $0\leq\Delta\omega<0.22\omega$, which is the one considered in the analysis reported in Fig.3 of the main text. Note that $\omega_{3}$ lies outside the mirror stop-band and thus the pump field does not resonate. ## Appendix E Output Fields In order to evaluate the PO output fields $E_{n}^{\left(t\right)}$, Eqs. (B) have to be solved for a given incident pump field $E_{3}^{\left(i\right)}$. POs are characterized by $Q_{1}\neq 0$ and $Q_{2}\neq 0$ for a given $Q_{3}\neq 0$. First note that Eqs. (B) are left invariant by the gauge transformation \[Q_{1} \to Q_{1}e^{i\theta},\] \[Q_{2} \to Q_{2}e^{-i\theta},\] (40) and this implies that for a given $P_{3}$ there are infinite pairs $(Q_{1},Q_{2})$, all with the same $\Psi=\arg Q_{1}+\arg Q_{2}$. In other words, the phase difference $\Phi=\arg Q_{1}-\arg Q_{2}$ is not set by the pump field $P_{3}$. Evidently, such symmetry is spontaneously broken in actual experiments where a single pair $(Q_{1},Q_{2})$ (i.e. a single value of $\Phi$) is selected by the specific way chosen to trigger POs. In the case of POs, the first and the complex conjugate of the second of Eqs. (B) can be casted as \[\frac{{Q_{1}}}{{Q_{2}^{}}}=-\frac{{\tilde{\sigma}_{23}Q_{3}}}{{ \Delta_{1}}},\] \[\frac{{Q_{1}}}{{Q_{2}^{}}}=-\frac{{\Delta_{2}^{}}}{{\tilde{ \sigma}_{13}^{}Q_{3}^{}}},\] (41) whose consistency requires their right and left hand sides to coincide or \[\left\|{Q_{3}}\right\|^{2}=\frac{{\Delta_{1}\Delta_{2}^{}}}{{\tilde{\sigma}_{23 }\tilde{\sigma}_{13}^{}}}\] (42) which is the PO oscillation condition of Section II. [see Eq. (33)]. The right hand side of Eq. (42) is a positive real number if and only if the complex numbers $\Delta_{1}/\tilde{\sigma}_{23}$ and $\Delta_{2}/\tilde{\sigma}_{13}$ have the same argument $\varphi$ or \[\Delta_{1} = \tilde{\sigma}_{23}\left\|{\frac{{\Delta_{1}}}{{\tilde{\sigma}_{23 }}}}\right\|e^{i\varphi},\] \[\Delta_{2} = \tilde{\sigma}_{13}\left\|{\frac{{\Delta_{2}}}{{\tilde{\sigma}_{13 }}}}\right\|e^{i\varphi},\] (43) which are equivalent to Eq. (34) of Section II so that, considering only those states for which Eqs. (E) are satisfied, Eqs. (B) yield \[\left\|{\frac{{\Delta_{1}}}{{\tilde{\sigma}_{23}}}}\right\|e^{i \varphi}Q_{1}+Q_{2}^{}Q_{3} = 0,\] \[\left\|{\frac{{\Delta_{2}}}{{\tilde{\sigma}_{13}}}}\right\|e^{i \varphi}Q_{2}+Q_{1}^{}Q_{3} = 0,\] \[\Delta_{3}Q_{3}+\tilde{\sigma}_{12}Q_{1}Q_{2} = P_{3}.\] (44) To solve this equation, after noting that Eqs. (E) require that $\left\|\frac{Q_{1}}{Q_{2}}\right\|^{2}=\left\|\frac{\Delta_{2}{\tilde{\sigma}_{23 }}}{{\Delta_{1}\tilde{\sigma}_{13}}}\right\|$ and exploiting the above discussed gauge symmetry, we set \[Q_{1} = \sqrt{\left\|{\frac{{\Delta_{2}}}{{\tilde{\sigma}_{13}}}}\right\|} \left\|Q\right\|e^{i\frac{1}{2}\left({\Psi+\Phi}\right)},\] \[Q_{2} = \sqrt{\left\|{\frac{{\Delta_{1}}}{{\tilde{\sigma}_{23}}}}\right\|} \left\|Q\right\|e^{i\frac{1}{2}\left({\Psi-\Phi}\right)},\] (45) where we have used the symbol $\|Q\|$ to stress than this quantity is real. Inserting Eqs. (E) into Eqs. (E) yield \[Q_{3} = -\sqrt{\left\|{\frac{{\Delta_{1}\Delta_{2}}}{{\tilde{\sigma}_{23} \tilde{\sigma}_{13}}}}\right\|}e^{i\left({\Psi+\varphi}\right)},\] \[\left\|Q\right\|^{2}+\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}} \sqrt{\left\|{\frac{{\tilde{\sigma}_{23}\tilde{\sigma}_{13}}}{{\Delta_{1}\Delta _{2}}}}\right\|}e^{-i\Psi}Q_{3} = \sqrt{\left\|{\frac{{\tilde{\sigma}_{23}\tilde{\sigma}_{13}}}{{ \Delta_{1}\Delta_{2}}}}\right\|}\frac{{P_{3}}}{{\tilde{\sigma}_{12}}}e^{-i\Psi}.\] (46) Note that the first two of Eqs. (E) both reduces to the first of Eqs. (E) through the change of variables given by Eqs. (E) and this is a consequence of the necessary Eq. (E). Substituting $Q_{3}$ from the first of Eqs. (E) into the second we get \[\left\|Q\right\|^{2}-\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{i\varphi}= \sqrt{\left\|\frac{{\tilde{\sigma}_{23}\tilde{\sigma}_{13}}}{{\Delta_{1}\Delta_ {2}}}\right\|}\frac{{P_{3}}}{{\tilde{\sigma}_{12}}}e^{-i\Psi},\] (47) which is a single complex equation for $\|Q\|$ and $\Psi$. After equating the square-moduli of the left and right sides of this equation we get \[\left\|Q\right\|^{4}-2{\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{ \tilde{\sigma}_{12}}}e^{i\varphi}}\right)\left\|Q\right\|^{2}+\left(\left\|{\frac {{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{i\varphi}}\right\|^{2}-\left\|\frac{{ \tilde{\sigma}_{23}\tilde{\sigma}_{13}}}{{\Delta_{1}\Delta_{2}}}\right\|\left\|{ \frac{{P_{3}}}{{\tilde{\sigma}_{12}}}}\right\|^{2}\right)=0,\] (48) which is a biquadratic equation for $\left\|Q\right\|$ whose solutions are \[\left\|Q\right\|=\sqrt{{\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{ \tilde{\sigma}_{12}}}e^{i\varphi}}\right)+\xi\sqrt{\left\|\frac{{\tilde{\sigma} _{23}\tilde{\sigma}_{13}}}{{\Delta_{1}\Delta_{2}}}\right\|\left\|{\frac{{P_{3}}} {{\tilde{\sigma}_{12}}}}\right\|^{2}-{\mathop{\rm Im}\nolimits}^{2}\left({\frac {{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{i\varphi}}\right)}},\] (49) where $\xi=\pm 1$. Note that, due to the $\xi$ factor, generally there are two $\|Q\|$ corresponding to a $\|P_{3}\|$ and hence bistable POs can in principle occur. In addition, it is fundamental stressing that $\|Q\|$ is real and hence Eq. (49) provides its value only if the arguments of the square roots are positive. Before discussing the range of $\|P_{3}\|$ where this is the case (see below), we assume $\|Q\|$ real and we deduce the output field amplitudes. Equation (47) yields \[e^{i\Psi}=\sqrt{\left\|{\frac{{\tilde{\sigma}_{23}\tilde{\sigma}_{13}}}{{\Delta _{1}\Delta_{2}}}}\right\|}\frac{{P_{3}}}{{\tilde{ \sigma}_{12}\left({\left\|Q\right\|^{2}-\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12} }}e^{i\varphi}}\right)}},\] (50) which is consistent since the modulus of its right hand side, due to Eq. (49), is equal to $1$. Hence Eqs. (E) and the first of Eqs. (E) eventually yield \[Q_{1} = \zeta\left\|{\frac{{\Delta_{2}\tilde{\sigma}_{23}}}{ {\Delta_{1}\tilde{\sigma}_{13}}}}\right\|^{1/4}\sqrt{\frac{ {P_{3}}}{{\tilde{\sigma}_{12}\left({\left\|Q\right\|^{ 2}-\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{i\varphi}}\right)}}}e^{i\frac{ \Phi}{2}}\left\|Q\right\|,\] \[Q_{2} = \zeta\left\|{\frac{{\Delta_{1}\tilde{\sigma}_{13}}}{ {\Delta_{2}\tilde{\sigma}_{23}}}}\right\|^{1/4}\sqrt{\frac{ {P_{3}}}{{\tilde{\sigma}_{12}\left({\left\|Q\right\|^{ 2}-\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{i\varphi}}\right)}}}e^{-i\frac {\Phi}{2}}\left\|Q\right\|,\] \[Q_{3} = \frac{{P_{3}}}{{\tilde{\sigma}_{12} \left({\left\|Q\right\|^{2}-\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{i \varphi}}\right)}}e^{i\left({\varphi+\pi}\right)},\] (51) where $\zeta=\pm 1$ and the principal branch is assumed for all the complex square-roots. Note that the output fields of Eqs. (E) satisfy Eq. (42) so that they describe all the possible cavity POs whenever they exist or, in other words, whenever $\|Q\|$ of Eq. (49) is a positive real number. Such requirement evidently sets a range for the input pump intensity $\|P_{3}\|^{2}$ and there are four different cases corresponding to the two values of $\xi$ and of the two signs of ${\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{ i\varphi}}\right)$. The results of this analysis are reported in the following table. Here we have set \[\left(\left\|{P_{3}}\right\|^{2}\right)_{-}=\left\|{\frac{{\Delta_{1}\Delta_{2}}} {{\tilde{\sigma}_{23}\tilde{\sigma}_{13}}}}\right\|{\mathop{\rm Im}\nolimits}^{ 2}\left({\Delta_{3}\frac{{\left\|{\tilde{\sigma}_{12}}\right\|}}{{\tilde{\sigma} _{12}}}e^{i\varphi}}\right)<{\left\|{\frac{{\Delta_{1}\Delta_{2}}}{{\tilde{ \sigma}_{23}\tilde{\sigma}_{13}}}}\right\|\left\|{\Delta_{3}}\right\|^{2}}=\left( \left\|{P_{3}}\right\|^{2}\right)_{th}.\] (52) Therefore at each state where PO can occur (i.e. at a each point of the surface $\Sigma$) the scenario is the following one. If ${\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{ i\varphi}}\right)<0$, there is only one PO (with $\xi=1$) that effectively starts when Eq. (35) is satisfied, thus confirming the analysis of Section III. On the other hand, if ${\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{ i\varphi}}\right)>0$ the scenario changes qualitatively since in this case there are two allowed POs (with $\xi=1$ and $\xi=-1$) when and a single PO (with $\xi=1$) when Eq. (35) is satisfied. As a consequence in this case POs also exist _below_ the threshold. In order to grasp the reason why the sub-threshold POs have not been entailed in Section III, note that in the case ${\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{ i\varphi}}\right)>0$, if $\left\|{P_{3}}\right\|^{2}=\left(\left\|{P_{3}}\right\|^{2}\right)_{-}$, Eq. (49) implies that $\left\|Q\right\|=\sqrt{{\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{ \tilde{\sigma}_{12}}}e^{i\varphi}}\right)}\neq 0$ so that $Q_{1}\neq 0$ and $Q_{2}\neq 0$. In other words in this situation the signal and idler fields do not vanish _at the threshold_ and accordingly this case is ruled out from the reasoning of Section III where the threshold has been obtained for PO oscillation starting from the linear regime. In a realistic experiment PO is switched on starting from the linear regime, with the intensity threshold given by Eq. (35). However, once PO ignites, by changing the pump intensity, incidence angle $\theta$, or the cavity length, we argue that one can in principle access sub-threshold PO states. In every design considered in the main paper we have focused on the case ${\mathop{\rm Re}\nolimits}\left({\frac{{\Delta_{3}}}{{\tilde{\sigma}_{12}}}e^{ i\varphi}}\right)<0$, where sub-threshold PO does not occur. <figure><img src="content_image/1707.08843/x6.png"><figcaption>Figure 6: Parametric Oscillation Thresholds. a,b Pump wavelength λ3 dependenceof the pump intensity thresholds for parametric oscillators embedding MoS2,WS2 (a) and MoSe2, WSe2 (b). c Pump intensity threshold of a parametricoscillator embedding MoS2 as a function of the Fermi level. The thresholds areassociated to non-degenerate (Δω=0) POs and they have been calculated forcavities whose length L is equal to the pump wavelength λ3.</figcaption></figure> ## Appendix F Pump intensity thresholds In Fig. 6 we compare the calculated pump intensity thresholds versus the pump wavelength $\lambda_{3}$ for parametric oscillators embedding MoS${}_{2}$, WS${}_{2}$ (Fig. 6a) and MoSe${}_{2}$, WSe${}_{2}$ (Fig. 6b). Note that, while the minimal pump intensity threshold occurs at $\lambda_{3}\approx 780$ nm for MoS${}_{2}$ and WS${}_{2}$, it shifts to $\lambda_{3}\approx 940$ nm for MoSe${}_{2}$ and WSe${}_{2}$. None of the ML-TMDs examined enables feasible PO with low pump intensity threshold at optical frequencies owing to the enlarged absorption in this frequency range, which is the main responsible for oscillation quenching. In addition, at optical frequencies such materials exhibit exciton resonances [41] (not taken into account in our theoretical approach) that are also detrimental for POs owing to the enhanced absorption they are accompanied with. In Fig. 6c we plot the pump intensity threshold as a function of the Fermi level of MoS${}_{2}$, showing that it can be increased efficiently. 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0711.1083	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 99549, "num_imgs": 20, "llama3_tokens_count": 27227 }	[ "content_image/0711.1083/x1.png", "content_image/0711.1083/x2.png", "content_image/0711.1083/x3.png", "content_image/0711.1083/x4.png", "content_image/0711.1083/x5.png", "content_image/0711.1083/x6.png", "content_image/0711.1083/x7.png", "content_image/0711.1083/x8.png", "content_image/0711.1083/x9.png", "content_image/0711.1083/x10.png", "content_image/0711.1083/x11.png", "content_image/0711.1083/x12.png", "content_image/0711.1083/x13.png", "content_image/0711.1083/x14.png", "content_image/0711.1083/x15.png", "content_image/0711.1083/x16.png", "content_image/0711.1083/x17.png", "content_image/0711.1083/x18.png", "content_image/0711.1083/x19.png", "content_image/0711.1083/x20.png" ]	# The Faint and Extremely Red K-band Selected Galaxy Population in the DEEP2/Palomar Fields C. J. Conselice${}^{1,2}$, K. Bundy${}^{2,3}$, Vivian U${}^{2,4}$, P. Eisenhardt${}^{5}$, J. Lotz${}^{6}$, J. Newman${}^{7}$ ${}^{1}$University of Nottingham, School of Physics & Astronomy, Nottingham, NG7 2RD UK ${}^{2}$Previously: Palomar Observatory, Caltech, MC 105-24, Pasadena, CA ${}^{3}$Department of Astronomy, University of Toronto, Canada ${}^{4}$Institute for Astronomy, University of Hawaii ${}^{5}$Jet Proportion Laboratory, Caltech, Pasadena, CA ${}^{6}$NOAO, Tuscon, AZ ${}^{7}$Lawrence Berkeley National Laboratory, Berkeley CA E-mail: conselice@nottingham.ac.uk Accepted ; Received ; in original form ###### Abstract We present in this paper an analysis of the faint and red near-infrared selected galaxy population found in near-infrared imaging from the Palomar Observatory Wide-Field Infrared Survey. This survey covers 1.53 deg${}^{2}$ to 5 $\sigma$ detection limits of K${}_{\rm vega}$ = 20.5-21 and J${}_{\rm vega}$ = 22.5, and overlaps with the DEEP2 spectroscopic redshift survey. We discuss the details of this NIR survey, including our J and K band counts. We show that the K-band galaxy population has a redshift distribution that varies with K-magnitude, with most K $<17$ galaxies at $z<1.5$ and a significant fraction (38.3$\pm 0.3$%) of $K>19$ systems at $z>1.5$. We further investigate the stellar masses and morphological properties of K-selected galaxies, particularly extremely red objects, as defined by $(R-K)>5.3$ and $(I-K)>4$. One of our conclusions is that the ERO selection is a good method for picking out galaxies at $z>1.2$, and within our magnitude limits, the most massive galaxies at these redshifts. The ERO limit finds 75% of all M${}_{}>10^{11}$ M${}_{\odot}\,$galaxies at $z\sim 1.5$ down to K${}_{\rm vega}=19.7$. We further find that the morphological break-down of $K<19.7$ EROs is dominated by early-types (57$\pm 3$%) and peculiars (34$\pm 3$%). However, about a fourth of the early-types are distorted ellipticals, and within CAS parameter space these bridge the early-type and peculiar population, suggesting a morphological evolutionary sequence. We also investigate the use of a $(I-K)>4$ selection to locate EROs, finding that it selects galaxies at slightly higher average redshifts ($<z>=1.43\pm 0.32$) than the $(R-K)>5.3$ limit with $<z>=1.28\pm 0.23$. Finally, by using the redshift distribution of $K<20$ selected galaxies, and the properties of our EROs, we are able to rule out all monolithic collapse models for the formation of massive galaxies. keywords: Galaxies: Evolution, Formation, Structure, Morphology, Classification ## 1 Introduction Deep imaging and spectroscopic surveys in the optical have become the standard method for determining the evolution of the galaxy population (e.g., Kron 1980; Steidel & Hamilton 1993; Williams et al. 1996; Giavalisco et al. 2004). These surveys have revolutionised galaxy formation studies, and have allowed us to characterise basic properties of galaxies, such as their luminosity functions, stellar masses, and morphologies, and how these properties have evolved (e.g., Lilly et al. 1995; Ellis 1997; Wolf et al. 2003; Conselice et al. 2005a). However, due to technological limitations with near-infrared arrays, most deep surveys have been conducted in optical light, typically between $\lambda=$ 4000-8000 Å. This puts limits on the usefulness of optical surveys, as they select galaxies in the rest-frame ultraviolet at higher redshifts where many of the galaxies contributing to the faint end of optical counts are located (e.g., Ellis et al. 1996). Galaxies selected in the rest-frame ultraviolet limit our ability to trace the evolution of the galaxy population in terms of stellar masses. As the properties of galaxies can be quite different between the rest-frame optical and UV (e.g., Windhorst et al. 2002; Papovich et al. 2003, 2005; Taylor-Manger et al. 2006; Ravindranath et al. 2006), it is desirable to trace galaxy evolution at wavelengths where most of the stellar mass in galaxies is visible. To make advances in our understanding of galaxy evolution and formation at high redshifts, ($z>1$), therefore requires us to search for, and investigate, galaxy properties in the near-infrared (NIR). Studying galaxies in the NIR has many advantages, including minimised K-corrections which are often substantial in the optical, as well as giving us a more direct probe of galaxy stellar mass up to $z\sim 3$ (e.g., Cowie et al. 1994). This has been recognised for many years, but most NIR surveys have been either deep pencil beam surveys (e.g., Gardner 1995; Djorgovski et al. 1995; Moustakas et al. 1997; Bershady, Lowenthal & Koo 1998; Dickinson et al. 2000; Saracco et al. 2001; Franx et al. 2003), or large-area, but shallow, surveys (e.g., Mobasher et al. 1986; Saracco et al. 1999; McCracken et al. 2000; Huang et al. 2001; Martini 2001; Drory et al. 2001; Elston et al. 2006). This is potentially a problem for understanding massive and evolved galaxies at high redshifts, as red objects are highly clustered (e.g., Daddi et al. 2000; Foucaud et al. 2007), as are the most massive galaxies (e.g., Coil et al. 2004a). Previous deep NIR surveys can detect these unique galaxy populations, but they typically do not have a large enough area to probe the range of galaxies selected in the near-infrared. Likewise, large area, but shallow surveys may not be deep enough to detect with a high enough signal to noise these unique populations. In this paper we overcome this problem by presenting a 1.5 deg${}^{2}$ survey down to 5 $\sigma$ depths of K${}_{\rm vega}$ = 20.2-21.5 and J${}_{\rm vega}=22.5$. This brings together the properties of both deep and wide surveys. In this paper we explore NIR galaxy counts, and study the properties of faint NIR galaxies, which are often red in near-infrared/optical colours. Unique galaxy populations have long been known to exist in near-infrared selected surveys. These include the extremely red objects (Elston, Reike, Reike 1988) and the distant red galaxies (Saracco et al. 2001; Franx et al. 2003; Conselice et al. 2007a; Foucaud et al. 2007), both of which are difficult to study at optical wavelengths. The existence of these galaxies reveals a large possible differential in the galaxy population between optical and near-infrared surveys. While these objects can be located in deep optical surveys, they are often very faint with $R>26$ (e.g., van Dokkum et al. 2006), making it difficult to understand these objects in any detail without NIR imaging or spectroscopy. In this paper we analyse the properties of galaxies selected in moderately deep K-band imaging. We also investigate the redshift distributions, structures and properties of the near-infrared selected galaxy population down to $K_{\rm vega}\sim 20$. One of our main conclusions are that the faint K-band population spans a range of redshift and properties. We find that galaxies with magnitudes $K_{\rm vega}<17$ are nearly all at $z<1.4$. Galaxies with magnitudes $K_{\rm vega}=17-21$ are found at high redshift, up to at least $z\sim 4$. The colours of these galaxies span a wide range, with in particular redder galaxies seen at higher redshifts. Finally, we investigate the properties of the extremely red objects in our sample, finding that they include most, but not all of, the highest mass galaxies at $z\sim 1.5$. The morphologies of these EROs, and the redshift distribution and dust properties of $K_{\rm vega}<20$ sources, show that hierarchical galaxy formation is the dominate method by which massive galaxies form. This paper is organised as follows: in §2 we discuss the data sources used in this paper, including our Palomar imaging, DEEP2 redshifts, and HST ACS imaging. This section also gives basic details of the Palomar survey. §3 includes our analysis, which contains information on the K and J-band counts, the redshift and colour distributions of K-selected galaxies. §4 includes an analysis of the extremely red galaxy population and its properties, including stellar mass, dust content, and redshift distributions. §5 includes a detailed discussion of our results in terms of galaxy models, while §6 is a summary of our findings. This paper uses Vega magnitudes unless otherwise specified, and assumes a cosmology with H${}_{0}=70$ km s${}^{-1}$ Mpc${}^{-1}$, $\Omega_{\rm m}=0.3$ and $\Omega_{\lambda}=0.7$. ## 2 Data, Reduction and Data Products ### Data Sources The objects we study in this paper consist of those found in the fields covered by the Palomar Observatory Wide-Field Infrared Survey (POWIR, Table 1). The POWIR survey was designed to obtain deep K-band and J-band data over a significant ($\sim$1.5 deg${}^{2}$) area. Observations were carried out between September 2002 and October 2005 over a total of $\sim 70$ nights. This survey covers the GOODS field North (Giavalisco et al. 2004; Bundy et al. 2005), the Extended Groth Strip (Davis et al. 2007), and three other fields the DEEP2 team has observed with the DEIMOS spectrograph (Davis et al. 2003). We however do not analyse the GOODS-North data in this paper given its much smaller area and deeper depth than the K-band imaging covering the DEEP2 fields. The total area we cover in the K-band in the DEEP2 fields is 5524 arcmin${}^{2}$ = 1.53 deg${}^{2}$, with half of this area imaged in the J-band. Our goal depth was K${}_{\rm vega}=21$, although not all fields are covered this deep, but all have 5 $\sigma$ depths between K = 20.2-21.5. Table 1 lists the DEEP2 fields, and the area we have covered in each. For our purposes we abbreviate the fields covered as: EGS (Extended Groth Strip), Field 2, Field 3, and Field 4. The K-band data were acquired utilising the WIRC camera on the Palomar 5 meter telescope. WIRC has an effective field of view of $8.1\arcmin\times 8.1\arcmin$, with a pixel scale of 0.25″pixel${}^{-1}$. In total, our K-band survey consists of 75 WIRC pointings. During observations of the K data we used 30 second integrations with four exposures per pointing. Longer exposure were utilised for the J-band data, with an exposure time of 120 seconds per pointing. Total exposure times in both K and J were between one to eight hours. The seeing FWHM in the K-band data ranges from 0.8” to 1.2”, and is on average 1.0”. Photometric calibration was carried out by referencing Persson standard stars during photometric conditions. The final K-band and J-band images were made by combining individual mosaics obtained over several nights. The K-band mosaics are comprised of co-additions of $4\times 30$ second exposures dithered over a non-repeating 7.0” pattern. The J-band mosaics were analysed in a similar way using single 120 second exposures per pointing. The images were processed using a double-pass reduction pipeline we developed specifically for WIRC. For galaxy detection and photometry we utilised the SExtractor package (Bertin & Arnouts 1996). Photometric errors, and the K-band detection limit for each image were estimated by randomly inserting fake objects of known magnitude into each image, and then measuring photometry with the same detection parameters used for real objects. The inserted objects were given Gaussian profiles with a FWHM of 1$\aas@@fstack{\prime\prime}$3 to approximate the shape of slightly extended, distant galaxies. We also investigated the completeness and retrievability of magnitudes for exponential and de Vaucouleurs profiles of various sizes and magnitudes. A more detailed discussion of this is included in §2.4 and Conselice et al. (2007b). Other data used in this paper consists of: optical imaging from the CFHT over all of the fields, imaging from the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope, and spectroscopy from the DEIMOS spectrograph on the Keck II telescope (Davis et al. 2003). A summary of these ancillary data sets, which are mostly within the Extended Groth Strip, are presented in Davis et al. (2007). The optical data from the CFHT 3.6-m includes imaging in the B, R and I bands taken with the CFH12K camera, which is a 12,288 $\times$ 8,192 pixel CCD mosaic with a pixel scale of 0.21″. The integration time for these observations are 1 hour in $B$ and $R$ and 2 hours in $I$, per pointing with 5 $\sigma$ depths of $\sim$ 25 in each band. For details of the data reduction see Coil et al. (2004b). From this imaging data a R${}_{\rm AB}$ = 24.1 magnitude limit was used for determining targets for the DEEP2 spectroscopy. The details for how these imaging data were acquired and reduced, see Coil et al. (2004b). The Keck spectra were obtained with the DEIMOS spectrograph (Faber et al. 2003) as part of the DEEP2 redshift survey. The EGS spectroscopic sample was selected based on a R-band magnitude limit only (with $R_{\rm AB}<24.1$), with no strong colour cuts applied to the selection. Objects in Fields 2-4 were selected for spectroscopy based on their position in $(B-K)$ vs. $(R-I)$ colour space, to locate galaxies at redshifts $z>0.7$. The total DEEP2 survey includes over 30,000 galaxies with a secure redshift, with about a third of these in the EGS field, and in total $\sim 11,000$ with a K-band detection (§3.1.1). In all fields the sampling rate for galaxies that meet the selection criteria is 60%. The DEIMOS spectroscopy was obtained using the 1200 line/mm grating, with a resolution R $\sim 5000$ covering the wavelength range 6500 - 9100 Å. Redshifts were measured through an automatic method comparing templates to data, and we only utilise those redshifts measured when two or more lines were identified, providing very secure redshift measurements. Roughly 70% of all targeted objects resulted in reliably measured redshifts. Many of the redshift failures are galaxies at higher redshift, $z>1.5$ (Steidel et al. 2004), where the [OII] $\lambda$3727 lines leaves the optical window. The ACS imaging over the EGS field covers a 10.1′$\times$ 70.5′ strip, for a coverage area of 0.2 deg${}^{2}$. The ACS imaging is discussed in Lotz et al. (2006), and is briefly described here, and in Conselice et al. (2007a,b). The imaging consists of 63 titles imaged in both the F606W (V) and F814W (I) bands. The 5-$\sigma$ depths reached in these images are V = 26.23 (AB) and I = 27.52 (AB) for a point source, and about two magnitudes brighter for extended objects. Our matching procedures for these catalogs progressed in the manner described in Bundy et al. (2006). The K-band catalog served as our reference catalog. We then matched the optical catalogs and spectroscopic catalogs to this, after correcting for any astrometry issues by referencing all systems to 2MASS stars. All magnitudes quoted in this paper are total magnitudes, while colours are measured through aperture magnitudes. Field \| RA \| Dec. \| # K \| # J \| K-area (arcmin2) ---\|---\|---\|---\|---\|--- EGS \| 14 17 00 \| +52 30 00 \| 33 \| 10 \| 2165 Field 2 \| 16 52 00 \| +34 55 00 \| 12 \| 0 \| 787 Field 3 \| 23 30 00 \| +00 00 00 \| 15 \| 15 \| 984 Field 4 \| 02 30 00 \| +00 00 00 \| 15 \| 12 \| 984 Total \| \| \| 75 \| 37 \| 4920 Table 1: The Palomar Fields, Number of WIRC pointings, and Area Covered ### Photometric Redshifts We calculate photometric redshifts for our K-selected galaxies, which do not have DEEP2 spectroscopy, in a number of ways. This sample is hereafter referred to as the ‘phot-z’ sample. Table 2 lists the number of spectroscopic and photometric redshifts within each of our K-band magnitude limits. These photometric redshifts are based on the optical+near infrared imaging, BRIJK (or BRIK for half the data) bands, and are fit in two ways, depending on the brightness of a galaxy in the optical. For galaxies that meet the spectroscopic criteria, $R_{\rm AB}<24.1$, we utilise a neural network photometric redshift technique to take advantage of the vast number of secure redshifts with similar photometric data. Most of the $R_{\rm AB}<24.1$ sources not targeted for spectroscopy should be within our redshift range of interest at $z<1.4$. The neural network fitting is done through the use of the ANNz (Collister & Lahav 2004) method and code. To train the code, we use the $\sim 5000$ redshifts in the EGS, which span our entire redshift range. The agreement between our photometric redshifts and our ANNz spectroscopic redshifts is very good using this technique, with $\delta z/z=0.07$ out to $z\sim 1.4$. The photometry we use for our photometric redshift measurements are done with a 2″ diameter aperture. For galaxies which are fainter than $R_{\rm AB}=24.1$ we utilise photometric redshifts using two methods, depending on whether the galaxy is detected in all optical bands or not. For systems which are detected at all wavelengths we use the Bayesian approach from Benitez (2000). For an object to have a photometric redshift using this method requires it to be detected at the 3 $\sigma$ level in all optical and near-infrared (BRIJK) bands, which in the R-band reaches $\sim 25.1$. We refer to these objects as having ‘full’ photometric redshifts. As described in Bundy et al. (2006) we optimised our results and corrected for systematics through the comparison with spectroscopic redshifts, resulting in a redshift accuracy of $\delta z/z=0.17$ for $R_{\rm AB}>24.1$ systems. Further details about our photometric redshifts are presented in Conselice et al. (2007b), including a lengthy discussion of biases that are potentially present in the measured values. Table 2 lists the number of galaxies with the various redshift types. As can be seen, the vast majority of our galaxies have either spectroscopic redshifts, or have measured photometric redshifts using the full optical SED. Only a small fraction ($<1$%) of our K-band sources are not detected in one optical band down to $K=21$. For completeness in the analysis of the $N(z)$ distribution of K-magnitudes discussed in §5, we calculate, using a $\chi^{2}$ minimisation through hyper-z (Bolzonella, Miralles & Pello 2000), the best fitting photometric redshifts for these faint systems. These galaxies however make up only a small fraction of the total K-band population, and their detailed redshift distribution, while not likely as accurate as our other photometric redshifts, do not influence the results at all significantly. ### Stellar Masses From our K-band/optical catalogs we compute stellar masses based on the methods outlined in Bundy, Ellis, Conselice (2005) and Bundy et al. (2006). The basic method consists of fitting a grid of model SEDs constructed from Bruzual & Charlot (2003) stellar population synthesis models, with different star formation histories. We use an exponentially declining model to characterise the star formation history, with various ages, metallicities and dust contents included. These models are parameterised by an age, and an e-folding time for characterising the star formation history. We also investigated how these stellar masses would change using the newest stellar population synthesis models with the latest prescriptions for AGB stars from Bruzual & Charlot (2007, in prep). We found stellar masses that were only slightly less, by 0.07 dex, compared to the earlier models (see Conselice et al. 2007b for a detailed discussion of this and other stellar mass issues.) Typical random errors for our stellar masses are 0.2 dex from the width of the probability distributions. There are also uncertainties from the choice of the IMF. Our stellar masses utilise the Chabrier (2003) IMF, which can be converted to Salpeter IMF stellar masses by adding 0.25 dex. There are additional random uncertainties due to photometric errors. The resulting stellar masses thus have a total random error of 0.2-0.3 dex, roughly a factor of two. However, using our method we find that stellar masses are roughly 10% of galaxy total masses at $z\sim 1$, showing their reliability (Conselice et al. 2005b). Details on the stellar masses we utilise, and how they are computed, are presented in Bundy et al. (2006) and Conselice et al. (2007b). ### K-band Completeness Limit Before we determine the properties of our K-selected galaxies, it is first important to characterise how our detection methods, and reduction procedures, influence the production of the final K-band catalog. While the major question we address is the nature of the faint and red galaxy population, it is important to understand what fraction of the faint population we are missing due to incompleteness. To understand this we investigate the K-band completeness of our sample in a number of ways. The first is through simulated detections of objects in our near-infrared imaging. As described in Bundy et al. (2006), Conselice et al. (2007b), and Trujillo et al. (2007) we placed artificial objects into our K-band images to determine how well we can retrieve and measure photometry for galaxies at a given magnitude. Our first simulations were performed from K = 18 to K = 22 using Gaussian profiles. We find that the completeness within our fields remains high at nearly all magnitudes, with a completeness of nearly 100% up to $K=19.5$ for all 75 fields combined together. The average completeness of these fields at $K=20$ is 94%, which drops to 70% at $K=20.5$ and 35% at $K=21.0$. If we take the 23 deepest fields we find a completeness at $K=21.0$ of 70%. However, galaxies are unlikely to have Gaussian light profiles, and as such, we investigate how the completeness would change in Conselice et al. (2007b), if our simulations were carried out with exponential and r${}^{1/4}$ light profiles. We find similar results as when using the Gaussian profiles up to $K=20$, but are less likely to detect faint galaxies with r${}^{1/4}$ profiles, and retrieve their total light output. As discussed in §3.1.1, these incompleteness corrections are critical for obtaining accurate galaxy counts, but the intrinsic profiles of galaxies of interest must be known to carry this out properly. As such, we utilise the Gaussian corrections as a fiducial estimate. In Figure 1 we plot our K-band counts with these corrections applied. We also plot the J-band counts up to their completeness limit, and do not apply any corrections for incompleteness. The 100% completeness for the optical data is $B=25.25$, $R=24.75$, and $I=24.25$ (Coil et al. 2004b). These limits are discussed in §4.1 where we consider our ability to retrieve a well defined population of extremely red objects (EROs). ## 3 Analysis ### Nature of the Faint K-band Population #### 3.1.1 K-band, J-band Counts and Incompleteness Within our total K-band survey area of 1.53 deg${}^{2}$ we detect 61,489 sources at all magnitudes, after removing false artifacts. Most of these objects (92%) are at K $<21$, while 68% are at K $<20$ and 37% are at K $<19$. In total there are 38,613 objects fainter than K $=19$ in our sample. Out of our total K-band population 10,693 objects (mostly galaxies) have secure spectroscopic DEIMOS redshifts from the DEEP2 redshift survey (Davis et al. 2003). We supplement these by 37,644 photometric redshifts within the range $0<z<2$ (Table 2). We remove stars from our catalogs, detected through their structures and colours, as described in Coil et al. (2005) and Conselice et al. (2007b). K Range \| Spec-z \| Full Photo-z \| Photo-z \| Total ---\|---\|---\|---\|--- 15<K<17 \| 353 \| 1541 \| 1 \| 1895 17<K<19 \| 4305 \| 11379 \| 30 \| 15714 19<K<21 \| 5483 \| 24405 \| 215 \| 30103 Table 2: Number of K-band selected galaxies with various redshift measurements We plot the differential number counts (Table 3) for galaxies in both the K and J-band for our K-selected sample in Figure 1 to test how our counts compare with those found in previous deep and wide near-infrared surveys. We carry this out to determine the reliability of our star and galaxy separation methods, as well as for determining how our incompleteness corrections in the K-band compare to others. As Figure 1 shows, we find little difference in our counts compared to previous surveys, and we are $\sim$100% complete up to magnitude K$\sim 20$ in all fields. As others have noted, we find a change in the slope of the galaxy counts at K = 17.5. We calculate that the slope at $K<17.5$ is dN/dK = $0.54\pm 0.07$, while at $K>17.5$ it is dN/dK = $0.26\pm 0.01$. Our counts, after correction, are slightly lower than those in the UKIDSS UDS survey (Simpson et al. 2006), and from studies by Cristobal-Hornillos et al. (2003) and Saracco et al. (1999). However, our counts are higher than those found in Iovino et al. (2005) and Kong et al. (2006). At brighter magnitudes these surveys all agree, with the exception of Maihara et al. (2001) who underpredict all surveys (not plotted). This difference at $K>20$ is likely the result of the different incompleteness correction methods used. As detailed in Bershady et al. (1998), using various intrinsic galaxy profiles when computing completeness can lead to over and underestimation of the correction factor. The only accurate way to determine the incompleteness is to know the detailed distribution of galaxy surface brightness profiles at the magnitude limits probed (Bershady et al. 1998). As this is difficult, and sometimes impossible to know, all corrected counts must be seen as best estimates. K-Magnitude \| log N (deg−2 mag−1) \| J-Magnitude \| log N (deg−2 mag−1) ---\|---\|---\|--- 14.0 \| 0.969+0.139−0.206 \| 14.5 \| 1.074+0.176−0.301 14.5 \| 1.270+0.102−0.135 \| 15.0 \| 0.949+0.198−0.374 15.0 \| 1.987+0.048−0.054 \| 15.5 \| 0.949+0.198−0.374 15.5 \| 2.349+0.032−0.035 \| 16.0 \| 1.852+0.081−0.010 16.0 \| 2.697+0.022−0.023 \| 16.5 \| 2.265+0.052−0.059 16.5 \| 3.000+0.016−0.016 \| 17.0 \| 2.540+0.038−0.042 17.0 \| 3.291+0.011−0.011 \| 17.5 \| 2.847+0.027−0.029 17.5 \| 3.517+0.009−0.009 \| 18.0 \| 3.152+0.019−0.020 18.0 \| 3.720+0.007−0.007 \| 18.5 \| 3.370+0.015−0.016 18.5 \| 3.889+0.006−0.006 \| 19.0 \| 3.631+0.011−0.012 19.0 \| 4.029+0.005−0.005 \| 19.5 \| 3.840+0.008−0.009 19.5 \| 4.171+0.004−0.004 \| 20.0 \| 4.004+0.007−0.008 20.0 \| 4.314+0.003−0.004 \| 20.5 \| 4.185+0.006−0.006 20.5 \| 4.396+0.003−0.003 \| 21.0 \| 4.288+0.005−0.005 21.0 \| 4.482+0.003−0.003 \| 21.5 \| 4.315+0.005−0.005 Table 3: K-band and J-band Counts for all Fields The J-band counts (Figure 1b) show a similar pattern as the K-band counts. These counts are not corrected for incompleteness and we are incomplete for very blue galaxies with $(J-K)<0$ at the faintest J-band limit due to our using the K-band detections as the basis for measuring J-band magnitudes. As can be seen, there is a larger variation in the J-band number counts when comparing the different surveys at a given magnitude compared to the K-band counts. Furthermore, there is no obvious slope change in the J-band counts, as seen in the K-band. We are complete overall in the J-band to J${}_{\rm vega}=22.0$ over the entire survey. Our J-band depth however varies between the three fields in which we have J-band coverage, and in fact varies between individual WIRC pointings. As we later discuss the properties of EROs in this paper, as defined with a $(R-K)$ colour cut, it is important to understand the corresponding depths of the $R$-band imaging. The depth and number counts for the $R$-band imaging is discussed in detail in Coil et al. (2004b). Based on aperture magnitudes the 5 $\sigma$ depth of the CFHT R-band imaging is roughly $R_{\rm AB}=25$. Our $R$-band photometry uses the same imaging as Coil et al. (2004b), however we retrieved our own magnitudes based on the $K$-band selected objects in our survey. Our $R$-band depth however is similar to Coil et al. (2004b), and we calculate a 50% incompleteness at $R=25.1$. <figure><img src="content_image/0711.1083/x1.png"><figcaption>Figure 1: Differential K-band and J-band counts for our survey compared toprevious published counts, including: Cristobal-Hornillos et al. (2003)(C.-H.), Huang et al. (2001), Saracco et al. (1999), Saracco et al. (2001),Teplitz et al. (1999), Bershady et al. (1998), Martini (2001), Kong et al.(2006), Iovino et al. (2005) and Simpson et al. (2006). Only the error barsare shown for the counts in our survey. The K-band counts at K>20 have beencorrected for incompleteness, while the J-band counts are only shown to theircompleteness limit. At the faintest magnitudes the other surveys, with theexception of Simpson et al. (2006), have a larger error range due to thesmaller areas used.</figcaption></figure> #### 3.1.2 Redshift Distributions of K-Selected Galaxies In this section we investigate the nature of galaxies selected in the K-band. The basic question we address is what are the properties of galaxies at various K-limits. This issue has been discussed earlier by Cimatti et al. (2002a), Somerville et al. (2004) and others. However, we are able to utilise the DEEP2 spectroscopic survey of these fields to determine the contribution of lower redshift galaxies to the K-band counts, and thus put limits on the contribution of high redshift $(z>1.5)$ galaxies to K-band counts at $K<20$. <figure><img src="content_image/0711.1083/x2.png"><figcaption>Figure 2: The spectroscopic redshift completeness for both the entire K-bandselected catalog, and for the the (R−K)>5.3 ERO selected sample. Note thatnone of our EROs at K>19.5 have a measured spectroscopic redshift, typical forspectroscopic surveys which are optically selected.</figcaption></figure> <figure><img src="content_image/0711.1083/x3.png"><figcaption>Figure 3: Redshift distributions for various K-band magnitude cuts. The solidblack line shows the redshift distribution for all galaxies with 19<K<21, theblued dashed line shows systems with 17<K<19, and the red hashed histogram isfor galaxies with 15<K<17. The levels at z=−1 show the number of galaxies whohad photo-zs not fit due to lacking significant optical detections.</figcaption></figure> The first, and most basic, method for understanding galaxies found in a K-band selection is to determine what fraction of the K-selected galaxies have a successfully measured spectroscopic redshift. The DEEP2 spectroscopic redshift success rate for our K-selected sample varies with K-band magnitude, from 10% to 30%, up to $K=21$. The highest selection fraction is 30% at K = 17.5. At the faintest limit, K = 21, the redshift selection is 10%, and the fraction is 15% at K = 15.5. Figure 2 shows our spectroscopic redshift completeness as a function of K-band magnitude for both the entire K-band selected sample, and the EROs (§4). The result of this plot is partially due to the fact that the DEEP2 selection deweights galaxies at $z<0.7$. The EGS and the other fields also have slightly different methods for choosing redshift targets (Davis et al. 2003), creating an inhomogeneous selection over the entire survey. When we include photometric redshifts to supplement our spectroscopic redshifts, we obtain total redshift distributions shown in Figure 3. Note that our photometric redshifts are only included in Figure 3 at $z>0$ if the object was significantly detected in all bands in the BRIJK photometry. In each K-band limit shown in Figure 3, and discussed below, there are a fraction of sources which do not meet this optical criteria, and these objects are counted at the $z=-1$ position on the redshift histograms. In Table 2 we list the number of K-band detections with and without spectroscopic redshifts, in each of the redshift ranges. It appears that nearly all bright K-band sources, with $15<K<17$, are located at $z<1.4$ (Figure 3). At fainter magnitudes, as shown by the plotted $17<K<19$ and $19<K<21$ ranges (Figure 3), we find a different distribution skewed toward high redshifts. While there are low redshift galaxies at these fainter K-limits, we also find a significant contribution of sources at higher redshifts, including those at $z>2$. The K-bright sources at these redshifts are potentially the highest mass galaxies in the early universe. Galaxies at the faintest magnitudes, at $19<K<21$, show a similar redshift distribution as the galaxies within the $17<K<19$ magnitude range, but there are a larger number of higher redshift galaxies. This demonstrates that faint K-band sources are just as likely to be at low redshift as at high redshift. This is shown in another way using Figure 4 where we plot the distribution of K-magnitude vs. redshift ($z$). As can be seen, at $K>19$ the entire redshift range is sampled, while a $K<17$ selection only finds galaxies at $z<1.4$. <figure><img src="content_image/0711.1083/x4.png"><figcaption>Figure 4: Contours of the redshift distribution for galaxies at variousK-magnitudes. The red dashed contours are for EROs defined as (R-K)>5.3(mostly at z>1), and the blue dotted contours are the distant red galaxies(DRGs) defined by (J−K)>2.3, which generally span 0.4<z<2.</figcaption></figure> #### 3.1.3 Colours of K-Selected Galaxies After examining the redshift distribution of our sample, the next step is determining the physical features of these galaxies. The easiest, and most traditional, way to do this is through the examination of colour-magnitude diagrams. Generally, galaxy colour is a mixture of at least three effects - redshift, stellar populations and dust. Galaxies generally become redder with redshift due to band-shifting effects, and become redder with age, and increased dust content. <figure><img src="content_image/0711.1083/x5.png"><figcaption>Figure 5: Colour-magnitude diagrams for our sample. The left panel shows (R-K)colour vs. K-band magnitude, while the right panel shows the (J-K) vs. Kdiagram. Objects with spectroscopic redshifts are coloured blue in bothdiagrams, while objects considered ‘red’ either through the extremely redobjects (EROs) or Distant Red Galaxies (DRGs) selection are labelled as red ineach diagram. The solid line in the (R-K) vs. K panel shows the spectroscopiclimit of R = 24.1, while the dashed line shows the 5σ limit for the R-bandphotometry of 25.1. Furthermore, we only plot points that are brighter than R= 26.5 and J = 23.5 in the two panels, respectively. The red triangles at thetop of each figure are galaxies which are undetected in R or J, but have ameasured K-band magnitude.</figcaption></figure> We can get an idea of the characteristics of our K-selected sample by examining the colour-magnitude diagram for the entire $K<21$ sample (Figure 5). Figure 5 plots the colours of our sample, as a function of $(B-R)$ and $(J-K)$ versus K-band magnitude. As can be seen, at fainter limits there are more red galaxies in each band. Since fainter/redder galaxies are more likely than brighter galaxies to be at higher redshifts, it is likely that these redder galaxies seen in Figure 5 are distant galaxies. The relation between $(R-K)$ and redshifts (Figure 6) shows this to be the case. As can be seen, at higher redshifts galaxies are redder in $(R-K)$, although even at these redshifts there are K-selected galaxies which are very blue. At the highest redshifts, where optical magnitudes are at $R>24$, we find that most galaxies hover around $(R-K)=5.3$. However, as can be seen, a significant fraction of the $K<19.7$ galaxies, which are massive systems at $z>1$, would be identified as EROs. <figure><img src="content_image/0711.1083/x6.png"><figcaption>Figure 6: The redshift distribution for galaxies within our sample at K<19.7.The left panel shows systems which are at R<24. The right panel shows thedistribution of (R−K) colours as a function of redshift for galaxies which arefainter than R=24. As can be seen, galaxies generally get redder at higherredshifts, but there still exists a scatter in colour at any redshift.</figcaption></figure> The detailed distribution of magnitudes, colours and stellar masses is shown in Figure 7 and Figure 8, divided into different redshift bins. Plotted with different symbols are the photometric redshift and the spectroscopic redshift samples. As can been seen, there are strong relations between stellar mass and K-band magnitude over the entire redshift range (Figure 7), with fainter K-selected galaxies having lower stellar masses. Also note that the galaxies with spectroscopic redshifts are generally brighter and bluer than the photometric redshift sample at a given stellar mass. This is particularly true at the highest redshift bins, and demonstrates that the spectroscopy is successfully measuring the brighter galaxies in the distant universe, while being less efficient in measuring redshifts for galaxies of the same mass, but at a fainter K-magnitude. Figure 8 furthermore shows how, as we go to higher redshifts, we obtain an overall redder distribution of colours. Within our lowest redshift bin, $0.5<z<0.75$, there are few galaxies which would be classified as EROs with $(R-K)>5.3$. It is also at this lowest redshift range where the overlap between the spectroscopic and photometric samples is highest. When we go to higher redshifts, such as at $0.75<z<1.0$, we find that the slope of the locus in the relation between stellar mass and $(R-K)$ colour steepens, such that galaxies at the same stellar mass become redder. This effect is dominated by the change in rest-frame wavelength sampled by the $R$ and $K$ filters. The fact that the higher mass galaxies become redder, while the lower mass galaxies tend to remain blue, is a sign that the spectral energy distributions of the lower mass galaxies are bluer than those for higher mass galaxies. This pattern evolves however, and at $z>1$, galaxies at every mass bin become redder with time. The upper envelope in the colour-stellar mass relation (Figure 8) is due to incompleteness, and is not a real limit. On Figure 7 and Figure 8 we plot hydrodynamic simulation results from Nagamine et al. (2005) using different dust extinctions. We over-plot the E(B-V) = 0, 0.4 and 1 models on these figures as contours. First, the fact that there is not a larger scatter in the log M${}_{}$ vs. $K$ relation (Figure 7) is an indication that these galaxies are not dominated by dust extinction. Very few galaxies overlap with the E(B-V) = 1 model, and most galaxies are better matched with the E(B-V) = 0 model. This can furthermore be seen in Figure 8 where only the lowest extinction model, with E(B-V) = 0, generally matches the location of galaxies in the $(R-K)$ vs. M${}_{}$ diagram which are not EROs. The (E-V) = 0.4 model does a good job of tracing the EROs, but these systems could also be composed of old stellar populations. We revisit the issue of dust extinction in these galaxies in §5.1. Finally, there are clearly two unique, and overlapping, samples identified in this near-infrared selected sample. The first are those galaxies in Figure 7 and 8 which are very massive, with masses M${}_{}>10^{11}$ M${}_{\odot}\,$. We discuss these objects, and their evolution in Conselice et al. (2007b). We investigate in the next section the properties of the extremely red objects (EROs), those with observed colours, $(R-K)>5.3$. <figure><img src="content_image/0711.1083/x7.png"><figcaption>Figure 7: The stellar mass vs. K-band magnitude relation for our sample ofgalaxies out to z∼2. These figures are divided up into different redshifts,increasing from left to right and top to bottom. Plotted on these figures areboth systems which have spectroscopically measured redshifts (open blackboxes) and those which have photometric redshifts (the red dots). As can beseen, there is generally a strong relationship between stellar mass andapparent K-band magnitude, with a small scatter. Note that generally galaxieswith spectroscopically measured redshifts are those which are brighter for agiven stellar mass. These same systems are furthermore on the blue edge of thestellar mass-colour relationship (Figure 8). This shows that the DEEP2spectroscopy is selecting primarily the bluer and brighter galaxies at a givenstellar mass. We also plot in the 1.0<z<1.25 bin models for how thesequantities relate, from SPH simulations from Nagamine et al. (2005). The blue,cyan and green contours (going from low mass to high mass at a given K) showthe location of model galaxies with E(B-V) = 0, 0.4 and 1, respectively.</figcaption></figure> <figure><img src="content_image/0711.1083/x8.png"><figcaption>Figure 8: The colour-stellar mass relation for our sample to z∼0. Similar toFigure 7, these galaxies have been divided up into different redshift bins.The horizontal and vertical lines show the limits for selecting unique galaxypopulations probed in the K-band. Galaxies to the right of the vertical lineare massive galaxies studied in Conselice et al. (2007b), while galaxies whichare above the horizontal line, defined by (R−K)>5.3 are the EROs studied inthis paper. Note that the final redshift panel with 1.5<z<2.0 consists solelyof photometric redshifts. Also, similar to Figure 7 we plot in the 1.0<z<1.25figure models results from Nagamine et al. (2005), although we utilise themodel results within the same limits as our data, K<20. As in Figure 7, theblue, cyan and green contours show the location of model galaxies with E(B-V)= 0, 0.4 and 1 (going from bluer to redder colours at a given stellar mass),respectively.</figcaption></figure> ## 4 Properties of Extremely Red Objects ### Sample Selection The ERO sample we construct is defined through a traditional colour cut to locate the reddest galaxies selected in near infrared/optical surveys with $(R-K)>5.3$. Galaxies selected in this way are often considered the progenitors of today’s most massive galaxies, as seen at roughly $z\sim 1-2$. Objects with these extremely red colours have remained a population of interest since their initial discovery (Elson, Rieke & Rieke 1988). Initially thought to be ultra-high redshift galaxies with $z>6$, it is now largely believed that EROs are a mix of galaxy types at $z>1$ (e.g., Daddi et al. 2000; McCarthy et al. 2001; Firth et al. 2002; Smail et al. 2002; Roche et al. 2003; Cimatti et al. 2002; Yan & Thompson 2003; Cimatti et al. 2003; Moustakas et al. 2004; Daddi et al. 2004; Wilson et al. 2007). Generally, it has also been thought that EROs should trace some aspect of the massive galaxy population (Daddi et al. 2000); an idea we can test further with our data. Furthermore, because EROs are an easily defined and observationally based population, there has been considerable observation and theoretical work done towards understanding these objects. Naturally, it is more desirable to work with mass selected sample (see Conselice et al. 2007b for this approach), but these samples rely on accurate redshifts and stellar mass measurements, while the EROs are simply observationally defined through a colour. The idea that EROs are red due to either an evolved galaxy population, or a dusty starburst is perhaps no longer the dominate way to think about these systems (e.g., Moustakas et al. 2004). However, there are properties of EROs that are still not understood, nor even constrained. For example, it is not clear why some EROs can have apparently normal galaxy morphologies, while others appear to be merging or peculiar galaxies (e.g., Yan & Thompson 2003; Moustakas et al. 2004). The questions we address include: what are the stellar mass, morphological and redshift distributions for these objects? In our analysis we study the properties of these traditionally colour selected EROs to determine these basic properties. Our sample of EROs however is not defined simply by a $(R-K)>5.3$ cut on our entire $R-$band and $K-$band catalogs. Due to the depth of both filters, we have to limit how deep we search for EROs to avoid false positives. As discussed in §2.4, we are 100% complete in our entire survey down to $K=20$. The $R-$band depth is however not well matched to the $K$-depth for finding EROs, and has a $>$ 5 $\sigma$ detection limit of $R=25.1$ (although 50% completeness). We therefore only select EROs which we are certain to within $>5$$\sigma$ have a colour $(R-K)>5.3$. This limits our analysis of EROs down to $K=19.7$. We divide our ERO sample into three types, depending on the origin of the redshift for each. The first type are those EROs with a high quality DEEP2 redshift, of which there are a total of 62 within our survey. The second type of ERO are those with $R<24.1$ which contain an ANNz photometric redshift (§2.2). There are 343 of these EROs. The third type are the EROs with magnitudes between $24.1<R<25.1$ which all have ‘full’ photometric redshifts (§2.2). There are 1122 of these objects. The entire $(R-K)>5.3$ sample therefore consists of 1527 EROs with some type of redshift. We examine the properties of $(I-K)>4$ selected EROs in §4.6. ### ERO Number Counts As with the number counts of the K-selected objects, we are interested in comparing the number counts of our ERO sample with measurements from previous work. In Figure 9 we plot the number counts for our ERO sample, as a function of K-magnitude. As can been seen, we are slightly under-dense at nearly all magnitudes compared to the UKIDSS UDS survey, but find similar results as Daddi et al. (2000). The differences between the counts in our survey and the UDS is likely due to several issues, including the slightly different filter sets used, and the correction for galactic extinction. The UDS survey uses Subaru Deep Field imaging utilising the Cousins $R_{\rm C}$ filter, while our R-band imaging is from the CFHT and utilises a Mould R filter, which has different characteristics. Another issue is that these previous surveys have not corrected for Galactic extinction, while we have. This can result in slight differences in the number counts. Another feature seen in previous surveys, which we also see, is a turn-over in the slope of the counts at about K = 18.5, towards a shallower slope at fainter magnitudes. <figure><img src="content_image/0711.1083/x9.png"><figcaption>Figure 9: The number counts for EROs in our fields plotted along with thecounts from previous surveys by Simpson et al. (2006) and Daddi et al. (2000).We also plot the number counts for the extremely red galaxies (ERGs) - the EROcounts with stars removed.</figcaption></figure> ### Redshift Distributions and Number Densities of EROs One of principle quantities needed to understand the properties of EROs is their redshift distribution and number densities. With our three different types of redshifts (§2.2) measured for our EROs, we can construct the redshift distribution and number density evolution for EROs down to a magnitude limit of $K=19.7$. Figure 10a shows the redshift distribution for our $(R-K)>5.3$ selected EROs. We have plotted this distribution with three different histograms: for the spectroscopic redshift sample (red diagonal hatch), a photometric redshift sample for galaxies at $R_{\rm AB}<24.1$ (black), and a photometric redshift sample at $R_{\rm AB}>24.1$ (blue horizontal dashed). It is clear that galaxies selected with the ERO criteria are at higher redshifts ($z>1$), with very few galaxies meeting this criteria at $z<0.8$, and all of those that do are photometrically derived redshifts. A similar pattern can be seen in Figure 6, which plots the colour-redshift distribution for our sample selected by $R_{\rm AB}<24$, and $R_{\rm AB}>24$. The average redshift for our $K<19.7$ ERO sample is $<z>=$ 1.28$\pm$0.23. This is at a higher redshift and has a smaller dispersion than the average redshift for all galaxies with $K<19.7$, $<z>=0.84\pm 0.31$. The above arguments show that the traditional $(R-K)>5.3$ ERO cut reliably locates galaxies at $z>1$, on average. There is also a fairly long tail of sources up to $z\sim 2$. ERO Selection \| Redshift \| log (ϕ) (h370 Mpc−3 dex−1) ---\|---\|--- (R−K)>5.3 \| 0.5 \| -6.00+0.31−0.33 \| 0.7 \| -5.42+0.16−0.27 \| 0.9 \| -4.64+0.10−0.13 \| 1.1 \| -3.90+0.09−0.11 \| 1.3 \| -4.10+0.09−0.11 \| 1.5 \| -4.24+0.10−0.11 \| 1.7 \| -4.70+0.14−0.12 \| 1.9 \| -5.13+0.13−0.15 (I−K)>4 \| 0.5 \| -5.10+0.15−0.23 \| 0.7 \| -5.19+0.14−0.21 \| 0.9 \| -4.82+0.11−0.14 \| 1.1 \| -4.16+0.09−0.11 \| 1.3 \| -4.12+0.09−0.11 \| 1.5 \| -4.11+0.10−0.11 \| 1.7 \| -4.32+0.13−0.11 \| 1.9 \| -4.47+0.11−0.11 Table 4: Extremely Red Object Number Densities for Systems at Kvega<19.7. For the most part it appears that EROs at $K<19.7$ are selecting high redshift galaxies at $z\sim 1.3$. However, we are missing a few galaxies from our ERO sample at $K<19.7$ which do not have a redshift due to non-detections in the optical bands. These galaxies could be at very high redshift, and will be discussed in a future paper. Figure 11 plots the number density evolution for our EROs at $K<19.7$ as a function of redshift, with tabulated values shown in Table 4. As can be seen, in agreement with Figure 10, there is a drop in the number density of EROs at $z<1$. The number density peak for EROs is also clearly found between $z=1-1.5$. We can compare this figure to previous measurements and models (e.g., Nagamine et al. 2005). Previously Moustakas et al. (2004) and Cimatti et al. (2002b) measured ERO number densities within various K-limits, but within $(R-K)>5$, as opposed to our $(R-K)>5.3$. However, when we compare our results to these papers, we find very similar results. Down to $K_{\rm vega}<20.12$, Moustakas et al. find a number density of EROs at $z=1$ of log($\phi$(Mpc${}^{-3}$)) = $-3.19$, whereas we find $-3.39$ in roughly good agreement. Similarly, Cimatti et al. (2002b) find down to $K_{\rm vega}<19.2$ a density of log($\phi$(Mpc${}^{-3}$)) = $-3.67$ at $z=1$, while we find $-3.60$, again in good agreement. When we compare our results to simulation results from Nagamine et al. (2005) we find again roughly good agreement, although the Lagrangian SPH results find a slightly higher number density. At $z=1$ these SPH simulations find log($\phi$(Mpc${}^{-3}$)) = $-2.96$, while the total variation diminishing (TVD) simulations give a slightly higher result. This density is a factor of 2.6 higher than the number density which we observe. These density are however the result of assuming a dust extinction of E(B-V) = 0.4, which might be too high in light of the results shown in Figures 7 and 8. Lower values of E(B-V) will make galaxies less red, and will produce a lower number density of EROs. ### Stellar Masses of EROs As shown in Figure 10b, our EROs generally have high stellar masses, and thus a fraction of the most massive galaxies at $z>0.8$ must be EROs. This is an important result, as it has been surmised from other criteria, such as clustering, that the EROs are contained within massive halos (e.g., Daddi et al. 2000; Roche, Dunlop & Almaini 2003). We have constructed a complete sample of M${}_{}>10^{11}$ M${}_{\odot}\,$galaxies up to $z\sim 1.4$ in our fields (Conselice et al. 2007b), from which we can directly test the idea that EROs are massive galaxies. Although Figure 10b demonstrates that our EROs at $K<19.7$ are selecting massive galaxies, this is likely due to the fact that our EROs are relatively bright, and we cannot constrain the masses or redshifts of fainter EROs, which must be either lower mass galaxies at the same redshifts as these, or higher redshift massive systems. There is also little difference in the distributions of stellar masses for our ERO sample at different redshifts. The peak mass is around 2$\times 10^{11}$ M${}_{\odot}\,$at all redshift selections. We find that most of our sample of $K<19.7$ EROs tend to have masses M${}_{}>10^{11}$ M${}_{\odot}\,$, which in the nearby universe are nearly all early-types (Conselice 2006a). This is a strong indication that $K<19.7$ EROs, regardless of their morphology or stellar population characteristics, are nearly certain to evolve into passive massive early-types in the nearby universe. #### 4.4.1 Are Massive Galaxies at $z>1$ EROs? <figure><img src="content_image/0711.1083/x10.png"><figcaption>Figure 10: The redshift and stellar mass histograms for our sample of EROs.Shown are three histograms created after dividing the sample three differentways, depending on the origin of the redshift. The dotted diagonal hatched redhistogram shows the redshift and stellar mass distributions for thespectroscopically confirmed EROs, while the non-hatched black histogram showsthe distributions for EROs with (R−K)>5.3, RAB<24.1, and whose redshifts arephotometric. The horizontal blue dashed hatched histogram shows thedistributions for EROs with RAB>24.1 with measured photometric redshifts. Ascan be seen at a K<19.7 limit, the ERO selection generally locates massivegalaxies at z>1.</figcaption></figure> While EROs are massive galaxies, the opposite of this is not necessarily true, as there are massive galaxies with M${}_{}>$$10^{11}$ M${}_{\odot}\,$, that are not EROs, some with very blue colours (Conselice et al. 2007b). Figure 12 plots the fraction of galaxies within the mass ranges M${}_{}>$$10^{11.5}$ M${}_{\odot}\,$and $10^{11}$ M${}_{\odot}\,$$<$ M${}_{}<$$10^{11.5}$ M${}_{\odot}\,$which are EROs between $z\sim 0-2$. Figure 11 plots the number density evolution for EROs selected in two ways and compares to galaxies selected with stellar masses M${}_{}>$$10^{11}$ M${}_{\odot}\,$. There are several interesting features in these figures. The first is that while massive galaxies exist throughout this redshift range, EROs only populate massive galaxies at $z>1$. Another interesting feature is that an ERO selection at $K<19.7$ will include a large fraction of the most massive galaxies with M${}_{}>$$10^{11.5}$ M${}_{\odot}\,$, at $1<z<2$. On average, between $1.0<z<1.4$, our ERO selection will find 36% of all M${}_{}>$$10^{11.5}$ M${}_{\odot}\,$galaxies. This increases to 75% within the redshift range $1.2<z<1.8$. However, the ERO colour limit does not do a good job in selecting massive galaxies with $10^{11}$ M${}_{\odot}\,$$<$ M${}_{}<$$10^{11.5}$ M${}_{\odot}\,$. In this mass range at $1.0<z<1.4$ the ERO selection finds only 35% of these systems. Similar to the M${}_{}>$$10^{11.5}$ M${}_{\odot}\,$mass range, we find a higher fraction of $10^{11}$ M${}_{\odot}\,$$<$ M${}_{}<$$10^{11.5}$ M${}_{\odot}\,$galaxies which are EROs at 44%. However, with a $K<19.7$ limit, we are obtaining a similar number density of EROs per co-moving volume as there are massive galaxies (Figure 11). The number densities of EROs however declines rapidly at $z<1$. While it appears that the traditional $(R-K)>5.3$ limit will find the most massive galaxies at $z>1$, this colour cut does not give a purely ultra-high mass sample, nor does it include all of the massive galaxies at these redshifts. At $1.2<z<1.8$ about 25% of galaxies with M${}_{}>$$10^{11.5}$ M${}_{\odot}\,$and 66% of systems with $10^{11}$ M${}_{\odot}\,$$<$ M${}_{}<$$10^{11.5}$ M${}_{\odot}\,$are not EROs. This is consistent with our finding in Conselice et al. (2007b) that $\sim 40$% of massive galaxies at $z>1$ are undergoing star formation and have blue $(U-B)_{0}$ colours. ### Structural Features We utilise visual estimates of Hubble types and the non-parametric CAS system to characterise the morphologies and structures of an ERO sample selected with $(R-K)>5$. While there are slightly more systems at $(R-K)>5.3$ than at $5<(R-K)<5.3$, we reduce our limit to aquire more systems for analysis and to better compare with previous work. A similar analysis is the focus of Conselice et al. (2007a,b), in which we examined the morphological properties of the most massive galaxies at $z>1.5$, as well as the Distant Red Galaxies (DRG), with $(J-K)>2.3$, which are also proposed to be the progenitors of today’s massive galaxies. <figure><img src="content_image/0711.1083/x11.png"><figcaption>Figure 11: The co-moving number densities of EROs selected by the(R−K)vega>5.3 and the (I−K)vega>4 criteria as a function of redshift. Alsoplotted for reference is the number density evolution for galaxies withstellar masses M∗> 1011 M⊙(see Conselice et al. 2007b).</figcaption></figure> The CAS (concentration, asymmetry, clumpiness) parameters are a non-parametric method for measuring the structures of galaxies on CCD images (e.g., Conselice 2007; Conselice et al. 2000a,b; Bershady et al. 2000; Conselice et al. 2002; Conselice 2003). The basic idea is that low redshift, nearby galaxies, have light distributions that reveal their past and present formation modes (Conselice 2003). Furthermore, well-known galaxy types in the nearby universe fall in well defined regions of the CAS parameter space (Conselice 2003). We apply the CAS system to our EROs to determine their structural features. There are two caveats to using the ACS imaging on these galaxies. The first is that there are redshift effects which will change the measured parameters, such that the asymmetry and clumpiness indices will decrease (Conselice et al. 2000a; Conselice 2003), and the concentration index will be less reliable (Conselice 2003). There is also the issue that for systems at $z>1.3$ we are viewing these galaxies in their rest-frame ultraviolet images, which means that there are complications when comparing their measured structures with the rest-frame optical calibration indices for the nearby galaxies. Our main purpose in using the CAS system is to identify relaxed massive ellipticals as well as any galaxies that are still involved in a recent major merger and are presumably dusty. The following structural and morphological analysis is based on the ACS imaging of the EGS field described in §2.1. The imaging we use covers 0.2 deg${}^{2}$ in the F814W (I) band, giving us coverage for $\sim 15$% of our ERO sample. <figure><img src="content_image/0711.1083/x12.png"><figcaption>Figure 12: Diagram showing the fraction of galaxies with extreme masses M∗>1011.5 M⊙(solid circles) and 1011 M⊙< M∗< 1011.5 M⊙(open boxes) which are also(R−K)>5.3 selected EROs. As can be seen, the ERO selection successfully findsgalaxies at z>1, yet does not locate all of these systems. A sample of EROs atK<19.7 will contain nearly all of the log M∗>11.5 systems at z∼1.5, but isless successful at finding the lower mass, log M∗>11 systems.</figcaption></figure> #### 4.5.1 Eye-Ball/Classical Morphologies We study the structures and morphologies of our sample using two different methods. The first is through simple visual estimate of morphologies based on the appearance of our ERO sample in CCD imaging. The outline of this process is given in Conselice et al. (2005a) and is also described in the companion paper (Conselice et al. 2007b). Our total sample of objects gives 436 unique EROs for which there is ACS imaging. Each of these galaxies were placed into one of six categories: compact, elliptical, lenticular (S0), early-type disk, late-type disk, edge-on disk, merger/peculiar, and unknown/too-faint. These classifications are very simple, and are only based on appearance. No other information, such as colour or redshift, was used to determine these types. An outline of these types is provided below, with the number in each class listed at the end of each description. 1. Ellipticals: A centrally concentrated galaxy with no evidence for a lower surface brightness, outer structure (152 systems). An additional 58 galaxies were classified as peculiar ellipticals, which appear similar to ellipticals, but have an unusual light distribution, or bulk asymmetry (see Conselice et al. 2007b). 2. Lenticular (S0): A galaxy was classified as an S0 if it appears as an elliptical but contained a disk-like outer structure with no evidence for spiral arms, or clumpy star forming knots or other asymmetries. (13 systems) 3. Compact - A galaxy was classified as compact if its structure was resolved, and is very similar to the elliptical classification in that these systems must be very smooth and symmetric. A compact galaxy differs from an elliptical in that it contain no obvious features, such as an extended light distribution or envelope. (24 systems) 4. Early-type disks: If a galaxy contained a central concentration with some evidence for lower surface brightness outer light, it was classified as an early-type disk. (3 systems) 5. Late-type disks: Late-type disks are galaxies that appear to have more outer low surface brightness light than inner concentrated light. (1 systems) 6. Edge-on disks: disk systems seen edge-on and whose face-on morphology cannot be determined but is presumably S0 or disk. (17 systems) 8. Peculiars/irregulars: Peculiars are systems that appear to be disturbed or peculiar looking, including elongated/tailed sources. These galaxies are possibly in some phase of a merger (Conselice et al. 2003a,b) and are common at high redshifts. (148 systems) 9. Unknown/too-faint: If a galaxy was too faint for any reliable classification it was placed in this category. Often these galaxies appear as smudges without any structure. These could be disks or ellipticals, but their extreme faintness precludes a reliable classification. (20 systems) #### 4.5.2 Morphological Distributions The morphological distribution of the EROs can help us address the question of the origin of these extremely red galaxies. In the past, this technique has been used to determine the fraction of EROs which are early-type, disk or peculiar. Previous studies on this topic include Yan & Soifer (2003) who study 115 EROs, and Moustakas et al. (2004) who studied 275 EROs in the GOODS fields. Our total population of EROs for which we have morphologies is 436. These earlier studies have found a mix of types, with generally half of the EROs early-types, and the other half appearing as star forming systems in the form of disks or peculiars/irregulars. In summary, we find that 57$\pm$3% of our EROs are early-type systems. This includes 58 systems, or 13% of the total, which are disturbed ellipticals. In classifications carried out in previous work some of these systems would be classified as peculiars. The bulk of the rest of the population consists of bonafide peculiars, which make up 34$\pm$3% of the ERO population. Only four EROs were found to be face on disk galaxies, while 17 systems (4%) of the ERO sample are made up of edge-on disk galaxies. Presumably these galaxies are red for different reasons - either evolved galaxy populations, or dusty star formation, or from dust absorption produced through orientation in the case of edge-on disks. Previous work has been somewhat inconsistent on the morphological break-down between peculiars and early-type galaxies (e.g., Yan & Soifer 2003; Moustakas et al. 2004). From our study, it is clear that much of this difference can be accounted for by the peculiar ellipticals. These systems appear in their large-scale morphology to be early-type, but have unusual features, such as offset nuclei that make them appear peculiar. The differences between previous findings can largely be accounted for by whether these peculiar ellipticals were included in the early-type or peculiar class. <figure><img src="content_image/0711.1083/x13.png"><figcaption>Figure 13: Our visual estimates of ERO morphological type as a function ofredshift. As labelled, the ellipticals, S0s and compact morphological typesare shown as a solid black line. The systems classified as a peculiar/mergersare shown as the dotted blue line and spirals are shown as the dashed redline.</figcaption></figure> We find that the relative number of peculiar and early-type EROs changes with redshifts (Figure 13), such that at the lowest redshifts ($z\sim 0.7$) the EROs are dominated by the E/S0/Compacts, with a type fraction of $\sim 65$%, but at $z>1.4$ the peculiar systems are more prominent. The fraction of EROs which are peculiars evolves from $\sim 45$% at $z\sim 1.7$ to $\sim 20$% at $z=0.9$. This mix between early types and peculiars evolves with redshift, although the exact reason for this evolution is not immediately clear. It is possible that some of the peculiars at $z>1.2$ only appear so because we are sampling their morphologies below the Balmer break that would produce more irregular/peculiar looking galaxies at these redshifts (Windhorst et al. 2002; Manger-Taylor et al. 2006; Conselice et al. 2007b). However, we are nearly always probing the rest-frame optical where the effects of the morphological k-correction are minimised both qualitatively and quantitatively (e.g., Conselice et al. 2005a; Conselice et al. 2007c in prep). We also find a higher fraction of peculiar systems within the ERO sample, than what is found for massive galaxies with M${}_{}>$$10^{11}$ M${}_{\odot}\,$at similar redshifts (Conselice et al. 2007b). Interestingly, we find that 10-15% of EROs at $z\sim 0.9$ are spiral/disks. For the most part however, it appears that most of the EROs are ellipticals, but we find that peculiars make up roughly a third of the systems, with the relative contribution coming from galaxies at the highest redshifts. We discuss the morphological break-down of these systems in §5.2 in terms of models of galaxy formation and evolution. #### 4.5.3 CAS Structural Parameters Another way to understand the structures of these systems is through their quantitative structural parameters as measured through the CAS system. The CAS parameters have a well-defined range of values and are computed using simple techniques. The concentration index is the logarithm of the ratio of the radius containing 80% of the light in a galaxy to the radius which contains 20% of the light (Bershady et al. 2000). The range in $C$ values is found from 2 to 5, with higher $C$ values for more concentrated galaxies, such as massive early types. The asymmetry is measured by rotating a galaxy’s image by 180${}^{\circ}\,$and subtracting this rotated image from the original galaxy’s image. The residuals of this subtraction are compared with the original galaxy’s flux to obtain a ratio of asymmetric light. The radii and centreing involved in this computation are well-defined and explained in Conselice et al. (2000a). The asymmetry ranges from 0 to $\sim 1$ with merging galaxies typically found at $A>0.35$. The clumpiness is defined in a similar way to the asymmetry, except that the amount of light in high frequency ‘clumps’ is compared to the galaxy’s total light (Conselice 2003). The values for $S$ range from 0 to $>2$, with most star forming galaxies having values, $S>0.3$. We show in Figure 14 the CA and AS projection of CAS space for EROs defined by $(R-K)>5$. As can be seen, the EROs, which are mostly early-types and peculiars, as defined by eye (Figure 14), fall along a large portion of the range of possible values in CAS space. As expected, the irregulars/peculiars have higher asymmetries, lower concentrations, and higher clumpiness values than the early types. This is similar to, but not exactly the same, as the structural parameter distribution for the most massive galaxies with M${}_{}>$$10^{11}$ M${}_{\odot}\,$at $z<1.4$. There is a larger fraction of peculiars within the ERO sample, and the redshift evolution with types is not identical between EROs and massive galaxies (cf. Conselice et al. 2007b). As with the massive galaxies found at high redshifts, the ERO visually determined types are slightly higher in asymmetry than their $z\sim 0$ counterparts (Conselice 2003). Figure 14 shows how the EROs classified as early-types have slightly larger asymmetries than their visual morphology would suggest. The distorted early-types have even higher asymmetries on average. Most systems also deviate from the asymmetry-clumpiness relation (Figure 14), showing that the production of these asymmetries is more likely due to dynamical effects, rather than star formation (Conselice 2003). Interestingly, we find that many of the peculiar EROs do not have a high clumpiness index, which is opposite to what we found for high asymmetry systems within the massive galaxies sample (Conselice et al. 2007b). The reason for this is that these red galaxies are likely dusty, and therefore bright star clusters are not seen within the ongoing star formation. This furthermore shows that the EROs are likely more dominated by galaxy merging than a pure mass selected sample. <figure><img src="content_image/0711.1083/x14.png"><figcaption>Figure 14: The CAS space for our ERO sample with (R−K)>5. Objects are labelledby their visual classification, discussed in §4.5.1. The solid black symbolsrepresent the normal (non-distorted) ellipticals, S0s and compacts. The openred triangles are the distorted ellipticals, while the blue crosses aregalaxies classified as peculiars/mergers, and the open cyan circles are forthe few disk galaxies found within the ERO sample.</figcaption></figure> We can understand the origin of these morphologies, and how they relate to the origin of the EROs, by comparing the CAS parameters of the EROs to their stellar masses. Figure 15 shows the comparison between ERO asymmetries and concentrations vs. their stellar masses. The concentration-stellar mass diagram shows a few interesting trends. The most obvious feature is that there appears to be a broad bimodality between EROs which are peculiar, and those which are early-types. The peculiars and early-types have similar masses, typically M${}_{}\sim 10^{11}$ M${}_{\odot}\,$, yet they have different light concentrations. The peculiars typically have lower concentrations, $C=2-3$, while the early-types are generally at $C>3$. The early-types also tend to show a correlation between concentration and stellar mass which is not seen for the peculiars. The distorted ellipticals fall in between these two populations suggesting an evolutionary connection in the passive evolution of galaxy structure. What we are potentially witnessing is the gradual transition from peculiar EROs at high redshifts, to passive ellipticals at lower redshifts, while on the way going through a distorted elliptical stage. This is consistent with the idea that the $z=1-2$ epoch is where massive galaxies final reach their passive morphology (Conselice et al. 2003; Conselice et al. 2005; Conselice 2006). <figure><img src="content_image/0711.1083/x15.png"><figcaption>Figure 15: The relationship between asymmetries and light concentrations ofour ACS EROs with their stellar masses. The symbols for each type are the sameas in Figure 14\. There appears to be a remarkable bimodal distribution inthese parameter spaces such that peculiars have low light concentration andhigher asymmetries than the elliptical-like objects, while having similarstellar masses. The distorted ellipticals appear to fall in the gap betweenthese two populations, and are likely a transitional phase between the two.This can be seen furthermore in Figure 13 where the fraction of early-typeEROs increases at the expense of the peculiars.</figcaption></figure> There also appears to be a bimodality within the stellar mass/asymmetry diagram (Figure 15). The peculiars in general have higher asymmetries than the early-types, but with the distorted ellipticals containing asymmetries mid-way between the peculiars and the ellipticals, suggesting again that the distorted ellipticals are a mid-way point in the evolution between the peculiar EROs and the passive EROs. As we do not see much mass evolution in the upper edge of the ERO mass distribution, it is likely that the peculiars are within their final merger phase, and transform into early-types relatively quickly over $1<z<2$. A similar pattern can be seen for a high mass selected sample of galaxies at similar redshifts (Conselice et al. 2007b). ### Other ERO Selection Criteria ERO selection through the $(R-K)>5.3$ criteria is only one way to find extremely red objects. Another popular method for finding EROs is a selection with $(I-K)>4$ (e.g., Moustakas et al. 1997). While both of these selection methods are used to find EROs, it is not clear how these two methods compare, and whether they are finding the same galaxy population. We investigate this issue briefly in this section. <figure><img src="content_image/0711.1083/x16.png"><figcaption>Figure 16: The (I−K) vs. (R−K) diagram for galaxies in our sample. Only shownare those galaxies which are considered EROs either through the (R−K)>5.3, orthe (I−K)>4 criteria. The points are plotted in terms of their redshifts, withgalaxies at z>1.5 shown as open red circles and galaxies at z<1.5 as bluedots. The (I−K)>4 limit appears to find galaxies at higher redshifts, onaverage, than the (R−K)>5.3 limit.</figcaption></figure> Figure 16 shows the relationship between $(I-K)$ and $(R-K)$ colours for galaxies within our sample. Those objects which have photometric redshifts $z>1.5$ are plotted as the red open symbols, while those at $z<1.5$ are plotted as the blue dots. What is obvious from this figure is that an ERO selection with $(I-K)>4$ is more likely to pick out galaxies at higher redshifts than the $(R-K)>5.3$ limit. This limit also appears to contain more contamination from lower redshift galaxies than the $(R-K)>5.3$ limit. Overall, we find an overlap of 767 EROs through both definitions down to $K=19.7$, this overlap constitutes 54% of the $(R-K)>5.3$ EROs and 60% of the $(I-K)>4$ EROs. In Figure 17 we show the redshift distribution for the EROs selected with $(I-K)>4$, which can be directly compared with the redshift distribution for $(R-K)>5.3$ systems in Figure 10. As can be seen, the redshift distribution for the $(I-K)>4$ systems is skewed towards higher redshifts than the $(R-K)>5.3$ selection. We find that, on average, the redshifts for the $(I-K)>4$ EROs is $<$z$>$ = 1.43$\pm 0.32$, compared with $<$z$>$ = $1.28\pm 0.23$ for EROs selected with the $(R-K)>5.3$ limit. There are also fewer systems at $z<1.4$ within the $(I-K)>4$ selection than for the $(R-K)>5.3$ limit, suggesting that the redder bands are finding galaxies at slightly higher redshifts. However, this does not appear to be the case for the $(J-K)>2.3$ ‘distant red galaxies’ (DRGs), as discussed in Conselice et al. (2007a) and Foucaud et al. (2007). It appears that these systems are picking up a significant fraction of massive galaxies at $z\sim 1$ up to $K=20$. <figure><img src="content_image/0711.1083/x17.png"><figcaption>Figure 17: Similar to Figure 10, but for ERO systems which have been selectedwith the (I−K)>4 limit. As can be seen by comparing this figure with Figure10, we are selecting, on average, higher redshift galaxies with these redderbands.</figcaption></figure> ## 5 Discussion Here we discuss the results from this paper in the context of galaxy formation models and scenarios. We include in this discussion the redshift distribution of K-selected galaxies, as well as the stellar mass and structural/morphological properties of EROs to address how the stellar mass assembly of galaxies is likely taking place. By examining these galaxies we are not relying on assumptions about stellar masses to find the most massive and evolved galaxies at high redshift. In this sense a K-band selected and colour selected population are an alternative approach from stellar mass selection (Conselice et al. 2007b) for understanding galaxy formation due to the simplicity, and reproducibility, of their selection. We first examine the redshift distribution of $K<20$ galaxies, and compare this to models. We use colour information of our faint K-selected galaxies to rule out all monolithic collapse formation scenarios for galaxies. We then examine the properties of the EROs themselves to further argue that these systems are, due to their stellar masses, likely an intrinsically homogeneous population, with the peculiar EROs evolving into the ellipticals at lower redshifts. ### K-band Redshift Distributions The number of K-band selected galaxies per redshift at a given K-magnitude limit is an important test of when the stellar masses of galaxies were put into place. In general, ideas for how massive galaxies form explicitly predict how much stellar mass galaxies would have in the past. In a rapid formation, such as with a monolithic collapse, the stellar mass for the most massive galaxies, which in a K-band limited sample will always probe the massive systems, the number of galaxies within a bright K-band limit, say $K<20$ at high redshift should be much larger than the number of sources seen in a hierarchical model, which predicts that the stellar masses of galaxies grows with time. Therefore the number of bright galaxies at higher redshifts in a hierarchical model is less than that predicted in a monolithic collapse. This test was first performed by Kauffmann & Charlot (1998) who concluded, based on an early hierarchical model, that the predicted counts exceed the observations, a result which has remained despite improvements in data and models (e.g., Kitzbichler & White 2006). <figure><img src="content_image/0711.1083/x18.png"><figcaption>Figure 18: Redshift distribution of our sample with a K<20 cut. Comparisons toprevious redshift distributions published in the K20 and GOODS surveys areshown. The two lines demonstrate predictions of fiducial monolithic collapse(Kitzbichler & White 2006), and hierarchical model predictions (Kitzbichler &White 2007) for this evolution within the same K-band limit.</figcaption></figure> In Figure 18 we show the number of $K<20$ galaxies as a function of redshift for systems at $z<4$. We further plot on Figure 18 model predictions for how $N(z)$ changes in a standard pure luminosity evolution monolithic collapse model from Kitzbichler & White (2006), and within a standard hierarchical formation scenario from Kitzbichler & White (2007). We have used in this figure our entire K-band distribution, including galaxies for which we had to measure photometric redshifts without an optical band (§2.2). We also plot on Figure 18 the $K<20$ magnitude distribution for galaxies seen in the GOODS and K20 surveys. While we generally agree with these previous results, we find a slightly lower number of systems at higher redshifts compared to GOODS and K20. The reason for this could simply be cosmic variance, as the GOODS and K20 samples at these redshifts also differ by a large amount. Otherwise, this difference is likely created by errors within our photometric redshifts. However, it must be noted that the integrated number of $K<20$ galaxies is similar for our survey and GOODS and K20, but is still much smaller than that predicted by the monolithic model. As can be seen in Figure 18 there is clearly a large difference between the observed distribution, and the predicted monolithic collapse distribution. We can rule out this basic monolithic collapse model, which does not include dust, at $>$ 10 $\sigma$, based on the comparisons to our $K<20$ redshift distribution. It is possible to match with a monolithic collapse model if extreme dust content is included in these galaxies, or if there are ‘hidden’ galaxies (Kitzbichler & White 2006). However, as we argue below, there is no evidence that dusty galaxies dominate our K-band selected sample. From Figure 18 it appears that a basic hierarchical model agrees better with the data (e.g., Somerville et al. 2004; Stanford et al. 2004; Kitzbichler & White 2007). There is perhaps a slight excess of galaxies in the hierarchical formation model, which has been noted before (e.g., Kauffman & Charlot 1998; Somerville et al. 2004; Kitzbichler & White 2007). The origin of this difference is not clear. It perhaps simply implies that too much mass is produced early in the hierarchical model, yet this would seem to contradict the fact that the most massive galaxies with M${}_{}>$$10^{11}$ M${}_{\odot}\,$are nearly all in place by $z\sim 1.5-2$ (Conselice et al. 2007b). We address this apparent conflict in more detail in §5.2. Although the basic hierarchical model agrees better with the observed redshift distribution, there are scenarios where the monolithic model can fit the data as well as the hierarchical scenario (see also Cimatti et al. 2002a). These scenarios require that the K-band selected galaxies have a significant amount of dust extinction in the observed K-band. The amount of extinction needed varies from 1.0 to 0.7 mag at rest-frame $z$ and $R$ bands from $z=1.5-2$ (Kitzbichler & White 2006). This extinction needed is even higher at $z=1-1.5$. By using the colours of galaxies that fit within the $K<20$ criteria, we can determine what the contribution of dusty galaxies are to these counts. Stellar population synthesis models show that only old stellar populations, and galaxies with dust extinctions with A${}_{\rm v}>1$, have a colour of $(R-K)>5$ at $z=1-4$. We can use this information, and the model results using various dust extinctions from Nagamine et al. (2005) (Figures 7 and 8), to argue that these galaxies are not dominated by dust extinction, which would need to be the case to reconcile the observed K-band distribution with a monolithic collapse model. First, the model results compared to data shown in Figures 7 and 8, and discussed in §4.3 clearly show that K-selected galaxies are not dominated by heavy dust extinction. At best, only models with E(B-V) = $0-0.4$ are able to reproduce the location of real galaxies. The E(B-V) = 0.4 models are however even too red to account for most galaxies in the $(R-K)$ vs. M${}_{}$ diagram (Figure 8). A representative value of R${}_{\rm V}$ = A${}_{\rm V}$/E(B-V) = 3.1 reveals values of A${}_{\rm V}=0-1.28$ for this range in E(B-V). For the Calzetti et al. (2000) extinction law, this gives lower values, with A${}_{\rm V}=0-0.25$. Thus, it does not appear likely that our $K<20$ galaxy sample is dominated by enough dust extinction (A${}_{\rm V}=1$ needed, on average) to match the monolithic collapse galaxy count model. We find that only 28.3$\pm 0.6$% of $K<20$ selected galaxies at $1<z<2$ have a colour $(R-K)>5$, required to meet the minimum condition for dust extinction. We have further argued in §4, that a significant fraction of these EROs are old passively evolving stellar populations, which are unlikely to have a screen of dust with A${}_{\rm V}>1$ (see e.g., White, Keel & Conselice 2000). As discussed in §4.5.2 over half of the EROs at $1<z<2$ appear as early-types, thus at most only $\sim 14\%$ of the $K<20$ galaxies at $1<z<2$ have a significant amount of dust extinction. The dusty pure-luminosity evolution monolithic collapse models from Kitzbichler & White (2006) predict that all $K<20$ galaxies have this amount of dust extinction, which clearly they do not. It is thus impossible to reconcile the K-band redshift distribution with any monolithic model. In summary our K-band redshift distribution, and those from the K20 and GOODS surveys, appear to be much closer to the basic hierarchical model of Kitzbichler & White (2007) than to the monolithic collapse model. We can also rule out pure luminosity evolution models with significant dust extinction. Therefore, down to $K=20$ at $z<4$ it appears that the monolithic model cannot be the dominate method for forming galaxies. ### Structural Evolution As discussed in §4.5, a large fraction of the EROs we have HST imaging appear to be undergoing some type of evolution based on their structural appearance. A logical conclusion is that many of the peculiar systems are in some phase of a merger (Conselice et al. 2003a). We can quantify this by examining the merger fraction as derived through the CAS definition of $A>0.35$ and $A>S$ (Conselice 2003). This is a strict definition, and will allow us to measure a lower limit on the merger fraction, even when observing in the rest-frame ultraviolet (Taylor-Mager et al. 2007). <figure><img src="content_image/0711.1083/x19.png"><figcaption>Figure 19: The fraction of galaxies identifiable as a merger using the CASsystem, and the fraction of early-types which appear to have a distortedstructure.</figcaption></figure> Figure 19 shows the evolution of the merger fraction derived through this CAS definition. We also include the fraction of early-types which appear to have a distorted structure. It appears from this that between $5-10\%$ of the EROs at $1<z<2$ would be identified as a major merger through the CAS parameters. This is a factor of $\sim 3$ less than the peculiar fraction (Figure 13), which is what we would expect for a population which is undergoing major mergers (Conselice 2006; Bridge et al. 2007). The reason for this is that the CAS approach is sensitive to $\sim 0.4$ Gyr of the merger process (Conselice 2006), while eye-ball estimates of merging last for $\sim 1.2$ Gyr (Conselice 2006). This is another indication that the peculiar galaxies we are seeing in the ERO population are in fact undergoing merging. Figure 20, as well as Figure 15 demonstrate what evolution is likely occurring within the ERO population. As can be seen, the EROs, within our $K<19.7$ selection, have the same upper range of masses at $z\sim 0.8-2$ (Figure 20). Therefore, little mass growth occurs within this population at this K-band limit. We also know that a large fraction of the EROs with M${}_{}>$$10^{11}$ M${}_{\odot}\,$are peculiar in some way. This varies from 64$\pm$24% at $z>1.5$ to 41$\pm$6% at $1<z<1.25$ (§4.5). However, at $z\sim 0$ the fraction of M${}_{}>$$10^{11}$ M${}_{\odot}\,$red galaxies which are morphologically early-types is $\sim 85$% (Conselice 2006a; Conselice et al. 2007b). It is clear that a significant fraction of EROs must have undergone morphological evolution as they cannot lose mass. What likely is occurring is that the distorted elliptical and peculiar galaxies which dominate the population at $z>1.5$ (Figure 13 and 20) transform into morphologically and spectrally evolved systems at $z\sim 1$. The reason this is likely the case is that there is no difference in the masses of the peculiar and the early-type EROs. This also explains why this population, despite having a mixed morphology clusters so strongly (e.g., Daddi et al. 2000; Roch et al. 2003). Calculations based on the merger rate in Conselice et al. (2007b) for the most massive galaxies suggest that on average about one or two major mergers are occurring for the M${}_{}>$$10^{11}$ M${}_{\odot}\,$population at $z<2$, but fewer at lower redshifts. Finally, these major mergers are what may make the K-band counts in the hierarchical model redshift distribution higher than the observed. The reason is that in the standard hierarchical model, the number and mass densities of the most massive galaxies with M${}_{}>$$10^{11}$ M${}_{\odot}\,$underpredict the observed number of massive systems by up to two orders of magnitude (Conselice et al. 2007b). Within these models the stellar masses for these systems are largely already formed, but are in distinct galaxies that have not yet merged (De Lucia et al. 2006). It is thus easy to see that if a single massive galaxy at $z\sim 1.5$ was in several pieces, all of which would still meet the criteria of $K<20$ based on the relation of stellar mass and K-mag (Figure 7), then the number of galaxies with $K<20$ at higher redshift in the hierarchical model would be higher than the observed number. This is consistent with the number densities of massive galaxies being higher than in the models, as well as for a rapid formation of massive systems through major mergers at $z>2$ (Conselice 2006b). <figure><img src="content_image/0711.1083/x20.png"><figcaption>Figure 20: The distribution of ERO galaxy mass with redshift. Labeled on thisfigure are the morphological types for each galaxy as determined through ourACS imaging. Note that massive EROs are detected all the way out to z=2.</figcaption></figure> ## 6 Summary In this paper we analyse the faint K-band selected galaxy population as found in the Palomar NIR survey/DEEP2 spectroscopic survey overlap. Our primary goal is to determine the nature of the faint $K>19$ galaxy population. While many of these galaxies are too faint for detailed spectroscopy, we can investigate their nature through spectroscopic and photometric redshifts, stellar masses, as well as photometric and structural features. Our major findings include: 1. The redshift distribution for K-selected galaxies depends strongly on apparent K-magnitude. Most systems at $K<17$ are at $z<1.4$, while a significant fraction of sources with $17<K<19$ are at $z>2$. These K-bright high$-z$ galaxies are the progenitors of today’s massive galaxies. 2. We find that a significant fraction (28.3$\pm 0.6$%) of the $K<20$ galaxy population consists of extremely red or massive galaxies at $z>1$. We characterise the population of log M${}_{}>11$ sources in Conselice et al. (2007b), while we analyse the extremely red objects (EROs) in this paper. 3. We find that EROs at $K<19.7$ are a well defined population in terms of redshifts and masses. Nearly all EROs are at $z>1$, and have stellar masses with M${}_{}>10^{11}$ M${}_{\odot}\,$. EROs are therefore certainly the progenitors of today’s massive galaxies. The corollary to this however is not necessarily true. There are massive galaxies at $z>1$ that would not be selected with the ERO criteria. We find that the ERO selection locates 35-75% of all ultra-massive, M${}_{}>10^{11.5}$ M${}_{\odot}\,$galaxies, at $z=1-2$, while only 25% of $10^{11}$ M${}_{\odot}\,$$<$ M${}_{}<10^{11.5}$ M${}_{\odot}\,$galaxies are located with this colour cut. 4. We examine the morphological and structural properties of our ERO sample and find, as others previously have, a mixed population of ellipticals and peculiars. In total, we find that the ERO population is dominated by early-type galaxies, with an overall fraction of 57$\pm 3$% of the total. Interestingly, we find that a significant fraction of the early-types ($\sim 25$%) are distorted ellipticals, which could be classified as peculiars, although these systems at a slightly lower resolution than ACS, or using a quantitative approach would be seen as early-types. Peculiars account for the remaining 34%, and many of these are likely in some merger phase. This fraction tends to evolve such that the peculiars are the dominant population at higher redshifts, $z>1$. 5. We investigate the structural parameters for our EROs using the CAS system. We find that visual estimates of galaxy class and position in CAS space roughly agree, although the asymmetries of these systems are higher than what their visual morphologies would suggest. We find a bimodality in the stellar mass-concentration diagram where the peculiar EROs are at a low concentration and the early-types are highly concentrated. The distorted ellipticals fall in between these two populations suggesting an evolutionary connection in the passive evolution of galaxy structure within the ERO population. 6. We compare the $(R-K)>5.3$ ERO selection with the $(I-K)>4$ ERO selection, and find that the $(I-K)>4$ ERO are at slightly higher redshifts than the $(R-K)>5.3$ selection, suggesting that it is a more useful criteria for finding evolved galaxies at $z>1.5$. 7. By examining the redshift distribution of $K<20$ galaxies, and comparing to monolithic collapse and hierarchical formation models, we are able to rule out all monolithic collapse models for the formation of massive galaxies. These monolithic collapse models predict a higher number of $K<20$ galaxies as a function of redshift at a significance of $\sim 10$$\sigma$. While some monolithic collapse models are able to reproduce our galaxy counts, these are dominated by dust. We are however able to show that only $\sim 14$% of the $K<20$ galaxies have potentially enough dust to match this model, the others being evolved galaxies, or blue star forming systems. The Palomar and DEEP2 surveys would not have been completed without the active help of the staff at the Palomar and Keck observatories. We particularly thank Richard Ellis and Sandy Faber for advice and their participation in these surveys. We thank Ken Nagamine and Manfred Kitzbichler for providing their models and comments on this paper. We also thank the referee for their careful reading and commenting on this paper. We acknowledge funding to support this effort from a National Science Foundation Astronomy & Astrophysics Fellowship, grants from the UK Particle Physics and Astronomy Research Council (PPARC), Support for the ACS imaging of the EGS in GO program 10134 was provided by NASA through NASA grant HST-G0-10134.13-A from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. JAN is supported by NASA through Hubble Fellowship grant HF-01182.01-A/HF-011065.01-A. The authors also wish to recognise and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. 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1211.6239	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 73967, "num_imgs": 8, "llama3_tokens_count": 23060 }	[ "content_image/1211.6239/x1.png", "content_image/1211.6239/x2.png", "content_image/1211.6239/x3.png", "content_image/1211.6239/x5.png", "content_image/1211.6239/x7.png", "content_image/1211.6239/x9.png", "content_image/1211.6239/x10.png", "content_image/1211.6239/x11.png" ]	# Optimal Power and Range Adaptation for Green Broadcasting † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] Shixin Luo, , Rui Zhang, , and Teng Joon Lim, ###### Abstract Improving energy efficiency is key to network providers maintaining profit levels and an acceptable carbon footprint in the face of rapidly increasing data traffic in cellular networks in the coming years. The energy-saving concept studied in this paper is the adaptation of a base station’s (BS’s) transmit power levels and coverage area according to channel conditions and traffic load. Cell coverage is usually pre-designed based on the estimated static (e.g. peak) traffic load. However, traffic load in cellular networks exhibits significant fluctuations in both space and time, which can be exploited, through cell range adaptation, for energy saving. In this paper, we design short- and long-term BS power control (STPC and LTPC respectively) policies for the OFDMA-based downlink of a single-cell system, where bandwidth is dynamically and equally shared among a random number of mobile users (MUs). STPC is a function of all MUs’ channel gains that maintains the required user-level quality of service (QoS), while LTPC (including BS on-off control) is a function of traffic density that minimizes the long-term energy consumption at the BS under a minimum throughput constraint. We first develop a power scaling law that relates the (short-term) average transmit power at BS with the given cell range and MU density. Based on this result, we derive the optimal (long-term) transmit adaptation policy by considering a joint range adaptation and LTPC problem. By identifying the fact that energy saving at BS essentially comes from two major energy saving mechanisms (ESMs), i.e. range adaptation and BS on-off power control, we propose low-complexity suboptimal schemes with various combinations of the two ESMs to investigate their impacts on system energy consumption. It is shown that when the network throughput is low, BS on-off power control is the most effective ESM, while when the network throughput is higher, range adaptation becomes more effective. Cellular network, cell zooming, power control, energy-efficient communication, broadcast channel, OFDMA. ## I Introduction Mobile data traffic is anticipated to grow many-fold between 2010 and 2020, inducing many technical challenges such as how to improve energy efficiency in order to limit growth in energy consumption to a factor smaller than that of data traffic growth. The drive to make cellular networks more “green” starts with base stations (BSs), since they make up a large proportion of the total energy consumed in any cellular network [1]. Cell planning, i.e. placement of BSs and coverage area of each one, is usually based on estimated static (e.g. peak) traffic load. Current research in cellular network planning mainly focus on the practical deployment algorithm design. For example, in [2], the authors used stochastic geometry to analyze the optimal macro/micro BS density for energy-efficient heterogeneous cellular networks with QoS constraints. The energy efficiency of heterogeneous networks and the effects of cell size on cell energy efficiency were investigated in [3] by introducing a new concept called area energy efficiency. However, traffic load in cellular networks fluctuates substantially over both space and time due to mobility and traffic burstiness. Therefore, there will always be some cells under light load, and others under heavy load, which suggests that static cell planning based on peak load will not be optimal. Load balancing schemes have thus been proposed in both academia and industry [11, 5, 6], which react to load variations across time and cells by adaptively re-allocating users to cells. In [11], a network-wide utility maximization problem was considered to jointly optimize partial frequency reuse and load-balancing in a multicell network. In [5, 6], the authors proposed the “cell breathing” technique, which shrinks (or expands) the coverage of congested (or under-loaded) cells by reducing (or raising) the power level, so that the load becomes more balanced. BSs consume a significant amount of energy (up to $60\%$ of the total network energy consumption [7]) due to their operational units, e.g., processing circuits, air conditioner, besides radio transmission. Therefore, selectively letting some BSs be switched off according to traffic load can yield substantial energy saving. There have been a few BS on-off switching schemes introduced in the literature. For example, energy saving as a function of the daily traffic pattern, i.e the traffic intensity as a function of time, was derived in [8], where it is shown through simulations that energy saving on the order of $25-30\%$ is possible. Centralized and distributed BS reconfiguration algorithms were proposed in [9], with simulations showing that the centralized algorithm outperforms the distributed one at the cost of increased complexity and overhead. In [10], the authors considered a wireless local area network (WLAN) consisting of a high density of access points (APs). The resource on-demand (RoD) strategy was introduced to power on or off WLAN APs dynamically, based on the volume and location of user demand. When some BSs are switched off, their coverage areas need to be served by the remaining active BSs in the network. Such a self-organized network (SON) has been introduced in 3GPP LTE [11]. A similar but more flexible method called “Cell Zooming” was proposed in [12], which adaptively adjusts the cell size according to traffic load, user requirements, and channel conditions, in order to balance the traffic load in the network and thereby reduce energy consumption. Energy-efficient cellular network planning with consideration of BSs’ ability of cell zooming, which is characterized as cell zooming ratio, was investigated in [13]. However, to the best of our knowledge, a scheme that adapts both coverage range and transmit power (including the possibility of turning off the BS) to minimize the total energy consumed has not been studied in the literature, even under the simple one-cell setup. This motivates our work, which studies the extreme case of one single-cell system in order to obtain useful insights that could be applied in a general multi-cell environment. In this paper, we consider the downlink transmission in an orthogonal frequency-division multiple access (OFDMA) based cellular network. Unlike traditional cellular networks using fixed time and/or bandwidth allocation, we consider that the available time-frequency transmission blocks are dynamically and equally allocated to a random number of active mobile users (MUs). Moreover, the BS is assumed to have two levels of power control: short-term power control (STPC) and long-term power control (LTPC), which correspond to the inherent difference in the time scales of the MUs’ average channel gain variations (in e.g. seconds) and traffic density variations (in e.g. hours). STPC sets the transmit power based on each MU’s distance from the BS to meet each MU’s outage probability requirement over fading, while LTPC (including BS on-off control)¹ is implemented according to traffic density variations such that the long-term energy consumption at the BS is minimized under a certain system-level throughput constraint. Under the above broadcast channel setup, a new power scaling law, which relates the (short-term) average transmit power of BS with the given cell coverage range and traffic density, is derived for the case of homogeneous Poisson point process (HPPP) distributed MU locations. Based on the derived power scaling law, we determine the optimal long-term cell adaptation policy by considering a joint range adaptation and LTPC problem. Since it is challenging to obtain closed-form expressions for the optimal policy, approximate solutions are derived in closed-form under a high spectrum efficiency (HSE) assumption, which provide further insights into the design of cellular networks of the future in which both power and spectral efficiency are important. [FOOTNOTE:1][ENDFOOTNOTE] By identifying that the energy saving at BS essentially comes from two major energy saving mechanisms (ESMs), i.e. range adaptation and BS on-off power control, we propose low-complexity suboptimal schemes with various combinations of the two ESMs to further investigate their effects on the system energy consumption. By numerical simulations, it is shown that significant energy saving can be achieved in OFDMA broadcast channels with the optimal cell adaptation policy. Furthermore, it is revealed that when the network throughput requirement is modest, the simple BS on-off control is nearly optimal for cell adaptation in terms of energy saving; however, when higher network throughput is required, a finer-grained strategy of range adaptation is needed. These results provide useful guidelines for designing energy-efficient cellular networks via cell power and/or range adaptations. The rest of this paper is organized as follows. Section II introduces the system model, and derives the BS power scaling law under STPC. Section III studies the joint cell range adaptation and LTPC problem. Section IV presents various low-complexity suboptimal schemes. Section V compares the performance of optimal and suboptimal schemes through numerical examples. Finally, Section VI concludes the paper. ## II System Model We consider an OFDMA downlink in one particular cell with bandwidth $W$ Hz. It is assumed that the BS can adaptively adjust its cell coverage according to MU density and power budget through admission control. In this section, we first introduce a spatial model of cellular traffic based on MUs distributed according to a HPPP. Then, we elaborate on the proposed bandwidth sharing scheme for the OFDMA-based broadcast channel. Finally, we describe the STPC, based on which a power scaling law relating the (short-term) average transmit power at a BS given a pair of coverage range and MU density is derived. ### _Traffic Model_ The two-dimensional Poisson process has been used to model the locations of MUs in a cellular network. In this paper, we assume that MUs form a HPPP $\Phi_{m}$ of density $\lambda_{m}$ in the Euclidean plane. Considering that every MU within the cell coverage requests connection (voice service or data application) randomly and independently with probability $q$, then according to the Marking Theorem [14], the active MUs (that need to communicate with a BS) form another HPPP $\Phi$ of density $\lambda$,² where $\lambda=q\lambda_{m}$. Since we are interested in active MUs, we refer to active MUs simply as MUs in the rest of this paper. The MU density $\lambda$ is assumed to be a non-negative random variable with finite support, i.e. $0\leq\lambda\leq\lambda_{\text{max}}$, with $f_{\lambda}(\cdot)$ and $F_{\lambda}(\cdot)$ denoting its probability density function (PDF) and cumulative distribution function (CDF), respectively. Let $N\triangleq\|\Phi(B)\|$ represent the total number of MUs within a cell, denoted by $B$. Then $N$ is a Poisson random variable with mean $\mu_{N}\triangleq\lambda\pi R^{2}$, where $R$ denotes the cell radius, and probability mass function (PMF) [FOOTNOTE:2][ENDFOOTNOTE] \[\mbox{Pr}[N=n]=\frac{\mu_{N}^{n}}{n!}e^{-\mu_{N}},~{}~{}n=0,1,\ldots.\] (1) ### _Equal Bandwidth Sharing_ <figure><img src="content_image/1211.6239/x1.png"><figcaption>Fig. 1: Equal bandwidth sharing (EBS)</figcaption></figure> Practically, dynamic bandwidth sharing (DBS) can be realized by users’ time-sharing the available sub-carriers in OFDMA. To be more specific, the available time-frequency resource is divided into Resource Blocks (RBs) over both time and frequency, which are allocated among MUs such that each MU can be ideally assigned an effective bandwidth with arbitrary value from $0$ to $W$ Hz. Note that in general, DBS allocates the available RBs dynamically among MUs in order to optimize certain system-level utility (e.g. throughput) based on the number of MUs, their channels from the BS, and their QoS requirements. For the purpose of exposition, in this paper we assume a simplified equal bandwidth sharing (EBS) scheme among MUs, i.e., the effective bandwidth allocated to MU $i$, $i=1,2...,N$, is $W/N$ Hz. An illustration of the EBS within a scheduled transmission frame $T_{F}$ is shown in Fig. 1. The available time-frequency resource is divided into RBs with dimensions $T_{\text{RB}}$ and $B_{\text{RB}}$ over time and frequency, respectively. $T_{\text{RB}}$ and $B_{\text{RB}}$ are assumed to be much smaller than the channel coherence time, $T_{c}$, and the channel coherence bandwidth, $B_{c}$, respectively; thus a flat-fading channel can be assumed in each RB. Let $N_{F}=\frac{W}{B_{\text{RB}}}$ and $N_{T}=\frac{T_{F}}{T_{\text{RB}}}$ be the number of frequency slices and time slices, respectively, within a transmission frame. The total number of available RBs within one frame can be computed as $U=N_{F}N_{T}$, which is assumed to be large enough such that each MU can be assigned a continuous effective bandwidth $\frac{U_{i}}{U}W$, where $U_{i}$ is the number of RBs allocated to MU $i$. For example, $4$ RB’s are allocated to MU $i$ as shown in Fig. 1. The total bandwidth allocated to MU $i$ is therefore $\frac{4}{N_{F}}W$, over a period of $N_{T}$ channel uses, where a channel use corresponds to $T_{\text{RB}}$ seconds. Therefore, MU $i$ is given $\frac{4W}{N_{F}N_{T}}=\frac{4W}{U}$ Hertz of bandwidth per channel use, which also implies that the BS is serving $N=\frac{U}{4}$ active MUs by EBS. With EBS, the achievable rate for MU $i$, given received signal power $S_{i}$, is \[V_{i}=\frac{W}{N}\log_{2}\left(1+\frac{NS_{i}}{\Gamma N_{0}W}\right)\] (2) where $\Gamma$ accounts for the gap from the channel capacity due to a practical coding and modulation scheme, and $N_{0}$ is the power spectral density of the additive white Gaussian noise (AWGN). Suppose that channel coding is performed over $L$ non-contiguous RBs allocated to a MU (c.f. Fig. 1 with $L=4$). Then from (2), the average achievable rate of MU $i$ over $L\geq 1$ RBs is given by [15] \[\bar{V}_{i}=\frac{1}{L}\sum_{l=1}^{L}\frac{W}{N}\log_{2}\left(1+ \frac{NS_{i,l}}{\Gamma N_{0}W}\right)\] (3) where $S_{i,l}$ is the received signal power at the $l$th allocated RB, $l=1,...,L$, and $S_{i,l}$’s are independent over $l$ due to independent channel fading if the $L$ RBs allocated to a MU are sufficiently far apart in time and/or frequency. ### _Power Scaling Law_ We assume a simplified channel model consisting of distance-dependent pathloss with path loss exponent $\alpha>2$ and an additional random term accounting for short-term fading of the channel from the BS to each MU. With the assumed channel model, the received signal power for the $l$th RB of MU $i$ is given by \[S_{i,l} =\left\{\begin{array}[]{cl} P_{i}h_{i,l}K\left(\frac {r_{i}}{r_{0}}\right)^{-\alpha}&\mbox{if }r_{i}\geq r_{0}\\ P_{i}h_{i,l}K&\mbox{otherwise}\end{array}\right.\] (4) where $r_{i}$ is a random variable representing the distance between MU $i$ and BS, $K$ is a constant equal to the pathloss at a reference distance $r_{0}$, $h_{i,l}$ is an exponential random variable with unit mean accounting for Rayleigh fading with $h_{i,l}$’s being independent and identically distributed (i.i.d) over both $i$ and $l$, and $P_{i}$ is the transmit power for MU $i$, which is assumed to be identical for all $l$’s since the realizations of $h_{i,l}$’s are not assumed to be known at BS. It is easy to verify that $S_{i,l}$’s are i.i.d over $l$ as previously assumed. To characterize the required minimum transmit power for MU $i$, $P_{i}$, outage performance is considered as the user-level QoS constraint. An outage event occurs when the link between MU $i$ and BS cannot support a desired target rate $\bar{v}$ bits/sec, which is assumed to be equal for all MUs for simplicity. According to (3), the outage probability for MU $i$ is given by \[\text{P}_{\text{out}}^{i}=\mbox{Pr}\left\{\sum_{l=1}^{L}\frac{W}{ N}\log_{2}\left(1+\frac{NS_{i,l}}{\Gamma N_{0}W}\right)<L\bar{v}\right\}.\] (5) Since outage typically occurs when none of the $L$ parallel channels can support the average rate $\bar{v}$[15], (5) can be properly approximated as \[\text{P}_{\text{out}}^{i} \approx\prod_{l=1}^{L}\mbox{Pr}\left\{\frac{W}{N}\log_{2}\left(1+ \frac{NS_{i,l}}{\Gamma N_{0}W}\right)<\bar{v}\right\}\] \[=\left(\mbox{Pr}\left\{S_{i,1}<\frac{\Gamma N_{0}W}{N}(2^{\frac{N \bar{v}}{W}}-1)\right\}\right)^{L}.\] (6) Given $r_{i}$, $S_{i,1}$ is an exponential random variable with mean $\bar{S}_{i,1}$, which is given by \[\bar{S}_{i,1} =\left\{\begin{array}[]{cl} P_{i}K\left(\frac{r_{i}} {r_{0}}\right)^{-\alpha}&\mbox{if }r_{i}\geq r_{0}\\ P_{i}K&\mbox{otherwise}.\end{array}\right.\] (7) Thus, the outage probability for MU $i$ given distance from BS $r_{i}$ can be simplified as \[\text{P}_{\text{out}}^{i}(r_{i})\approx\left[1-\exp\left(-\frac{ \Gamma N_{0}W}{N\bar{S}_{i,1}}(2^{\frac{N\bar{v}}{W}}-1)\right)\right]^{L}.\] (8) Let $\bar{\text{P}}_{\text{out}}$ denote the maximum allowable outage probability for all MUs. Then the inequality \[\text{P}_{\text{out}}^{i}\leq\bar{\text{P}}_{\text{out}}\] (9) needs to be maintained for all $i$’s. From (7), (8) and (9), we can obtain $P_{i}$ given $r_{i}$ and $N$ for the BS’s STPC as³ [FOOTNOTE:3][ENDFOOTNOTE] \[P_{i}(r_{i},N)\] (10) where $C_{1}=-\ln(1-\bar{\text{P}}_{\text{out}}^{1/L})$ and $C_{2}=\frac{\bar{v}}{W}$. With $P_{i}(r_{i},N)$, the total transmit power $P_{t}$ at the BS can be expressed as \[P_{t}=\sum_{i=1}^{N}P_{i}(r_{i},N).\] (11) Note that $P_{t}$ is a random variable due to the randomness in the number of MUs, $N$, and their random distances from the BS, $r_{i}$’s. In this paper, we assume that the BS can perform a slow LTPC based on the MU density variation, in addition to the more rapid STPC, for the purpose of minimizing the long-term energy consumption (more details will be given in Section III). Considering the fluctuations of $P_{t}$ given coverage range $R$ and MU density $\lambda$, according to (11), a power scaling law that averages the random effects of the number of MUs and their locations is desired to facilitate the LTPC design to be studied in Section III. This motivates us to find the (short-term) average transmit power $\bar{P}_{t}\triangleq\mathbb{E}[P_{t}]$ at BS for a given pair of $R$ and $\lambda$, where the expectation is taken over $N$ and $r_{i}$’s. The approach for finding $\bar{P}_{t}$ is to apply the law of iterated expectations, i.e., \[\bar{P}_{t}=\mathbb{E}_{N}\left[\mathbb{E}[P_{t}\|N]\right]\] (12) where the inner expectation is taken over the random user locations given $N=n$ number of MUs, and the outer expectation is performed over the Poisson distributed $N$. This method works because $\mathbb{E}[P_{t}\|N=n]$ in (12) can be obtained using the following property of conditioned HPPP [14]: \[\mathbb{E}[P_{t}\|N=n] =\mathbb{E}\left[\sum_{i=1}^{n}P_{i}(r_{i},n)\right]\] \[=n\mathbb{E}[P_{i}(r_{i},n)]\] (13) where $P_{i}(r_{i},n)$ represents the required transmit power from the BS to any MU $i$ with distance $r_{i}$ given that $N=n$ number of MUs equally share the total bandwidth $W$ by EBS. It can be further verified that given $N=n$, MU $i$ is uniformly distributed within a circular coverage area with radius $R$. Thus, $\mathbb{E}[P_{i}(r_{i},n)]$ is identical for all $i$’s, and computed as \[\mathbb{E}[P_{i}(r_{i},n)]=\int_{0}^{R}P_{i}(r_{i},n)f(r_{i})dr_{i}\] (14) where $f(r_{i})=\frac{2r_{i}}{R^{2}}$, $0\leq r_{i}\leq R$, is the PDF of $r_{i}$. Using (14) and averaging $\mathbb{E}[P_{t}\|N=n]$ in (II-C) over the Poisson distribution of $N$, we obtain a closed-form expression for $\bar{P}_{t}$, which is given in the following theorem. Theorem ii.1: _Consider an OFDMA-based broadcast channel, where the available bandwidth $W$ Hz is equally shared among all MUs with STPC to support a target rate $\bar{v}$ bits/sec with outage constraint $\bar{\text{P}}_{\text{out}}$. Suppose that the channels from the BS to all MUs experience independent Rayleigh fading, then the transmit power at the BS averaged over MU population $N$ and BS-MU distance $r_{i}$, given a coverage range $R$ and a MU intensity $\lambda$, is approximated by_ \[\bar{P}_{t}(R,\lambda)=D_{1}R^{\alpha}\left(2^{D_{2}\pi\lambda R^ {2}}-1\right)\] (15) _where $D_{1}=\frac{2\Gamma N_{0}W}{K(-\ln(1-\bar{\text{P}}_{\text{out}}^{1/L}))( \alpha+2)r_{0}^{\alpha}}$ and $D_{2}=\frac{\bar{v}}{W}$ is the per-user spectrum efficiency in bps/Hz._ See Appendix A. Remark ii.1: _Theorem II.1 relates the average BS transmit power $\bar{P}_{t}$ with cell range $R$ and MU density $\lambda$. Given $R$, $\bar{P}_{t}$ grows exponentially with increasing $\lambda$ due to the reduced bandwidth equally allocated among (on average) $\mu_{N}=\lambda\pi R^{2}$ MUs. On the other hand, given $\lambda$, besides the exponential increment in $\bar{P}_{t}$ with respect to $R^{2}$ due to the similar effect of per-user bandwidth reduction, there exists an extra polynomial term $R^{\alpha}$ in $\bar{P}_{t}$, due to the increased power consumption needed to compensate for more significant path loss with growing $R$. Since $\bar{P}_{t}$ is a strictly increasing function of both $R$ and $\lambda$, to maintain a constant $\bar{P}_{t}$, $R$ needs to be reduced when $\lambda$ increases and vice versa. Theorem II.1 therefore quantifies the relationship among BS transmit power, cell size and MU density, which enables the design of the (long-term) cell adaptation strategies introduced in the rest of this paper._ ## III Optimal Power and Range Adaptation Power and range adaptation is the combined task of cell range adaptation and BS LTPC (including on-off control), which are both assumed to be performed on the time scale of MU density variation. Since MU’s density variation is much slower as compared with MU’s channel variation (which is taken care of by STPC studied in Section II-C), LTPC is implemented over $\bar{P}_{t}$ given in (15) for the purpose of minimizing the BS’s long-term energy consumption. In this section, we first present a practical energy consumption model for BS by considering both transmission and non-transmission related power consumptions. Based on the presented energy consumption model, we study a joint cell range adaptation and LTPC problem to minimize the long-term power consumption at BS under a system-level throughput constraint. ### _Energy Consumption Model at BS_ The energy consumption of a BS in general includes two parts: transmit power $\bar{P}_{t}$ and a constant power $P_{c}$ accounting for all non-transmission related power consumption of e.g. electronic hardware and air conditioning. When the BS does not need to support any user, it can switch to a “sleep” mode [16], by turning off the power amplifier to reduce energy consumption. We note that the two cases of $R>0$ and $R=0$ correspond to “on” and “off (sleep)” modes of BS, respectively. A power consumption model for the BS is thus given by \[\bar{P}_{\text{BS}}(R,\lambda) =\left\{\begin{array}[]{cl} a\bar{P}_{t}(R,\lambda)+ P_{c},&~{}R>0\\ P_{\text{sleep}},&~{}R=0\end{array}\right.\] (16) where $\bar{P}_{\text{BS}}(R,\lambda)$ represents the (short-term) average power consumption at BS given a pair of $R$ and $\lambda$, $P_{\text{sleep}}$ denotes the power consumed during the off mode, and $a\geq 1$ corresponds to the scaling of the actual power consumed with the radiated power due to amplifier and feeder losses. In practice, $P_{\text{sleep}}$ is generally much smaller than $P_{c}$[7] and thus in this paper, we assume $P_{\text{sleep}}=0$ for simplicity. Since $a$ is only a scaling constant, we further assume $a=1$ in our subsequent analysis unless stated otherwise. ### _Optimal Cell Adaptation_ According to (15), $\bar{P}_{t}(R,\lambda)$ is determined by $R$ and $\lambda$. LTPC is thus equivalent to range adaptation over $\lambda$, i.e., by first finding the range adaptation function $R(\lambda)$ and then obtaining $\bar{P}_{t}(R,\lambda)$ as $\bar{P}_{t}(R(\lambda),\lambda)$, the LTPC policy $\bar{P}_{\text{BS}}(R(\lambda),\lambda)$ follows from (16). The joint cell range adaptation and LTPC problem can thus be formulated as \[\mathrm{(P0)}:~{}\mathop{\mathtt{Min.}}\limits_{R(\lambda)\geq 0} ~{}~{}\mathbb{E}_{\lambda}\left[\bar{P}_{\text{BS}}(R(\lambda), \lambda)\right]\] (17) \[\mathtt{s.t.} ~{}~{}\mathbb{E}_{\lambda}\left[U(R(\lambda),\lambda)\right]\geq U _{\text{avg}}\] (18) \[~{}~{}\bar{P}_{\text{BS}}(R(\lambda),\lambda)\leq P_{\text{max}}, ~{}~{}\forall\lambda\] (19) where $U(R(\lambda),\lambda)=\pi\lambda R^{2}(\lambda)$ corresponds to the (short-term) average number of supported MUs, $U_{\text{avg}}$ represents the (long-term) system throughput⁴ constraint, and $P_{\text{max}}$ is the (short-term) power constraint at BS. For convenience, in the rest of this paper, $\bar{P}_{t}(R(\lambda),\lambda)$ and $\bar{P}_{\text{BS}}(R(\lambda),\lambda)$ are referred to as (short-term average) transmit power and power consumption at BS for a given $\lambda$, respectively, while $\mathbb{E}_{\lambda}\left[\bar{P}_{t}(R(\lambda),\lambda)\right]$ and $\mathbb{E}_{\lambda}\left[\bar{P}_{\text{BS}}(R(\lambda),\lambda)\right]$ are called the (long-term) average transmit power and average power consumption at BS, respectively. [FOOTNOTE:4][ENDFOOTNOTE] Note that if choosing $R(\lambda)$ such that $\bar{P}_{\text{BS}}(R(\lambda),\lambda)=P_{\text{max}}$ for all $\lambda>0$ still leads to a violation of constraint (18), then Problem (P0) is infeasible. For analytical tractability, we only consider the case where $U_{\text{avg}}$ yields a feasible (P0). (P0) is not convex due to the non-convexity of both the objective function (at $R=0$) and the throughput constraint (18) since $U(R(\lambda),\lambda)$ is a non-concave function over $R(\lambda)$. We start with reformulating (P0) via a change of variable: $x=R^{2}$, and making the constraint (19) implicit, which yields an equivalent problem \[\mathrm{(P1)}:~{}\mathop{\mathtt{Min.}}\limits_{x(\lambda)\in \mathcal{X}_{a}} ~{}~{}\mathbb{E}_{\lambda}\left[\bar{P}_{\text{BS}}(x(\lambda), \lambda)\right]\] (20) \[\mathtt{s.t.} ~{}~{}\mathbb{E}_{\lambda}\left[U(x(\lambda),\lambda)\right]\geq U _{\text{avg}}\] (21) where $\mathcal{X}_{a}\triangleq\left\{x(\lambda):x(\lambda)\geq 0,\bar{P}_{\text{BS} }(x(\lambda),\lambda)\leq P_{\text{max}},\forall\lambda\right\}$. In (P1), the constraint (21) becomes convex since $U(x(\lambda),\lambda)=\pi\lambda x(\lambda)$ is affine over $x(\lambda)$. Furthermore, $\mathcal{X}_{a}$ is a convex set, and $\mathbb{E}_{\lambda}\left[\bar{P}_{\text{BS}}(x(\lambda),\lambda)\right]$ is the affine mapping of an infinite number of quasi-convex functions $\bar{P}_{\text{BS}}(x(\lambda),\lambda)$ and can be shown to be quasi-convex. Therefore, (P1) is a quasi-convex optimization problem and it can be verified that Lagrangian duality method can be applied to solve (P1) globally optimally [17]. The Lagrangian of Problem (P1) is \[\mathcal{L}(x(\lambda),\mu)=\mathbb{E}_{\lambda}\left[\bar{P}_{ \text{BS}}(x(\lambda),\lambda)\right]-\mu\left(\mathbb{E}_{\lambda}\left[U(x( \lambda),\lambda)\right]-U_{\text{avg}}\right)\] (22) where $\mu\geq 0$ is the dual variable associated with the throughput constraint (21). Then it can be shown that solving (P1) is equivalent to solving parallel subproblems all having the same structure and each for a different value of $\lambda$. For a particular $\lambda$, the associated subproblem is expressed as \[\mathop{\mathtt{Min.}}\limits_{x(\lambda)\in\mathcal{X}_{a}} ~{}~{}L_{\lambda}(x(\lambda),\mu)\] (23) where $L_{\lambda}(x(\lambda),\mu)=\bar{P}_{\text{BS}}(x(\lambda),\lambda)-\mu U(x( \lambda),\lambda)$. To tackle the non-continuity of $\bar{P}_{\text{BS}}(x(\lambda),\lambda)$ at $x(\lambda)=0$ (due to $P_{c}>P_{\text{sleep}}\triangleq 0$) and the power constraint $\bar{P}_{\text{BS}}(x(\lambda),\lambda)\leq P_{\text{max}}$, we first consider the case where BS is always on, i.e., $x(\lambda)>0$ (thus, $\bar{P}_{\text{BS}}(x(\lambda),\lambda)$ is always differentiable) and there is no power constraint, i.e., $P_{\text{max}}=+\infty$. The power constraint and the non-continuity at $x(\lambda)=0$ will be incorporated into the solution later without loss of optimality. Denote $x_{1}^{}(\lambda)$ and $x_{2}^{}(\lambda)$ as the roots of the following two equations: \[\frac{\partial L_{\lambda}(x(\lambda),\mu)}{\partial x(\lambda)} =0,~{}~{}x(\lambda)>0\] (24) \[\bar{P}_{\text{BS}}(x(\lambda),\lambda) =P_{\text{max}},\] (25) respectively, where (24) is the optimality condition for $x(\lambda)$ in the case where BS is always on with infinite power budget and (25) gives the maximum coverage range due to finite $P_{\text{max}}$ for any given $\lambda$. Note that it is difficult to obtain closed-form solutions for $x_{1}^{}(\lambda)$ and $x_{2}^{}(\lambda)$ due to the complex form of $\bar{P}_{\text{BS}}(x(\lambda),\lambda)$ in (16). However, since $\bar{P}_{\text{BS}}(x(\lambda),\lambda)$ is a strictly increasing function of $x(\lambda)$, and furthermore is convex in $x(\lambda)$ when $x(\lambda)>0$, $x_{1}^{}(\lambda)$ and $x_{2}^{}(\lambda)$ can both be obtained numerically by a simple bisection search given $\mu$ and/or $\lambda$. Let $x^{}(\lambda)$ denote the optimal solution of Problem (23) with finite $P_{c}$ and $P_{\text{max}}$. Then $x^{}(\lambda)$ has three possible values: $x_{1}^{}(\lambda)$, $x_{2}^{}(\lambda)$ and 0, where $x_{2}^{}(\lambda)$ is taken when $x_{1}^{}(\lambda)$ violates the power constraint of $P_{\text{max}}$, i.e., $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)>P_{\text{max}}$. In the case of $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)\leq P_{\text{max}}$, a comparison between $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ and $L_{\lambda}(0,\mu)=0$ is needed to tackle the non-continuity due to $P_{c}>0$. If $L_{\lambda}(x_{1}^{}(\lambda),\mu)<0$, $x_{1}^{}(\lambda)$ indeed gives the optimal solution; otherwise, we have $x^{}(\lambda)=0$ since it minimizes $L_{\lambda}(x(\lambda),\mu)$ over $x(\lambda)\geq 0$. On the other hand, if $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)>P_{\text{max}}$, a similar comparison between $L_{\lambda}(x_{2}^{}(\lambda),\mu)$ and $L_{\lambda}(0,\mu)=0$ is needed to verify the optimality between $x_{2}^{}(\lambda)$ and $0$. Thus, the signs of $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ and $L_{\lambda}(x_{2}^{}(\lambda),\mu)$ as well as the value of $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)$ jointly determine $x^{}(\lambda)$, as summarized below: \[x^{}(\lambda) =\left\{\begin{array}[]{cl} x_{1}^{}(\lambda)&\mbox {if }\left.\begin{array}[]{cl}\bar{P}_{\text{BS}}(x_{1}^{}( \lambda),\lambda)\leq P_{\text{max}},\\ L_{\lambda}(x_{1}^{}(\lambda),\mu)<0\end{array}\right.\\ x_{2}^{}(\lambda)&\mbox{if }\left.\begin{array}[]{cl}\bar{P}_{ \text{BS}}(x_{1}^{}(\lambda),\lambda)>P_{\text{max}},\\ L_{\lambda}(x_{2}^{}(\lambda),\mu)<0\end{array}\right.\\ 0&\mbox{otherwise}.\end{array}\right.\] (26) To avoid checking the conditions in (26) for all $\lambda$’s and gain more insights to the optimal power and range adaptation scheme, we proceed to characterize some critical values of $\lambda$, based on which the BS can determine $x^{}(\lambda)$ with only the knowledge of the current density $\lambda$, through the following lemmas. Lemma* iii.1: _There exists $\lambda_{1}$, where $L_{\lambda}(x_{1}^{}(\lambda_{1}),\mu)=0$, such that $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ is positive for all $\lambda<\lambda_{1}$ and negative for all $\lambda>\lambda_{1}$._ See Appendix B. Lemma iii.2: $x_{1}^{}(\lambda)$ _is a strictly decreasing function of $\lambda$; $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)$ and $U(x_{1}^{}(\lambda),\lambda)$ are all strictly increasing functions of $\lambda$._ See Appendix C. Lemma* iii.3: $x_{2}^{}(\lambda)$ _is a strictly decreasing function of $\lambda$; $U(x_{2}^{}(\lambda),\lambda)$ is a strictly increasing function of $\lambda$._ The monotonicity of $x_{2}^{}(\lambda)$ can be directly obtained from Remark II.1. The proof for $U(x_{2}^{}(\lambda),\lambda)$ is similar to that of Lemma III.2 in Appendix C, and is thus omitted for brevity. Since $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)$ is a strictly increasing function of $\lambda$, there exists $\lambda_{2}$ with $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda_{2}),\lambda_{2})=P_{\text{max}}$, above which $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)>P_{\text{max}}$. Furthermore, since $U(x_{2}^{}(\lambda),\lambda)$ strictly increases with $\lambda$, $L_{\lambda}(x_{2}^{}(\lambda),\mu)=P_{\text{max}}-\mu U(x_{2}^{}(\lambda),\lambda)$ is thus a strictly decreasing function of $\lambda$ and there exists $\lambda_{3}$ with $L_{\lambda}(x_{2}^{}(\lambda_{3}),\mu)=0$, such that $L_{\lambda}(x_{2}^{}(\lambda),\mu)<0$ for all $\lambda>\lambda_{3}$. Therefore, the conditions in (26) can be simplified as the inequalities among $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$, which is presented in the following theorem. Theorem iii.1: _The optimal solution of Problem (P1) is given by_ * _If_ $\lambda_{2}\geq\lambda_{1}$__ \[x^{}(\lambda) =\left\{\begin{array}[]{cl} 0&\mbox{if }\lambda\leq \lambda_{1}\\ x_{1}^{}(\lambda)&\mbox{if }\lambda_{1}<\lambda\leq\lambda_{2}\\ x_{2}^{}(\lambda)&\mbox{otherwise}.\end{array}\right.\] (27) _If_ $\lambda_{2}<\lambda_{1}$__ \[x^{}(\lambda) =\left\{\begin{array}[]{cl} 0&\mbox{if }\lambda\leq \lambda_{3}\\ x_{2}^{}(\lambda)&\mbox{otherwise}.\end{array}\right.\] (28) See Appendix D. Note that Problem (P1) needs to be solved by iteratively solving $x^{}(\lambda)$ with a fixed $\mu$ based on Theorem III.1, and updating $\mu$ via the bisection search until the throughput constraint (21) is met with equality. The optimal solution of Problem (P0), $R^{}(\lambda)$, can then be obtained as $R^{}(\lambda)=\sqrt{x^{}(\lambda)}$. From Theorem III.1, Lemma III.2 and Lemma III.3, we obtain the following corollary. Corollary 1: $R^{}(\lambda)$ _and $U(R^{}(\lambda),\lambda)$ are strictly decreasing and increasing functions of $\lambda$, respectively, if $R^{}(\lambda)>0$; $\bar{P}_{\text{BS}}(R^{}(\lambda),\lambda)$ is a non-decreasing function of $\lambda$ if $R^{}(\lambda)>0$._ The proof directly follows from Lemmas III.2 and III.3, and thus is omitted for brevity. Next, we illustrate the optimal solution $R^{}(\lambda)$ to Problem (P0) to gain more insights to the optimal cell adaptation scheme. It is observed that there exists a cut-off value of $\lambda$ for each of the two cases in Theorem III.1, below which the BS is switched off. This on-off behavior implies that allowing BS be switched off under light load is essentially optimal for energy saving. Since $x_{2}^{}(\lambda)$ is the root of (25), which corresponds to the maximum coverage range with finite $P_{\text{max}}$ for any given $\lambda$, it is worth noticing that when $\lambda_{2}<\lambda_{1}$, constant power transmission with $P_{\text{max}}$ is optimal. The reason is that when $P_{\text{max}}$ is relatively small for the given throughput constraint $U_{\text{avg}}$, BS has to transmit at its maximum power at all the “on” time. According to Corollary 1, the average number of supported MUs $U(x^{}(\lambda),\lambda)$ strictly increases with $\lambda$. This is because that under the optimal scheme, BS should support more MUs when the density is larger to optimize energy-efficiency. ### _High Spectrum-Efficiency Regime_ Although Theorem III.1 reveals the structure of the optimal cell adaptation solution, which can be efficiently obtained numerically, the solution is expressed in terms of critical values of $\lambda$, namely $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$, for which closed-form expressions are difficult to be obtained. In this subsection, we obtain closed-form expressions of the solution in Theorem III.1 under a high spectrum-efficiency (HSE) assumption. It is observed from (15) that $D_{2}\pi\lambda R^{2}=\frac{\bar{v}\pi\lambda R^{2}}{W}=\frac{\bar{v}\mu_{N}}{W}$, which can be interpreted as the average network throughput in bps divided by the total bandwidth, and is thus the system spectrum-efficiency in bps/Hz. Therefore, the HSE assumption is equivalent to letting $D_{2}\pi\lambda R^{2}\gg 1$. Under this condition, (15) in Theorem II.1 can be simplified as \[\bar{P}_{t}(R,\lambda)=D_{1}R^{\alpha}2^{D_{2}\pi\lambda R^{2}}.\] (29) Lemma iii.4: _Under the HSE assumption of $D_{2}\pi\lambda R^{2}\gg 1$, $x_{1}^{}(\lambda)$ and $x_{2}^{}(\lambda)$ in Theorem III.1 are given by_ \[x_{1}^{}(\lambda) =\frac{\alpha}{2D_{3}\pi\lambda}\mathcal{W}\left(\frac{2D_{3}\pi \lambda}{\alpha}\left(\frac{\mu}{D_{1}D_{3}}\right)^{\frac{2}{\alpha}}\right)\] (30) \[x_{2}^{}(\lambda) =\frac{\alpha}{2D_{3}\pi\lambda}\mathcal{W}\left(\frac{2D_{3}\pi \lambda}{\alpha}\left(\frac{P^{t}_{\text{max}}}{D_{1}}\right)^{\frac{2}{\alpha }}\right)\] (31) _where $D_{3}=(\ln 2)D_{2}$, $P^{t}_{\text{max}}=P_{\text{max}}-P_{c}$, and $\mathcal{W}(\cdot)$ is the Lambert W function defined as $y=\mathcal{W}(y)e^{\mathcal{W}(y)}$[18]._ See Appendix E. The accuracy of the above HSE approximation will be verified by numerical results in Section V. With (30) and (31), closed-form expressions of $U(x_{1}^{}(\lambda),\lambda)$, $U(x_{2}^{}(\lambda),\lambda)$ and $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)$ under the HSE assumption can be easily obtained, which can be verified to preserve the properties given in Lemmas III.1-III.3 by using properties of the Lambert W function. For brevity, we omit the details here. Moreover, we obtain the following corollary from Lemma III.4. Corollary 2: _Under the HSE assumption of $D_{2}\pi\lambda R^{2}\gg 1$, $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ in Theorem III.1 are given by_ \[\lambda_{1} =\left(\frac{1}{\pi D_{3}}+\frac{P_{c}}{\mu\pi}\right)\left(\frac {D_{1}D_{3}}{\mu}\right)^{\frac{2}{\alpha}}\exp\left(\frac{2}{\alpha}+\frac{2D _{3}P_{c}}{\mu\alpha}\right)\] (32) \[\lambda_{2} =\frac{\alpha P^{t}_{\text{max}}}{2\pi(\mu-D_{3}P^{t}_{\text{max} })}\left(\frac{D_{1}D_{3}}{\mu}\right)^{\frac{2}{\alpha}}\exp\left(\frac{D_{3} P^{t}_{\text{max}}}{\mu-D_{3}P^{t}_{\text{max}}}\right)\] (33) \[\lambda_{3} =\frac{P_{\text{max}}}{\mu\pi}\left(\frac{D_{1}}{P^{t}_{\text{max }}}\right)^{\frac{2}{\alpha}}\exp\left(\frac{2D_{3}P_{\text{max}}}{\mu\alpha} \right).\] (34) The proof is similar to that of Lemma III.4, and thus omitted for brevity. Remark* iii.1: $\lambda_{1}$_, $\lambda_{2}$ and $\lambda_{3}$ in Corollary 2 can be verified to be all strictly decreasing functions of the dual variable $\mu$ as follows. Let $\mu^{}$ be the optimal dual solution of Problem (P1), $\lambda^{}_{1}$, $\lambda^{}_{2}$ and $\lambda^{}_{3}$ be the corresponding critical values of $\lambda$ when $\mu=\mu^{}$. Since $\mu^{}$ strictly increases as the throughput constraint $U_{\text{avg}}$ increases, it follows from (32)-(34) that $\lambda^{}_{1}$, $\lambda^{}_{2}$ and $\lambda^{}_{3}$ are all strictly decreasing functions of $U_{\text{avg}}$. Since in Theorem III.1, $\lambda_{1}$ and $\lambda_{3}$ are the thresholds of the MU density above which BS switches from off to on mode, their decrease with increasing $U_{\text{avg}}$ implies that BS needs to be stay on for more time if large system throughput is required._ ## IV Suboptimal schemes The optimal power and range adaptation policy presented in Section III combines cell range adaptation and BS LTPC (including on-off control), suggesting that the energy saving at BS essentially comes from two major energy saving mechanisms (ESMs): range adaptation and BS on-off control. In this section, we propose four low-complexity suboptimal schemes, which can be considered as suboptimal solutions of (P0) with various combinations of these two ESMs, to investigate their effects on the system energy consumption. 1. Fixed range with BS on-off control (FRw/OFC): In this scheme, BS is switched off when MU density is lower than a cutoff value $\lambda_{c}$, while the coverage range $R$ is fixed as $R_{f}$ whenever BS is on. For a given $\lambda_{c}$, since from (15) the BS transmission power is a strictly increasing function of $R$, $R_{f}$ should be chosen as the minimum value, denoted by $R_{f}(\lambda_{c})$, to satisfy the throughput constraint $U_{\text{avg}}$ by applying BS power control with fixed coverage based on $\lambda$ according to (15). Furthermore, $\lambda_{c}$ should be optimized to minimize the average BS power (including both transmission and non-transmission related portions) consumption. The optimal cutoff value $\lambda^{}_{c}$ and its corresponding coverage range $R_{f}(\lambda^{}_{c})$ can be found via solving Problem (P0) by assuming the following (suboptimal) range adaptation policy: \[R(\lambda) =\left\{\begin{array}[]{cl} R_{f}(\lambda_{c})&\mbox {if }\lambda\geq\lambda_{c}\\ 0&\mbox{otherwise}.\end{array}\right.\] (35) Specifically, we have \[\lambda^{}_{c}=\arg\mathop{\mathtt{min.}}\limits_{\lambda_{c}< \lambda_{\text{max}}}\mathbb{E}_{\lambda_{c}}\left[\bar{P}_{\text{BS}}(R_{f}( \lambda_{c}),\lambda)\right]\] (36) where \[R_{f}(\lambda_{c})=\mathop{\mathtt{min.}} ~{}R_{f}\] (37) \[\mathtt{s.t.} ~{}\mathbb{E}_{\lambda_{c}}\left[U(R_{f},\lambda)\right]\geq U_{ \text{avg}}\] \[~{}\bar{P}_{\text{BS}}(R_{f},\lambda)\leq P_{\text{max}},\forall \lambda\geq\lambda_{c}.\] where $\mathbb{E}_{\lambda_{c}}\left[f(\lambda)\right]\triangleq\mathbb{E}_{\lambda} \left[\left.f(\lambda)\right\|\lambda\geq\lambda_{c}\right]\mbox{Pr}\left\{ \lambda\geq\lambda_{c}\right\}$. For a given $\lambda_{c}$, since $\mathbb{E}_{\lambda_{c}}\left[U(R_{f},\lambda)\right]$ is a strictly increasing function of $R_{f}$, Problem (37) can be solved efficiently through the bisection search. Then, the optimal cut-off threshold in (36) can be found by a line search over $[0,\lambda_{\text{max}}]$. 2. Fixed range without BS on-off control (FRw/oOFC): In this scheme, BS is not allowed to be switched off during operation. The coverage range is fixed as $R_{f}$, which is chosen as the minimum value of $R$ to satisfy the throughput constraint $U_{\text{avg}}$ by applying BS power control only based on $\lambda$ according to (15). Note that FRw/oOFC can be treated as a special case of FRw/OFC with $\lambda_{c}$ in (35) set to be $0$. Thus, the fixed coverage $R_{f}$ can be directly determined by solving Problem (37) with $\lambda_{c}=0$. 3. Adaptive range with BS on-off control (ARw/OFC): In this scheme, BS is switched off when MU density is lower than a cutoff value $\lambda_{c}$, while BS transmits with constant power $P_{f}-P_{c}$ whenever it is powered on by applying range adaptation only based on $\lambda$ according to (15). Given $P_{f}$, the corresponding $\lambda_{c}$ is chosen as the maximum value of $\lambda$, denoted by $\lambda_{c}(P_{f})$, to satisfy the throughput constraint $U_{\text{avg}}$, in order to minimize the BS average power consumption $\mathbb{E}_{\lambda_{c}(P_{f})}\left[P_{f}\right]$; $P_{f}$ is then optimized to further minimize the average power consumption at BS. The optimal transmit power $P^{}_{f}-P_{c}$ and its corresponding cutoff value $\lambda_{c}(P^{}_{f})$ can be obtained via solving Problem (P0) by assuming the following (suboptimal) range adaptation policy: \[R(\lambda) =\left\{\begin{array}[]{cl}\bar{P}^{-1}_{\text{BS}}( P_{f},\lambda)&\mbox{if }\lambda\geq\lambda_{c}(P_{f})\\ 0&\mbox{otherwise},\end{array}\right.\] (38) where $\bar{P}^{-1}_{\text{BS}}(P_{f},\lambda)$ is the inverse function of (16) which computes the coverage range with given BS power consumption $P_{f}$ and MU density $\lambda$. Specifically, we have \[P^{}_{f}=\arg\mathop{\mathtt{min.}}\limits_{P_{f}\leq P_{\text{ max}}}\mathbb{E}_{\lambda_{c}(P_{f})}\left[P_{f}\right]\] (39) where \[\lambda_{c}(P_{f})=\mathop{\mathtt{max.}} ~{}\lambda_{c}\] (40) \[\mathtt{s.t.} ~{}\mathbb{E}_{\lambda_{c}}\left[U(R(\lambda),\lambda)\right]\geq U _{\text{avg}}\] \[~{}\bar{P}_{\text{BS}}(R(\lambda),\lambda)=P_{f},\forall\lambda \geq\lambda_{c}.\] Note that from (38) and Remark II.1, $R(\lambda)$ increases strictly with $P_{f}$ given $\lambda$, $U(R(\lambda),\lambda)=\pi\lambda R^{2}(\lambda)$ is thus a strictly increasing function of $P_{f}$. Therefore, Problem (40) can be solved efficiently through the bisection search. Then, the optimal constant BS power consumption in (39) can be found by a line search over $[0,P_{\text{max}}]$. 4. Adaptive range without BS on-off control (ARw/oOFC): In this scheme, BS transmits with constant power $P_{f}-P_{c}$ and is not allowed to be switched off during operation, i.e., no BS power control is applied. The constant transmit power $P_{f}-P_{c}$ is chosen as the minimum value to satisfy the throughput constraint $U_{\text{avg}}$ by applying range adaptation only based on $\lambda$ according to (15). Note that ARw/oOFC is a special case of ARw/OFC with $\lambda_{c}$ in (38) set to be $0$. Thus, $P_{f}$ can be obtained by solving Problem (39) with $\lambda_{c}=0$. The suboptimal schemes presented above all yield feasible and in general suboptimal solutions of Problem (P0). In particular, FRw/OFC and ARw/oOFC apply only BS power control (including on-off control) and only range adaptation, respectively; ARw/OFC applies both BS on-off control and range adaptation, while FRw/oOFC does not apply any of them for lowest complexity. By comparing the performance of these suboptimal schemes with the optimal scheme presented in Section III, we can investigate the effect of each individual ESM, namely, BS power control and range adaptation on the BS energy saving, as will be shown in the next section through numerical examples. ## V Numerical Results To obtain numerical results, we assume a time-varying traffic density with PDF: $f(\lambda)=\frac{4\lambda}{\lambda^{2}_{\text{max}}},~{}0\leq\lambda\leq\frac{ \lambda_{\text{max}}}{2}$; $f(\lambda)=\frac{4}{\lambda_{\text{max}}}-\frac{4\lambda}{\lambda^{2}_{\text{ max}}},~{}\frac{\lambda_{\text{max}}}{2}<\lambda\leq\lambda_{\text{max}}$, where $\lambda_{\text{max}}=1\times 10^{-4}$ MUs/$\mbox{m}^{2}$ is the peak traffic load. We consider pathloss and Rayleigh fading for channels between BS and MUs, where the pathloss exponent $\alpha$ is 3 and the outage probability threshold $\bar{\text{P}}_{\text{out}}$ is $10^{-3}$. The bandwidth $W$ and the rate requirement $\bar{v}$ of each MU are set to be 5 MHz and $150$ kbits/sec, respectively, if not specified otherwise. We also set a short-term power constraint at BS as $P_{\text{max}}=160$ W. Other parameters are set as $\Gamma=1$, $N_{0}=-174$ dBm/Hz, $r_{0}=10$ m, and $K=-60$ dB. <figure><img src="content_image/1211.6239/x2.png"><figcaption>Fig. 2: Average transmit power ¯Pt(R,λ) in Theorem II.1.</figcaption></figure> Fig. 2 verifies the power scaling law in Theorem II.1. For a given MU density $\lambda$, it is observed that the simulation results match well with our analytical result in (15). <figure><img src="content_image/1211.6239/x3.png"><figcaption></figcaption></figure> <figure><img src="content_image/1211.6239/x5.png"><figcaption></figcaption></figure> <figure><img src="content_image/1211.6239/x7.png"><figcaption></figcaption></figure> Fig. 3 and Fig. 3 show the optimal range adaptation in Theorem III.1 and the approximate range adaptation in Lemma III.4 under the HSE assumption as functions of MU density, i.e., $R^{}(\lambda)=\sqrt{x^{}(\lambda)}$, for the two cases of $\lambda_{2}\geq\lambda_{1}$ and $\lambda_{2}<\lambda_{1}$, respectively. Fig. 4 and Fig. 5 show the corresponding optimal BS power adaptation and the resulting system throughput (in terms of average number of supported MUs), respectively⁵. For Fig. 3, Fig. 4 and Fig. 5, it is assumed that $P_{c}=120$ W and the corresponding optimal dual solution for Problem (P1) is $\mu^{}=1.05$, with which it can be verified that $\lambda_{2}>\lambda_{1}$, i.e., corresponding to the first case in Theorem III.1. For Fig. 3, Fig. 4 and Fig. 5, it is assumed that $P_{c}=140$ W and $\mu^{}=0.8$; thus the critical values of $\lambda$ satisfy $\lambda_{3}>\lambda_{1}>\lambda_{2}$, which is in accordance with the second case of Theorem III.1. It is observed that the numerical examples validate our theoretical results. As shown in Fig. 3, a cut-off value of $\lambda$ exists (note that $\bar{\lambda}_{i},i=1,2,3$, represent the approximate critical values of $\lambda$ obtained by Corollary 2) in either of the two cases of Theorem III.1, which implies that allowing BS to be switched off under light load is optimal for energy saving. Note that from Fig. 3, the approximate range adaptation is observed to match well with the optimal range adaptation for both cases. Fig. 4 shows the optimal BS power adaptation versus the MU density. It is observed that once the BS is on, it transmits near or at the maximum power budget, which implies that constant power transmission at “on” mode is near or even optimal. This also explains the observation in Fig. 3 that the deviation of the approximated value of $\lambda_{2}$ or $\bar{\lambda}_{2}$ from $\lambda_{2}$ does not affect the accuracy of the approximate range adaptation policy, since the accuracy of $\lambda_{1}$ and $\lambda_{3}$ that control BS’s on-off behavior is more crucial. The variations of the system throughput $U(R^{}(\lambda),\lambda)$ with MU density $\lambda$ under the optimal scheme is shown in Fig. 5. As discussed in Corollary 1, $U(R^{}(\lambda),\lambda)$ is observed to increase strictly with $\lambda$ indicating that the optimal adaptation scheme takes advantage of higher MU density to maximize the system throughput. [FOOTNOTE:5][ENDFOOTNOTE] <figure><img src="content_image/1211.6239/x9.png"><figcaption>Fig. 6: Performance comparison with Pc=60 W and ¯v=150 Kbps</figcaption></figure> Next, we compare the suboptimal schemes in Section IV with the optimal scheme. With $P_{c}=60$ W, Fig. 6 shows the average power consumption $\bar{P}_{\text{BS}}$ at BS versus the system throughput $U_{\text{avg}}$. From Fig. 6, we observe that ARw/OFC performs almost the same as the optimal scheme over the entire range of values of $U_{\text{avg}}$. This is because that constant power transmission at BS “on” mode is near or even optimal (c.f. Fig. 4) and ARw/OFC differs from the optimal scheme only in that the (long-term) transmit power control when BS is on (c.f. Fig. 4 with $\lambda_{1}<\lambda<\lambda_{2}$) is not implemented. It is also observed that when $U_{\text{avg}}$ is small, FRw/OFC has similar energy consumption as the optimal scheme and ARw/OFC; however, their performance gap is enlarged as $U_{\text{avg}}$ increases. A similar observation can be made by comparing ARw/oOFC and FRw/oOFC. From these observations, it follows that BS on-off control is the most effective ESM when the network throughput is low, while range adaptation plays a more important role when the network throughput becomes higher. Finally, we observe that ARw/OFC and FRw/OFC converge to ARw/oOFC and FRw/oOFC, respectively, as $U_{\text{avg}}$ increases. This is because that to achieve higher network throughput, BS needs to be “on” for more time to support larger number of MUs; as a result, BS on-off control is less useful for energy saving. <figure><img src="content_image/1211.6239/x10.png"><figcaption>Fig. 7: Performance comparison with Pc=100 W and ¯v=150 Kbps</figcaption></figure> In Fig. 7, we set $P_{c}=100$ W to further evaluate the performances of different schemes under a higher non-transmission related power consumption at BS. Similar observations can be made from Fig. 7 as in Fig. 6. However, it is worth noticing that BS on-off control plays a more dominant role for energy saving when $U_{\text{avg}}$ is small, since a higher $P_{c}$ is required. It is also interesting to observe that the performance gaps among different schemes with and without range adaptation are almost invariant to the change of $P_{c}$ at high network throughput, which is around $45$ W in both Figs. 6 and 7 with $U_{\text{avg}}=220$. In Fig. 8, $P_{c}$ is reset as $60$ W but the transmission rate for each MU $\bar{v}$ is increased to $500$ kbits/sec to model the case with high-rate multimedia traffic. The simulation result shows that the convergence between different schemes with and without BS on-off control is much faster, which implies that range adaptation becomes more effective. <figure><img src="content_image/1211.6239/x11.png"><figcaption>Fig. 8: Performance comparison with Pc=60 W and ¯v=500 Kbps</figcaption></figure> To summarize, we draw the following key conclusions on the effects of different ESMs on the BS energy saving performance: BS on-off control is the most effective ESM when the network throughput is not high; * Cell range adaptation plays a more important role in BS energy saving when the network throughput is higher; * Finer-grained transmit power control at BS does not introduce significant benefit, i.e. constant power transmission at BS “on” mode is practically optimal. ## VI Conclusion In this paper, under an OFDMA-based broadcast channel setup, we investigate optimal power and range adaptation polices with time-varying traffic to minimize the BS average power consumption subject to the throughput and QoS constraints. A new power scaling law that relates the (short-term) average transmit power at BS with the given cell range and MU density is derived, based on which we obtain the optimal power and range adaptation policy by solving a joint cell range adaptation and (long-term) power control problem. By exploiting the fact that energy saving at BS essentially comes from two major mechanisms, namely BS on-off power control and range adaptation, suboptimal schemes are proposed to achieve efficient performance-complexity tradeoffs. It is shown by simulation results that when the network throughput is modest, BS on-off power control is the most effective energy saving mechanism, while when the network throughput is higher, range adaptation becomes more effective. The results of this paper provide a preliminary unified framework for evaluating the performance of existing cell adaptation schemes such as BS’s on-off switching and cell zooming, and for designing cell adaptation strategies for optimal energy saving. In this paper, we focus on the extreme case of a one-cell system for the purpose of obtaining useful insights, which needs to be extended to the more practical multi-cell scenario. It is thus interesting as well as important to investigate the optimal cell adaptation policy in a cooperative multi-cell setup by balancing between the cellular network energy consumption and its coverage performance by extending the mathematical framework developed in this paper. ## Appendix A Proof of Theorem ii.1 First, $\mathbb{E}[P_{i}(r_{i},n)]$ is computed based on (14) as follows, where $P_{i}(r_{i},n)$ is given by (10) with $N$ replaced by $n$. \[\mathbb{E}[P_{i}(r_{i},n)]=\frac{2\Gamma N_{0}W(2^{nC_{2}}-1)}{KC _{1}(\alpha+2)r_{0}^{\alpha}n}\left(R^{\alpha}+\frac{\alpha r_{0}^{\alpha+2}}{ 2R^{2}}\right).\] (41) Since $\mathbb{E}[P_{i}(r_{i},n)]$ is identical for all $i$’s, according to (II-C), $\mathbb{E}[P_{t}\|N]$ can be simply obtained through multiplying $\mathbb{E}[P_{i}(r_{i},n)]$ by the number of MUs $n$, i.e. \[\mathbb{E}[P_{t}\|N] =n\mathbb{E}[P_{i}(r_{i},n)]\] \[=\frac{2\Gamma N_{0}W(2^{nC_{2}}-1)}{KC_{1}(\alpha+2)r_{0}^{ \alpha}}\left(R^{\alpha}+\frac{\alpha r_{0}^{\alpha+2}}{2R^{2}}\right).\] (42) Averaging (A) over the Poisson distribution of $N$, we finally obtain $\bar{P_{t}}$ as \[\bar{P_{t}} =\sum_{n=0}^{\infty}\frac{2\Gamma N_{0}W(2^{nC_{2}}-1)}{KC_{1}( \alpha+2)r_{0}^{\alpha}}\left(R^{\alpha}+\frac{\alpha r_{0}^{\alpha+2}}{2R^{2} }\right)\frac{\mu_{N}^{n}}{n!}e^{-\mu_{N}}\] (43) \[=D_{1}\left(R^{\alpha}+\frac{\alpha r_{0}^{\alpha+2}}{2R^{2}} \right)\left(\sum_{n=0}^{\infty}\frac{(\mu_{N}2^{C_{2}})^{n}}{n!}e^{-\mu_{N}}- 1\right)\] (44) \[=D_{1}\left(R^{\alpha}+\frac{\alpha r_{0}^{\alpha+2}}{2R^{2}} \right)\left(e^{D^{{}^{\prime}}_{1}\pi\lambda R^{2}}-1\right)\] (45) \[\approx D_{1}R^{\alpha}\left(e^{D^{{}^{\prime}}_{1}\pi\lambda R^{ 2}}-1\right)\] (46) where $D_{1}=\frac{2\Gamma N_{0}W}{KC_{1}(\alpha+2)r_{0}^{\alpha}}$ and $D^{{}^{\prime}}_{1}=2^{\frac{\bar{v}}{W}}-1$. Note that since cell radius $R$ is practically much larger than the reference distance $r_{0}$, we have ignored the term $\frac{\alpha r_{0}^{\alpha+2}}{2R^{2}}$ in (45). It is worth noting that \[D^{{}^{\prime}}_{1}=(2^{\frac{\bar{v}}{W}}-1)=(2^{\frac{r_{se}}{ \bar{N}}}-1)\] (47) where $r_{se}$ is the system spectrum efficiency in bps/Hz and $\bar{N}$ is the nominal number of supported users, both of which are pre-designed system parameters. In practice, $r_{se}=2\sim 6$ bps/Hz and $\bar{N}$ is a couple of hundreds and even thousands. Therefore, $\frac{r_{se}}{\bar{N}}$ is generally a very small number such that \[D^{{}^{\prime}}_{1}\approx\frac{\bar{v}}{W}\ln 2.\] (48) Thus, (46) can be further simplified as \[\bar{P_{t}}\approx D_{1}R^{\alpha}\left(2^{D_{2}\pi\lambda R^{2}} -1\right)\] (49) where $D_{2}=\frac{\bar{v}}{W}$. Theorem II.1 is thus proved. ## Appendix B Proof of Lemma iii.1 To prove Lemma III.1, the following two facts are first verified: 1. For any $P_{c}$, which yields feasible (P0), there always exist some $\lambda$ such that $L_{\lambda}(x_{1}^{}(\lambda),\mu)<0$; 2. If $L_{\lambda}(x_{1}^{}(\lambda_{a}),\mu)\leq 0$, then $L_{\lambda}(x_{1}^{}(\lambda_{b}),\mu)<0$ for all $\lambda_{b}>\lambda_{a}$. The first fact can be shown by contradiction as follows. Suppose that $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ is always non-negative, i.e. \[L_{\lambda}(x_{1}^{}(\lambda),\mu)\geq 0,~{}~{}~{}~{}\forall x> 0,\lambda\geq 0.\] (50) Then, according to (26) we have \[x^{}(\lambda)=0,~{}~{}~{}~{}\forall\lambda\geq 0\] (51) which violates the throughput constraint $\mathbb{E}_{\lambda}\left[U(x(\lambda),\lambda)\right]\geq U_{\text{avg}}$. The first fact is thus proved. Next, we verify the second fact. According to the first fact, there always exists a $\lambda$ such that $L_{\lambda}(x_{1}^{}(\lambda),\mu)<0$. Therefore, without loss of generality, we can assume $L_{\lambda}(x_{1}^{}(\lambda_{a}),\mu)\leq 0$, i.e. \[\min\limits_{x(\lambda_{a})>0}\bar{P}_{\text{BS}}(x(\lambda_{a}), \lambda_{a})-\mu U(x(\lambda_{a}),\lambda_{a})\leq 0.\] (52) Then there exists at least one $x_{a}(\lambda_{a})>0$ such that \[\bar{P}_{\text{BS}}(x_{a}(\lambda_{a}),\lambda_{a})-\mu U(x_{a}( \lambda_{a}),\lambda_{a})\leq 0\] (53) or equivalently, \[D_{1}x_{a}(\lambda_{a})^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda_{a}x_{a}(\lambda_{a})}-1\right)+P_{c}\leq\mu\pi\lambda_{a}x_{a}( \lambda_{a}).\] (54) For any given $\lambda_{b}>\lambda_{a}$, by letting $x_{b}(\lambda_{b})=x_{a}(\lambda_{a})\frac{\lambda_{a}}{\lambda_{b}}$, then \[D_{1}x_{b}(\lambda_{b})^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda_{b}x_{b}(\lambda_{b})}-1\right)+P_{c}\] (55) \[= D_{1}x_{b}(\lambda_{b})^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda_{a}x_{a}(\lambda_{a})}-1\right)+P_{c}\] (56) \[< D_{1}x_{a}(\lambda_{a})^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda_{a}x_{a}(\lambda_{a})}-1\right)+P_{c}\] (57) \[\leq \mu\pi\lambda_{a}x_{a}(\lambda_{a})=\mu\pi\lambda_{b}x_{b}( \lambda_{b}).\] (58) Thus for any $\lambda_{b}>\lambda_{a}$, we can always find an $x_{b}(\lambda_{b})$ such that $\bar{P}_{\text{BS}}(x_{b}(\lambda_{b}),\lambda_{b})-\mu U(x_{b}(\lambda_{b}), \lambda_{b})<0$, which implies $L_{\lambda}(x_{1}^{}(\lambda_{b}),\mu)<0$. The second fact is thus proved. We are now ready to prove Lemma III.1. The proof is by first showing the fact that $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ is positive for sufficiently small $\lambda$’s, and then combining this result with the two facts previously shown. According to the first-order Taylor expansion, we have \[D_{1}x(\lambda)^{\frac{\alpha}{2}}\left(2^{D_{2}\pi\lambda x( \lambda)}-1\right)+P_{c}\] (59) \[> (\ln 2)D_{1}D_{2}\pi\lambda x(\lambda)^{\frac{\alpha+2}{2}}+P_{c} ,~{}~{}\forall x>0.\] (60) Let $h(x(\lambda))=(\ln 2)D_{1}D_{2}\pi\lambda x(\lambda)^{\frac{\alpha+2}{2}}+P_{c }-\mu\pi\lambda x(\lambda)$; then the minimum value of $h(x(\lambda))$ could be easily found by its first-order differentiation, given by \[h(x(\lambda))_{\text{min}}=P_{c}-x_{\text{min}}\lambda\mu\pi \frac{\alpha}{\alpha+2}\] (61) where $x_{\text{min}}=\left(\frac{2\mu}{(\alpha+2)(\ln 2)D_{1}D_{2}}\right)^{\frac{2} {\alpha}}$. It is easy to verify that if $\lambda<\frac{(\alpha+2)P_{c}}{\alpha\mu\pi x_{\text{min}}}$, $h(x(\lambda))_{\text{min}}>0$. Since $L_{\lambda}(x(\lambda),\mu)$ is an upper bound of $h(x(\lambda))$, we have \[L_{\lambda}(x(\lambda),\mu)>0,~{}~{}\forall x(\lambda)>0~{}\text {and}~{}\lambda<\frac{(\alpha+2)P_{c}}{\alpha\mu\pi x_{\text{min}}}\] (62) which implies that \[L_{\lambda}(x_{1}^{}(\lambda),\mu)>0,~{}~{}\forall\lambda<\frac {(\alpha+2)P_{c}}{\alpha\mu\pi x_{\text{min}}}.\] (63) We thus show that $L_{\lambda}(x_{1}^{}(\lambda_{b}),\mu)$ is positive for $\lambda$’s satisfying (63). With the two facts given earlier, it follows that $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ cannot be positive for all $\lambda$’s and $L_{\lambda}(x_{1}^{}(\lambda),\mu)$ will remain negative once it turns to be negative for the first time as $\lambda$ increases; thus, we conclude that there must exist a critical value for $\lambda$, i.e., $\lambda_{1}>0$ as given in Lemma III.1. Lemma III.1 is thus proved. ## Appendix C Proof of Lemma iii.2 Using the series expansion $2^{x}=\sum\limits_{k=0}^{\infty}\frac{(x(\ln 2))^{k}}{k!}$, (24) is expanded as \[x_{1}^{}(\lambda)^{\frac{\alpha}{2}}\sum_{k=1}^{\infty}\frac{(k +\frac{\alpha}{2})((\ln 2)D_{2}\pi)^{k}(\lambda x_{1}^{}(\lambda))^{k-1}}{k!} =\frac{\mu\pi}{D_{1}}.\] (64) It can be verified that the left-hand-side (LHS) of (64) is a strictly increasing function of both $\lambda$ and $x_{1}^{}(\lambda)$. Thus, to maintain the equality in (64), $x_{1}^{}(\lambda)$ needs to be decreased when $\lambda$ increases and vice versa. Since $U(x_{1}^{}(\lambda),\lambda)=\pi\lambda x_{1}^{}(\lambda)$, checking the monotonicity of $U(x_{1}^{}(\lambda),\lambda)$ is equivalent to checking that of $\lambda x_{1}^{}(\lambda)$. It is observed that if $\lambda$ increases, decreasing $x_{1}^{}(\lambda)$ with $\lambda x_{1}^{}(\lambda)$ being a constant will decrease the LHS of (64) due to the term $x_{1}^{}(\lambda)^{\frac{\alpha}{2}}$. Therefore, $\lambda x_{1}^{}(\lambda)$ needs to be an increasing function of $\lambda$ and so does $U(x_{1}^{}(\lambda),\lambda)$. To prove the monotonicity of $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)$, we expand (24) as \[+ (\ln 2)D_{1}D_{2}\pi\lambda x_{1}^{}(\lambda)^{\frac{\alpha}{2}} 2^{D_{2}\pi\lambda x_{1}^{}(\lambda)}=\mu\pi\lambda\] (65) which can be rearranged as \[D_{1}x_{1}^{}(\lambda)^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda x_{1}^{}(\lambda)}-1\right)\frac{\alpha}{2\lambda x_{1}^{}(\lambda)}\] \[+ D_{1}x_{1}^{}(\lambda)^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda x_{1}^{}(\lambda)}-1\right)(\ln 2)\pi D_{2}\] \[+ (\ln 2)D_{1}D_{2}\pi x_{1}^{}(\lambda)^{\frac{\alpha}{2}}=\mu\pi\] (66) or equivalently, \[+\] \[+ (\ln 2)D_{1}D_{2}\pi x_{1}^{}(\lambda)^{\frac{\alpha}{2}}=\mu\pi.\] (67) Suppose that $x_{1}^{}(\lambda_{1})$ and $x_{1}^{}(\lambda_{2})$ are the two roots of (24) when $\lambda=\lambda_{1}$ and $\lambda=\lambda_{2}$, respectively, where $\lambda_{2}>\lambda_{1}$. Based on the monotonicity of $x_{1}^{}(\lambda)$ and $U(x_{1}^{}(\lambda),\lambda)$ proved above, we have \[x_{1}^{}(\lambda_{1})\lambda_{1} <x_{1}^{}(\lambda_{2})\lambda_{2},\] (68) \[x_{1}^{}(\lambda_{1})^{\frac{\alpha}{2}} >x_{1}^{}(\lambda_{2})^{\frac{\alpha}{2}}.\] (69) Due to the equality in (C) for all $\lambda>0$, we have \[\bar{P}_{\text{BS}}(x_{1}^{}(\lambda_{1}),\lambda_{1})<\bar{P}_{ \text{BS}}(x_{1}^{}(\lambda_{2}),\lambda_{2}),\forall\lambda_{2}>\lambda_{1}.\] (70) Lemma III.2 is thus proved. ## Appendix D Proof of Theorem iii.1 First, we consider the case of $\lambda_{2}\geq\lambda_{1}$, in which three subcases are addressed as follows: 1. If $\lambda\leq\lambda_{1}$, according to the definition of $\lambda_{1}$ given in Lemma III.1, $L_{\lambda}(x_{1}^{}(\lambda),\mu)\geq 0$ for $\lambda\leq\lambda_{1}$, which corresponds to the third condition in (26). Therefore, we have \[x^{}(\lambda)=0.\] 2. If $\lambda_{1}<\lambda\leq\lambda_{2}$, we have $L_{\lambda}(x_{1}^{}(\lambda),\mu)<0$. Since $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda_{2}),\lambda_{2})=P_{\text{max}}$ and $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda_{2})$ increases with $\lambda$ from Lemma III.2, it can be easily verified that $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda_{1})<P_{\text{max}}$ for the assumed range of $\lambda$, which is in accordance with the first condition in (26). Therefore, we have \[x^{}(\lambda)=x_{1}^{}(\lambda).\] 3. Otherwise, if $\lambda>\lambda_{2}\geq\lambda_{1}$, similar to the previous subcase, we know that $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda_{1})>P_{\text{max}}$. Next, we need to check the sign of $L_{\lambda}(x_{2}^{}(\lambda),\mu)=P_{\text{max}}-\mu\pi\lambda x_{2}^{}(\lambda)$. Note that $L_{\lambda}(x_{2}^{}(\lambda_{2}),\mu)=L_{\lambda}(x_{1}^{}(\lambda_{2}),\mu)$, which is non-positive due to $\lambda_{2}\geq\lambda_{1}$. Since $U(x_{2}^{}(\lambda),\lambda)$ strictly increases with $\lambda$, $L_{\lambda}(x_{2}^{}(\lambda),\mu)$ is thus a strictly decreasing function of $\lambda$. Therefore $L_{\lambda}(x_{2}^{}(\lambda),\mu)<0$ for $\lambda>\lambda_{2}$, which implies \[x^{}(\lambda)=x_{2}^{}(\lambda).\] Second, consider the case of $\lambda_{2}<\lambda_{1}$. It is first verified that $\lambda_{3}>\lambda_{1}>\lambda_{2}$ in this case as follows: since $x_{1}^{}(\lambda_{1})$ minimizes $L_{\lambda}(x(\lambda),\mu)$ when $\lambda=\lambda_{1}$ to attain a zero value, and $L_{\lambda}(x(\lambda),\mu)$ is strictly convex in $x(\lambda)$, it follows that $L_{\lambda}(x_{2}^{}(\lambda_{1}),\mu)>0$. Since $L_{\lambda}(x_{2}^{}(\lambda_{3}),\mu)=0$ and $L_{\lambda}(x_{2}^{}(\lambda),\mu)$ is a strictly decreasing function of $\lambda$, we conclude that $\lambda_{3}>\lambda_{1}$. Next, we consider the following three subcases: 1. If $\lambda\leq\lambda_{1}$, according to Lemma III.1, it is easy to verify that $L_{\lambda}(x_{2}^{}(\lambda),\mu)>L_{\lambda}(x_{1}^{}(\lambda),\mu)\geq 0$. Therefore, we have \[x^{}(\lambda)=0.\] 2. If $\lambda_{1}<\lambda\leq\lambda_{3}$, we have $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)>P_{\text{max}}$ and $L_{\lambda}(x_{2}^{}(\lambda),\mu)\geq 0$, which implies \[x^{}(\lambda)=0.\] 3. Otherwise, if $\lambda>\lambda_{3}$, we have $\bar{P}_{\text{BS}}(x_{1}^{}(\lambda),\lambda)>P_{\text{max}}$ and $L_{\lambda}(x_{2}^{}(\lambda),\mu)<0$, which is in accordance with the second condition in (26). Therefore, we have \[x^{}(\lambda)=x_{2}^{}(\lambda).\] Combining the above two cases, Theorem III.1 is thus proved. ## Appendix E Proof of Lemma iii.4 From (15) and (24), we obtain the following equation \[D_{1}x_{2}^{}(\lambda)^{\frac{\alpha}{2}}\left(2^{D_{2}\pi \lambda x_{2}^{}(\lambda)}-1\right)=P_{\text{max}}-P_{c}.\] (71) With the HSE assumption of $D_{2}\pi\lambda x_{2}^{}(\lambda)\gg 1$, (71) is simplified as \[D_{1}x_{2}^{}(\lambda)^{\frac{\alpha}{2}}2^{D_{2}\pi\lambda x_{ 2}^{}(\lambda)}=P_{\text{max}}-P_{c}\] (72) which can be rearranged as \[2^{-\frac{2D_{2}\pi\lambda}{\alpha}x_{2}^{}(\lambda)}=\left( \frac{D_{1}}{P_{\text{max}}-P_{c}}\right)^{\frac{2}{\alpha}}x_{2}^{}(\lambda).\] (73) By utilizing \[p^{ax+b}=cx+d\Rightarrow x=-\frac{\mathcal{W}\left(-\frac{a\ln p }{c}p^{b-\frac{ad}{c}}\right)}{a\ln p}-\frac{d}{c}\] (74) with $p>0$, $a,c\neq 0$, it is easy to verify that $a=-\frac{2D_{2}\pi\lambda}{\alpha}$, $b=0$, $c=\left(\frac{D_{1}}{P_{\text{max}}-P_{c}}\right)^{\frac{2}{\alpha}}$, $d=0$ and $p=2$ in (73). Thus, $x_{2}^{}(\lambda)$ is given by \[x_{2}^{}(\lambda)=\frac{\alpha}{2D_{3}\pi\lambda}\mathcal{W} \left(\frac{2D_{3}\pi\lambda}{\alpha}\left(\frac{P_{\text{max}}-P_{c}}{D_{1}} \right)^{\frac{2}{\alpha}}\right).\] (75) We then proceed to derive the expression of $x_{1}^{}(\lambda)$. Note that $x_{1}^{}(\lambda)$ is the root of equation (C), which can be expressed as \[x_{1}^{}(\lambda)^{\frac{\alpha-2}{2}}2^{D_{2}\pi\lambda x_{1}^ {}(\lambda)}\left[\frac{\alpha}{2}+(\ln 2)D_{2}\pi\lambda x_{1}^{}(\lambda) \right]=\frac{\mu\pi\lambda}{D_{1}}\] (76) by applying the HSE assumption of $D_{2}\pi\lambda x_{1}^{}(\lambda)\gg 1$. Furthermore, it is observed that (76) can be simplified as \[(\ln 2)D_{1}D_{2}x_{1}^{}(\lambda)^{\frac{\alpha}{2}}2^{D_{2}\pi \lambda x_{1}^{}(\lambda)}=\mu\] (77) due to the fact that $(\ln 2)D_{2}\pi\lambda x_{1}^{}(\lambda)\gg\frac{\alpha}{2}$, where $\alpha=2\sim 6$ in practice. Similar to the case for obtaining $x_{2}^{}(\lambda)$, $x_{1}^{}(\lambda)$ can be solved from (77) and given by \[x_{1}^{}(\lambda)=\frac{\alpha}{2D_{3}\pi\lambda}\mathcal{W} \left(\frac{2D_{3}\pi\lambda}{\alpha}\left(\frac{\mu}{D_{1}D_{3}}\right)^{ \frac{2}{\alpha}}\right).\] (78) Lemma III.4 is thus proved. ## References * [1] F. Richter, A. J. Fehske, and G. P. Fettweis, “Energy efficiency aspects of base station deployment strategies for cellular networks,” in _Proc. IEEE Vehic. Tech. Conf. (VTC)_, Sep. 2009. * [2] S. V. Hanly and R. Mathar, “On the optimal base-station density for CDMA cellular networks,” _IEEE Trans. Commun._, vol. 50, no. 8, pp. 1274-1281, Aug. 2002. * [3] W. Wang and G. Shen, “Energy efficiency of heterogeneous cellular network,” in _Proc. IEEE Vehic. Tech. Conf. (VTC)_, Sept. 2010. * [4] K. Son, S. Chong, and G. de Veciana, “Dynamic association for load balacing and interference avoidance in multi-cell networks,” _IEEE Trans. Wireless Commun._, vol. 8, no. 7, pp. 3566-3576, Jul. 2009. * [5] S. V. Hanly, “An algorithm for combined cell-site selection and power control to maximize cellular spread spectrum capacity,” _IEEE J. Select. Areas Commun._, vol. 13, no. 7, pp. 1332-1340, Sep. 1995. * [6] S. Das, H. Viswanathan, and G. Rittenhouse, “Dynamic load balancing through coordinated scheduling in packet data systems,” in _Proc. IEEE INFOCOM_, pp. 786-796, Mar. 2003. * [7] 3GPP TR 32.826, Telecommunication management: Study on energy savings management (ESM), (Release 10), Mar. 2010. * [8] M. Marsan, L. Chiaraviglio, D. Ciullo, and M. Meo, “Optimal energy savings in cellular access networks,” _IEEE ICC GreenCom Wkshps._, Jun. 2009. * [9] K. Samdanis, D. Kutscher, and M. Brunner, “Dynamic energy-aware network re-configuration for cellular urban infrastructures,” _IEEE GLOBECOM GreenCom Wkshps._, pp. 1448-1452, Dec. 2010. * [10] A. P. Jardosh, K. Papagiannaki, E. M. Belding, K. C. Almeroth, G. Iannaccone, and B. 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Vandenberghe, _Convex Optimization_, Cambidge University Press, 2004. * [18] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” _Advances in Computational Mathematics_, vol 5, no. 1, pp. 329-359, 1996.
1103.4591	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 80741, "num_imgs": 3, "llama3_tokens_count": 28668 }	[ "content_image/1103.4591/x1.png", "content_image/1103.4591/x2.png", "content_image/1103.4591/x4.png" ]	# Quantitative version of Kipnis-Varadhan’s theorem and Monte-Carlo approximation of homogenized coefficients Antoine Gloria & Jean-Christophe Mourrat Projet SIMPAF, INRIA Lille-Nord Europe, France antoine.gloria@inria.fr EPFL, institut de mathématiques, station 8, 1015 Lausanne, Switzerland jean-christophe.mourrat@epfl.ch ###### Abstract. This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time $t>0$, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions rescaled by $t$ of $n$ independent random walks in $n$ independent environments. Relying on a new quantitative version of Kipnis-Varadhan’s theorem (which is of independent interest), we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the rescaled final position of the random walk in terms of $t$. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of $n$ and $t$, and prove a large-deviation estimate. Compared to other numerical strategies, this Monte-Carlo approach has the advantage to be dimension-independent in terms of convergence rate and computational cost. Keywords: random walk, random environment, stochastic homogenization, effective coefficients, Monte-Carlo method, quantitative estimates. 2010 Mathematics Subject Classification: 35B27, 60K37, 60H25, 65C05, 60H35, 60G50. ## 1. Introduction This article is part of a larger program, which consists in devising and quantitatively analyzing numerical methods to approximate effective coefficients in stochastic homogenization of linear elliptic equations. More precisely we tackle here the case of a discrete elliptic equation with independent and identically distributed coefficients (see however the end of this introduction for more general statistics), and present and fully analyze an approximation procedure based on a Monte-Carlo method. A first possibility to approximate effective coefficients is to directly solve the so-called corrector equation. In this approach, a first step towards the derivation of error estimates is a quantification of the qualitative results proved by Künnemann [15] (and inspired by Papanicolaou and Varadhan’s treatment of the continuous case [19]) and Kozlov [14]. In the stochastic case, such an equation is posed on the whole $\mathbb{Z}^{d}$, and we need to localize it on a bounded domain, say the hypercube $Q_{R}$ of side $R>0$. As shown in a series of papers by Otto and the first author [7, 8], and the first author [6], there are three contributions to the $\mathbb{L}^{2}$-error in probability between the true homogenized coefficients and its approximation. The dominant error in small dimensions takes the form of a variance: it measures the fact that the approximation of the homogenized coefficients by the average of the energy density of the corrector on a box $Q_{R}$ fluctuates. This error decays at the rate of the central limit theorem $R^{-d}$ in any dimension (with a logarithmic correction for $d=2$). The second error is the so-called systematic error: it is due to the fact that we have modified the corrector equation by adding a zero-order term of strength $T^{-1}>0$ (as standard in the analysis of the well-posedness of the corrector equation). The scaling of this error depends on the dimension and saturates at dimension $4$. It is of higher order than the random error up to dimension $8$. The last error is due to the use of boundary conditions on the bounded domain $Q_{R}$. Provided there is a buffer region, this error is exponentially small in the distance to the buffer zone measured in units of $\sqrt{T}$. This approach has two main drawbacks. First the numerical method only converges at the central limit theorem scaling in terms of $R$ up to dimension $8$, which is somehow disappointing from a conceptual point of view (although this is already fine in practice). Second, although the size of the buffer zone is roughly independent of the dimension, its cost with respect to the central limit theorem scaling dramatically increases with the dimension (recall that in dimension $d$, the CLT scaling is $R^{-d}$, so that in high dimension, we may consider smaller $R$ for a given precision, whereas the use of boundary conditions requires $R\gg\sqrt{T}$ in any dimension). Based on ideas of the second author in [18], we have taken advantage of the spectral representation of the homogenized coefficients (originally introduced by Papanicolaou and Varadhan to prove their qualitative homogenization result) in order to devise and analyze new approximation formulas for the homogenized coefficients in [10]. In particular, this has allowed us to get rid of the restriction on dimension, and exhibit refinements of the numerical method of [6] which converge at the central limit theorem scaling in any dimension (thus avoiding the first mentioned drawback). Unfortunately, the second drawback is inherent to the type of method used: if the corrector equation has to be solved on a bounded domain $Q_{R}$, boundary conditions need to be imposed on the boundary $\partial Q_{R}$. Since their values are actually also part of the problem, a buffer zone seems mandatory — with the notable exception of the periodization method, whose analysis is yet still unclear to us, especially when spatial correlations are introduced in the coefficients. In order to avoid the issue of boundary conditions, we adopt here another point of view on the problem: the random walk in random environment approach. This other point of view on the same homogenization problem has been analyzed in the celebrated paper [12] by Kipnis and Varadhan, and then extended by De Masi, Ferrari, Goldstein, and Wick [5]. The strategy of the present paper is to obtain an approximation of the homogenized coefficients by the numerical simulation of this random walk up to some large time. As we did in the case of the approach based on the corrector equation, a first step towards the analysis of this numerical method is to quantify the corresponding qualitative result, namely here Kipnis-Varadhan’s convergence theorem. Compared to the deterministic approach based on the approximate corrector equation, the advantage of the present approach is that its convergence rate and computational costs are dimension-independent. As we shall also see, as opposed to the approach based on the corrector equation, the environment only needs to be generated along the trajectory of the random walker, so that much less information has to be stored during the calculation. This may be quite an important feature of the Monte Carlo method in view of the discussion of [6, Section 4.3]. We consider the discrete elliptic operator $-\nabla^{}\cdot A\nabla$, where $\nabla^{}\cdot$ and $\nabla$ are the discrete backward divergence and forward gradient, respectively. For all $x\in\mathbb{Z}^{d}$, $A(x)$ is the diagonal matrix whose entries are the conductances $\omega_{x,x+\mathbf{e}_{i}}$ of the edges $(x,x+\mathbf{e}_{i})$ starting at $x$, where $(\mathbf{e}_{i})_{i\in\{1,\dots,d\}}$ denotes the canonical basis of $\mathbb{Z}^{d}$. Let $\mathbb{B}$ denote the set of edges of $\mathbb{Z}^{d}$. We call the family of conductances $\omega=(\omega_{e})_{e\in\mathbb{B}}$ the _environment_. The environment $\omega$ is random, and we write $\mathbb{P}$ for its distribution (with corresponding expectation $\mathbb{E}$). We make the following assumptions : * the measure $\mathbb{P}$ is invariant under translations, * the conductances are i. i. d.¹, [FOOTNOTE:1][ENDFOOTNOTE] * there exists $0<\alpha<\beta$ such that $\alpha\leqslant\omega_{e}\leqslant\beta$ almost surely. Under these conditions, standard homogenization results ensure that there exists some _deterministic_ symmetric matrix $A_{\mathrm{hom}}$ such that the solution operator of the deterministic continuous differential operator $-\nabla\cdot A_{\mathrm{hom}}\nabla$ describes the large scale behavior of the solution operator of the random discrete differential operator $-\nabla^{}\cdot A\nabla$ almost surely (for this statement, (H2) can in fact be replaced by the weaker assumption that the measure $\mathbb{P}$ is ergodic with respect to the group of translations, see [15]). The operator $-\nabla^{}\cdot A\nabla$ is the infinitesimal generator of a stochastic process $(X(t))_{t\in\mathbb{R}_{+}}$ which can be defined as follows. Given an environment $\omega$, it is the Markov process whose jump rate from a site $x\in\mathbb{Z}^{d}$ to a neighbouring site $y$ is given by $\omega_{x,y}$. We write $\mathbf{P}^{\omega}_{x}$ for the law of this process starting from $x\in\mathbb{Z}^{d}$. It is proved in [12] that under the averaged measure $\mathbb{P}\mathbf{P}^{\omega}_{0}$, the rescaled process $\sqrt{\varepsilon}X(\varepsilon^{-1}t)$ converges in law, as $\varepsilon$ tends to $0$, to a Brownian motion whose infinitesimal generator is $-\nabla\cdot A_{\mathrm{hom}}\nabla$, or in other words, a Brownian motion with covariance matrix $2A_{\mathrm{hom}}$ (see also [1, 15, 13] for prior results). We will use this fact to construct computable approximations of $A_{\mathrm{hom}}$. As proved in [5], this invariance principle holds as soon as (H1) is true, (H2) is replaced by the ergodicity of the measure $\mathbb{P}$, and (H3) by the integrability of the conductances. Under the assumptions (H1-H3), [20] strengthens this result in another direction, showing that for almost every environment, $\sqrt{\varepsilon}X(\varepsilon^{-1}t)$ converges in law under $\mathbf{P}^{\omega}_{0}$ to a Brownian motion with covariance matrix $2A_{\mathrm{hom}}$. This has been itself extended to environments which do not satisfy the uniform ellipticity condition (H3), see [3, 17, 4, 16, 2]. Let $(Y(t))_{t\in\mathbb{N}}$ denote the sequence of consecutive sites visited by the random walk $(X(t))_{t\in\mathbb{R}_{+}}$ (note that the “times” are different in nature for $X(t)$ and $Y(t)$). This sequence is itself a Markov chain that satisfies for any two neighbours $x,y\in\mathbb{Z}^{d}$: \[\mathbf{P}^{\omega}_{x}[Y(1)=y]=\frac{\omega_{x,y}}{p_{\omega}(x)},\] where $p_{\omega}(x)=\sum_{\|z\|=1}\omega_{x,x+z}$. We simply write $p(\omega)$ for $p_{\omega}(0)$. Let us introduce a “tilted” version of the law $\mathbb{P}$ on the environments, that we write $\tilde{\mathbb{P}}$ and define by (1.1) \[{\mathrm{d}}\tilde{\mathbb{P}}(\omega)=\frac{p(\omega)}{\mathbb{E}[p]}\ { \mathrm{d}}\mathbb{P}(\omega).\] The reason why this measure is natural to consider is that it makes the environment seen from the position of the random walk $Y$ a stationary process (see (3.2) for a definition of this process). Interpolating between two integers by a straight line, we can think of $Y$ as a continuous function on $\mathbb{R}_{+}$. With this in mind, it is also true that there exists a matrix $A_{\mathrm{hom}}^{\mathrm{disc}}$ such that, as $\varepsilon$ tends to $0$, the rescaled process $\sqrt{\varepsilon}Y(\varepsilon^{-1}t)$ converges in law under $\tilde{\mathbb{P}}\mathbf{P}^{\omega}_{0}$ to a Brownian motion with covariance matrix $2A_{\mathrm{hom}}^{\mathrm{disc}}$. Moreover, $A_{\mathrm{hom}}^{\mathrm{disc}}$ and $A_{\mathrm{hom}}$ are related by (see [5, Theorem 4.5 (ii)]) : (1.2) \[A_{\mathrm{hom}}=\mathbb{E}[p]\ A_{\mathrm{hom}}^{\mathrm{disc}}.\] Given that the numerical simulation of $Y$ saves some operations compared to the simulation of $X$ (there is no waiting time to compute, and the running time is equal to the number of steps), we will focus on approximating $A_{\mathrm{hom}}^{\mathrm{disc}}$. More precisely, we fix once and for all some $\xi\in\mathbb{R}^{d}$ with $\|\xi\|=1$, and define (1.3) \[\sigma_{t}^{2}=t^{-1}\ \tilde{\mathbb{E}}\mathbf{E}^{\omega}_{0}[(\xi\cdot Y(t ))^{2}],\qquad\sigma^{2}=2\xi\cdot A_{\mathrm{hom}}^{\mathrm{disc}}\xi.\] It follows from results of [12] (or [5, Theorem 2.1]) that $\sigma_{t}^{2}$ tends to $\sigma^{2}$ as $t$ tends to infinity. Our first contribution is to give a quantitative estimate of this convergence. In particular we shall show that, with i. i. d. coefficients and up to a logarithmic correction in dimension $2$, the difference between $\sigma_{t}^{2}$ and $\sigma^{2}$ is of order $1/t$. We now describe a Monte-Carlo method to approximate $\sigma_{t}^{2}$. Using the definition of the tilted measure (1.1), one can see that (1.4) \[\sigma_{t}^{2}=\frac{\tilde{\mathbb{E}}\mathbf{E}^{\omega}_{0}[(\xi\cdot Y(t)) ^{2}]}{t}=\frac{\mathbb{E}\mathbf{E}^{\omega}_{0}[p(\omega)(\xi\cdot Y(t))^{2} ]}{t\mathbb{E}[p]}.\] Assuming that we have easier access to the measure $\mathbb{P}$ than to the tilted $\tilde{\mathbb{P}}$, we prefer to base our Monte-Carlo procedure on the r. h. s. of the second identity in (1.4). Let $Y^{(1)},Y^{(2)},\ldots$ be independent random walks evolving in the environments $\omega^{(1)},\omega^{(2)},\ldots$ respectively. We write $\mathbf{P}^{\overline{\omega}}_{0}$ for their joint distribution, all random walks starting from $0$, where $\overline{\omega}$ stands for $(\omega^{(1)},\omega^{(2)},\ldots)$. The family of environments $\overline{\omega}$ is itself random, and we let $\mathbb{P}^{\otimes}$ be the product distribution with marginal $\mathbb{P}$. In other words, under $\mathbb{P}^{\otimes}$, the environments $\omega^{(1)},\omega^{(2)},\ldots$ are independent and distributed according to $\mathbb{P}$. Our computable approximation of $\sigma_{t}^{2}$ is defined by (1.5) \[\hat{A}_{n}(t)=\frac{p(\omega^{(1)})(\xi\cdot Y^{(1)}(t))^{2}+\cdots+p(\omega^ {(n)})(\xi\cdot Y^{(n)}(t))^{2}}{t(p(\omega^{(1)})+\cdots+p(\omega^{(n)}))}.\] The following step in the analysis is to quantify the random fluctuations of $\hat{A}_{n}(t)$ in terms of $n$ — the number of random walks considered in the empirical average to approximate $\sigma_{t}^{2}$ — and $t$. We shall prove a large deviation result which ensures that the $\mathbb{P}^{\otimes}\mathbf{P}^{\overline{\omega}}_{0}$-probability that the difference between $\hat{A}_{n}(t)$ and $\sigma_{t}^{2}$ exceeds $1/t$ is exponentially small in the ratio $n/t^{2}$. The rest of this article is organized as follows. In Section 2, which can be read independently of the rest of the paper, we consider a general discrete or continuous-time reversible Markov process. Kipnis-Varadhan’s theorem (and its subsequent development due to [5]) gives conditions for additive functionals of this process to satisfy an invariance principle. We show that, under additional conditions written in terms of a spectral measure, the statement can be made quantitative. More precisely, Kipnis-Varadhan’s theorem relies on writing the additive functional under consideration as the sum of a martingale plus a remainder. This remainder, after suitable normalization, is shown to converge to $0$ in $\mathbb{L}^{2}$. Under our additional assumptions, we give explicit bounds on the rate of decay. In Section 3, we make use of this result, in the context of the approximation of homogenized coefficients, to estimate the systematic error $\|\sigma_{t}^{2}-\sigma^{2}\|$. The central achievement of this section is to prove that the relevant spectral measure satisfies the conditions of our quantitative version of Kipnis-Varadhan’s theorem. Section 4 is dedicated to the estimate of the random fluctuations. These are controlled through large deviations estimates. Relying on these results, we give in Section 5 a complete error analysis of the Monte-Carlo method to approximate the homogenized matrix $A_{\mathrm{hom}}^{\mathrm{disc}}$, which we illustrate by numerical tests. Let us quickly discuss the sharpness of these results. If $A$ was a periodic matrix (or even a constant matrix) the systematic error would also be of order $1/t$ (without logarithmic correction for $d=2$), and the fluctutations would decay exponentially fast in the ratio $n/t^{2}$ as well. This shows that our analysis is optimal (the additional logarithm seems unavoidable for $d=2$, as discussed in the introduction of [7]). Let us also point out that although the results of this paper are proved under assumptions (H1)-(H3), the assumption (H2) on the statistics of $\omega$ is only used to obtain the variance estimate of [7, Lemma 2.3]. In particular, (H2) can be weakened as follows: * the distribution of $\omega_{(z,z+\mathbf{e}_{i})}$ may in addition depend on $\mathbf{e}_{i}$, * independence can be replaced by finite correlation length $C_{L}>0$, that is for all $e,e^{\prime}\in\mathbb{B}$, $\omega_{e}$ and $\omega_{e^{\prime}}$ are independent if $\|e-e^{\prime}\|\geqslant C_{L}$, * independence can be replaced by mixing in the sense of Dobrushin and Shlosman — we refer the reader to work in progress by Otto and the first author for this issue [9]. Notation. So far we have already introduced the probability measures $\mathbf{P}^{\omega}_{0}$ (distribution of $Y$), $\mathbf{P}^{\overline{\omega}}_{0}$ (distribution of $Y^{(1)},Y^{(2)},\ldots$), $\mathbb{P}$ (i.i.d. distribution for $\omega=(\omega_{e})_{e\in\mathbb{B}}$), $\tilde{\mathbb{P}}$ (tilted measure defined in (1.1)) and $\mathbb{P}^{\otimes}$ (product distribution of $\overline{\omega}$ with marginal $\mathbb{P}$). It will be convenient to define $\tilde{\mathbb{P}}^{\otimes}$ the product distribution of $\overline{\omega}$ with marginal $\tilde{\mathbb{P}}$. For convenience, we write $\mathbb{P}_{0}$ as a short-hand notation for $\mathbb{P}\mathbf{P}^{\omega}_{0}$, $\tilde{\mathbb{P}}_{0}$ for $\tilde{\mathbb{P}}\mathbf{P}^{\omega}_{0}$, $\mathbb{P}^{\otimes}_{0}$ for $\mathbb{P}^{\otimes}\mathbf{P}^{\overline{\omega}}_{0}$, and $\tilde{\mathbb{P}}^{\otimes}_{0}$ for $\tilde{\mathbb{P}}^{\otimes}\mathbf{P}^{\overline{\omega}}_{0}$. The corresponding expectations are written accordingly, replacing “P” by “E” with the appropriate typography. Finally, we write $\|\cdot\|$ for the Euclidian norm of $\mathbb{R}^{d}$. ## 2. Quantitative version of Kipnis-Varadhan’s theorem Kipnis-Varadhan’s theorem [12] concerns additive functionals of reversible Markov processes. It gives conditions for such additive functionals to satisfy an invariance principle. The proof of the result relies on a decomposition of the additive functional as the sum of a martingale term plus a remainder term, the latter being shown to be negligible. In this section, which can be read independently of the rest of the paper, we give conditions that enable to obtain some quantitative bounds on this remainder term. We consider discrete and continuous times simultaneously. Let $(\eta_{t})_{t\geqslant 0}$ be a Markov process defined on some measurable state space $\aleph$ (here, $t\geqslant 0$ stands either for $t\in\mathbb{N}$ or for $t\in\mathbb{R}_{+}$). We denote by $P_{x}$ the distribution of the process started from $x\in\aleph$, and by $E_{x}$ the associated expectation. We assume that this Markov process is reversible and ergodic with respect to some probability measure $\nu$. We write $P_{\nu}$ for the law of the process started from the distribution $\nu$, and $E_{\nu}$ for the associated expectation. To the Markov process is naturally associated a semi-group $(P_{t})_{t\geqslant 0}$ defined, for any $f\in\mathbb{L}_{2}(\nu)$, by \[P_{t}f(x)=E_{x}[f(\eta_{t})].\] Each $P_{t}$ is a self-adjoint contraction of $\mathbb{L}^{2}(\nu)$. In the continuous-time case, we assume further that the semi-group is strongly continuous, that is to say, that $P_{t}f$ converges to $f$ in $\mathbb{L}^{2}(\nu)$ as $t$ tends to $0$, for any $f\in\mathbb{L}^{2}(\nu)$. We let $L$ be the $\mathbb{L}^{2}(\nu)$-infinitesimal generator of the semi-group. It is self-adjoint in $\mathbb{L}^{2}(\nu)$, and we fix the sign convention so that it is a positive operator (i.e., $P_{t}=e^{-tL}$). Note that in general, one can see using spectral analysis that there exists a projection $\overline{P}$ such that $P_{t}f$ converges to $\overline{P}f$ as $t$ tends to $0$, $t>0$. Changing $\mathbb{L}^{2}(\nu)$ to the image of the projection $\overline{P}$, and $P_{0}$ for $\overline{P}$, one recovers a strongly continuous semigroup of contractions, and one can still carry the analysis below replacing $\mathbb{L}^{2}(\nu)$ by the image of $\overline{P}$ when necessary. In discrete time, we set $L=\mathrm{Id}-P_{1}$. Again, $L$ is a positive self-adjoint operator on $\mathbb{L}^{2}(\nu)$. Note that we slightly depart from the custom of defining the generator as $P_{1}$ in order to match more closely the continuous time situation. We denote by $\langle\cdot,\cdot\rangle$ the scalar product in $\mathbb{L}^{2}(\nu)$. For any function $f\in\mathbb{L}^{2}(\nu)$ we define the _spectral measure_ of $L$ projected on the function $f$ as the measure $e_{f}$ on $\mathbb{R}_{+}$ that satisfies, for any bounded continuous $\Psi:\mathbb{R}_{+}\to\mathbb{R}$, the relation (2.1) \[\langle f,\Psi(L)f\rangle=\int\Psi(\lambda)\ {\mathrm{d}}e_{f}(\lambda).\] The Dirichlet form associated to $L$ is given by (2.2) \[\\|f\\|_{1}^{2}=\int\lambda\ {\mathrm{d}}e_{f}(\lambda).\] We denote by $H^{1}$ the completion of the space $\{f\in\mathbb{L}^{2}(\nu):\\|f\\|_{1}<+\infty\}$ with respect to this $\\|\cdot\\|_{1}$ norm, taken modulo functions of zero $\\|\cdot\\|_{1}$ norm. This turns $(H^{1},\\|\cdot\\|_{1})$ into a Hilbert space, and we let $H^{-1}$ denote its dual. One can identify $H^{-1}$ with the completion of the space $\{f\in\mathbb{L}^{2}(\nu):\\|f\\|_{-1}<+\infty\}$ with respect to the norm $\\|\cdot\\|_{-1}$ defined by \[\\|f\\|_{-1}^{2}=\int\lambda^{-1}\ {\mathrm{d}}e_{f}(\lambda).\] Indeed, for all $f\in\mathbb{L}^{2}(\nu)$, the linear form \[\left\{\begin{array}[]{ccc}(\mathbb{L}^{2}(\nu)\cap H^{1},\\|\cdot\\|_{1})&\to& \mathbb{R}\\ \phi&\mapsto&\langle f,\phi\rangle\end{array}\right.\] has norm $\\|f\\|_{-1}$, and thus defines an element of $H^{-1}$ (with norm $\\|f\\|_{-1}$) iff $\\|f\\|_{-1}$ is finite. The notion of spectral measure introduced in (2.1) for functions of $\mathbb{L}^{2}(\nu)$ can be extended to elements of $H^{-1}$. Indeed, let $\Psi:\mathbb{R}_{+}\to\mathbb{R}$ be a continuous function such that $\Psi(\lambda)=O(\lambda^{-1})$ as $\lambda\to+\infty$. One can check that the map \[\left\{\begin{array}[]{ccc}(\mathbb{L}^{2}(\nu)\cap H^{-1},\\|\cdot\\|_{-1})&\to &H^{1}\\ f&\mapsto&\Psi(L)f\end{array}\right.\] extends to a bounded linear map on $H^{-1}$. One can then define the spectral measure of $L$ projected on the function $f$ as the measure $e_{f}$ such that for any continuous $\Psi$ with $\Psi(\lambda)=O(\lambda^{-1})$, (2.1) holds. With a slight abuse of notation, for all $f\in H^{-1}$ and $g\in H^{1}$, we write $\langle f,g\rangle$ for the $H^{-1}-H^{1}$ duality product between $f$ and $g$. For any $f\in H^{-1}$, we define $(Z_{f}(t))_{t\geqslant 0}$ as (2.3) \[Z_{f}(t)=\int_{0}^{t}f(\eta_{s})\ {\mathrm{d}}s\qquad\text{or}\qquad Z_{f}(t)= \sum_{s=0}^{t-1}f(\eta_{s}),\] according to whether we consider the continuous or the discrete time cases. In the continuous case, the meaning of (2.3) is unclear a priori. Yet it is proved in [5, Lemma 2.4] that for any $t\geqslant 0$ the map \[\left\{\begin{array}[]{ccc}\mathbb{L}^{2}(\nu)\cap H^{-1}&\to&\mathbb{L}^{2}(P _{\nu})\\ f&\mapsto&Z_{f}(t)\end{array}\right.\] can be extended by continuity to a bounded linear map on $H^{-1}$, and moreover, that (2.3) coincides with the usual integral as soon as $f\in\mathbb{L}^{1}(\nu)$. The following theorem is due to [5], building on previous work of [12]. Theorem 2.1.: _(i) For all $f\in H^{-1}$, there exists $(M_{t})_{t\geqslant 0}$, $(\xi_{t})_{t\geqslant 0}$ such that $Z_{f}(t)$ defined in (2.3) satisfies the identity $Z_{f}(t)=M_{t}+\xi_{t}$, where $(M_{t})$ is a square-integrable martingale with stationary increments under $P_{\nu}$ (and the natural filtration), and $(\xi_{t})$ is such that :_ (2.4) \[t^{-1}E_{\nu}[(\xi_{t})^{2}]\xrightarrow[t\to+\infty]{}0.\] _As a consequence, $t^{-1/2}Z_{f}(t)$ converges in law under $P_{\nu}$ to a Gaussian random variable of variance $\sigma^{2}(f)<+\infty$ as $t$ goes to infinity, and_ (2.5) \[t^{-1}E_{\nu}[(Z_{f}(t))^{2}]\xrightarrow[t\to+\infty]{}\sigma^{2}(f).\] _(ii) If, moreover, $f\in\mathbb{L}^{1}(\nu)$ and, for some $\overline{t}>0$, $\sup_{0\leqslant t\leqslant\overline{t}}\|Z_{f}(t)\|$ is in $\mathbb{L}^{2}(\nu)$, then the process $t\mapsto\sqrt{\varepsilon}Z_{f}(\varepsilon^{-1}t)$ converges in law under $P_{\nu}$ to a Brownian motion of variance $\sigma^{2}(f)$ as $\varepsilon$ goes to $0$._ Remarks. The additional conditions appearing in statement (ii) are automatically satisfied in discrete time, due to the fact that $H^{-1}\subseteq\mathbb{L}^{2}(\nu)$ in this case. In the continuous-time setting and when $f\in\mathbb{L}^{1}(\nu)$, the process $t\mapsto Z_{f}(t)$ is almost surely continuous, and $\sup_{0\leqslant t\leqslant\overline{t}}\|Z_{f}(t)\|$ is indeed a well-defined random variable. Under some additional information on the spectral measure of $f$, we can estimate the rates of convergence in the limits (2.4) and (2.5). For any $\gamma>1$ and $q\geqslant 0$, we say that the spectral exponents of a function $f\in H^{-1}$ are at least $(\gamma,-q)$ if (2.6) \[\int_{0}^{\mu}{\mathrm{d}}e_{f}(\lambda)=O\big{(}\mu^{\gamma}\ln^{q}(\mu^{-1}) \big{)}\qquad(\mu\to 0).\] Note that the phrasing is consistent, since if $(\gamma^{\prime},-q^{\prime})\leqslant(\gamma,-q)$ for the lexicographical order, and if the spectral exponents of $f$ are at least $(\gamma,-q)$, then they are at least $(\gamma^{\prime},-q^{\prime})$. In [18], it was found more convenient to consider, instead of (2.6), a condition of the following form : (2.7) \[\int_{0}^{\mu}\lambda^{-1}\ {\mathrm{d}}e_{f}(\lambda)=O\big{(}\mu^{\gamma-1} \ln^{q}(\mu^{-1})\big{)}\qquad(\mu\to 0).\] One can easily check that conditions (2.6) and (2.7) are equivalent. Indeed, on the one hand, one has the obvious inequality \[\int_{0}^{\mu}{\mathrm{d}}e_{f}(\lambda)\leqslant\mu\int_{0}^{\mu}\lambda^{-1} \ {\mathrm{d}}e_{f}(\lambda),\] which shows that (2.7) implies (2.6). On the other hand, one may perform a kind of integration by parts, use Fubini’s theorem : \[\int_{0}^{\mu}\lambda^{-1}\ {\mathrm{d}}e_{f}(\lambda) = \int_{0}^{\mu}\int_{\delta=\lambda}^{+\infty}\delta^{-2}\ { \mathrm{d}}\delta\ {\mathrm{d}}e_{f}(\lambda)\] \[= \int_{\delta=0}^{+\infty}\delta^{-2}\int_{\lambda=0}^{\delta \wedge\mu}{\mathrm{d}}e_{f}(\lambda)\ {\mathrm{d}}\delta,\] and obtain the converse implication by examining separately the integration over $\delta$ in $[0,\mu)$ and in $[\mu,+\infty)$. For all $\gamma>1$ and $q\geqslant 0$, we set (2.8) \[\psi_{\gamma,q}(t)=\left\|\begin{array}[]{ll}t^{1-\gamma}\ln^{q}(t)&\text{if } \gamma<2,\\ t^{-1}\ln^{q+1}(t)&\text{if }\gamma=2,\\ t^{-1}&\text{if }\gamma>2.\end{array}\right.\] The quantitative version of Theorem 2.1 is as follows. Theorem 2.2.: _If the spectral exponents of $f\in H^{-1}$ are at least $(\gamma,-q)$, then the decomposition $Z_{f}(t)=M_{t}+\xi_{t}$ of Theorem 2.1 holds with the additional property that_ \[t^{-1}E_{\nu}[(\xi_{t})^{2}]\ =O(\psi_{\gamma,q}(t))\qquad(t\to+\infty).\] _Moreover,_ \[\sigma^{2}(f)-\frac{E_{\nu}[Z_{f}(t)^{2}]}{t}=O(\psi_{\gamma,q}(t))\qquad(t\to +\infty).\] Proof.: In the continuous-time setting, the argument for the first estimate is very similar to the one of [18, Proposition 8.2], and we do not repeat the details here. It is based on the observation that (2.9) \[\frac{1}{t}E_{\nu}[(\xi_{t})^{2}]=2\int\frac{1-e^{-\lambda t}}{\lambda^{2}t}\ {\mathrm{d}}e_{f}(\lambda).\] One needs to take into account the possible logarithmic terms that appear in (2.7) and which are not considered in [18]. Some care is also needed because we do not assume that $f\in\mathbb{L}^{2}(\nu)$. Yet one can easily replace the bound involving the $\mathbb{L}^{2}(\nu)$ norm of $f$ by its $H^{-1}$ norm. The second part of the statement is given by [18, Proposition 8.3]. We now turn to the discrete time setting. In this context, identity (2.9) should be replaced by \[\frac{1}{t}E_{\nu}[(\xi_{t})^{2}]=2\int\frac{1-(1-\lambda)^{t}}{\lambda^{2}t} \ {\mathrm{d}}e_{f}(\lambda).\] By definition, $L=\mathrm{Id}-P_{1}$, where $P_{1}$ is the semi-group at time $1$. Hence the spectrum of $L$ is contained in $[0,2]$. One can then follow the same computations as before to prove the first part of Theorem 2.2. Somewhat surprisingly, the second part of the statement requires additional attention in the discrete time setting. Indeed, in the continuous case, the argument of [18, Proposition 8.3] (which already appears in [5]) is that $Z_{f}(t)$ and $\xi(t)$ are orthogonal in $\mathbb{L}^{2}(P_{\nu})$, a fact obtained using the invariance under time symmetry. This orthogonality is only approximately valid in the discrete-time setting. Indeed, let us recall that $Z_{f}(t)$ is given by (2.3), while $\xi_{t}$ is obtained as the limit in $\mathbb{L}^{2}(P_{\nu})$ of \[-u_{\varepsilon}(\eta_{t})+u_{\varepsilon}(\eta_{0}),\] where $u_{\varepsilon}=(\varepsilon+L)^{-1}f$. Using time symmetry, what we obtain is that $\xi_{t}$ is orthogonal to $(Z_{f}(t)+f(\eta_{t}))$. As a consequence, the cross-product $E_{\nu}[Z_{f}(t)\xi_{t}]$, which is equal to $0$ in the proof of [18, Proposition 8.3], is in the present case equal to $-E_{\nu}[f(\eta_{t})\xi_{t}]$. Yet spectral analysis ensures that this term is equal to \[\int\frac{1-(1-\lambda)^{t}}{\lambda}\ {\mathrm{d}}e_{f}(\lambda)=O(1)\qquad(t \to+\infty),\] which is what we need to obtain the second claim of the theorem. ∎ ## 3. The systematic error We now come back to the analysis of the Monte-Carlo approximation of the homogenized coefficients within assumptions (H1)-(H3). The aim of this section is to estimate the difference between $\sigma_{t}^{2}$ and the quantity $\sigma^{2}$ we wish to approximate (both being defined in (1.3)). This difference, that we refer to as the _systematic error_ after [7], is shown to be of order $1/t$ as $t$ tends to infinity, up to a logarithmic correction in dimension $2$. Theorem 3.1.: _Under assumptions (H1)-(H3), there exists $q\geqslant 0$ such that, as $t$ tends to infinity,_ (3.1) \[\sigma_{t}^{2}-\sigma^{2}=\left\|\begin{array}[]{ll}O\big{(}t^{-1}\ln^{q}(t) \big{)}&\text{if }d=2,\\ O\big{(}t^{-1}\big{)}&\text{if }d>2.\end{array}\right.\] Theorem 3.1 is a discrete-time version of [18, Corollary 2.6]. Its proof makes use of an auxiliary process that we now introduce. Let $(\theta_{x})_{x\in\mathbb{Z}^{d}}$ be the translation group that acts on the set of environments as follows: for any pair of neigbhours $y,z\in\mathbb{Z}^{d}$, $(\theta_{x}\ \omega)_{y,z}=\omega_{x+y,x+z}$. The _environment viewed by the particle_ is the process defined by (3.2) \[\omega(t)=\theta_{Y(t)}\ \omega.\] One can check that $(\omega(t))_{t\in\mathbb{N}}$ is a Markov chain, whose generator is given by (3.3) \[-\mathcal{L}f(\omega)=\frac{1}{p(\omega)}\sum_{\|z\|=1}\omega_{0},z(f(\theta_{z} \ \omega)-f(\omega)),\] so that $\mathbf{E}^{\omega}_{0}[f(\omega(1))]=(I-\mathcal{L})f(\omega)$. Moreover, the measure $\tilde{\mathbb{P}}$ defined in (1.1) is reversible and ergodic for this process [5, Lemma 4.3 (i)]. As a consequence, the operator $\mathcal{L}$ is (positive and) self-adjoint in $\mathbb{L}^{2}(\tilde{\mathbb{P}})$. The proof of Theorem 3.1 relies on spectral analysis. For any function $f\in L^{2}(\tilde{\mathbb{P}})$, let $e_{f}$ be the spectral measure of $\mathcal{L}$ projected on the function $f$. This measure is such that, for any positive continuous function $\Psi:[0,+\infty)\to\mathbb{R}_{+}$, one has \[\tilde{\mathbb{E}}[f\ \Psi(\mathcal{L})f]=\int\Psi(\lambda)\ {\mathrm{d}}e_{f} (\lambda).\] For any $\gamma>1$ and $q\geqslant 0$, we recall that we say that the spectral exponents of a function $f$ are at least $(\gamma,-q)$ if (2.6) holds. Let us define the local drift $\mathfrak{d}$ in direction $\xi$ as \[\mathfrak{d}(\omega)=\mathbf{E}^{\omega}_{0}[\xi\cdot Y(1)]=\frac{1}{p(\omega) }\sum_{\|z\|=1}\omega_{0,z}\ \xi\cdot z.\] As we shall prove at the end of this section, we have the following bounds on the spectral exponents of $\mathfrak{d}$. Proposition 3.2.: _Under assumptions (H1)-(H3), there exists $q\geqslant 0$ such that the spectral exponents of the function $\mathfrak{d}$ are at least_ (3.4) \[\left\|\begin{array}[]{ll}(2,-q)&\text{if }d=2,\\ (d/2+1,0)&\text{if }3\leqslant d\leqslant 5,\\ (4,-1)&\text{if }d=6,\\ (4,0)&\text{if }d\geqslant 7.\end{array}\right.\] Let us see how this result implies Theorem 3.1. In order to do so, we also need the following information, that is a consequence of Proposition 3.2. Corollary 3.3.: _Let_ (3.5) \[\mathfrak{d}_{t}(\omega)=\mathbf{E}^{\omega}_{0}[\mathfrak{d}(\omega(t))]\] _be the image of $\mathfrak{d}$ by the semi-group at time $t$ associated with the Markov chain $(\omega(t))_{t\in\mathbb{N}}$. There exists $q\geqslant 0$ such that_ \[\tilde{\mathbb{E}}[(\mathfrak{d}_{t})^{2}]=\left\|\begin{array}[]{ll}O\big{(}t^ {-2}\ \ln^{q}(t)\big{)}&\text{if }d=2,\\ O\big{(}t^{-(d/2+1)}\big{)}&\text{if }3\leqslant d\leqslant 5,\\ O\big{(}t^{-4}\ \ln(t)\big{)}&\text{if }d=6,\\ O\big{(}t^{-4}\big{)}&\text{if }d\geqslant 7.\end{array}\right.\] Proof.: This result is the discrete-time analog of [10, Corollary 1]. It is obtained the same way, noting that \[\tilde{\mathbb{E}}[(\mathfrak{d}_{t})^{2}]=\int(1-\lambda)^{2t}\ {\mathrm{d}}e _{f}(\lambda),\] and that the support of the measure $e_{f}$ is contained in $[0,2]$. ∎ We are now in position to prove Theorem 3.1. Proof of Theorem 3.1.: The proof has the same structure as for the continuous-time case of [18, Proposition 8.4]. Note that [5, Theorem 2.1] ensures that (3.6) \[\lim_{t\to\infty}\sigma_{t}^{2}\,\stackrel{{\text{(def)}}}{{=}}\, \lim_{t\to\infty}t^{-1}{\mathbb{E}}_{0}[(\xi\cdot Y(t))^{2}]\,=\,\sigma^{2}.\] The starting point is the observation that, under $\tilde{\mathbb{P}}_{0}$, the process defined by (3.7) \[N_{t}=\xi\cdot Y(t)-\sum_{s=0}^{t-1}\mathfrak{d}(\omega(s))\] is a square-integrable martingale with stationary increments. On the one hand, following (2.3), we denote by $Z_{\mathfrak{d}}(t)$ the sum appearing in the r. h. s. of (3.7). From Proposition 3.2 and Theorem 2.2, we learn that there exist $\overline{\sigma}$ and $q\geqslant 0$ such that (3.8) On the other hand, since $N_{t}$ is a martingale with stationary increments, (3.9) \[\tilde{\mathbb{E}}_{0}[(N_{t})^{2}]=t\tilde{\mathbb{E}}_{0}[(N_{1})^{2}].\] As in the proof of Theorem 2.2 in the discrete time case, we then use that $\xi\cdot Y(t)$ is orthogonal to $(Z_{\mathfrak{d}}(t)+\mathfrak{d}(\omega(t)))$ to turn (3.7) into (3.10) \[t^{-1}\tilde{\mathbb{E}}_{0}[(N_{t})^{2}]=t^{-1}\tilde{\mathbb{E}}_{0}[(\xi \cdot Y(t))^{2}]+t^{-1}\tilde{\mathbb{E}}_{0}[(Z_{\mathfrak{d}}(t))^{2}]+2t^{- 1}\tilde{\mathbb{E}}_{0}[\mathfrak{d}(\omega(t))(\xi\cdot Y(t))].\] We already control the l. h. s. and the second term of the r. h. s. of (3.10). In order to quantify the convergence of $t^{-1}\tilde{\mathbb{E}}_{0}[(\xi\cdot Y(t))^{2}]$ it remains to control the last term. In particular, provided we show that (3.11) (3.10), (3.8), (3.9), and (3.6) imply first that $\sigma^{2}=\tilde{\mathbb{E}}_{0}[(N_{1})^{2}]-\overline{\sigma}^{2}$, and then the desired quantitative estimate (3.1). We now turn to (3.11) and write \[\tilde{\mathbb{E}}_{0}[\mathfrak{d}(\omega(t))(\xi\cdot Y(t))] = \sum_{s=0}^{t-1}\tilde{\mathbb{E}}_{0}[\mathfrak{d}(\omega(t))( \xi\cdot(Y(s+1)-Y(s))]\] \[= \sum_{s=0}^{t-1}\tilde{\mathbb{E}}_{0}[\mathfrak{d}_{t-s-1}( \omega(s+1))(\xi\cdot(Y(s+1)-Y(s))],\] where we have used the Markov property at time $s+1$, together with the definition (3.5) of $\mathfrak{d}_{t-s-1}$. Using Cauchy-Schwarz inequality and the stationarity of the process $(\omega(t))_{t\in\mathbb{N}}$ under $\tilde{\mathbb{E}}_{0}$, this sum is bounded by \[\|\xi\|^{2}\sum_{s=0}^{t-1}{\tilde{\mathbb{E}}[(\mathfrak{d}_{t-s-1})^{2}]}^{1/2}.\] Esimate (3.11) then follows from Corollary 3.3. This concludes the proof of the theorem. ∎ We finally turn to the proof of Proposition 3.2, which is a discrete-time counterpart of [10, Theorem 5]. In [10, Theorem 5] however, we had proved in addition that the spectral exponents are at least $(d/2-2,0)$, which is sharper than the exponents of Proposition 3.2 for $d>10$. In particular for $d>10$ the bounds of [10, Theorem 5] follow from results of [18], whose adaptation to the discrete time setting is not straightforward. As shown above, the present statement is sufficient to prove the optimal scaling of the systematic error, and we do not investigate further this issue. The proof of Proposition 3.2 is rather involved and one may wonder whether this is worth the effort in terms of the application we have in mind — namely Theorem 3.1. In order to obtain the optimal convergence rate in Theorem 3.1 we need the spectral exponents to be larger than $(2,0)$. Proving that the exponents are at least $(2,0)$ is rather direct using results of [7] (see the first three steps of the proof of Proposition 3.2). Yet proving that they are larger than $(2,0)$ for $d>2$ is as involved as proving Proposition 3.2 itself. This is the reason why we display the complete proof of Proposition 3.2 — although the precise values of the spectral exponents are not that important in the context of this paper. Proof of Proposition 3.2.: The present proof is a direct version of the proof of [10, Theorem 5]. In particular, the proof of [10, Theorem 5] relies on a nontrivial (weaker) estimate of the spectral exponents obtained in [8] using a covariance estimate. Yet if one wants to extend these results to more general statistics of the conductivity function — for instance to mixing coefficients in the sense of Dobrushin and Shlosman — one has to give up the covariance estimate (which we are not able to prove any longer, see [9]). With this in mind we only rely on the variance estimate of [7, Lemma 2.3]. Our strategy is similar to the strategy used for (continuous) elliptic equations in [9] to prove the estimate of the systematic error. In particular we directly focus on the spectral exponents rather than on some other quantity like the systematic error itself. Yet we will go slightly further. Starting point is the inequality: (3.12) \[\int_{0}^{T^{-1}}{\mathrm{d}}e_{\mathfrak{d}}(\lambda)\,\lesssim\,T^{-4}\int_{ 0}^{\infty}\frac{1}{(T^{-1}+\lambda)^{4}}\ {\mathrm{d}}e_{\mathfrak{d}}( \lambda),\] which follows from the fact that for $\lambda\leqslant T^{-1}$, $\frac{T^{-4}}{(T^{-1}+\lambda)^{4}}\gtrsim 1$ (here and below $\lesssim$ and $\gtrsim$ stand respectively for $\leqslant$ and $\geqslant$ up to multiplicative constants). The variable $T^{-1}$ for $T$ large plays the role of $\mu$ in (2.6). In what follows we make the standard identification between stationary functions $(z,\omega)\mapsto f(z,\omega)$ of both the space variable $z\in\mathbb{Z}^{d}$ and the environment $\omega$ and their translated versions at 0 $\omega\mapsto f(0,\theta_{z}\omega)$ depending on the environment only. We define $\phi_{T}$ as the unique stationary solution to (3.13) \[T^{-1}\phi_{T}(x)-\frac{1}{p_{\omega}(x)}\nabla^{}\cdot A(x)\nabla\phi_{T}\,= \,\frac{1}{p_{\omega}(x)}\nabla^{}\cdot A(x)\xi,\] whose existence and uniqueness follow from Lax-Milgram’s theorem in $\mathbb{L}^{2}(\tilde{\mathbb{P}})$ using the identification between the stationary function $\phi_{T}$ and its version defined on the environment only (see a similar argument of [15]). In particular, with the notation $\mathfrak{d}=\frac{1}{p_{\omega}(x)}\nabla^{}\cdot A(x)\xi$, \[\phi_{T}\,=\,(T^{-1}+\mathcal{L})^{-1}\mathfrak{d},\] where $\mathcal{L}$ is the operator defined in (3.3), and the spectral theorem ensures that \[\tilde{\mathbb{E}}(\phi_{T}^{2})\,=\,\tilde{\mathbb{E}}(\mathfrak{d}(T^{-1}+ \mathcal{L})^{-2}\mathfrak{d})\,=\,\int_{0}^{\infty}\frac{1}{(T^{-1}+\lambda)^ {2}}de_{\mathfrak{d}}(\lambda)\] where $e_{\mathfrak{d}}$ is the spectral measure of $\mathcal{L}$ projected on the drift $\mathfrak{d}$. We also let $\psi_{T}$ be the unique stationary solution to (3.14) \[T^{-1}\psi_{T}(x)-\frac{1}{p_{\omega}(x)}\nabla^{}\cdot A(x)\nabla\psi_{T}(x) \,=\,\phi_{T}(x),\] whose existence and uniqueness also follows from Lax-Milgram’s theorem in the probability space as well. This time, \[\psi_{T}\,=\,(T^{-1}+\mathcal{L})^{-2}\mathfrak{d},\] and the spectral theorem yields \[\tilde{\mathbb{E}}(\psi_{T}^{2})\,=\,\tilde{\mathbb{E}}(\mathfrak{d}(T^{-1}+ \mathcal{L})^{-4}\mathfrak{d})\,=\,\int_{0}^{\infty}\frac{1}{(T^{-1}+\lambda)^ {4}}de_{\mathfrak{d}}(\lambda)\] From now on, we shall use the shorthand notation $\left\langle u\right\rangle:=\tilde{\mathbb{E}}(u)$ and $\mathrm{var}\left[u\right]=\left\langle(u-\left\langle u\right\rangle)^{2}\right\rangle$ for all $u\in\mathbb{L}^{2}(\tilde{\mathbb{P}})$. In particular the identity above turns into (3.15) \[\int_{0}^{\infty}\frac{1}{(T^{-1}+\lambda)^{4}}\ {\mathrm{d}}e_{\mathfrak{d}}( \lambda)\,=\,\mathrm{var}\left[\psi_{T}\right],\] since $\left\langle\psi_{T}\right\rangle=\frac{1}{\mathbb{E}[p]}\int\psi_{T}p\ { \mathrm{d}}\mathbb{P}=\frac{T}{\mathbb{E}[p]}\int\phi_{T}p\ {\mathrm{d}} \mathbb{P}=0$ using equations (3.14) and (3.13). The rest of the proof, which is dedicated to the estimate of $\mathrm{var}\left[\psi_{T}\right]$, is divided in six steps. Starting point is the application of the variance estimate of [7, Lemma 2.3] to $\psi_{T}$, which requires to estimate the susceptibility of $\psi_{T}$ with respect to the random coefficients. In view of (3.14) it is not surprising that we will have to estimate not only the susceptibility of $\psi_{T}$ but also of $\phi_{T}$ and of some Green’s function with respect to the random coefficients. In the first step, we establish the susceptibility estimate for the Green’s function. In Step 2 we turn to the susceptibility estimate for the approximate corrector $\phi_{T}$. We then show in Step 3 that, relying on [7], this implies that the spectral exponents are at least (3.16) \[\left\|\begin{array}[]{rcl}(2,-q)&\mbox{ for }&d=2,\\ (2,0)&\mbox{ for }&d>2.\end{array}\right.\] Following step is to estimate the susceptibility of $\psi_{T}$. In Step 5 we show that this, combined with the suboptimal estimates (3.17) \[\left\langle\|\nabla\psi_{T}\|^{2}\right\rangle\,\lesssim\,\left\|\begin{array}[] {rcl}T\ln^{q}T&\mbox{ for }&d=2,\\ T&\mbox{ for }&d>2,\end{array}\right.\quad\mbox{ and }\left\langle\psi_{T}^{2} \right\rangle\,\lesssim\,\left\|\begin{array}[]{rcl}T^{2}\ln^{q}T&\mbox{ for }& d=2,\\ T^{2}&\mbox{ for }&d>2\end{array}\right.\] obtained using (3.16), allow us to improve the spectral exponents to (3.18) \[\left\|\begin{array}[]{rcl}(2,-q)&\mbox{ for }&d=2,\\ (5/2,0)&\mbox{ for }&d=3,\\ (3,-1)&\mbox{ for }&d=4,\\ (3,0)&\mbox{ for }&d>4.\end{array}\right.\] In the last step, we quickly argue that in turn these spectral exponents yield the optimal and suboptimal estimates (3.19) \[\left\langle\|\nabla\psi_{T}\|^{2}\right\rangle\,\lesssim\,\left\|\begin{array}[] {rcl}T\ln^{q}T&\mbox{ for }&d=2,\\ \sqrt{T}&\mbox{ for }&d=3,\\ \ln T&\mbox{ for }&d=4,\\ 1&\mbox{ for }&d>4,\end{array}\right.\quad\mbox{ and }\left\langle\psi_{T}^{2} \right\rangle\,\lesssim\,\left\|\begin{array}[]{rcl}T^{2}\ln^{q}T&\mbox{ for }& d=2,\\ T^{3/2}&\mbox{ for }&d=3,\\ T\ln T&\mbox{ for }&d=4,\\ T&\mbox{ for }&d>4,\end{array}\right.\] which finally bootstrap (3.18) to the desired estimates of the spectral exponents, and consequently yield the following optimal estimate of $\mathrm{var}\left[\psi_{T}\right]$: \[\mathrm{var}\left[\psi_{T}\right]\,\lesssim\,\left\|\begin{array}[]{rcl}T^{2} \ln^{q}T&\mbox{ for }&d=2,\\ T^{3/2}&\mbox{ for }&d=3,\\ T&\mbox{ for }&d=4,\\ \sqrt{T}&\mbox{ for }&d=5,\\ \ln T&\mbox{ for }&d=6,\\ 1&\mbox{ for }&d>6.\end{array}\right.\] We do not need Step 6 to prove Theorem 3.1. Yet this step is interesting in itself since it may be the starting point of a subtle “multi-level” induction to obtain the sharp spectral exponents in any dimension. _Step_ 1. Susceptility of the Green’s function. For all $y\in\mathbb{Z}^{d}$ we define the Green’s function $G_{T}(\cdot,y)$ with singularity at $y$ as the unique solution in $L^{2}(\mathbb{Z}^{d})$ to the equation (3.20) \[T^{-1}p_{\omega}(x)G_{T}(x,y)-\nabla_{x}^{}\cdot A(x)\nabla_{x}G_{T}(x,y)\,= \,\delta(x-y),\] using Lax-Milgram’s theorem. We shall prove for all $e=(z,z^{\prime})\in\mathbb{B}$, $z\in\mathbb{Z}^{d}$, $z^{\prime}=z+\mathbf{e}_{i}$, (3.21) \[\frac{\partial G_{T}}{\partial\omega_{e}}(x,y)\,=\,-T^{-1}\Big{(} G_{T}(z,y)G_{T}(x,z)+G_{T}(z^{\prime},y)G_{T}(x,z^{\prime})\Big{)}\\ -\nabla_{z_{i}}G_{T}(x,z)\nabla_{z_{i}}G_{T}(z,y),\] and (3.22) \[\begin{array}[]{rcl}\sup_{\omega_{e}}\|\nabla_{z_{i}}G_{T}(z,y)\|& \lesssim&\|\nabla_{z_{i}}G_{T}(z,y)\|+T^{-1}g_{T}(y-z),\\ \sup_{\omega_{e}}\|\nabla_{z_{i}}G_{T}(y,z)\|&\lesssim&\|\nabla_{z_{ i}}G_{T}(y,z)\|+T^{-1}g_{T}(y-z),\end{array}\] where $g_{T}:\mathbb{Z}^{d}\to\mathbb{R}^{+}$ satisfies for some constant $c>0$ (depending on $\alpha,\beta,d$) (3.23) \[g_{T}(x)\,=\,(1+\|x\|)^{2-d}\exp\big{(}-c\frac{\|x\|}{\sqrt{T}}\big{)}\] for $d>2$, and (3.24) \[\begin{array}[]{l}g_{T}(x)\,=\,\left\|\ln\Big{(}\frac{\sqrt{T}}{1+ \|x\|}\Big{)}\right\|\exp\big{(}-c\frac{\|x\|}{\sqrt{T}}\big{)}\end{array}\] for $d=2$. We define the elliptic operator $L_{T}$ as \[(L_{T}u)(x)\,=\,\sum_{x^{\prime},\|x-x^{\prime}\|=1}\omega_{(x,x^{\prime})}T^{-1 }u(x)+\sum_{x^{\prime},\|x-x^{\prime}\|=1}\omega_{(x,x^{\prime})}\big{(}u(x)-u(x ^{\prime})\big{)},\] so that (3.20) takes the form (3.25) \[(L_{T}G_{T}(\cdot,y))(x)\,=\,\delta(x-y).\] Formally differentiating this equation with respect to $\omega_{e}$ yields \[L_{T}\Big{(}\frac{\partial G_{T}}{\partial\omega_{e}}(\cdot,y) \Big{)}(x)+T^{-1}G_{T}(x,y)(\delta(x-z)+\delta(x-z^{\prime}))\\ +\big{(}G_{T}(z,y)-G_{T}(z^{\prime},y)\big{)}\delta(x-z)+\big{(}G _{T}(z^{\prime},y)-G_{T}(z^{\prime},y)\big{)}\delta(x-z^{\prime})\,=\,0.\] Using (3.25) this identity turns into \[L_{T}\bigg{(}\frac{\partial G_{T}}{\partial\omega_{e}}(\cdot,y)+ T^{-1}\big{(}G_{T}(z,y)G_{T}(\cdot,z)+G_{T}(z^{\prime},y)G_{T}(\cdot,z^{\prime })\big{)}\\ +\nabla_{z_{i}}G_{T}(\cdot,z)\nabla_{z_{i}}G_{T}(z,y)\bigg{)}(x) \,=\,0.\] Formally erasing the operator $L_{T}$ on the l. h. s. then yields the desired identity (3.21). To turn this into a rigorous argument, we may proceed as in [7, Proof of Lemma 2.5], first consider finite differences instead of derivative w. r. t. $\omega_{e}$, use that $L_{T}$ is bijective on $L^{2}(\mathbb{Z}^{d})$, and then pass to the limit. We leave the details to the reader and directly turn to (3.22). From (3.21) we infer that \[\frac{\partial\nabla_{z_{i}}G_{T}(z,y)}{\partial\omega_{e}}\,=\,- \nabla_{z_{i}}\nabla_{z_{i}}G_{T}(z,z)\nabla_{z_{i}}G_{T}(z,y)\\ -T^{-1}\big{(}G_{T}(z,y)\nabla_{z_{i}}G_{T}(z,z)+G_{T}(z^{\prime} ,y)\nabla_{z_{i}}G_{T}(z,z^{\prime})\big{)}.\] Using then the uniform pointwise estimate $\|\nabla G_{T}\|\lesssim 1$ (see [7, Corollary 2.3], whose proof holds as well in the present case with a non-constant zero order term since $2^{d}\alpha T^{-1}\leqslant T^{-1}p_{\omega}(x)\leqslant 2^{d}\beta T^{-1}$), the uniform pointwise estimate of the Green’s function from [6, Lemma 3], and considering this identity as an ODE for $\nabla_{z_{i}}G_{T}(z,y)$ in function of $\omega_{e}$, we obtain (3.22). _Step_ 2. Susceptibility of the approximate corrector $\phi_{T}$. In this step we shall prove that for $e=(z,z^{\prime})\in\mathbb{B}$, $z\in\mathbb{Z}^{d}$ and $z^{\prime}=z+\mathbf{e}_{i}$, (3.26) \[\frac{\partial\phi_{T}}{\partial\omega_{e}}(x)\,=\,-(\nabla_{i} \phi_{T}(z)+\xi_{i})\nabla_{z_{i}}G_{T}(x,z)\\ -T^{-1}\phi_{T}(z)\big{(}G_{T}(x,z)+G_{T}(x,z^{\prime})\big{)},\] (3.27) \[\sup_{\omega_{e}}\|\phi_{T}(x)\|\,\lesssim\,\|\phi_{T}(x)\|\\ +(\|\nabla_{i}\phi_{T}(z)\|+1)\Big{(}\|\nabla_{z_{i}}G_{T}(x,z)\|+T^{ -1/2}g_{T}(x-z)\Big{)},\] (3.28) \[\sup_{\omega_{e}}\left\|\frac{\partial\phi_{T}}{\partial\omega_{e} }(x)\right\|\,\lesssim\,(\|\nabla_{i}\phi_{T}(z)\|+1)\Big{(}\|\nabla_{z_{i}}G_{T}( x,z)\|+T^{-1/2}g_{T}(x-z)\Big{)},\] and for all $n\in\mathbb{N}$, (3.29) \[\sup_{\omega_{e}}\left\|\frac{\partial(\phi_{T}(x)^{n+1})}{ \partial\omega_{e}}\right\|\,\lesssim\,(\|\nabla_{i}\phi_{T}(z)\|+1)\Big{(}\| \nabla_{z_{i}}G_{T}(x,z)\|+T^{-1/2}g_{T}(x-z)\Big{)}\\ \times\Big{(}\|\phi_{T}(x)\|+(\|\nabla_{i}\phi_{T}(z)\|+1)(\|\nabla_{z _{i}}G_{T}(z,x)\|+T^{-1/2}g_{T}(x-z)\Big{)}^{n}.\] As for the Green’s function, we rewrite the defining equation for $\phi_{T}$ as (3.30) \[(L_{T}G_{T}(\cdot,y))(x)-\nabla^{}\cdot A(x)\xi\,=\,0.\] Formally differentiating (3.30) w. r. t. $\omega_{e}$ yields \[L_{T}\frac{\partial\phi_{T}}{\partial\omega_{e}}(x)-(\nabla_{i} \phi_{T}(z)+\xi_{i})\big{(}\delta(x-z)-\delta(x-z^{\prime})\big{)}\\ +T^{-1}\phi_{T}(z)\big{(}\delta(x-z)+\delta(x-z^{\prime})\big{)} \,=\,0,\] which using (3.25) turns into \[L_{T}\bigg{(}\frac{\partial\phi_{T}}{\partial\omega_{e}}-(\nabla _{i}\phi_{T}(z)+\xi_{i})\big{(}G_{T}(\cdot,z)-G_{T}(\cdot,z^{\prime})\big{)}\\ +T^{-1}\phi_{T}(z)\big{(}G_{T}(\cdot,z)+G_{T}(\cdot,z^{\prime}) \big{)}\bigg{)}(x)\,=\,0.\] This (formally) shows (3.26). To turn this into a rigorous argument, we may use the Green representation formula \[\phi_{T}(x)\,=\,\int_{\mathbb{Z}^{d}}G_{T}(x,y)\nabla^{}\cdot A(y)\xi\ { \mathrm{d}}y,\] where $\int_{\mathbb{Z}^{d}}\ {\mathrm{d}}y$ denotes the sum over all $y\in\mathbb{Z}^{d}$, and proceed as in [7, Proof of Lemma 2.4]. We now turn to (3.28). This estimate follows from (3.26), (3.22), and the following two facts: (3.31) \[\|\phi_{T}\|\,\lesssim\,\sqrt{T}\] and (3.32) \[\sup_{\omega_{e}}\|\nabla_{i}\phi_{T}(z)\|\,\lesssim\,\|\nabla_{i}\phi_{T}(z)\|+1.\] Starting point to prove (3.31) is the Green representation formula in the form of \[\|\phi_{T}(x)\| = \left\|\int_{\mathbb{Z}^{d}}G_{T}(x,y)\nabla^{}\cdot A(y)\xi\ { \mathrm{d}}y\right\|\] \[= \left\|\int_{\mathbb{Z}^{d}}\nabla_{y}G_{T}(x,y)\cdot A(y)\xi\ { \mathrm{d}}y\right\|\] \[\lesssim \int_{\mathbb{Z}^{d}}\|\nabla_{y}G_{T}(0,y)\|\ {\mathrm{d}}y,\] from which we deduce the claim by a dyadic decomposition of space combined with Cauchy-Schwarz’ inequality and [7, Lemma 2.9] (a similar calculation is detailed in [6, Proof of Lemma 4]). For (3.32), we first note that (3.26) implies \[\frac{\partial\nabla_{i}\phi_{T}(z)}{\partial\omega_{e}}\,=\,-( \nabla_{i}\phi_{T}(z)+\xi_{i})\big{(}\nabla_{z_{i}}G_{T}(z^{\prime},z)-\nabla_ {z_{i}}G_{T}(z,z)\big{)}\\ +T^{-1}\phi_{T}(z)\big{(}\nabla_{z_{i}}G_{T}(z,z)+\nabla_{z_{i}}G _{T}(z,z^{\prime})\big{)},\] which — seen as an ODE w. r. t. $\omega_{e}$ — yields the claim using the uniform bound $\|\nabla G_{T}\|\lesssim 1$ of [7, Corollary 2.3] and (3.31). Estimate (3.27) is a direct consequence of (3.28), whereas (3.29) follows from the Leibniz’ rule combined with (3.26), (3.27), and (3.28). _Step_ 3. Proof of (3.16). The estimates (3.16) of the spectral exponents follow from the more general estimates: for all $q>0$ there exists $\gamma(q)>0$ such that (3.33) \[\left\langle\|\phi_{T}\|^{q}\right\rangle\,\lesssim\,\left\|\begin{array}[]{rcl} \ln^{\gamma(q)}T&\mbox{ for }&d=2,\\ 1&\mbox{ for }&d>2,\end{array}\right.\] combined with the fact that \[\int_{0}^{T^{-1}}\ {\mathrm{d}}e_{\mathfrak{d}}(\lambda)\,\lesssim\,T^{-2}\int _{0}^{\infty}\frac{1}{(T^{-1}+\lambda)^{2}}\ {\mathrm{d}}e_{\mathfrak{d}}( \lambda)\,=\,T^{-2}\left\langle\phi_{T}^{2}\right\rangle.\] The proof of (3.33) is an adaptation of [7, Proof of Proposition 2.1] which already covers the case of a constant coefficient in the zero order term of $L_{T}$, that is for $T^{-1}\phi_{T}$ instead of $T^{-1}p_{\omega}\phi_{T}$ (no randomness in the zero order term). The first step to apply the variance estimate of [7, Lemma 2.3] is to show that $\phi_{T}$ is measurable with respect to cylindrical topology associated with the random variables. This can be proved exactly as in [7, Lemma 2.6]. The estimates (3.22), (3.27), (3.28), and (3.29) of Steps 1 and 2 are like in the auxiliary lemmas of [7] provided we replace the terms $\|\nabla_{z_{i}}G_{T}(x,z)\|$ in [7, Lemmas 2.4 & 2.5] by $\|\nabla_{z_{i}}G_{T}(x,z)\|+T^{-1/2}g_{T}(x-z)$. A close look at the proof of [7, Proposition 2.1] shows that these terms $\|\nabla_{z_{i}}G_{T}(x,z)\|$ are either estimated by the Green’s function itself (in which case the additional term $T^{-1/2}g_{T}(x-z)$ is of higher order), or they are controlled on dyadic annuli by the Meyers’ estimate [7, Lemma 2.9]. This lemma shows in particular that there exists $p>2$ such that for all $p\geqslant q\geqslant 2$, $k>$ and $R\gg 1$, \[\int_{R\leqslant\|x-z\|<2R}\|\nabla_{z}G_{T}(x,z)\|^{q}dz\,\lesssim\,R^{d}(R^{1-d} )^{q}\min\{1,\sqrt{T}R^{-1}\}^{k}.\] By the properties (3.23) for $d>2$ and (3.24) for $d=2$ of the function $g_{T}$, it is easy to see that for $d>2$ \[\int_{R\leqslant\|x-z\|<2R}\big{(}T^{-1/2}g_{T}(x-z)\big{)}^{q}dz\,\lesssim\,R^{ d}(R^{1-d})^{q}\min\{1,\sqrt{T}R^{-1}\}^{k}\] as well, whereas for $d=2$ \[\int_{R\leqslant\|x-z\|<2R}\big{(}T^{-1/2}g_{T}(x-z)\big{)}^{q}dz\,\lesssim\,R^{ 2}(R^{-1})^{q}\ln^{q}T\min\{1,\sqrt{T}R^{-1}\}^{k}.\] Hence the proof of [7, Proposition 2.1] adapts mutadis mutandis to the present case (with possibly an additional logarithmic correction for $d=2$), and we have (3.33). _Step_ 4. Susceptibility of $\psi_{T}$. In this step we shall prove that for $e=(z,z^{\prime})$, $z\in\mathbb{Z}^{d}$ and $z^{\prime}=z+\mathbf{e}_{i}$, \[\frac{\partial\psi_{T}}{\partial\omega_{e}}(x) = -\nabla_{z_{i}}G_{T}(x,z)\nabla_{i}\psi_{T}(z)-T^{-1}G_{T}(x,z) \psi_{T}(z)\] \[-T^{-1}G_{T}(x,z^{\prime})\psi_{T}(z^{\prime})\] \[-(\nabla_{i}\phi_{T}(z)+\xi_{i})\int_{\mathbb{Z}^{d}}G_{T}(x,y)p_ {\omega}(y)\nabla_{z_{i}}G_{T}(y,z)\ {\mathrm{d}}y\] \[-T^{-1}\phi_{T}(z)\int_{\mathbb{Z}^{d}}G_{T}(x,y)p_{\omega}(y) \big{(}G_{T}(y,z)+G_{T}(y,z^{\prime})\big{)}\ {\mathrm{d}}y\] \[+G_{T}(x,z)\phi_{T}(z)+G_{T}(x,z^{\prime})\phi_{T}(z^{\prime}),\] and \[\sup_{\omega_{e}}\left\|\frac{\partial\psi_{T}}{\partial\omega_{e} }(0)\right\|\] \[\lesssim g_{T}(z)\big{(}\|\nabla_{i}\psi_{T}(z)\|+T^{-1}\|\psi_{T}(z)\|+\nu_{ d}(T)(1+\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime})\|)\big{)}\] \[+(1+\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime})\|)\int_{\mathbb{Z}^{d}}g_{ T}(y)\big{(}\|\nabla_{z_{i}}G_{T}(y,z)\|+T^{-1}g_{T}(y-z)\big{)}\ {\mathrm{d}}y,\] where (3.36) \[\nu_{d}(T)\,=\,\left\|\begin{array}[]{rcl}T&\mbox{ for }&d=2,\\ \sqrt{T}&\mbox{ for }&d=3,\\ \ln T&\mbox{ for }&d=4,\\ 1&\mbox{ for }&d>4.\end{array}\right.\] Starting point is again the Green representation formula \[\psi_{T}(x)\,=\,\int_{\mathbb{Z}^{d}}G_{T}(x,y)p_{\omega}(y)\phi_{T}(y)\ { \mathrm{d}}y,\] associated with (3.14) in the form \[T^{-1}p\psi_{T}-\nabla^{}\cdot A\nabla\psi_{T}\,=\,p\phi_{T}.\] Differentiated w. r. t. $\omega_{e}$ it turns into \[\frac{\partial\psi_{T}(x)}{\partial\omega_{e}}\,=\,\int_{\mathbb{ Z}^{d}}\frac{\partial G_{T}(x,y)}{\partial\omega_{e}}p_{\omega}(y)\phi_{T}(y) \ {\mathrm{d}}y\\ +\int_{\mathbb{Z}^{d}}G_{T}(x,y)\frac{\partial p_{\omega}(y)}{ \partial\omega_{e}}\phi_{T}(y)\ {\mathrm{d}}y+\int_{\mathbb{Z}^{d}}G_{T}(x,y)p _{\omega}(y)\frac{\partial\phi_{T}(y)}{\partial\omega_{e}}\ {\mathrm{d}}y.\] Combined with (3.21), (3.26), and the Green representation formula itself, this shows (3). We now turn to (3) and treat each term of the r. h. s. of (3) separately. We begin with the third line of (3), appeal to (3.32), then bound the gradient of the approximate corrector $\|\nabla_{i}\phi_{T}(z)\|$ by the approximate corrector $\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime})\|$ itself, control the Green’s function $G_{T}$ by $g_{T}$, and use (3.22) to estimate the supremum in $\omega_{e}$ of the gradient of the Green’s function $\|\nabla_{z_{i}}G_{T}(y,x)\|$. This term is thus controlled by the second term of the r. h. s. of (3). The term in the fourth line of (3) is also estimated by the second term of the r. h. s. of (3) using (3.27) (and the uniform bounds $\|\nabla G_{T}\|,T^{-1/2}G_{T}\lesssim 1$), whereas the last two terms of (3) are bounded by the first term of the r. h. s. of (3) using (3.27). The subtle terms are the first three ones, for which we have to estimate the suprema of $\|\nabla_{i}\psi_{T}(z)$, $\|\psi_{T}(z)\|$, and $\|\psi_{T}(z^{\prime})\|$ w. r. t. $\omega_{e}$. We begin with the following two estimates \[\sup_{\omega_{e}}\|\psi_{T}(z)\| \lesssim\] \[+\sup_{\omega_{e}}\|\nabla_{i}\psi_{T}(z)\|,\] \[\sup_{\omega_{e}}\|\nabla_{i}\psi_{T}(z)\| \lesssim \|\nabla_{i}\psi_{T}(z)\|+\big{(}\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime} )\|+1\big{)}\nu_{d}(T)\] \[+T^{-1}\sup_{\omega_{e}}\|\psi_{T}(z)\|,\] which — considered as a system of two coupled ODEs— show that there exists some $T_{}>0$ such that for all $T\geqslant T^{}$, \[\sup_{\omega_{e}}\|\psi_{T}(z)\| \lesssim\] \[+\|\nabla_{i}\psi_{T}(z)\|,\] \[\sup_{\omega_{e}}\|\nabla_{i}\psi_{T}(z)\| \lesssim \|\nabla_{i}\psi_{T}(z)\|+\big{(}\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime} )\|+1\big{)}\nu_{d}(T)\] \[+T^{-1}\|\psi_{T}(z)\|.\] To prove (3) we consider (3) as an ODE on $\psi_{T}(z)$, bound $\psi_{T}(z^{\prime})$ by $\psi_{T}(z)+\|\nabla_{i}\psi_{T}(z)\|$, and use that the four last terms of the r. h. s. of (3) are bounded by the second term of the r. h. s. of (3), as discussed above. Hence (3) turns into \[\left\|\frac{\partial\psi_{T}}{\partial\omega_{e}}(z)\right\| \lesssim \sup_{\omega_{e}}\{\|\nabla_{z_{i}}G_{T}(x,z)\|\|\nabla_{i}\psi_{T}( z)\}+T^{-1}G_{T}(z,z)\|\psi_{T}(z)\|\] \[+T^{-1}G_{T}(z,z^{\prime})\big{(}\|\psi_{T}(z)\|+\sup_{\omega_{e}}\| \nabla_{i}\psi_{T}(z)\big{)}\] \[+(1+\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime})\|)\] \[\qquad\times\int_{\mathbb{Z}^{d}}g_{T}(y)\big{(}\|\nabla_{z_{i}}G_ {T}(y,z)\|+T^{-1}g_{T}(y-z)\big{)}\ {\mathrm{d}}y.\] Using the uniform bounds $\|\nabla G_{T}\|,T^{-1}G_{T}\lesssim 1$, and replacing the gradient of the Green’s function by the Green’s function itself in the integral — which we then control by $\nu_{d}(T)$ —, this estimate yields (3) by integrating the ODE. We now turn to (3) and infer from (3) that \[\frac{\partial\nabla_{i}\psi_{T}(z)}{\partial\omega_{e}} = -\nabla_{z_{i}}\nabla_{z_{i}}G_{T}(z,z)\nabla_{i}\psi_{T}(z)-T^{- 1}\nabla_{z_{i}}G_{T}(z,z)\psi_{T}(z)\] \[-T^{-1}\nabla_{z_{i}}G_{T}(z,z^{\prime})\psi_{T}(z^{\prime})\] \[-(\nabla_{i}\phi_{T}(z)+\xi_{i})\int_{\mathbb{Z}^{d}}\nabla_{z_{i }}G_{T}(z,y)p_{\omega}(y)\nabla_{z_{i}}G_{T}(y,z)\ {\mathrm{d}}y\] \[-T^{-1}\phi_{T}(z)\int_{\mathbb{Z}^{d}}\nabla_{z_{i}}G_{T}(z,y)p_ {\omega}(y)\big{(}G_{T}(y,z)+G_{T}(y,z^{\prime})\big{)}\ {\mathrm{d}}y\] \[+\nabla_{z_{i}}G_{T}(z,z)\phi_{T}(z)+\nabla_{z_{i}}G_{T}(z,z^{ \prime})\phi_{T}(z^{\prime}).\] Proceeding as from (3) to (3), this implies (3), and therefore (3) and (3). Combining the inequality $\|\psi_{T}(z^{\prime})\|\leqslant\|\psi_{T}(z)\|+\|\nabla_{i}\psi_{T}(z)\|$ with (3) and (3) yields the last estimate we need: (3.41) \[\sup_{\omega_{e}}\|\psi_{T}(z^{\prime})\|\,\lesssim\,\|\psi_{T}(z)\|+\big{(}\|\phi_ {T}(z)\|+\|\phi_{T}(z^{\prime})\|+1\big{)}\nu_{d}(T)+\|\nabla_{i}\psi_{T}(z)\|.\] We are finally in position to conclude the proof of (3): the first three terms of the r. h. s. of (3) are estimated by (3), (3), and (3.41). Replacing then the gradient of the Green’s function by the Green’s function itself, and the Green’s function by $g_{T}$, we obtain (3), as desired. _Step_ 5. Proof of (3.18). This is an application of the variance estimate [7, Lemma 2.3] on $\psi_{T}$ based on (3). In particular, (3.42) \[\mathrm{var}\left[\psi_{T}\right]\,\lesssim\,\sum_{e\in\mathbb{B}}\left\langle \sup_{\omega_{e}}\left\|\frac{\partial\psi_{T}(0)}{\partial\omega_{e}}\right\|^{ 2}\right\rangle.\] We distinguish the contributions of the two terms of the r. h. s. of (3) in this sum and use the notation \[A_{e} := g_{T}(z)\big{(}\|\nabla_{i}\psi_{T}(z)\|+T^{-1}\|\psi_{T}(z)\|+\nu_{ d}(T)(1+\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime})\|)\big{)},\] \[B_{e} := (1+\|\phi_{T}(z)\|+\|\phi_{T}(z^{\prime})\|)\int_{\mathbb{Z}^{d}}g_{T }(y)\big{(}\|\nabla_{z_{i}}G_{T}(y,z)\|+T^{-1}g_{T}(y-z)\big{)}\ {\mathrm{d}}y.\] The contribution of the first term is estimated as follows: \[\sum_{e\in\mathbb{B}}\left\langle A_{e}^{2}\right\rangle\] \[\lesssim \sum_{e\in\mathbb{B}}\left\langle g_{T}(z)^{2}\big{(}\|\nabla_{i} \psi_{T}(z)\|^{2}+T^{-2}\|\psi_{T}(z)\|^{2}+\nu_{d}(T)^{2}(1+\|\phi_{T}(z)\|^{2}+\| \phi_{T}(z^{\prime})\|^{2})\big{)}\right\rangle\] \[\lesssim\] \[\lesssim \nu_{d}(T)\big{(}\left\langle\|\nabla\psi_{T}\|^{2}\right\rangle+T^ {-2}\left\langle\psi_{T}^{2}\right\rangle+\nu_{d}(T)^{2}(1+\left\langle\phi_{T }^{2}\right\rangle)\big{)},\] by stationarity of $\phi_{T}$, $\psi_{T}$, and $\nabla\psi_{T}$. Combined with (3.33) for $q=2$, and (3.17) (which is proved below), this turns into (3.43) \[{\sum_{e\in\mathbb{B}}\left\langle A_{e}^{2}\right\rangle} \lesssim \left\|\begin{array}[]{rcl}T^{3}\ln^{q}T&\mbox{ for }&d=2,\\ T^{3/2}&\mbox{ for }&d=3,\\ T\ln T&\mbox{ for }&d=4,\\ T&\mbox{ for }&d>4.\end{array}\right.\] Estimate (3.17) is a consequence of the estimate (3.16) of the spectral exponents and of the spectral representations \[\left\langle\psi_{T}^{2}\right\rangle = \int_{0}^{\infty}\frac{1}{(T^{-1}+\lambda)^{4}}\ {\mathrm{d}}e_{ \mathfrak{d}}(\lambda)\] \[\leqslant\] \[\alpha\left\langle\|\nabla\psi_{T}\|^{2}\right\rangle \leqslant \left\langle\nabla\psi_{T}\cdot A\nabla\psi_{T}\right\rangle\] \[= \int_{0}^{\infty}\frac{\lambda}{(T^{-1}+\lambda)^{4}}\ {\mathrm{d }}e_{\mathfrak{d}}(\lambda)\] \[\leqslant T\int_{0}^{1}\frac{1}{\lambda^{2}}\ {\mathrm{d}}e_{\mathfrak{d}} (\lambda)+\int_{0}^{\infty}\frac{1}{\lambda}\ {\mathrm{d}}e_{\mathfrak{d}}( \lambda),\] since $\int_{0}^{\infty}\frac{1}{\lambda}\ {\mathrm{d}}e_{\mathfrak{d}}(\lambda)\lesssim 1$, and by definition (2.6) of the spectral exponents. We now turn to the second term of the r. h. s. of (3), and note that it coincides with the term treated in [10, Step 3, Proof of Lemma 5] so that we have (3.44) \[{\sum_{e\in\mathbb{B}}\left\langle B_{e}^{2}\right\rangle} \lesssim \left\|\begin{array}[]{rcl}T^{2}\ln^{q}T&\mbox{ for }&d=2,\\ T^{3/2}&\mbox{ for }&d=3,\\ T&\mbox{ for }&d=4,\\ \sqrt{T}&\mbox{ for }&d=5,\\ \ln T&\mbox{ for }&d=6,\\ 1&\mbox{ for }&d>6.\\ \end{array}\right.\] Hence, (3.42), (3), (3.43) and (3.44) yield (3.18) for $d>2$ using (3.12) and (3.15). Note that (3.43) is not optimal for $d=2$ in view of the estimate of $\left\langle\psi_{T}^{2}\right\rangle$ in (3.17) (which is sharper). This comes from our estimate of $\sup_{\omega_{e}}\psi_{T}(z)$ where we have replaced a gradient of the Green’s function by the Green’s function itself and therefore given up some potential better decay (at least integrated on dyadic annuli). In high dimensions however, there is no loss because $g_{T}$ is then square-integrable itself. Anyway, the optimal spectral exponents for $d=2$ have been already obtained in (3.16). _Step_ 6. Proof of (3.4). We quickly show how (3.18) allows to get the sharper spectral exponents (3.4). This is the same argument as in Step 5, except that (3.17) can now be boostrapped to (3.19) using (3.18) and \[\left\langle\psi_{T}^{2}\right\rangle = \int_{0}^{\infty}\frac{1}{(T^{-1}+\lambda)^{4}}\ {\mathrm{d}}e_{ \mathfrak{d}}(\lambda)\] \[\leqslant T\int_{0}^{T_{}^{-1}}\frac{1}{\lambda^{3}}\ {\mathrm{d}}e_{ \mathfrak{d}}(\lambda)+T_{}^{3}\int_{0}^{\infty}\frac{1}{\lambda}\ {\mathrm{d }}e_{\mathfrak{d}}(\lambda)\] \[\left\langle\nabla\psi_{T}\cdot A\nabla\psi_{T}\right\rangle = \int_{0}^{\infty}\frac{\lambda}{(T^{-1}+\lambda)^{4}}\ {\mathrm{d }}e_{\mathfrak{d}}(\lambda)\] \[\leqslant \int_{0}^{T_{}^{-1}}\frac{1}{\lambda^{3}}\ {\mathrm{d}}e_{ \mathfrak{d}}(\lambda)+T_{}^{2}\int_{0}^{\infty}\frac{1}{\lambda}\ {\mathrm{d }}e_{\mathfrak{d}}(\lambda),\] where $T_{}<\infty$ has been defined in Step 4. Proceeding as in the end of Step 5 this yields the spectral exponents (3.4), and concludes the proof of the proposition. ∎ ## 4. The random fluctuations In this section, we show that the computable quantity $\hat{A}_{n}(t)$ defined in (1.5) is a good approximation of $\sigma_{t}^{2}$, in the sense that its random fluctuations are small as soon as $n/t^{2}$ is large. We write $\mathbb{N}^{}$ for $\mathbb{N}\setminus\{0\}$. Theorem 4.1.: _There exists $c>0$ such that, for any $n\in\mathbb{N}^{}$, $\varepsilon>0$ and $t$ large enough,_ \[\mathbb{P}^{\otimes}_{0}\left[\big{\|}\hat{A}_{n}(t)-\sigma_{t}^{2}\big{\|} \geqslant\varepsilon/t\right]\leqslant\exp\left(-\frac{n\varepsilon^{2}}{ct^{2 }}\right).\] In order to prove this result, we rewrite $\hat{A}_{n}(t)$ as ${A_{n}(t)}/{\hat{p}_{n}},$ where \[{A}_{n}(t)=\frac{p(\omega^{(1)})(\xi\cdot Y^{(1)}(t))^{2}+\cdots+p(\omega^{(n) })(\xi\cdot Y^{(n)}(t))^{2}}{nt\mathbb{E}[p]},\] and \[\hat{p}_{n}=\frac{p(\omega^{(1)})+\cdots+p(\omega^{(n)})}{n\mathbb{E}[p]}.\] Both are sums of independent and identically distributed random variables under $\mathbb{P}^{\otimes}_{0}$, thus enabling us to use standard tools from large deviations theory. We start with the following classical but instructing fact. Proposition 4.2.: _There exists $c_{0}>0$ such that, for any $n\in\mathbb{N}$ and any small enough $\varepsilon>0$:_ \[\mathbb{P}^{\otimes}\left[\big{\|}\hat{p}_{n}-1\big{\|}\geqslant\varepsilon \right]\leqslant\exp\left(-n\varepsilon^{2}/c_{0}\right).\] Proof.: Let $\overline{p}(\omega)=p(\omega)-\mathbb{E}[p]$. It suffices to show that, for some $C>0$, (4.1) \[\mathbb{P}^{\otimes}\left[\big{\|}\overline{p}(\omega^{(1)})+\cdots+\overline{p }(\omega^{(n)})\big{\|}\geqslant n\varepsilon\right]\leqslant\exp(-n\varepsilon ^{2}/C).\] Chebyshev’s inequality implies that, for any $\lambda\geqslant 0$, \[\mathbb{P}^{\otimes}\left[\overline{p}(\omega^{(1)})+\cdots+ \overline{p}(\omega^{(n)})\geqslant n\varepsilon\right] \leqslant e^{-n\varepsilon\lambda}\mathbb{E}^{\otimes}\left[\exp(\lambda( \overline{p}(\omega^{(1)})+\cdots+\overline{p}(\omega^{(n)}))\right]\] \[\leqslant e^{-n\varepsilon\lambda}\mathbb{E}\left[\exp(\lambda\overline{p} (\omega))\right]^{n}.\] Using a series expansion of the exponential, one can check that there exists $C>0$ such that, for any $\lambda$ small enough, (4.2) \[\ln\mathbb{E}\left[\exp(\lambda\overline{p}(\omega))\right]\leqslant C\lambda^ {2}.\] As a consequence, for any $\lambda$ small enough, \[\mathbb{P}^{\otimes}\left[\overline{p}(\omega^{(1)})+\cdots+\overline{p}( \omega^{(n)})\geqslant\varepsilon\right]\leqslant\exp\left(-n(\lambda \varepsilon-C\lambda^{2})\right),\] and for $\lambda=\varepsilon/(2C)$, the latter term becomes $\exp(-n\varepsilon^{2}/(4C))$. The event $\overline{p}(\omega^{(1)})+\cdots+\overline{p}(\omega^{(n)})\leqslant-n\varepsilon$ can be handled the same way, and we thus obtain (4.1). ∎ What makes the proof of Proposition 4.2 work is the observation (4.2) that the log-Laplace transform of $\overline{p}=p-\mathbb{E}[p]$ is quadratic close to the origin. In order to prove the corresponding result for $A_{n}(t)$, we will need to control the log-Laplace transform of $\frac{(\xi\cdot Y(t))^{2}}{t}-\sigma_{t}^{2}$ uniformly in $t$. To that end, we use a sharp upper bound on the transition probabilities of the random walk recalled in the following theorem. We refer the reader to [11] or [21, Theorem 14.12] for a proof. Theorem 4.3.: _There exists a constant $c_{1}>0$ such that, for any environement $\omega$ with conductances in $[\alpha,\beta]$, any $t\in\mathbb{N}^{}$ and $x\in\mathbb{Z}^{d}$,_ \[\mathbf{P}^{\omega}_{0}[Y(t)=x]\leqslant\frac{c_{1}}{t^{d/2}}\exp\left(-\frac{ \|x\|^{2}}{c_{1}t}\right).\] From Theorem 4.3 we deduce the following result. Corollary 4.4.: _Let $c_{1}$ be given by Theorem 4.3. For all $\lambda<1/c_{1}$, one has_ \[\sup_{t\in\mathbb{N}^{}}\tilde{\mathbb{E}}_{0}\left[\exp\left(\lambda\frac{\|Y (t)\|^{2}}{t}\right)\right]<+\infty.\] Proof.: Let $\delta=1/c_{1}-\lambda$. By Theorem 4.3, \[\mathbf{E}^{\omega}_{0}\left[e^{\lambda\|Y(t)\|^{2}/t}\right]\leqslant c_{1}{t^{ -d/2}}\sum_{x\in\mathbb{Z}^{d}}e^{-\delta\|x\|^{2}/t}.\] If the sum ranges over all $x\in(\mathbb{N}^{})^{d}$, it is easy to bound it by a convergent integral: \[{t^{-d/2}}\sum_{x\in(\mathbb{N}^{})^{d}}e^{-\delta\|x\|^{2}/t}\leqslant{t^{-d/2 }}\int_{\mathbb{R}_{+}^{d}}e^{-\delta\|x\|^{2}/t}\ {\mathrm{d}}x=\int_{\mathbb{R }_{+}^{d}}e^{-\delta\|x\|^{2}}\ {\mathrm{d}}x.\] By symmetry, the estimate carries over to the sum over all $x\in(\mathbb{Z}^{})^{d}$. The same argument applies for the sum over all $x=(x_{1},\ldots,x_{d})$ having exactly one component equal to $0$, and so on. ∎ The following lemma contains the required uniform control on the log-Laplace transform of $\frac{(\xi\cdot Y(t))^{2}}{t}-\sigma_{t}^{2}$. Lemma 4.5.: _There exist $\lambda_{1}>0$ and $c_{2}$ such that, for any $\lambda<\lambda_{1}$ and any $t\in\mathbb{N}^{}$,_ \[\ln\tilde{\mathbb{E}}_{0}\left[\exp\left(\lambda\left(\frac{(\xi\cdot Y(t))^{2 }}{t}-\sigma_{t}^{2}\right)\right)\right]\leqslant c_{2}\lambda^{2}.\] Proof.: It is sufficient to prove that there exists $c_{3}$ such that, for any $\lambda$ small enough and any $t$, \[\tilde{\mathbb{E}}_{0}\left[\exp\left(\lambda\left(\frac{(\xi\cdot Y(t))^{2}}{ t}-\sigma_{t}^{2}\right)\right)\right]\leqslant 1+c_{3}\lambda^{2}.\] We use the series expansion of the exponential to rewrite this expectation as \[\sum_{k=0}^{+\infty}\frac{\lambda^{k}}{k!}\ \tilde{\mathbb{E}}_{0}\left[\left( \frac{(\xi\cdot Y(t))^{2}}{t}-\sigma_{t}^{2}\right)^{k}\right].\] The term corresponding to $k=0$ is equal to $1$, whereas the term for $k=1$ vanishes. The remaining sum, for $k$ ranging from $2$ to infinity, can be controlled using Corollary 4.4 combined with the bound which follows from the definition of $\sigma_{t}^{2}$ and Jensen’s inequality. ∎ We are now in position to prove Theorem 4.1. Proof of Theorem 4.1.: Starting point is the inequality (4.3) \[\mathbb{P}^{\otimes}_{0}\left[\hat{A}_{n}(t)-\sigma_{t}^{2} \geqslant 2\varepsilon/t\right]\\ \leqslant\mathbb{P}^{\otimes}_{0}\left[A_{n}(t)-\sigma_{t}^{2} \geqslant\varepsilon/t\right]+\mathbb{P}^{\otimes}_{0}\left[A_{n}(t)(\hat{p}_{ n}^{-1}-1)\geqslant\varepsilon/t\right].\] We treat both terms of the r. h. s. separately. For the first one, the key observation is that (recalling the definition of $\tilde{\mathbb{P}}$ given in (1.1)) \[\mathbb{P}^{\otimes}_{0}\left[A_{n}(t)-\sigma_{t}^{2}\geqslant\varepsilon/t \right]=\tilde{\mathbb{P}}^{\otimes}_{0}\left[\frac{(\xi\cdot Y^{(1)}(t))^{2}+ \cdots+(\xi\cdot Y^{(n)}(t))^{2}}{nt}-\sigma_{t}^{2}\geqslant\varepsilon/t \right].\] Let $\lambda>0$. As in the proof of Proposition 4.2, we bound this term using Chebyshev’s inequality: (4.4) \[\begin{split}&\mathbb{P}^{\otimes}_{0}\left[A_{n}(t)-\sigma_{t}^{ 2}\geqslant\varepsilon/t\right]\\ &\qquad\leqslant\tilde{\mathbb{E}}^{\otimes}_{0}\left[\exp\left( \lambda\left(\frac{(\xi\cdot Y^{(1)}(t))^{2}+\cdots+(\xi\cdot Y^{(n)}(t))^{2}} {t}-n\sigma_{t}^{2}\right)\right)\right]\ \exp\left(-\frac{n\lambda\varepsilon }{t}\right)\\ &\qquad\leqslant{\tilde{\mathbb{E}}_{0}\left[\exp\left(\lambda \left(\frac{(\xi\cdot Y(t))^{2}}{t}-\sigma_{t}^{2}\right)\right)\right]}^{n}\ \exp\left(-\frac{n\lambda\varepsilon}{t}\right).\end{split}\] By Lemma 4.5, the r. h. s. of (4.4) is bounded by \[\exp\left(n\left(c_{2}\lambda^{2}-\frac{\lambda\varepsilon}{t}\right)\right),\] for all $\lambda$ small enough. Choosing $\lambda=\varepsilon/2c_{2}t$ (which is small enough for $t$ large enough), we obtain (4.5) \[\mathbb{P}^{\otimes}_{0}\left[A_{n}(t)-\sigma_{t}^{2}\geqslant\varepsilon/t \right]\leqslant\exp\left(-\frac{n\varepsilon^{2}}{4c_{2}t^{2}}\right),\] as needed. We now turn to the second term of the r. h. s. of (4.3). From inequality (4.5), we infer that there exists $M>0$ such that \[\mathbb{P}^{\otimes}_{0}\left[A_{n}(t)\geqslant M\right]\leqslant\exp\left(- \frac{n\varepsilon^{2}}{4c_{2}t^{2}}\right).\] Since $\hat{p}_{n}$ is almost surely bounded by a constant, it is enough to evaluate the probability \[\mathbb{P}^{\otimes}_{0}[\hat{p}_{n}^{-1}-1\geqslant\varepsilon/(Mt)],\] which is controlled by Proposition 4.2. We have thus obtained the required control of the l. h. s. of (4.3). The probability of the symmetric event \[\mathbb{P}^{\otimes}_{0}\left[\sigma_{t}^{2}-\hat{A}_{n}(t)\geqslant 2 \varepsilon/t\right]\] can be handled the same way. ∎ ## 5. Numerical validation In this section, we illustrate on a simple two-dimensional example the sharpness of the estimates of the systematic error and of the random fluctuations obtained in Theorems 3.1 and 4.1. In the numerical tests, each conductivity of $\mathbb{B}$ takes the value $\alpha=1$ or $\beta=4$ with probability $1/2$. In this simple case, the homogenized matrix is given by Dykhne’s formula, namely $A_{\mathrm{hom}}=\sqrt{\alpha\beta}\mathrm{Id}=2\mathrm{Id}$ (see for instance [6, Appendix A]). For the simulation of the random walk, we generate — and store — the environment along the trajectory of the walk. In particular, this requires to store up to a constant times $t$ data. In terms of computational cost, the expansive part of the computations is the generation of the randomness. In particular, to compute one realization of $\hat{A}_{t^{2}}(t)$ costs approximately the generation of $t^{2}\times 4t=4t^{3}$ random variables. A natural advantage of the method is its full scalability: the $t^{2}$ random walks used to calculate a realization of $\hat{A}_{t^{2}}(t)$ are completely independent. We first test the estimate of the systematic error: up to a logarithmic correction, the convergence is proved to be linear in time. In view of Theorem 4.1, typical fluctuations of $t(\hat{A}_{n(t)}(t)-\sigma_{t}^{2})$ are of order no greater than $t/\sqrt{n(t)}$, and thus become negligible when compared with the systematic error as soon as the number $n(t)$ of realizations satisfies $n(t)\gg t^{2}$. We display in Table 1 an estimate of the systematic error obtained with $n(t)=K(t)t^{2}$ realizations. The systematic error is plotted on Figure 1 in function of the time in logarithmic scale. The apparent convergence rate (linear fitting) is $-.85$, which is consistent with Theorem 3.1, which predicts $-1$ and a logarithmic correction. t \| 10 \| 20 \| 40 \| 80 \| 160 \| 320 \| 640 ---\|---\|---\|---\|---\|---\|---\|--- K(t) \| 105 \| 3000 \| 3000 \| 1000 \| 500 \| 100 \| 20 Systematic error \| 1.27E-01 \| 7.43E-02 \| 4.17E-02 \| 2.46E-02 \| 1.26E-02 \| 6.96E-03 \| 3.72E-03 Table 1. Systematic error in function of the final time t for K(t)t2 realizations. <figure><img src="content_image/1103.4591/x1.png"><figcaption>Figure 1. Systematic error in function of the final time t for n(t)=K(t)t2realizations</figcaption></figure> We now turn to the random fluctuations of $\hat{A}_{n(t)}(t)$. Theorem 4.1 gives us a large deviation estimate which essentially says that the fluctuations of $t(\hat{A}_{n(t)}(t)-\sigma_{t}^{2})$ have a Gaussian tail, measured in units of $t/\sqrt{n(t)}$. The Figures 3-5 display the histograms of $t(\hat{A}_{t^{2}}(t)-\sigma_{t}^{2})$ for $t=10,20,40$ and $80$ (with 10000 realizations of $\hat{A}_{t^{2}}(t)$ in each case). As expected, they look Gaussian. <figure><img src="content_image/1103.4591/x2.png"><figcaption>Figure 2. Histogram of the rescaled fluctuations for t=10</figcaption></figure> <figure><img src="content_image/1103.4591/x4.png"><figcaption>Figure 4. Histogram of the rescaled fluctuations for t=40</figcaption></figure> ## Acknowledgements The authors acknowledge the support of INRIA through the “Action de recherche collaborative” DISCO. This work was also supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the “Contrat de Projets Etat Region (CPER) 2007-2013”. ## References * [AKS82] V.V. Anshelevich, K.M. Khanin, and Ya.G. Sinaĭ. Symmetric random walks in random environments. _Comm. Math. Phys._, 85(3), 449–-470, 1982. * [BD10] M.T. Barlow, and J.-D. Deuschel. 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1212.6934	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 24379, "num_imgs": 4, "llama3_tokens_count": 7028 }	[ "content_image/1212.6934/x1.png", "content_image/1212.6934/x2.png", "content_image/1212.6934/x3.png", "content_image/1212.6934/x4.png" ]	# Elementary excitations of ultracold soft-core bosons across the superfluid-supersolid phase transition T. Macrì, F. Maucher, F. Cinti and T. Pohl Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany February 23, 2024 ###### Abstract We investigate the zero-temperature excitation spectrum of two-dimensional soft-core bosons for a wide range of parameters and across the phase transition from a superfluid to a supersolid state. Based on mean field calculations and recent Quantum Monte Carlo results, we demonstrate the applicability of the Bogoliubov-de Gennes equations, even at high interaction strengths where the system forms an insulating cluster crystal. Interestingly, our study reveals that the maximum energy of the longitudinal phonon band in the supersolid phase connects to the maxon energy of the superfluid at the phase transition. pacs: 03.75.Kk,67.85.De,05.30.Jp,67.80.K- A supersolid is a phase of matter that simultaneously accommodates diagonal as well as off-diagonal long-range order, which means that particles self-assemble into a rigid, regular crystal but at the same time can flow superfluidly through the formed solid. More than forty years ago, this peculiar state has been conjectured to emerge in pressurized solid Helium [1; 2; 3], which led to an intense search for supersolidity in such systems [4]. In 2004, experimental evidence for superfluidity in toroidal oscillator measurements [5; 6] has greatly revived interest in supersolid Helium and initiated a recent surge of activity on this problem. Yet, theoretical work has not reached a general consensus [7; 8; 9; 10; 11] as to whether solid ${}^{4}$He can display superfluidity, and very recent experiments [12] are casting doubt on the original interpretation of the measurements [5; 6]. At the same time, ultracold atomic gases have emerged as a promising alternative platform to realize and study continuous-space supersolids in an unambiguous and controlled fashion. Recent work has demonstrated that long-range soft-core interactions between bosonic atoms can be engineered via optical coupling to highly-lying electronic Rydberg states [13; 14; 15; 16; 17; 18; 19; 20] or via light-induced interactions in an optical cavity [21]. Such soft-core interactions can give rise to so-called cluster-solids [22] and cluster-supersolids [13; 15], i.e. crystalline arrangements of atomic clusters or droplets where superfluidity can arise from particle-hopping between the self-assembled droplets. On a more formal level, supersolidity can be understood in terms of the simultaneous breaking of fundamentally different symmetries: (_i_) the breaking of translational symmetry responsible for the crystalline ordering and (_ii_) the breaking a global gauge symmetry that enables long-range phase coherence and thereby superfluidity of the system. A direct consequence is the emergence of Goldstone bosons, i.e gapless modes in the excitation spectrum for each of the broken symmetries. Therefore, the excitation spectrum provides a powerful experimental way to probe supersolidity, e.g. via Bragg scattering [23; 24] and has recently attracted considerable theoretical interest [25; 26; 27; 28; 29; 30; 31]. Most recently, Quantum Monte Carlo [32] simulations have been used to determine the dynamical structure factor of two-dimensional soft-core Bosons [25], while mean field approaches have been applied in numerous works [33; 34; 35; 26; 27; 36], aiming at a simplified description of the zero-temperature physics of these systems. <figure><img src="content_image/1212.6934/x1.png"><figcaption>Figure 1: Superfluid fraction of two dimensional Bosons as a function of thedimensionless interaction strength α=mρV0R40/ℏ2. The points show results ofMonte Carlo simulations for 320 particles with a density of R20ρ=4.4, whereasthe continuous line is a guide for the eye. The superfluid fraction dropsabruptly at α≈13.4 (dotted line), marking a first-order superfluid-supersolidphase transition, in close agreement with the mean field prediction of α=12.7(dashed line). Around α≈22 the systems enters an insulating phase. The insetsshow PIMC snapshots for different α, illustrating the particle density profilein the three different phases. We checked that the results do not change for atemperature range 0.1−0.75 ℏ2/mR20 and particle number 160, 320, 640.</figcaption></figure> Here we present a thorough comparison between these two approaches and show that the excitation spectrum can be accurately described by the Bogoliubov-de Gennes (BdG) equations. Surprisingly, we find quantitative agreement with Monte Carlo simulations, not only in the superfluid and supersolid phases but also in the strongly interacting regime, where superfluidity is destroyed by quantum fluctuations and mean field approaches are generally expected to fail. This finding should prove valuable for future work, as it justifies the use of more efficient mean field calculations, which moreover enable investigations of dynamical processes, such as relaxation phenomena or externally driven systems. Here, we exploit this fact to study the excitation spectrum for a wide range of parameters, and find unexpected features at the superfluid-supersolid phase transition. ## I Ground state properties We consider an ensemble of Bosons confined to two dimensions with with mass $m$ and positions ${\bf q}_{i}$, as described by the Hamiltonian \[\hat{H}=\sum_{i}-\frac{\hbar^{2}}{2m}\nabla^{2}_{i}+\sum_{i<j}V({\bf q}_{i}-{ \bf q}_{j}).\] (1) As a prototype example for soft-core interactions we chose a simple step function potential $V({\bf r})=V_{0}\Theta(R_{0}-r)$, where $\Theta(r)$ denotes the Heaviside function and $V_{0}$ and $R_{0}$ define the strength and range of the interaction potential. Upon scaling lengths by $R_{0}$ and energies by $\hbar^{2}/mR_{0}^{2}$, the zero temperature physics, determined by eq.(1), depends only on two dimensionless parameters: an effective interaction strength $\alpha^{\prime}=V_{0}mR_{0}^{2}/\hbar^{2}$ and the dimensionless density $\rho R_{0}^{2}$. At the mean field level, this set of parameters can be further reduced. In this case, the system dynamics is described by a non-local Gross-Pitaevskii equation (GPE), which, in terms of the rescaled variables, can be written as \[i\partial_{t}\psi_{0}({\bf r})=\left[-\frac{\nabla^{2}}{2}+\alpha\int{\rm d}{ \bf r}^{\prime}U({\bf r}-{\bf r}^{\prime})\|\psi_{0}({\bf r}^{\prime})\|^{2} \right]\psi_{0}({\bf r})\;,\] (2) where ${\bf r}={\bf q}/R_{0}$, $U({\bf r})=\Theta(1-r)$, and $\alpha=m\rho V_{0}R_{0}^{4}/\hbar^{2}$ is a dimensionless interaction strength that solely determines the dynamics and ground state properties. Such a mean field treatment is expected to be valid in the limit of weak interactions, reached by decreasing $\alpha^{\prime}$ and increasing the density $\rho$ such that the product $\alpha^{\prime}\rho$ stays finite. In this regime the zero temperature physics is determined only by the effective interaction strength $\alpha$, which greatly simplifies the analysis of the underlying phase diagram. For small $\alpha$ the system is in a homogenous superfluid phase whose energy $\pi\alpha$ follows directly from eq.(2). We describe the modulated supersolid state by a variational wave function that is composed of localized Gaussians, arranged on a triangular lattice. Their dispersion $\sigma$ and the lattice constant $a$ is obtained by minimizing the total energy. This simple analysis shows that density modulations become energetically favorable for $\alpha\geq 12.65$, marking a first order phase transition to a cluster supersolid state composed of small superfluid droplets [13; 14; 32]. The droplets become more localized as the interaction strength increases where both their size as well as the lattice spacing decreases from $\sigma=0.39$ and $a=1.51$ at the phase transition to $\sigma=0.22$ and $a=1.40$ for $\alpha=40$. <figure><img src="content_image/1212.6934/x2.png"><figcaption>Figure 2: Excitation spectrum in the superfluid phase according to eq.(6)(line) compared to the PIMC data (circles) of Ref. smb12 .</figcaption></figure> In order to test these predictions we additionally performed path integral Monte Carlo (PIMC) simulations at finite temperature based on the continuous-space Worm algorithm [37; 38], carefully extrapolating the zero temperature behavior. Fig.1 shows the obtained superfluid fraction as a function of $\alpha$ for a total number of $320$ particles with a density of $\rho R_{0}^{2}=4.4$ corresponding to eight particles per droplet. One finds a first order phase transition at $\alpha\approx 13.4$, signaled by an abrupt drop of the superfluid fraction from unity in the homogenous superfluid phase to $\sim 0.6$ at $\alpha=13.6$ in remarkable agreement with the above mean field prediction. Also consistent with the above discussion, a further increase of $\alpha$ leads to a stronger localization of the droplets, and, hence, a drop of the superfluid fraction [3]. Around $\alpha\approx 22$ the particle density between the droplets decreases to a point where quantum fluctuations destroy phase coherence between individual clusters such that the system enters an insulating crystal of superfluid droplets without long range off diagonal order [15]. While this transition can evidently not be captured by eq.(2), the meanfield theory nevertheless yields an accurate description of the excitation spectrum in the insulating phase, as discussed below. <figure><img src="content_image/1212.6934/x3.png"><figcaption>Figure 3: (color online) Mean field spectra (lines) at α=16.93 (a) andα=30.62 (b) obtained from eqs.(II) along the three symmetry directions of theBrillouin zone [see inset of panel (b)]. The symbols represent the PIMC dataof Ref. smb12 for longitudinal excitations computed along the direction Γ−M−Γin the first two Brillouin zones. (c-e) Density fluctuationsΔρnk(r)=\|un,k(r)−vn,k(r)\|2 and (f-h) phase fluctuationsΔφnk(r)=\|un,k(r)+vn,k(r)\|2 computed at α=16.93 and k=→ΓK/10 (directed alongthe horizontal axis) for the lowest (continuous line) band (c,f), middle(dashed) band (d,g), and higher (dot-dashed) gapless band (e,h).</figcaption></figure> ## II Excitations The excitation spectrum is obtained by expanding the field, $\hat{\psi}=\psi_{0}+\delta\hat{\psi}$, around the groundstate. Substituting small perturbations of the form \[\delta\hat{\psi}(\mathbf{r},t)=e^{-i\mu t}\sum_{n}\left[u_{n}(\mathbf{r})e^{-i \omega t}a_{n}-v_{n}^{}(\mathbf{r})e^{i\omega t}a_{n}^{\dagger}\right]\;,\] (3) where $n$ labels the bands, into the GPE (2) yields to leading order in $\delta\hat{\psi}$ the familiar BdG equations for the Bogoliubov modes $u_{n}$ and $v_{n}$. We solve these equations in real space by expanding the modes into Bloch waves \[u_{n}({\bf r}) = u_{n,\bf k}({\bf r})e^{i{\bf k}\cdot{\bf r}}\] \[v_{n}({\bf r}) = v_{n,\bf k}({\bf r})e^{i{\bf k}\cdot{\bf r}}\;,\] (4) where the functions $u_{n,\bf k}({\bf r})$ and $v_{n,\bf k}({\bf r})$ obey the translational symmetry of the underlying groundstate, i.e. a continuous translational symmetry in the superfluid phase and a discrete triangular lattice periodicity in the supersolid phase. This ansatz leads to the following set of equations \[\left(\frac{k^{2}}{2}-i\mathbf{k}\cdot\mathbf{\nabla}-\frac{1}{2} \nabla^{2}+\alpha A({\bf r})-\mu\right)u_{n,\bf k}(\mathbf{r})+\alpha\psi_{0}( \mathbf{r})\int d\mathbf{r^{\prime}}U(\mathbf{r-r^{\prime}})\psi_{0}(\mathbf{r ^{\prime}})e^{i{\bf k}\cdot({\bf r}^{\prime}-{\bf r})}[u_{n,\bf k}({\bf r}^{ \prime})-v_{n,\bf k}({\bf r}^{\prime})] = \omega u_{n,\bf k}(\mathbf{r})\] \[-\left(\frac{k^{2}}{2}-i\mathbf{k}\cdot\mathbf{\nabla}-\frac{1}{2 }\nabla^{2}+\alpha A({\bf r})-\mu\right)v_{n,\bf k}(\mathbf{r})+\alpha\psi_{0} (\mathbf{r})\int d\mathbf{r^{\prime}}U(\mathbf{r-r^{\prime}})\psi_{0}(\mathbf{ r^{\prime}})e^{i{\bf k}\cdot({\bf r}^{\prime}-{\bf r})}[u_{n,\bf k}({\bf r}^{ \prime})-v_{n,\bf k}({\bf r}^{\prime})] = \omega v_{n,\bf k}(\mathbf{r})\] for $u_{n,\bf k}$ and $v_{n,\bf k}$, where $A({\bf r})=\int d\mathbf{r^{\prime}}U(\mathbf{r^{\prime}-r})\left\|\psi_{0}( \mathbf{r^{\prime}})\right\|^{2}$. Recently, the excitation spectrum has been investigated by calculating the dynamical structure factor via PIMC simulations [25], using the so-called GIFT approach [39]. In the following, these first-principle results will be employed to assess the predictive power of eqs.(II). In the homogeneous superfluid phase, eqs.(II) can be solved analytically and yield the familiar Bogoliubov spectrum \[\omega=\sqrt{\frac{k^{2}}{2}\left(\frac{k^{2}}{2}+2\alpha\tilde{U}({\bf k}) \right)},\] (6) where $\tilde{U}({\bf k})=2\pi J_{1}(k)/k$ and $J_{1}$ denotes the Bessel function of the first kind. For $\alpha\geq 5.03$ the spectrum develops a roton-maxon structure and roton softening occurs at $\alpha=14.74$, preceded by the supersolid phase transition at $\alpha=12.7$. Fig. 2 illustrates the roton-maxon spectrum of the superfluid phase for an interaction strength of $\alpha=11.86$, i.e. slightly below the superfluid-supersolid phase transition. Eq.(6) is in excellent agreement with the numerical PIMC results of Ref. [25], indicating that the system can indeed be described as a weakly coupled fluid. In the supersolid phase, a reliable calculation of the excitation spectrum requires accurate knowledge of the ground state and its chemical potential. To this end we iterate the time-independent GPE [40], $\mu\psi_{0}({\bf r})=[-\nabla^{2}/2+\alpha A({\bf r})]\psi_{0}({\bf r})$, starting from our optimized variational wave function and employing the same grid used to solve eqs.(II). In Fig.(3) we show the obtained spectrum of low-energy excitations for two values of the interaction strength that lie in the supersolid ($\alpha=16.93$) and in the insulating crystal phase ($\alpha=30.62$), respectively. The figure shows the excitation energies along the three symmetry axes of the Brillouin zone corresponding to the underlying triangular lattice. We find three gapless bands, i.e. three Goldstone modes reflecting the symmetries that are broken in the supersolid phase [26]. In addition to the ”superfluid band” due to the breaking of global gauge symmetry, there are two bands corresponding to longitudinal and transverse phonon excitations of the two-dimensional lattice. While the latter were not accessible by the PIMC calculations of Ref. [25], we find good agreement for the two longitudinal modes in the supersolid phase ($\alpha=16.92$). Somewhat surprisingly, even in the insulating phase, eq.(II) yields excellent agreement for the longitudinal phonon mode, despite its evident inability to describe the break-down of global superfluidity. This indicates that each individual droplet maintains a high condensate fraction despite the apparent lack of global phase coherence between the crystalline ordered droplets (see Fig.1). A proper identification of each band can be done by computing local fluctuations on top of the mean field solution $\psi_{0}(r)$[43]. The substitution $\hat{\psi}(\mathbf{r})=e^{i\delta\hat{\varphi}(\mathbf{r})}\sqrt{\|\psi_{0}( \mathbf{r})\|^{2}+\delta\hat{\rho}(\mathbf{r})}$ allows us to identify local density and phase fluctuations: \[\begin{array}[]{ccl}\left<\delta\hat{\rho}^{\dagger}(\bf r)\delta\hat{\rho}( \bf r)\right>/\left\|\psi_{0}(\mathbf{r})\right\|^{2}&=&\sum_{n,\bf k}\left\|u_{n ,\bf{k}}(\mathbf{r})-v_{n,\bf{k}}(\mathbf{r})\right\|^{2}\\ \left<\delta\hat{\varphi}^{\dagger}(\bf r)\delta\hat{\varphi}(\bf r)\right> \cdot 4\left\|\psi_{0}(\mathbf{r})\right\|^{2}&=&\sum_{n,\bf k}\left\|u_{n,\bf{k} }(\mathbf{r})+v_{n,\bf{k}}(\mathbf{r})\right\|^{2}\end{array}.\] (7) Fig.3 (c-h) shows the contributions to (7) for one specific value of $\mathbf{k}$ for each of the three gapless bands at $\alpha=16.93$. One clearly distinguishes the transverse band from the direction of the fluctuations, orthogonal to the perturbing vector $\bf k$. The contribution of this band to phase fluctuations is strongly suppressed. The first and third band both contribute to density and phase fluctuations with different weight tough. The first band is mostly responsible for phase whereas the third to density fluctuations. Therefore the lower band can be associated to the superfluid response of the system, whereas the other two to the classical collective excitations of the crystal. <figure><img src="content_image/1212.6934/x4.png"><figcaption>Figure 4: (a) Linear dispersion v=∂ω/∂k\|k=0(ℏ2/mR0) of the gapless modes and(b) their energy ω(ℏ2/mR20) at distinct momenta as a function of α. In thesuperfluid phase (α<12.7) panel (b) shows the maxon energy (line) and theexcitation energies (circles) of the three gapless modes at the M-point (ωM)and of the longitudinal and transverse phonon band at the K-point (ωK) in thesupersolid phase (see Fig.3).</figcaption></figure> Having demonstrated the accuracy of the mean field description we can now exploit its efficiency to study the mode structure over a wider range of interaction strengths and, in particular, across the superfluid-supersolid phase transition. The results are summarized in Fig.4, which shows the linear dispersion $v=\partial\omega/\partial k\|_{k=0}$ as well as the excitation frequency at several distinct momenta. As expected for a first order phase transition, the linear dispersion exhibits a discontinuous jump (Fig.4a), reflecting the sudden onset of finite density modulations at the phase transition. On the other hand, the excitation energy shows a different and rather unexpected behavior. Fig.4b shows the $\alpha$-dependence of the maxon energy in the superfluid phase along with the excitation energy of the three gapless bands at the $M$-symmetry-point of the Brillouin zone (see Fig.3). At this symmetry point the longitudinal phonon mode provides the maximum energy in the entire reciprocal space. Interestingly, we observe that the maxon energy of the superfluid merges into the latter at the superfluid-supersolid phase transition, i.e. does not show a discontinuous jump. Remarkably, at the phase transition, this energy also coincides with the excitation energy at the other symmetry point ($K$-point) where, in addition, the longitudinal and transverse phonon modes are degenerate. ## III Conclusion Based on PIMC simulations and meanfield calculations, we studied the zero-temperature physics of two-dimensional soft-core bosons and, in particular, their excitation spectrum at the phase transition to a supersolid droplet phase. The close match between both methods not only demonstrates the predictive power of the mean field approach but also attests to the accuracy of the GIFT method [39] for extracting dynamical properties from Quantum Monte Carlo simulations. Our spectra are consistent with recent calculations of Refs. [26; 27], but differ qualitatively from the results of Ref. [41], where only one gapless mode has been found in the supersolid phase of soft-core dipoles. A careful scan of the particle interaction revealed that the maxon energy of the superfluid phase merges into the energy of longitudinal phonons at both reciprocal symmetry points of the supersolid phase. This degeneracy at the phase transition may suggest that the maxon part of the superfluid excitations plays a more significant role than thus far anticipated – an implication which calls for further investigation. Future studies of other types of soft-core interactions [13; 15; 21] will clarify whether this behavior is of generic nature or a mere consequence of the step-function potential considered in this work. Moreover, PIMC simulations [25] would allow to elucidate the effects of correlations, i.e. to address the question whether the found connection between maxon and phonon excitations persists in the strong coupling regime beyond the validity of the BdG eqs.(II). Along these lines, the behavior of Goldstone modes for related scenarios, e.g., at the crystallization point of dipolar systems [42] or at the superfluid-solid transition of Helium [11; 44], suggests itself as an interesting question for future studies. 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1610.06986	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 28831, "num_imgs": 1, "llama3_tokens_count": 11235 }	[ "content_image/1610.06986/x1.png" ]	# ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays with perturbative QCD approach Junfeng Sun Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China Qingxia Li Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China Yueling Yang Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China Haiyan Li Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China Qin Chang Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China Zhiqing Zhang Department of Physics, Henan University of Technology, Zhengzhou 450001, China ###### Abstract With the potential prospects of the ${\Upsilon}(1S)$ at high-luminosity dedicated heavy-flavor factories, the bottom-changing ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ weak decays are studied with the pQCD approach. It is found that branching ratio for the color-favored and CKM-favored ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$ decay can reach up to ${\cal O}(10^{-11})$. So the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$ decay might be measured promisingly by the future experiments. pacs: 13.25.Gv 12.39.St 14.40.Pq ## I Introduction The ${\Upsilon}(1S)$ particle is the ground vector bottomonia (bound states of $b\bar{b}$) with well established quantum number of $I^{G}J^{PC}$$=$$0^{-}1^{--}$[1]. The mass of the ${\Upsilon}(1S)$ particle, $m_{{\Upsilon}(1S)}$$=$$9460.30{\pm}0.26$ MeV [1], is less than the kinematic $B\bar{B}$ threshold. The ${\Upsilon}(1S)$ particle, in a close analogy with $J/{\psi}$, decay primarily through the annihilation of the $b\bar{b}$ pairs into three gluons, followed by evolution of gluons into hadrons, glueballs, hybrid, multiquark and other exotic states. The hadronic ${\Upsilon}(1S)$ decay offers an ideal place to reap the properties of the invisible gluons and of the quark-gluon coupling [2]. It is well known that strong decay of the ${\Upsilon}(1S)$ particle is suppressed by the phenomenological Okubo-Zweig-Iizuka (OZI) rules [3; 4; 5], so electromagnetic and radiative transitions become competitive. Besides, the ${\Upsilon}(1S)$ particle can also decay via the weak interactions within the standard model, although the branching ratio is very small, about $2/{\tau}_{B}{\Gamma}_{{\Upsilon}(1S)}$${\sim}$${\cal O}(10^{-8})$[1], where ${\tau}_{B}$ and ${\Gamma}_{{\Upsilon}(1S)}$ are the lifetime of the $B_{u,d,s}$ meson and the decay width of the ${\Upsilon}(1S)$ particle, respectively. In this paper, we will estimate the branching ratios for the bottom-changing nonleptonic ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ weak decays with perturbative QCD (pQCD) approach [6; 7; 8]. The motivation is listed as follows. From the experimental point of view, (1) over $10^{8}$${\Upsilon}(1S)$ samples have been accumulated at Belle [9]. Many more upsilons could be collected with great precision at the forthcoming SuperKEKB and the running upgraded LHC. The abundant ${\Upsilon}(1S)$ samples provide a golden opportunity to search for the ${\Upsilon}(1S)$ weak decays which in some cases might be detectable. Theoretical studies on the ${\Upsilon}(1S)$ weak decays are just necessary to offer a ready reference. (2) For the two-body ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays, final states with opposite charges have definite energies and momenta in the rest frame of the ${\Upsilon}(1S)$ particle. Besides, identification of a single explicitly flavored $B_{c}$ meson is free from inefficiently double tagging above the $B\bar{B}$ threshold [10], and can provide a conclusive evidence of the ${\Upsilon}(1S)$ weak decay. Of course, small branching ratios make the observation of the ${\Upsilon}(1S)$ weak decays extremely challenging, and evidences of an abnormally large production rate of single $B_{c}$ mesons in the ${\Upsilon}(1S)$ decay might be a hint of new physics [10]. From the theoretical point of view, the bottom-changing upsilon weak decays permit one to cross check parameters obtained from $B$ meson decay. The color-favored ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays have been estimated with the naive factorization (NF) approximation in previous works [10; 11; 12]. An obvious deficiency of NF approximation is the absence of strong phases and the renormalization scale from hadronic matrix elements (HME). Recently, several attractive methods have been developed to evaluate HME, such as pQCD [6; 7; 8], the QCD factorization (QCDF) [13; 14; 15] and soft and collinear effective theory [16; 17; 18; 19], which could give reasonable explanation for many measurements on $B_{u,d}$ hadronic decays. The ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays are calculated at the next-to-leading (NLO) order with the QCDF approach [691261 ]. In this paper, the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ weak decays will be evaluated with the pQCD approach to check the consistency of prediction on branching ratios among different models. This paper is organized as follows. In section II, we present the theoretical framework and the amplitudes for the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays with the pQCD approach. Section III is devoted to numerical results and discussion. The last section is our summary. ## II theoretical framework ### The effective Hamiltonian The effective Hamiltonian responsible for the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays is [9512380 ] (1) where $G_{F}$$=$$1.166{\times}10^{-5}\,{\rm GeV}^{-2}$[1] is the Fermi coupling constant; the Cabibbo-Kabayashi-Maskawa (CKM) factors are expanded as a power series in the Wolfenstein parameter ${\lambda}$${\sim}$$0.2$[1], \[V_{cb}V_{ud}^{\ast} = A{\lambda}^{2}-\frac{1}{2}A{\lambda}^{4}-\frac{1}{8}A{\lambda}^{ 6}+{\cal O}({\lambda}^{8}),\] (2) \[V_{cb}V_{us}^{\ast} = A{\lambda}^{3}+{\cal O}({\lambda}^{8}).\] (3) The Wilson coefficients $C_{1,2}(\mu)$ summarize the physical contributions above scales of ${\mu}$, and have properly been calculated to the NLO order with the renormalization group improved perturbation theory. The local operators are defined as follows. \[Q_{1} = [\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\alpha}][\bar{q }_{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})u_{\beta}],\] (4) \[Q_{2} = [\bar{c}_{\alpha}{\gamma}_{\mu}(1-{\gamma}_{5})b_{\beta}][\bar{q} _{\beta}{\gamma}^{\mu}(1-{\gamma}_{5})u_{\alpha}],\] (5) where ${\alpha}$ and ${\beta}$ are color indices and the sum over repeated indices is understood. From the effective Hamiltonian Eq.(1), it can be easily seen that only tree operators with coupling strength proportional to the CKM element $V_{cb}$ contribute to the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays, and there is no pollution from penguin and annihilation contributions. ### Hadronic matrix elements To obtain the decay amplitudes, the remaining works are how to calculate accurately hadronic matrix elements of local operators. Using the Lepage-Brodsky approach for exclusive processes [22], HME could be expressed as the convolution of hard scattering subamplitudes containing perturbative contributions with the universal wave functions reflecting the nonperturbative contributions. Sometimes the high-order corrections to HME produce collinear and/or soft logarithms based on collinear factorization approximation, for example, the spectator scattering amplitudes within the QCDF framework [15]. The pQCD approach advocates that [6; 7; 8] this inconsistent treatment on HME could be smeared by retaining the transverse momentum of quarks and introducing the Sudakov factor. The decay amplitudes could be factorized into three parts: the “harder” effects incorporated into the Wilson coefficients $C_{i}$, the process-dependent heavy quark decay subamplitudes $H$, and the universal wave functions ${\Phi}$; and are written as \[{\int}dx\,db\,C_{i}(t)H(t,x,b){\Phi}(x,b)e^{-S},\] (6) where $t$ is a typical scale, $x$ is the longitudinal momentum fraction of the valence quark, $b$ is the conjugate variable of the transverse momentum, and $e^{-S}$ is the Sudakov factor. ### Kinematic variables The light cone kinematic variables in the ${\Upsilon}(1S)$ rest frame are defined as follows. \[p_{{\Upsilon}}\,=\,p_{1}\,=\,\frac{m_{1}}{\sqrt{2}}(1,1,0),\] (7) \[p_{B_{c}}\,=\,p_{2}\,=\,(p_{2}^{+},p_{2}^{-},0),\] (8) \[p_{\pi(K)}\,=\,p_{3}\,=\,(p_{3}^{-},p_{3}^{+},0),\] (9) \[k_{i}\,=\,x_{i}\,p_{i}+(0,0,\vec{k}_{i{\perp}}),\] (10) \[{\epsilon}_{\Upsilon}^{\parallel}\,=\,\frac{1}{\sqrt{2}}(1,-1,0),\] (11) \[n_{+}=(1,0,0),\quad n_{-}=(0,1,0),\] (12) where $x_{i}$ and $\vec{k}_{i{\perp}}$ are the longitudinal momentum fraction and transverse momentum of the light valence quark, respectively; ${\epsilon}_{\Upsilon}^{\parallel}$ is the longitudinal polarization vector of the ${\Upsilon}(1S)$ particle; $n_{+}$ and $n_{-}$ are the positive and negative null vectors, respectively. The notation of momentum is displayed in Fig.1(a). The relations among these kinematic variables are \[p_{i}^{\pm}\,=\,\frac{E_{i}\,{\pm}\,p}{\sqrt{2}},\] (13) \[p\,=\,\frac{\sqrt{{\lambda}(m_{1}^{2},m_{2}^{2},m_{3}^{2})}}{2\,m_{1}},\] (14) \[s\,=\,2\,p_{2}{\cdot}p_{3}\,=\ m_{1}^{2}-m_{2}^{2}-m_{3}^{2},\] (15) \[t\,=\,2\,p_{1}{\cdot}p_{2}\,=\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2},\] (16) \[u\,=\,2\,p_{1}{\cdot}p_{3}\,=\ m_{1}^{2}-m_{2}^{2}+m_{3}^{2},\] (17) \[t+u-s\,=\,m_{1}^{2}+m_{2}^{2}+m_{3}^{2},\] (18) \[s\,t+s\,u-u\,t\,=\,{\lambda}(m_{1}^{2},m_{2}^{2},m_{3}^{2}),\] (19) \[{\lambda}(a,b,c)\,=\,a^{2}+b^{2}+c^{2}-2\,a\,b-2\,a\,c-2\,b\,c,\] (20) where $p$ is the common momentum of final states; $m_{1}$, $m_{2}$ and $m_{3}$ denote the masses of the ${\Upsilon}(1S)$, $B_{c}$ and ${\pi}(K)$ mesons, respectively. ### Wave functions With the notation in [23; 24; 25], the explicit definitions of matrix elements of diquark operators sandwiched between vacuum and the longitudinally polarized ${\Upsilon}(1S)$, the double-heavy pseudoscalar $B_{c}$, the light pseudoscalar $P$ ($=$${\pi}$, $K$) are \[{\langle}0{\|}b_{i}(z)\bar{b}_{j}(0){\|}{\Upsilon}(p_{1},{\epsilon}_{\parallel}) {\rangle}\,=\,\frac{1}{4}f_{\Upsilon}{\int}dx_{1}\,e^{-ix_{1}p_{1}{\cdot}z} \Big{\{}\!\!\not{\epsilon}_{\parallel}\Big{[}m_{1}\,{\phi}_{\Upsilon}^{v}(x_{1 })-\!\!\not{p}_{1}\,{\phi}_{\Upsilon}^{t}(x_{1})\Big{]}\Big{\}}_{ji},\] (21) \[{\langle}B_{c}^{+}(p_{2}){\|}\bar{c}_{i}(z)b_{j}(0){\|}0{\rangle}\,=\,\frac{i}{4 }f_{B_{c}}{\int}dx_{2}\,e^{ix_{2}p_{2}{\cdot}z}\,\Big{\{}{\gamma}_{5}\Big{[}\! \!\not{p}_{2}+m_{2}\Big{]}{\phi}_{B_{c}}(x_{2})\Big{\}}_{ji},\] (22) \[{\langle}P(p_{3}){\|}u_{i}(z)\bar{q}_{j}(0){\|}0{\rangle}\] (23) \[= \frac{i}{4}f_{P}{\int}dx_{3}\,e^{ix_{3}p_{3}{\cdot}z}\Big{\{}{ \gamma}_{5}\Big{[}\!\!\not{p}_{3}\,{\phi}_{P}^{a}(x_{3})+{\mu}_{P}{\phi}_{P}^{ p}(x_{3})-{\mu}_{P}(\!\not{n}_{-}\!\!\not{n}_{+}\!-\!1)\,{\phi}_{P}^{t}(x_{3}) \Big{]}\Big{\}}_{ji},\] where $f_{\Upsilon}$, $f_{B_{c}}$, $f_{P}$ are decay constants, ${\mu}_{P}$$=$$m_{3}^{2}/(m_{u}+m_{q})$ and $q$$=$$d(s)$ for ${\pi}(K)$ meson. The leading twist distribution amplitudes of light pseudoscalar ${\pi}$, $K$ mesons are defined in terms of Gegenbauer polynomials [25]: \[{\phi}_{P}^{a}(x)=6\,x\bar{x}\Big{\{}1+\sum\limits_{n=1}^{\infty}a_{n}^{P}\,C_ {n}^{3/2}(x-\bar{x})\Big{\}},\] (24) where $\bar{x}$$=$$1$$-$$x$; $a_{n}^{P}$ and $C_{n}^{3/2}(z)$ are Gegenbauer moment and polynomials, respectively; $a^{\pi}_{i}$$=$$0$ for $i$$=$ 1, 3, 5, ${\cdots}$ due to the $G$-parity invariance of the pion distribution amplitudes. Because of $m_{{\Upsilon}(1S)}$${\simeq}$$2m_{b}$ and $m_{B_{c}}$${\simeq}$$m_{b}$$+$$m_{c}$, both ${\Upsilon}(1S)$ and $B_{c}$ systems are nearly nonrelativistic, which can play the same role in understanding hadronic dynamics as the positronium and hydrogen atom in understanding the atomic physics [26]. Nonrelativistic quantum chromodynamics (NRQCD) [27; 28; 29] and Schrödinger equation can be used to describe their spectrum and thus one can learn about the interquark binding forces responsible for these states [2]. The eigenfunction of the time-independent Schrödinger equation with scalar harmonic oscillator potential corresponding to the quanta $nL$$=$$1S$ is written as \[{\phi}(\vec{k})\ {\sim}\ e^{-\vec{k}^{2}/2{\beta}^{2}},\] (25) where the parameter ${\beta}$ determines the average transverse momentum, i.e., ${\langle}1S{\|}\vec{k}^{2}_{\perp}{\|}1S{\rangle}$$=$${\beta}^{2}$. According to the NRQCD power counting rules [27], the characteristic magnitude of the momentum is order of $Mv$, where $M$ is the mass of the heavy quark with typical velocity $v$${\sim}$${\alpha}_{s}(M)$. Thus we will take ${\beta}$$=$$M{\alpha}_{s}(M)$ in our calculation. Employing the substitution ansatz [30], \[\vec{k}^{2}\ {\to}\ \frac{1}{4}\sum\limits_{i}\frac{\vec{k}_{i\perp}^{2}+m_{q_ {i}}^{2}}{x_{i}},\] (26) where $x_{i}$, $\vec{k}_{i\perp}$, $m_{q_{i}}$ are the longitudinal momentum fraction, transverse momentum, mass of the light valence quark, respectively, with the relations ${\sum}x_{i}$$=$$1$ and $\sum\vec{k}_{i\perp}$$=$$0$. Integrating out $\vec{k}_{i\perp}$ and combining with their asymptotic forms, one can obtain \[{\phi}_{B_{c}}(x)=A\,x\bar{x}\,{\exp}\Big{\{}-\frac{\bar{x}\,m_{c}^{2}+x\,m_{b }^{2}}{8\,{\beta}_{2}^{2}\,x\,\bar{x}}\Big{\}},\] (27) \[{\phi}_{\Upsilon}^{v}(x)=B\,x\bar{x}\,{\exp}\Big{\{}-\frac{m_{b}^{2}}{8\,{ \beta}_{1}^{2}\,x\,\bar{x}}\Big{\}},\] (28) \[{\phi}_{\Upsilon}^{t}(x)=C\,(x-\bar{x})^{2}\,{\exp}\Big{\{}-\frac{m_{b}^{2}}{8 \,{\beta}_{1}^{2}\,x\,\bar{x}}\Big{\}},\] (29) where ${\beta}_{i}$$=$${\xi}_{i}{\alpha}_{s}({\xi}_{i})$ with ${\xi}_{i}$$=$$m_{i}/2$; parameters $A$, $B$, $C$ are the normalization coefficients satisfying the conditions \[{\int}_{0}^{1}dx\,{\phi}_{B_{c}}(x)=1,\quad{\int}_{0}^{1}dx\,{\phi}_{\Upsilon} ^{v,t}(x)=1.\] (30) ### Decay amplitudes The Feynman diagrams for ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$ decay are shown in Fig.1, where (a) and (b) are factorizable topology; (c) and (d) are nonfactorizable topology. <figure><img src="content_image/1610.06986/x1.png"><figcaption>Figure 1: Feynman diagrams for the Υ → Bcπ decay with the pQCD approach.</figcaption></figure> With the master formula Eq.(6), the decay amplitudes of ${\Upsilon}(1S)$${\to}$$B_{c}P$ ($P$$=$${\pi}$ and $K$) decay can be written as \[{\cal A}({\Upsilon}(1S){\to}B_{c}P)=\sqrt{2}G_{F}\frac{{\pi}C_{F}}{N}V_{cb}V_{ uq}^{\ast}\,m_{\Upsilon}^{3}\,pf_{\Upsilon}f_{B_{c}}f_{P}\!\!\!\!\sum\limits_{ i=a,b,c,d}\!\!\!\!{\cal A}_{\rm Fig.\ref{fig1}(i)},\] (31) where $C_{F}$$=$$4/3$ and the color number $N$$=$$3$. The explicit expressions of ${\cal A}_{\rm Fig.\ref{fig1}(i)}$ are \[{\cal A}_{\rm Fig.\ref{fig1}(a)}\,=\,{\int}_{0}^{1}dx_{1}{\int}_{ 0}^{\infty}b_{1}db_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{2}db_{2}\,{ \alpha}_{s}(t_{a})\,a_{1}(t_{a})\,E_{a}(t_{a})\] (32) \[{\cal A}_{\rm Fig.\ref{fig1}(b)}\,=\,{\int}_{0}^{1}dx_{1}{\int}_{ 0}^{\infty}b_{1}db_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{2}db_{2}\,{ \alpha}_{s}(t_{b})\,a_{1}(t_{b})\,E_{b}(t_{b})\] (33) \[{\times}{\phi}_{B_{c}}(x_{2})\,H_{b}(x_{1},x_{2},b_{2},b_{1})\, \Big{\{}{\phi}_{\Upsilon}^{v}(x_{1})\,\Big{[}2r_{2}r_{c}-r_{2}^{2}x_{1}-r_{3}^ {2}\bar{x}_{1}\Big{]}\] \[+{\phi}_{\Upsilon}^{t}(x_{1})\,\Big{[}2r_{2}x_{1}-r_{c}\Big{]} \Big{\}},\] \[{\cal A}_{\rm Fig.\ref{fig1}(c)}\,=\,{\int}_{0}^{1}dx_{1}{\int}_{ 0}^{\infty}db_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^ {1}dx_{3}{\int}_{0}^{\infty}b_{3}db_{3}\,{\delta}(b_{1}-b_{2})\] (34) \[{\times}{\alpha}_{s}(t_{c})\,\frac{C_{2}(t_{c})}{N}\,E_{c}(t_{c}) \,{\phi}_{B_{c}}(x_{2})\,{\phi}_{P}^{a}(x_{3})\,H_{c}(x_{1},x_{2},x_{3},b_{2}, b_{3})\] \[{\times}\Big{\{}{\phi}_{\Upsilon}^{v}(x_{1})\Big{[}\frac{t\,(x_{1 }-x_{3})}{m_{1}^{2}}+2\,r_{2}^{2}\,(x_{3}-x_{2})\Big{]}+{\phi}_{\Upsilon}^{t}( x_{1})\,r_{2}\,(x_{2}-x_{1})\Big{\}},\] \[{\cal A}_{\rm Fig.\ref{fig1}(d)}\,=\,{\int}_{0}^{1}dx_{1}{\int}_{ 0}^{\infty}db_{1}{\int}_{0}^{1}dx_{2}{\int}_{0}^{\infty}b_{2}db_{2}{\int}_{0}^ {1}dx_{3}{\int}_{0}^{\infty}b_{3}db_{3}\,{\delta}(b_{1}-b_{2})\] (35) \[{\times}{\alpha}_{s}(t_{d})\,\frac{C_{2}(t_{d})}{N}\,E_{d}(t_{d}) \,{\phi}_{B_{c}}(x_{2})\,{\phi}_{P}^{a}(x_{3})\,H_{d}(x_{1},x_{2},x_{3},b_{2}, b_{3})\] \[{\times}\Big{\{}{\phi}_{\Upsilon}^{v}(x_{1})\frac{s\,(\bar{x}_{2} -x_{3})}{m_{1}^{2}}+{\phi}_{\Upsilon}^{t}(x_{1})\,r_{2}\,(x_{2}-x_{1})\Big{\}},\] where ${\alpha}_{s}$ is the QCD coupling; $a_{1}$$=$$C_{1}$$+$$C_{2}/N$; $C_{1,2}$ is the Wilson coefficients; $r_{i}$$=$$m_{i}/m_{1}$. It can be easily seen that the nonfactorizable contributions ${\cal A}_{\rm Fig.\ref{fig1}(c,d)}$ are color-suppressed with respect to the factorizable contributions ${\cal A}_{\rm Fig.\ref{fig1}(a,b)}$. The typical scales $t_{i}$ and the Sudakov factor $E_{i}$ are defined as \[t_{a(b)}={\max}(\sqrt{-{\alpha}_{g}},\sqrt{-{\beta}_{a(b)}},1/b_{1},1/b_{2}),\] (36) \[t_{c(d)}={\max}(\sqrt{-{\alpha}_{g}},\sqrt{{\|}{\beta}_{c(d)}{\|}},1/b_{1},1/b_{ 2},1/b_{3}),\] (37) \[E_{a(b)}(t)={\exp}\{-S_{B_{c}}(t)\},\] (38) \[E_{c(d)}(t)={\exp}\{-S_{B_{c}}(t)-S_{P}(t)\},\] (39) \[{\alpha}_{g}=\bar{x}_{1}^{2}m_{1}^{2}+\bar{x}_{2}^{2}m_{2}^{2}-\bar{x}_{1}\bar {x}_{2}t,\] (40) \[{\beta}_{a}=m_{1}^{2}-m_{b}^{2}+\bar{x}_{2}^{2}m_{2}^{2}-\bar{x}_{2}t,\] (41) \[{\beta}_{b}=m_{2}^{2}-m_{c}^{2}+\bar{x}_{1}^{2}m_{1}^{2}-\bar{x}_{1}t,\] (42) \[{\beta}_{c}=x_{1}^{2}m_{1}^{2}+x_{2}^{2}m_{2}^{2}+x_{3}^{2}m_{3}^{2}-x_{1}x_{2 }t-x_{1}x_{3}u+x_{2}x_{3}s,\] (43) \[{\beta}_{d}=\bar{x}_{1}^{2}m_{1}^{2}+\bar{x}_{2}^{2}m_{2}^{2}+x_{3}^{2}m_{3}^{ 2}-\bar{x}_{1}\bar{x}_{2}t-\bar{x}_{1}x_{3}u+\bar{x}_{2}x_{3}s,\] (44) \[S_{B_{c}}(t)=s(x_{2},p_{2}^{+},1/b_{2})+2{\int}_{1/b_{2}}^{t}\frac{d{\mu}}{\mu }{\gamma}_{q},\] (45) \[S_{P}(t)=s(x_{3},p_{3}^{+},1/b_{3})+s(\bar{x}_{3},p_{3}^{+},1/b_{3})+2{\int}_{ 1/b_{3}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q},\] (46) where ${\alpha}_{g}$ and ${\beta}_{i}$ are the virtuality of the internal gluon and quark, respectively; ${\gamma}_{q}$$=$$-{\alpha}_{s}/{\pi}$ is the quark anomalous dimension; the expression of $s(x,Q,1/b)$ can be found in the appendix of Ref.[6]. The scattering functions $H_{i}$ in the subamplitudes ${\cal A}_{\rm Fig.\ref{fig1}(i)}$ are defined as \[H_{a(b)}(x_{1},x_{2},b_{i},b_{j})\,=\,K_{0}(\sqrt{-{\alpha}_{g}}b_{i})\Big{\{} {\theta}(b_{i}-b_{j})K_{0}(\sqrt{-{\beta}}b_{i})I_{0}(\sqrt{-{\beta}}b_{j})+(b _{i}{\leftrightarrow}b_{j})\Big{\}},\] (47) \[H_{c(d)}(x_{1},x_{2},x_{3},b_{2},b_{3}) = \Big{\{}{\theta}(-{\beta})K_{0}(\sqrt{-{\beta}}b_{3})+\frac{{\pi} }{2}{\theta}({\beta})\Big{[}iJ_{0}(\sqrt{{\beta}}b_{3})-Y_{0}(\sqrt{{\beta}}b_ {3})\Big{]}\Big{\}}\] (48) \[{\times} \Big{\{}{\theta}(b_{2}-b_{3})K_{0}(\sqrt{-{\alpha}_{g}}b_{2})I_{0 }(\sqrt{-{\alpha}_{g}}b_{3})+(b_{2}{\leftrightarrow}b_{3})\Big{\}},\] where $J_{0}$ and $Y_{0}$ ($I_{0}$ and $K_{0}$) are the (modified) Bessel function of the first and second kind, respectively. ## III Numerical results and discussion In the rest frame of the ${\Upsilon}(1S)$ particle, branching ratio for the ${\Upsilon}(1S)$${\to}$$B_{c}P$ weak decays can be written as \[{\cal B}r({\Upsilon}(1S){\to}B_{c}P)\ =\ \frac{1}{12{\pi}}\,\frac{p}{m_{{ \Upsilon}}^{2}{\Gamma}_{{\Upsilon}}}\,{\|}{\cal A}({\Upsilon}(1S){\to}B_{c}P){\| }^{2},\] (49) where the decay width ${\Gamma}_{\Upsilon}$$=$$54.02{\pm}1.25$ keV [1]. The values of input parameters are listed as follows. (1) Wolfenstein parameters [1]: $A$$=$$0.814^{+0.023}_{-0.024}$ and ${\lambda}$$=$$0.22537{\pm}0.00061$. (2) Masses of mesons [1]: $m_{B_{c}}$$=$$6275.6{\pm}1.1$ MeV and $m_{{\Upsilon}(1S)}$$=$$9460.30{\pm}0.26$ MeV. (3) Masses of quarks [1]: $m_{c}$$=$$1.67{\pm}0.07$ GeV and $m_{b}$$=$$4.78{\pm}0.06$ GeV. (4) Gegenbauer moments at the scale of ${\mu}$$=$ 1 GeV: $a_{2}^{\pi}$$=$$0.17{\pm}0.08$ and $a_{4}^{\pi}$$=$$0.06{\pm}0.10$[31] for twist-2 pion distribution amplitudes, $a_{1}^{K}$$=$$0.06{\pm}0.03$ and $a_{2}^{K}$$=$$0.25{\pm}0.15$[25] for twist-2 kaon distribution amplitudes. (5) Decay constants: $f_{\pi}$$=$$130.41{\pm}0.20$ MeV [1], $f_{K}$$=$$156.2{\pm}0.7$ MeV [1], $f_{B_{c}}$$=$$489{\pm}5$ MeV [32]. As for the decay constant $f_{{\Upsilon}}$, one can use the definition of decay constant $f_{V}$ for vector meson $V$ with mass $m_{V}$ and polarization vector ${\epsilon}_{V}$, \[{\langle}0{\|}\bar{\psi}{\gamma}^{\mu}{\psi}{\|}V{\rangle}=f_{V}m_{V}{\epsilon}_ {V}^{\mu}.\] (50) The decay constant $f_{V}$ is related to the experimentally measurable leptonic branching ratio: (51) where ${\alpha}_{\rm QED}$ is the fine-structure constant, $m_{\ell}$ is the lepton mass, $Q_{q}$ is the electric charge of the quark in the unit of ${\|}e{\|}$, and $Q_{b}$$=$$-1/3$ for the bottom quark. The experimental measurements on leptonic ${\Upsilon}(1S)$ decays give the weighted average decay constant $f_{{\Upsilon}}$$=$$(676.4{\pm}10.7)$ MeV (see Table.1). decay mode \| branching ratio \| decay constant ---\|---\|--- Υ(1S) → e+e− \| (2.38±0.11)% \| (664.2±23.1) MeV \| Υ(1S) → μ+μ− \| (2.48±0.05)% \| (677.9±14.7) MeV \| (676.4±10.7) MeV Υ(1S) → τ+τ− \| (2.60±0.10)% \| (683.3±21.1) MeV \| Table 1: Branching ratios for leptonic Υ(1S) decays and decay constants fΥ, where the last column is the weighted average, and errors come from mass, width and branching ratios. \| Ref. ijma14 \| Ref. adv2013 \| Ref. 691261 \| this work ---\|---\|---\|---\|--- 1011×Br(Υ(1S)→Bcπ) \| 6.91 \| 2.8 \| 5.03 \| 7.04+0.48+0.80+0.83+0.47−0.46−0.52−0.98−0.44 1012×Br(Υ(1S)→BcK) \| 5.03 \| 2.3 \| 3.73 \| 5.41+0.40+0.63+0.64+0.39−0.38−0.41−0.74−0.37 Table 2: Branching ratios for the Υ(1S) → Bcπ, BcK decays, where the results of Refs. ijma14 ; adv2013 ; 691261 are calculated with the coefficient a1 = 1.05. The uncertainties of the last column come from the CKM parameters, the renormalization scale μ = (1±0.1)ti, masses of b and c quarks, hadronic parameters (decay constants and Gegenbauer moments), respectively. If not specified explicitly, we will take their central values as the default inputs. Our numerical results on the $CP$-averaged branching ratios for the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays are displayed in Table 2, where the uncertainties come from the CKM parameters, the renormalization scale ${\mu}$$=$$(1{\pm}0.1)t_{i}$, masses of $b$ and $c$ quarks, hadronic parameters (decay constants and Gegenbauer moments), respectively. The following are some comments. (1) The pQCD’s results on branching ratios for the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays have the same magnitude as those of Refs. [11; 12; 691261 ]. The estimation of Refs. [11; 12] is based on the NF approximation, where nonfactorizable corrections to HME are not considered, and the form factors for the transition between ${\Upsilon}(1S)$ and $B_{c}$ mesons are calculated with the heavy quark effective theory in Ref. [11] and the Wirbel-Stech-Bauer [33] model in Ref. [12], respectively. The coefficient $a_{1}$ containing the NLO nonfactorizable contributions to HME are used in Ref. [691261 ] within the QCDF framework, where the form factors are written as the overlap integrals of nonrelativistic wave functions for ${\Upsilon}(1S)$ and $B_{c}$ mesons based on the Wirbel-Stech-Bauer model. Compared with the NF and QCDF approach, there are more contributions from the nonfactorizable decay amplitudes ${\cal A}_{\rm Fig.\ref{fig1}(c,d)}$ with the pQCD approach. This may be why the pQCD’s results are slightly larger than previous ones. (2) Because the CKM factors ${\|}V_{cb}V_{us}^{\ast}{\|}$$<$${\|}V_{cb}V_{ud}^{\ast}{\|}$, there is a relation between branching ratios, ${\cal B}r({\Upsilon}(1S){\to}B_{c}{\pi})$$>$${\cal B}r({\Upsilon}(1S){\to}B_{c}K)$. (3) Branching ratio for the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$ decay can reach up to $10^{-11}$. So the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$ decay should be sought for with high priority and first observed at the running LHC and forthcoming SuperKEKB. For example, the ${\Upsilon}(1S)$ production cross section in p-Pb collision can reach up to a few ${\mu}b$ with the LHCb [34] and ALICE [35] detectors at LHC. Over $10^{11}$${\Upsilon}(1S)$ particles per 100 $fb^{-1}$ data collected at LHCb and ALICE are in principle available, corresponding to a few tens of ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$ events. (4) There are many uncertainties on our results. Other factors, such as the contributions of higher order corrections to HME, relativistic effects and so on, which are not considered here, deserve the dedicated study. Our results just provide an order of magnitude estimation. ## IV Summary The ${\Upsilon}(1S)$ weak decay is legal within the standard model, although branching ratio is tiny compared with the strong and electromagnetic decays. With the potential prospects of the ${\Upsilon}(1S)$ at high-luminosity dedicated heavy-flavor factories, the bottom-changing ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ weak decays are studied with the pQCD approach. It is found that with the nonrelativistic wave functions for ${\Upsilon}(1S)$ and $B_{c}$ mesons, branching ratios for the ${\Upsilon}(1S)$${\to}$$B_{c}{\pi}$, $B_{c}K$ decays have the same order as previous works, and ${\cal B}r({\Upsilon}(1S){\to}B_{c}{\pi})$${\gtrsim}$$10^{-11}$. 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1705.10853	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 57937, "num_imgs": 11, "llama3_tokens_count": 16816 }	[ "content_image/1705.10853/x1.png", "content_image/1705.10853/x2.png", "content_image/1705.10853/x4.png", "content_image/1705.10853/x5.png", "content_image/1705.10853/x6.png", "content_image/1705.10853/x8.png", "content_image/1705.10853/x9.png", "content_image/1705.10853/x10.png", "content_image/1705.10853/x11.png", "content_image/1705.10853/x12.png", "content_image/1705.10853/x13.png" ]	# Phase Slips in Superconducting Weak Links Gregory Kimmel Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60202, USA Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Andreas Glatz Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Igor S. Aranson Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60202, USA March 1, 2024 ###### Abstract Superconducting vortices and phase slips are primary mechanisms of dissipation in superconducting, superfluid, and cold atom systems. While the dynamics of vortices is fairly well described, phase slips occurring in quasi-one dimensional superconducting wires still elude understanding. The main reason is that phase slips are strongly non-linear time-dependent phenomena that cannot be cast in terms of small perturbations of the superconducting state. Here we study phase slips occurring in superconducting weak links. Thanks to partial suppression of superconductivity in weak links, we employ a weakly nonlinear approximation for dynamic phase slips. This approximation is not valid for homogeneous superconducting wires and slabs. Using the numerical solution of the time-dependent Ginzburg-Landau equation and bifurcation analysis of stationary solutions, we show that the onset of phase slips occurs via an infinite period bifurcation, which is manifested in a specific voltage-current dependence. Our analytical results are in good agreement with simulations. ## I Introduction The motion of Abrikosov vortices is recognized as the main cause of dissipation in type-II superconductors Blatter _et al._ (1994). Conversely, in thin nanowires, the motion of vortices is impeded and phase-slip events are responsible for the dissipation. Phase slips, changing the phase difference of the superconducting order parameter by $2\pi$, may be caused by different physical mechanisms. Thermally activated phase slips at high temperatures and small applied currents are well understood Tinkham (1996). At very low temperatures, phase slips can be caused by quantum fluctuations (aptly called quantum phase slips) Mooij and Nazarov (2006); Lau _et al._ (2001); Glatz and Nattermann (2002). Phase slips are not unique to superconductors, they also occur in superfluid systems Anderson (1966); Langer and Fisher (1967); Schwarz (1990), and more recently, dissipation due to phase slips were studied in cold atom systems McKay _et al._ (2008); Scherpelz _et al._ (2014, 2015). In particular, phase slips can be triggered in a superfluid cold atom system by a rotating weak link Wright _et al._ (2013). Even without thermal and quantum fluctuations, the phase slip phenomena and dissipative (or resistive) states can be induced by an applied current Skocpol _et al._ (1974); Meyer (1973). Magnetic field penetrates type-II superconductors in the form of Abrikosov vortices. If an external current is applied, the Lorentz force induces motion of the vortices. This motion is the main cause of dissipation in 2D and 3D superconductors. However, in quasi-one dimensional nanowires with the coherence length $\xi(T)$ and the penetration depth $\lambda(T)$ large compared to the wire diameter, vortex motion is suppressed. In this situation the transition to the normal state was made through successive voltage jumps which are attributed to the appearance of phase slip centersSkocpol _et al._ (1974); Meyer (1973). A study of this phenomenon was given first by Kramer and Baratoff who found that slightly below the depairing current, there is a dissipative state which consists of localized phase slips occurring in the superconducting filament Kramer and Baratoff (1977). In a narrow range of currents close to the depairing current, the material is superconducting except in narrow regions where phase slip centers (PSCs) occur. The period of these PSCs diverge as the external current approaches the lower bound in this narrow region. It was also shown that random thermal fluctuations allow for phase slips Little (1967), but these did not persist indefinitely. Further numerical study of the one-dimensional time-dependent Ginzburg-Landau equation revealed periodic phase slips existing in a narrow range of currents close to the depairing current Kramer and Rangel (1984); Rangel and Kramer (1989). Follow-up numerical studies of narrow two-dimensional superconducting strips discovered a transition from a phase-slip-line to vortex pairs Weber and Kramer (1991). Periodic lattices of the phase slip centers were studied in the context of vortex penetration in thin superconducting films near the third critical magnetic field Aranson and Vinokur (1998). Using a saddle-point approximation for the Ginzburg-Landau energy in narrow superconducting strips, the dependence of voltage drop vs temperature and bias current (neglecting thermal fluctuations) was studied in [Ovchinnikov and Varlamov, 2015]. The situation is different, however, for spatially inhomogeneous systems, such as superconductors with macroscopic defects or weak links Langer and Ambegaokar (1967). Perhaps the most famous examples are Dayem bridges and Josephson junctions Josephson (1962); Anderson and Dayem (1964). The mechanism for dissipation in these cases is the quantum tunneling of Cooper pairs between the two superconductors, which is caused by a phase difference between the weakly-linked superconductors. When the current is below some threshold $j_{c}$, the phase difference is fixed in time and a stationary superconducting state persists. Above this threshold, the solution exhibits oscillations, which lead to a finite voltage. In a review paper by Ivlev and Kopnin, inhomogeneities were analyzed, but in regards to the stability of the normal state Ivlev and Kopnin (1984). Thus, their analysis involved currents much closer to the GL critical threshold $j_{GL}=2/\sqrt{27}$. A lower bound $j_{1}$ at which the normal state was globally unstable (i.e. arbitrary small perturbation lead to instability of the normal state), and above which there was a critical-sized perturbation which separated the normal and superconducting states was estimated. Also, an upper critical current $j_{2}$ such that the normal state was absolutely stable for an external current $j_{0}>j_{2}$ was found. An inhomogeneity much smaller than the coherence length, $\xi(T)$, was used and was approximated by a $\delta$ function, simplifying the algebra. Here we consider a more realistic situation for the type-II high-temperature superconductors: an inclusion on the scale of $\xi(T)$. The transition we are interested in analyzing, occurs between the non-uniform superconducting state and the oscillatory state with phase slips. Therefore, the steady state and linearization in this paper are much more complex then in analyzing the normal state. The authors of [Van Dover _et al._, 1981] have shown experimental results of weak-links with non-hysteric behavior. The phase slip state of homogenous systems have recently been analyzed in much greater detail Baranov _et al._ (2011). Using bifurcation analysis, Baranov et. al. extract the normal form of a saddle-node bifurcation when the current is near the critical current. They then correctly determine the characteristic scaling law and show its agreement with numerical simulations. The period diverges in an infinite-period saddle-node bifurcation as $j_{0}\to j_{c}$. These authors further expanded upon their analysis by showing the important role that the material parameter $u$ plays in the type of bifurcation that can occur Baranov _et al._ (2013) ($u$ is related to the electric field penetration depth). They observed that for finite lengths and values of $u$ above some critical threshold $u_{c2}$, numerical simulations showed hysteresis in the I-V curve. However, our work focuses on analytical methods for the inhomogeneous system, which as stated previously makes the steady state and linearization to much more difficult to handle. We show that a simplified system can be obtained through weakly nonlinear analysis and that this system contains the normal form obtained in [Baranov _et al._, 2011] as the size of the weak link shrinks to zero. We also demonstrate that in addition to the infinite period bifurcation for small $u$, a hysteresis exists in our system for large $u$ values, similar to that in Ref. [Baranov _et al._, 2011]. However, in contrast to previous studies, our reduced two-dimensional nonlinear system exhibits evolution of periodic orbits and a transition between superconducting and normal states that are not properly captured by the one-dimensional model in Ref [Baranov _et al._, 2011]. A work by Michotte et. al. in [Michotte _et al._, 2004] have found that the condition for PSCs to occur is based on the competition between two relaxation times: the relaxation time for the magnitude of the order parameter $t_{\|\Psi\|}$ and the relaxation time for the phase of the order parameter $t_{\phi}$. They observed that phase slips are possible only when $t_{\phi}<t_{\|\Psi\|}$. A linearized Eilenberger equation in the dirty limit was studied, resembling a generalized TDGL equation with additional parameters related to inelastic electron-phonon collisions, which was first derived in [Kramer and Watts-Tobin, 1978]. They derived an approximate critical current via this equation and their results implied that there was a finite maximal oscillation period for the order parameter. In contrast, for weak links all oscillation periods diverged. The generalized GL equation used contained an additional parameter $\gamma$ characterizing relative superconducting phase relaxation time (for us, $\gamma=0$). For large $\gamma$ values hysteresis was observed in the I-V curve. On a qualitative level, the effect of increasing parameter $\gamma$ is similar to an increase in parameter $u$Baranov _et al._ (2011). Correspondingly, we observed hysteresis when $u\gg 1$. The authors of [Berdiyorov _et al._, 2012] have done numerical analysis of a periodic array of weak links using the generalized TDGL equation. They showed I-V curves for different magnetic fields, however no analysis of the divergence of the period of vortices was presented. We focus on a 1D superconductor, separated by a normal or weakly superconducting inhomogeneity. The complete system is modeled by a spatially dependent critical temperature $T_{c}(x)$. The weak link is created by a lower transition temperature inside an interval $I=[-r,r]$, which leads to a suppression of the order parameter. Here $r$ is the inclusion radius. Below some critical current, this system relaxes to a stationary superconducting state, but above it, the superconductor exhibits a finite voltage with oscillatory behavior. Thermal fluctuations are initially not considered in this model and therefore does not cause a finite voltage in the superconducting state. The Josephson junction analysis is not applicable here. Indeed, since there is no dielectric contact between the two superconductor pieces, the phase should always be the same, implying zero voltage. We will show via simulations of the time-dependent Ginzburg-Landau equation, that the oscillations in the voltage is caused by phase slips in the center of the inclusion. The system approaches this state via a saddle-node bifurcation of two superconducting states, which occur at the critical current (at a saddle-node bifurcation stable and unstable stationary superconducting states annihilate and a periodic resistive state appears). The suppression of the order parameter in and near the weak link allows us to employ analytical methods in the vicinity of the critical current. We derive a reduced two-dimensional system governing the time evolution of the phase slip solution and describe a sequence of transitions between superconducting and dissipative states. The paper is organized as follows: section II describes the model, section III deals with the stationary case and estimates the critical current which is obtained from the saddle-node bifurcation condition. Sections IV-VII deal with the time periodic solutions, extracting a time-dependent system via weakly nonlinear analysis and then studying the simplified model to show that it exhibits the same qualitative behavior. In section VIII, we interpret our analytical results, show the correspondence to the parameters of the superconductor and its effects on the phase slip state. Finally, section IX gives closing remarks and ideas for further study. ## II Governing equations The time-dependent Ginzburg-Landau equations (TDGLE) are obtained by minimization of the GL free energyAranson and Kramer (2002). In the absence of a magnetic field this results in \[\Gamma\left(\partial_{t}+i\frac{2e}{\hbar}\mu\right)\Psi=a_{0}\nu (x)\Psi-b\|\Psi\|^{2}+\Psi+\frac{\hbar^{2}}{4m}\partial_{x}^{2}\Psi,\] (1) where $\Gamma,a_{0},b$ are phenomenological parameters that can be found from the microscopic theory Bardeen _et al._ (1957), $e,m$ are the electron charge and mass, $\mu$ is the scalar potential, and $\nu(x)$ a spatially dependent linear coefficient modeling inhomogeneities in the system. Following Sadovskyy et al.Sadovskyy _et al._ (2015), we define the $+x$ direction be the direction of the external current and obtain the following dimensionless form: \[u(\partial_{t}+i\mu)\Psi=\partial_{x}^{2}\Psi+[\nu(x)-\|\Psi\|^{2}]\Psi\] (2a) with the total current \[j_{0}\] \[j_{0}=\Im(\Psi^{}\partial_{x}\Psi)-\partial_{x}\mu.\] (2b) Here $\Psi$ is the complex order parameter, satisfying $\|\Psi\|=1$ in the purely superconducting state, and $\|\Psi\|=0$ in the normal state. The parameter $u=\Gamma/a_{0}\tau_{\text{GL}}$ with time $\tau_{\text{GL}}=4\pi\sigma\lambda_{0}^{2}/c^{2}$, $\lambda_{0}=\sqrt{\frac{mc^{2}}{8\pi e^{2}\psi_{0}^{2}}}$ is the magnetic penetration depth ($c$ the speed of light) and $\psi_{0}=\sqrt{a_{0}/b}$ is the equilibrium value of the order parameter when spatial variations are neglected, i.e., $\nu(x)=1$. The zero temperature coherence length $\xi_{0}=\sqrt{\frac{\hbar^{2}}{4ma_{0}}}$ is used for the unit of length. For more details see [Sadovskyy _et al._, 2015]. We apply periodic boundary conditions for $\Psi$. Since $\mu$ is on average an increasing function of $x$, there is necessarily a discontinuity at the boundary. This is resolved by making the following transformations: \[\Psi =\tilde{\Psi}e^{iK(t)x}\] (3a) \[\mu =-Ax+\tilde{\mu}.\] (3b) Here, $\tilde{\mu}$ is a periodic function in $x$. Essentially, we are moving the growth of $\mu$ to the phase of $\Psi$. The growth in $K$ now does not affect the magnitude. Indeed, this also allows us to _rewind_$K$ through $K\to K-(2\pi/\Delta x)\lfloor K\Delta_{x}/2\pi\rfloor$ which will remove any error from rapid phase oscillations Sadovskyy _et al._ (2015). Inserting this into (2a) gives \[u[\partial_{t}+ix(\partial_{t}K-A)+i\tilde{\mu}]\tilde{\Psi}=(\partial_{x}+iK) ^{2}\tilde{\Psi}+[\nu(x)-\|\tilde{\Psi}\|^{2}]\tilde{\Psi}.\] Setting $\partial_{t}K=A$ eliminates the linear term. Now inserting this into (2b), we have \[j_{0}=\Im(\tilde{\Psi}^{}\partial_{x}\tilde{\Psi})+\|\tilde{\Psi}\|^{2}K+ \partial_{t}K-\partial_{x}\tilde{\mu}.\] Averaging this equation over space and noting that $\langle\tilde{\mu}_{x}\rangle=0$ results in an ordinary differential equation (ODE) for $K$ For clearer notation, we now suppress the tildes, and we arrive at our modified TDGLE \[u(\partial_{t}+i\mu)\Psi =(\partial_{x}+iK)^{2}\Psi+[\nu(x)-\|\Psi\|^{2}]\Psi\] (4a) \[\mu_{x} =\Im(\Psi^{}\partial_{x}\Psi)+\partial_{t}K+\|\Psi\|^{2}K-j_{0}\] (4b) \[j_{n} =\partial_{t}K+\left\langle\|\Psi\|^{2}\right\rangle K.\] (4c) The integration domain is periodic with the period $L$. For the numerical integration, we generally took $L=20$ and $u=1$, however this was relaxed to see if the qualitative behavior changed. We verified that increasing $L$ does not affect the results, however changing $u$ can have a large effect (see section VIII.3). To make the analysis simpler, we placed the weak link of length $2r$ symmetrically at the origin in the interval $I$. The inclusion’s effect enters through the term $\nu(x)\Psi$ defined by \[\nu(x)\equiv\begin{cases}1,&x\not\in I\\ -C,&x\in I\end{cases}\,.\] (5) Numerical analysis has shown that for $L\gg r$ there exists a critical current $j_{c}$, which is a function of $r$ that separates the dynamics of this system. For $j_{0}<j_{c}$, the system goes to a stationary superconducting state, while for $j_{0}>j_{c}$ the system exhibits a dissipative state represented by periodic phase slips occurring in the center of the inclusion via a stable limit cycle. In the following sections, we explain these results analytically. We first provide an analytical approximation of the critical current. Next, we extract a coupled two-dimensional nonlinear system of ODEs from (2a) which describes qualitatively, the correct behavior for suitable choices of the coefficients of the simplified system. ## III The stationary case $j_{0}<j_{c}$ In the superconducting state with an applied current of $j_{0}<j_{c}$, it can be shown that $\mu=0$, (see appendix .1 for details). To proceed, we rewrite (2a) in terms of amplitude and phase of the order parameter, i.e., $\Psi=Fe^{i\phi}$. Inserting this into (2a) and (2b) gives for the stationary equation \[0 =\partial_{x}^{2}F+[\nu(x)-(\partial_{x}\phi)^{2}-F^{2}]F\] (6a) \[j_{0} =F^{2}\partial_{x}\phi.\] (6b) Plugging (6b) into (6a) gives the nonlinear ODE \[0=\partial_{x}^{2}F+[\nu(x)-j_{0}^{2}F^{-4}-F^{2}]F.\] (7) ### Large $C$ approximation We now assume a large $C$ approximation, that is, the weak link strongly suppresses superconductivity in the inclusion (i.e. $C\gg j_{0}^{2}F^{-4}$). This allows us to neglect the nonlinear term and obtain a first order approximation of the solution of (6). From this we notice that (6a) has a first integral for both the inclusion domain and the superconducting domain. Asymptotic analysis of the size of these coefficients gives us a condition for $j_{c}$ given by \[j_{c}=\frac{1}{2\sqrt{C}}e^{-2r\sqrt{C}},\] (8) for details see appendix .2. Setting $C=1$, we have that \[j_{c}=\frac{1}{2}e^{-2r}.\] (9) Comparing this approximation with numerical simulations, we see that the large $C$ approximation with $C=1$ is in good agreement with the numerical solution (see Fig. 1). Thus, we derived that a weak link results in a exponential suppression of the critical current as a function of the inclusion width $2r$ and strength $C$. A similar result was obtained through a different method in Ref. Rangel and Kramer, 1989. However, our method is appealing for the simple generalization to multiple inclusions. ### Multiple inclusions Let $r_{1},\dots r_{k}$ be the radii of $k$ inclusions in the domain. We have $k+1$ superconducting domains and $k$ normal domains, each with their own first integral constant. The analysis from appendix .2 carries over and we expect the inclusion domain’s first integral constant $E_{I_{k}}$ to be approximately 0 for each $k$. This holds at the center of each respective inclusion, which each give different critical currents. However, when one is no longer satisfied, the system will no longer be satisfied and the global $j_{c}$ is determined by the lowest local $j_{c}$, which appears at the longest inclusion: \[j_{c}\approx\frac{1}{2}e^{-2\max\limits_{k}r_{k}}.\] (10) <figure><img src="content_image/1705.10853/x1.png"><figcaption>Figure 1: The critical current as a function of inclusion size using (9) (e.g.C=1 with (8)). For the two inclusions, one inclusion is held fixed at r=2.Above the curves the superconducting order parameter Ψ oscillates.</figcaption></figure> ### Linear stability analysis of the stationary state Consider now a perturbation $\eta$ of the stable state in the form $\Psi=(F+\eta)e^{i\phi}$. Inserting this into (2a)-(2b) and linearizing in $\eta$, we obtain with (6a) and (6b) \[u\partial_{t}\eta =\partial_{x}^{2}\eta+\left(\nu-(\partial_{x}\phi)^{2}-2F^{2} \right)\eta+\] \[\phantom{blah}i(2\partial_{x}\phi\partial_{x}\eta+\partial_{x}^{2 }\phi\eta-uF\mu)-F^{2}\eta^{}\] \[0 =\Im(F\partial_{x}\eta+2iF\partial_{x}\phi\eta+\partial_{x}F\eta^ {})-\partial_{x}\mu.\] Separating $\eta(x,t)=(U+iV)e^{\lambda t}$ we obtain the following system (here $\lambda$ is the growth rate) \[\begin{split} 0&=\partial_{x}^{2}U+ \left(\nu-(\partial_{x}\phi)^{2}-3F^{2}-\lambda\right)U-\\ &\phantom{blah}(2\partial_{x}\phi\partial_{x}V+V\partial_{x}^{2} \phi)\end{split}\] (11a) \[\begin{split} 0&=\partial_{x}^{2}V+ \left(\nu-(\partial_{x}\phi)^{2}-F^{2}-\lambda\right)V+\\ &\phantom{0=}(2\partial_{x}\phi\partial_{x}U+U\partial_{x}^{2} \phi)-uF\mu\end{split}\] (11b) \[\partial_{x}\mu =F\partial_{x}V-V\partial_{x}F+2FU\partial_{x}\phi.\] (11c) This system along with (6a)-(6b) represents a 7 dimensional boundary-value eigenvalue problem which must be solved with appropriate boundary conditions. First, we note from (6a) that replacing $x\to-x$ leaves the differential equation unchanged. This with the reflection symmetry implies that $F$ is an even function in $x$. This symmetry implies from (2b) that $\partial_{x}\phi$ and $\partial_{x}\mu$ are even in $x$. Thus $x\to-x$ changes $\Psi\to\Psi^{}$. The action of this must be retained in the linearization implying that $\eta(-x)$ and $\eta^{}(x)$ are both solutions. Hence $U$ is even and $V$ is odd in $x$. Furthermore, by symmetry it suffices to solve the equations only on the half interval $(0,L/2)$ with the obtained natural boundary conditions from symmetry and the remaining conditions to be found by matching-shooting algorithm. To solve this we used a technique developed in Tsimring and Aranson (1997); Aranson and Vinokur (1998). In order to do so, we used a numerical matching-shooting solver for ODEs by beginning with a small domain (typically $L\sim 3$). We extracted the appropriate shooting boundary conditions and approximation for $\lambda$ and used these as guesses for a larger system size. Iterating this process, we continued to $L$ sufficiently large until the boundary conditions and $\lambda$ were not changing significantly. The results are plotted in Fig. 2. We note here that $j_{c}\approx 0.0637$ obtained by the solver is only 6% away from the value obtained through direct numerical solution of the Ginzburg-Landau model. The step size used in the dynamic simulations were much larger ($\Delta x=0.05$ compared to shooting solver with $\Delta x=0.001$) and each had an associated numerical error. Therefore, $j_{c}\approx 0.0637$ is more accurate. We checked if the error is independent of the solvers by analyzing the dynamic simulations $j_{c}$ as a function of $\Delta x$ in appendix .3. We found that as $\Delta x\to 0$, we approached a similar value to that found from shooting. Thus, from Fig. 2 one sees that at the critical current, when stable ($\lambda<0$) and unstable ($\lambda>0$) solutions merge and annihilate, the corresponding linear system becomes degenerate. At the critical point it possesses two zero eigenvalues $\lambda_{1,2}=0$. This degeneracy is taken into account through weakly nonlinear analysis. <figure><img src="content_image/1705.10853/x2.png"><figcaption>Figure 2: (a) Amplitude \|Ψ\| and linearized solutions U,V,μ with j0=0.061, r=1.plots (b) and (c) shows the value of \|Ψ(0)\| and location of the smallesteigenvalue respectively, for stable (solid line) and unstable (dashed line)solutions of eqs. (7), (11a)-(11c) for varying current. At the criticalcurrent the stable and unstable stationary (i.e. superconducting) solutionsmerge and annihilate.</figcaption></figure> ## IV Analysis of time-periodic solutions for $j_{0}>j_{c}$ When the current is above the critical threshold, the above analysis breaks down. Numerical simulations indicate that the superconductor exhibits oscillations in the order parameter, where phase slips are now present (i.e. $\|\Psi(0,t)\|=0$ for some $t$). In figure 3 we have estimated the period of oscillation $T$ as a function of $j_{0}-j_{c}\ll 1$. Numerical simulations indicate that the period $T\sim O(\|j_{0}-j_{c}\|^{-1/2})$, which is indicative of an infinite-period bifurcation (IPB) at the point $j_{0}=j_{c}$. In general for a bifurcation parameter $R$ (e.g. current $j$) the period of oscillations $T\sim O(\|R-R_{c}\|^{-1/2})$ for $\|(R/R_{c})-1\|\ll 1$ for an IPBStrogatz (2014). We can see from figure 3 that an IPB is occurring at the critical value. In section VIII.3, we show that for $u\gg 1$, we also observe hysteresis, behavior which is typical of a homoclinic bifurcation, a different mechanism through which a limit cycle can be destroyedStrogatz (2014). <figure><img src="content_image/1705.10853/x4.png"><figcaption>Figure 3: IPB analysis with L=20 and r=1. The critical current jc≈0.067 wasobtained via stable state calculation from Section II. The simplified systemderived in section VII from weakly nonlinear analysis at γ=−0.13 withcIP≈−0.565 also exhibits an IPB. As expected, period T∝1√R−Rc near thebifurcation point in both cases. Here R is current j in the TDGLE andparameter c in the simplified system.</figcaption></figure> <figure><img src="content_image/1705.10853/x5.png"><figcaption>Figure 4: Plots (a)-(c) show dependence of voltage vs time above the criticalcurrent where j0=1.045jc, j0=1.015jc and j0=(1+10−6)jc, respectively andjc≈0.067. System size L=20, with an inclusion r=1 in the center.</figcaption></figure> Figure 4 shows time-voltage curves for $j_{0}>j_{c}$. One clearly sees the period diverging as we approach the critical value. To calculate the current-period relationship, we ramped the current from an initial amount (typically $j_{\text{init}}<j_{c}$). If the system was stationary for a certain number of iterations, we increased the current. Once the system started oscillating, we calculated peaks in voltage, while skipping the first few to account for system equilibration. Then we averaged over the remaining peaks to obtain the period. We then used linear extrapolation to find the new current. For example, at the $n^{\text{th}}$ step, we have the current $j_{n}$ and corresponding period $T_{n}$. Let $m_{n}=\Delta T_{n}/\Delta j_{n}$, then suppose we want to find the current corresponding to a new period $T_{n+1}=(1+\alpha)T_{n}$, with $\alpha>0$. This is given by $j_{n+1}=j_{n}+\frac{\alpha T_{n}}{m_{n}}$. Figure 5 shows a similar period divergence of the oscillations of $\Psi$ and the simplified model (see section VII). <figure><img src="content_image/1705.10853/x6.png"><figcaption>Figure 5: Plots (a)-(c) show dependence of \|Ψ(0)\| vs time above the criticalcurrent where j0=1.045jc, j0=1.015jc and j0=(1+10−6)jc, respectively andjc≈0.067. System size L=20, with an inclusion r=1 in the center. Plots (d)-(f)correspond to the simplified system Eqs. (16) where γ=−0.13 with cIP≈−0.565 isthe IPB threshold, with c=0.955cIP, c=0.985cIP and c=(1−10−5)cIP.</figcaption></figure> ## V Weakly nonlinear analysis We now extract a coupled ODE system, which exhibits two dynamical possibilities. In the case $j_{0}<j_{c}$, we show that the stationary (fixed) solution is stable, while in the opposite case, a stable limit cycle exists. It is of course possible that a bistability region can exist, which would lead to hysteric effects. Such effects have been observed in homogeneous superconductors Weber and Kramer (1991); Baranov _et al._ (2011, 2013). For large $u$, we have also observed hysteric I-V curves and we show that our extracted system contains both possibilities. The process is standard and is broken into these steps: Find stationary (basic) state $\Psi_{0}=Fe^{i\phi}$ (it is already shown in Fig. 2) * Perturb solution and solve linearized system. * Extract weakly nonlinear effects from orthogonality condition. * Show that certain conditions allow for a stable limit cycle to exist. Though standard, the difficulty in this problem is that the basic state and linearization cannot be solved in closed form. Though we can approximate it to a certain degree, its region of validity is dependent on the radius of the inclusion $r$, the current $j_{0}$ and to a smaller extent, the system size $L$. Indeed, it is impractical to obtain it numerically since the solutions are sensitive to these choices. However, our analysis will assume that these are all known _a_ priori and proceed through the framework. The simplified system is then obtained generally, and we show that the system exhibits the appropriate behavior for certain values in parameter space. We expand Eqs. (4a)-(4c) near the stationary solution and near the critical point $j_{0}=j_{c}+\epsilon$ with $\epsilon\ll 1$. The first order solution will be given by $\Psi_{0}=Fe^{i\phi}$ (since $K=0$, there is no electric potential in the super conducting state), in fact the initial transient would show exponential decay of $K\to e^{-\left\langle\|\Psi\|^{2}\right\rangle t}$ and so $\mu=0$ as expected. Let $\Psi=(F+\eta)e^{i\phi}$, where $\eta$, and time will now both slowly vary and be controlled by a small parameter $0<\delta\ll 1$, whose size will be related to $\epsilon$. The proper scaling will be determined from the ODE for $K$. Based on numerical simulations, we assume $K=O(\delta^{2})$. We claim that we may regard $K$ as constant in the relevant order of the perturbation method by the following argument. The perturbation $\eta$ at first order is highly localized inside the inclusion and from this we argue that \[\langle\|\Psi\|^{2}\rangle =\frac{1}{L}\int_{0}^{L}F^{2}+2F(\eta+\eta^{})+O(\eta^{2})\,dx\] \[\approx\frac{1-j_{0}^{2}}{L}(L-r)+\frac{1}{L}\int_{0}^{r}F^{2}+2F (\eta+\eta^{})\,dx\] \[\approx 1-j_{0}^{2}+O\left(\frac{r}{L}\right).\] For $L\gg r$, we can regard $\langle\|\Psi\|^{2}\rangle$ as a constant. In a similar way all averaged quantities in the voltage equation can be neglected in the large superconductor domain limit. This analysis shows that the time-dependence of the voltage is slaved to the behavior of the order parameter $\Psi$. Therefore, we set $K$ to a constant by \[K=\frac{\epsilon}{1-j_{0}^{2}}+O\left(\frac{r}{L}\right).\] (12) From this, we extract the relation $\epsilon=\alpha\delta^{2}$ where $\alpha=\pm 1$. The linearized system at $\epsilon=0$ has a degenerate eigenvalue as was shown previously in Fig. 2. Therefore we expand $\eta(x,\tau)=A(\tau)\eta_{1}(x)+\sqrt{\delta}[B(\tau)\eta_{2}(x)+z_{1}(x,\tau) ]+\delta z_{2}(x,\tau)$ where $\mathcal{L}\eta_{1}=0$, $\mathcal{L}\eta_{2}=\eta_{1}$ and $\mathcal{L}$ is the linear operator from (11). Using orthogonality conditions, we arrive at the coupled system \[\begin{split} uA_{\tau}&=B+c_{1}A^{2}\\ uB_{\tau}&=c_{2}AB+c_{3}A^{3},\end{split}\] (13) where the coefficients $c_{k}$ can be found through evaluating the integrals (see appendix .4). We will show in section VI why we chose to not include the constant $K$ at this order. The general behavior is only captured correctly at $\epsilon=0$. When $\epsilon\neq 0$ (i.e. $K\neq 0$) we do not see a saddle-node bifurcation. To correct for this deficiency, higher order terms will be included. However, we can still gain some insight by analyzing this simplified system first. ## VI Dynamical System Analysis We begin with (13) by making a dimensionless system to analyze it more easily. We introduce the dimensionless variables \[x=\frac{A}{L_{A}},Y=\frac{B}{L_{B}},t^{\prime}=\frac{t}{uL_{t}}.\] Inserting this into the system and defining the characteristic variables \[L_{A}=\frac{1}{c_{2}L_{t}},\quad L_{B}=\frac{1}{c_{2}L_{t}^{2}},\] we arrive at the dimensionless system \[\begin{split}\dot{X}&=Y+aX^{2}\\ \dot{Y}&=XY+bX^{3},\end{split}\] (14) where $a\equiv c_{1}/c_{2}$ and $b\equiv c_{3}/c_{2}^{2}$. The characteristic scale for time is arbitrary and is a consequence of the degeneracy in the system. The culprits are the $X^{2}$ term and $XY$ terms whose combination of characteristic scales simultaneously vanish. ### Fixed points and stability There is only one fixed point located at the origin, provided that $a\neq b$. In this case there is a family of non-isolated fixed points along the parabola $Y=-aX^{2}$, however this case is not physical so we omit it. Next, we note the symmetry $t\to-t$ and $X\to-X$ of (14), which implies that the linearized center located at the origin is robust. We wish to see if this system exhibits closed orbits. The system is conservative if $a=-1/2$. In this case, a first integral can be obtained \[H(X,Y)=\frac{1}{2}Y^{2}-\frac{1}{2}X^{2}Y-\frac{1}{4}bX^{4}.\] This has closed orbits provided that $b<-1/2$. So now that we have established the existence of closed orbits, we seek to gain insight if $a\neq-1/2$. We replace $Y$ via the transformation \[Y=\frac{U}{2a+1}-aX^{2},\] and rescale $X\to\frac{X}{2a+1}$ and obtain \[\dot{X} =U\] (15a) \[\dot{U} =UX+\frac{b-a}{(2a+1)^{2}}X^{3}\equiv UX+\gamma X^{3}.\] (15b) This leaves us with one independent parameter $\gamma$. We have already analyzed the case where $a=-1/2$ which, if $b<-1/2$ corresponds to $\gamma\to-\infty$ and has a family of closed orbits. If $b>-1/2$ then $\gamma\to\infty$ and we know this does not have closed orbits. Therefore, there must be some critical value of $\gamma$ where this behavior changes. We seek a solution of (15) of the form $X=\tilde{C}t^{-1}$ with $\tilde{C}$ to be determined. Plugging this into the equation gives the condition \[\tilde{C}=\frac{1\pm\sqrt{1+8\gamma}}{2\gamma}.\] These two solutions form a saddle-type connection only when they are equal which occurs at $\gamma_{c}=-1/8$ or in the original coefficients \[b_{c}=-\frac{1}{8}(2a_{c}-1)^{2}.\] <figure><img src="content_image/1705.10853/x8.png"><figcaption>Figure 6: Plots (a)-(c) represent the solutions to (15) in the phase plane(X,Y) with γ=−0.25,−0.15,−0.13, respectively. There is a dimple near theorigin where the trajectories are being squeezed down due to the homoclinicorbit at γc=−1/8. In plot (d), we display this dimple as a function of γ bytaking 150 initial conditions and taking the average maximum.</figcaption></figure> This critical curve separates closed orbit solutions in the $(a,b)$ parameter space. We have shown that the simplest (first order) system obtained, demonstrates a saddle-type infinite period bifurcation, however this creates an infinite family of closed orbits and a unique stable limit cycle is not obtained. The bottleneck is created near the origin (see Fig 6). Additionally, it does not have a saddle-node bifurcation which we expect to occur at $j_{0}=j_{c}$. We note also that introducing $K$ at this order, which adds a nonzero constant term to the second ODE would still only have one fixed point and a constant at this order would destroy the degeneracy (and also any closed orbits) in a degenerate Hopf-type bifurcation when that constant crosses through zero. This should be corrected by including the next higher order cubic terms which will saturate and force the system to select one unique closed orbit. The bottleneck created near the emergence of the saddle-node bifurcation is apparent in both the physical and simplified system (see figure 5). Note that the time scales need not be the same and careful treatment of the parameters in the simplified system (see section IV) would lead to the relation between the GL time and the time scale of the simplified system. ## VII Full Dynamical System We modify the system to include the next order cubic terms. In principle, we could obtain the next order terms by continuing the perturbation expansion, however, we chose to include the generic next higher order terms $X^{3},X^{2}Y,XY^{2}$, and so on. We then found that the removal of some cubic terms e.g. $XY^{2},Y^{3}$ slightly shifts the transitions boundaries but does not qualitatively change the bifurcation sequence. Therefore, we chose to keep the following system for our analysis: \[\dot{X} =Y+aX^{2}+w_{1}X^{3}\] (16a) \[\dot{Y} =XY+bX^{3}+c+w_{2}X^{2}Y,\] (16b) where we have introduced the new coefficients $c,w_{1},w_{2}$. We will enforce $w_{1},w_{2}<0$ to ensure the phase flow cannot escape to infinity, which would be a nonphysical state for this system. ### Analysis The fixed points cannot be found analytically in general since the equation involves a quintic polynomial. Instead we look to find the two critical curves which correspond to our system. We wish to find a saddle-node bifurcation curve and an infinite-period bifurcation as the current is varied. The saddle-node bifurcation involves the merging and annihilation of the stable and unstable stationary solutions. An infinite-period bifurcation is a saddle-node bifurcation which occurs on the limit cycle in the phase plane Strogatz (2014). We first find the fixed points of (16). Using (16a), we obtain $Y^{}=-(X^{})^{2}(a+w_{1}X^{})$, which leads to the quintic equation \[f(X)\equiv w_{1}w_{2}X^{5}+(w_{1}+aw_{2})X^{4}+(a-b)X^{3}-c=0.\] A saddle-node bifurcation occurs provided that $f(X^{})=f^{\prime}(X^{})=0$. The curve exists only if $X^{}$ is real which leads to the requirement that \[b\geq a-\frac{4(w_{1}+aw_{2})^{2}}{15w_{1}w_{2}}.\] To motivate our choice of parameters, we write this in terms of $\gamma$ \[\gamma\geq-\frac{4w_{1}}{15w_{2}}\left(\frac{\frac{w_{2}}{w_{1}}a+1}{2a+1} \right)^{2}.\] If we set $w_{2}=2w_{1}$ we can eliminate $a$ from the dependence on $\gamma$. Thus, we have that the saddle-node bifurcation exists only if $\gamma\geq-\frac{2}{15}$. Writing the quintic now with $a=-1$ allows us to cast the quintic function solely in terms of $w_{1},\gamma\text{ and }c$. \[2w_{1}^{2}X^{5}-w_{1}X^{4}-\gamma X^{3}-c=0.\] The saddle node bifurcation then occurs along the curve \[c_{\text{SN}}(X^{})=\frac{1}{5}(X^{})^{3}\left[2\gamma+w_{1}(X^{})^{2} \right],\] where $X^{}$ is given by \[X^{}=\frac{1\pm\sqrt{1+\frac{15}{2}\gamma}}{5w_{1}}.\] The Jacobian of this system is \[J=\left[\begin{matrix}2aX^{}+3w_{1}(X^{})^{2}&1\\ Y^{}+3b(X^{})^{2}+2w_{2}X^{}Y^{}&X^{}+w_{2}(X^{})^{2}\end{matrix}\right].\] A necessary condition for a Hopf bifurcation to occur is for a (un)stable spiral to change stability. This occurs when the trace of the Jacobian $\tau=X^{}[2a+1+(3w_{1}+w_{2})X^{}]=0$ and the determinant $\Delta>0$. For our analysis this implies that $X^{}=0$ or $X^{}=(5w_{1})^{-1}$. Of course our fixed point $X^{}$ must also satisfy the quintic equation. Inserting this gives a necessary condition and curve in $(\gamma,c)$ space for a Hopf bifurcation \[c_{\text{Hopf}}=-\frac{1}{125w_{1}^{3}}\left(\gamma+\frac{3}{25}\right),\text{ or }c_{\text{Hopf}}=0.\] The determinant is \[\Delta=-\frac{1}{125w_{1}^{2}}(2+15\gamma).\] Thus, the first Hopf bifurcation curve exists only when $\gamma<-2/15$. The second Hopf bifurcation is more complicated since $\Delta=0$ and so nonlinear terms are important. The existence of that curve was found numerically. ### Phase Diagram In general, this system has many different ways in which a limit cycle is destroyed. Numerical experiments indicate that this can occur via a Hopf, cycle bifurcation, infinite period or homoclinic bifurcation. Slightly changing the parameters can change which bifurcation we obtain. From the preceding section, we motivated the choices $w_{1}=-0.05,w_{2}=-0.1,a=-1$ to keep our parameter space $(\gamma,c)$. This leads to a generalized phase diagram of section VI. <figure><img src="content_image/1705.10853/x9.png"><figcaption>Figure 7: Phase diagram with a=−1, γ=b+1, w1=−0.05, w2=−0.1. There is a stablelimit cycle, i.e. periodic phase slips, (green) only in region I. Region IIhas one stable fixed point and region III has three fixed points. The saddle-node bifurcation (SNB) is boundary of the superconducting region. There is anIPB occurring along the yellow line. Possible trajectories in phase space aremapped with purple lines and the dashed yellow line corresponds to increasingr. Note that this phase diagram does not have a bistability region (with u≫1,we observed hysteresis, see section VIII.3).</figcaption></figure> The Hopf and saddle-node bifurcation curves of figure 7 were obtained analytically. The IPB curve $c_{\text{IP}}=c_{\text{IP}}(\gamma_{\text{IP}})$ was found numerically and for comparison is compared to the observed physical limit cycle in figures 3 and 5. Additionally, it was found numerically that the HB in region III, did not exhibit the birth of a stable limit cycle. Possible trajectories of the superconductor through this phase diagram is shown with purple lines. A more generic phase diagram with $w_{2}\neq 2w_{1}$ is given in figure 8. Here, both an IPB and homoclinic bifurcation can destroy the limit cycle. The existence of the homoclinic bifurcation changes the morphology of the phase diagram to now include a bistability region in which the limit cycle (phase slips) and fixed point (superconducting state) coexist. This is particularly encouraging since we also found hysteresis for $u\gg 1$ (see section VIII.3). Possible trajectories of the superconductor through this phase diagram is shown with purple lines. <figure><img src="content_image/1705.10853/x10.png"><figcaption>Figure 8: Phase diagram with a=−1, γ=b+1, w1=−0.09, w2=−0.08. There is astable limit cycle (green) in region I. Region II has one stable fixed pointand region III has three fixed points. Region IV is a bistability region wherea limit cycle and distant attractor coexist. The limit cycle is destroyedalong the yellow line via a homoclinic bifurcation (a saddle point movingtowards the limit cycle), and the dashed yellow line corresponds to increasingu. This homoclinic bifurcation line eventually merges with the SNB line(boundary of region III) and becomes an IPB (similar to Fig. 7).</figcaption></figure> ## VIII Discussion ### Sensitivity to temperature To test the sensitivity of these phase slips to small thermal noise we modified (4) to include a small random noise term uniformly distributed between $[-T_{f},T_{f}]$ at each point in space. Numerical simulations indicate that the system is stable to small fluctuations. The qualitative change is the existence of finite voltage in the superconducting state, however the critical current at which phase slips begin is unchanged. ### Effect of parameter $u$ The parameter $u$ characterizes the penetration of the electric field. In homogeneous wires, it has been found that hysteresis of the phase slip state exists for finite domains with $u\gg 1$Baranov _et al._ (2013). We analyzed $u=0.01,1,10,100$ with $L=20$ and $r=1$ (see figure 9). Another important quantity not yet discussed is that of the retrapping current $j_{r}$. The authors of [Baranov _et al._, 2013] discuss the effect of $u$, numerically simulating the GL equation and finding a curve separating the hysteresis region of the I-V curve through some length dependent critical curve $u_{c2}(L)$. For our simulations of weak links, $u$ small (for $r=1$, $u<30$ is small enough), $j_{r}=j_{c}$. However for $u\gg 1$, $j_{r}<j_{c}$, this leads to hysteresis in the I-V curve (see figure 10). <figure><img src="content_image/1705.10853/x11.png"><figcaption>Figure 9: I-V curve with different u=0.01,1,10, (color online). The criticalcurrent does not change, however the slope as j0→jc increases as u→0.Additionally, jc=jr (the reentrance current) for all u shown (no hysteresis).</figcaption></figure> <figure><img src="content_image/1705.10853/x12.png"><figcaption>Figure 10: I-V curve for u=100. Hysteresis is present, the saddle-nodebifurcation still occurs at jc≈0.067, however jr≈0.0614 below which thesuperconducting state reappears.</figcaption></figure> ### Physical quantities in simplified system The phase diagram is in $(\gamma,c)$ space. We can relate the important physical quantities $u,r,j_{0},L$ to $\gamma,c$ by using appendix .4. The coefficient $c$ is strongly affected by the parameter $u$ and the current $j_{0}$. Consider $j_{0}<j_{c}$ and $u\to 0$, then we know that there is no voltage (i.e. $K=0$), and $\alpha=-1$. This implies that $c\sim-u\zeta^{2}$ for some $\zeta(r,j_{0},L)$ for small $u$. At a significantly large enough $u$ we expect our initial trajectory to begin from a region in figure 8 where hysteresis is possible. Increasing the current $j_{0}>j_{c}$ switches $\alpha=1$ and $K\neq 0$, as $j_{0}$ continues to increase, $F$ decreases and we expect $c$ to change sign as we continue to increase it, which explains our motivation for the direction of trajectories. Increasing $r$ lowers $j_{c}$ and so we expect the trajectories to spend more time in the phase slip state, which leads us to expect that $c$ decreases. A similar argument, leads us to assume the same holds for $\gamma$ (see figure 7 and 8). The effects on $\gamma$ are more complicated for the current and probably non-monotonous in a general case. From physical arguments we know that the trajectories must begin in the superconducting state and move into the phase slip state via either an IPB or homoclinic bifurcation. Comparing this to the phase diagrams, we see that as $j_{0}$ increases, $\gamma$ must decrease. We also attempt to justify this from the terms in appendix .4. We consider the scaling from section VI, which implied $b=c_{3}/c_{2}^{2}$. We noted that $F$ is decreasing as $j_{0}$ increases (where $F^{\prime}$ is relatively unchanged). Again, employing appendix .4, we see that $c_{3}$ is decreasing with the current since the positive terms involve $F$ and the negative terms involve $F^{\prime}$. Finally we use the fact that $b=c_{3}/c_{2}^{2}$ to deduce that $b$ must be decreasing and since $\gamma=(b-a)/(2a+1)^{2}$, we see that $\gamma$ is also decreasing with the current. ## IX Conclusion We have considered a weak-link superconductor qualitatively similar to other weak-link systems, but fundamentally different in mechanism. We demonstrated the existence of a superconducting state and a PSC periodic state separated by a critical current $j_{c}$. This current was calculated asymptotically and agrees very well with numerical simulations. We then extracted a coupled ODE system from the TDGL equations using weakly nonlinear theory and showed under certain choices of parameters, an infinite period bifurcation and homoclinic bifurcations can occur. This demonstrates that the dynamics of phase-slip behavior in weak links described by the TDGL equations can be correctly captured by a simpler system of two coupled ordinary differential equations. Further research is to extend this analysis to two dimensions. We anticipate additional transitions from phase slips occurring instantly inside the weak link to a more complicated dynamic regime involving phase slips and nucleation of vortex pairs, similar to that in [Weber and Kramer, 1991]. Another interesting generalization is to include disorder in the transverse direction inside the weak link. Possibly, some of the vortices will be pinned in the weak link. It may. in turn, lead to further suppression of the critical current. The work was supported by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy for computations. The Office of Science, Advanced Scientific Computing Research and Basic Energy Science, Division of Materials Science and Engineering for analysis. ## X Appendix ### No voltage in the superconducting state We begin by multiplying (2a) by $\Psi^{}$ and we differentiate (2b) with respect to $x$. This gives \[u(i\|\Psi\|^{2}\mu+\Psi^{}\partial_{t}\Psi) =\Psi^{}\partial_{x}^{2}\Psi+[\nu(x)-\|\Psi\|^{2}]\|\Psi\|^{2}\] (17) \[0 =\Im{(\Psi^{}\partial_{x}^{2}\Psi)}-\partial_{x}^{2}\mu.\] (18) Taking the imaginary part of (17) and substituting this result into (18), we obtain \[\partial_{x}^{2}\mu-u\|\Psi\|^{2}\mu=u\Im{(\Psi^{}\partial_{t}\Psi)}.\] (19) Far from the inclusion, all the applied current is supercurrent and so if $L\gg r$, we expect $j_{0}=\Im{(\Psi^{}\partial_{x}^{2}\Psi\|)_{x=\pm L}}$, which implies that $\partial_{x}\mu(\pm L)=0$. Multiplying (19) by $\mu$ and integrating over the domain gives \[\int_{-L}^{L}\left[(\partial_{x}\mu)^{2}+u\|\Psi\|^{2}\mu^{2}\right)\,dx=\mu \partial_{x}\mu\bigg{\|}_{-L}^{L}+u\int_{-L}^{L}\Im{(\Psi^{}\partial_{t}\Psi)} \,dx.\] Noting the boundary conditions for $\mu$ and the fact that $\partial_{t}\Psi=0$ (stationary state), we see that $\mu\equiv 0$. ### Critical current calculation We separate (7) by region (superconducting vs. normal metal) and then take the first integral to obtain the equations \[E_{S} =(\partial_{x}F)^{2}+F^{2}+j_{0}^{2}F^{-2}-\frac{1}{2}F^{4}, x\not\in I\] (20) \[E_{I} =(\partial_{x}F)^{2}-CF^{2}+j_{0}^{2}F^{-2}-\frac{1}{2}F^{4}, x\in I.\] (21) Now, far from the inclusion (near the boundary of the superconductor), $F\to F_{\infty}$ a constant. Assuming the relevant approximation that $j_{0}\ll 1$, we see that $F_{\infty}^{2}\approx 1-j_{0}^{2}$. Inserting this into (20), implies that $E_{S}\approx\frac{1}{2}+j_{0}^{2}$. We now use the large $C$ approximation that $C\gg j_{0}^{2}F^{-4}$. Proceeding, we obtain \[F_{I}(x)=K_{1}e^{(\|x\|-r)\sqrt{C}},\] where we have introduced the radius $r$ of the inclusion. Solving the outer region at first order is given by \[F_{S}(x)=\tanh\left(\frac{\|x\|-K_{2}}{\sqrt{2}}\right).\] The two constants are determined by the continuity conditions at the boundary of the inclusion. By symmetry, we may analyze just one side of the boundary, then our conditions are \[K_{1} =\tanh\left(\frac{r-K_{2}}{\sqrt{2}}\right)\] (22a) \[K_{1} =\frac{1}{\sqrt{2C}}\operatorname{\text{sech}}^{2}\left(\frac{r-K _{2}}{\sqrt{2}}\right).\] (22b) Solving for $K_{1}$ and $K_{2}$, we obtain \[K_{1} =\frac{1}{\sqrt{2C}}+O\left(\frac{1}{C}\right)\] (23a) \[K_{2} =r-\frac{1}{\sqrt{C}}+O\left(\frac{1}{C}\right).\] (23b) Note the identity $E_{S}-E_{I}=(1+C)F^{2}(r)\geq 0$. This implies that \[E_{I}\approx j_{0}^{2}-\frac{1}{2C}\ll 1.\] Motivated by this, we assume that $E_{I}$ is a small parameter. At first order then $E_{I}=0$ and looking at $x=0$ we see that \[E_{I}=0=-CF^{2}(0)+j_{0}^{2}F^{-2}(0)-\frac{1}{4}F^{4}(0),\] where the derivative has vanished by symmetry. Since $F$ is small in the inclusion, the last term can be neglected and we are left with $j_{0}\approx\sqrt{C}F^{2}(0)$. This leads to Eq. (8). ### Numerical analysis of $j_{c}$ To analyze the error associated with calculating $j_{c}$ numerically, we took $L=20$ and varied $\Delta x$. The results are shown in figure 11. Assuming the error is linear, we extrapolate the critical current to be $j_{c}\approx 0.06366$, which is in excellent agreement with the linear system solved using the shooting method with $\Delta x=0.001$. <figure><img src="content_image/1705.10853/x13.png"><figcaption>Figure 11: Convergence of jc as a function of Δx. As Δx→0, jc approaches thetrue value. Dynamic simulations took place with Δx=0.05.</figcaption></figure> For fixed $\Delta x=0.05$, we measured the sensitivity of $L$ on $j_{c}$ and found no significant change for $L\gg r$ (typically $L>5r$ was sufficient). ### Weakly nonlinear calculation To obtain the weakly nonlinear system, we analyze near $j_{0}=j_{c}+\epsilon$ where $\|\epsilon\|\ll 1$. Linearizing about the base state near $\epsilon=0$ with $\Psi=(F+\eta)e^{i\phi}$. From before, we saw that $\epsilon=0$ leads to a degenerate zero eigenvalue implying that the linearized system has a generalized eigenvector solution where $\mathcal{L}\eta_{1}=0$ and $\mathcal{L}\eta_{2}=\eta_{1}$. We use Ansatz $\eta=A\delta\eta_{1}+\delta^{2}B(\eta_{2}+z)+\delta^{3}\zeta$ where $\eta_{k}=\left(\begin{matrix}U_{k}\\ V_{k}\end{matrix}\right)$ and $\epsilon=\alpha\delta^{2}$. Inserting this into (4a)–(4c) with the aid of mathematica and obtain at first order the ODE for $A$ \[\mathcal{L}z =\] At next order, we obtain the ODE for $B$ (where we have already projected onto the eigenvector) \[u\partial_{\tau}B\langle U_{1}^{\dagger},U_{2}\rangle = \bigg{\langle}U_{1}^{\dagger},A^{3}\left\{uV_{1}\int_{-L/2}^{x}[ \phi^{\prime}(U_{1}^{2}+V_{1}^{2})+U_{1}V_{1}^{\prime}-U_{1}^{\prime}V_{1}]\, ds-U_{1}^{3}-U_{1}V_{1}^{2}\right\}-2KF\phi^{\prime}+\] \[AB\left[uV_{2}\int_{-L/2}^{x}(2F\phi^{\prime}U_{1}+FV_{1}^{ \prime}-F^{\prime}V_{1})\,ds-uV_{1}\int_{-L/2}^{x}(2R\phi^{\prime}U_{2}+FV_{2} ^{\prime}-R^{\prime}V_{2})\,ds-2R(3U_{1}U_{2}+V_{1}V_{2})\right]\bigg{\rangle}\] \[u\partial_{\tau}B\langle V_{1}^{\dagger},V_{2}\rangle = \bigg{\langle}V_{1}^{\dagger},-A^{3}\left\{U_{1}^{2}V_{1}+V_{1}^{ 3}+uU_{1}\int_{-L/2}^{x}[\phi^{\prime}(U_{1}^{2}+V_{1}^{2})+U_{1}V_{1}^{\prime }-U_{1}^{\prime}V_{1}]\,ds\right\}-\] \[AB\bigg{\{}U_{2}\left[2FV_{1}+u\int_{-L/2}^{x}(FV_{2}^{\prime}-F ^{\prime}V_{2}+2F\phi^{\prime}U_{2})\,ds\right]+U_{1}\bigg{[}2RV_{2}+\] \[u\int_{-L/2}^{x}(FV_{2}^{\prime}-F^{\prime}V_{2}+2F\phi^{\prime} U_{2})\,ds\bigg{]}+uF\int_{-L/2}^{x}[U_{2}V_{1}^{\prime}-V_{2}U_{1}^{\prime}+U _{1}V_{2}^{\prime}-V_{1}U_{2}^{\prime}+2\phi^{\prime}(U_{1}U_{2}+V_{1}V_{2})] \,ds\bigg{\}}\] \[+K\left(2F^{\prime}-uF\int_{-L/2}^{x}F^{2}\,ds\right)+u\alpha xF \bigg{\rangle}\] ## References Blatter _et al._ (1994)G. 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1705.07012	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 110613, "num_imgs": 0, "llama3_tokens_count": 40772 }	[]	# A note on the metastability in three modifications of the standard Ising model K. Bashiri ¹ [FOOTNOTE:1][ENDFOOTNOTE] May 1, 2017 ###### Abstract We consider three extensions of the standard 2D Ising model with Glauber dynamics on a finite torus at low temperature. The first model (see Chapter 2) is an anisotropic version, where the interaction energy takes different values on vertical and on horizontal bonds. The second model (Chapter 3) adds next-nearest-neighbor attraction to the standard Ising model. And the third model (Chapter 4) associates different alternating signs for the magnetic fields on even and odd rows. All these models have already been studied, and results concerning metastability have been established using the so-called _pathwise approach_ (see [7],[8],[9]). In this text, we extend these earlier results, and apply the _potential-theoretic approach_ to metastability to obtain more precise asymptotic information on the transition time from the metastable phase to the stable phase. ††† [FOOTNOTE:†][ENDFOOTNOTE] [FOOTNOTE:†][ENDFOOTNOTE] [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction Section 1.1 provides basic background and motivation for metastability. In Section 1.2 we give a rough overview of this text, and Section 1.3 introduces quickly the main method that is used in this text in a general context. In Section 1.4, we add some further definitions that will be used for the remaining part of this work. ### Background In many physical, biological or chemical evolutions, one can observe a phenomenon called _metastability_. If the states of the system are associated to an energy functional, this phenomenon can be described as follows. For a relatively long time, the state of the system resides around a local minimum of the energy landscape, which is not the global minimum. This state is called the _metastable state_. However, under thermal fluctuations and after many unsuccessful attempts the system can finally free itself from this valley in the energy landscape and it manages the crossover to the global minimum, which is called the _stable state_. Often, this crossover is triggered by reaching a _critical state_ in the system. An example is _over-saturated water vapour_, where below critical temperature, the formation of a critical droplet is needed to achieve the transition from the gas-phase to the liquid-phase. An analogue situation holds for _over-cooled liquids_ and for _magnetic hysteresis_. Through the last decades many mathematical models have been built to study this phenomenon. One is mostly interested in three topics: * The average transition time from the metastable to the stable state, * The exponential distribution of this transition time, * The typical paths for the transition from the metastable to the stable state. Mainly two methods have been crystallized to be very fruitful to tackle these problems. The first one is the _pathwise approach_, initiated by Cassandro, Galves, Olivieri and Vares in [6]. Motivated by the _Freidlin and Wentzell theory_, one uses large deviation estimates on the path space to identify the most likely paths of the system for the transition from the metastable to the stable state. The advantage is a very detailed description of the tube of typical paths for the transition, but at the same time the average transition time can only be computed up to a multiplicative factor of the order $e^{\varepsilon\beta}$ as $\beta\rightarrow\infty$, where $\beta$ is the inverse temperature and $\varepsilon>0$ is independent of $\beta$ and can be chosen arbitrary small. For an extensive discussion on the pathwise approach to metastability, the reader is referred to the monograph [11] by Olivieri and Vares. The second method is the _potential-theoretic approach to metastabilty_, which was initiated by Bovier, Eckhoff, Klein and Gayrard in [2]. Here, one uses _potential theory_ to rewrite the average transition time in terms of quantities from _electric networks_, namely _capacities_. Now, using variational principles for the capacity, the average transition time can be computed up to a multiplicative error that tends to one as $\beta\rightarrow\infty$, which provides a sharp estimate. This method is also the basis of this text, and in Section 1.3, we will shortly review a general recipe to obtain metastability results for a stochastic process on a finite graph, whose dynamics are given through a metropolis algorithm. For a more detailed overview, we refer to the 2015 monograph [4] by Bovier and den Hollander (especially Chapter 16). Probably the easiest application of this methods, where one can rigorously investigate metastable behavior, is the two-dimensional standard Ising model on a finite torus in the low temperature regime. Neves and Schonmann applied 1991 in [10] the pathwise approach to this model, and in the year 2002 the potential-theoretic approach was used in [5] by Bovier and Manzo. In this paper, we study three modifications of the Ising model that are defined in chapters 2–4. The pathwise approach has already been applied to all these models in [7], [8] and [9], respectively. In Chapters 7.7 – 7.10 of [11], a brief overview on these three papers is given. Here, we complement these results and apply the potential-theoretic approach to obtain more precise information on topic i). All three models differ from the standard Ising model mainly in the fact that we lose the applicability of _isoperimetrical inequalities_. Namely, in the Ising case, for a given number of _up-spins_, the configuration with minimal energy is a droplet of up-spins with its shape being a square (or a quasi-square) with a possible bar of up-spins attached to one of its (longer) sides. Here, we do not have this property. Instead we need to look at the _stability_ of certain classes of configurations separately in order to specify the metastable and the critical state rigorously. ### Outline We first give a structural outline of this work. The next subsection is a quick overview of the setting and the results of Chapter 16 in [4].¹ A dynamical spin-flip model on the two-dimensional lattice is introduced, which is driven by a general energy function. Also definitions concerning the geometrical properties of the energy landscape are given, which is crucial for the study of metastability. At the end of that section, we state the so-called _metastability theorems_ that cover the topics i) and ii) of those listed in Section 1.1. Topic iii) has already been solved in [7],[9] and [8], respectively. Hence, in order to apply the potential-theoretic approach to the three above mentioned models with their specific energy functionals, we need to verify the conditions of the metastability theorems and to compute the model-dependent parameters in that cases. This is the content of chapters 2–4. Finally, Section 1.4 introduces some additional objects that will be needed for all three cases. [FOOTNOTE:1][ENDFOOTNOTE] In Chapter 2 we look at the same model as in [7], where the interaction between neighboring spins is anisotropic in the sense that the attraction on horizontal bonds is stronger than on vertical bonds. In Chapter 3 we allow next-nearest-neighbor attraction, i.e. two spins that have euclidean distance of $\sqrt{2}$ feel an interaction energy, which is strictly less than the interaction energy between nearest-neighbor bonds. This has the physical intuition that next-nearest-neighbor attraction is seen as a perturbation of nearest-neighbor attraction. An interesting fact is that the local minima of the energy landscape are given by droplets of _octogonal shape_. For the pathwise approach to this model we refer to [8]. Chapter 4 modifies the standard Ising model by allowing the magnetic field to take alternating signs and absolute values on even and on odd rows. The pathwise approach has been applied to this model in [9]. ### Potential-theoretic approach to metastability Let $\Lambda\subset{\mathbb{Z}}^{2}$ be a finite, square box with periodic boundary conditions, centered at the origin. $S=\{-1,1\}^{\Lambda}$ will be called the _configuration space_. An element $\sigma\in S$ is called _configuration_, and at each _site_$x\in\Lambda$, $\sigma(x)\in\{-1,1\}$ is called the _spin-value_ at $x$. By abuse of notation, we often identify each configuration $\sigma\in S$ with the sites that have spin $+1$, i.e. \[\sigma\equiv\{x\in\Lambda\ \|\ \sigma(x)=+1\}.\] (1.1) Moreover, we represent $\sigma$ geometrically by identifying to each $x\in\sigma$ a unit square centered at $x$. See Figure 1 for an example. [FIGURE:S1.F1][ENDFIGURE] The energy of the system is given by a _Hamiltonian_$H:S\rightarrow{\mathbb{R}}$. If $\beta>0$ is the _inverse temperature_, the _Gibbs measure_ associated with $H$ and $\beta$ is given by \[\mu_{\beta}(\sigma)=\frac{1}{Z_{\beta}}e^{-\beta H(\sigma)}, \qquad\text{for }\sigma\in S,\] (1.2) where $Z_{\beta}$ is a normalization constant called _partition function_. For $\sigma\in S$ and $x\in\Lambda$, we define $\sigma^{x}\in S$ by \[\sigma^{x}(y)=\begin{cases}\sigma(y)&:y\neq x,\\ -\sigma(x)&:y=x.\end{cases}\] (1.3) For all $\sigma,\sigma^{\prime}\in S$, we write $\sigma\sim\sigma^{\prime}$, if there exists $x\in\Lambda$ such that $\sigma^{x}=\sigma^{\prime}$. This induces a graph structure on $S$ by defining an edge between each $\sigma,\sigma^{\prime}\in S$, whenever $\sigma\sim\sigma^{\prime}$. The dynamics of the system is given by the continuous time Markov Chain $(\sigma_{t})_{t\geq 0}$ on $S$, whose generator ${\mathcal{L}}_{\beta}$ is given by \[({\mathcal{L}}_{\beta}f)(\sigma)=\sum_{x\in\Lambda}c_{\beta}( \sigma,\sigma^{x})(f(\sigma^{x})-f(\sigma)),\] (1.4) where $f:S\rightarrow{\mathbb{R}}$ and \[c_{\beta}(\sigma,\sigma^{\prime})=\begin{cases}e^{-\beta\max\{0, H(\sigma^{\prime})-H(\sigma)\}}&:\sigma\sim\sigma^{\prime},\\ 0&:\text{else}.\end{cases}\] (1.5) Notice that for $\beta=\infty$, only moves to configurations with lower or equal energy are permitted. Moreover, one can immediately see that the following _detailed balance condition_ holds: \[\mu_{\beta}(\sigma)c_{\beta}(\sigma,\sigma^{\prime})=\mu_{\beta}(\sigma^{ \prime})c_{\beta}(\sigma^{\prime},\sigma)\qquad\forall\,\sigma,\sigma^{\prime} \in{\mathcal{X}}_{\beta}^{(n_{\beta})}.\] (1.6) Hence, the dynamics is _reversible_ with respect to the Gibbs measure. The law of $(\sigma_{t})_{t\geq 0}$ given that $\sigma_{0}=\sigma\in S$ will be denoted by ${\mathbb{P}}_{\sigma}$, and for a set $A\subset S$, we denote its _first hitting time after the starting configuration has been left_ by $\tau_{A}$, i.e. \[\tau_{A}=\inf\{t>0\|\ \sigma_{t}\in A,\,\exists\,0<s<t:\sigma_{s} \neq\sigma_{0}\}.\] (1.7) A few definitions are needed to describe the geometry of the energy landscape of the system. Definition 1.1: * _Let_ $\sigma,\sigma^{\prime}\in S$_. The_ communication height _between_ $\sigma$ _and_ $\sigma^{\prime}$ _is defined by_ \[\Phi(\sigma,\sigma^{\prime})=\min_{\gamma:\sigma\rightarrow\sigma ^{\prime}}\max_{\eta\in\gamma}H(\eta),\] (1.8) _where the minimum is taken over all finite paths_ $\gamma$ _of allowed moves in_ $S$ _going from_ $\sigma$ _to_ $\sigma^{\prime}$_._ * _Let_ $\sigma,\sigma^{\prime}\in S$_. A finite path_ $\gamma:\sigma\rightarrow\sigma^{\prime}$ _is called_ optimal path _between_ $\sigma$ _and_ $\sigma^{\prime}$_, if_ \[\Phi(\sigma,\sigma^{\prime})=\max_{\eta\in\gamma}H(\eta).\] (1.9) _The set of all optimal paths between_ $\sigma$ _and_ $\sigma^{\prime}$ _is denoted by_ $(\sigma\rightarrow\sigma^{\prime})_{\mathrm{opt}}$_._ * _Let_ $\sigma\in S$_. The_ stability level _of_ $\sigma$ _is defined by_ \[V_{\sigma}=\min_{\eta\in S:H(\eta)<H(\sigma)}\Phi(\sigma,\eta)-H (\sigma).\] (1.10) * _The set of_ stable states _in_ $S$ _is defined by:_ \[S_{\mathrm{stab}}=\{\sigma\in S\ \|\ H(\sigma)=\min_{\eta\in S}H( \eta)\}.\] (1.11) * _The set of_ metastable states _in_ $S$ _is defined by:_ \[S_{\mathrm{meta}}=\{\sigma\in S\ \|\ V_{\sigma}=\max_{\eta\in S \setminus S_{\mathrm{stab}}}V_{\eta}\,\}.\] (1.12) According to the motivation in Section 1.1, to study metastable behavior, it will be crucial to define the notion of a _critical state_ in a mathematically precise way. This is done in the following Definition 1.2: _Let $(m,s)\in S_{\mathrm{meta}}\times S_{\mathrm{stab}}$ and let_ \[\Gamma^{}=\Phi(m,s)-H(m).\] (1.13) _Then $(\mathcal{P}^{\star}(m,s),{\mathcal{C}}^{\star}(m,s))$ is defined as the maximal subset of $S\times S$ such that_ 1. _, and_ _,_ 2. $\forall\sigma\in{\mathcal{P}}^{\star}(m,s)\ :\ \Phi(m,\sigma)<\Phi(\sigma,s)$_,_ 3. $\forall\sigma^{\prime}\in{\mathcal{C}}^{\star}(m,s)\ \exists\gamma:\sigma^{ \prime}\to s\ :\ \max_{\eta\in\gamma}H(\eta)-\ H(m)\leq\Gamma^{}, \forall\eta\in\gamma:\Phi(m,\eta)\geq\Phi(\eta,s)$_._ _We call ${\mathcal{P}}^{\star}(m,s)$ the set of protocritical states, and ${\mathcal{C}}^{\star}(m,s)$ the set of critical states._ We are now in the position to formulate the metastability theorems (see Theorem 16.4 – 16.6 in [4]). These will hold subject to the following hypothesis * $S_{\mathrm{meta}}=\{m\}$ and $S_{\mathrm{stab}}=\{s\}$, where $m,s\in S$. One challenge is to verify this hypothesis in the Chapters 2–4 for the three models. Under (H1), it would not lead to confusions, if we abbreviate ${\mathcal{P}}^{\star}={\mathcal{P}}^{\star}(m,s)$ and ${\mathcal{C}}^{\star}={\mathcal{C}}^{\star}(m,s)$. Theorem 1.3: _Subject to (H1), it holds that_ * $\lim_{\beta\rightarrow\infty}{\mathbb{P}}_{m}[\tau_{{\mathcal{C}}^{\star}}< \tau_{s}\ \|\ \tau_{s}<\tau_{m}]=1$_,_ * _If, moreover, the following assumption holds_ * $\sigma^{\prime}\rightarrow\|\{\sigma\in{\mathcal{P}}^{\star}(m,s)\ :\ \sigma \sim\sigma^{\prime}\}\|$ _is constant on_ ${\mathcal{C}}^{\star}$_,_ _then for all_ $\chi\in{\mathcal{C}}^{\star}$_, it holds:_ $\lim_{\beta\rightarrow\infty}{\mathbb{P}}_{m}[\sigma_{\tau_{{\mathcal{C}}^{ \star}}}=\chi]=\frac{1}{\|{\mathcal{C}}^{\star}\|}.$__ Theorem 1.4: _Subject to (H1), it holds that_ * $\lim_{\beta\rightarrow\infty}\lambda_{\beta}{\mathbb{E}}_{m}[\tau_{s}]=1$_, where_ $\lambda_{\beta}$ _is the second largest eigenvalue of_ $-{\mathcal{L}}_{\beta}$_._ * $\lim_{\beta\rightarrow\infty}{\mathbb{P}}_{m}[\tau_{\sigma}>t\cdot{\mathbb{E}} _{m}[\tau_{s}]]=e^{-t}$ _for all_ $t\geq 0$_._ Theorem 1.5: _Subject to (H1), it holds that_ * _There exists a constant_ $K\in(0,\infty)$ _such that_ $\lim_{\beta\rightarrow\infty}e^{-\beta\Gamma^{\star}}{\mathbb{E}}_{m}[\tau_{s} ]=K$_,_ * _Let_ * $S^{\star}\subset S$ _be the subgraph obtained by removing all vertices_ $\eta$ _with_ $H(\eta)>\Gamma^{\star}+H(m)$ _and all edges incident to these vertices,_ * $S^{\star\star}\subset S^{\star}$ _be the subgraph obtained by removing all vertices_ $\eta$ _with_ $H(\eta)=\Gamma^{\star}+H(m)$ _and all edges incident to these vertices,_ * $S_{m}=\{\eta\in S\ \|\ \Phi(m,\eta)<\Phi(\eta,s)=\Gamma^{\star}+H(m)\}$_,_ * $S_{s}=\{\eta\in S\ \|\ \Phi(\eta,s)<\Phi(m,\eta)=\Gamma^{\star}+H(m)\}$_,_ * $S_{1},\dots,S_{I}\subset S^{\star\star}$ _be such that_ $S^{\star\star}\setminus(S_{m}\cup S_{m})=\cup_{i=1}^{I}S_{i}$ _and each_ $S_{i}$ _is a maximal set of communicating configurations._ _Then:_ \[\frac{1}{K}=\min_{C_{1},\dots,C_{I}\in[0,1]}\min_{ \begin{subarray}{c}h:S^{\star}\rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{S_{m}}}=1,{\left.\kern-1.2pth\vphantom {\|}\right\|_{S_{s}}}=0,{\left.\kern-1.2pth\vphantom{\|}\right\|_{S_{i}}}=C_{i}\, \forall i\end{subarray}}\frac{1}{2}\sum_{\eta,\eta^{\prime}\in S^{\star}} \mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[h(\eta)-h(\eta^{\prime})]^{2}.\] (1.14) Theorem 1.3 says that the set of critical states is a _gate_ for the transition, i.e. that ${\mathcal{C}}^{\star}$ has to be reached in order to cross over from the metastable to the stable state. If the additional assumption holds, then part b) of Theorem 1.3 says that the entrance into ${\mathcal{C}}^{\star}$ is uniformly distributed. Theorem 1.4 represents the average transition time of the system in terms of the spectrum of its generator and part b) covers topic ii) of section 1.1. Finally, Theorem 1.5 covers topic i) and gives a variational formula to compute the prefactor. ### Further definitions We conclude this chapter with some definitions hat are used in all three situations in Chapters 2–4. * For $x\in{\mathbb{R}}$, $\lceil x\rceil$ denotes the smallest integer greater than $x$. * For $l_{1},l_{2}\in{\mathbb{N}}$, $R(l_{1}\times l_{2})$ denotes the set of all configurations consisting of a single rectangle with horizontal length $l_{1}$ and vertical length $l_{2}$ somewhere on the torus $\Lambda$. An element $\sigma\in R(l_{1}\times l_{2})$ is called _rectangle_ and will often be denoted by $l_{1}\times l_{2}$, since usually we can ignore the position of the rectangle in the torus. For this reason, by abuse of notation, we often identify the whole set $R(l_{1}\times l_{2})$ with $l_{1}\times l_{2}$. We also define $R(l_{1},l_{2})=R(l_{1}\times l_{2})\cup R(l_{2}\times l_{1})$, since sometimes the arguments are also symmetric with respect to rotation of the rectangle. If $\|l_{1}-l_{2}\|=1$ or $\|l_{1}-l_{2}\|=0$, then $l_{1}\times l_{2}$ is called _quasi-square_ or _square_, respectively. $1\times l_{2}$ is called _vertical bar_ or _column_ and $l_{1}\times 1$ is called _horizontal bar_ or _row_. * For a rectangle $R\in S$, we denote by $P_{H}R\in{\mathbb{N}}$ its _horizontal length_, and by $P_{V}R\in{\mathbb{N}}$ its _vertical length_. * Two droplets on the torus are called _isolated_, if their Euclidean distance is greater or equal to $\sqrt{2}$. * Let $\sigma\in S$ and $x\in\Lambda$ be such that $\sigma(x)=+1$. Then $x$ is called _protuberance_, if $\sum_{y\in\Lambda:\|y-x\|=1}\sigma(y)=-2$. * If $\sigma\in S$ consists of a single, connected droplet, then $R(\sigma)$ is the smallest rectangle that contains $\sigma$. * A _row or column_ of a connected configuration $\sigma\in S$ is defined as the intersection of a row or a column of $\Lambda$ with $\sigma$. * For $\sigma\in S$, let $\|\sigma\|$ be the _area_ of $\sigma$, i.e. its number of $(+1)$–spins. Further, $\partial(\sigma)$ is the Euclidean boundary of $\sigma$ in its geometric representation and $\|\partial(\sigma)\|$ denotes the _perimeter_, i.e. the length of $\partial(\sigma)$. * For $A\subset S$, let $\partial^{+}A=\{\sigma\in S\setminus A\,\|\,\exists\sigma^{\prime}\in S:\sigma \sim\sigma^{\prime}\}$ denote the _outer boundary_$A$. We also define $A^{+}=A\cup\partial^{+}A$. Moreover, if $\eta\in S$, then $A\sim\eta\subset S$ is defined by $A\sim\eta=\{\sigma\in A\,\|\,\sigma\sim\eta\}$. * In all three situations in the following chapters, we will have to show that \[m=\boxminus,\qquad\text{and}\qquad s=\boxplus,\] (1.15) where $\boxminus\in S$ is the configuration, where all spin values are $-1$ and $\boxplus$ is the configuration with all spin values being $+1$. ## 2 Anisotropic Ising model In the same setting as in Section 1.3 let the _Hamiltonian_ be given by \[H^{\mathrm{A}}(\sigma)=-\frac{J_{H}}{2}\sum_{(x,y)\in\Lambda_{H}^{\star}} \sigma(x)\sigma(y)-\frac{J_{V}}{2}\sum_{(x,y)\in\Lambda_{V}^{\star}}\sigma(x) \sigma(y)-\frac{h}{2}\sum_{x\in\Lambda}\sigma(x),\] (2.1) where $\sigma\in S$, $J_{H},J_{V},h>0$, $\Lambda_{H}^{\star}$ is the set of _unordered horizontal nearest-neighbor bonds_ in $\Lambda$ and $\Lambda_{V}^{\star}$ is the set of _unordered vertical nearest-neighbor bonds_ in $\Lambda$. Using the geometric representation of $\sigma$, one can rewrite $H^{\mathrm{A}}(\sigma)$ as \[H^{\mathrm{A}}(\sigma)=H^{\mathrm{A}}(\boxminus)-h\|\sigma\|+J_{H}\|\partial_{V}( \sigma)\|+J_{V}\|\partial_{H}(\sigma)\|,\] (2.2) where $\|\partial_{V}(\sigma)\|$ is the length of the vertical part of $\partial(\sigma)$ and $\|\partial_{H}(\sigma)\|$ is the length of the horizontal part of $\partial(\sigma)$. In Figure 1, we get that $\|\partial_{V}(\sigma)\|=34$ and $\|\partial_{H}(\sigma)\|=40$. The critical length in this model will be given by \[L_{V}^{\star}=\left\lceil\frac{2J_{V}}{h}\right\rceil.\] (2.3) The following assumptions will be made for this chapter. Assumption 2.1: 1. $J_{H}>J_{V}$_,_ 2. $2J_{V}>h$_,_ 3. $\frac{2J_{V}}{h}\notin{\mathbb{N}}$_,_ 4. $\|\Lambda\|>\left(\max\{\frac{2J_{H}}{hL_{V}^{\star}-2J_{V}},\frac{2J_{H}(L_{V}^ {\star}-1)}{2J_{V}-h(L_{V}^{\star}-1)}+L_{V}^{\star}\}\right)^{2}$_._ By symmetry, assumption a) could have also been chosen the other way around. Assumption c) is made to avoid degeneracy in the arguments, and b) induces that the dynamics prefers aligned neighboring spins than $(+1)$–spins. In that way the dynamics, starting from $\boxminus$, will have to increase the energy in order to obtain $(+1)$–spins and to reach $\boxplus$, since it needs to break bonds in $\boxminus$. This is essential to obtain the metastable behavior of the system. Assumption d) implies that after the critical state has reached for an optimal path, it can avoid to reach that energy level again and that it is not profitable to enlarge a droplet such that one side is subcritical and the other side wraps around the torus. Moreover, d) assures that the torus is large enough to contain a critical droplet. More details on these facts will be seen later. It immediately follows from Assumption 2.1 c) that \[(L_{V}^{\star}-1)h<2J_{V}<L_{V}^{\star}h.\] (2.4) Before stating the main result of this chapter, we need the following definition. Definition 2.2: _• Let $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$ denote the set of all configurations consisting only of a rectangle from $R(L_{V}^{\star}-1,L_{V}^{\star})$ and with an additional protuberance attached to one of its longer sides. The right droplet in Figure 2 provides an example._ _• Let_ $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{2pr}}$ _be the set of all configurations that are obtained from a configuration in_ $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$ _by adding a second_ $(+1)$_–spin next to the protuberance._ Now we can formulate the main result of this chapter. Theorem 2.3: _Under Assumption 2.1, the pair $(\boxminus,\boxplus)$ satisfies (H1) and (H2) so that Theorems 1.3–1.5 hold for the anisotropic Ising model._ _Moreover, the quadruple_ $({\mathcal{P}}^{\star},{\mathcal{C}}^{\star},\Gamma^{\star},K)$ _is given by_ * ${\mathcal{P}}^{\star}=R(L_{V}^{\star}-1,L_{V}^{\star})$_,_ * ${\mathcal{C}}^{\star}=R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$_,_ * $\Phi(\boxminus,\boxplus)-H^{\mathrm{A}}(\boxminus)=2L_{V}^{\star}(J_{H}+J_{V}) -h(1+(L_{V}^{\star}-1)L_{V}^{\star})=:\Gamma^{\star}_{A}=:E_{A}^{\star}-H^{ \mathrm{A}}(\boxminus)$_,_ * $K^{-1}=\frac{4(2L_{V}^{\star}-1)}{3}\|\Lambda\|$_._ _Proof_. The proof is divided into the Sections 2.1–2.6. $\square$ [FIGURE:S2.F2][ENDFIGURE] ### Proof of $\Phi(\boxminus,\boxplus)-H^{\mathrm{A}}(\boxminus)\leq\Gamma^{\star}_{A}$ It will be enough to construct a path $\gamma=(\gamma(n))_{n\geq 0}:\boxminus\rightarrow\boxplus$ such that \[\max_{\eta\in\gamma}H^{\mathrm{A}}(\eta)\leq H^{\mathrm{A}}(\boxminus)+\Gamma^ {\star}_{A}=E_{A}^{}.\] (2.5) This path will be called _reference path_. _Construction of $\gamma$._ Let $\gamma(0)=\boxminus$. In the first step an arbitrary spin is flipped from $-1$ to $+1$. Then $\gamma$ will first pass through a sequence of squares and quasi-squares as follows. If at some step $i$, $\gamma(i)$ is a square, then a $(+1)$–spin is added somewhere above the droplet. Afterwards, this row is filled by successively flipping in this row adjacent $(-1)$–spins until the droplet has the shape of a quasi-square. Now the same thing as before is done but on the right of the droplet. Hence, at any step $i$, $\gamma(i)$ is either a square or a quasi square and possibly with an attached horizontal or vertical bar. This procedure is stopped, when $R((L_{V}^{\star}-1)\times L_{V}^{\star})$ is reached. Now the same adding structure is continued but solely on the right side of the droplet. If the droplet winds around the torus, then one adds a $(+1)$–spin above the droplet and fills this row until it also winds around the torus. This is repeated until $\boxplus$ is reached. _Inequality (2.5) holds._ Let $k^{\star}$ be such that $\gamma(k^{\star})\in R((L_{V}^{\star}-1)\times L_{V}^{\star})$. Then $H^{\mathrm{A}}(\gamma(k^{\star}))=E^{\star}_{A}-2J_{V}+h<E^{\star}_{A}$. If we go backwards in the path from that point on, then we will have to cut the top row of $R((L_{V}^{\star}-1)\times L_{V}^{\star})$, which has the length $L_{V}^{\star}-1$. This is an increase of the energy in each step until the top row turns into a protuberance. At this point the energy equals to \[H^{\mathrm{A}}(\gamma(k^{\star}-(L_{V}^{\star}-2)))=E^{\star}_{A }-2J_{V}+(L_{V}^{\star}-1)h<E^{\star}_{A}\] (2.6) by (2.4). Cutting the last protuberance decreases the energy even more. With the same reasoning, if we keep on going backwards in the path of $\gamma$, we will always stay below $E^{\star}_{A}$, since the size of the above and right bars of the droplets will be at most $L_{V}^{\star}-1$. Hence, we get that \[\max_{i=1,\dots,k^{\star}}H^{\mathrm{A}}(\gamma(i))<E^{\star}_{A}.\] (2.7) Let us look at the remaining path of $\gamma$ after the step $k^{\star}+2$. It holds that $H^{\mathrm{A}}(\gamma(k^{\star}+2))=E^{\star}_{A}-h<E^{\star}_{A}$. Until the right column is filled, the energy is decreased in every step. Afterwards, a protuberance is added on the right side and the energy increases by $2J_{V}-h$. Again by (2.4), we get \[H^{\mathrm{A}}(\gamma(k^{\star}+(L_{V}^{\star}+1)))=E^{\star}_{A }+2J_{V}-L_{V}^{\star}h<E^{\star}_{A}\] (2.8) Repeating this until the droplet wraps around the torus, the following energy level is reached \[E^{\star}_{A}-(hL_{V}^{\star}-2J_{V})(\sqrt{\|\Lambda\|}-L_{V}^{ \star})-h(L_{V}^{\star}-1)-2J_{V}L_{V}^{\star}.\] (2.9) Now we add a protuberance above the droplet and the energy increases by $2J_{H}-h$. Assumption 2.1 d) and (2.4) imply \[E^{\star}_{A} -(hL_{V}^{\star}-2J_{V})(\sqrt{\|\Lambda\|}-L_{V}^{\star})-h(L_{V}^ {\star}-1)-2J_{V}L_{V}^{\star}+2J_{H}-h\] \[\leq E^{\star}_{A}+(hL_{V}^{\star}-2J_{V})L_{V}^{\star}-hL_{V}^{ \star}-2J_{V}L_{V}^{\star}\] (2.10) \[<E^{\star}_{A}-2J_{V}L_{V}^{\star}.\] Filling this row, decreases the energy by $(\sqrt{\|\Lambda\|}-1)h+2J_{V}$. In the same way, one can show that the remaining part of the path stays below $E^{\star}_{A}$. Together with (2.7), we infer (2.5). ### Proof of $\Phi(\boxminus,\boxplus)-H^{\mathrm{A}}(\boxminus)\geq\Gamma^{\star}_{A}$ It is enough to show that every optimal path from $\boxminus$ to $\boxplus$ has to pass through $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$. The following observations will be very useful. Lemma 2.4: _Let $\sigma\in S$ be a local minimum of the energy $H^{\mathrm{A}}$, i.e. $H^{\mathrm{A}}(\sigma^{x})>H^{\mathrm{A}}(\sigma)$ for all $x\in\Lambda$. Then $\sigma$ is a union of isolated rectangles._ _Proof_. Assume $\sigma$ has a connected component $\sigma_{1}$ that is not a rectangle. Consider a connected component $\gamma_{1}$ of $R(\sigma_{1})\cap({\mathbb{Z}}^{2}\setminus\sigma_{1})$. Let $l_{1}$ be the maximal component of the boundary of $\gamma_{1}$ that does not belong to the boundary of $R(\sigma_{1})$. An example would be: Then, since $\sigma_{1}$ is connected and $l_{1}$ lies inside $R(\sigma_{1})$, $l_{1}$ has both a horizontal part and a vertical part. Let $x\in\Lambda$ be a site with $\sigma(x)=-1$, where both such parts come together. Obviously, $\sigma^{x}$ has strictly lower energy than $\sigma$, since $x$ has at least two nearest-neighbor $(+1)$–spins. $\square$ As an immediate consequence, we get: Corollary 2.5: _Assume that $\sigma\in S$ consists of only one connected component. Then_ \[H^{\mathrm{A}}(\sigma)\geq H^{\mathrm{A}}(R(\sigma)),\] (2.11) _and equality holds, if and only if $\sigma=R(\sigma)$._ We first show that every optimal path has to cross $R(L_{V}^{\star}-1,L_{V}^{\star})$. Lemma 2.6: _Let $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. Then $\gamma$ has to cross $R(L_{V}^{\star}-1,L_{V}^{\star})$._ _Proof_. Assume the contrary, i.e. $\gamma\cap R(L_{V}^{\star}-1,L_{V}^{\star})=\emptyset$. Let us first assume that throughout its whole path $\gamma$ consists of a single connected component. On its way to $\boxplus$, $\gamma$ has to cross a configuration, whose rectangular envelope has both horizontal and vertical length greater or equal to $L_{V}^{\star}$. Let \[t=\min\{l\geq 0\,\|\,P_{H}R(\gamma(l)),P_{V}R(\gamma(l))\geq L_{V }^{\star}\}.\] (2.12) Since $\gamma$ is assumed to be connected, we observe that either $P_{H}R(\gamma(t-1))=L_{V}^{\star}-1$ holds or $P_{V}R(\gamma(t-1))=L_{V}^{\star}-1$. In the following we analyze both cases and show that the assumption $\gamma\cap R(L_{V}^{\star}-1,L_{V}^{\star})=\emptyset$ leads to a contradiction. • Case 1: [$P_{V}R(\gamma(t-1))=L_{V}^{\star}-1$]. From the definition of $t$, it is clear that $R(\gamma(t-1))\in R((L_{V}^{\star}+m)\times(L_{V}^{\star}-1))$ for some $m\geq 0$. • Case 1.1: [$m=0$]. Since $\gamma$ does not cross $R(L_{V}^{\star}\times(L_{V}^{\star}-1))$, we have by Corollary 2.5 that \[H^{\mathrm{A}}(\gamma(t-1))>H^{\mathrm{A}}(L_{V}^{\star}\times(L _{V}^{\star}-1))=H^{\mathrm{A}}(\boxminus)+\Gamma^{\star}_{A}-2J_{H}+h=E_{A}^{ \star}-2J_{H}+h.\] (2.13) The minimal increase of energy to enlarge the vertical length of the rectangular envelope of a configuration is $2J_{H}-h$. Hence, \[H^{\mathrm{A}}(\gamma(t))\geq H^{\mathrm{A}}(\gamma(t-1))+2J_{H} -h>E_{A}^{\star}.\] (2.14) This contradicts $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$, since we already know from Section 2.1 that $\Phi(\boxminus,\boxplus)\leq E^{\star}_{A}$. • Case 1.2: [$m\in[1,\sqrt{\|\Lambda\|}-L_{V}^{\star})$]. Again, by Corollary 2.5 we have that \[H^{\mathrm{A}}(\gamma(t-1)) \geq H^{\mathrm{A}}((L_{V}^{\star}+m)\times(L_{V}^{\star}-1))\] \[=H^{\mathrm{A}}(L_{V}^{\star}\times(L_{V}^{\star}-1))+m(2J_{V}-h( L_{V}^{\star}-1))\] (2.15) \[>E_{A}^{\star}-2J_{H}+h,\] where we used inequality (2.4) in the last step. As before, this leads to a contradiction, since \[H^{\mathrm{A}}(\gamma(t))\geq H^{\mathrm{A}}(\gamma(t-1))+2J_{H} -h>E_{A}^{\star}.\] (2.16) • Case 1.3: [$m=\sqrt{\|\Lambda\|}-L_{V}^{\star}$]. In this case, $\gamma(t-1)$ wraps around the torus. Using Assumption 2.1 d), we infer that \[H^{\mathrm{A}} (\gamma(t-1))\geq H^{\mathrm{A}}(\sqrt{\|\Lambda\|}\times(L_{V}^{ \star}-1))\] \[=H^{\mathrm{A}}(L_{V}^{\star}\times(L_{V}^{\star}-1))+(\sqrt{\| \Lambda\|}-L_{V}^{\star})(2J_{V}-h(L_{V}^{\star}-1))-2J_{V}(L_{V}^{\star}-1)\] (2.17) \[>H^{\mathrm{A}}(L_{V}^{\star}\times(L_{V}^{\star}-1))=E_{A}^{ \star}-2J_{H}+h.\] Finally, \[H^{\mathrm{A}}(\gamma(t))\geq H^{\mathrm{A}}(\gamma(t-1))+2J_{H} -h>E_{A}^{\star},\] (2.18) which is a contradiction. • Case 2: [$P_{H}R(\gamma(t-1))=L_{V}^{\star}-1$]. Here we have that $R(\gamma(t-1))\in R((L_{V}^{\star}-1)\times(L_{V}^{\star}+m^{\prime}))$ for some $m^{\prime}\geq 0$. • Case 2.1: [$m^{\prime}=0$]. Since $\gamma$ does not cross $R((L_{V}^{\star}-1)\times L_{V}^{\star})$, we have by Corollary 2.5 that \[H^{\mathrm{A}}(\gamma(t-1))>H^{\mathrm{A}}((L_{V}^{\star}-1) \times L_{V}^{\star})=E_{A}^{\star}-2J_{V}+h.\] (2.19) The minimal increase of energy to enlarge the horizontal length of the rectangular envelope of a configuration is $2J_{V}-h$. Hence, \[H^{\mathrm{A}}(\gamma(t))\geq H^{\mathrm{A}}(\gamma(t-1))+2J_{V} -h>E_{A}^{\star}.\] (2.20) As before, this contradicts $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. • Case 2.2: [$m^{\prime}\in[1,\sqrt{\|\Lambda\|}-L_{V}^{\star})$]. As before, this case also leads to a contradiction, since \[H^{\mathrm{A}}(\gamma(t)) \geq H^{\mathrm{A}}(\gamma(t-1))+2J_{V}-h\geq H^{\mathrm{A}}((L_{ V}^{\star}-1)\times(L_{V}^{\star}+m^{\prime}))+2J_{V}-h\] \[=H^{\mathrm{A}}((L_{V}^{\star}-1)\times L_{V}^{\star})+m^{\prime} (2J_{H}-h(L_{V}^{\star}-1))+2J_{V}-h\] (2.21) \[>E_{A}^{\star},\] where we have used inequality (2.4) and Assumption 2.1 a) in the last step. • Case 2.3: [$m^{\prime}=\sqrt{\|\Lambda\|}-L_{V}^{\star}$]. Using Assumption 2.1 d), we infer that \[H^{\mathrm{A}} (\gamma(t-1))\geq H^{\mathrm{A}}((L_{V}^{\star}-1)\times\sqrt{\| \Lambda\|})\] \[=H^{\mathrm{A}}((L_{V}^{\star}-1)\times L_{V}^{\star})+(\sqrt{\| \Lambda\|}-L_{V}^{\star})(2J_{H}-h(L_{V}^{\star}-1))-2J_{H}(L_{V}^{\star}-1)\] (2.22) \[>H^{\mathrm{A}}((L_{V}^{\star}-1)\times L_{V}^{\star})=E_{A}^{ \star}-2J_{V}+h.\] Finally, \[H^{\mathrm{A}}(\gamma(t))\geq H^{\mathrm{A}}(\gamma(t-1))+2J_{V} -h>E_{A}^{\star},\] (2.23) which is a contradiction. Now assume that $\gamma$ can consist of several connected components. Let \[\ell=\min\{j\in{\mathbb{N}}\,\|\,\exists\sigma\subset\gamma(j): \sigma\text{ is connected and }P_{V}(\sigma),P_{H}(\sigma)\geq L_{V}^{\star}\}.\] (2.24) There are three possible cases. 1. Let $\gamma$ consist of a single connected component at the steps $\ell$ and $\ell-1$. Then it can be seen easily that the above proof can be carried over to this case. 2. Let $\gamma$ consist of several _isolated_ droplets $\gamma^{1},\dots,\gamma^{n}$ for some $n\geq 2$ at the steps $\ell$ and $\ell-1$. Let $\gamma^{i}$ be the component that reaches at time $\ell$ the configuration, whose rectangular envelope has both horizontal and vertical length greater or equal to $L_{V}^{\star}$. One immediately observes that $H^{\mathrm{A}}(\gamma^{i}(k))\leq H^{\mathrm{A}}(\gamma(k))$ for all $k\leq\ell$, since all other components contribute nonnegative energy, which follows from Assumption 2.1 and the definition of $\ell$ by an easy computation. Hence, applying the same arguments from above to $\gamma^{i}$ concludes the proof in this case. 3. Let $\gamma$ be such that at the steps $\ell-1$ and $\ell$, there are two connected components that touch each other at their corners. Of course, this case can only differ from the previous cases, if $\gamma(\ell)$ is obtained from $\gamma(\ell-1)$ by flipping a $(-1)$–spin at these corners. However, also here it can easily be seen from the arguments above that this contradicts $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$, since the resulting droplet will have horizontal and vertical length greater or equal to $L_{V}^{\star}$, but a large part in the rectangular envelope of the droplet are $(-1)$–spins. This concludes the proof. $\square$ As a byproduct, we obtain the following lemma that concludes the proof of $\Phi(\boxminus,\boxplus)-H^{\mathrm{A}}(\boxminus)\geq\Gamma^{\star}_{A}$. Lemma 2.7: _Let $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. In order to cross a configuration whose rectangular envelope has both vertical and horizontal length greater or equal to $L_{V}^{\star}$, $\gamma$ has to pass through $R(L_{V}^{\star}-1,L_{V}^{\star})$ and $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$. In particular, each optimal path between $\boxminus$ and $\boxplus$ has to cross $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$._ _Proof_. Consider the time step $t$ defined in the proof of Lemma 2.6. It was shown there that necessarily $\gamma(t-1)$ needs to belong to $R(L_{V}^{\star}-1,L_{V}^{\star})$. Since $P_{V}R(\gamma(t)),P_{H}R(\gamma(t))\geq L_{V}^{\star}$, $\gamma(t)$ must be obtained from $\gamma(t-1)$ by flipping a $(-1)$–spin at a site that is attached to a longer side of the droplet. This implies that $\gamma(t)$ needs to belong to $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$. $\square$ ### Identification of ${\mathcal{P}}^{\star}$ and ${\mathcal{C}}^{\star}$ From Section 2.1, it is clear that $R(L_{V}^{\star}-1,L_{V}^{\star})\subset{\mathcal{P}}^{\star}$. Now let $\sigma\in{\mathcal{P}}^{\star}$ and $x\in\Lambda$ be such that $\sigma^{x}\in{\mathcal{C}}^{\star}$. If follows from the definition of ${\mathcal{P}}^{\star}$ and ${\mathcal{C}}^{\star}$ that there exists $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$ and $\ell\in{\mathbb{N}}$ such that 1. $\gamma(\ell)=\sigma$ and $\gamma(\ell+1)=\sigma^{x}$, 2. $H^{\mathrm{A}}(\gamma(k))<E^{\star}_{A}$ for all $k\in\{0,\dots,\ell\}$, 3. $\Phi(\boxminus,\gamma(k))\geq\Phi(\gamma(k),\boxplus)$ for all $k\geq\ell+1$. By Lemma 2.7, (ii) implies that $\min(P_{H}R(\sigma),P_{V}R(\sigma))\leq L_{V}^{\star}-1$, since otherwise the energy level $E^{\star}_{A}$ would have been reached. There are two possible cases. • Case 1: [$P_{H}R(\sigma^{x}),P_{V}R(\sigma^{x})\geq L_{V}^{\star}$]. Lemma 2.7 implies that we necessarily have that $\sigma\in R(L_{V}^{\star}-1,L_{V}^{\star})$ and $\sigma^{x}\in R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$. • Case 2: [$\min(P_{H}R(\sigma^{x}),P_{V}R(\sigma^{x}))\leq L_{V}^{\star}-1$]. Also by Lemma 2.7, there must exist some $k^{\star}\geq\ell+2$ such that $\gamma(k^{\star})\in R(L_{V}^{\star}-1,L_{V}^{\star})$. But this contradicts (iii), since $\Phi(\boxminus,\gamma(k^{\star}))<\Phi(\gamma(k^{\star}),\boxplus)=E_{A}^{\star}$. Therefore, such a path $\gamma$ can not exist, which contradicts the fact $\sigma^{x}\in{\mathcal{C}}^{\star}$. Hence, only Case 1 can hold true. We conclude that ${\mathcal{P}}^{\star}=R(L_{V}^{\star}-1,L_{V}^{\star})$ and ${\mathcal{C}}^{\star}=R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{1pr}}$. ### Verification of (H1) Obviously, $S_{\mathrm{stab}}=\{\boxplus\}$, since $\boxplus$ minimizes all three sums in (2.1). It remains to show that $S_{\mathrm{meta}}=\{\boxminus\}.$ Let $\sigma\in S\setminus\{\boxminus,\boxplus\}$. We have to show that $V_{\sigma}<\Gamma^{\star}_{A}$, i.e. there exists $\sigma^{\prime}\in S$ such that $H^{\mathrm{A}}(\sigma^{\prime})<H^{\mathrm{A}}(\sigma)$ and $\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{A}}(\sigma)<\Gamma_{A}^{\star}$. There are four possible cases. • Case 1: [$\sigma$ contains a connected component, which is not a rectangle]. Lemma 2.4 implies that $\sigma$ is not a local minimum, i.e. there exists $x\in\Lambda$ such that $H^{\mathrm{A}}(\sigma^{x})<H^{\mathrm{A}}(\sigma)$. Moreover, $\Phi(\sigma,\sigma^{x})-H^{\mathrm{A}}(\sigma)=0<\Gamma_{A}^{\star}$. • Case 2: [$\sigma$ contains a connected component, which is a rectangle $R=l_{1}\times l_{2}$ with $l_{2}\geq L_{V}^{\star}$ and $l_{1}<\sqrt{\|\Lambda\|}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by attaching on the right of $R$ a new column of length $l_{2}$. Then: \[H^{\mathrm{A}}(\sigma^{\prime}) \leq H^{\mathrm{A}}(\sigma)+2J_{V}-l_{2}h\leq H^{\mathrm{A}}( \sigma)+2J_{V}-L_{V}^{\star}h<H^{\mathrm{A}}(\sigma),\quad\text{and}\] \[\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{A}}(\sigma) =2J_{V}-h<\Gamma_{A}^{\star}.\] (2.25) • Case 3: [$\sigma$ contains a connected component, which is a rectangle $R=l_{1}\times l_{2}$ with $l_{2}\leq L_{V}^{\star}-1$ and $l_{1}<\sqrt{\|\Lambda\|}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by cutting the right column of $R$. Then: \[H^{\mathrm{A}}(\sigma^{\prime}) =H^{\mathrm{A}}(\sigma)-2J_{V}+l_{2}h\leq H^{\mathrm{A}}(\sigma)- 2J_{V}+(L_{V}^{\star}-1)h<H^{\mathrm{A}}(\sigma),\quad\text{and}\] \[\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{A}}(\sigma) =(l_{2}-1)h<\Gamma_{A}^{\star}.\] (2.26) • Case 4: [$\sigma$ contains a connected component, which is a rectangle $R=l_{1}\times l_{2}$ with $l_{1}=\sqrt{\|\Lambda\|}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by attaching above $R$ a row that also wraps around the torus. Then, by Assumption 2.1 d): \[H^{\mathrm{A}}(\sigma^{\prime}) =H^{\mathrm{A}}(\sigma)+2J_{H}-l_{1}h<H^{\mathrm{A}}(\sigma), \quad\text{and}\] \[\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{A}}(\sigma) =2J_{H}-h<\Gamma_{A}^{\star}.\] (2.27) We conclude that $S_{\mathrm{meta}}=\{\boxminus\}.$ ### Verification of (H2) Obviously, $\|\{\sigma\in{\mathcal{P}}^{\star}\,\|\,\sigma\sim\sigma^{\prime}\}\|=1$ for all $\sigma^{\prime}\in{\mathcal{C}}^{\star}$. Therefore, (H2) holds. ### Computation of $K$ We start the computation of $K^{-1}$ from the variational formula given in Theorem 1.5. • _Lower bound_. Since the sum in the variational formula of Theorem 1.5 has only nonnegative summands, we can bound $K^{-1}$ from below by \[\frac{1}{K}\geq\min_{C_{1},\dots,C_{I}\in[0,1]}\min_{ \begin{subarray}{c}h:S^{\star}\rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{S_{\boxminus}}}=1,{\left.\kern-1.2pth \vphantom{\|}\right\|_{S_{\boxplus}}}=0,{\left.\kern-1.2pth\vphantom{\|}\right\|_{ S_{i}}}=C_{i}\,\forall i\end{subarray}}\frac{1}{2}\sum_{\eta,\eta^{\prime}\in( {\mathcal{C}}^{\star})^{+}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[h(\eta)-h( \eta^{\prime})]^{2}.\] (2.28) Obviously, it holds that $\partial^{+}{\mathcal{C}}^{\star}\cap S^{\star}=R(L_{V}^{\star}-1,L_{V}^{\star })\cup R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{2pr}}$. Moreover, similar computations as in Section 2.1 show that $R(L_{V}^{\star}-1,L_{V}^{\star})\subset S_{\boxminus}$ and $R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{2pr}}\subset S_{\boxplus}$. This leads to \[\frac{1}{K} \geq\min_{h:{\mathcal{C}}^{\star}\rightarrow[0,1]}\sum_{\eta\in{ \mathcal{C}}^{\star}}\left(\sum_{\eta^{\prime}\in R(L_{V}^{\star}-1,L_{V}^{ \star}),\eta^{\prime}\sim\eta}[1-h(\eta)]^{2}+\sum_{\eta^{\prime}\in R(L_{V}^{ \star}-1,L_{V}^{\star})^{\mathrm{2pr}},\eta^{\prime}\sim\eta}h(\eta)^{2}\right)\] \[=\sum_{\eta\in{\mathcal{C}}^{\star}}\min_{h\in[0,1]}\Big{(}\|R(L_{ V}^{\star}-1,L_{V}^{\star})\sim\eta\|\ [1-h]^{2}+\|R(L_{V}^{\star}-1,L_{V}^{ \star})^{\mathrm{2pr}}\sim\eta\|\ h^{2}\Big{)}\] (2.29) \[=\sum_{\eta\in{\mathcal{C}}^{\star}}\frac{\|R(L_{V}^{\star}-1,L_{V }^{\star})\sim\eta\|\cdot\|R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{2pr}}\sim \eta\|}{\|R(L_{V}^{\star}-1,L_{V}^{\star})\sim\eta\|+\|R(L_{V}^{\star}-1,L_{V}^{ \star})^{\mathrm{2pr}}\sim\eta\|}.\] For all $\eta\in{\mathcal{C}}^{\star}$ we have that $\|R(L_{V}^{\star}-1,L_{V}^{\star})\sim\eta\|=1$. If the protuberance in $\eta$ is attached at a corner of $(L_{V}^{\star}-1)\times L_{V}^{\star}$, then $\|R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{2pr}}\sim\eta\|=1$, otherwise $\|R(L_{V}^{\star}-1,L_{V}^{\star})^{\mathrm{2pr}}\sim\eta\|=2$. Taking into account that there are $\|\Lambda\|$ possible locations for each shape of a critical droplet and 2 possible rotations, we obtain: \[\frac{1}{K} \geq\Big{(}2(L_{V}^{\star}-2)\frac{2}{3}+4\frac{1}{2}\Big{)}2\| \Lambda\|=\frac{4(2L_{V}^{\star}-1)}{3}\|\Lambda\|.\] (2.30) • _Upper bound_. Let \[R^{-}_{0}=\{\sigma\in S^{\star} \|\,\sigma\text{ is not connected and }\min(P_{H}R(\eta),P_{V}R( \eta))\leq L_{V}^{\star}-1\] \[\ \text{ for all its connected components }\eta\},\] \[R^{+}_{0}=\{\sigma\in S^{\star} \|\,\sigma\text{ is not connected and }P_{H}R(\eta),P_{V}R(\eta) \geq L_{V}^{\star}\] \[\ \text{ for at least one of its connected components }\eta\},\] (2.31) \[R^{-}_{1}=\{\sigma\in S^{\star} \|\,\sigma\text{ is connected and }\min(P_{H}R(\sigma),P_{V}R( \sigma))\leq L_{V}^{\star}-1\},\] \[R^{+}_{1}=\{\sigma\in S^{\star} \|\,\sigma\text{ is connected and }P_{V}R(\sigma),P_{H}R(\sigma) \geq L_{V}^{\star}\}.\] Set $R^{-}=R^{-}_{0}\cup R^{-}_{1}$ and $R^{+}=R^{+}_{0}\cup R^{+}_{1}$. The following lemma is very important. Lemma 2.8: _Let $\sigma\in R^{-}$ and $\sigma^{\prime}\in R^{+}$. Then $\sigma\sim\sigma^{\prime}$, if and only if $\sigma\in{\mathcal{P}}^{\star}$ and $\sigma^{\prime}\in{\mathcal{C}}^{\star}$._ _Proof_. We skip the details of this proof, since they walk along the same lines as the proof of Lemma 2.6. One should only notice that, if $\sigma$ is connected and $\sigma^{\prime}$ is not connected or vice versa, then $\|H^{\mathrm{A}}(\sigma^{\prime})-H^{\mathrm{A}}(\sigma)\|=2J_{H}+2J_{V}-h$. $\square$ Finally, notice that for all $i$ either $S_{i}\subset R^{-}$ holds or $S_{i}\subset R^{+}$, since the $S_{i}$ are connected. For the same reason and by Section 2.1, $S_{\boxminus}\subset R^{-}$ and $S_{\boxplus}\subset R^{+}$ holds. Therefore, (2.32) ## 3 Ising model with next-nearest-neighbor attraction In this chapter, let the _Hamiltonian_ be given by \[H^{\mathrm{NN}}(\sigma)=-\frac{\tilde{J}}{2}\sum_{(x,y)\in\Lambda^{\star}} \sigma(x)\sigma(y)-\frac{K}{2}\sum_{(x,y)\in\Lambda^{\star\star}}\sigma(x) \sigma(y)-\frac{h}{2}\sum_{x\in\Lambda}\sigma(x),\] (3.1) where $\sigma\in S$, $\tilde{J},K,h>0$, $\Lambda^{\star}$ is the set of _unordered nearest-neighbor bonds_ in $\Lambda$ and $\Lambda^{\star\star}$ is the set of _unordered next-nearest-neighbor bonds_ in $\Lambda$, i.e \[\Lambda^{\star\star}=\big{\{}\{x,y\}\in\Lambda^{2}\ \big{\|}\ \|x-y \|=\sqrt{2}\big{\}}.\] (3.2) Using the geometric representation of $\sigma$, one can rewrite $H^{\mathrm{NN}}(\sigma)$ as \[H^{\mathrm{NN}}(\sigma)=H^{\mathrm{NN}}(\boxminus)-h\|\sigma\|+J\|\partial(\sigma )\|-K\|A(\sigma)\|,\] (3.3) where $J=\tilde{J}+2K$ and $\|A(\sigma)\|$ is the number of corners (or right angles) of $\sigma$. Note that in the following situation, we count 4 corners. The critical length in this model will be given by \[\ell^{\star}=\left\lceil\frac{2K}{h}\right\rceil\qquad\text{and} \qquad D^{\star}=\left\lceil\frac{2J}{h}\right\rceil\qquad\text{and}\qquad L^{ \star}=D^{\star}-2(\ell^{\star}-1).\] (3.4) The following assumptions will be made for this chapter. Assumption 3.1: 1. $2K>h$_,_ 2. $\tilde{J}\geq 2K+h$_,_ 3. $\frac{2J}{h}\notin{\mathbb{N}}$_,_ $\frac{2K}{h}\notin{\mathbb{N}}$_,_ 4. $\|\Lambda\|>\left(\frac{2J(D^{\star}-1)}{2J-h(D^{\star}-1)}+D^{\star}\right)^{2}$_._ Similar as in Chapter 2, a) and b) induce a hierarchy in the sense that for the system it is most important to align nearest-neighbors-neighbors, then next-nearest-neighbors and then to align the spin values with the sign of the magnetic field. Assumption c) is made for non-degeneracy reasons. Assumption d) implies that it is not profitable to enlarge a droplet such that one side is subcritical and the other side wraps around the torus. This will become clear later in Lemma 3.6. Moreover, d) assures that the torus is large enough to contain a critical droplet. It immediately follows from Assumption 3.1 c) that \[(\ell^{\star}-1)h<2K<\ell^{\star}h\qquad\text{and}\qquad(D^{\star }-1)h<2J<D^{\star}h.\] (3.5) We need a few definitions that are mostly carried over from [8]. Definition 3.2: _•_ $A\subset{\mathbb{Z}}^{2}$ _is called an_ oblique bar_, if_ $A=\{x_{1},\dots,x_{n}\}$ _for some_ $n\in{\mathbb{N}}$ _and it holds that either_ $x_{i}=x_{i-1}+(1,1)^{T}$ _for all_ $i\leq n$ _or_ $x_{i}=x_{i-1}+(1,-1)^{T}$ _for all_ $i\leq n$_._ _•_ $\sigma\in S$ _is called_ octagon of side lengths $D_{n},D_{w}\in{\mathbb{N}}\cap[1,\sqrt{\|\Lambda\|}-1]$ and oblique edge lengths__$\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se}\in{\mathbb{N}}$_, if_ $R(\sigma)\in R(D_{n},D_{w})$ _and if_ $\sigma$ _is obtained from_ $R(\sigma)$ _by cutting_ $\ell_{ne}-1$ _oblique bars from the upper right corner of_ $R(\sigma)$_,_ $\ell_{nw}-1$ _oblique bars from the upper left corner,_ $\ell_{sw}-1$ _oblique bars from the down left corner,_ $\ell_{se}-1$ _oblique bars from the down right corner. The set of all such octagons and their rotations is denoted by_ $Q(D_{n},D_{w};\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se})$_. We often abuse the notation by denoting configurations in this set in the same way._ _• Any_ $Q\in Q(D_{n},D_{w};\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se})$ _has 8 edges. The upper right edge of length_ $\ell_{ne}$ _is called_ NE-edge_, the upper left edge of length_ $\ell_{nw}$__NW-edge_, the down left edge of length_ $\ell_{sw}$__SW-edge _and the down right edge of length_ $\ell_{se}$ _is called_ SE-edge_. These four edges are also called_ oblique edges_. The four remaining horizontal or vertical edges are called_ coordinate edges_. We call the upper coordinate edge_ N-edge_, the left one_ W-edge_, the bottom one_ S-edge _and the right coordinate edge_ E-edge_. An example with_ $D_{n}=15$_,_ $D_{w}=12$_,_ $\ell_{ne}=5$_,_ $\ell_{nw}=6$_,_ $\ell_{sw}=4$_,_ $\ell_{se}=3$ _is given by_ _• The length of the N-edge is_ $D_{n}-(\ell_{ne}-1)-(\ell_{nw}-1)$ _and will be denoted by_ $L_{n}$_. In the same way, we define_ $L_{w},L_{s}$ _and_ $L_{e}$ _as the lengths of the W-edge, S-edge and E-edge, respectively._ _•_ $Q(D_{n},D_{w};\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se})$ _is called_ stable octagon_, if_ $L_{n},L_{w},L_{s},L_{e},\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se}\geq 2$_._ _• If_ $\ell_{ne}=\ell_{nw}=\ell_{sw}=\ell_{se}=\ell$_, we write_ $Q(D_{n},D_{w};\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se})=Q(D_{n},D_{w};\ell)$_._ _• If_ $\ell=\ell^{\star}$_, we write_ $Q(D_{n},D_{w};\ell^{\star})=Q(D_{n},D_{w})$_._ _• If_ $L_{n}=L_{w}=L_{s}=L_{e}=\ell_{ne}=\ell_{nw}=\ell_{sw}=\ell_{se}=\ell$_, we write_ $Q(3\ell-2,3\ell-2;\ell)=Q(\ell)$_._ _•_ $Q(D_{n},D_{w};\ell)^{\mathrm{1pr}}$ _denotes the set of all configurations consisting only of an octagon from_ $Q(D_{n},D_{w};\ell)$ _and with an additional protuberance attached at the interior of one of its longest coordinate edges. Here, the interior of a coordinate edge are all sites except of the two sites at both ends of the edge. The right droplet in Figure_ 3 _provides an example._ _•_ $Q(D_{n},D_{w};\ell)^{\mathrm{2pr}}$ _denotes the set of all configurations that are obtained from a configuration in_ $Q(D_{n},D_{w};\ell)^{\mathrm{1pr}}$ _by adding a second_ $(+1)$_–spin adjacent to the protuberance at the_ interior _of the coordinate edge._ Note that the energy of an octagon $Q\in Q(D_{n},D_{w};\ell_{ne},\ell_{nw},\ell_{sw},\ell_{se})$ is given by \[H^{\mathrm{NN}}(Q)=H^{\mathrm{NN}}(\boxminus)-hD_{n}D_{w}+2J(D_{ n}+D_{w})+\sum_{a\in\{ne,nw,sw,se\}}F(\ell_{a}),\] (3.6) where $F(\ell)=-K(2\ell-1)+\frac{1}{2}h(\ell-1)\ell$. Now we can formulate the main result of this chapter. Theorem 3.3: _Under Assumption 3.1, the pair $(\boxminus,\boxplus)$ satisfies (H1) and (H2) so that Theorems 1.3–1.5 hold for the Ising model with next-nearest-neighbor attraction._ _Moreover, the quadruple_ $({\mathcal{P}}^{\star},{\mathcal{C}}^{\star},\Gamma^{\star},K)$ _is given by_ ${\mathcal{P}}^{\star}=Q(D^{\star}-1,D^{\star})$_,_ * ${\mathcal{C}}^{\star}=Q(D^{\star}-1,D^{\star})^{\mathrm{1pr}}$_,_ * $\Phi(\boxminus,\boxplus)-H^{\mathrm{NN}}(\boxminus)=H^{\mathrm{NN}}(Q(D^{\star }-1,D^{\star}))+2J-4K-h=:\Gamma^{\star}_{\mathrm{NN}}=:E_{\mathrm{NN}}^{\star} -H^{\mathrm{NN}}(\boxminus)$_,_ * $K^{-1}=\frac{4(2L^{\star}-5)}{3}\|\Lambda\|$_._ _Proof_. The proof is divided into the Sections 3.1–3.6. $\square$ [FIGURE:S3.F3][ENDFIGURE] ### Proof of As in Section 2.1, we need to construct a reference path $\gamma:\boxminus\rightarrow\boxplus$ such that \[\max_{\eta\in\gamma}H^{\mathrm{NN}}(\eta)\leq H^{\mathrm{NN}}(\boxminus)+ \Gamma^{\star}_{NN}=E_{NN}^{}.\] (3.7) _Construction of $\gamma$._ The construction of $\gamma$ will be similar to [8] but simplified, since we are mainly interested in the part of the path around the critical state. We quickly sketch this critical part of the path and give the exact reference for the remaining steps. • [From $\boxminus$ to $Q(2)$.] See Scheme 5.1 of [8]. • [From $Q(\ell)$ to $Q(\ell+1)$ for all $\ell=2,\dots,\ell^{\star}-1$.] See Scheme 5.2 of [8]. • [From $Q(D,D)$ to $Q(D+1,D+1)$ for all $D=\ell^{\star},\dots,\sqrt{\|\Lambda\|}-2$.] This is Scheme 5.5 of [8], and it goes as follows. A $(+1)$–spin is added somewhere at the interior of the E-edge of $Q(D,D)$. Afterwards, this row is filled by successively flipping in this column adjacent $(-1)$–spins until $Q(D,D+1;\ell^{\star}+1,\ell^{\star},\ell^{\star},\ell^{\star}+1)$ is reached. Then a $(-1)$–spin is flipped at the upper end of the SE-edge. Now $(-1)$–spins are flipped until $Q(D,D+1;\ell^{\star}+1,\ell^{\star},\ell^{\star},\ell^{\star})$ is reached. Next, the same this is done at the NE-edge such that $Q(D,D+1;\ell^{\star},\ell^{\star},\ell^{\star},\ell^{\star})=Q(D,D+1)$ is reached. Then this procedure is repeated below $Q(D,D+1)$, i.e. first $(-1)$–spins are flipped at the S-edge until $Q(D+1,D+1;\ell^{\star},\ell^{\star},\ell^{\star}+1,\ell^{\star}+1)$ is reached, then an oblique bar is added at the SW-edge to reach $Q(D+1,D+1;\ell^{\star},\ell^{\star},\ell^{\star},\ell^{\star}+1)$, and finally, $(-1)$–spins are flipped at the SE-edge, until we arrive at $Q(D+1,D+1)$. • Lastly, flip all remaining $(-1)$–spins outside of $Q(\sqrt{\|\Lambda\|}-1,\sqrt{\|\Lambda\|}-1)$ until $\boxplus$ is reached. _Inequality (3.7) holds._ Let $k^{\star}$ be such that $\gamma(k^{\star})\in Q(D^{\star}-1,D^{\star})$. Then $H^{\mathrm{NN}}(\gamma(k^{\star}))=E^{\star}_{\mathrm{NN}}-2\tilde{J}+h<E^{ \star}_{\mathrm{NN}}$. If we go backwards in the path from that point on, then we will have to cut the NE-edge of $Q(D^{\star}-1,D^{\star})$. This is an increase of the energy in each step until only one $(+1)$–spin remains on this edge. At this point the energy equals to \[E^{\star}_{\mathrm{NN}}-2\tilde{J}+\ell^{\star}h<E^{\star}_{ \mathrm{NN}},\] (3.8) where we have used Assumption 3.1 b) and (3.5). Cutting the last $(+1)$–spin on this edge lowers the energy even more. Next we do the same thing on the SE-edge, i.e. we cut all but one $(+1)$–spins on this edge and arrive at the energy \[E^{\star}_{\mathrm{NN}}-2\tilde{J}-2K+2\ell^{\star}h<E^{\star}_{ \mathrm{NN}}.\] (3.9) Cutting the last $(+1)$–spin on this edge, we arrive at $E^{\star}_{\mathrm{NN}}-2J+(2\ell^{\star}+1)h$. Finally, we need to cut all but one $(+1)$–spins on the E-edge, which leads to the energy level \[E^{\star}_{\mathrm{NN}}-2J+(D^{\star}-1)h<E^{\star}_{\mathrm{NN}},\] (3.10) and cutting the last protuberance, we arrive at the energy $E^{\star}_{\mathrm{NN}}-4J+4K+D^{\star}h$. With the same reasoning, if we keep on going backwards in the path of $\gamma$, we will always stay below $E^{\star}_{\mathrm{NN}}$, since the size of the cutted sides of the octagon will be at most $D^{\star}-1$. Hence, we get that \[\max_{i=1,\dots,k^{\star}}H^{\mathrm{NN}}(\gamma(i))<E^{\star}_{\mathrm{NN}}.\] (3.11) Let us look at the remaining path of $\gamma$ after the step $k^{\star}+2$. It holds that $H^{\mathrm{NN}}(\gamma(k^{\star}+2))=E^{\star}_{\mathrm{NN}}-h<E^{\star}_{ \mathrm{NN}}$. First, $L^{\star}-4$$(+1)$–spins are attached at the interior of the S-edge. The energy is decreased to $E^{\star}_{\mathrm{NN}}-(L^{\star}-3)h$. Afterwards, a $(+1)$–spin is added at the SW-edge, which leads to the energy \[E^{\star}_{\mathrm{NN}}+2K-(L^{\star}-2)h<E^{\star}_{\mathrm{NN}},\] (3.12) where we have used the inequality $L^{\star}\geq 2\ell^{\star}+1$, which follows immediately from Assumption 3.1 b). Filling the SW-edge decreases the energy by $(\ell^{\star}-1)h$. Then we do the same things for the SE-edge by attaching first a $(+1)$–spin on this edge, which increases the energy to \[E^{\star}_{\mathrm{NN}}+4K-(L^{\star}+\ell^{\star}-2)h<E^{\star} _{\mathrm{NN}},\] (3.13) and then filling up this edge, which decreases the energy to $E^{\star}_{\mathrm{NN}}+4K-(D^{\star}-1)h$. Next, a protuberance is added at the interior of the E-edge. We arrive at the energy level \[E^{\star}_{\mathrm{NN}}+2J-D^{\star}h<E^{\star}_{\mathrm{NN}}.\] (3.14) This configuration is now “over the hill”, since, if we keep on following the path of $\gamma$, we will always stay below $E^{\star}_{\mathrm{NN}}$, since the size of the added sides to the octagon will be at least $D^{\star}$. Together with (3.11), we infer (3.7). ### Proof of We first list a few observations taken from [8]. Lemma 3.4: _Let $\sigma\in S$ be a local minimum of the energy $H^{\mathrm{NN}}$, i.e. $H^{\mathrm{NN}}(\sigma^{x})>H^{\mathrm{NN}}(\sigma)$ for all $x\in\Lambda$. Then $\sigma$ is either a union of isolated and stable octagons or $\sigma$ is a rectangle that wraps around the torus._ _Proof_. The first fact was proven in Lemma 2.1 of [8]. Now assume that $\sigma$ wraps around the torus. Then a straightforward adaptation of the proof of Lemma 2.4 yields the claim. $\square$ Lemma 3.5: _Assume that $\sigma\in S$ consists of only one connected component that does not wrap around the torus. Let $R(\sigma)\in R(D_{n},D_{w})$ with $D_{n}\geq D_{w}$._ _• If $D_{w}\geq 2\ell^{\star}-1$, then_ \[H^{\mathrm{NN}}(\sigma)\geq H^{\mathrm{NN}}(Q(D_{n},D_{w})),\] (3.15) _and equality holds, if and only if $\sigma=Q(D_{n},D_{w})$._ _• If $D_{w}<2\ell^{\star}-1\text{ and }D_{w}\text{ is odd}$, then_ \[H^{\mathrm{NN}}(\sigma)\geq H^{\mathrm{NN}}(Q(D_{n},D_{w};\tfrac {1}{2}(D_{w}+1))),\] (3.16) _and equality holds, if and only if $\sigma=Q(D_{n},D_{w};\tfrac{1}{2}(D_{w}+1))$._ _• If $D_{w}<2\ell^{\star}-1\text{ and }D_{w}\text{ is even}$, then_ \[H^{\mathrm{NN}}(\sigma)\geq H^{\mathrm{NN}}(Q(D_{n},D_{w};\tfrac {1}{2}D_{w},\tfrac{1}{2}D_{w},\tfrac{1}{2}D_{w}+1,\tfrac{1}{2}D_{w}+1)),\] (3.17) _and equality holds, if and only if $\sigma=Q(D_{n},D_{w};\tfrac{1}{2}D_{w},\tfrac{1}{2}D_{w},\tfrac{1}{2}D_{w}+1, \tfrac{1}{2}D_{w}+1)$._ _Proof_. See Lemma 3.2 and the proof of Lemma 4.1A in [8]. The main step is to notice that the function $l\mapsto F(l)$ is minimized in $\ell^{\star}$. $\square$ In the following lemma we show that every optimal path has to cross $Q(D^{\star}-1,D^{\star})$. Lemma 3.6: _Let $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. Then $\gamma$ has to cross $Q(D^{\star}-1,D^{\star})$._ _Proof_. Assume the contrary, i.e. $\gamma\cap Q(D^{\star}-1,D^{\star})=\emptyset$. Using the same arguments as in the end of the proof of Lemma 2.6, we can restrict ourselves to the case that throughout its whole path $\gamma$ consists only of a single connected component. On its way to $\boxplus$, $\gamma$ has to cross a configuration, whose rectangular envelope has both horizontal and vertical length greater or equal to $D^{\star}$. Let \[t=\min\{l\geq 0\,\|\,P_{H}R(\gamma(l)),P_{V}R(\gamma(l))\geq D^{ \star}\}.\] (3.18) Notice that from the definition of $t$, it is clear that $R(\gamma(t-1))\in R(D^{\star}+m,D^{\star}-1)$ for some $m\geq 0$. • Case 1: [$m=0$]. Obviously, $D^{\star}\geq 2\ell^{\star}-1$. Hence, by Lemma 3.5 and since $\gamma$ does not cross $Q(D^{\star}-1,D^{\star})$, we have that \[H^{\mathrm{NN}}(\gamma(t-1))>H^{\mathrm{NN}}(Q(D^{\star}-1,D^{ \star}))=E_{\mathrm{NN}}^{\star}-2\tilde{J}+h.\] (3.19) The minimal increase of energy to enlarge the rectangular envelope of a configuration is $2\tilde{J}-h$. Hence, \[H^{\mathrm{NN}}(\gamma(t))\geq H^{\mathrm{NN}}(\gamma(t-1))+2 \tilde{J}-h>E_{\mathrm{NN}}^{\star}.\] (3.20) This contradicts $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$, since we already know from Section 3.1 that $\Phi(\boxminus,\boxplus)\leq E^{\star}_{\mathrm{NN}}$. • Case 2: [$m\in[1,\sqrt{\|\Lambda\|}-D^{\star})$]. Again, by Corollary 3.5 we have that \[\begin{split} H^{\mathrm{NN}}(\gamma(t-1))& \geq H^{\mathrm{NN}}(Q(D^{\star}+m,D^{\star}-1))\\ &=H^{\mathrm{NN}}(Q(D^{\star},D^{\star}-1))+m(2J-h(D^{\star}-1)) \\ &>E_{\mathrm{NN}}^{\star}-2\tilde{J}+h.\end{split}\] (3.21) As before, this leads to a contradiction, since \[H^{\mathrm{NN}}(\gamma(t))\geq H^{\mathrm{NN}}(\gamma(t-1))+2 \tilde{J}-h>E_{\mathrm{NN}}^{\star}.\] (3.22) • Case 3: [$m=\sqrt{\|\Lambda\|}-D^{\star}$]. In this case, $\gamma(t-1)$ wraps around the torus. One can easily observe that $H^{\mathrm{NN}}(\gamma(t-1))\geq H^{\mathrm{NN}}(R(\sqrt{\|\Lambda\|},D^{\star}- 1))$. We infer that \[H^{\mathrm{NN}} (\gamma(t-1))\geq H^{\mathrm{NN}}(R(\sqrt{\|\Lambda\|},D^{\star}-1))\] \[=H^{\mathrm{NN}}(Q(D^{\star},D^{\star}-1))+(\sqrt{\|\Lambda\|}-D^{ \star})(2J-h(D^{\star}-1))-2J(D^{\star}-1)-4F(\ell^{\star})\] \[>H^{\mathrm{NN}}(Q(D^{\star},D^{\star}-1))=E_{\mathrm{NN}}^{\star }-2\tilde{J}+h,\] (3.23) where we have used $F(\ell^{\star})<0$ and Assumption 3.1 d). Finally, \[H^{\mathrm{NN}}(\gamma(t))\geq H^{\mathrm{NN}}(\gamma(t-1))+2 \tilde{J}-h>E_{\mathrm{NN}}^{\star}.\] (3.24) This concludes the proof. $\square$ As a byproduct, we obtain the following lemma that concludes the proof of . Lemma 3.7: _Let $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. In order to cross a configuration whose rectangular envelope has both vertical and horizontal length greater or equal to $D^{\star}$, $\gamma$ has to pass through $Q(D^{\star}-1,D^{\star})$ and $Q(D^{\star}-1,D^{\star})^{\mathrm{1pr}}$. In particular, each optimal path between $\boxminus$ and $\boxplus$ has to cross $Q(D^{\star}-1,D^{\star})^{\mathrm{1pr}}$._ _Proof_. Consider the time step $t$ defined in the proof of Lemma 3.6. It was shown there that necessarily $\gamma(t-1)$ needs to belong to $Q(D^{\star}-1,D^{\star})$. Since $P_{V}R(\gamma(t)),P_{H}R(\gamma(t))\geq D^{\star}$, $\gamma(t)$ must be obtained from $\gamma(t-1)$ by flipping a $(-1)$–spin at a site that is attached at the coordinate edge of a longer side of the droplet. If it would not attach at the interior of the coordinate edge, then the energy level $E_{\mathrm{NN}}^{\star}+2K$ would be reached. Hence, the protuberance must be added at the interior of the coordinate edge, which implies that $\gamma(t)$ needs to belong to $Q(D^{\star}-1,D^{\star})^{\mathrm{1pr}}$. $\square$ ### Identification of ${\mathcal{P}}^{\star}$ and ${\mathcal{C}}^{\star}$ From Section 3.1, it is clear that $Q(D^{\star}-1,D^{\star})\subset{\mathcal{P}}^{\star}$. Now let $\sigma\in{\mathcal{P}}^{\star}$ and $x\in\Lambda$ be such that $\sigma^{x}\in{\mathcal{C}}^{\star}$. If follows from the definition of ${\mathcal{P}}^{\star}$ and ${\mathcal{C}}^{\star}$ that there exists $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$ and $\ell\in{\mathbb{N}}$ such that 1. $\gamma(\ell)=\sigma$ and $\gamma(\ell+1)=\sigma^{x}$, 2. $H^{\mathrm{NN}}(\gamma(k))<E^{\star}_{\mathrm{NN}}$ for all $k\in\{0,\dots,\ell\}$, 3. $\Phi(\boxminus,\gamma(k))\geq\Phi(\gamma(k),\boxplus)$ for all $k\geq\ell+1$. By Lemma 3.7, (ii) implies that $\min(P_{H}R(\sigma),P_{V}R(\sigma))\leq D^{\star}-1$, since otherwise the energy level $E^{\star}_{\mathrm{NN}}$ would have been reached. There are two possible cases. • Case 1: [$P_{H}R(\sigma^{x}),P_{V}R(\sigma^{x})\geq D^{\star}$]. Lemma 3.7 implies that necessarily $\sigma\in Q(D^{\star}-1,D^{\star})$ and $\sigma^{x}\in Q(D^{\star}-1,D^{\star})^{\mathrm{1pr}}$. • Case 2: [$\min(P_{H}R(\sigma^{x}),P_{V}R(\sigma^{x}))\leq D^{\star}-1$]. Also by Lemma 3.7, there must exist some $k^{\star}\geq\ell+2$ such that $\gamma(k^{\star})\in Q(D^{\star}-1,D^{\star})$. But this contradicts (iii), since $\Phi(\boxminus,\gamma(k^{\star}))<\Phi(\gamma(k^{\star}),\boxplus)=E_{\mathrm{ NN}}^{\star}$. Therefore, such a path $\gamma$ can not exist, which contradicts the fact $\sigma^{x}\in{\mathcal{C}}^{\star}$. Hence, only Case 1 can hold true. We conclude that ${\mathcal{P}}^{\star}=Q(D^{\star}-1,D^{\star})$ and ${\mathcal{C}}^{\star}=Q(D^{\star}-1,D^{\star})^{\mathrm{1pr}}$. ### Verification of (H1) Obviously, $S_{\mathrm{stab}}=\{\boxplus\}$, since $\boxplus$ minimizes all three sums in (3.1). It remains to show that $S_{\mathrm{meta}}=\{\boxminus\}$. Let $\sigma\in S\setminus\{\boxminus,\boxplus\}$. As in Section 2.4, we have to show that there exists $\sigma^{\prime}\in S$ such that $H^{\mathrm{NN}}(\sigma^{\prime})<H^{\mathrm{NN}}(\sigma)$ and $\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{NN}}(\sigma)<\Gamma_{\mathrm{NN}}^{\star}$. • Case 1: [$\sigma$ contains a connected component, which is not a stable octagon and not a rectangle that wraps around the torus]. Lemma 3.4 implies that $\sigma$ is not a local minimum, i.e. there exists $x\in\Lambda$ such that $H^{\mathrm{NN}}(\sigma^{x})<H^{\mathrm{NN}}(\sigma)$ and $\Phi(\sigma,\sigma^{x})-H^{\mathrm{NN}}(\sigma)=0<\Gamma_{\mathrm{NN}}^{\star}$. • Case 2: [$\sigma$ contains a connected component $Q$, which is a stable octagon with $P_{V}R(Q)\geq D^{\star}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by attaching at $Q$ an oblique bar at its NE-edge and its SE-edge respectively, and a vertical bar at its E-edge in the same way that was described in the third step of the construction of $\gamma$ given in Section 3.1. Then we obtain: \[\begin{split} H^{\mathrm{NN}}(\sigma^{\prime})-H^{ \mathrm{NN}}(\sigma)&\leq 2J-P_{V}R(Q)h\leq 2J-D^{\star}h<0,\quad \text{and}\\ \Phi(\sigma,\sigma^{\prime})-H^{\mathrm{NN}}(\sigma)& \leq 2\tilde{J}-h<\Gamma_{\mathrm{NN}}^{\star}.\end{split}\] (3.25) • Case 3: [$\sigma$ contains a connected component $Q$, which is a stable octagon with $P_{V}R(Q)\leq D^{\star}-1$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by detaching the NE-edge, the SE-edge and the E-edge of $Q$ respectively, similar as in the construction of $\gamma$ given in Section 3.1 but in the reverse way. Then: \[\begin{split} H^{\mathrm{NN}}(\sigma^{\prime})-H^{ \mathrm{NN}}(\sigma)&=-2J+P_{V}R(Q)h\leq-2J+(D^{\star}-1)h<0, \quad\text{and}\\ \Phi(\sigma,\sigma^{\prime})-H^{\mathrm{NN}}(\sigma)& \leq(P_{V}R(Q)-1)h<\Gamma_{\mathrm{NN}}^{\star}.\end{split}\] (3.26) • Case 4: [$\sigma$ contains a connected component $R$ that is a rectangle that wraps around the torus.]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by attaching at $R$ a bar that also wraps around the torus. Then, by Assumption 2.1 d): \[\begin{split} H^{\mathrm{NN}}(\sigma^{\prime})-H^{ \mathrm{NN}}(\sigma)&=2\tilde{J}-\sqrt{\|\Lambda\|}h<0,\quad\text{ and}\\ \Phi(\sigma,\sigma^{\prime})-H^{\mathrm{NN}}(\sigma)& =2\tilde{J}-h<\Gamma_{\mathrm{NN}}^{\star}.\end{split}\] (3.27) We conclude that $S_{\mathrm{meta}}=\{\boxminus\}.$ ### Verification of (H2) Obviously, $\|\{\sigma\in{\mathcal{P}}^{\star}\,\|\,\sigma\sim\sigma^{\prime}\}\|=1$ for all $\sigma^{\prime}\in{\mathcal{C}}^{\star}$. Therefore, (H2) holds. ### Computation of $K$ The computation of $K^{-1}$ here will be done analogously to Section 2.6. • _Lower bound_. Note that $\partial^{+}{\mathcal{C}}^{\star}\cap S^{\star}=Q(D^{\star}-1,D^{\star})\cup Q (D^{\star}-1,D^{\star})^{\mathrm{2pr}}$, $Q(D^{\star}-1,D^{\star})\subset S_{\boxminus}$ and $Q(D^{\star}-1,D^{\star})^{\mathrm{2pr}}\subset S_{\boxplus}$. Hence, \[\begin{split}\frac{1}{K}&\geq\min_{C_{1 },\dots,C_{I}\in[0,1]}\min_{\begin{subarray}{c}h:S^{\star}\rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{S_{\boxminus}}}=1,{\left.\kern-1.2pth \vphantom{\|}\right\|_{S_{\boxplus}}}=0,{\left.\kern-1.2pth\vphantom{\|}\right\|_{ S_{i}}}=C_{i}\,\forall i\end{subarray}}\frac{1}{2}\sum_{\eta,\eta^{\prime}\in( {\mathcal{C}}^{\star})^{+}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[h(\eta)-h( \eta^{\prime})]^{2}\\ &=\min_{h:{\mathcal{C}}^{\star}\rightarrow[0,1]}\sum_{\eta\in{ \mathcal{C}}^{\star}}\left(\sum_{\eta^{\prime}\in Q(D^{\star}-1,D^{\star}), \eta^{\prime}\sim\eta}[1-h(\eta)]^{2}+\sum_{\eta^{\prime}\in Q(D^{\star}-1,D^{ \star})^{\mathrm{2pr}},\eta^{\prime}\sim\eta}h(\eta)^{2}\right)\\ &=\sum_{\eta\in{\mathcal{C}}^{\star}}\min_{h\in[0,1]}\Big{(}\|Q(D^ {\star}-1,D^{\star})\sim\eta\|\ [1-h]^{2}+\|Q(D^{\star}-1,D^{\star})^{\mathrm{2 pr}}\sim\eta\|\ h^{2}\Big{)}\\ &=\sum_{\eta\in{\mathcal{C}}^{\star}}\frac{\|Q(D^{\star}-1,D^{ \star})\sim\eta\|\cdot\|Q(D^{\star}-1,D^{\star})^{\mathrm{2pr}}\sim\eta\|}{\|Q(D^{ \star}-1,D^{\star})\sim\eta\|+\|Q(D^{\star}-1,D^{\star})^{\mathrm{2pr}}\sim\eta\| }.\end{split}\] (3.28) For all $\eta\in{\mathcal{C}}^{\star}$ we have that $\|Q(D^{\star}-1,D^{\star})\sim\eta\|=1$. Notice that there are 4 possible seats at a longer coordinate edge of a critical droplet such that $\|Q(D^{\star}-1,D^{\star})^{\mathrm{2pr}}\sim\eta\|=1$, and $2(L^{\star}-4)$ possible seats such that $\|Q(D^{\star}-1,D^{\star})^{\mathrm{2pr}}\sim\eta\|=2$. Moreover, we observe that there are $\|\Lambda\|$ possible locations for a configuration in ${\mathcal{C}}^{\star}$, and that there are two analogue rotations for each critical droplet. Therefore, we obtain \[\frac{1}{K} \geq\Big{(}2(L^{\star}-4)\frac{2}{3}+4\frac{1}{2}\Big{)}2\|\Lambda \|=\frac{4(2L^{\star}-5)}{3}\|\Lambda\|.\] (3.29) • _Upper bound_. Define \[\begin{split} R^{-}_{0}=\{\sigma\in S^{\star}& \|\,\sigma\text{ is not connected and }\min(P_{H}R(\eta),P_{V}R( \eta))\leq D^{\star}-1\\ &\ \text{ for all its connected components }\eta\},\\ R^{+}_{0}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is not connected and }P_{H}R(\eta),P_{V}R(\eta)\geq D^{\star}\\ &\ \text{ for at least one of its connected components }\eta\},\\ R^{-}_{1}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }\min(P_{H}R(\sigma),P_{V}R(\sigma))\leq D^{\star}-1\},\\ R^{+}_{1}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{V}R(\sigma),P_{H}R(\sigma)\geq D^{\star}\}.\end{split}\] (3.30) Set $R^{-}=R^{-}_{0}\cup R^{-}_{1}$ and $R^{+}=R^{+}_{0}\cup R^{+}_{1}$. The following lemma is very important. Lemma 3.8: _Let $\sigma\in R^{-}$ and $\sigma^{\prime}\in R^{+}$. Then $\sigma\sim\sigma^{\prime}$, if and only if $\sigma\in{\mathcal{P}}^{\star}$ and $\sigma^{\prime}\in{\mathcal{C}}^{\star}$._ _Proof_. We skip the details of this proof, since they walk along the same lines as the proof of Lemma 3.6. One should only notice that, if $\sigma$ is connected and $\sigma^{\prime}$ is not connected or vice versa, then $\|H^{\mathrm{NN}}(\sigma^{\prime})-H^{\mathrm{NN}}(\sigma)\|\geq 4\tilde{J}+2K-h$. $\square$ Finally, notice that for all $i$ either $S_{i}\subset R^{-}$ holds or $S_{i}\subset R^{+}$, since the $S_{i}$ are connected. For the same reason and by Section 3.1, $S_{\boxminus}\subset R^{-}$ and $S_{\boxplus}\subset R^{+}$ holds. Therefore, \[\frac{1}{K} \leq\min_{\begin{subarray}{c}h:S^{\star}\rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{R^{-}}}=1,{\left.\kern-1.2pth\vphantom {\|}\right\|_{R^{+}\setminus{\mathcal{C}}^{\star}}}=0\end{subarray}}\frac{1}{2} \sum_{\eta,\eta^{\prime}\in S^{\star}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[ h(\eta)-h(\eta^{\prime})]^{2}\] (3.31) \[=\min_{\begin{subarray}{c}h:({\mathcal{C}}^{\star})^{+} \rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{R^{-}\cap\partial^{+}{\mathcal{C}}^{ \star}}}=1,{\left.\kern-1.2pth\vphantom{\|}\right\|_{R^{+}\cap\partial^{+}{ \mathcal{C}}^{\star}}}=0\end{subarray}}\frac{1}{2}\sum_{\eta,\eta^{\prime}\in( {\mathcal{C}}^{\star})^{+}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[h(\eta)-h( \eta^{\prime})]^{2}\] (3.32) \[=\min_{h:{\mathcal{C}}^{\star}\rightarrow[0,1]}\sum_{\eta\in{ \mathcal{C}}^{\star}}\left(\sum_{\eta^{\prime}\in Q(D^{\star}-1,D^{\star}), \eta^{\prime}\sim\eta}[1-h(\eta)]^{2}+\sum_{\eta^{\prime}\in Q(D^{\star}-1,D^{ \star})^{\mathrm{2pr}},\eta^{\prime}\sim\eta}h(\eta)^{2}\right)\] (3.33) \[=\frac{4(2L^{\star}-5)}{3}\|\Lambda\|.\] (3.34) ## 4 Ising model with alternating magnetic field We adapt the same strategy as in the Chapters 2 and 3 to a third modification of the Ising model. Here, the _Hamiltonian_ is given by \[H^{\mathrm{\pm}}(\sigma)=-\frac{J}{2}\sum_{(x,y)\in\Lambda^{\star}}\sigma(x) \sigma(y)+\frac{h_{\mathrm{odd}}}{2}\sum_{x\in\Lambda_{\mathrm{odd}}}\sigma(x) -\frac{h_{\mathrm{even}}}{2}\sum_{x\in\Lambda_{\mathrm{even}}}\sigma(x),\] (4.1) where $\sigma\in S$, $J,h_{\mathrm{odd}},h_{\mathrm{even}}>0$, $\Lambda_{\mathrm{odd}}=\{(x_{1},x_{2})\in\Lambda\,\|\,x_{2}\text{ is odd}\}$ are the _odd rows_ in $\Lambda$, $\Lambda_{\mathrm{even}}=\Lambda\setminus\Lambda_{\mathrm{odd}}$ are the _even rows_ and $\Lambda^{\star}$ is the set of _unordered nearest-neighbor bonds_ in $\Lambda$. One can rewrite $H^{\mathrm{\pm}}(\sigma)$ geometrically as \[H^{\mathrm{\pm}}(\sigma)=H^{\mathrm{\pm}}(\boxminus)+h_{\mathrm{odd}}\|\sigma \cap\Lambda_{\mathrm{odd}}\|-h_{\mathrm{even}}\|\sigma\cap\Lambda_{\mathrm{even} }\|+J\|\partial(\sigma)\|.\] (4.2) Under the assumptions below, the critical lengths in this model will be given by \[l_{b}^{\star}=\left\lceil\frac{\mu}{\varepsilon}\right\rceil \qquad\text{and}\qquad l_{h}^{\star}=2l_{b}^{\star}-1,\] (4.3) where \[\begin{split}&\varepsilon=h_{\mathrm{even}}-h_{ \mathrm{odd}},\quad\text{and}\\ &\mu=2J-h_{\mathrm{odd}}.\end{split}\] (4.4) $\ell_{b}^{\star}$ will be the length of the basis of the critical droplet, and $\ell_{h}^{\star}$ will be its height. The following assumptions will be made for this chapter. Assumption 4.1: 1. $h_{\mathrm{even}}>h_{\mathrm{odd}}$_,_ 2. $J>h_{\mathrm{even}}$_,_ 3. $\frac{\mu}{\varepsilon}\notin{\mathbb{N}}$_,_ 4. $\|\Lambda\|>\left(2\left\lceil\frac{2J(l_{h}^{\star}-1)+h_{\mathrm{odd}}}{4J- \varepsilon(l_{b}^{\star}-1)}\right\rceil+l_{h}^{\star}\right)^{2}$_._ Assumption a) ensures that $\boxplus$ is the _stable state_ in this system. Assumptions b), c) and d) are made due to similar reasons as in the Chapters 2 and 3. Assumption b) can also be modified in various ways. E.g. one can take $J<h_{\mathrm{even}}<2J$. We refer to [9], page 10, where several other regimes are listed. In contrast to [9], in this text, we only consider the regime given in Assumption 4.1, since all other regimes can be handled in a similar way without using new ideas. It immediately follows from Assumption 4.1 c) that \[(l_{b}^{\star}-1)\varepsilon<\mu<l_{b}^{\star}\varepsilon.\] (4.5) The protocritical and the critical set in this model are given through the following sets. Figure 4 below provides an example. Definition 4.2: _the set of all configurations consisting only of a rectangle from_ $R((l_{b}^{\star}-1)\times l_{h}^{\star})$ _that start and end in_ $\Lambda_{\mathrm{even}}$ _(i.e. the bottom and the top row belong to_ $\Lambda_{\mathrm{even}}$_) and with an additional protuberance attached at one of its longer sides on a row in_ $\Lambda_{\mathrm{even}}$_,_ * _the set of all configurations that are built from a configuration in_ ${\mathcal{P}}_{1}$ _by adding a second_ $(+1)$_–spin in_ $\Lambda_{\mathrm{odd}}$ _adjacent to the protuberance,_ * _the set of all configurations consisting only of a rectangle from_ $R(l_{b}^{\star}\times(l_{h}^{\star}-2))$ _that start and end in_ $\Lambda_{\mathrm{even}}$ _and with an additional vertical or horizontal bar of length_ $2$ _attached above or below the droplet,_ * _the set of all configurations that are built from a configuration in_ ${\mathcal{P}}_{2}$ _by adding a third_ $(+1)$_–spin above or next to the bar of length_ $2$ _in_ ${\mathcal{P}}_{2}$_, i.e. above or below_ $R(l_{b}^{\star}\times(l_{h}^{\star}-2))$ _there is a vertical bar of length_ $2$ _with a_ $(+1)$_–spin next to it in an odd row._ We now state the main result of this chapter. Theorem 4.3: _Under Assumption 4.1, the pair $(\boxminus,\boxplus)$ satisfies (H1) so that Theorem 1.3 a), Theorem 1.4 and Theorem 1.5 hold for the Ising model with alternating magnetic field._ _Moreover, the quadruple_ $({\mathcal{P}}^{\star},{\mathcal{C}}^{\star},\Gamma^{\star},K)$ _is given by_ * ${\mathcal{P}}^{\star}={\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$_,_ * ${\mathcal{C}}^{\star}={\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$_,_ * $\Phi(\boxminus,\boxplus)-H^{\mathrm{\pm}}(\boxminus)=4J\,l_{b}^{\star}+\mu(l_{ b}^{\star}-1)-\varepsilon(l_{b}^{\star}(l_{b}^{\star}-1)+1)=:\Gamma^{\star}_{ \pm}=:E_{\pm}^{\star}-H^{\mathrm{\pm}}(\boxminus)$_,_ * $K^{-1}=\frac{14\,(l_{b}^{\star}-1)}{3}\|\Lambda\|$_._ _Proof_. The proof is divided into the Sections 4.1–4.5. $\square$ [FIGURE:S4.F4][ENDFIGURE] ### Proof of $\Phi(\boxminus,\boxplus)-H^{\mathrm{\pm}}(\boxminus)\leq\Gamma^{\star}_{\pm}$ As in Chapters 2 and 3, we construct a reference path $\gamma:\boxminus\rightarrow\boxplus$ such that \[\max_{\eta\in\gamma}H^{\mathrm{\pm}}(\eta)\leq H^{\mathrm{\pm}}(\boxminus)+ \Gamma^{\star}_{\pm}=E_{\pm}^{}.\] (4.6) _Construction of $\gamma$._$\gamma$ is given through the following scheme. • Let $\gamma(0)=\boxminus$. • In the first step an arbitrary spin in $\Lambda_{\mathrm{even}}$ is flipped from $(-1)$ to $(+1)$. • [From $R(l\times(2l-1))$ to $R((l+1)\times(2(l+1)-1))$ for $l\leq l_{b}^{\star}-1$.] A protuberance is added on the right of the droplet at a row that belongs to $\Lambda_{\mathrm{even}}$. Then successively adjacent $(-1)$–spins are flipped until the droplet belongs to $R\left((l+1)\times(2l-1)\right)$. Now a protuberance is added in the row above the droplet, which is an odd row. Afterwards, a second $(+1)$–spin is added above the protuberance on the even row. Then one adds successively vertical bars of length $2$ above the droplet, until $R((l+1)\times(2(l+1)-1))$ is reached. • [From $R(l\times l_{h}^{\star})$ to $R((l+1)\times l_{h}^{\star})$ for $l\geq l_{b}^{\star}$.] A protuberance is added on the right of the droplet at a row that belongs to $\Lambda_{\mathrm{even}}$, and successively adjacent $(-1)$–spins are flipped until the droplet belongs to $R((l+1)\times l_{h}^{\star})$. This procedure is repeated until the droplet wraps around the torus. • [From $R(\sqrt{\|\Lambda\|}\times l_{h}^{\star})$ to $\boxplus$.] A protuberance is added in the odd row above the droplet. Afterwards, a second $(+1)$–spin is added above the protuberance on the even row. Then one adds successively vertical bars of length $2$ above the droplet, until a configuration in $R(\sqrt{\|\Lambda\|}\times(l_{h}^{\star}+2))$ is reached. This is repeated until $\boxplus$ is reached. _Inequality (4.6) holds._ Let $k^{\star}$ be such that $\gamma(k^{\star})\in R(l_{b}^{\star}\times(l_{h}^{\star}-2))$. Then $H^{\mathrm{\pm}}(\gamma(k^{\star}))=E^{\star}_{\pm}-4J+\varepsilon-h_{\mathrm{ odd}}<E^{\star}_{\pm}$. If we go backwards in the path from that point on, then we will have to cut the right bar of the droplet. Doing that, the highest energy level is reached when only two adjacent $(+1)$–spins remain, one in $\Lambda_{\mathrm{even}}$ and one in $\Lambda_{\mathrm{odd}}$. Indeed, at that point the energy equals \[E^{\star}_{\pm}-4J+\varepsilon(l_{b}^{\star}-1)<E^{\star}_{\pm},\] (4.7) and cutting the last $(+1)$–spins, we reach at $R((l_{b}^{\star}-1)\times(l_{h}^{\star}-2))$ and the energy decreases to $E^{\star}_{\pm}-6J+\varepsilon l_{b}^{\star}$. Next, we have to cut the above two rows by successively cutting vertical bars of length $2$. Doing that, the highest energy point is the stage, where only one vertical bar of length $2$ and a single $(+1)$–spin in $\Lambda_{\mathrm{odd}}$ next to it have remained. At this point the energy equals to \[E^{\star}_{\pm}-6J+2\varepsilon(l_{b}^{\star}-1)+h_{\mathrm{even }}<E^{\star}_{\pm}.\] (4.8) With the same reasoning, if we keep on going backwards in the path of $\gamma$, we will always stay below $E^{\star}_{\pm}$, since the sizes of the cutted columns and rows further decreases. Hence, we get that \[\max_{i=1,\dots,k^{\star}}H^{\mathrm{\pm}}(\gamma(i))<E^{\star}_{\pm}.\] (4.9) Let us look at the path of $\gamma$ after the step $k^{\star}+2$. First, one has to fill the two rows above the droplet. This lowers the energy to $E^{\star}_{\pm}-\varepsilon(l_{b}^{\star}-1)-h_{\mathrm{odd}}$. Afterwards, a protuberance is attached on the right of the droplet in a row that belongs to $\Lambda_{\mathrm{even}}$. The energy increases to \[E^{\star}_{\pm}+\mu-\varepsilon l_{b}^{\star}-h_{\mathrm{odd}}<E ^{\star}_{\pm}.\] (4.10) Adding a second $(+1)$–spin adjacent to the protuberance further increases the energy by $h_{\mathrm{odd}}$. But by (4.5), we still get \[E^{\star}_{\pm}+\mu-\varepsilon l_{b}^{\star}<E^{\star}_{\pm}.\] (4.11) If we fill this column, we further decrease the energy so that the energy still remains below $E^{\star}_{\pm}$. In the following, in the same way, columns are added successively on the right of the droplet and each column decreases the energy by $\mu-\varepsilon l_{b}^{\star}$. This is repeated until the droplet wraps around the torus. It is easy to see that the remaining part of $\gamma$ stays below $E^{\star}_{\pm}$. Together with (4.9), we conclude (4.6). ### Proof of $\Phi(\boxminus,\boxplus)-H^{\mathrm{\pm}}(\boxminus)\geq\Gamma^{\star}_{\pm}$ We first list two useful facts that were already proven in [9].² The first one characterizes all _$h_{\mathrm{odd}}$–stable configurations_, i.e. all configurations $\sigma\in S$ such that there exists $\sigma^{\prime}\in S$ with [FOOTNOTE:2][ENDFOOTNOTE] $H^{\pm}(\sigma^{\prime})<H^{\pm}(\sigma)$, and * $\Phi(\sigma^{\prime},\sigma)-H^{\pm}(\sigma)\leq h_{\mathrm{odd}}$. Lemma 4.4: $\sigma\in S$ _is $h_{\mathrm{odd}}$–stable, if and only if $\sigma$ is a union of isolated rectangles of the form $R(l_{1}\times l_{2})$ that start and end in $\Lambda_{\mathrm{even}}$ and where $l_{1}\geq 2$, $l_{2}\geq 3$, $l_{2}$ is odd and the rectangle can possibly wrap around the torus. Such rectangles are called stable rectangles._ The second fact is the analogue of Corollary 2.5 for this model. Lemma 4.5: _Let $\sigma\in S$ be such that $R(\sigma)$ is a stable rectangle. Then_ \[H^{\mathrm{\pm}}(\sigma)\geq H^{\mathrm{\pm}}(R(\sigma)),\] (4.12) _and equality holds, if and only if $\sigma=R(\sigma)$._ For a rectangle $R\in R(l_{1}\times l_{2})$, we write $R_{\mathrm{odd}}$, if it starts in $\Lambda_{\mathrm{odd}}$, i.e. its bottom row belongs to $\Lambda_{\mathrm{odd}}$, otherwise we still write $R$. Note that $H^{\mathrm{\pm}}(R_{\mathrm{odd}})\geq H^{\mathrm{\pm}}(R)$. We will use this fact tacitly several times to prove the following lemma. Lemma 4.6: _Let $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. Then $\gamma$ has to cross ${\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$._ _Proof_. Assume the contrary, i.e. $\gamma\cap\{{\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}\}=\emptyset$. By the same arguments as in the end of the proof of Lemma 2.6, we can assume without restriction that throughout its whole path $\gamma$ has only one connected component.³ Since $\gamma$ leads to $\boxplus$, there exists some time $t$ such that $P_{V}R(\gamma(j))\geq l_{h}^{\star}$ and $P_{H}R(\gamma(j))\geq l_{b}^{\star}$ for all $j\geq t$, i.e. [FOOTNOTE:3][ENDFOOTNOTE] \[t-1=\max\big{\{}j\geq 0\ \|\ P_{V}R(\gamma(j))<l_{h}^{\star}\text { or }P_{H}R(\gamma(j))<l_{b}^{\star}\}.\] (4.13) Note that $\gamma(t-1)$ has to satisfy either 1. $P_{H}R(\gamma(t-1))=l_{b}^{\star}-1$ and $P_{V}R(\gamma(t-1))=l_{h}^{\star}+n$ for some $n\geq 0$, or 2. $P_{V}R(\gamma(t-1))=l_{h}^{\star}-1$ and $P_{H}R(\gamma(t-1))=l_{b}^{\star}+m$ for some $m\geq 0$. • Case 1: [$P_{H}R(\gamma(t-1))=l_{b}^{\star}-1$ and $P_{V}R(\gamma(t-1))=l_{h}^{\star}+n$ for some $n\geq 0$]. • Case 1.1: [$n=0$]. Let $\tau$ be the first time that a second $(+1)$–spin is added outside of $R(\gamma(t-1))=R((l_{b}^{\star}-1)\times l_{h}^{\star})$, i.e. \[\tau=\min\big{\{}j\geq t+1\ \big{\|}\ \|\gamma(j)\setminus R(\gamma (t-1))\|=2\big{\}}\] (4.14) Note that $\|\gamma(\tau-1)\setminus R(\gamma(t-1))\|=1$ and that this protuberance is either on the right or on the left of $R(\gamma(t-1))$, since $\gamma(t-1)$ was the last configuration with the property $P_{H}R(\gamma(t-1))=l_{b}^{\star}-1$. Analogously, $P_{V}R(\gamma(\tau-1))=l_{h}^{\star}$, otherwise, this would also contradict the definition of $t-1$. Now if $\gamma(\tau-1)\setminus R(\gamma(t-1))\in\Lambda_{\mathrm{odd}}$, we have that \[H^{\mathrm{\pm}}(\gamma(\tau-1))\geq H^{\mathrm{\pm}}((l_{b}^{ \star}-1)\times l_{h}^{\star})+2J+h_{\mathrm{odd}}=E_{\pm}^{\star}+h_{\mathrm{ even}}>E_{\pm}^{\star}.\] (4.15) This contradicts $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$, since we already know from Section 4.1 that $\Phi(\boxminus,\boxplus)\leq E^{\star}_{\pm}$. But if $\gamma(\tau-1)\setminus R(\gamma(t-1))\in\Lambda_{\mathrm{even}}$, then, since $\gamma$ does not cross ${\mathcal{P}}_{1}$ and since the minimal increase of energy to enlarge the rectangular envelope is $2J-h_{\mathrm{even}}$, we have by Lemma 4.5 that \[H^{\mathrm{\pm}}(\gamma(\tau-1))>H^{\mathrm{\pm}}((l_{b}^{\star} -1)\times l_{h}^{\star})+2J-h_{\mathrm{even}}=E_{\pm}^{\star}-h_{\mathrm{odd}}.\] (4.16) $\gamma(\tau)$ is obtained from $\gamma(\tau-1)$ by flipping a $(-1)$–spin outside of $R(\gamma(t-1))$. One can easily see that the most profitable way is to flip a $(-1)$–spin at a site that is adjacent to the protuberance of $\gamma(\tau-1)$, which consequently must belong to $\Lambda_{\mathrm{odd}}$. Hence, \[H^{\mathrm{\pm}}(\gamma(\tau))\geq H^{\mathrm{\pm}}(\gamma(k-1)) +h_{\mathrm{odd}}>E_{\pm}^{\star},\] (4.17) which leads to a contradiction. • Case 1.2: [$n=2k$ for some $k>1$]. According to Lemma 4.5, we have that \[\begin{split} H^{\mathrm{\pm}}(\gamma(t))& \geq H^{\mathrm{\pm}}(\gamma(t-1))+2J-h_{\mathrm{even}}\geq H^{ \mathrm{\pm}}((l_{b}^{\star}-1)\times(l_{h}^{\star}+2k))+2J-h_{\mathrm{even}} \\ &=H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times l_{h}^{\star})+k(4J- \varepsilon(l_{b}^{\star}-1))+2J-h_{\mathrm{even}}\\ &>E_{\pm}^{\star}.\end{split}\] (4.18) As before, this leads to a contradiction. • Case 1.3: [$n=2k+1$ for some $k\geq 0$]. It holds that either the top or bottom row of $\gamma(t)$ must belong to $\Lambda_{\mathrm{odd}}$. Similar to Case 1.2, we obtain a contradiction, since \[H^{\mathrm{\pm}}(\gamma(t)) \geq H^{\mathrm{\pm}}(\gamma(t-1))+2J-h_{\mathrm{even}}\geq H^{ \mathrm{\pm}}((l_{b}^{\star}-1)\times(l_{h}^{\star}+2k))+4J+h_{\mathrm{odd}}-h _{\mathrm{even}}\] \[\geq H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times l_{h}^{\star})+4J+h _{\mathrm{odd}}-h_{\mathrm{even}}\] (4.19) \[>E_{\pm}^{\star}.\] • Case 1.4: [$P_{V}R(\gamma(t-1))=\sqrt{\|\Lambda\|}$]. Using Assumption 4.1 d), we observe that \[\begin{split} H^{\mathrm{\pm}}(\gamma(t-1))& \geq H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times\sqrt{\|\Lambda\|})\\ &\geq H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times l_{h}^{\star})+ \lfloor(\sqrt{\|\Lambda\|}-l_{h}^{\star})/2\rfloor(4J-\varepsilon(l_{b}^{\star}- 1))-2J(l_{b}^{\star}-1)\\ &>H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times l_{h}^{\star})+h_{ \mathrm{odd}}.\end{split}\] (4.20) This leads to a contradiction, since \[\begin{split} H^{\mathrm{\pm}}(\gamma(t))& \geq H^{\mathrm{\pm}}(\gamma(t-1))+2J-h_{\mathrm{even}}\\ &>H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times l_{h}^{\star})+h_{ \mathrm{odd}}+2J-h_{\mathrm{even}}=E_{\pm}^{\star}.\end{split}\] (4.21) • Case 2: [$P_{V}R(\gamma(t-1))=l_{h}^{\star}-1$ and $P_{H}R(\gamma(t-1))=l_{b}^{\star}+m$ for some $m\geq 0$]. Assume first that $\gamma(t)$ starts in $\Lambda_{\mathrm{odd}}$. Then by cutting the top and the bottom row of $\gamma(t)$, which belong to $\Lambda_{\mathrm{odd}}$, we easily obtain a contradiction, since \[\begin{split} H^{\mathrm{\pm}}(\gamma(t))& \geq H^{\mathrm{\pm}}((l_{b}^{\star}+m)\times(l_{h}^{\star}-2))+4 J+2h_{\mathrm{odd}}\\ &=H^{\mathrm{\pm}}(l_{b}^{\star}\times(l_{h}^{\star}-2))+m(\mu- \varepsilon(L_{b}^{\star}-1))+4J+2h_{\mathrm{odd}}>E_{\pm}^{\star}.\end{split}\] (4.22) So from now on assume that $\gamma(t)$ starts in $\Lambda_{\mathrm{even}}$. Moreover, $\gamma(t)$ is obtained from $\gamma(t-1)$ either by flipping a $(-1)$–spin above $R(\gamma(t-1))$ or below. Without restriction, we assume that above $R(\gamma(t-1))$ a $(+1)$–spin is added. Finally, let $P_{H}R(\gamma(t))\times(l_{h}^{\star}-2)$ denote the rectangle of horizontal length $P_{H}R(\gamma(t))$ and vertical length $l_{h}^{\star}-2$ that start from the same row as the bottom of $R(\gamma(t))$. • Case 2.1: [$\|\gamma(t)\setminus\{P_{H}R(\gamma(t))\times(l_{h}^{\star}-2)\}\|>2$]. Note that we necessarily have that $\gamma(t-1)$ has at least two $(+1)$–spins in its uppermost row. If $m=0$, then, since $\gamma$ does not cross ${\mathcal{P}}_{2}$, we have by Lemma 4.5 that \[H^{\mathrm{\pm}}(\gamma(t-1))>H^{\mathrm{\pm}}(l_{b}^{\star} \times(l_{h}^{\star}-2))+2J+2h_{\mathrm{odd}}=E_{\pm}^{\star}-2J+h_{\mathrm{ even}}.\] (4.23) This leads to a contradiction, since \[H^{\mathrm{\pm}}(\gamma(t))\geq H^{\mathrm{\pm}}(\gamma(t-1))+2J -h_{\mathrm{even}}>E_{\pm}^{\star}.\] (4.24) If $m>0$ and $P_{H}R(\gamma(t-1))<\sqrt{\|\Lambda\|}$, then similarly, we observe \[\begin{split} H^{\mathrm{\pm}}(\gamma(t))& \geq H^{\mathrm{\pm}}(\gamma(t-1))+2J-h_{\mathrm{even}}\\ &\geq H^{\mathrm{\pm}}((l_{b}^{\star}+m)\times(l_{h}^{\star}-2))+ 2J+2h_{\mathrm{odd}}+2J-h_{\mathrm{even}}\\ &=H^{\mathrm{\pm}}(l_{b}^{\star}\times(l_{h}^{\star}-2))+m(\mu- \varepsilon(l_{b}^{\star}-1))+4J+2h_{\mathrm{odd}}-h_{\mathrm{even}}\\ &>E_{\pm}^{\star},\end{split}\] (4.25) which is a contradiction. Finally, if $P_{H}R(\gamma(t-1))=\sqrt{\|\Lambda\|}$, it holds that \[H^{\mathrm{\pm}} (\gamma(t))\geq H^{\mathrm{\pm}}(\gamma(t-1))+2J-h_{\mathrm{even}}\] \[\geq H^{\mathrm{\pm}}(\sqrt{\|\Lambda\|}\times(l_{h}^{\star}-2))+2J +2h_{\mathrm{odd}}-2J+2J-h_{\mathrm{even}}\] \[=H^{\mathrm{\pm}}(l_{b}^{\star}\times(l_{h}^{\star}-2))+(\sqrt{\| \Lambda\|}-l_{b}^{\star})(\mu-\varepsilon(l_{b}^{\star}-1))-2J(l_{h}^{\star}-1) +4J+2h_{\mathrm{odd}}-h_{\mathrm{even}}\] \[>E_{\pm}^{\star},\] (4.26) • Case 2.2: [$\|\gamma(t)\setminus\{P_{H}R(\gamma(t))\times(l_{h}^{\star}-2)\}\|=2$]. Define \[T=\max\Big{\{}j\geq t\,\Big{\|}\,\|\gamma(j)\setminus\{P_{H}R( \gamma(t))\times(l_{h}^{\star}-2)\}\|\leq 2\Big{\}}\] (4.27) i.e. the last time that a configuration has only two $(+1)$–spins outside of $P_{H}R(\gamma(t))\times(l_{h}^{\star}-2)$. From the maximality property of $t$, we have that $P_{V}R(\gamma(T))=l_{h}^{\star}$ and $P_{H}R(\gamma(T))=l_{b}^{\star}+m^{\prime}$ for some $m^{\prime}\geq 0$. Moreover, we easily observe that $H^{\mathrm{\pm}}(\gamma(T+1))\geq H^{\mathrm{\pm}}(\gamma(T))+h_{\mathrm{odd}}$. As in the Case 2.1, every possible value of $m^{\prime}$ leads to a contradiction. Indeed, if $m^{\prime}=0$, then, since $\gamma$ does not cross ${\mathcal{P}}_{2}$, Lemma 4.5 implies that \[H^{\mathrm{\pm}}(\gamma(T))>H^{\mathrm{\pm}}(l_{b}^{\star}\times (l_{h}^{\star}-2))+4J-h_{\mathrm{even}}+h_{\mathrm{odd}}=E_{\pm}^{\star}-h_{ \mathrm{odd}},\] (4.28) and therefore \[H^{\mathrm{\pm}}(\gamma(T+1)\geq H^{\mathrm{\pm}}(\gamma(T))+h_{ \mathrm{odd}}>E_{\pm}^{\star}.\] (4.29) If $m^{\prime}>0$ and $P_{H}R(\gamma(T))<\sqrt{\|\Lambda\|}$, then \[\begin{split} H^{\mathrm{\pm}}(\gamma(T+1))& \geq H^{\mathrm{\pm}}(\gamma(T))+h_{\mathrm{odd}}\\ &\geq H^{\mathrm{\pm}}((l_{b}^{\star}+m)\times(l_{h}^{\star}-2))+ 4J-h_{\mathrm{even}}+h_{\mathrm{odd}}+h_{\mathrm{odd}}\\ &>E_{\pm}^{\star}.\end{split}\] (4.30) Finally, if $P_{H}R(\gamma(T))=\sqrt{\|\Lambda\|}$, it holds that \[\begin{split} H^{\mathrm{\pm}}(\gamma(T+1))& \geq H^{\mathrm{\pm}}(\gamma(T))+h_{\mathrm{odd}}\\ &\geq H^{\mathrm{\pm}}(\sqrt{\|\Lambda\|}\times(l_{h}^{\star}-2))+4 J-h_{\mathrm{even}}+2h_{\mathrm{odd}}\\ &>E_{\pm}^{\star}.\end{split}\] (4.31) This concludes the proof. $\square$ The following observation concludes the proof of $\Phi(\boxminus,\boxplus)-H^{\pm}(\boxminus)\geq\Gamma^{\star}_{\pm}$. Lemma 4.7: _Let $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. In order to cross at a time $t$ a configuration $\gamma(t)$ such that $P_{V}R(\gamma(j))\geq l_{h}^{\star}$ and $P_{H}R(\gamma(j))\geq l_{b}^{\star}$ for all $j\geq t$ , $\gamma$ has to pass through ${\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$ and ${\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$ at some time $t^{\prime}\geq t-1$. In particular, every optimal path between $\boxminus$ and $\boxplus$ has to cross ${\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$._ _Proof_. Consider the time step $t$ defined in the proof of Lemma 4.6. It was shown that there necessarily needs to exist a time $t^{\prime}\geq t-1$ such that $\gamma(t^{\prime})\in{\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$, otherwise $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$ can not hold true. By the definition of $t$, $P_{V}R(\gamma(t^{\prime}+1))\geq l_{h}^{\star}$ and $P_{H}R(\gamma(t^{\prime}+1))\geq l_{b}^{\star}$. Hence, by a straightforward analysis of all the possible cases, we observe that $\gamma(t^{\prime}+1)$ must belong to ${\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$, since otherwise the energy level would exceed $E_{\pm}^{\star}$, which violates the fact that $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$. This proves the lemma. $\square$ ### Identification of ${\mathcal{P}}^{\star}$ and ${\mathcal{C}}^{\star}$ Repeating similar computations as in Section 4.1, it is clear that ${\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}\subset{\mathcal{P}}^{\star}$. Now let $\sigma\in{\mathcal{P}}^{\star}$ and $x\in\Lambda$ be such that $\sigma^{x}\in{\mathcal{C}}^{\star}$. Then there exists $\gamma\in(\boxminus,\boxplus)_{\mathrm{opt}}$ and $\ell\in{\mathbb{N}}$ such that 1. $\gamma(\ell)=\sigma$ and $\gamma(\ell+1)=\sigma^{x}$, 2. $H^{\mathrm{\pm}}(\gamma(k))<E^{\star}_{\pm}$ for all $k\in\{0,\dots,\ell\}$, 3. $\Phi(\boxminus,\gamma(k))\geq\Phi(\gamma(k),\boxplus)$ for all $k\geq\ell+1$. As in the proof of Lemma 4.6 and in Lemma 4.7, let \[t-1=\max\big{\{}j\geq 0\ \|\ P_{V}R(\gamma(j))<l_{h}^{\star}\text { or }P_{H}R(\gamma(j))<l_{b}^{\star}\}.\] (4.32) We know from Lemma 4.7 that there exists $t^{\prime}\geq t$ such that $\gamma(t^{\prime})\in{\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$ and $\gamma(t^{\prime}+1)\in{\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$. We get from fact (ii) that $\ell\leq t^{\prime}$. • If $\ell=t^{\prime}$, then we have that $\sigma\in{\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$ and $\sigma^{x}\in{\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$. • If $\ell<t^{\prime}$, then fact (iii) is violated, since $\Phi(\boxminus,\gamma(t^{\prime}))<\Phi(\gamma(t^{\prime}),\boxplus)=E_{\pm}^{\star}$. Therefore, such a path $\gamma$ can not exist, which contradicts the fact $\sigma^{x}\in{\mathcal{C}}^{\star}$. Hence, it must be the case that $\ell=t^{\prime}$. We conclude that ${\mathcal{P}}^{\star}={\mathcal{P}}_{1}\cup{\mathcal{P}}_{2}$ and ${\mathcal{C}}^{\star}={\mathcal{C}}_{1}\cup{\mathcal{C}}_{2}$. ### Verification of (H1) Obviously, $S_{\mathrm{stab}}=\{\boxplus\}$, since $h_{\mathrm{even}}>h_{\mathrm{odd}}$. It remains to show that $S_{\mathrm{meta}}=\{\boxminus\}.$ Let $\sigma\in S$. There are four cases. • Case 1: [$\sigma$ contains a component, which is not a stable rectangle]. Lemma 4.4 implies that $\sigma$ is not $h_{\mathrm{odd}}$–stable, i.e. there exists $\sigma^{\prime}\in S$ such that $H^{\mathrm{\pm}}(\sigma^{\prime})<H^{\mathrm{\pm}}(\sigma)$ and $\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{\pm}}(\sigma)\leq h_{\mathrm{odd}}< \Gamma_{\pm}^{\star}$. • Case 2: [$\sigma$ contains a connected component $R$, which is a stable rectangle with $P_{V}R\geq l_{h}^{\star}$ and $P_{H}R<\sqrt{\|\Lambda\|}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by attaching on the right of $R$ a column of length $P_{V}R$. We start to attach on an even row on the right of $R$ and then successively flip adjacent spins until the column is filled. Then \[H^{\mathrm{\pm}}(\sigma^{\prime}) \leq H^{\mathrm{\pm}}(\sigma)+\mu-\frac{P_{V}R+1}{2}\varepsilon \leq H^{\mathrm{\pm}}(\sigma)+\mu-l_{b}^{\star}\varepsilon<H^{\mathrm{\pm}}( \sigma),\] (4.33) and \[\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{\pm}}(\sigma)\leq 2J-h_{ \mathrm{even}}<\Gamma_{\pm}^{\star}.\] (4.34) • Case 3: [$\sigma$ contains a connected component $R$ which is a stable rectangle with $P_{V}R\leq l_{h}^{\star}-2$ and $P_{H}R<\sqrt{\|\Lambda\|}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by cutting the right column of $R$. Then \[H^{\mathrm{\pm}}(\sigma^{\prime}) =H^{\mathrm{\pm}}(\sigma)-\mu+\frac{P_{V}R+1}{2}\varepsilon\leq H ^{\mathrm{\pm}}(\sigma)-\mu+(l_{b}^{\star}-1)\varepsilon<H^{\mathrm{\pm}}( \sigma),\] (4.35) and \[\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{\pm}}(\sigma)\leq\frac{P_ {V}R-1}{2}\varepsilon+h_{\mathrm{odd}}<\Gamma_{\pm}^{\star}.\] (4.36) • Case 4: [$\sigma$ contains a connected component $R$, which is a stable rectangle with $P_{H}R=\sqrt{\|\Lambda\|}$]. Let $\sigma^{\prime}$ be obtained from $\sigma$ by attaching above $R$ successively vertical bars of length $2$ until the two rows above $R$ wrap around the torus. Then: \[\begin{split}& H^{\mathrm{\pm}}(\sigma^{\prime})=H^{ \mathrm{\pm}}(\sigma)+4J-l_{1}\varepsilon<H^{\mathrm{\pm}}(\sigma),\quad\text{ and}\\ &\Phi(\sigma,\sigma^{\prime})-H^{\mathrm{\pm}}(\sigma)\leq 4J- \varepsilon<\Gamma_{\pm}^{\star}.\end{split}\] (4.37) This proves that $S_{\mathrm{meta}}=\{\boxminus\}.$ ### Computation of $K$ Before estimating $K^{-1}$ from below and above, we define $\bar{{\mathcal{C}}}=\bar{{\mathcal{C}}_{1}}\cup\bar{{\mathcal{C}}_{2}}$, where * $\bar{{\mathcal{C}}_{1}}$ is the set of all configurations $\sigma$ that are obtained from a configuration $\sigma^{\prime}\in{\mathcal{C}}_{1}$ as follows. There is a column in $\sigma^{\prime}$ that has length 2. $\sigma$ is obtained from $\sigma^{\prime}$ by adding a third $(+1)$–spin on the even row adjacent to this column, and * $\bar{{\mathcal{C}}_{2}}$ is the set of all configurations $\sigma$ that are obtained from a configuration $\sigma^{\prime}\in{\mathcal{C}}_{2}$ as follows. There is a component of three $(+1)$–spins above or below the $l_{b}^{\star}\times(l_{h}^{\star}-2)$-rectangle in $\sigma^{\prime}$. $\sigma$ is obtained from $\sigma^{\prime}$ by adding a $(+1)$–spin such that this component becomes a $2\times 2$-square. [FIGURE:S4.F5][ENDFIGURE] Note that and ${\mathcal{P}}^{\star}\subset S_{\boxminus}$ and $\bar{{\mathcal{C}}}\subset S_{\boxplus}$. • _Lower bound_. Using these definitions and facts, we can estimate $K^{-1}$ as follows. \[\begin{split}\frac{1}{K}&\geq\min_{C_{1 },\dots,C_{I}\in[0,1]}\min_{\begin{subarray}{c}h:S^{\star}\rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{S_{\boxminus}}}=1,{\left.\kern-1.2pth \vphantom{\|}\right\|_{S_{\boxplus}}}=0,{\left.\kern-1.2pth\vphantom{\|}\right\|_{ S_{i}}}=C_{i}\,\forall i\end{subarray}}\frac{1}{2}\sum_{\eta,\eta^{\prime}\in( {\mathcal{C}}^{\star})^{+}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[h(\eta)-h( \eta^{\prime})]^{2}\\ &=\min_{h:{\mathcal{C}}^{\star}\rightarrow[0,1]}\sum_{\eta\in{ \mathcal{C}}^{\star}}\left(\sum_{\eta^{\prime}\in{\mathcal{P}}^{\star},\eta^{ \prime}\sim\eta}[1-h(\eta)]^{2}+\sum_{\eta^{\prime}\in\bar{{\mathcal{C}}},\eta ^{\prime}\sim\eta}h(\eta)^{2}\right)\\ &=\sum_{\eta\in{\mathcal{C}}_{1}}\frac{\|{\mathcal{P}}^{\star}\sim \eta\|\cdot\|\bar{{\mathcal{C}}}\sim\eta\|}{\|{\mathcal{P}}^{\star}\sim\eta\|+\|\bar {{\mathcal{C}}}\sim\eta\|}+\sum_{\eta\in{\mathcal{C}}_{2}}\frac{\|{\mathcal{P}}^ {\star}\sim\eta\|\cdot\|\bar{{\mathcal{C}}}\sim\eta\|}{\|{\mathcal{P}}^{\star}\sim \eta\|+\|\bar{{\mathcal{C}}}\sim\eta\|}.\end{split}\] (4.38) For all $\eta\in{\mathcal{C}}_{1}$ we have that $\|{\mathcal{P}}^{\star}\sim\eta\|=1$, whereas for all $\eta\in{\mathcal{C}}_{2}$ we have that $\|{\mathcal{P}}^{\star}\sim\eta\|=2$. Moreover, $\|\bar{{\mathcal{C}}}\sim\eta\|=1$ for all $\eta\in{\mathcal{C}}^{\star}$. Finally, it can be seen easily that $\|{\mathcal{C}}_{1}\|=\|{\mathcal{C}}_{2}\|=4\|\Lambda\|(l_{b}^{\star}-1)$ Hence, \[\frac{1}{K}\geq\|{\mathcal{C}}_{1}\|\frac{1}{2}+\|{\mathcal{C}}_{2}\| \frac{2}{3}=\frac{14\,(l_{b}^{\star}-1)}{3}\|\Lambda\|.\] (4.39) • _Upper bound_. We say that a row or a column of a configuration is a _singleton_, if it consists only of a single $(+1)$– spin. Define the following subsets of $S^{\star}$ \[\begin{split} R^{-}_{0}=\{\sigma\in S^{\star}& \|\,\sigma\text{ is not connected and either }P_{V}R(\eta)<l_{h}^{ \star}\text{ or }P_{H}R(\eta)<l_{b}^{\star}\\ &\ \text{ for all its connected components }\eta\},\\ R^{+}_{0}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is not connected and }P_{V}R(\eta)\geq l_{h}^{\star}\text{ and }P_{H}R(\eta)\geq l _{b}^{\star}\\ &\ \text{ for at least one of its connected components }\eta\},\\ R^{-}_{1}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{V}R(\sigma)<l_{h}^{\star}\},\\ R^{-}_{2}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)<l_{b}^{\star}\},\\ R^{-}_{3}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)\geq l_{b}^{\star}\text{ and }P_{V}R(\sigma)=l_{h }^{\star}\\ &\ \text{ and at least two rows of $\sigma$ are singletons}\},\\ R^{-}_{4}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)=l_{b}^{\star}\text{ and }P_{V}R(\sigma)=l_{h}^{ \star}\\ &\ \text{ and at least one column of $\sigma$ is a singleton}\}, \\ R^{+}_{1}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)=l_{b}^{\star}\text{ and }P_{V}R(\sigma)=l_{h}^{ \star}\\ &\ \text{ and no column of $\sigma$ is a singleton}\\ &\ \text{ and at most one row of $\sigma$ is a singleton}\},\\ R^{+}_{2}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)>l_{b}^{\star}\text{ and }P_{V}R(\sigma)=l_{h}^{ \star}\\ &\ \text{ and at most one row of $\sigma$ is a singleton}\},\\ R^{+}_{3}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)=l_{b}^{\star}\text{ and }P_{V}R(\sigma)>l_{h}^{ \star}\},\\ R^{+}_{4}=\{\sigma\in S^{\star}&\|\,\sigma\text{ is connected and }P_{H}R(\sigma)>l_{b}^{\star}\text{ and }P_{V}R(\sigma)>l_{h}^{ \star}\}.\end{split}\] (4.40) Moreover, let $R^{-}=R^{-}_{0}\cup R^{-}_{1}\cup R^{-}_{2}\cup R^{-}_{3}\cup R^{-}_{4}$ and $R^{+}=R^{+}_{0}\cup R^{+}_{1}\cup R^{+}_{2}\cup R^{+}_{3}\cup R^{+}_{4}$. Notice that $S^{\star}=R^{-}\cup R^{+}$, $R^{-}\cap R^{+}=\emptyset$, ${\mathcal{P}}^{\star}\subset R^{-}$ and ${\mathcal{C}}^{\star}\subset R^{+}$. Lemma 4.8: _Let $\sigma\in R^{-}$ and $\sigma^{\prime}\in R^{+}$. Then $\sigma\sim\sigma^{\prime}$, if and only if $\sigma\in{\mathcal{P}}^{\star}$ and $\sigma^{\prime}\in{\mathcal{C}}^{\star}$._ _Proof_. We show for each case separately that, if $\sigma\notin{\mathcal{P}}^{\star}$ or $\sigma^{\prime}\notin{\mathcal{C}}^{\star}$, then $\sigma,\sigma^{\prime}\in S^{\star}$ is contradicted. • Case 0: [$\sigma\in R^{-}_{0}$ or $\sigma^{\prime}\in R^{+}_{0}$]. We only treat the case $\sigma\in R^{-}_{0}$ and $\sigma^{\prime}\in R^{+}$. The other case is analogue. There must be a component $\eta\subset\sigma$ such that either (i) $P_{V}R(\eta)=l_{h}^{\star}+m\text{ for some }m\geq 0\text{ and }P_{H}R(\eta)=l _{b}^{\star}-1$ or (ii) $P_{V}R(\eta)=l_{h}^{\star}-1\text{ and }P_{H}R(\eta)=l_{b}^{\star}+m$ for some $m\geq 0$, and a component $\eta^{\prime}\subset\sigma^{\prime}$ such that $P_{V}R(\eta^{\prime})\geq l_{h}^{\star}\text{ and }P_{H}R(\eta^{\prime})\geq l _{b}^{\star}$ and $\eta\sim\eta^{\prime}$. We only consider case (ii), because the other one is analogue. It is easy to see that \[H^{\mathrm{\pm}}(\sigma)\geq H^{\mathrm{\pm}}(\eta)+4J-h_{ \mathrm{even}}.\] (4.41) Then \[\begin{split} H^{\mathrm{\pm}}(\sigma)& \geq H^{\mathrm{\pm}}(\eta)+4J-h_{\mathrm{even}}\geq H^{\mathrm{\pm}}((l_{b}^{ \star}+m)\times(l_{h}^{\star}-2))+6J-\varepsilon\\ &=E_{\pm}^{\star}+m(\mu-\varepsilon(l_{b}^{\star}-1))+2J-h_{ \mathrm{odd}}>E_{\pm}^{\star}.\end{split}\] (4.42) This contradicts $\sigma\in S^{\star}$. Hence, this case is not possible. • Cases 1, 2, 3, 4, 5: [($\sigma\in R^{-}_{1},\sigma^{\prime}\in R^{+}_{3}$) or ($\sigma\in R^{-}_{1},\sigma^{\prime}\in R^{+}_{4}$) or ($\sigma\in R^{-}_{2},\sigma^{\prime}\in R^{+}_{2}$) or ($\sigma\in R^{-}_{2},\sigma^{\prime}\in R^{+}_{4}$) or ($\sigma\in R^{-}_{4},\sigma^{\prime}\in R^{+}_{4}$)]. Due to the fact that both configurations are connected, all these cases are not possible. • Cases 6, 7: [($\sigma\in R^{-}_{1}\setminus{\mathcal{P}}^{\star}$, $\sigma^{\prime}\in R^{+}_{1}\setminus{\mathcal{C}}^{\star}$) or ($\sigma\in R^{-}_{1}$, $\sigma^{\prime}\in R^{+}_{2}$)]. Necessarily, each row of $\sigma$ needs to have more than two $(+1)$–spins, since to increase $P_{V}R(\sigma)$, a row must be added to $\sigma$ that consists of a single $(+1)$–spin only. Now the same computations as in Case 2.1 from the proof of Lemma 4.6 show that both cases are not possible. • Case 8: [$\sigma\in R^{-}_{2}$ and $\sigma^{\prime}\in R^{+}_{1}$]. To increase $P_{H}R(\sigma)$, it is necessary to add a column to $\sigma$ that consists of a single $(+1)$–spin only. But since no column of $\sigma^{\prime}$ is a singleton, $\sigma\sim\sigma^{\prime}$ can not hold true, which implies that this case is not possible. • Case 9: [$\sigma\in R^{-}_{2}$ and $\sigma^{\prime}\in R^{+}_{3}$]. The same computations as in the Cases 1.2, 1.3 and 1.4 from the proof of Lemma 4.6 show that this case is not possible. • Cases 10, 11: [($\sigma\in R^{-}_{3}\setminus{\mathcal{P}}^{\star}$, $\sigma^{\prime}\in R^{+}_{1}\setminus{\mathcal{C}}^{\star}$) or ($\sigma\in R^{-}_{3}$, $\sigma^{\prime}\in R^{+}_{2}$)]. $\sigma^{\prime}$ is obtained from $\sigma$ by adding a $(+1)$–spin to one of the two rows of $\sigma$ that are singletons. The same computations as in the Case 2.2 from the proof of Lemma 4.6 show that this case is not possible. • Cases 12, 13: [($\sigma\in R^{-}_{3}$, $\sigma^{\prime}\in R^{+}_{3}$) or ($\sigma\in R^{-}_{3}$, $\sigma^{\prime}\in R^{+}_{4}$)]. $\sigma^{\prime}$ is obtained from $\sigma$ by attaching a protuberance above or below of $\sigma$. Let $P_{H}R(\sigma)=l_{b}^{\star}+m$ for some $m\geq 0$. Obviously, $H^{\mathrm{\pm}}(\sigma)\geq H^{\mathrm{\pm}}((l_{b}^{\star}+m)\times(l_{h}^{ \star}-2))+4J-\varepsilon$. This implies that (4.43) Hence, this case is not possible. • Case 14: [$\sigma\in R^{-}_{4}\setminus{\mathcal{P}}^{\star}$ and $\sigma^{\prime}\in R^{+}_{1}\setminus{\mathcal{C}}^{\star}$]. The same computations as in the Case 1.1 from the proof of Lemma 4.6 show that this case is not possible. • Cases 15, 16: [($\sigma\in R^{-}_{4}$, $\sigma^{\prime}\in R^{+}_{2}$) or ($\sigma\in R^{-}_{4}$, $\sigma^{\prime}\in R^{+}_{3}$)]. $\sigma^{\prime}$ is obtained from $\sigma$ by attaching a protuberance outside of $\sigma$. Obviously, $H^{\mathrm{\pm}}(\sigma)\geq H^{\mathrm{\pm}}((l_{b}^{\star}-1)\times(l_{h}^{ \star}-2))+2J-h_{\mathrm{even}}$. This implies that (4.44) Hence, this case is not possible. $\square$ Moreover, we have that for all $i$ either $S_{i}\subset R^{-}$ holds or $S_{i}\subset R^{+}$, since the $S_{i}$ are connected. For the same reason and by Section 4.1, $S_{\boxminus}\subset R^{-}$ and $S_{\boxplus}\subset R^{+}$ holds. Therefore, \[\begin{split}\frac{1}{K}&\leq\min_{ \begin{subarray}{c}h:S^{\star}\rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{R^{-}}}=1,{\left.\kern-1.2pth\vphantom {\|}\right\|_{R^{+}\setminus{\mathcal{C}}^{\star}}}=0\end{subarray}}\frac{1}{2} \sum_{\eta,\eta^{\prime}\in S^{\star}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[ h(\eta)-h(\eta^{\prime})]^{2}\\ &=\min_{\begin{subarray}{c}h:({\mathcal{C}}^{\star})^{+} \rightarrow[0,1]\\ {\left.\kern-1.2pth\vphantom{\|}\right\|_{R^{-}\cap\partial^{+}{\mathcal{C}}^{ \star}}}=1,{\left.\kern-1.2pth\vphantom{\|}\right\|_{R^{+}\cap\partial^{+}{ \mathcal{C}}^{\star}}}=0\end{subarray}}\frac{1}{2}\sum_{\eta,\eta^{\prime}\in( {\mathcal{C}}^{\star})^{+}}\mathbbm{1}_{\{\eta\sim\eta^{\prime}\}}[h(\eta)-h( \eta^{\prime})]^{2}\\ &=\min_{h:{\mathcal{C}}^{\star}\rightarrow[0,1]}\sum_{\eta\in{ \mathcal{C}}^{\star}}\left(\sum_{\eta^{\prime}\in{\mathcal{P}}^{\star},\eta^{ \prime}\sim\eta}[1-h(\eta)]^{2}+\sum_{\eta^{\prime}\in\bar{{\mathcal{C}}},\eta ^{\prime}\sim\eta}h(\eta)^{2}\right)\\ &=\frac{14\,(l_{b}^{\star}-1)}{3}\|\Lambda\|.\end{split}\] (4.45) ## Acknowledgment The author would like to give many thanks to Anton Bovier and Muhittin Mungan for a lot of useful discussions and suggestions. ## References * [1] L. 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0708.0080	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 17896, "num_imgs": 0, "llama3_tokens_count": 5448 }	[]	# Farey Statistics in Time $n^{2/3}$ and Counting Primitive Lattice Points in Polygons Mihai Pǎtraşcu mip@mit.edu ###### Abstract We present algorithms for computing ranks and order statistics in the Farey sequence, taking time $\widetilde{O}(n^{2/3})$. This improves on the recent algorithms of Pawlewicz [16], running in time $\widetilde{O}(n^{3/4})$. We also initiate the study of a more general algorithmic problem: counting primitive lattice points in planar shapes. Since the publication of this technical report, this work has been extended and merged with the paper of Pawlewicz. The merged version is available at: http://web.mit.edu/~mip/www/papers/farey2/paper.pdf ## 1 An Improved Algorithm for the Farey Sequence The Farey sequence of order $n$, denoted $\mathcal{F}_{n}$, is the ordered list of irreducible fractions $\frac{a}{b}$ with $a\leq b\leq n$. This sequence is a well-studied mathematical object, with fascinating properties. See [3] for a discussion at length. The sequence has $\Theta(n^{2})$ terms, and there are many algorithms for generating it entirely in $O(n^{2})$ time. Perhaps the best known ones are based on the Stern-Brocot tree, and the properties of the mediant. One can ask, however, for more local access to the sequence. Two natural questions arise: * given a number $x\in[0,1]$ find $\mbox{\sc Rank}(x,n)=\big{\|}\mathcal{F}_{n}\cap[0,x]\big{\|}$. * given an index $k\leq\|\mathcal{F}_{n}\|$ find $\mbox{\sc Statistic}(k,n)=$ the $k$-th value in $\mathcal{F}_{n}$ (in sorted order). The Statistic problem can be solved with $O(\lg n)$ calls to the Rank problem [16], so below only bounds for the Rank version are discussed. To the best of my knowledge, the question was first formulated in 2003, when I proposed it as a contest problem at the 11th Balkan Olympiad in Informatics. The official solution consisted of an $O(n\lg n)$ algorithm, and several contestant also found this algorithm. We [17] later described a solution with a slightly better running time of $O(n)$. Quite recently, Pawlewicz [16] broke the linear time barrier, and provided an algorithm with running time $O(n^{3/4})$. In the present, I describe an improved $O(n^{2/3}\lg^{1/3}n)$ algorithm. ### Review: The Algorithm of Pawlewicz Let $S_{n}(x)=\left\|\left\{\frac{a}{b}\mid b\leq n~{}\land~{}\frac{a}{b}\leq x~{} \land~{}\gcd(a,b)=1\right\}\right\|$. This provides the quantity we want to compute. It can be seen that: \[S_{n}(x)=\sum_{b=1}^{n}\lfloor bx\rfloor-\sum_{d\geq 2}S_{\lfloor\frac{n}{d} \rfloor}(x)\] (1) It is shown in [16] that $A_{n}(x)=\sum_{b=1}^{n}\lfloor bx\rfloor$ can be computed in $O(\lg n)$ time. Thus, the only challenge is to estimate the recursive component of the sum. The recursion will only need $S_{\lfloor n/d\rfloor}(x)$ for all $d$, since $\big{\lfloor}\lfloor\frac{n}{d_{1}}\rfloor/d_{2}\big{\rfloor}=\big{\lfloor} \frac{n}{d_{1}d_{2}}\big{\rfloor}$. The crux of the algorithm is the following observation: given all relevant $S_{i}(x)$, for $i<k$, then $S_{k}(x)$ can be computed in $O(\sqrt{k})$ time. Indeed, $\sum_{d\geq 2}S_{\lfloor k/d\rfloor}(x)$ only contains at most $2\sqrt{k}$ distinct terms: $\sqrt{k}$ terms corresponding to $d\leq\sqrt{k}$, and at most $\sqrt{k}$ terms for $d\geq\sqrt{k}$ because then we have $k/d\leq\sqrt{k}$. The latter terms have multiplicities, but the multiplicity of each term can be computed easily in $O(1)$ time. Applying this observation to compute all needed $S$ terms recursively, the running time is: \[\sum_{d=1}^{n}\sqrt{\frac{n}{d}}~{}\leq~{}\sum_{d=1}^{\sqrt{n}}\sqrt{\frac{n}{ d}}+\sum_{j=1}^{\sqrt{n}}\sqrt{j}~{}\leq~{}\sqrt{n}\cdot\sum_{d=1}^{\sqrt{n}} \frac{1}{\sqrt{d}}+\sum_{j=1}^{\sqrt{n}}\sqrt{\sqrt{n}}~{}\leq~{}\sqrt{n}\cdot O (\sqrt[4]{n})+\sqrt{n}\cdot\sqrt[4]{n}~{}=~{}O(n^{3/4})\] ### Our Improved Algorithm The key to our improved algorithm is to show that $S_{1},\dots,S_{k}$ can be computed in time $O(k\lg k)$. Then, the remaining terms can be computed by the old algorithm in time $\sum_{d=1}^{n/k}\sqrt{n/d}=\sqrt{n}\cdot O(\sqrt{n/k})=O(n/\sqrt{k})$. The total running time is then $O(k\lg k+n/\sqrt{k})$, so it is optimized by picking $k=(n/\lg n)^{2/3}$. Thus, the running time is $O(n^{2/3}\lg^{1/3}n)$. To compute $S_{1},\dots,S_{k}$ efficiently, we make the observation that the composition (with respect to recursive terms) of $S_{i}$ and $S_{i-1}$ are not too different. Specifically, $\sum_{d\geq 2}S_{\lfloor i/d\rfloor}(x)$ differs from $\sum_{d\geq 2}S_{\lfloor(i-1)/d\rfloor}(x)$ only for the values of $d$ that $i$ is a multiple of. To maintain understanding of divisibility by all $d$’s, and compute $S_{i}(x)$ values in order, we use an algorithm similar in flavor to Eratosthene’s sieve [7]. We first create an array $D[1\mathinner{\ldotp\ldotp}k]$, where $D[i]$ holds a list of all divisors of $i$. To create the array, simple consider all $d$, and add $d$ to $D[md]$, for all $m$. This takes time $\sum_{d\geq 1}\frac{k}{d}=O(k\lg k)$. Now, iterate $i$ from $1$ to $k$, maintaining $\sum_{d\geq 2}S_{\lfloor i/d\rfloor}(x)$ at all times. The sum is updated by considering all $d\in D[i]$, subtracting $S_{\lfloor(i-1)/d\rfloor}(x)$ and adding $S_{\lfloor i/d\rfloor}(x)$. The complexity is linear in the size of $D[1\mathinner{\ldotp\ldotp}k]$, and is thus $O(k\lg k)$. To compute $S_{i}(x)$ from this running sum, we only need $A_{i}(x)$, which takes $O(\lg i)$, giving an additive $O(k\lg k)$. ## 2 Counting Primitive Lattice Points A primitive lattice point is a point $(x,y)$ in the plane, with $x,y\in\mathbb{Z}$ and $\gcd(x,y)=1$. We observe that computing $\mbox{\sc Rank}(x,n)$ in the Farey sequence is equivalent to couting primitive lattice points inside the right triangle defined by $(0,0)$, $(n,0)$ and $(n,xn)$. Let us now generalize the algorithm to counting primitive lattice points in more general shapes. Counting primitive lattice points in planar shape is a relatively new topic in mathematics, but one that is gathering significant momentum [10, 12, 4, 13, 6, 11, 14, 8, 19, 2, 18, 20, 15]. In the mathematical sense, “counting” refers to estimating the number of primitive points with a small error, as the size of the shape goes to infinity. In this paper, we initiate the study of the _algorithmic_ problem of counting (exactly) the number of primitive lattice points inside a given shape. More precisely, we study this problem for polygons containing the origin. The condition that the shape should contain the origin also appears in the mathematical works referenced above, and is natural given that we are counting points visible from the origin. Theorem 1.: _Let $P$ be a polygon containing the origin, defined by $k$ vertices at $b$-bit rational coordinates. If $D$ is the diameter of the polygon, one can count the number of primitive lattice points inside $P$ in time $D^{6/7}\cdot k\cdot b^{O(1)}$._ A very pertinent question is how efficient this running time actually is. Remember that [17] shows that the Farey rank problem can be used to factor integers. Since counting primitive lattice points is a generalization, we conclude that a polynomial-time algorithm, i.e. $\mathop{\rm poly}\nolimits(k\cdot b)$, is likely impossible. Thus, the algorithm needs to depend on some parameter describing the polygon, which can be exponential in $b$. One such parameter is the diameter $D$. Clearly, however, this is not the only choice, and it is conceivable that other measures lead to better results. For right triangles such as those in the Farey rank problem, the diameter is $n$, yet we know a better algorithm with time essentially $n^{2/3}$. A trivial alternative to Theorem 1 is the algorithm which iterates over all lattice points inside the polygon, and runs Euclid’s algorithm on each point. It is possible [1, 9, 5] to list all lattice points with a polynomial $\mathop{\rm poly}\nolimits(k\cdot b)$ cost per point. If the polygon has integral coordinates, Pick’s formula shows that the number of lattice points inside is asymptotically equal to the area. Thus, the exhaustive algorithm has complexity proportional to the area, times polynomial factors. Unfortunately, the area and the diameter are not related in the worst-case (e.g., for very skinny shapes). However, in the more “typical” case when the polygon is fat, the area is $A=\Theta(D^{2})$. Thus our running time of $D^{6/7}$ can be rewritten as $A^{3/7}$, which gives a significant saving over the exhaustive algorithm. It is an interesting open problem to construct an algorithm which beats exhaustive search for _any_ polygon. A standard idea.Let $P$ be a polygon, defined by rational points $(x_{1},y_{1}),\dots,(x_{k},y_{k})$. Then, let $P_{/d}$ be the polygon defined by $(\frac{x_{1}}{d},\frac{y_{1}}{d}),\dots,(\frac{x_{n}}{d},\frac{y_{n}}{d})$. Define $A(P)$ to be the number of lattice points inside polygon $P$, and $S(P)$ the number of primitive lattice points inside $P$. We first observe the following recursive formula: \[S(P)=A(P)-\sum_{d\geq 2}S(P_{/d})\] (2) Indeed, every point in $A(P)$, but not in $S(P)$ is a nonprimitive lattice point $(x,y)$. If $\gcd(x,y)=d>1$, then $(\frac{x}{d},\frac{y}{d})$ is a primitive lattice point. Furthermore, such a point is inside $P_{/d}$, so we can remove all points with a greater common divisor equal to $d$ by subtracting $P_{/d}$. We can bound the recursion depth in (2), by appealing to the diameter $D$. We note that the diameter of $P_{/(D+1)}$ is less than 1, so it does not contain any lattice point outside the origin. Thus, $S(P_{/d})=0$ for all $D<d<\infty$, and $S(P_{/\infty})=1$ (the origin). Then, it suffices to consider only $P,P_{/2},\dots,P_{/D}$ in the algorithm. To compute the “constants” $A(P_{/i})$ in the recursion, one needs to compute the number of lattice points inside polygons with rational coordinates. This is a well-studied problem, and [1, 9, 5] given polynomial-time algorithms. Observe that for $i\leq D\leq 2^{b}$, coordinates of $P_{/i}$ have at most $2b$ bits of precision. Thus, computing $A(P_{/i})$ takes time $O(k\cdot\mathop{\rm poly}\nolimits(b))$; the dependence on $k$ is linear by triangulating the polygon. Note that formula (2) is very similar to the recursive formula for the Farey rank problem (1). Indeed, (1) is simply a transcription of (2), where the polygons are the relevant right triangles. It is thus tempting to conjecture that we can use the same dynamic program to evaluate (2). Unfortunately, this is not the case, for a somewhat subtle reason. Due to the geometry of the rank problem, $S_{n/d}$ was the same as $S_{\lfloor n/d\rfloor}$. By taking floors, we concluded that among $S_{\lfloor n/d\rfloor}$’s with $d\in\{\sqrt{n},\dots,n\}$, there are only $\sqrt{n}$ distinct quantities. Unfortunately, in general we cannot round the vertices of $P_{/d}$ to lattice points, and thus we cannot conclude that for $d\geq\sqrt{D}$ there are only $\sqrt{D}$ distinct cases. A more careful analysis.Since we are not interested in factors of $k\cdot b^{O(1)}$ in the running time, let us define $O^{}(f)=f\cdot k\cdot b^{O(1)}$. To use ideas similar to the previous dynamic program, we begin by obtaining a weaker bound for computing the small terms of the recurrence: Lemma 2.: _We can (implicitly) compute $S(P_{/\tau}),S(P_{/(\tau+1)}),\dots,S(P_{/D})$ in time $O^{}\big{(}\frac{D^{2}}{\tau^{2}}\big{)}$._ Proof.: Note that $P_{/D}\subseteq P_{/(D-1)}\subseteq\cdots\subseteq P_{/\tau}$. Also, the diameter of $P_{/\tau}$ is $D/\tau$, implying that $P_{/\tau}$, and any smaller polygon only contain $O\big{(}(\frac{D}{\tau})^{2}\big{)}$ lattice points. Since $S(P_{/i})$ can only grow between $1$ and $O\big{(}(\frac{D}{\tau})^{2}\big{)}$ when $i$ goes from $D$ to $\tau$, there are only so many distinct values that can appear. To actually compute these values, we perform an exhaustive enumeration of all lattice points in $P_{/\tau}$. Points which are not primitive are discarded. For every primitive point $(x,y)$, we compute a value $\varphi(x,y)$ which is the minimum $i$ such that $P_{/i}$ does not contain it. This is done by a binary search for $i$. Every comparison is a point-in-polygon test, which takes $O(k)$ time. We now sort the $\phi(x,y)$ values from largest to smallest. The sorted list gives us an _implicit_ representation for $S(P_{/i})$ for every $i\geq\tau$. Indeed, it suffices to binary search for the first occurrence of $i$; the elements to the left correspond to primitive points which are inside $P_{/i}$. ∎ Let us now consider the problem of computing a term using our recursion: $S(P_{/i})=A(P_{/i})-\sum_{d\geq 2}S(P_{/id})$. We assume terms $S(P_{/j})$ for $j\geq i$ have already been computed; in particular, for $j\geq\tau$ we have the implicit representation from Lemma 2. In principle, the expression for $S(P_{/i})$ has $\lfloor\frac{D}{i}\rfloor$ terms. However, as in Lemma 2, we can observe that for fixed $\delta$, there are only $O\big{(}\frac{D^{2}}{\delta^{2}}\big{)}$ distinct terms among all $S(P_{/id})$’s with $id\geq\delta$. These terms can actually be summed up in $O^{}\big{(}\frac{D^{2}}{\delta^{2}}\big{)}$ time. Indeed, we begin with $d_{0}=\lceil\frac{\delta}{i}\rceil$, and binary search for the minimum $d_{1}$ such that $S(P_{/id_{1}})<S(P_{/id_{0}})$. We add to the sum $S(P_{/id_{0}})\cdot(d_{1}-d_{0})$, then binary search for the minimum $d_{2}$ leading to a different value, and so on. After dealing like this with all terms $d\geq\frac{\delta}{i}$, we can simply add the remaining terms. The running time for computing $S(P_{/i})$ is therefore $O^{}\big{(}\frac{D^{2}}{\delta^{2}}+\frac{\delta}{i}\big{)}$. We can optimize by setting $\delta=D^{2/3}i^{1/3}$, yielding a running time of $O^{}\big{(}(\frac{D}{i})^{2/3}\big{)}$. We now want to optimize the parameter $\tau$ in Lemma 2. A choice of $\tau$ implies that we spend $O^{}\big{(}\frac{D^{2}}{\tau^{2}}\big{)}$ time by Lemma 2, and then run the dynamic program for terms $S(P_{/i})$ with $i<\tau$. This second part will take time: \[\sum_{i=1}^{\tau}O^{}\bigg{(}\frac{D}{i}\bigg{)}^{2/3}=O^{}(D^{2/3}\tau^{1/3})\] Then, we can optimize by setting $\frac{D^{2}}{\tau^{2}}=D^{2/3}\tau^{1/3}$, i.e. $\tau=D^{4/7}$. The final running time is therefore $O^{}(D^{6/7})$. ## References [Bar94] Alexander I. Barvinok. A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. _Mathematics of Operations Research_, 19(4):769–779, 1994. * [BCZ00] Florin P. Boca, Cristian Cobeli, and Alexandru Zaharescu. Distribution of lattice points visible from the origin. _Communications in Mathematical Physics_, 213(2):433–470, 2000. * [GKP94] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. _Concrete Mathematics: A Foundation for Computer Science_. 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1808.10122	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 74387, "num_imgs": 2, "llama3_tokens_count": 22446 }	[ "content_image/1808.10122/x1.png", "content_image/1808.10122/x1.png" ]	# Learning Neural Templates for Text Generation Sam Wiseman Stuart M. Shieber Alexander M. Rush School of Engineering and Applied Sciences Harvard University Cambridge, MA, USA {swiseman,shieber,srush}@seas.harvard.edu ###### Abstract While neural, encoder-decoder models have had significant empirical success in text generation, there remain several unaddressed problems with this style of generation. Encoder-decoder models are largely (a) uninterpretable, and (b) difficult to control in terms of their phrasing or content. This work proposes a neural generation system using a hidden semi-markov model (HSMM) decoder, which learns latent, discrete templates jointly with learning to generate. We show that this model learns useful templates, and that these templates make generation both more interpretable and controllable. Furthermore, we show that this approach scales to real data sets and achieves strong performance nearing that of encoder-decoder text generation models. ## 1 Introduction With the continued success of encoder-decoder models for machine translation and related tasks, there has been great interest in extending these methods to build general-purpose, data-driven natural language generation (NLG) systems (Mei et al., 2016; Dušek and Jurcıcek, 2016; Lebret et al., 2016; Chisholm et al., 2017; Wiseman et al., 2017). These encoder-decoder models (Sutskever et al., 2014; Cho et al., 2014; Bahdanau et al., 2015) use a neural encoder model to represent a source knowledge base, and a decoder model to emit a textual description word-by-word, conditioned on the source encoding. This style of generation contrasts with the more traditional division of labor in NLG, which famously emphasizes addressing the two questions of “what to say” and “how to say it” separately, and which leads to systems with explicit content selection, macro- and micro-planning, and surface realization components (Reiter and Dale, 1997; Jurafsky and Martin, 2014). <figure><img src="content_image/1808.10122/x1.png"><figcaption>Figure 2: An example from the WikiBio dataset (Lebret et al., 2016), with adatabase x (top) for Frederick Parker-Rhodes and corresponding referencegeneration y (bottom).</figcaption></figure> Encoder-decoder generation systems appear to have increased the fluency of NLG outputs, while reducing the manual effort required. However, due to the black-box nature of generic encoder-decoder models, these systems have also largely sacrificed two important desiderata that are often found in more traditional systems, namely (a) interpretable outputs that (b) can be easily controlled in terms of form and content. This work considers building interpretable and controllable neural generation systems, and proposes a specific first step: a new data-driven generation model for learning discrete, _template-like_ structures for conditional text generation. The core system uses a novel, neural hidden semi-markov model (HSMM) decoder, which provides a principled approach to template-like text generation. We further describe efficient methods for training this model in an entirely data-driven way by backpropagation through inference. Generating with the template-like structures induced by the neural HSMM allows for the explicit representation of what the system intends to say (in the form of a learned template) and how it is attempting to say it (in the form of an instantiated template). We show that we can achieve performance competitive with other neural NLG approaches, while making progress satisfying the above two desiderata. Concretely, our experiments indicate that we can induce explicit templates (as shown in Figure 1) while achieving competitive automatic scores, and that we can control and interpret our generations by manipulating these templates. Finally, while our experiments focus on the data-to-text regime, we believe the proposed methodology represents a compelling approach to learning discrete, latent-variable representations of conditional text. ## 2 Related Work A core task of NLG is to generate textual descriptions of knowledge base records. A common approach is to use hand-engineered templates (Kukich, 1983; McKeown, 1992; McRoy et al., 2000), but there has also been interest in creating templates in an automated manner. For instance, many authors induce templates by clustering sentences and then abstracting templated fields with hand-engineered rules (Angeli et al., 2010; Kondadadi et al., 2013; Howald et al., 2013), or with a pipeline of other automatic approaches (Wang and Cardie, 2013). There has also been work in incorporating probabilistic notions of templates into generation models (Liang et al., 2009; Konstas and Lapata, 2013), which is similar to our approach. However, these approaches have always been conjoined with discriminative classifiers or rerankers in order to actually accomplish the generation (Angeli et al., 2010; Konstas and Lapata, 2013). In addition, these models explicitly model knowledge base field selection, whereas the model we present is fundamentally an end-to-end model over generation segments. Recently, a new paradigm has emerged around neural text generation systems based on machine translation (Sutskever et al., 2014; Cho et al., 2014; Bahdanau et al., 2015). Most of this work has used unconstrained black-box encoder-decoder approaches. There has been some work on discrete variables in this context, including extracting representations (Shen et al., 2018), incorporating discrete latent variables in text modeling (Yang et al., 2018), and using non-HSMM segmental models for machine translation or summarization (Yu et al., 2016; Wang et al., 2017; Huang et al., 2018). Dai et al. (2017) develop an approximate inference scheme for a neural HSMM using RNNs for continuous emissions; in contrast we maximize the exact log-marginal, and use RNNs to parameterize a discrete emission distribution. Finally, there has also been much recent interest in segmental RNN models for non-generative tasks in NLP (Tang et al., 2016; Kong et al., 2016; Lu et al., 2016). The neural text generation community has also recently been interested in “controllable” text generation (Hu et al., 2017), where various aspects of the text (often sentiment) are manipulated or transferred (Shen et al., 2017; Zhao et al., 2018; Li et al., 2018). In contrast, here we focus on controlling either the content of a generation or the way it is expressed by manipulating the (latent) template used in realizing the generation. ## 3 Overview: Data-Driven NLG Our focus is on generating a textual description of a knowledge base or meaning representation. Following standard notation (Liang et al., 2009; Wiseman et al., 2017), let $x\niceq\{r_{1}\ldots r_{J}\}$ be a collection of records. A record is made up of a _type_ ($r.t$), an _entity_ ($r.e$), and a _value_ ($r.m$). For example, a knowledge base of restaurants might have a record with $r.t$ = Cuisine, $r.e$ = Denny’s, and $r.m$ = American. The aim is to generate an adequate and fluent text description $\hat{y}_{1:T}\niceq\hat{y}_{1},\ldots,\hat{y}_{T}$ of $x$. Concretely, we consider the E2E Dataset (Novikova et al., 2017) and the WikiBio Dataset (Lebret et al., 2016). We show an example E2E knowledge base $x$ in the top of Figure 1. The top of Figure 2 shows an example knowledge base $x$ from the WikiBio dataset, where it is paired with a _reference_ text $y\niceq y_{1:T}$ at the bottom. The dominant approach in neural NLG has been to use an encoder network over $x$ and then a conditional decoder network to generate $y$, training the whole system in an end-to-end manner. To generate a description for a given example, a black-box network (such as an RNN) is used to produce a distribution over the next word, from which a choice is made and fed back into the system. The entire distribution is driven by the internal states of the neural network. While effective, relying on a neural decoder makes it difficult to understand what aspects of $x$ are correlated with a particular system output. This leads to problems both in controlling fine-grained aspects of the generation process and in interpreting model mistakes. As an example of why controllability is important, consider the records in Figure 1. Given these inputs an end-user might want to generate an output meeting specific constraints, such as not mentioning any information relating to customer rating. Under a standard encoder-decoder style model, one could filter out this information either from the encoder or decoder, but in practice this would lead to unexpected changes in output that might propagate through the whole system. <figure><img src="content_image/1808.10122/x1.png"><figcaption>Figure 2: An example from the WikiBio dataset (Lebret et al., 2016), with adatabase x (top) for Frederick Parker-Rhodes and corresponding referencegeneration y (bottom).</figcaption></figure> As an example of the difficulty of interpreting mistakes, consider the following actual generation from an encoder-decoder style system for the records in Figure 2: ”frederick parker-rhodes (21 november 1914 - 2 march 1987) was an english mycology and plant pathology, mathematics at the university of uk.” In addition to not being fluent, it is unclear what the end of this sentence is even attempting to convey: it may be attempting to convey a fact not actually in the knowledge base (e.g., where Parker-Rhodes studied), or perhaps it is simply failing to fluently realize information that _is_ in the knowledge base (e.g., Parker-Rhodes’s country of residence). Traditional NLG systems (Kukich, 1983; McKeown, 1992; Belz, 2008; Gatt and Reiter, 2009), in contrast, largely avoid these problems. Since they typically employ an explicit planning component, which decides which knowledge base records to focus on, and a surface realization component, which realizes the chosen records, the intent of the system is always explicit, and it may be modified to meet constraints. The goal of this work is to propose an approach to neural NLG that addresses these issues in a principled way. We target this goal by proposing a new model that generates with template-like objects induced by a neural HSMM (see Figure 1). Templates are useful here because they represent a fixed plan for the generation’s content, and because they make it clear what part of the generation is associated with which record in the knowledge base. ## 4 Background: Semi-Markov Models What does it mean to learn a template? It is natural to think of a template as a sequence of typed text-segments, perhaps with some segments acting as the template’s “backbone” (Wang and Cardie, 2013), and the remaining segments filled in from the knowledge base. A natural probabilistic model conforming with this intuition is the hidden semi-markov model (HSMM) (Gales and Young, 1993; Ostendorf et al., 1996), which models latent segmentations in an output sequence. Informally, an HSMM is much like an HMM, except emissions may last multiple time-steps, and multi-step emissions need not be independent of each other conditioned on the state. We briefly review HSMMs following Murphy (2002). Assume we have a sequence of observed tokens $y_{1}\ldots y_{T}$ and a discrete, latent state $z_{t}\nicein\{1,\ldots,K\}$ for each timestep. We additionally use two per-timestep variables to model multi-step segments: a length variable $l_{t}\nicein\{1,\ldots,L\}$ specifying the length of the current segment, and a deterministic binary variable $f_{t}$ indicating whether a segment finishes at time $t$. We will consider in particular _conditional_ HSMMs, which condition on a source $x$, essentially giving us an HSMM decoder. An HSMM specifies a joint distribution on the observations and latent segmentations. Letting $\theta$ denote all the parameters of the model, and using the variables introduced above, we can write the corresponding joint-likelihood as follows \[p(y,z,l,f\given x;\theta)= \prod_{t=0}^{T-1}p(z_{t+1},l_{t+1}\given z_{t},l_{t},x)^{f_{t}}\] \[\times \prod_{t=1}^{T}p(y_{t-l_{t}+1:t}\given z_{t},l_{t},x)^{f_{t}},\] where we take $z_{0}$ to be a distinguished start-state, and the deterministic $f_{t}$ variables are used for excluding non-segment log probabilities. We further assume $p(z_{t+1},l_{t+1}\given z_{t},l_{t},x)$ factors as $p(z_{t+1}\given z_{t},x)\times p(l_{t+1}\given z_{t+1})$. Thus, the likelihood is given by the product of the probabilities of each discrete state transition made, the probability of the length of each segment given its discrete state, and the probability of the observations in each segment, given its state and length. ## 5 A Neural HSMM Decoder We use a novel, neural parameterization of an HSMM to specify the probabilities in the likelihood above. This full model, sketched out in Figure 3, allows us to incorporate the modeling components, such as LSTMs and attention, that make neural text generation effective, while maintaining the HSMM structure. ### Parameterization Since our model must condition on $x$, let $\boldr_{j}\nicein\reals^{d}$ represent a real embedding of record $r_{j}\nicein x$, and let $\boldx_{a}\nicein\reals^{d}$ represent a real embedding of the entire knowledge base $x$, obtained by max-pooling coordinate-wise over all the $\boldr_{j}$. It is also useful to have a representation of just the unique _types_ of records that appear in $x$, and so we also define $\boldx_{u}\nicein\reals^{d}$ to be the sum of the embeddings of the unique types appearing in $x$, plus a bias vector and followed by a ReLU nonlinearity. Transition DistributionThe transition distribution $p(z_{t+1}\given z_{t},x)$ may be viewed as a $K\,{\times}\,K$ matrix of probabilities, where each row sums to 1. We define this matrix to be \[p(z_{t+1}\given z_{t},x)\propto\boldA\boldB+\boldC(\boldx_{u}) \boldD(\boldx_{u}),\] where $\boldA\nicein\reals^{K\times m_{1}}$, $\boldB\nicein\reals^{m_{1}\times K}$ are state embeddings, and where $\boldC:\reals^{d}\rightarrow\reals^{K\times m_{2}}$ and $\boldD:\reals^{d}\rightarrow\reals^{m_{2}\times K}$ are parameterized non-linear functions of $\boldx_{u}$. We apply a row-wise $\mathrm{softmax}$ to the resulting matrix to obtain the desired probabilities. Length DistributionWe simply fix all length probabilities $p(l_{t+1}\given z_{t+1})$ to be uniform up to a maximum length $L$.¹ [FOOTNOTE:1][ENDFOOTNOTE] [FIGURE:S5.F3][ENDFIGURE] Emission DistributionThe emission model models the generation of a text segment conditioned on a latent state and source information, and so requires a richer parameterization. Inspired by the models used for neural NLG, we base this model on an RNN decoder, and write a segment’s probability as a product over token-level probabilities, \[p (y_{t-l_{t}+1:t}\given z_{t}\niceq k,l_{t}\niceq l,x)=\] \[\prod_{i=1}^{l_{t}}p(y_{t-l_{t}+i}\given y_{t-l_{t}+1:t-l_{t}+i-1 },z_{t}\niceq k,x)\] \[\times p({<}\text{/seg}{>}\given y_{t-l_{t}+1:t},z_{t}\niceq k,x) \times\mathbf{1}_{\{l_{t}\niceq l\}},\] where ${<}\text{/seg}{>}$ is an end of segment token. The RNN decoder uses attention and copy-attention over the embedded records $\boldr_{j}$, and is conditioned on $z_{t}\niceq k$ by concatenating an embedding corresponding to the $k$’th latent state to the RNN’s input; the RNN is also conditioned on the entire $x$ by initializing its hidden state with $\boldx_{a}$. More concretely, let $\boldh^{k}_{i-1}\nicein\reals^{d}$ be the state of an RNN conditioned on $x$ and $z_{t}\niceq k$ (as above) run over the sequence $y_{t-l_{t}+1:t-l_{t}+i-1}$. We let the model attend over records $\boldr_{j}$ using $\boldh^{k}_{i-1}$ (in the style of Luong et al. (2015)), producing a context vector $\boldc^{k}_{i-1}$. We may then obtain scores $\boldv_{i-1}$ for each word in the output vocabulary, \[\boldv_{i-1}\niceq\boldW\tanh(\boldg_{1}^{k}\circ[\boldh^{k}_{i-1 },\boldc^{k}_{i-1}]),\] with parameters $\boldg_{1}^{k}\nicein\reals^{2d}$ and $\boldW\nicein\reals^{V\times 2d}$. Note that there is a $\boldg_{1}^{k}$ vector for each of $K$ discrete states. To additionally implement a kind of slot filling, we allow emissions to be directly copied from the value portion of the records $r_{j}$ using copy attention (Gülçehre et al., 2016; Gu et al., 2016; Yang et al., 2016). Define copy scores, \[\rho_{j}=\boldr_{j}^{\trans}\tanh(\boldg_{2}^{k}\circ\boldh^{k}_{ i-1}),\] where $\boldg_{2}^{k}\nicein\reals^{d}$. We then normalize the output-vocabulary and copy scores together, to arrive at \[\widetilde{\boldv}_{i-1}\niceq\mathrm{softmax}([\boldv_{i-1},\rho _{1},\ldots,\rho_{J}]),\] and thus \[p(y_{t-l_{t}+i} \niceq w\given y_{t-l_{t}+1:t-l_{t}+i-1},z_{t}\niceq k,x)=\] \[\widetilde{\boldv}_{i-1,w}+\sum_{j:r_{j}.m\niceq w}\widetilde{ \boldv}_{i-1,V+j}.\] An Autoregressive VariantThe model as specified assumes segments are independent conditioned on the associated latent state and $x$. While this assumption still allows for reasonable performance, we can tractably allow interdependence between tokens (but not segments) by having each next-token distribution depend on all the previously generated tokens, giving us an autoregressive HSMM. For this model, we will in fact use $p(y_{t-l_{t}+i}\niceq w\given y_{1:t-l_{t}+i-1},z_{t}\niceq k,x)$ in defining our emission model, which is easily implemented by using an additional RNN run over all the preceding tokens. We will report scores for both non-autoregressive and autoregressive HSMM decoders below. ### Learning The model requires fitting a large set of neural network parameters. Since we assume $z$, $l$, and $f$ are unobserved, we marginalize over these variables to maximize the log marginal-likelihood of the observed tokens $y$ given $x$. The HSMM marginal-likelihood calculation can be carried out efficiently with a dynamic program analogous to either the forward- or backward-algorithm familiar from HMMs (Rabiner, 1989). It is actually more convenient to use the backward-algorithm formulation when using RNNs to parameterize the emission distributions, and we briefly review the backward recurrences here, again following Murphy (2002). We have: \[\beta_{t}(j) =p(y_{t+1:T}\given z_{t}\niceq j,f_{t}\niceq 1,x)\] \[=\sum_{k=1}^{K}\beta^{}_{t}(k)\,p(z_{t+1}\niceq k\given z_{t}=j)\] \[\beta^{}_{t}(k) =p(y_{t+1:T}\given z_{t+1}=k,f_{t}=1,x)\] \[=\sum_{l=1}^{L}\Big{[}\beta_{t+l}(k)\,p(l_{t+1}\niceq l\given z_{ t+1}\niceq k)\] \[\quad\quad\quad\;p(y_{t+1:t+l}\given z_{t+1}\niceq k,l_{t+1} \niceq l)\Big{]},\] with base case $\beta_{T}(j)\niceq 1$. We can now obtain the marginal probability of $y$ as $p(y\given x)\niceq\sum_{k=1}^{K}\beta^{}_{0}(k)\,p(z_{1}\niceq k)$, where we have used the fact that $f_{0}$ must be 1, and we therefore train to maximize the log-marginal likelihood of the observed $y$: \[\ln p(y\given x;\theta)=\ln\sum_{k=1}^{K}\beta^{}_{0}(k)\,p(z_{1 }\niceq k).\] (1) Since the quantities in (1) are obtained from a dynamic program, which is itself differentiable, we may simply maximize with respect to the parameters $\theta$ by back-propagating through the dynamic program; this is easily accomplished with automatic differentiation packages, and we use pytorch (Paszke et al., 2017) in all experiments. ### Extracting Templates and Generating After training, we could simply condition on a new database and generate with beam search, as is standard with encoder-decoder models. However, the structured approach we have developed allows us to generate in a more template-like way, giving us more interpretable and controllable generations. First, note that given a database $x$ and reference generation $y$ we can obtain the MAP assignment to the variables $z$, $l$, and $f$ with a dynamic program similar to the Viterbi algorithm familiar from HMMs. These assignments will give us a typed segmentation of $y$, and we show an example Viterbi segmentation of some training text in Figure 4. Computing MAP segmentations allows us to associate text-segments (i.e., phrases) with the discrete labels $z_{t}$ that frequently generate them. These MAP segmentations can be used in an exploratory way, as a sort of dimensionality reduction of the generations in the corpus. More importantly for us, however, they can also be used to guide generation. [FIGURE:S5.F4][ENDFIGURE] In particular, since each MAP segmentation implies a sequence of hidden states $z$, we may run a _template extraction_ step, where we collect the most common “templates” (i.e., sequences of hidden states) seen in the training data. Each “template” $z^{(i)}$ consists of a sequence of latent states, with $z^{(i)}\niceq z^{(i)}_{1},\ldots z^{(i)}_{S}$ representing the $S$ distinct segments in the $i$’th extracted template (recall that we will technically have a $z_{t}$ for each time-step, and so $z^{(i)}$ is obtained by collapsing adjacent $z_{t}$’s with the same value); see Figure 4 for an example template (with $S\niceq 17$) that can be extracted from the E2E corpus. The bottom of Figure 1 shows a visualization of this extracted template, where discrete states are replaced by the phrases they frequently generate in the training data. With our templates $z^{(i)}$ in hand, we can then restrict the model to using (one of) them during generation. In particular, given a new input $x$, we may generate by computing \[\hat{y}^{(i)}=\argmax_{y^{\prime}}p(y^{\prime},z^{(i)}\given x),\] (2) which gives us a generation $\hat{y}^{(i)}$ for each extracted template $z^{(i)}$. For example, the generation in Figure 1 is obtained by maximizing (2) with $x$ set to the database in Figure 1 and $z^{(i)}$ set to the template extracted in Figure 4. In practice, the $\argmax$ in (2) will be intractable to calculate exactly due to the use of RNNs in defining the emission distribution, and so we approximate it with a constrained beam search. This beam search looks very similar to that typically used with RNN decoders, except the search occurs only over a segment, for a particular latent state $k$. ### Discussion Returning to the discussion of controllability and interpretability, we note that with the proposed model (a) it is possible to explicitly force the generation to use a chosen template $z^{(i)}$, which is itself automatically learned from training data, and (b) that every _segment_ in the generated $\hat{y}^{(i)}$ is typed by its corresponding latent variable. We explore these issues empirically in Section 7.1. We also note that these properties may be useful for other text applications, and that they offer an additional perspective on how to approach latent variable modeling for text. Whereas there has been much recent interest in learning continuous latent variable representations for text (see Section 2), it has been somewhat unclear what the latent variables to be learned are intended to capture. On the other hand, the latent, template-like structures we induce here represent a plausible, probabilistic latent variable story, and allow for a more controllable method of generation. Finally, we highlight one significant possible issue with this model – the assumption that segments are independent of each other given the corresponding latent variable and $x$. Here we note that the fact that we are allowed to condition on $x$ is quite powerful. Indeed, a clever encoder could capture much of the necessary interdependence between the segments to be generated (e.g., the correct determiner for an upcoming noun phrase) in its encoding, allowing the segments themselves to be decoded more or less independently, given $x$. ## 6 Data and Methods Our experiments apply the approach outlined above to two recent, data-driven NLG tasks. ### Datasets Experiments use the E2E (Novikova et al., 2017) and WikiBio (Lebret et al., 2016) datasets, examples of which are shown in Figures 1 and 2, respectively. The former dataset, used for the 2018 E2E-Gen Shared Task, contains approximately 50K total examples, and uses 945 distinct word types, and the latter dataset contains approximately 500K examples and uses approximately 400K word types. Because our emission model uses a word-level copy mechanism, any record with a phrase consisting of $n$ words as its value is replaced with $n$ positional records having a single word value, following the preprocessing of Lebret et al. (2016). For example, “type[coffee shop]” in Figure 1 becomes “type-1[coffee]” and “type-2[shop].” For both datasets we compare with published encoder-decoder models, as well as with direct template-style baselines. The E2E task is evaluated in terms of BLEU (Papineni et al., 2002), NIST (Belz and Reiter, 2006), ROUGE (Lin, 2004), CIDEr (Vedantam et al., 2015), and METEOR (Banerjee and Lavie, 2005).² The benchmark system for the task is an encoder-decoder style system followed by a reranker, proposed by Dušek and Jurcıcek (2016). We compare to this baseline, as well as to a simple but competitive non-parametric template-like baseline (‘‘SUB’’ in tables), which selects a training sentence with records that maximally overlap (without including extraneous records) the unseen set of records we wish to generate from; ties are broken at random. Then, word-spans in the chosen training sentence are aligned with records by string-match, and replaced with the corresponding fields of the new set of records.³ [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] The WikiBio dataset is evaluated in terms of BLEU, NIST, and ROUGE, and we compare with the systems and baselines implemented by Lebret et al. (2016), which include two neural, encoder-decoder style models, as well as a Kneser-Ney, templated baseline. ### Model and Training Details We first emphasize two additional methodological details important for obtaining good performance. Constraining LearningWe were able to learn more plausible segmentations of $y$ by constraining the model to respect word spans $y_{t+1:t+l}$ that appear in some record $r_{j}\nicein x$. We accomplish this by giving zero probability (within the backward recurrences in Section 5) to any segmentation that splits up a sequence $y_{t+1:t+l}$ that appears in some $r_{j}$, or that includes $y_{t+1:t+l}$ as a subsequence of another sequence. Thus, we maximize (1) subject to these hard constraints. Increasing the Number of Hidden StatesWhile a larger $K$ allows for a more expressive latent model, computing $K$ emission distributions over the vocabulary can be prohibitively expensive. We therefore tie the emission distribution between multiple states, while allowing them to have a different transition distributions. We give additional architectural details of our model in the Supplemental Material; here we note that we use an MLP to embed $\boldr_{j}\nicein\reals^{d}$, and a 1-layer LSTM (Hochreiter and Schmidhuber, 1997) in defining our emission distributions. In order to reduce the amount of memory used, we restrict our output vocabulary (and thus the height of the matrix $\boldW$ in Section 5) to only contain words in $y$ that are _not_ present in $x$; any word in $y$ present in $x$ is assumed to be copied. In the case where a word $y_{t}$ appears in a record $r_{j}$ (and could therefore have been copied), the input to the LSTM at time $t+1$ is computed using information from $r_{j}$; if there are multiple $r_{j}$ from which $y_{t}$ could have been copied, the computed representations are simply averaged. For all experiments, we set $d\niceq 300$ and $L\niceq 4$. At generation time, we select the 100 most common templates $z^{(i)}$, perform beam search with a beam of size 5, and select the generation with the highest overall joint probability. For our E2E experiments, our best non-autoregressive model has 55 “base” states, duplicated 5 times, for a total of $K\niceq 275$ states, and our best autoregressive model uses $K\niceq 60$ states, without any duplication. For our WikiBio experiments, both our best non-autoregressive and autoregressive models uses 45 base states duplicated 3 times, for a total of $K\niceq 135$ states. In all cases, $K$ was chosen based on BLEU performance on held-out validation data. Code implementing our models is available at https://github.com/harvardnlp/neural-template-gen. ## 7 Results \| BLEU \| NIST \| ROUGE \| CIDEr \| METEOR ---\|---\|---\|---\|---\|--- \| \| \| Validation \| \| D&J \| 69.25 \| 8.48 \| 72.57 \| 2.40 \| 47.03 SUB \| 43.71 \| 6.72 \| 55.35 \| 1.41 \| 37.87 NTemp \| 64.53 \| 7.66 \| 68.60 \| 1.82 \| 42.46 NTemp+AR \| 67.07 \| 7.98 \| 69.50 \| 2.29 \| 43.07 \| \| \| Test \| \| D&J \| 65.93 \| 8.59 \| 68.50 \| 2.23 \| 44.83 SUB \| 43.78 \| 6.88 \| 54.64 \| 1.39 \| 37.35 NTemp \| 55.17 \| 7.14 \| 65.70 \| 1.70 \| 41.91 NTemp+AR \| 59.80 \| 7.56 \| 65.01 \| 1.95 \| 38.75 Table 1: Comparison of the system of Dušek and Jurcıcek (2016), which forms the baseline for the E2E challenge, a non-parametric, substitution-based baseline (see text), and our HSMM models (denoted “NTemp” and “NTemp+AR” for the non-autoregressive and autoregressive versions, resp.) on the validation and test portions of the E2E dataset. “ROUGE” is ROUGE-L. Models are evaluated using the official E2E NLG Challenge scoring scripts. \| BLEU \| NIST \| ROUGE-4 ---\|---\|---\|--- Template KN † \| 19.8 \| 5.19 \| 10.7 NNLM (field) † \| 33.4 \| 7.52 \| 23.9 NNLM (field & word) † \| 34.7 \| 7.98 \| 25.8 NTemp \| 34.2 \| 7.94 \| 35.9 NTemp+AR \| 34.8 \| 7.59 \| 38.6 Seq2seq (Liu et al., 2018) \| 43.65 \| - \| 40.32 Table 2: Top: comparison of the two best neural systems of Lebret et al. (2016), their templated baseline, and our HSMM models (denoted “NTemp” and “NTemp+AR” for the non-autoregressive and autoregressive versions, resp.) on the test portion of the WikiBio dataset. Models marked with a † are from Lebret et al. (2016), and following their methodology we use ROUGE-4. Bottom: state-of-the-art seq2seq-style results from Liu et al. (2018). Our results on automatic metrics are shown in Tables 1 and 2. In general, we find that the templated baselines underperform neural models, whereas our proposed model is fairly competitive with neural models, and sometimes even outperforms them. On the E2E data, for example, we see in Table 1 that the SUB baseline, despite having fairly impressive performance for a non-parametric model, fares the worst. The neural HSMM models are largely competitive with the encoder-decoder system on the validation data, despite offering the benefits of interpretability and controllability; however, the gap increases on test. Table 2 evaluates our system’s performance on the test portion of the WikiBio dataset, comparing with the systems and baselines implemented by Lebret et al. (2016). Again for this dataset we see that their templated Kneser-Ney model underperforms on the automatic metrics, and that neural models improve on these results. Here the HSMMs are competitive with the best model of Lebret et al. (2016), and even outperform it on ROUGE. We emphasize, however, that recent, sophisticated approaches to encoder-decoder style database-to-text generation have since surpassed the results of Lebret et al. (2016) and our own, and we show the recent seq2seq style results of Liu et al. (2018), who use a somewhat larger model, at the bottom of Table 2. ### Qualitative Evaluation We now qualitatively demonstrate that our generations are controllable and interpretable. Travellers Rest Beefeater --- \| name[Travellers Rest Beefeater], customerRating[3 out of 5], --- area[riverside], near[Raja Indian Cuisine] \| 1\. [Travellers Rest Beefeater]55 [is a]59 [3 star]43 --- [restaurant]11 [located near]25 [Raja Indian Cuisine]40 [.]53 2\. [Near]31 [riverside]29 [,]44 [Travellers Rest Beefeater]55 [serves]3 [3 star]50 [food]1 [.]2 3\. [Travellers Rest Beefeater]55 [is a]59 [restaurant]12 [providing]3 [riverside]50 [food]1 [and has a]17 [3 out of 5]26 [customer rating]16 [.]2 [It is]8 [near]25 [Raja Indian Cuisine]40 [.]53 4\. [Travellers Rest Beefeater]55 [is a]59 [place to eat]12 [located near]25 [Raja Indian Cuisine]40 [.]53 5\. [Travellers Rest Beefeater]55 [is a]59 [3 out of 5]5 [rated]32 [riverside]43 [restaurant]11 [near]25 [Raja Indian Cuisine]40 [.]53 Table 3: Impact of varying the template z(i) for a single x from the E2E validation data; generations are annotated with the segmentations of the chosen z(i). Results were obtained using the NTemp+AR model from Table 1. kenny warren --- \| name: kenny warren, birth date: 1 april 1946, birth name: kenneth warren deutscher, birth place: brooklyn, new york, --- occupation: ventriloquist, comedian, author, notable work: book - the revival of ventriloquism in america \| 1\. [kenneth warren deutscher]132 [ ( ]75 [born]89 [april 1, 1946]101 [ ) ]67 [is an american]82 [author]20 [and]1 --- [ventriloquist and comedian]69 [.]88 2\. [kenneth warren deutscher]132 [ ( ]75 [born]89 [april 1, 1946]101 [ ) ]67 [is an american]82 [author]20 [best known for his]95 [the revival of ventriloquism]96 [.]88 3\. [kenneth warren]16 [“kenny” warren]117 [ ( ]75 [born]89 [april 1, 1946]101 [ ) ]67 [is an american]127 [ventriloquist, comedian]28 [.]133 4\. [kenneth warren]16 [“kenny” warren]117 [ ( ]75 [born]89 [april 1, 1946]101 [ ) ]67 [is a]104 [new york]98 [author]20 [.]133 5\. [kenneth warren deutscher]42 [is an american]82 [ventriloquist, comedian]118 [based in]15 [brooklyn, new york]84 [.]88 Table 4: Impact of varying the template z(i) for a single x from the WikiBio validation data; generations are annotated with the segmentations of the chosen z(i). Results were obtained using the NTemp model from Table 2. Controllable DiversityOne of the powerful aspects of the proposed approach to generation is that we can manipulate the template $z^{(i)}$ while leaving the database $x$ constant, which allows for easily controlling aspects of the generation. In Table 3 we show the generations produced by our model for five different neural template sequences $z^{(i)}$, while fixing $x$. There, the segments in each generation are annotated with the latent states determined by the corresponding $z^{(i)}$. We see that these templates can be used to affect the word-ordering, as well as which fields are mentioned in the generated text. Moreover, because the discrete states align with particular fields (see below), it is generally simple to automatically infer to which fields particular latent states correspond, allowing users to choose which template best meets their requirements. We emphasize that this level of controllability is much harder to obtain for encoder-decoder models, since, at best, a large amount of sampling would be required to avoid generating around a particular mode in the conditional distribution, and even then it would be difficult to control the sort of generations obtained. Interpretable StatesDiscrete states also provide a method for interpreting the generations produced by the system, since each segment is explicitly typed by the current hidden state of the model. Table 4 shows the impact of varying the template $z^{(i)}$ for a single $x$ from the WikiBio dataset. While there is in general surprisingly little stylistic variation in the WikiBio data itself, there is variation in the information discussed, and the templates capture this. Moreover, we see that particular discrete states correspond in a consistent way to particular pieces of information, allowing us to align states with particular field types. For instance, birth names have the same hidden state (132), as do names (117), nationalities (82), birth dates (101), and occupations (20). To demonstrate empirically that the learned states indeed align with field types, we calculate the average purity of the discrete states learned for both datasets in Table 5. In particular, for each discrete state for which the majority of its generated words appear in some $r_{j}$, the _purity_ of a state’s record type alignment is calculated as the percentage of the state’s words that come from the most frequent record type the state represents. This calculation was carried out over training examples that belonged to one of the top 100 most frequent templates. Table 5 indicates that discrete states learned on the E2E data are quite pure. Discrete states learned on the WikiBio data are less pure, though still rather impressive given that there are approximately 1700 record types represented in the WikiBio data, and we limit the number of states to 135. Unsurprisingly, adding autoregressiveness to the model decreases purity on both datasets, since the model may rely on the autoregressive RNN for typing, in addition to the state’s identity. \| NTemp \| NTemp+AR ---\|---\|--- E2E \| 81.7 (17.9) \| 81.2 (15.7) WikiBio \| 37.5 (18.9) \| 36.3 (20.2) Table 5: Empirical analysis of the average purity of discrete states learned on the E2E and WikiBio datasets, for the NTemp and NTemp+AR models. Average purities are given as percents, and standard deviations follow in parentheses. See the text for full description of this calculation. ## 8 Conclusion and Future Work We have developed a neural, template-like generation model based on an HSMM decoder, which can be learned tractably by backpropagating through a dynamic program. The method allows us to extract template-like latent objects in a principled way in the form of state sequences, and then generate with them. This approach scales to large-scale text datasets and is nearly competitive with encoder-decoder models. More importantly, this approach allows for controlling the diversity of generation and for producing interpretable states during generation. We view this work both as the first step towards learning discrete latent variable template models for more difficult generation tasks, as well as a different perspective on learning latent variable text models in general. Future work will examine encouraging the model to learn maximally different (or minimal) templates, which our objective does not explicitly encourage, templates of larger textual phenomena, such as paragraphs and documents, and hierarchical templates. ## Acknowledgments SW gratefully acknowledges the support of a Siebel Scholars award. 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In _Proceedings of the 35th International Conference on Machine Learning, ICML 2018_, pages 5897–5906. ## Appendix A Supplemental Material ### Additional Model and Training Details Computing $\boldr_{j}$A record $r_{j}$ is represented by embedding a feature for its type, its position, and its word value in $\reals^{d}$, and applying an MLP with ReLU nonlinearity (Nair and Hinton, 2010) to form $\boldr_{j}\nicein\reals^{d}$, similar to Yang et al. (2016) and Wiseman et al. (2017). LSTM DetailsThe initial cell and hidden-state values for the decoder LSTM are given by $\boldQ_{1}\boldx_{a}$ and $\tanh(\boldQ_{2}\boldx_{a})$, respectively, where $\boldQ_{1},\boldQ_{2}\nicein\reals^{d\times d}$. When a word $y_{t}$ appears in a record $r_{j}$, the input to the LSTM at time $t+1$ is computed using an MLP with ReLU nonlinearity over the concatenation of the embeddings for $r_{j}$’s record type, word value, position, and a feature for whether it is the final position for the type. If there are multiple $r_{j}$ from which $y_{t}$ could have been copied, the computed representations are averaged. At test time, we use the MAP $r_{j}$ to compute the input, even if there are multiple matches. For $y_{t}$ which could not have been copied, the input to the LSTM at time $t+1$ is computed using the same MLP over $y_{t}$ and three dummy features. For the autoregressive HSMM, an additional 1-layer LSTM with $d$ hidden units is used. We experimented with having the autoregressive HSMM consume either tokens $y_{1:t}$ in predicting $y_{t+1}$, or the average embedding of the field _types_ corresponding to copied tokens in $y_{1:t}$. The former worked slightly better for the WikiBio dataset (where field types are more ambiguous), while the latter worked slightly better for the E2E dataset. Transition DistributionThe function $\boldC(\boldx_{u})$, which produces hidden state embeddings conditional on the source, is defined as $\boldC(\boldx_{u})\niceq\boldU_{2}(\mathrm{ReLU}(\boldU_{1}\boldx_{u}))$, where $\boldU_{1}\nicein\reals^{m_{3}\times d}$ and $\boldU_{2}\nicein\reals^{K\times m_{2}\times m_{3}}$; $\boldD(x)$ is defined analogously. For all experiments, $m_{1}\niceq 64$, $m_{2}\niceq 32$, and $m_{3}\niceq 64$. OptimizationWe train with SGD, using a learning rate of 0.5 and decaying by 0.5 each epoch after the first epoch in which validation log-likelihood fails to increase. When using an autoregressive HSMM, the additional LSTM is optimized only after the learning rate has been decayed. We regularize with Dropout (Srivastava et al., 2014). ### Additional Learned Templates In Tables 6 and 7 we show visualizations of additional templates learned on the E2E and WikiBio data, respectively, by both the non-autoregressive and autoregressive HSMM models presented in the paper. For each model, we select a set of five dissimilar templates in an iterative way by greedily selecting the next template (out of the 200 most frequent) that has the highest percentage of states that do not appear in the previously selected templates; ties are broken randomly. Individual states within a template are visualized using the three most common segments they generate. 1. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The WatermanThe Golden PalaceBrowns Cambridge… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is ais anis a family friendly… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ItalianFrenchfast food… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantpubplace… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| with awithwith an… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| averagehighlow… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| customer ratingprice rangerating…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. ---\|--- 2. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| There is aThere is a cheapThere is an… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantcoffee shopFrench restaurant… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The MillBibimbap HouseThe Twenty Two… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| located in thelocated on thelocated north of the… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| centre of the cityrivercity centre… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| that servesservingthat provides… \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| fast foodsushitake-away deliveries…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 3. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The Olive GroveThe PunterThe Cambridge Blue… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantpub… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| servesoffershas… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| fast foodsushitake-away deliveries…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 4. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| TheChild friendlyThe average priced… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantcoffee shopFrench restaurant… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The MillBibimbap HouseThe Twenty Two… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| servesoffershas… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| EnglishIndianItalian… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| foodcuisinedishes…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 5. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The StradaThe Dumpling TreeAlimentum… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| providesservesoffers… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| IndianChineseEnglish… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| food in thefood at afood and has a… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| customer rating ofprice range ofrating of… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 1 out of 5average5 out of 5…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 1. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The EagleThe Golden CurryZizzi… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| providesprovidingserves… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| IndianChineseEnglish… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| foodcuisineFood… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| in thewith aand has a… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| highmoderateaverage… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| price rangecustomer ratingrating…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| It isThey areIt's… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| nearlocated in thelocated near… \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| riversidecity centreCafe Sicilia… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Its customer rating isIt has aThe price range is… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 1 out of 5averagehigh…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 2. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Located nearLocated in theNear… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The Portland Armsriversidecity centre… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is anis a family friendlythere is a… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Italianfast foodFrench… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurant calledplace calledrestaurant named… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The WatermanCocumLoch Fyne…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 3. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| AAnA family friendly… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Italianfast foodFrench… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantpubcoffee shop… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| iscallednamed… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The WatermanCocumLoch Fyne…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 4. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Located nearLocated in theNear… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The Portland Armsriversidecity centre…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| , \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The EagleThe Golden CurryZizzi… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is ais a family friendlyis an… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| cheapfamily-friendlyfamily friendly… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Italianfast foodFrench… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantpubcoffee shop…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 5. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| AAnA family friendly… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| Italianfast foodFrench… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| restaurantpubcoffee shop… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| nearlocated in thelocated near… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| riversidecity centreCafe Sicilia… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| iscallednamed… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| The WatermanCocumLoch Fyne…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. Table 6: Five templates extracted from the E2E data with the NTemp model (top) and the Ntemp+AR model (bottom). 1. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| william henrygeorge augustus frederickmarie anne de bourbon… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| (was (;… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| bornborn onborn 1… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 196819601970… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| )])]… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is an americanis a russianwas an american… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| politicianactorfootball player…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. ---\|--- 2. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| sircaptainlieutenant… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| john herberthartleydonald charles cameron… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| was awas a britishwas an english… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| world war iworld warfirst world war… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| national teamorganizationsuper league…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 3. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| john herberthartleydonald charles cameron… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is awas ais an… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| indie rockdeath metalska… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| bandmidfielderdefenceman… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| fromforbased in… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| australialos angeles, californiachicago…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 4. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| john herberthartleydonald charles cameron… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| was ais ais a former… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| americanmajor league baseballaustralian… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| footballprofessional baseballprofessional ice hockey… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| midfielderdefendergoalkeeper…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 5. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| jameswilliam johnwilliam… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| `` billy '' wilsonsmith`` jack '' henry… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ( \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 1900c. 18941913… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| – \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| francebudapestbuenos aires… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ) \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is an americanis an englishwas an american… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| footballerprofessional footballerrules footballer… \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| who plays forwho currently plays forwho played with… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| paganesesouth melbournefc dynamo kyiv… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| in theof theand the… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| vicotiral football leaguenational football leagueaustralian football league… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ( \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| vflnflafl… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ) \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 1. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| aftab ahmedanderson da silvadavid jones… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| (;… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| bornborn onborn 1… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 195119701974… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| )]… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is an americanwas an americanis an english… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| actoractresscricketer…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 2. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| aftab ahmedanderson da silvadavid jones… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| was ais a formeris a… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| world war iliberalbaseball… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| member of theparty member of therecipient of the… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| austrianpennsylvaniamontana… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| house of representativeslegislaturesenate…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 3. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| adjutantlieutenantcaptain… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| aftab ahmedanderson da silvadavid jones… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| was ais a formeris a… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| world war iliberalbaseball… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| member of theparty member of therecipient of the… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| knessetscottish parliamentfc lokomotiv liski…\definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 4. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| williamjohn williamjames ``… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| `` billy '' watsonsmithjim '' edward… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ( \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 1913c. 19001913… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| \--in-… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| 1917surrey, englandbritish columbia… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ) \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| was an americanwas an australianis an american… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| football playerrules footballerdefenceman… \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| who plays forwho currently plays forwho played with… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| collingwoodst kildacarlton… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| in theof theand the… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| victorial football leaguenational football leagueaustralian football league… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ( \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| vflaflnfl… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| ) \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. 5. \| \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| aftab ahmedanderson da silvadavid jones… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| is ais a formeris a female… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| member of theparty member of therecipient of the… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\| knessetscottish parliamentfc lokomotiv liski… \definecolor[named]pgfstrokecolorrgb.5,.5,.5\pgfsys@color@gray@stroke.5\pgfsys@color@gray@fill.5\|. Table 7: Five templates extracted from the WikiBio data with the NTemp model (top) and the Ntemp+AR model (bottom).
0710.4550	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 34198, "num_imgs": 3, "llama3_tokens_count": 10378 }	[ "content_image/0710.4550/x1.png", "content_image/0710.4550/x2.png", "content_image/0710.4550/x3.png" ]	# Ginzburg-Landau theory for the conical cycloid state in multiferroics: applications to CoCr${}_{2}$O${}_{4}$ Chuanwei Zhang${}^{1,2}$ Sumanta Tewari${}^{1,3}$ John Toner${}^{4}$ S. Das Sarma${}^{1}$ ${}^{1}$Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742 ${}^{2}$Department of Physics and Astronomy, Washington State University, Pullman, WA 99164 ${}^{3}$Department of Physics and Astronomy, Clemson University, Clemson, SC 29634 ${}^{4}$Department of Physics and Institute of Theoretical Science, University of Oregon, Eugene, OR 97403 ###### Abstract We show that the cycloidal magnetic order of a multiferroic can arise in the absence of spin and lattice anisotropies, for e.g., in a cubic material, and this explains the occurrence of such a state in CoCr${}_{2}$O${}_{4}$. We discuss the case when this order coexists with ferromagnetism in a so called ‘conical cycloid’ state, and show that a direct transition to this state from the ferromagnet is necessarily first order. On quite general grounds, the reversal of the direction of the _uniform_ magnetization in this state can lead to the reversal of the electric polarization as well, without the need to invoke ‘toroidal moment’ as the order parameter. pacs: 75.80.+q,77.80.Fm,75.30.Fv,75.10.-b ## I Introduction Ferromagnetism and ferroelectricity are two of the most well-known and technologically relevant types of long range ordering that can occur in solids. It is therefore of paramount interest and importance that in a class of ternary oxides, known as “multiferroics”, both types of order seem to coexist with the possibility of interplay between long range magnetism and long range electric polarization [1; 2; 3; 4]. The recently discovered new class of multiferroics with strong magnetoelectric effects often display the coexistence of a spatially modulated magnetic order, called ‘cycloidal’ order, and uniform polarization ($\mathbf{P}$), which is induced by the broken inversion symmetry due to the modulation of the magnetization [5; 6]. Since $\mathbf{P}$ is inherently of magnetic origin, unusual magnetoelectric effects, as displayed by the ability to tune the polarization by a magnetic field which acts on the cycloidal order parameter, are possible, opening up many applications [2; 7; 8; 9; 10; 11; 12; 13; 14; 15]. Among this exciting class of materials, the cubic spinel oxide CoCr${}_{2}$O${}_{4}$ is even more unusual, since it displays not only a non-zero $\mathbf{P}$ and a spatially _modulated_ magnetic order, but also a _uniform_ magnetization [15] ($\mathbf{M}$) in a so-called ‘conical cycloid’ state (see below). The uniform component of $\mathbf{M}$ provides an extra handle [2] with which to tune $\mathbf{P}$, as has been recently demonstrated [15]. The low value of the required tuning magnetic field $\sim.5$ T, makes this material even more experimentally appealing. The ability to tune $\mathbf{P}$ by tuning the uniform part of $\mathbf{M}$ poses a theoretical puzzle, since, in existing theories, the uniform piece of $\mathbf{M}$ should not influence the polarization at all [16; 5; 6; 17]. This has lead to the introduction of the ‘toroidal moment’, $\mathbf{T}=\mathbf{P}\times\mathbf{M}$, as the real order parameter characterizing the conical cycloid state of CoCr${}_{2}$O${}_{4}$[15]. In this Letter, we explain this unique phenomenon and the other interesting aspects of the physics of the conical cycloid state by developing a phenomenological Ginzburg-Landau (GL) theory. Additionally, the rotationally invariant form of the theory proves that both the ordinary and the conical cycloidal orders, with the resulting multiferroicity, are possible even in systems _without_ easy plane spin and easy axis lattice anisotropies. This is important since earlier models [2; 5; 18] of the cycloidal state depend crucially on such anisotropies. However, such anisotropic models can _not_ explain the presence of the cycloidal state in cubic systems like CoCr${}_{2}$O${}_{4}$, where such phases are also observed despite the fact that their cubic symmetry forbids such easy plane and easy axis anisotropies. CoCr${}_{2}$O${}_{4}$, with the lattice structure of a cubic spinel, enters into a state with a uniform magnetization at a temperature $T_{m}=93$ K. Microscopically, the magnetization is of ferrimagnetic origin [15], and in what follows we will only consider the ferromagnetic component, $\mathbf{M}$, of the magnetization of a ferrimagnet. At a lower critical temperature, $T_{c}=26$ K, the system develops a special helical modulation of the magnetization in a plane transverse to the large uniform component. Such a state can be described by an order parameter, \[\mathbf{M}_{h}=m_{1}\hat{e}_{1}\cos(\mathbf{q}\cdot\mathbf{r})+m_{2}\hat{e}_{2 }\sin(\mathbf{q}\cdot\mathbf{r})+m_{3}\hat{e}_{3}+h.h.,\] (1) where $\{\hat{e}_{i}\}$ form an orthonormal triad and $h.h.$ denotes “higher harmonics” such as terms proportional to sines and cosines of $(2n+1)\mathbf{q}\cdot\mathbf{r}$ with integer $n$. When the pitch vector, $\mathbf{q}$, is normal to the plane of the rotating components, the rotating components form a conventional helix [19]. For $m_{3}=0$ such a state, which we call an ‘ordinary helix’ state, is observed in many rare-earth metals [20], e.g. MnSi [21; 22], and FeGe [23]. We call a helix state with $m_{3}\neq 0$, which is observed in some heavy rare-earth metals [20], a ‘conical helix’ state because the tip of the magnetization falls on the edge of a cone. A more complicated modulation arises when $\mathbf{q}$ lies _in the plane_ of the rotating components. For $m_{3}=0$, we call such a state an ‘ordinary cycloid’ state because the profile of the magnetization resembles the shape of a cycloid. The state with $m_{3}\neq 0$ is called a ‘conical cycloid’ state. It is easy to see that the helical, but not the cycloidal, modulation preserves a residual symmetry under translations and suitable simultaneous rotations about the pitch vector. Since $\mathbf{M}$ and $\mathbf{P}$ respectively break time reversal and spatial inversion symmetry, the leading $\mathbf{P}$-dependent piece in a GL Hamiltonian density, $h_{P}$, for a centrosymmetric, time reversal invariant system with cubic symmetry is [5], \[h_{P}=\mathbf{P}^{2}/2\chi+\alpha\mathbf{P}\cdot\mathbf{M}\times\nabla\times \mathbf{M},\] (2) where $\chi>0$ and $\alpha$ are coupling constants. We assume that $\mathbf{P}$ is a slave of $\mathbf{M}$, in the sense that a non-zero $\mathbf{P}$ only occurs due to the spontaneous development of a magnetic state with a non-zero $\mathbf{M}\times\nabla\times\mathbf{M}$, which then, through the linear coupling to $\mathbf{P}$ in (2), induces a non-zero $\mathbf{P}$. For an order parameter ansatz given by Eq. 1, the macroscopic polarization, $\mathbf{\bar{P}}$, is given by minimizing the Hamiltonian density (2) over $\mathbf{P}$, $\mathbf{\bar{P}}=\chi\alpha m_{1}m_{2}[\hat{e}_{3}\times\mathbf{q}]$. So $\mathbf{\bar{P}}$ is normal to both $\mathbf{q}$ and the axis of rotation, $\hat{e}_{3}$. Note that in a conventional spin density wave state ($m_{1}\hskip 4.267913pt\text{or}\hskip 4.267913ptm_{2}=0$), as in the helix states, $\mathbf{\bar{P}}$ is zero. However, for a cycloid state, $\mathbf{q}\perp\hat{e}_{3}$, so there is a non-zero $\mathbf{\bar{P}}$. Note that $\mathbf{\bar{P}}$ is entirely due to the cycloidal components $m_{1}$ and $m_{2}$, and is independent of the uniform magnetization $m_{3}$. Thus, while it is conceivable that magnetic fields strong enough to ‘flop’ the spins and the axis of rotation of the cycloidal components will alter $\mathbf{\bar{P}}$[5; 6; 7; 8], no explanation of how tuning the uniform component of $\mathbf{M}$ can affect the induced polarization has been offered. We will do so later in this paper. The paper is organized as follows: Section II lays out the Ginzburg-Landau Hamiltonian and the parameter regions which exhibits the cycloidal phase. Section III and V are devoted to the phase diagrams of ordinary cycloidal state and conical cycloidal state respectively. In Section V, we explain why the reversal of the direction of the uniform magnetization in the conical cycloidal state can lead to the reversal of electric polarization. Section VI consists of conclusions. ## II Ginzburg-Landau Hamiltonian We consider a Hamiltonian that is _completely_ invariant under simultaneous rotations of positions and magnetization. This guarantees that any phase that can occur in our model is _necessarily_ allowed in a crystal of _any_ symmetry. The full Hamiltonian is given by, $H=\int(h_{M}+h_{P})d\mathbf{r}\equiv\int hd\mathbf{r}$. Using $\mathbf{P=-}\chi\alpha\mathbf{M}\times\nabla\times\mathbf{M}$ to eliminate $\mathbf{P}$, we can write the total Hamiltonian density $h$ entirely in terms of $\mathbf{M}$, \[h = t\mathbf{M}^{2}+u\mathbf{M}^{4}+K_{0}\left(\nabla\cdot\mathbf{M} \right)^{2}+K_{1}\left(\nabla\times\mathbf{M}\right)^{2}\] (3) \[+K_{2}\mathbf{M}^{2}\left(\nabla\cdot\mathbf{M}\right)^{2}+K_{3} \left(\mathbf{M}\cdot\nabla\times\mathbf{M}\right)^{2}\] \[+K_{4}\left\|\mathbf{M}\times\nabla\times\mathbf{M}\right\|^{2}\] \[+D_{L}\|\nabla\left(\nabla\cdot\mathbf{M}\right)\mathbf{\|}^{2}+D_{ T}\|\nabla\left(\nabla\times\mathbf{M}\right)\mathbf{\|}^{2},\] where we have $u$, $D_{L,T}>0$ for stability. In Eq. 3, where the Landau expansion of the free energy is truncated at the fourth order, the usual gradient-squared term, $c\left\|\nabla\mathbf{M}\right\|^{2}$, is omitted since, $\left\|\nabla\mathbf{M}\right\|^{2}=(\nabla\cdot\mathbf{M})^{2}+\|\nabla\times \mathbf{M}\|^{2}$, plus an unimportant surface term which can be neglected. Notice that, for $K_{0}=K_{1}$ and $K_{2}=K_{3}=K_{4}$, $h$ is rotationally invariant in the spin space alone, so the $K_{i}$’s themselves are not proportional to the spin-orbit coupling constant (for e.g., via the above identity, $K_{0},K_{1}\sim c$). However, the _difference_ among the $K_{i}$’s should be small due to the smallness of the spin-orbit coupling. The effects of the competing magnetic interactions, which are present in the multiferroics and are responsible for the spatial modulation of $\mathbf{M}$[2; 5; 17; 18], are embodied in $K_{0},K_{1}$, which can be negative leading to a spatially modulated order parameter. For decoupled spin and coordinate spaces ($K_{i}$’s equal), the energies of the helical and the cycloidal modulations of the spins are identical. In a system where the spin anisotropy constrains the spins to lie on a plane, and the lattice anisotropy forces $\mathbf{q}$ to be also on that plane, the energy of the cycloidal modulation can be lower than that of the helical modulation [5; 18]. Such anisotropies have been implicitly taken as the driving force behind the cycloidal order by Mostovoy [5], and Katsura _et al._[18]. For cubic crystals, however, no such anisotropy exists among the principal directions. We argue below that, in this case, the magnetoelectric couplings themselves, leading to the difference among the $K_{i}$’s, can lower the energy of the cycloidal state than that of any other state with an arbitrary angle between $\mathbf{q}$ and the plane of the magnetization. Rather than exploring the complete parameter space of this model, we limit ourselves to two different parameter regions, which exhibit all the phases described above: Region I: $K_{0,1}<0,K_{i>1}$ small, $t>0$ , and Region II: $t<0$, $K_{3}<0$, $K_{1}>0$, $K_{2}=K_{4}=0$. We have checked that our results are robust against allowing small non-zero values of the various $K_{i}$’s that we take to be zero. In that sense our results, in particular the topology of the phase diagrams shown in Figs. 1a and 2a for Regions I and II, respectively, and the orders of the various phase transitions that we predict, are generic. As usual, our theoretical phase diagrams can be related to experimental ones by noting that _all_ of the phenomenological parameters $(t,K_{i},D_{L,T},u)$ in our model should depend on experimental parameters like, e.g., temperature ($T$). Thus, an experiment in which, e.g., $T$ is varied with all other parameters held fixed will map out a locus of points through our theoretical phase diagrams. In Landau theories, $t$ is expected to vary from large positive values, corresponding to disordered phases with $\mathbf{M}(\mathbf{r})=\mathbf{0}$, at high $T$, to smaller values at which $\mathbf{M}(\mathbf{r})\neq\mathbf{0}$ become possible. In order to access the conical cycloid state, we must also allow $K_{0}(T)$ and $K_{1}(T)$ to change sign as $T$ is decreased. For the most part we will work in mean field theory, which is simply finding a magnetization configuration $\mathbf{M}(\mathbf{r})$ that minimizes the Hamiltonian (3). Clearly, the task of finding the _global_ minimum is a formidable one. Instead, we restrict ourselves to ansatzes of the form: \[\mathbf{M}=m_{1}\hat{e}_{1}\cos(\mathbf{q}\cdot\mathbf{r})+m_{2}\hat{e}_{2} \sin(\mathbf{q}\cdot\mathbf{r})+\mathbf{M}_{0},\] (4) where the spatially constant vector $\mathbf{M}_{0}$ is allowed to point in _any_ direction. (Given the global rotation invariance under simultaneous rotations of magnetization and space, an infinity of other solutions trivially related to (4) by such rotations, and with exactly the same energy, also exist, of course.) In the special case of $\mathbf{q}$ along $x$ direction (or, equivalently, anywhere in the $x-y$-plane), this is a cycloid state with a uniform background magnetization $\mathbf{M}_{0}=\left(M_{01},M_{02},M_{03}\right)$. When $\mathbf{q}$ is along $z$ direction, it is a helix state. Inserting this ansatz (4) into the Hamiltonian (3), and integrating over the volume $V$ of the system, we can obtain the energy of the system. Through the minimization of the energy, we find the conical cycloid state is the _only_ state with a non-zero $\mathbf{M}_{0}$ when $K_{3}<K_{4}$. In addition, the optimal direction for $\mathbf{q}$ is _always_ either in the ($x-y$) plane, or orthogonal to it. Putting these facts together means that _all_ of the minimum energy configurations are of the form (1). Furthermore, when $\mathbf{q}$ lies in the ($x-y$) plane, we can always use the global rotation invariance of our model to rotate $\mathbf{q}$ to lie along the $x$-axis, and will henceforth do so. ## III Ordinary Cycloid State In Region I, the dominant terms in the Hamiltonian involving the uniform component $m_{3}$ are $tm_{3}^{2}+um_{3}^{4}$, therefore the lowest energy states have $m_{3}=0$. Small negative $K_{i>1}$ clearly cannot change this fact. The energy for the ordinary cycloid (OC) state is obtained by inserting (1) with $m_{3}=0$ into the Hamiltonian \[{E/V}=\Gamma_{L}\left(q\right)m_{1}^{2}+\Gamma_{T}\left(q\right)m_{2}^{2}+u \Phi(m_{1}^{2},m_{2}^{2}),\] (5) where $\Gamma_{L}\left(q\right)=\left(t+K_{0}q^{2}+D_{L}q^{4}\right)/2$, $\Gamma_{T}\left(q\right)=\left(t+K_{1}q^{2}+D_{T}q^{4}\right)/2$, and $\Phi(m_{1}^{2},m_{2}^{2})=3\left(m_{1}^{4}+m_{2}^{4}\right)/8+m_{1}^{2}m_{2}^{ 2}/4$. In writing this, we have neglected the higher harmonics in Eq. (1), whose amplitude vanishes much faster (specifically, as fast or faster than $\|m_{i}\|^{3}$) than the magnitude of the order parameter itself, and thus have negligible effects on the phase boundaries. For large positive $t$, all the terms in this energy are positive, and, hence, the lowest energy state is $m_{1}=m_{2}=0$; i.e., the paramagnet. As $T$ decreases, $t$ becomes smaller and the first phase transition that will occur depends on whether the minimum over $q$ of $\Gamma_{L}\left(q\right)$ or $\Gamma_{T}\left(q\right)$ becomes negative first. For $r\equiv K_{1}/K_{0}<\sqrt{D_{L}/D_{T}}$, $\Gamma_{L}\left(q\right)$ becomes negative first at $t_{OLS}=K_{0}^{2}/4D_{L}$, and $m_{1}$ starts to be nonzero. This boundary between paramagnet and the ordinary longitudinal spin density wave (OLS) phase ($m_{2}=m_{3}=0$, $m_{1}\neq 0$) is the horizontal (solid blue ) line in the phase diagram Fig. 1a in the ($r,t$) plane for fixed negative $K_{0}$ and all $K_{i>1}=0$. <figure><img src="content_image/0710.4550/x1.png"><figcaption>Figure 1: (Color online) (a) Phase diagram in Region I for the ordinarycycloid state. Solid (blue) lines represent second order phase transitions.Dotted (green) line indicates the first order transition to the helix state.The dotted (red) arrow represents one possible schematic locus of theexperimental points obtained by varying T. r0≡(1+3DT/DL)/6. (b) The sequenceof phases with decreasing T along the locus shown.</figcaption></figure> The OLS phase will, as we continue lowering $t$, eventually become unstable to a non-zero $m_{2}$; this is the OC state. By minimizing the energy (5) in the OLS phase, we find $q^{2}=q_{L,min}^{2}=-K_{0}/2D_{L}$ and $m_{1}^{2}={2}(t_{OLS}-t)/3u$. Inserting these into (5) we find that the coefficient of $m_{2}^{2}$ becomes negative below $t_{LOC}=t_{OLS}\left[3r-\left(1+3D_{T}/D_{L}\right)/2\right]$. This value $t_{LOC}$ of $t$ therefore defines the locus of a continuous OLS-OC phase transition, and is the non-horizontal straight (solid (blue)) line in the $r-t$ plane shown in Fig. 1a. For $r>\sqrt{{\frac{D_{L}}{D_{T}}}}$, $\Gamma_{T}$ becomes non-zero first, which seems to imply that one enters the ordinary transverse spin density wave (OTS) phase ($m_{1}=m_{3}=0$, $m_{2}\neq 0$) first for large $r$. However, it is not true because the OTS phase always has higher energy than the ordinary helical (OH) phase. The energy for the ordinary helix state is \[E/V=\Gamma_{T}(m_{1}^{2}+m_{2}^{2})+\Phi(m_{1}^{2},m_{2}^{2}).\] (6) The minimization of the energy over the direction of $\left(m_{1},m_{2}\right)$ vector yields $\|m_{1}\|=\|m_{2}\|=m_{H}/\sqrt{2}$, that is, a _circular_ helix. Further minimization over $m_{H}$ and $q$ gives the energy $E_{OH}$ of the ordinary helix state $E_{OH}/V=-(t_{OH}-t)^{2}/4u{\ }$for $t<t_{OH}$, where $t_{OH}=r^{2}D_{L}t_{OLS}/D_{T}$. The energy for the OTS state is $E_{OTS}/V=-(t_{OH}-t)^{2}/6u$, which is obtained from equation (5) by setting $m_{1}=0$ and $q^{2}=q_{T,min}^{2}\equiv-\frac{K_{1}}{2D_{T}}$, and then minimizing over $m_{2}$. $E_{OTS}$ is clearly higher than $E_{OH}$. Hence, the helical state is always favored over the OTS state throughout Region I of the phase diagram. Note that $t_{OH}$ defines the boundary for the second order transition from the paramagnet to the OH state. There is also a direct first order phase transition between the OH and the OLS states along the line where $E_{OH}=E_{OLS}$. Here $E_{OLS}/V=-(t_{OLS}-t)^{2}/6u$ is the energy for OLS state obtained from equation (5). This equality yields the first order phase boundary $t_{OLH}=(\sqrt{3/2}t_{OH}-t_{OLS})/(\sqrt{3/2}-1)$ between the OH and the OLS states (the dotted (green) line). The line for the OLS-OC transition always intersects the first order OLS-OH phase boundary before crossing the paramagnet-OLS boundary. This therefore always yields the topology shown in Fig. 1a. A typical experimental locus through this phase diagram, namely one in which $t$ decreases as temperature $T$ does, with $r$ constant, is shown in Fig. 1a. The sequence of phases that results is illustrated in Fig. 1b. We see that the paramagnet to ordinary cycloid phase transition is always preempted by a paramagnet to OLS phase transition, and the cycloid state is always elliptical. Both of these predictions are borne out by recent experiments on TbMnO${}_{3}$[13; 14]. On the other hand, a direct transition to the circular helix state is predicted by our theory, and has indeed been observed experimentally [21; 22]. All of the above statements are based on mean field theory, that is theory without considering the fluctuations. Going beyond mean field theory, very general arguments due to Brazovskii [24] imply that, in rotation invariant models, _any_ direct transition from a homogeneous state (paramagnet) to a translationally ordered one (OLS and OH) _must_ be driven first order by fluctuations. Consideration of topological defects and orientational order [25; 26; 27] supports this conclusion, but raises the additional possibility that direct transition between the homogeneous and the translationally ordered phases could split into two, with an intermediate orientationally ordered phase, analogous to the 2D “hexatic” phase [28]. In the present context, this implies that both the paramagnet to OLS and OH phase transitions are either driven first order by fluctuations, or split into two transitions with an intermediate orientationally ordered phase. Crystal symmetry breaking fields neglected in our model could invalidate this conclusion, if strong enough. ## IV Conical Cycloid State In Region II, we can show that conical cycloid (CC) state of the form $\mathbf{M}=\left(m_{1}\cos(qx),m_{2}\sin(qx),m_{3}\right)$ is the lowest energy state among all the possible states with arbitrary mutual angles between the uniform magnetization, $\mathbf{q}$, and the cycloid plane. The energy $E$ for this state takes the form \[E/V = \left(t+K_{0}q^{2}+D_{L}q^{4}+2um_{3}^{2}\right)m_{1}^{2}/2\] \[+\left(t+K_{1}q^{2}+D_{T}q^{4}+2um_{3}^{2}+K_{3}q^{2}m_{3}^{2} \right)m_{2}^{2}/2\] \[+u\Phi(m_{1}^{2},m_{2}^{2})+tm_{3}^{2}+um_{3}^{4},\] where we have again neglected the higher harmonics in Eq. (1). In this region, the _h.h._ terms do not vanish as the conical longitudinal spin density wave (CLS) ($m_{2}=0$, $m_{1,3}\neq 0$) or conical transverse spin density wave (CTS) ($m_{1}=0$, $m_{2,3}\neq 0$) to FM transition in Fig. 2 is approached. However, we have verified that amplitudes of the _h.h._ terms are only a very small fraction of the cycloidal components $m_{1}$ and $m_{2}$ (not of the uniform component $m_{3}$), therefore their neglect below (but close to) the lower cycloidal transition temperature of 26 K is justified. They have little or no quantitative effect on our phase diagram or the orders of the transition. Since $t<0$, we can minimize Eq. (IV) over $m_{3}$ with $m_{1}=m_{2}=0$, and find a ferromagnetic (FM) state with $m_{3}=\sqrt{-t/2u}$. For large positive $K_{0}$ and $K_{1}$, this ferromagnetic state is clearly stable against the development of non-zero $m_{1}$ and $m_{2}$. It also clearly becomes _unstable_ against the development of a non-zero $m_{1}$ if $K_{0}$ is lowered to negative values, because then the coefficient $(K_{0}q^{2}+D_{L}q^{4})$ of $m_{1}^{2}$ becomes negative for sufficiently small $q$. This instability (which is clearly into the CLS state) will occur at $K_{0}=0$, at a wavevector $q$ satisfying $q_{L,min}^{2}=-K_{0}/2D_{L}$. Note, however, that now, because $K_{0}$ is being varied _through_ zero, this wavevector will now _vanish_ as the transition is approached from below. The order parameter $m_{1}^{2}=K_{0}^{2}/2uD_{L}$_also_ vanishes as this transition is approached. Thus, this transition is, like the $\beta$ - incommensurate transition in quartz and berlinite [29], simultaneously a _nucleation_ transition ($q$ vanishes), and an _instability_ transition (order parameter vanishes). Indeed, this transition and the FM$\rightarrow$ CTS transition, which is of the same type and will be discussed below, are, to our knowledge, the _first_ examples of transitions that exhibit such a dual character in a model _without_ terms linear in the gradient operator. <figure><img src="content_image/0710.4550/x2.png"><figcaption>Figure 2: (Color online) (a) Phase diagram in Region II for the conicalcycloid state. Solid (blue) lines are the boundaries between different phases.The dotted (red) arrows represent possible paths for transition to the CCstate via continuous transitions. (b) The succession of the phases withdecreasing T. The (green) arrow represents a direct first order transitionbetween the FM and the CC state.</figcaption></figure> We can find the loci of instability between the CLS phase and the CC state by calculating the coefficient of $m_{2}^{2}$ in (IV) in the CLS phase, and finding where it becomes negative. The minimization of the energy (IV) over $q$, $m_{3}$ and $m_{1}$ yields $q^{2}=-K_{0}/2D_{L}$, $m_{3}^{2}=-\left(t+K_{0}^{2}/2D_{L}\right)/2u$ and $m_{1}^{2}=K_{0}^{2}/2uD_{L}$. Inserting these expressions into (IV) and taking the coefficient of $m_{2}^{2}$ to be zero, we find the CLS to CC phase boundary as: \[K_{1}=\frac{K_{0}}{2}\left(\frac{D_{T}}{D_{L}}-1+\frac{K_{0}K_{3}}{2uD_{L}} \right)+\frac{tK_{3}}{2u}.\] (8) Similar analysis of the sequence of the phase transition, FM $\rightarrow$ CTS $\rightarrow$ CC, yields the schematic phase diagram on the $K_{0}-K_{1}$ plane given in Fig. 2a. The phase boundary between FM and CTS is given by $K_{1}=tK_{3}/2u$. The phase boundary between the CTS and the CC phase at small $K_{0}$ is $K_{1}=2K_{0}/\left(D_{L}/D_{T}-1\right)$, which is also shown in Fig. 2a. Fig. 2 shows that it is not possible to go from the FM to the CC state via a continuous transition, except at a single special point. Generic paths like the diagonal dashed lines in Fig. 2a _must_ go through either the CLS or the CTS state, so two transitions are required to reach the CC state, which, additionally, must be elliptical. Hence the only way there can be a direct transition from the FM state to the CC state is via a first order phase transition, which is not addressed by our theory. This prediction is borne out by experiments of CoCr${}_{2}$O${}_{4}$, where the direct FM to CC transition is indeed first order [15]. ## V Magnetic Reversal of the electric polarization: The polarization $\bar{\mathbf{P}}=\chi\alpha m_{1}m_{2}\hat{y}$ in the CC state is in the $xy$ plane, normal to $\hat{e}_{3}$ and $\mathbf{q}$. It is independent of the uniform magnetization, $m_{3}$. Experimentally [15], the sample is cooled through $T_{c}$ in the presence of a small electric field, $\mathbf{E}=E_{0}\hat{y}$, and a small magnetic field, $\mathbf{H}=H_{0}\hat{z}$. The direction of the pitch vector, $\hat{x}$, or, equivalently, the axis of rotation, $\hat{z}$, are set by the direction of $\bar{\mathbf{P}}$ ($\mathbf{E}$), which determines the ‘helicity’ of the cycloid [7]. It is found, at first, that $\bar{\mathbf{P}}$ is uniquely determined by $\mathbf{E}$ alone, independent of the _initial_ direction of $\mathbf{H}$, as expected. However, once $\bar{\mathbf{P}}$ and $m_{3}$ have set in, changing $H_{0}$ to $-H_{0}$ not only reverses the direction of $m_{3}$, but also, quite unexpectedly, reverses the direction of $\bar{\mathbf{P}}$ as well. In the literature [2; 15], this has lead to the definition of the ‘toroidal moment’, $\mathbf{T}=\mathbf{P}\times\mathbf{M}$, as the order parameter. It is clear that the experimental system is in the conical cycloid state, where $m_{3},\mathbf{q}$ and $\bar{\mathbf{P}}$ are always in mutually orthogonal directions [15]. Further, as expected for this state, the directions of $m_{3}$ and $\bar{\mathbf{P}}$ are uniquely determined by the small cooling fields, $\mathbf{H}$ and $\mathbf{E}$, respectively, which add terms to the Hamiltonian that split the degeneracy between the minima corresponding to the different directions. Now assume that the direction of $\mathbf{H}$ is reversed, $H_{0}\rightarrow-H_{0}$, reversing the direction of $m_{3}$ once it has well developed. There are two ways the uniform magnetization can reverse its direction. First, $m_{3}$ may continue to remain along the $z$-axis and its magnitude may pass through zero to become $-m_{3}$ for $\mathbf{H}=-H_{0}\hat{z}$. If this is the case, $\bar{\mathbf{P}}$ will remain fixed in the direction $\hat{y}$, since the mutual orthogonality of $m_{3},\mathbf{q}$ and $\bar{\mathbf{P}}$ can always be maintained and there is no direct coupling between $m_{3}$ and $\bar{\mathbf{P}}$. However, since $m_{3}$ is already well developed and large ($T_{m}=93$ K), due to the magnetic exchange energy cost it may be energetically more favorable to leave the magnitude of $m_{3}$ unchanged, and its direction may _rotate in space_ to $-\hat{z}$. If this is the case, then $m_{3}$ must rotate staying on the $y-z$ plane, since that way it always remains perpendicular to $\mathbf{q}$, whose direction fluctuations cost the crystalline anisotropy energy. It is then clear, see Fig. 3, that the cycloid plane itself, which is always perpendicular to $m_{3}$ to maintain the lowest energy configuration, must rotate about $\hat{x}$ by a total angle $\pi$. It follows that $\bar{\mathbf{P}}$, always on the cycloid plane, reverses its direction to $-\hat{y}$. This way, even though there is no dynamical coupling between $m_{3}$ and $\bar{\mathbf{P}}$, the latter can also _rotate_ by an angle $\pi$ as a result of the former reversing its direction in space. Based on this, we predict that, at some intermediate $\mathbf{H}\sim-H^{\prime}\hat{z}$, where $H^{\prime}<H_{0}$, $\bar{\mathbf{P}}$ points in the direction $-\hat{z}$, which can be experimentally tested. <figure><img src="content_image/0710.4550/x3.png"><figcaption>Figure 3: The reversal of the polarization (¯P) by the reversal of themagnetization (m3). (a) If m3 rotates to −m3, remaining perpendicular to q,the cycloidal (xy) plane must rotate accordingly to always remain transverseto m3, which is the lowest energy configuration. Since ¯P is in the cycloidalplane, it will rotate by a total angle π. (b) An intermediate stage when m3has rotated by an angle π2 and points in the ^y direction. At this stage, ¯Ppoints in the −^z direction.</figcaption></figure> ## VI Conclusions To conclude, we’ve shown that the magnetic cycloidal orders, and the resulting multiferroicity, can naturally arise due to the magnetoelectric couplings even in rotationally invariant systems, or in cubic crystals. This explains such orders in CoCr${}_{2}$O${}_{4}$, which lack easy plane anisotropies, and are hence outside the realm of the previous theoretical studies on multiferroics. We also predict that a second order transition from the ferromagnet to the conical cycloid state can only occur through an intervening conical longitudinal or transverse spin density wave state with the ultimate cycloidal state being elliptical. A direct such transition, then, must be first order. 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0706.3185	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 17861, "num_imgs": 3, "llama3_tokens_count": 5202 }	[ "content_image/0706.3185/x1.png", "content_image/0706.3185/x2.png", "content_image/0706.3185/x3.png" ]	# Resistivity effects in surface superconductivity of thin films in strong magnetic fields A. A. Zyuzin and A. Yu. Zyuzin A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia ###### Abstract Phase slips creation in the thin film in perpendicular magnetic filed with edge superconductivity is studied. These centers are due to thermal activation of the order parameter below superconducting temperature transition leading to the suppression of the superconductivity. The corresponding resistance is calculated. The Alsamazov- Larkin correction to the conductivity above the critical magnetic field destroying the surface superconductivity is studied. Such structures could be applied as a new system for the study of the phase slip phenomenon in one- dimensional superconducting wires. pacs: 74.25.Fy, 74.25.Ha, 74.40.+k, 74.25.Op As it was first shown by Saint- James and de Gennes [1], superconductivity can nucleate in the thin surface superconducting sheath to magnetic fields higher than the bulk critical field $H_{c2}<H<H_{c3}\approx 1.69H_{c2}$. Thin films reveal the most simple picture of surface superconductivity. In particular, this case was studied experimentally in papers [2; 3], where the temperature dependencies of resistivity of thin $\mathrm{Nb}$ films were measured. Effects of the surface inhomogeneities and the sample shape, properties of mesoscopic size superconductors in surface superconductivity regime were investigated both theoretically and experimentally in papers [4; 5]. However, it is of definite interest to consider the limit of the extremely thin superconducting film, where superconductivity persists in the quasi- one- dimensional edge layer. It is well known that the fluctuations of the order parameter play an important role in the physics of low- dimensional superconductors (thin films, wires). At temperatures above the critical temperature of the superconducting transition $T_{c}$ fluctuations lead to the enhancement of the conductivity [6], while below $T_{c}$ they destroy the long- range order and lead to the finite resistance of the system. In the vicinity of $T_{c}$ for example in thin superconducting wires with the diameter smaller than the coherence length thermal activation of the phase slips centers locally destroys the superconductivity [7; 8; 9; 10]. Phase slip event is of the order of the coherence length $\xi(T)$ where the amplitude of the order parameter vanishes at one point while the phase difference between the opposite sites of this point is $\pi$. The observed resistance of the extremely thin superconducting wires at temperatures $T<<T_{c}$[11; 12] is argued to be caused by the phase slip events due to quantum tunneling of the order parameter [13]. In the present paper we will show that similar fluctuation effects appear in the edge superconductivity of thin superconducting film in the magnetic field perpendicular to the plane of the film. To the best of our knowledge the question of the fluctuations of the order parameter in the vicinity of the phase transition in thin films with edge superconductivity still remains open. We will give the detailed analysis of the sample resistivity dependencies on temperature and magnetic field. The equation for Aslamazov- Larkin correction to the conductivity at fields higher than $H_{c3}(T)$ will be also obtained. Let us consider a thin superconducting film with the magnetic field applied perpendicular to the surface of the film (see Fig.1). We will study the case of the 2-type superconductor under the surface superconductivity condition, when the Ginzburg- Landau parameter $\kappa\gg 1$. Thus, we will not take into account the magnetic field modulations due to supercurrents. The size of the film is such that $d\ll\xi(T)\ll L$, where $d$ and $L$ is the width and length of the film. The Ginzburg- Landau equation in dimensionless variables could be written in the form \[(i\mathbf{\nabla}+\mathbf{A})^{2}\Psi=\Psi(1-\|\Psi\|^{2})\] (1) where $\Psi$ is the complex order parameter, length is measured in units of the coherence length $\xi(T)=(\hbar\pi D/8(T_{c}(H)-T))^{1/2}$, $\mathbf{A}$- vector potential measured in units $\frac{c\hbar}{2e\xi(T)}$, $D$- diffusion coefficient. Since the superconductivity in this regime exists only in the thin edge layer of the film we can treat each edge independently. Then let $y$ axis to be applied along the corresponding edge, $x$ axis directed to the bulk of the film. Magnetic filed is applied along the $z$-axis. Vector potential is chosen in the Landau gauge, $\mathbf{A}=(0,Hx,0)$. The boundary condition for the order parameter at the $(0,y)$ edge of the film is given as \[\frac{d\Psi}{dx}\mid_{x=0}=0\] (2) at the same time, the order parameter vanishes at the bulk of the film. We will search for the solution of the nonlinear Ginzburg- Landau equation (1) in the form \[\Psi_{0}(x,y)=\gamma g(x)e^{iky}\] (3) where $\gamma$ is some constant, $g(x)$ is a function subject to the condition $\int_{0}^{\infty}g^{2}(x)dx=1$. Then the equation (1) can be reformulated as \[V(x)=\|\gamma g_{0}(x)\|^{2},E_{0}=0\] (4) where $g_{0}(x)$ is the eigenfunction corresponding to the lowest eigenvalue $E_{0}$ of the equation \[\left(-\frac{d^{2}}{dx^{2}}+\left(Hx-k\right)^{2}-1+V(x)\right)g(x)=Eg(x)\] (5) In order to solve equation (5) we can use the perturbation theory for $\gamma\ll 1$. The eigenfunctions are then expressed through the parabolic cylinder function [16]$g_{n}(x)\propto D_{a}\left(x\sqrt{2H}-k\sqrt{\frac{2}{H}}\right)$, where $a=\frac{E_{n}-H+1}{2H}$. <figure><img src="content_image/0706.3185/x1.png"><figcaption>Figure 1: Thin film of thickness d in perpendicular magnetic field H, magneticlength lH is the width of the superconducting layer, L\- the length of thelayer along the edge of the film</figcaption></figure> The eigenvalue of the linear equation (5) corresponding to the ground state of the system is a function of $k$ and has a minimum where \[E_{0}=\left[k-k_{0}(H)\right]^{2}-\epsilon+\gamma^{2}\int_{0}^{\infty}g_{0}^{4 }(x)dx\] (6) where $\epsilon=1-H/H_{c3}$ and $k_{0}(H)\sim\sqrt{H}$. First two terms in the equation (6) can be obtained from the boundary condition $\frac{d}{dx}D_{a}\left(x\sqrt{2H}-k\sqrt{\frac{2}{H}}\right)\|_{x=0}$. The ground state eigenvalue condition $\min E_{0}=0$ describes the superconducting transition point. The choice of $k=k_{0}(H)$ corresponds to the zero current case, further we will treat only this case. Finally, we obtain \[\gamma^{2}=\epsilon\left(\int_{0}^{\infty}dxg_{0}^{4}(x)\right)^{-1}\] (7) We also provide the numerical solution for the nonlinear Ginzburg- Landau equation. Fig. 2 shows the zero current case $k=k_{0}(H)$ solution for different values of the magnetic field. Indeed, the order parameter is localized in the vicinity of the edge of the film at distance of the order of the magnetic length. The amplitude of the order parameter vanishes with increasing the magnetic field. The deviations from the solution obtained by the perturbation theory are small up to $H\approx H_{c2}$. For practical purpose [16] it is convenient to approximate the solution for the $g(x)$ by the function \[\tilde{g}(x)=\left(\frac{4bH}{\pi}\right)^{1/4}e^{-\frac{bHx^{2}}{2}}\] (8) where $b=\sqrt{1-2/\pi}$ and $H_{c3}=1/b\simeq 1.66$, that differs from the exact one only by the 2%, while $k_{0}(H)=\sqrt{H/b\pi}$. Using equations (3) and (8) we find the approximate solution to the nonlinear Ginzburg -Landau equation \[\tilde{\Psi}_{0}(x,y)=(\epsilon\sqrt{2})^{1/2}e^{-\frac{H}{2H_{c3}}x^{2}}e^{ik _{0}y}\] (9) It is well known that phase slip events in 1D superconducting wires [7] as well as vortices creation in superfluid liquid [14] at $T\leq T_{c}$ are due to thermal activation of the order parameter. The order parameter switches between the metastable states of the superconductor changing phase by $2\pi$. The probability of such process is governed by the Arrhenius law $\propto\exp{(-\Delta F/T)}$, where $\Delta F$ is the energy barrier separating these states. Solution for the phase slip event corresponds to the saddle point of the barrier. We have found the numerical solution for the phase slip center in the edge layer of the thin film by analyzing the time- dependent Ginzburg- Landau equation (for the review see [19]) with the periodic boundary conditions on the order parameter. The solution for the amplitude of the order parameter is shown in Fig. 3 for different values of $x$ at the magnetic field $H=1.6$, $k=0.93$, which corresponds to $\epsilon=0.1$. <figure><img src="content_image/0706.3185/x2.png"><figcaption>Figure 2: The amplitude of the order parameter \|Ψ0\| for the H=1, k=0.73(dashed line); H=1.3, k=0.85 (dash- dot); H=1.6, k=0.93 (solid line)</figcaption></figure> In order to derive the approximate analytical solution for the phase slip center we will search for the order parameter in the form $\Psi_{1}(x,y)=\gamma e^{ik_{0}y}\sum_{n\geq 0}C_{n}(y)g_{n}(x)$, where the summation is over the set of eigenfunctions of the equation (5). In the zero-mode regime taking into account only $n=0$, we find \[\frac{d^{2}C}{dy^{2}}+\epsilon C(1-C^{2})=0\] (10) This equation is similar to the Ginzburg- Landau equation for the one- dimensional wire at zero applied current case. Solving equation (10) we obtain the expression for the order parameter corresponding to phase slip event in surface superconductivity \[\tilde{\Psi}_{1}(x,y)=\tilde{\Psi}_{0}(x,y)\tanh\left(y\sqrt{\epsilon/2}\right)\] (11) Taking into account solution (11) we conclude according to [7] with the expression for the resistivity of the thin film superconducting edge layer. \[R=\frac{\pi\hbar^{2}\Omega}{2e^{2}T}\exp\left(-\Delta F/T\right)\] (12) where \[\Delta F=\frac{b\sqrt{2}H_{c3}^{2}(T)}{16\pi\kappa^{2}}[\ell^{2}_{H_{c3}}d] \int dxdy\left(\|\Psi_{0}\|^{4}-\|\Psi_{1}\|^{4}\right)\] (13) is the saddle-point free energy barrier increment, $\Omega=(L/\xi(T))(\epsilon^{3/2}/\tau_{GL})\left(\Delta F/T\right)^{1/2}$ is the attempt frequency, $L$ is the length of the edge superconducting layer, $\tau_{GL}=[\pi\hbar/8(T_{c3}(H)-T)]$ is the relaxation time and $\kappa$ is the GL parameter. Using equations (9) and (11) at $H<H_{c3}(T)$ we find \[\Delta F=\frac{bH_{c3}^{2}(T)}{12\sqrt{\pi}\kappa^{2}}\left[\ell_{H_{c3}}\ell_ {H}d\right]\epsilon^{3/2}\] (14) where $d$- is the film thickness, $\ell_{H_{c3}}=\sqrt{\hbar c/eH_{c3}(T)}$- is the magnetic length. This expression is simply the condensation energy of the superconductivity in volume $\ell_{H_{c3}}\ell_{H}d$ of thin film superconducting edge layer. Notice that the width of the edge superconducting layer $\ell_{H}$ at $T\to T_{c}(H)$ is much smaller than the length of the normal part of the layer $\ell_{H_{c3}}/\sqrt{\epsilon}$ caused by the thermal activation of the phase slip event, pointing the applicability of the thin wire approximation to the surface superconductivity of thin film. Phase slip events in 1D superconducting wires at low temperatures are argued to be due to quantum tunneling [11; 12; 13]. The resistivity of the wire is then $R\propto\exp{(-2S)}$, where the exponent of the tunneling amplitude is $S=A\frac{\hbar}{e^{2}}G_{\xi}$ and $A$ is a constant of the order of unity, $G_{\xi}$ is the conductance of the wire of length $\xi(T)$[13]. In our case of the edge superconducting layer it is the magnetic length $\ell_{H_{c3}}$ that governs the tunneling amplitude. <figure><img src="content_image/0706.3185/x3.png"><figcaption>Figure 3: The amplitude of the order parameter \|Ψ1(x,y)\| for the case H=1.6,k=0.93; solid line corresponds to the solution on the boundary x=0, dashedline corresponds to x=1, x=1.6, x=2.3</figcaption></figure> The tunneling process is accompanied by the creation of the acoustic plasmons [17] which are responsible for the interaction between the phase slip centers. As a result, this interaction suppresses the tunneling probability and this effect is stronger as smaller the plasmons velocity, i.e. as stronger the Coulomb screening. The dissipation effects are the second factor that leads to decreasing of the probability of the phase slip centers due to quantum tunneling [15]. In contrast to the case of isolated superconducting wire, the Coulomb interactions in the regime of edge superconductivity is screened by the charge of the normal part of the film. This immediately leads to decreasing of the acoustic plasmons velocity. The interaction with the normal part of the film results in the effective relaxation of the order parameter phase fluctuations. Consequently, the resistivity of the surface superconducting layer of the thin film should be lower than the resistivity of the wire, taken under the same conditions. Let us consider the fluctuations of the order parameter in the case of surface superconductivity at magnetic fields $H>H_{c3}(T)$. The corresponding Aslamazov- Larkin correction to the conductivity at magnetic fields higher than the superconducting transition field $H_{c3}$ was studied in the paper [18]. It was shown that for the case of two- dimensional surface superconducting layer this correction has the same temperature dependence as for the thin film at $T>T_{c}$. However, in present work we focus on the extreme case of surface superconductivity, when the order parameter is concentrated in the quasi-one- dimensional layer of the thin film. Fluctuations of the order parameter at magnetic fields $H>H_{c3}(T)$ result in additional correction to the conductance which according to [19] is \[G=\frac{(2e)^{2}}{2m}\sum_{\nu}\langle\|\phi_{\nu}\|^{2}\rangle\frac{\tau_{\nu}} {2}\] (15) where summation goes over the set $\nu=n,k$. The value of fluctuation of the order parameter $\Psi(x,y)=\sum_{\nu}\phi_{\nu}g_{\nu}(x)e^{iky}$ is written as \[\langle\|\phi_{\nu}\|^{2}\rangle=\left[\frac{2m\ell^{2}_{H_{c3}}}{b\hbar^{2}} \right]\frac{T}{E_{n}(k)}\] (16) while $\tau_{\nu}=\tau_{GL}/E_{n}(k)$ is the characteristic decay time of the fluctuation. The lowest eigenvalue with $n=0$ gives the main contribution to the sum and coming from summation to the integration over $k$ we obtain the correction to the conductance of unit length of the layer \[G\simeq 0.28\frac{e^{2}}{\hbar}\frac{H_{c3}(0)}{H_{c3}(T)}\frac{\ell_{H_{c3}}} {\|\epsilon\|^{3/2}}\] (17) Notice that the functional dependence of the correction $G\propto\|\epsilon\|^{-3/2}$ is similar to the case of one- dimensional superconducting wire. It is seen also, with decreasing the temperature $H_{c3}(T)$ increases and the value of the Aslamazov- Larkin correction decreases. However, at the same time equation (17) still valid in the interval of the magnetic fields $H-H_{c3}(T)$ that increases with decreasing the temperature. For the numerical estimations we take typical values $H_{c3}\sim 1$T then $\ell_{H_{c3}}\sim 25$nm. For ultrathin film $d\sim 10$nm, $\kappa=10$, we estimate at $T_{c3}\sim 1$K for the value $\Delta F/T\sim 10^{3}\epsilon^{3/2}$. The probability of the phase slip event becomes negligibly small unless the parameter $\epsilon=1-H/H_{c3}$ is of the order of $10^{-2}$. According [16] the critical current destroying the surface superconductivity could be written as $J_{c}\simeq j_{c}\ell_{H_{c3}}d$, where $j_{c}=\frac{1}{3\pi\sqrt{6}}\frac{cH_{c}(T)}{\kappa\xi(T)}$ is the critical current density of thin wire, we estimate $J_{c}\sim 20$$\mu A$. To summarize, we have shown that the phase slip phenomenon reveals in edge superconducting layer of thin film in perpendicular magnetic field at $H<H_{c3}(T)$. The corresponding resistance was calculated. The Aslamazov- Larkin correction to the edge superconductivity of thin film at $H>H_{c3}(T)$ have also been obtained. We conclude that such structures could be applied as a new system for the study of the phase slip phenomenon in one- dimensional superconducting wires. We thank V.I. Kozub for helpful discussions and valuable questions. The research was supported by Dynasty foundation, INTAS Grant 05- 109- 4829 and RFFI Grant 06- 02- 17047. ## References * (1) D. Saint- James and P. deGennes, Phys. Lett. 7, 306 (1963). * (2) D. Stamopoulos, M. Pissas, V. Karanasos et. al., Phys. Rev. B 70, 054512 (2004). * (3) J. Scola, A. Pautrat, C. Goupil et. al., Phys. Rev. B 72, 012507 (2005). * (4) B.J. Baelus and F.M. Peeters, Phys. Rev. B 65, 104515 (2002). * (5) D.A. Dikin, V. Chandrasekhar, V.R. Misko et. al., Eur. Phys. J. B 34, 231-235 (2003). * (6) L.G. Aslamazov, A.I. Larkin, Phys. Tv. Tela 10, 1104 (1968). * (7) J.S Langer, A. Ambegoakar, Phys. Rev. 164, 498 (1967). * (8) D.E. McCumber and B.I. Halperin, Phys. Rev. B 1, 1054 (1970). * (9) J.E. Lukens, R.J. Warburton, and W. W. Webb, Phys. Rev. Lett. 25, 1180 (1970). * (10) R.S. Newbower, M.R. Beasley, and M. Tinkham, Phys. Rev. B 5, 864 (1972). * (11) N. Giordano, Phys. Rev. Lett. 61, 2137 (1988). * (12) C.N. Lau, M. Markovich, M. Bockrath, A. Bezryadin, and M. Tinkham, Phys. Rev. Lett. 87, 217003 (2001). * (13) D.S. Golubev, A.D. Zaikin, Phys. Rev. B 64, 014504 (2001). * (14) S.V. Iordanski JETP 21, 467 (1965) * (15) H.P. B$\ddot{\texttt{u}}$chler, V.B. Geshkenbein, and G. Blatter, Phys. Rev. Lett. 92, 067007 (2004). * (16) A.A. Abrikosov, _Fundamentals of the Theory of Metals_ (Elseiver, New York, 1988). * (17) J.E. Mooij and G. Sch$\ddot{\texttt{o}}$n, Phys. Rev. B 55, 114 (1985). * (18) H. Schmidt and H.J. Mikeska, J. Low Temp. Phys. 3, 123 (1970). * (19) M. Tinkham, _Introduction to Superconductivity_ (McGraw- Hill, New York, 1996).
0912.3607	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 69563, "num_imgs": 7, "llama3_tokens_count": 23938 }	[ "content_image/0912.3607/x1.png", "content_image/0912.3607/x2.png", "content_image/0912.3607/x3.png", "content_image/0912.3607/x4.png", "content_image/0912.3607/x5.png", "content_image/0912.3607/x6.png", "content_image/0912.3607/x7.png" ]	# Diffusive versus local spin currents in dynamic spin pumping systems Akihito Takeuchi atake@phys.metro-u.ac.jp Kazuhiro Hosono Gen Tatara Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan February 22, 2024 ###### Abstract Using microscopic theory, we investigate the properties of a spin current driven by magnetization dynamics. In the limit of smooth magnetization texture, the dominant spin current induced by the spin pumping effect is shown to be the diffusive spin current, i.e., the one arising from only a diffusion associated with spin accumulation. That is to say, there is no effective field that locally drives the spin current. We also investigate the conversion mechanism of the pumped spin current into a charge current by spin-orbit interactions, specifically the inverse spin Hall effect. We show that the spin-charge conversion does not always occur and that it depends strongly on the type of spin-orbit interaction. In a Rashba spin-orbit system, the local part of the charge current is proportional to the spin relaxation torque, and the local spin current, which does not arise from the spin accumulation, does not play any role in the conversion. In contrast, the diffusive spin current contributes to the diffusive charge current. Alternatively, for spin-orbit interactions arising from random impurities, the local charge current is proportional to the local spin current that constitutes only a small fraction of the total spin current. Clearly, the dominant spin current (diffusive spin current) is not converted into a charge current. Therefore, the nature of the spin current is fundamentally different depending on its origin and thus the spin transport and the spin-charge conversion behavior need to be discussed together along with spin current generation. pacs: 72.25.Ba, 76.50.+g, 72.20.My, 75.75.-c ## I introduction Spintronics [1; 2] is a novel technology that enables control of both charge and spin of electrons. To accomplish this aim, establishing methods for generation and observation of spin currents is an urgent issue. For generation in nonmagnetic conductors, several methods have been proposed. The standard way is to use the spin pumping effect in ferromagnetic-normal metal junctions. [3; 4; 5; 6; 7] In those systems, a spin current can be induced by the precession of the magnetization caused by an external alternating field and is then pumped into the normal metal. This spin pumping effect is widely used in experiments. A second technique is non-local spin injection in ferromagnetic-normal metal junctions. [8; 9] In this case, a spin-polarized current is induced in a ferromagnet by applying an electric field. The spin-polarized current results in a spin imbalance at the interface that gives rise to a diffusive flow of spin without a charge current in the normal metal. A third technique is to use the spin Hall effect, [10; 11; 12; 13; 14; 15] where the spin current flows in a transverse direction to an applied electric voltage in the presence of a spin-orbit interaction. Very recently, the spin Seebeck effect was discovered enabling a generation of a spin current from thermal gradients in ferromagnets. [16] Thus, spin current generation can be realized by several mechanisms using various magnetic, electric, thermoelectric, and quantum relativistic (spin-orbit) effects. In contrast, direct measurement of spin currents is still an open issue. Spin current detection has so far been performed by measuring subsidiary observables that arise from spin currents. The first such observation was accomplished by Kato _et al._ [14] by measuring optically the spin accumulation that appears as a result of spin currents at the edges of samples (GaAs and InGaAs) in spin Hall systems. The critical issue, however, is that spin currents are not conserved. Therefore, spin accumulation and spin currents are not in direct correspondence, in sharp contrast to charge currents that are conserved. The inverse spin Hall effect was proposed as a method for measuring spin currents electrically. [17; 18; 19] The idea is based on the argument that spin currents flowing in the presence of spin-orbit interactions induce an electric voltage through the inverse process associated with the spin Hall effect. [17; 20; 21] (The electric detection of spin transport was first demonstrated at the interface between ferromagnetic and paramagnetic metals by Johnson and Silsbee. [8]) Being electric, the inverse spin Hall effect has been widely utilized to detect spin currents. [16; 22; 23; 24] Theoretical justification for the inverse spin Hall effect has been carried out on various systems. [20; 21; 25; 26; 27; 28; 29; 30; 31] Takahashi and Maekawa [20; 21] investigated the inverse spin Hall effect because of nonlocal spin injection in ferromagnetic-nonmagnetic metal junctions, and demonstrated phenomenologically the relationship between charge current (${\bm{j}}_{\rm c}$) and spin current (${\bm{j}}_{\rm s}$), i.e., ${\bm{j}}_{\rm c}\propto{\bm{\sigma}}\times{\bm{j}}_{\rm s}$, where ${\bm{\sigma}}$ is the spin polarization direction. Generation of the charge current by magnetization dynamics in magnetic junctions was studied phenomenologically by Wang _et al._ [25] and Xiao _et al._ [26] By using microscopic theory, the direct connection between spin currents pumped by magnetization dynamics and induced charge currents was revealed in disordered metallic systems [27; 28] and in disordered Rashba systems. [29; 30; 31] The result for metallic systems followed phenomenological predictions, ${\bm{j}}_{\rm c}\propto{\bm{\sigma}}\times{\bm{j}}_{\rm s}$, whereas pumped charge currents in Rashba systems deviated from this simple relation. The naive picture that all spin currents are converted into charge currents is, therefore, incorrect. What has been missing in this picture is the distinction between spin currents induced by some effective field and those arising from spin accumulation diffusion. The first one, local spin current, is a contribution proportional to a local value of the magnetization texture. The other contribution is a diffusive spin current which contains a long-range diffusion factor. As we will show below, this difference is crucial in discussing the inverse spin Hall effect but can not be addressed by phenomenological schemes. For applications, these two spin currents need to be considered separately since the direction of the local (field-driven) spin current is controlled by the field, while there is no way to control the direction of the diffusive spin current. The first aim of the present paper is to study these spin currents arising from magnetization dynamics (spin pumping effect) in the absence of spin-orbit interactions by providing a fully quantum-mechanical treatment of the conduction electrons. (We note that spin-orbit interactions are not essential in discussing the spin pumping effect.) Generation of spin currents because of a precessing magnetization was first studied by Silsbee _et al._ [3] They showed that this precession generates an accumulation of spin at the interface between ferromagnetic and nonmagnetic domains and that the spin diffusion, i.e., spin current, arises from this spin accumulation. Tserkovnyak _et al._ [4] argued that the pumped spin current can be expressed in the form ${\bm{j}}_{\rm s}=a\dot{{\bm{S}}}+b{\bm{S}}\times\dot{{\bm{S}}}$, introducing phenomenological parameters $a$ and $b$ ($a$ and $b$ are proportional to their mixing conductances and ${\bm{S}}$ denotes the magnetization direction). They assumed that the spin-flip relaxation rate is sufficiently larger than their spin injection rate and the spin accumulation does not build up at the interface. Therefore, their pumping mechanism is different from that of Silsbee _et al._ Later, Brataas _et al._ considered the effect of a back flow arising from the spin accumulation at the interface. [5] There have been, therefore, predicted two types of spin currents: one arising from a nonlocal diffusion of the spin accumulation and the other a local contribution driven by an effective field. We show in the present paper that the diffusive spin current originating from spin accumulation dominates in the case of the slow-varying magnetization (disordered system). This result is in agreement with the prediction by Silsbee _et al._ [3] The scenario of the spin pumping effect without spin accumulation by Tserkovnyak _et al._ [4] does not hold in the present situation, but it may apply to cases where rapid-varying magnetization occurs at the interface of a ferromagnetic-normal metal junctions. The second aim of the present paper is to clarify the spin-charge conversion mechanism based on microscopic theory and derive its conversion formula. By calculating a spin current in the absence of spin-orbit interactions and a charge current with spin-orbit interactions (treated as linear-order perturbation), we will reveal the spin-to-charge current conversion phenomena via the spin-orbit interaction. We will demonstrate that this conversion mechanism differs depending on the type of spin-orbit interaction, smoothness of the magnetization profile, and the disorder (electron mean free path). In particular, the origin of spin currents, namely, whether spin currents are generated by effective fields or arise from spin accumulation, turns out to be crucially important for spin-charge conversion efficiency. Depending on the spin-orbit interaction causing the conversion, we consider two cases: Rashba spin-orbit interactions and spin-orbit interaction originating from random impurity scattering in metals. In the former, the dominant spin current is converted into a charge current. In contrast, in the latter, the correlation between the dominant spin current and pumped charge currents is very weak. Instead, the local spin current (a small fraction of the pumped spin current) is converted. Identifying the origin of spin currents is, therefore, crucial for realizing high conversion efficiency. ## II spin pumping effect ### Model <figure><img src="content_image/0912.3607/x1.png"><figcaption>Figure 1: (Color online) Schematic illustration of spin current generation.The localized spin S is assumed to be slowly varying in space (compared to theelectron mean-free path) and time (compared to the electron lifetime).Precession of the localized spin S in the ferromagnet generates a flow of thespin in the contiguous normal metal.</figcaption></figure> We consider a disordered electron system coupled with localized spin ${\bm{S}}(\bm{x},t)$ (Fig. 1). The localized spin can have spatial and temporal dependences, but we consider only the slowly varying case [namely, that the spatial correlation length is larger than the electron mean free path (see below)]. The total Hamiltonian of the system is given as $H(t)=H_{0}+H_{\rm ex}(t)$, where \[\begin{split} H_{0}&=\sum_{\bm{k}}\varepsilon_{\bm{k }}c^{\dagger}_{\bm{k}}c_{\bm{k}}+\frac{u_{\rm i}}{V}\sum_{n}\sum_{{\bm{k}},{ \bm{p}}}e^{i{\bm{p}}\cdot{\bm{r}}_{n}}c^{\dagger}_{{\bm{k}}-{\bm{p}}}c_{\bm{k} },\\ H_{\rm ex}(t)&=-J\sum_{{\bm{k}},{\bm{q}}}\big{(}c^{ \dagger}_{{\bm{k}}-{\bm{q}}}\hat{\bm{\sigma}}c_{\bm{k}}\big{)}\cdot{\bm{S}}_{ \bm{q}}(t).\end{split}\] (1) The first term $H_{0}$ describes free electrons in the presence of spin-independent impurity scatterers and $H_{\rm ex}$ represents the exchange interaction between the conducting electron and the local spin. We have introduced in momentum space annihilation (creation) operators $c_{\bm{k}}^{(\dagger)}$ for conduction electrons with kinetic energy $\varepsilon_{\bm{k}}\equiv\hbar^{2}{\bm{k}}^{2}/2m$, while $u_{\rm i}$ is the strength of the impurity scatterers, $V$ is the system volume, ${\bm{r}}_{n}$ represents the position of the $n$th impurity, $J$ is the exchange coupling constant, $\hat{\bm{\sigma}}$ characterizes a vector of Pauli matrices, the caret signifies a matrix, and ${\bm{S}}_{\bm{q}}$ denotes the Fourier transform of the localized spin (or magnetization). We note that impurity scattering gives rise to an elastic electron lifetime $\tau$. Let us stress that spin-orbit interactions are not considered in this section, since it is not essential to spin current generation and to spin-charge conversion. The electron-spin density is defined as $\rho_{\rm s}^{\alpha}({\bm{x}},t)\equiv(\hbar/2){\rm tr}\big{\langle}{\psi^{ \dagger}({\bm{x}},t)\hat{\sigma}^{\alpha}\psi({\bm{x}},t)}\big{\rangle}_{H}$, where $\psi^{(\dagger)}({\bm{x}})$ is the Fourier transform of $c^{(\dagger)}_{\bm{k}}$, ${\rm tr}$ denotes the trace over spin indices, and $\langle{\cdots}\rangle_{H}$ represents the expectation value estimated for the total Hamiltonian $H$. The spin current density is defined (without spin-orbit interaction) as \[{\bm{j}}_{\rm s}^{\alpha}({\bm{x}},t)=-\frac{i\hbar^{3}}{2mV}\sum_{{\bm{k}},{ \bm{q}}}e^{-i{\bm{q}}\cdot{\bm{x}}}{\rm tr}\Big{[}{\bm{k}}\hat{\sigma}^{\alpha }\hat{G}^{<}_{{\bm{k}}-\frac{{\bm{q}}}{2},{\bm{k}}+\frac{{\bm{q}}}{2}}(t,t) \Big{]},\] (2) where the spin current $j_{{\rm s}i}^{\alpha}$ has two direction components, one associated with the flow of spin in direction $i$ and the other associated with a spin polarization in direction $\alpha$. Here we used the lesser component of the path-ordered Green’s function defined as $G^{<}_{{\bm{k}}\alpha,{\bm{k}}^{\prime}\alpha^{\prime}}(t,t^{\prime})\equiv(i/ \hbar)\big{\langle}{c^{\dagger}_{{\bm{k}}^{\prime}\alpha^{\prime}}(t^{\prime}) c_{{\bm{k}}\alpha}(t)}\big{\rangle}_{H}$. ### Pumped spin current <figure><img src="content_image/0912.3607/x2.png"><figcaption>Figure 2: Diagrammatic representations of the spin current density js.Diagram (a) describes first-order contributions in J and (b) second-ordercontributions. The wavy lines denote the interaction with the local spin S andthe gray shaded region represents the vertex correction from impurityscatterers.</figcaption></figure> We carry out the calculation in a weak exchange coupling regime, $J\ll\hbar/\tau$. This assumption would be satisfied in a junction of normal metal and a ferromagnet, as shown in Fig. 1, since the interface is usually disordered. [32] The Feynman diagrams contributing to the spin current at first and second orders in the exchange coupling $J$ are shown in Fig. 2. We assume a spatially smooth magnetization structure $q\ell\ll 1$, where $q$ is the momentum of the magnetization and $\ell$ denotes the mean free path for conduction electrons, and assume a slow precession of magnetization $\Omega\tau\ll 1$, where $\Omega$ is a frequency of magnetization dynamics. Contributions from Fig. 2(a) reads \[{\bm{j}}_{\rm s}^{(1)\alpha}({\bm{x}},t)=-\frac{2\hbar^{3}J}{3mV}\sum_{{\bm{k} },{\bm{k}}^{\prime},{\bm{q}}}\sum_{\omega,\Omega}e^{-i{\bm{q}}\cdot{\bm{x}}+i \Omega t}S^{\alpha}_{{\bm{q}},\Omega}{\bm{q}}\Omega f^{\prime}_{\omega}\Big{[} {\rm Im}\varepsilon_{\bm{k}}g^{\rm r}_{{\bm{k}},\omega}(g^{\rm a}_{{\bm{k}}, \omega})^{2}\Big{]}\bigg{(}1+\frac{n_{\rm i}u_{\rm i}^{2}}{V}\Pi^{\rm ra}_{{ \bm{q}};\omega,\Omega}g^{\rm r}_{{\bm{k}}^{\prime},\omega}g^{\rm a}_{{\bm{k}}^ {\prime},\omega}\bigg{)}.\] (3) Here $n_{\rm i}$ is the impurity concentration, $f_{\omega}=\theta(\varepsilon_{\rm F}-\hbar\omega)$ denotes the Fermi distribution function at zero temperature [$\theta(\omega)$ is the step function and $\varepsilon_{\rm F}$ is the Fermi energy], $g^{\rm a}$ ($g^{\rm r}$) denotes the advanced (retarded) free Green’s function given by $g^{\rm a}_{{\bm{k}},\omega}=(g^{\rm r}_{{\bm{k}},\omega})^{}=\big{[}\hbar \omega-\varepsilon_{\bm{k}}-(i\hbar/2\tau)\big{]}^{-1}$, and $\Pi^{\rm ra}$ describes the diffusion ladder defining the vertex correction, \[\Pi^{\rm ra}_{{\bm{q}};\omega,\Omega}\equiv\sum_{n=0}^{\infty}\bigg{(}\sum_{ \bm{k}}\frac{n_{\rm i}u_{\rm i}^{2}}{V}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2}, \omega-\frac{\Omega}{2}}g^{\rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+\frac{ \Omega}{2}}\bigg{)}^{n}.\] (4) The dominant first-order contribution to the spin current is calculated as \[{\bm{j}}_{\rm s}^{(1)\alpha}({\bm{x}},t)=\frac{\hbar\nu J\tau{\rm D}}{V}{\bm{ \nabla}}\big{\langle}{\dot{S}^{\alpha}({\bm{x}},t)}\big{\rangle},\] (5) where $\nu$ denotes the density of states, ${\rm D}$ denotes a diffusion coefficient given as $2\varepsilon_{\rm F}\tau/3m$, and $\langle{\cdots}\rangle$ describes long-range diffusion because of random impurity scattering \[\big{\langle}{A({\bm{x}},t)}\big{\rangle}\equiv\frac{1}{V}\int{d^{3}x^{\prime} }\int{dt^{\prime}}\sum_{\bm{q}}\sum_{\Omega}e^{-i{\bm{q}}\cdot({\bm{x}}-{\bm{x }}^{\prime})+i\Omega(t-t^{\prime})}\frac{A({\bm{x}}^{\prime},t^{\prime})}{{\rm D }{\bm{q}}^{2}\tau+i\Omega\tau},\] (6) where $A({\bm{x}},t)$ is an arbitrary function of both ${\bm{x}}$ and $t$. (We show details of the calculations in Appendix A.) Similarly, we obtain second-order contributions in $J$ [Fig. 2(b)] as ${\bm{j}}_{\rm s}^{(2)}={\bm{j}}_{\rm s}^{\rm p(2)}+{\bm{j}}_{\rm s}^{\rm sc(2)}$, where the dynamic component (pumped spin current) ${\bm{j}}_{\rm s}^{\rm p(2)}$ is given by \[{\bm{j}}_{\rm s}^{\rm p(2)\alpha}({\bm{x}},t)= -\frac{4\hbar^{2}J^{2}\tau}{3mV}\sum_{{\bm{k}},{\bm{k}}^{\prime}, {\bm{q}},{\bm{q}}^{\prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}} \cdot{\bm{x}}+i\Omega t}\big{(}{\bm{S}}_{{\bm{q}}^{\prime},\Omega^{\prime}} \times{\bm{S}}_{{\bm{q}}-{\bm{q}}^{\prime},\Omega-\Omega^{\prime}}\big{)}^{ \alpha}\Omega^{\prime}f^{\prime}_{\omega}\] \[\times\Big{[}{\rm Im}\varepsilon_{\bm{k}}g^{\rm r}_{{\bm{k}}, \omega}(g^{\rm a}_{{\bm{k}},\omega})^{2}\Big{]}\bigg{[}({\bm{q}}+{\bm{q}}^{ \prime})+\frac{n_{\rm i}u_{\rm i}^{2}{\bm{q}}}{V}\Pi^{\rm ra}_{{\bm{q}};\omega ,\Omega}g^{\rm r}_{{\bm{k}}^{\prime},\omega}g^{\rm a}_{{\bm{k}}^{\prime}, \omega}\bigg{]}\] \[\simeq -\frac{2\nu J^{2}\tau^{2}{\rm D}}{V}{\bm{\nabla}}\big{\langle}{ \big{[}{\bm{S}}({\bm{x}},t)\times\dot{{\bm{S}}}({\bm{x}},t)\big{]}^{\alpha}} \big{\rangle},\] (7) and the equilibrium component (spin super-current) ${\bm{j}}_{\rm s}^{\rm sc(2)}$ is given by \[{\bm{j}}_{\rm s}^{\rm sc(2)\alpha}({\bm{x}},t)= \frac{i\hbar^{2}J^{2}}{3mV}\sum_{{\bm{k}},{\bm{q}},{\bm{q}}^{ \prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{\bm{x}}+i\Omega t }\big{(}{\bm{S}}_{{\bm{q}}^{\prime},\Omega^{\prime}}\times{\bm{S}}_{{\bm{q}}-{ \bm{q}}^{\prime},\Omega-\Omega^{\prime}}\big{)}^{\alpha}{\bm{q}}^{\prime}f^{ \prime}_{\omega}{\rm Im}(g^{\rm a}_{{\bm{k}},\omega})^{2}\] \[\simeq -\frac{\hbar^{2}\nu J^{2}}{12m\varepsilon_{\rm F}V}\big{[}{\bm{S} }({\bm{x}},t)\times{\bm{\nabla}}{\bm{S}}({\bm{x}},t)\big{]}^{\alpha}.\] (8) This equilibrium flow is a spin super-current and arises from the angular difference between two localized spins (or magnetizations) ${\bm{S}}_{1}\times{\bm{S}}_{2}$, i.e., from the spin Josephson effect. [33] In Eqs. (5) and (7), the dynamic component without vertex correction does not contribute because it is smaller than that with vertex correction by 1 order of magnitude $(q\ell)^{2}\ll 1$. As a result, the pumped spin current for a smooth-varying magnetization texture is a long-range diffusive flow (Eqs. (5) and (7)). From Eqs. (5) and (7), the pumped spin current, ${\bm{j}}_{\rm s}^{\rm(p)}\equiv{\bm{j}}_{\rm s}^{(1)}+{\bm{j}}_{\rm s}^{\rm p( 2)}$, can be written as a gradient of an effective potential $\mu_{\rm s}$, \[{\bm{j}}_{\rm s}^{\rm(p)\alpha}({\bm{x}},t)=-{\bm{\nabla}}\mu_{\rm s}^{\alpha} ({\bm{x}},t),\] (9) where (10) This effective spin potential arises from dissipations of magnetization energy ${\bm{S}}\times\dot{{\bm{S}}}$ and $\dot{{\bm{S}}}$. Our result, Eqs. (9) and (10), indicates that the sources of the spin current are $\dot{{\bm{S}}}$ and ${\bm{S}}\times\dot{{\bm{S}}}$ and this result appears to agree with phenomenological predictions for the spin pumping effect. [4] However, the spin current here has been diffusion-averaged and is not simply a local function of the source. Similarly, spin density is given by \[\rho_{\rm s}^{\alpha}({\bm{x}},t)=-\frac{\hbar\nu J\tau{\rm D}}{V}{\bm{\nabla} }^{2}\big{\langle}{S^{\alpha}({\bm{x}},t)}\big{\rangle}+\frac{2\nu J^{2}\tau^{ 2}}{V}\big{\langle}{\big{[}{\bm{S}}({\bm{x}},t)\times\dot{{\bm{S}}}({\bm{x}},t )\big{]}^{\alpha}}\big{\rangle}.\] (11) It satisfies the spin diffusion equation: \[\dot{\rho}_{\rm s}^{\alpha}({\bm{x}},t)-{\rm D}{\bm{\nabla}}^{2}\rho_{\rm s}^{ \alpha}({\bm{x}},t)=\tau_{\rho}^{\alpha}({\bm{x}},t),\] (12) where $\tau_{\rho}$ represents the spin relaxation given as \[\tau_{\rho}^{\alpha}({\bm{x}},t)\equiv-\frac{\hbar\nu J{\rm D}}{V}{\bm{\nabla} }^{2}S^{\alpha}({\bm{x}},t)+\frac{2\nu J^{2}\tau}{V}\big{[}{\bm{S}}({\bm{x}},t )\times\dot{{\bm{S}}}({\bm{x}},t)\big{]}^{\alpha}.\] (13) The effective spin potential is written as \[\mu_{\rm s}^{\alpha}({\bm{x}},t)={\rm D}\rho_{\rm s}^{\alpha}({\bm{x}},t)- \frac{\hbar\nu J{\rm D}}{V}S^{\alpha}({\bm{x}},t).\] (14) Specifically, it is the spin density excluding the direct spin polarization because of the exchange interaction, $\rho_{\rm s}-(\hbar\nu J/V)S$, multiplied by ${\rm D}$. Therefore, the total spin current is reducible to \[{\bm{j}}_{\rm s}^{\alpha}({\bm{x}},t) ={\bm{j}}_{\rm s}^{\rm sc(2)\alpha}({\bm{x}},t)+{\bm{j}}_{\rm s}^ {\rm(p)\alpha}({\bm{x}},t)\] \[=-\frac{\hbar^{2}\nu J^{2}}{12m\varepsilon_{\rm F}V}\big{[}{\bm{S }}({\bm{x}},t)\times{\bm{\nabla}}{\bm{S}}({\bm{x}},t)\big{]}^{\alpha}-{\bm{ \nabla}}\mu_{\rm s}^{\alpha}({\bm{x}},t).\] (15) The last term, a diffusive contribution, can be written in terms of the spin density by using Eq. (14) and we finally obtain \[{\bm{j}}_{\rm s}^{\alpha}({\bm{x}},t)={\bm{j}}_{\rm s}^{\rm(sc)\alpha}({\bm{x} },t)-{\rm D}{\bm{\nabla}}\rho_{\rm s}^{\alpha}({\bm{x}},t),\] (16) where we have defined the total equilibrium spin current as \[{\bm{j}}_{\rm s}^{\rm(sc)\alpha}({\bm{x}},t)=\frac{\hbar\nu J{\rm D}}{V}{\bm{ \nabla}}S^{\alpha}({\bm{x}},t)-\frac{\hbar^{2}\nu J^{2}}{12m\varepsilon_{\rm F }V}\big{[}{\bm{S}}({\bm{x}},t)\times{\bm{\nabla}}{\bm{S}}({\bm{x}},t)\big{]}^{ \alpha}.\] (17) As seen in Eq. (16), the dominant contribution of the dynamic spin current derives completely from a diffusion of the spin polarization. Equations (16) and (17) are the main results describing the spin pumping phenomena. For a charge current ${\bm{j}}_{\rm c}$, it is described generally as \[{\bm{j}}_{\rm c}({\bm{x}},t)=\sigma_{\rm c}{\bm{E}}({\bm{x}},t)+{ \bm{j}}_{\rm sc}({\bm{x}},t)-{\rm D}{\bm{\nabla}}\rho_{\rm c}({\bm{x}},t),\] (18) where $\sigma_{\rm c}$ is an electrical conductivity, ${\bm{E}}$ represents an electric field, ${\bm{j}}_{\rm sc}$ is a super-current which exists in an equilibrium situation (corresponding to ${\bm{j}}_{\rm s}^{\rm(sc)}$ of the spin current), and $\rho_{\rm c}$ denotes a charge density. Comparing the two expressions, Eqs. (16) and (18), we immediately see a striking difference between the charge and the spin, namely, there is no field that induces a spin current at least in the present perturbative regime without spin-orbit interactions. In other words, the spin pumping effect does not directly generate the spin current itself, but causes a spin imbalance or spin accumulation that then gives rise to a diffusive spin current. This result is in agreement with the study by Silsbee _et al._, [3] while the prediction by Tserkovnyak _et al._, [4] stating that spin pumping is not associated with spin accumulation, does not hold in the smooth spin case. The diffusive spin current represented by the vertex correction found here would correspond to the phenomenologically discussed back-flow because of spin accumulation at the interface. [5] Absence of an effective field for the spin current is crucial in spintronics. In fact, in charge electronics, the charge and its current are independently controllable and can be measured by different mechanisms, for instance, by capacitance means for charges and Ampère’s law for currents. This is not the case in spin transport phenomena. If we use the spin pumping mechanism in a disordered system, the spin current is always accompanied by spin accumulation according to Eq. (16). Fortunately, we know that there is an effective field for a spin current if we use spin-orbit interactions, specifically, as in the spin Hall system. By including spin-orbit interactions, the spin current is generalizable to \[j_{{\rm s}i}^{\alpha}({\bm{x}},t)=\sigma_{\rm SH}\epsilon_{ij \alpha}E_{j}({\bm{x}},t)+j_{{\rm s}i}^{\rm(sc)\alpha}({\bm{x}},t)-{\rm D} \nabla_{i}\rho_{\rm s}^{\alpha}({\bm{x}},t),\] (19) where $\sigma_{\rm SH}$ represents the spin Hall conductivity proportional to the spin-orbit interaction and $\epsilon_{ij\alpha}$ denotes the Levi-Civita antisymmetric tensor. Therefore, charge and spin currents, Eqs. (18) and (19), now look symmetric. However, there appears a crucial difference when one includes spin-orbit interactions, namely, the violation of the conservation law for spin. In fact, spin and its current in the presence of spin-orbit interactions satisfy the identity \[\dot{\rho}_{\rm s}^{\alpha}({\bm{x}},t)+{\bm{\nabla}}\cdot{\bm{j}}_{\rm s}^{ \alpha}({\bm{x}},t)=\mathcal{T}_{\rm s}^{\alpha}({\bm{x}},t),\] (20) where $\mathcal{T}_{\rm s}$ represents the spin-relaxation torque arising from the spin-orbit interaction. [34] (This equation is equivalent to the diffusion equation for the spin density. [35; 36; 37]) The non-conservation of spin causes definitional ambiguity of a spin current. Since a definition of spin current is absolutely related to a definition of spin relaxation torque, spin currents cannot be defined uniquely under the spin-orbit interaction. (As a possible solution, a gauge covariant derivative was proposed. [38]) Because of the spin relaxation torque $\mathcal{T}_{\rm s}$, the measurement of the spin current can never be carried out by simply measuring the spin density induced at the edge since the induced spin current can disappear because of the term $\mathcal{T}_{\rm s}$ while being transported. (In this sense, the observation of the spin Hall effect in Ref. [14] cannot be considered as a direct observation of the spin current.) At present, there has been no indication of an emission of an observable field (either electric or magnetic) from the spin current and, therefore, in contrast to charge currents which are observable by detecting an Ampère’s field, electromagnetic detection of spin currents is not possible. [The absence of electromagnetic fields induced by spin currents is reasonable from Maxwell’s equations because the equations as determined by special relativity and U(1) gauge invariance cannot be modified by the spin current.] Here a serious dilemma for spintronics arises. Specifically, spin currents and spin densities are independently controllable only if one switches on spin-orbit interactions, but such interactions make it impossible to detect the spin current by measuring the spin accumulation. ### Diffusive spin current vs. local spin current <figure><img src="content_image/0912.3607/x3.png"><figcaption>Figure 3: (Color online) Relationship between pumped spin current andinterface condition in a ferromagnetic-normal metal junction. (a) Thediffusive spin current is dominant where the magnetization structure S variesslowly as compared to the electron mean free path ℓ in normal metal. (b) Thelocal spin current is dominant where the magnetization structure variesrapidly at the interface.</figcaption></figure> In the spin pumping effect, we have demonstrated that the diffusive spin current is dominant in a slow-varying magnetization structure, subject to $\ell\ll\lambda$ ($\ell$ is the electron mean free path and $\lambda$ represents the length scale of the magnetization structure) as depicted in Fig. 3(a). In an actual spin pumping system, this condition would be satisfied since the mean-free path in such experimental systems is known to be very short, for example, less than 1 nm size in Ni${}_{81}$Fe${}_{19}$/Pt sample. [32] Strictly speaking, a local spin current can also exist but this contribution is negligibly small compared with the diffusive spin current. Therefore, in this situation, it is impossible that local and nonlocal spin currents coexist, as found in a study by Brataas _et al._ [5] The situation can be different with a clean sample or very sharp magnetization profile, subject to $\ell\gg\lambda$, as shown in Fig. 3(b). In this case, the local spin current without spin accumulation could become dominant and the diffusive spin current arising from spin accumulation is small. We therefore expect that the mechanism of Tserkovnayk _et al._ without spin accumulation may be valid. The spin current induced by the spin pumping effect, therefore, depends much on the disorder or the electron mean-free path. In this section, we did not take into account spin-orbit interactions. It is essential when we need to discuss spin-charge conversions, which we do in the following section. ## III spin-charge conversion We now consider the conversion mechanism of the pumped spin current into the charge current through the inverse spin Hall effect. In this section, we consider two types of spin-orbit interactions, the Rashba type and such interactions because of random impurity scattering, and we finally obtain the exact spin-charge conversion formula. ### Rashba spin-orbit interaction systems We first consider charge currents driven by Rashba spin-orbit interactions. The Rashba spin-orbit coupling was first found as a peculiar effect in a two-dimensional electron-gas system. [39] However recent studies have revealed that the Rashba system appears quite generally at surfaces of nonmagnetic materials without inversion symmetry and that this Rashba coupling can be quite large. [40; 41; 42; 43] Therefore, there is a possibility that these surface effects contribute greatly to the inverse spin Hall effect in ferromagnetic-normal metal junction systems. The microscopic theory in the presence of Rashba spin-orbit interactions was demonstrated by Ohe _et al._ [29] They considered a two-dimensional electron-gas system, where the maximum number of diffusion ladder needs to be taken into account. The result was extended to a three-dimensional and spatially dependent Rashba coupling case. [30] However, the account payed little attention to the charge conservation law because the analyses were intended only to see whether the spin-charge current conversion indeed occurs or not. In the following, we consider a three-dimensional system and evaluate the dominant contribution including one diffusion ladder as a vertex correction. By considering the vertex correction, we maintain charge conservation throughout the calculation. The Rashba spin-orbit interaction is given by \[H_{\rm so}=-\sum_{\bm{k}}{\bm{E}}_{\rm so}\cdot\big{[}{\bm{k}}\times\big{(}c^{ \dagger}_{\bm{k}}\hat{\bm{\sigma}}c_{\bm{k}}\big{)}\big{]},\] (21) where ${\bm{E}}_{\rm so}$ describes the spin-orbit field (or strength of the Rashba coupling). In the presence of this interaction, the anomalous velocity resulting from the spin-orbit interaction modifies the charge current. By defining the charge density as $\rho_{\rm c}({\bm{x}},t)\equiv-e{\rm tr}\big{\langle}{\psi^{\dagger}({\bm{x}}, t)\psi({\bm{x}},t)}\big{\rangle}_{H}$, the charge current density is given by ${\bm{j}}_{\rm c}={\bm{j}}_{\rm c}^{\rm n}+{\bm{j}}_{\rm c}^{\rm so}$, where the normal charge current ${\bm{j}}_{\rm c}^{\rm n}$ is \[{\bm{j}}_{\rm c}^{\rm n}({\bm{x}},t)=\frac{ie\hbar^{2}}{mV}\sum_{{\bm{k}},{\bm {q}}}e^{-i{\bm{q}}\cdot{\bm{x}}}{\rm tr}\Big{[}{\bm{k}}\hat{G}^{<}_{{\bm{k}}- \frac{{\bm{q}}}{2},{\bm{k}}+\frac{{\bm{q}}}{2}}(t,t)\Big{]},\] (22) and the correction from the Rashba coupling ${\bm{j}}_{\rm c}^{\rm so}$ is defined as \[{\bm{j}}_{\rm c}^{\rm so}({\bm{x}},t)=\frac{ie}{V}\sum_{{\bm{k}},{\bm{q}}}e^{- i{\bm{q}}\cdot{\bm{x}}}{\rm tr}\Big{[}\big{(}{\bm{E}}_{\rm so}\times\hat{\bm{ \sigma}}\big{)}\hat{G}^{<}_{{\bm{k}}-\frac{{\bm{q}}}{2},{\bm{k}}+\frac{{\bm{q} }}{2}}(t,t)\Big{]}.\] (23) We treat the Rashba spin-orbit interaction perturbatively by imposing the constraint $E_{\rm so}k_{\rm F}\ll\hbar/\tau$, with $k_{\rm F}$ being the Fermi wavelength. For slow-varying magnetization textures, the contribution from both the first-order Rashba and the first-order exchange interactions vanishes identically. [29; 30] (Strictly speaking, a contribution proportional to $\nabla E_{\rm so}\nabla\dot{S}$ arises if Rashba spin-orbit interactions are inhomogeneous. [30]) We now consider the charge current to first order in the Rashba coupling and second order in the exchange coupling. <figure><img src="content_image/0912.3607/x4.png"><figcaption>Figure 4: Dominant contributions to the pumped charge current bymagnetization dynamics in a Rashba system: (a) is the normal charge current,jnc, and (b) describes corrections to the charge current arising from theRashba coupling, jsoc. Double lines represent Rashba spin-orbit interactions(SOIs). The Rashba spin-orbit coupling gives the anomalous velocity to theconduction electrons, thereby modifying the definition of the charge current.</figcaption></figure> In Fig. 4, we present the Feynman diagrams for the dominant contribution which is calculated (see details in Appendix B) as \[j_{{\rm c}i}({\bm{x}},t)= \frac{ieJ^{2}\tau}{mV}\sum_{{\bm{k}},{\bm{k}}^{\prime},{\bm{q}},{ \bm{q}}^{\prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{\bm{x} }+i\Omega t}\Big{[}{\bm{E}}_{\rm so}\times\big{(}{\bm{S}}_{{\bm{q}}^{\prime}, \Omega^{\prime}}\times{\bm{S}}_{{\bm{q}}-{\bm{q}}^{\prime},\Omega-\Omega^{ \prime}}\big{)}\Big{]}_{j}\Omega^{\prime}f^{\prime}_{\omega}g^{\rm r}_{{\bm{k} },\omega}g^{\rm a}_{{\bm{k}},\omega}\] \[\times\bigg{\{}-\frac{m}{\hbar}\delta_{ij}+\frac{\hbar q_{i}q_{j} }{3\pi\nu}\Big{[}{\rm Im}\varepsilon_{{\bm{k}}^{\prime}}g^{\rm r}_{{\bm{k}}^{ \prime},\omega}(g^{\rm a}_{{\bm{k}}^{\prime},\omega})^{2}\Big{]}\Pi^{\rm ra}_{ {\bm{q}};\omega,\Omega}\bigg{\}}\] \[\simeq\] \[-\frac{4e\nu J^{2}\tau^{3}{\rm D}}{\hbar^{2}V}\nabla_{i}\Big{\{}{ \bm{\nabla}}\cdot\big{\langle}{{\bm{E}}_{\rm so}\times\big{[}{\bm{S}}({\bm{x}} ,t)\times\dot{{\bm{S}}}({\bm{x}},t)\big{]}}\big{\rangle}\Big{\}}.\] (24) This pumped charge current is expressible in terms of the dynamic component of the spin relaxation torque $\mathcal{T}_{\rm s}^{\rm(dy)}$ and the charge density [31] \[j_{{\rm c}i}({\bm{x}},t)=\epsilon_{ij\alpha}a^{\rm R}_{j}\mathcal{T}_{\rm s}^{ \rm(dy)\alpha}({\bm{x}},t)-{\rm D}\nabla_{i}\rho_{\rm c}({\bm{x}},t),\] (25) where ${\bm{a}}^{\rm R}\equiv-2e\tau{\bm{E}}_{\rm so}/\hbar^{2}$ and \[\begin{split}\mathcal{T}_{\rm s}^{\rm(dy)\alpha}({\bm{x}},t)& =\frac{2\nu J^{2}\tau}{V}\big{[}{\bm{S}}({\bm{x}},t)\times\dot{{ \bm{S}}}({\bm{x}},t)\big{]}^{\alpha},\\ \rho_{\rm c}({\bm{x}},t)&=\frac{2\nu J^{2}\tau^{2}}{ V}{\bm{a}}^{\rm R}\cdot\big{\langle}{{\bm{\nabla}}\times\big{[}{\bm{S}}({\bm{x }},t)\times\dot{{\bm{S}}}({\bm{x}},t)\big{]}}\big{\rangle}.\end{split}\] (26) We first note that Eqs. (25) and (26) indicate that only the damping of the local spin, ${\bm{S}}\times\dot{{\bm{S}}}$, is converted into a charge current by Rashba coupling. The equilibrium spin current, Eq. (17), does not contribute, as is reasonable from energy conservation considerations. Since ${\bm{j}}_{\rm c}$ is expressible in terms of $\mathcal{T}_{\rm s}^{\rm(dy)}$ and $\rho_{\rm c}$, the naive formula for the inverse spin Hall effect, i.e. $j_{\rm c}$ proportional locally to $j_{\rm s}$, does not hold in disordered Rashba systems. Instead, Eq. (25) indicates that the generation mechanism of the local charge current in a Rashba system is the inverse effect of the spin-relaxation torque. The non-local part in Eq. (25) arises from a diffusion of the charge density, which is written in terms of the dynamic component of the pumped spin current [Eq. (7)], \[\rho_{\rm c}({\bm{x}},t)=-\frac{1}{{\rm D}}\epsilon_{ij\alpha}a^{\rm R}_{i}j_{ {\rm s}j}^{\rm p(2)\alpha}({\bm{x}},t).\] (27) Therefore, the (diffusive) spin current induces a charge polarization via Rashba spin-orbit interactions, but not a charge current itself. Since diffusive spin currents dominate in slow-varying magnetization structures, we expect that the ratio of the charge current to the pumped spin current is high [see Fig. 5(a)]. <figure><img src="content_image/0912.3607/x5.png"><figcaption>Figure 5: (Color online) Conversion mechanism of the pumped spin current intoa charge current via spin-orbit interactions (SOIs). (a) In the Rashba system,the diffusive spin current is just converted into a charge current. (b) Withspin-orbit interactions caused by random impurity scattering, the diffusivespin current is present but not converted. The local spin current is convertedinto a charge current.</figcaption></figure> ### Random impurity-induced spin-orbit interaction systems We now focus on spin-orbit interactions caused by random impurities. This interaction is defined as \[H_{\rm so}=\frac{iu_{\rm i}\lambda_{\rm so}}{V}\sum_{n}\sum_{{\bm{k}},{\bm{p}} }e^{i{\bm{p}}\cdot{\bm{r}}_{n}}({\bm{k}}\times{\bm{p}})\cdot\big{(}c^{\dagger} _{{\bm{k}}-{\bm{p}}}\hat{\bm{\sigma}}c_{\bm{k}}\big{)},\] (28) where $\lambda_{\rm so}$ is the spin-orbit strength. We have here assumed that the random impurity spin-orbit interaction arises from the same impurities giving rise to the electron lifetime $\tau$. In this case, the correction current resulting from the spin-orbit interaction is given by (29) As shown in Ref. [28], the charge current is derived in the form \[j_{{\rm c}i}({\bm{x}},t)=a^{\rm imp}\epsilon_{ij\alpha}j_{{\rm s}j}^{\rm(local )\alpha}({\bm{x}},t)-{\rm D}\nabla_{i}\rho_{\rm c}({\bm{x}},t),\] (30) where $a^{\rm imp}\equiv 2e\lambda_{\rm so}k_{\rm F}^{2}/\varepsilon_{\rm F}\tau$ and (31) We have here denoted the local spin current as ${\bm{j}}_{\rm s}^{\rm(local)}$. Equation (30) seems consistent with the naive formula of the inverse spin Hall effect, ${\bm{j}}_{\rm c}\propto{\bm{\sigma}}\times{\bm{j}}_{\rm s}$. However, one should note that this local spin current is a very small correction to the dominant spin current [given by $-{\rm D}{\bm{\nabla}}\rho_{\rm s}$ in Eq. (16)]. Therefore, most of the spin currents generated by the spin pumping effect are not converted into a charge current; only a small fraction [of order $(q\ell)^{2}\ll 1$] develops into a charge current [Fig 5(b)]. Thus, the spin-charge conversion efficiency is small in the presence of random impurity-induced spin-orbit interactions. ### Conversion mechanism By comparing Eqs. (25) and (30) to the general formulation of the charge current [Eq. (18)], the spin relaxation torque $\mathcal{T}_{\rm s}^{\rm(dy)}$ and the local spin current ${\bm{j}}_{\rm s}^{\rm(local)}$ act as effective electric fields for Rashba and random impurity-induced spin-orbit interactions, respectively. Therefore, as we mentioned above, the spin current changes only the constitutive relations associated with electromagnetic fields but does not change the Maxwell’s equations themselves. Correctly, there is a spin current caused by spin-orbit interactions. This spin current, however, produces a second-order charge current with respect to spin-orbit interactions at least. It should be negligible compared to the above results. From the above results, we see that spin accumulation at the interface plays a crucial role in determining spin pumping and spin-charge conversion mechanisms. In fact, the pumped charge current is proportional to the spin accumulation when Rashba interactions are present, but does not occur with the spin accumulation for random impurity-induced spin-orbit interactions. Therefore, measuring spin accumulation at the interface would provide impetus to determine the dominant spin-orbit interaction and to clarify the spin-charge conversion mechanism. ## IV conclusions We have studied aspects of spin pumping and the spin-charge conversion mechanism through spin-orbit interactions in the disordered electron system. We showed that the spin current generated by the spin dynamics is a diffusive process arising from a dynamics-induced spin accumulation. There is, therefore, no effective field that drives the spin current directly in the disordered case. We have confirmed that a charge current is induced by these spin-orbit interactions. This process involves the conversion of a pumped spin current into a charge transport, but the mechanism has turned out not to be as simple as an earlier phenomenological proposal, ${\bm{j}}_{\rm c}\propto({\bm{\sigma}}\times{\bm{j}}_{\rm s})$, [17; 20; 21] had anticipated. In fact, the spin-charge conversion depends largely on the type of spin-orbit interaction. For Rashba spin-orbit interactions, the charge current is given by a local contribution proportional to the spin relaxation torque and a diffusive contribution arising from the diffusive spin current. Therefore, the naive formula for the inverse spin Hall effect does not hold in the Rashba systems. In contrast, for random impurity-induced spin-orbit interactions, the local part of the charge current is written as a very small fraction of the spin current [smaller by $\mathcal{O}(\ell^{2}/\lambda^{2})$, where $\ell$ and $\lambda$ are the electron mean-free path and the coherence length scale of the magnetization, respectively]. However, the dominant spin current is not converted into a charge current in the presence of impurities. Thus, the naive inverse spin Hall effect does not occur either. Our result indicates that the spin-charge conversion formula as proposed earlier using phenomenological arguments is too simple and the whole phenomenon needs discussing together with the origin of spin currents. ## Appendix A details of calculations for the spin pumping effect We perform here calculations of the pumped spin current using standard perturbation expansion techniques. We treat the exchange coupling up to the second order. The electron spin and its relaxation torque densities are defined in terms of Green’s function as \[\begin{split}\rho_{\rm s}^{\alpha}({\bm{x}},t)&=- \frac{i\hbar}{2V}\sum_{{\bm{k}},{\bm{q}}}e^{-i{\bm{q}}\cdot{\bm{x}}}{\rm tr} \Big{[}\hat{\sigma}^{\alpha}\hat{G}^{<}_{{\bm{k}}-\frac{{\bm{q}}}{2},{\bm{k}}+ \frac{{\bm{q}}}{2}}(t,t)\Big{]},\\ \mathcal{T}_{\rm s}^{\alpha}({\bm{x}},t)&=-\frac{i \hbar J}{V}\sum_{{\bm{k}},{\bm{q}}}e^{-i{\bm{q}}\cdot{\bm{x}}}{\rm tr}\Big{\{} \big{[}\hat{\bm{\sigma}}\times{\bm{S}}({\bm{x}},t)\big{]}^{\alpha}\hat{G}^{<}_ {{\bm{k}}-\frac{{\bm{q}}}{2},{\bm{k}}+\frac{{\bm{q}}}{2}}(t,t)\Big{\}},\end{split}\] (32) respectively. Before carrying out the calculation, we introduce the Dyson equation: \[G_{{\bm{k}}\alpha,{\bm{k}}^{\prime}\alpha^{\prime}}(t,t^{\prime})= \delta_{{\bm{k}}{\bm{k}}^{\prime}}\delta_{\alpha\alpha^{\prime}}g _{{\bm{k}}\alpha}(t,t^{\prime})\] \[+\frac{u_{\rm i}}{V}\int_{\rm C_{K}}{dT}\sum_{n}\sum_{\bm{p}}e^{i {\bm{p}}\cdot{\bm{r}}_{n}}g_{{\bm{k}}\alpha}(t,T)G_{{\bm{k}}+{\bm{p}}\alpha,{ \bm{k}}^{\prime}\alpha^{\prime}}(T,t^{\prime})\] \[-J\int_{\rm C_{K}}{dT}\sum_{\beta}\sum_{\bm{q}}g_{{\bm{k}}\alpha} (t,T)\big{[}{\bm{\sigma}}_{\alpha\beta}\cdot{\bm{S}}_{\bm{q}}(T)\big{]}G_{{\bm {k}}+{\bm{q}}\beta,{\bm{k}}^{\prime}\alpha^{\prime}}(T,t^{\prime}),\] (33) where $G_{{\bm{k}}\alpha,{\bm{k}}^{\prime}\alpha^{\prime}}(t,t^{\prime})\equiv-(i/ \hbar)\big{\langle}{{\rm T}_{\rm C_{K}}\big{[}c_{{\bm{k}}\alpha}(t)c^{\dagger} _{{\bm{k}}^{\prime}\alpha^{\prime}}(t^{\prime})\big{]}}\big{\rangle}_{H}$ (${\rm T}_{\rm C_{K}}$ being the path-ordering operator defined on the Keldysh contour ${\rm C_{K}}$) and $g$ is the free Green’s function which is obtained from the free Hamiltonian $H_{0}$. The Dyson equation is very useful in carrying out the perturbation expansion because this equation can be solved iteratively. Here we assume a weak exchange coupling regime, $J\ll\hbar/\tau$, and therefore we can treat the exchange interaction perturbatively. To evaluate the lesser component of $G(t,t^{\prime})=\int_{\rm C_{K}}{dT}G_{1}(t,T)G_{2}(T,t^{\prime})$, we use the following [44] \[\begin{split} G^{<}(t,t^{\prime})&=\int_{-\infty}^{ \infty}{dT}\Big{[}G_{1}^{\rm r}(t,T)G_{2}^{<}(T,t^{\prime})+G_{1}^{<}(t,T)G_{2 }^{\rm a}(T,t^{\prime})\Big{]},\\ G^{\rm a(r)}(t,t^{\prime})&=\int_{-\infty}^{\infty} {dT}\Big{[}G_{1}^{\rm a(r)}(t,T)G_{2}^{\rm a(r)}(T,t^{\prime})+G_{1}^{\rm a(r) }(t,T)G_{2}^{\rm a(r)}(T,t^{\prime})\Big{]}.\end{split}\] (34) The lesser component of the free Green’s function satisfies $g_{{\bm{k}},\omega}^{<}=f_{\omega}\big{(}g^{\rm a}_{{\bm{k}},\omega}-g^{\rm r} _{{\bm{k}},\omega}\big{)}$. ### First-order calculations in $J$ <figure><img src="content_image/0912.3607/x6.png"><figcaption>Figure 6: The lesser component of the Green’s function involving the vertexcorrection can be divided into the three terms. The diagram is partitioned offby defining the diffusion ladder as the boundary. We denote contributions fromthe left-hand side as Γ, the middle representing the diffusion ladder as Π,and the right-hand side as Λ. The spiral line represents all interactions andtherefore the contribution from Λ depends on the interactions.</figcaption></figure> First, we show the calculation of the spin current to first-order in the exchange interaction. The diagram in Fig. 2(a) is written as \[{\bm{j}}_{\rm s}^{(1)\alpha}({\bm{x}},t)= \frac{i\hbar^{3}J}{2mV}\sum_{n=0}^{\infty}\bigg{(}\frac{n_{\rm i} u_{\rm i}^{2}}{V}\bigg{)}^{n}\sum_{\bm{q}}e^{-i{\bm{q}}\cdot{\bm{x}}}\sum_{\{{ \bm{k}}_{i}\}_{i=0}^{n}}{\rm tr}\Bigg{\{}{\bm{k}}_{0}\hat{\sigma}^{\alpha} \Bigg{[}\prod_{i=0}^{n}\int_{{\rm C}_{\rm K}^{i+1}}{dt_{i+1}}\hat{g}_{{\bm{k}} _{i}-\frac{{\bm{q}}}{2}}(t_{i},t_{i+1})\Bigg{]}\] (35) By using ${\rm tr}\big{(}\hat{\sigma}^{\alpha}\hat{\sigma}^{\beta}\big{)}=2\delta_{ \alpha\beta}$ and taking the lesser component, the equation reads \[{\bm{j}}_{\rm s}^{(1)\alpha}({\bm{x}},t)= \frac{i\hbar^{3}J}{mV}\sum_{{\bm{k}},{\bm{k}}^{\prime},{\bm{q}}} \sum_{\omega,\Omega}e^{-i{\bm{q}}\cdot{\bm{x}}+i\Omega t}S^{\alpha}_{{\bm{q}}, \Omega}{\bm{k}}\Big{(}\Lambda^{\rm aa(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}- \Lambda^{\rm rr(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}+\Lambda^{\rm ra(1)}_{{ \bm{k}},{\bm{q}};\omega,\Omega}\] \[+\Gamma^{\rm aa}_{{\bm{k}},{\bm{q}};\omega,\Omega}\Pi^{\rm aa}_{{ \bm{q}};\omega,\Omega}\Lambda^{\rm aa(1)}_{{\bm{k}}^{\prime},{\bm{q}};\omega, \Omega}-\Gamma^{\rm rr}_{{\bm{k}},{\bm{q}};\omega,\Omega}\Pi^{\rm rr}_{{\bm{q} };\omega,\Omega}\Lambda^{\rm rr(1)}_{{\bm{k}}^{\prime},{\bm{q}};\omega,\Omega} +\Gamma^{\rm ra}_{{\bm{k}},{\bm{q}};\omega,\Omega}\Pi^{\rm ra}_{{\bm{q}}; \omega,\Omega}\Lambda^{\rm ra(1)}_{{\bm{k}}^{\prime},{\bm{q}};\omega,\Omega} \Big{)}.\] (36) A diagram involving the vertex correction is divided into three parts: the left-hand side, the diffusion ladder (middle), and the right-hand side of the diagram shown in Fig. 6. The contribution from the left hand side is given as \[\begin{split}\Gamma^{\rm aa(rr)}_{{\bm{k}},{\bm{q}};\omega,\Omega }&\equiv\frac{n_{\rm i}u_{\rm i}^{2}}{V}g^{\rm a(r)}_{{\bm{k}}- \frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}g^{\rm a(r)}_{{\bm{k}}+\frac{{\bm{q }}}{2},\omega+\frac{\Omega}{2}},\\ \Gamma^{\rm ra}_{{\bm{k}},{\bm{q}};\omega,\Omega}& \equiv\frac{n_{\rm i}u_{\rm i}^{2}}{V}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2}, \omega-\frac{\Omega}{2}}g^{\rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+\frac{ \Omega}{2}}.\end{split}\] (37) The diffusion ladder arising from the vertex correction is written as \[\begin{split}\Pi^{\rm aa(rr)}_{{\bm{q}};\omega,\Omega}& \equiv\sum_{n=0}^{\infty}\bigg{(}\sum_{\bm{k}}\Gamma^{\rm aa(rr)} _{{\bm{k}},{\bm{q}};\omega,\Omega}\bigg{)}^{n},\\ \Pi^{\rm ra}_{{\bm{q}};\omega,\Omega}&\equiv\sum_{n= 0}^{\infty}\bigg{(}\sum_{\bm{k}}\Gamma^{\rm ra}_{{\bm{k}},{\bm{q}};\omega, \Omega}\bigg{)}^{n}.\end{split}\] (38) The contribution from the right-hand side of a diagram depends on the diagram and in Fig. 2(a) is given by \[\begin{split}\Lambda^{\rm aa(rr)(1)}_{{\bm{k}},{\bm{q}};\omega, \Omega}&\equiv f_{\omega-(+)\frac{\Omega}{2}}g^{\rm a(r)}_{{\bm{k }}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}g^{\rm a(r)}_{{\bm{k}}+\frac{{ \bm{q}}}{2},\omega+\frac{\Omega}{2}},\\ \Lambda^{\rm ra(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}& \equiv\big{(}f_{\omega+\frac{\Omega}{2}}-f_{\omega-\frac{\Omega}{ 2}}\big{)}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}g^{ \rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+\frac{\Omega}{2}}.\end{split}\] (39) Assuming slow dynamics $\Omega\tau\ll 1$ and a spatially smooth local spin structure $q\ell\ll 1$, we obtain the result \[{\bm{j}}_{\rm s}^{(1)\alpha}({\bm{x}},t)\] \[\simeq\frac{\hbar\nu J\tau{\rm D}}{V}{\bm{\nabla}}\big{\langle}{ \dot{S}^{\alpha}({\bm{x}},t)}\big{\rangle}.\] (40) Here we note simplifications in the $k$ summation \[\sum_{\bm{k}}g^{\rm r}_{\bm{k}}g^{\rm a}_{\bm{k}} \simeq\frac{2\pi\nu\tau}{\hbar},\] (41) \[\sum_{\bm{k}}\varepsilon_{\bm{k}}g^{\rm r}_{\bm{k}}(g^{\rm a}_{ \bm{k}})^{2} \simeq\frac{i2\pi\nu\varepsilon_{\rm F}\tau^{2}}{\hbar^{2}},\] (42) where we have put $g_{\bm{k}}\equiv g_{{\bm{k}},\hbar\omega=\varepsilon_{\rm F}}$. The diffusion ladder arising from the vertex correction is also given as \[\Pi^{\rm aa(rr)}_{{\bm{q}};0,\Omega}\simeq 1,\] (43) \[\Pi^{\rm ra}_{{\bm{q}},0,\Omega}\simeq \sum_{n=0}^{\infty}\Bigg{\{}\frac{n_{\rm i}u_{\rm i}^{2}}{V}\sum_ {\bm{k}}\Bigg{[}\big{(}1-i\tau\Omega\big{)}g^{\rm r}_{\bm{k}}g^{\rm a}_{\bm{k} }-\frac{2\hbar\tau{\bm{q}}^{2}}{3m}{\rm Im}\varepsilon_{\bm{k}}g^{\rm r}_{\bm{ k}}(g^{\rm a}_{\bm{k}})^{2}\Bigg{]}\Bigg{\}}^{n}\] \[\simeq \sum_{n=0}\Big{(}1-{\rm D}{\bm{q}}^{2}\tau-i\Omega\tau\Big{)}^{n}\] \[= \frac{1}{{\rm D}{\bm{q}}^{2}\tau+i\Omega\tau}.\] (44) Since the product of only $g^{\rm a}$ (or $g^{\rm r}$) gives a very small contribution which is of order $1/\varepsilon_{\rm F}$ compared to the coefficient of $g^{\rm a}$ and $g^{\rm r}$, the diffusion ladder reduces approximately to unity. We cannot, however, ignore the product of only $g^{\rm a}$ (or $g^{\rm r}$) completely because that contribution corresponds to an equilibrium current. Similarly, the spin density is also calculated in the form \[\rho_{\rm s}^{(1)\alpha}({\bm{x}},t) =\frac{i\hbar^{2}J}{V}\sum_{{\bm{k}},{\bm{q}}}\sum_{\omega,\Omega }e^{-i{\bm{q}}\cdot{\bm{x}}+i\Omega t}S^{\alpha}_{{\bm{q}},\Omega}\Big{(} \Lambda^{\rm aa(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}-\Lambda^{\rm rr(1)}_{{ \bm{k}},{\bm{q}};\omega,\Omega}+\Pi^{\rm ra}_{{\bm{q}};\omega,\Omega}\Lambda^{ \rm ra(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}\Big{)}\] \[\simeq\frac{i\hbar^{2}J}{V}\sum_{{\bm{k}},{\bm{q}}}\sum_{\omega, \Omega}e^{i{\bm{q}}\cdot{\bm{x}}+i\Omega t}S^{\alpha}_{{\bm{q}},\Omega}f^{ \prime}_{\omega}\bigg{(}\frac{i}{\tau}+\Omega\Pi^{\rm ra}_{{\bm{q}};\omega, \Omega}\bigg{)}g^{\rm r}_{{\bm{k}},\omega}g^{\rm a}_{{\bm{k}},\omega}\] (45) To first order in the exchange coupling, the spin-relaxation torque corresponding to diagram Fig. 7(a) vanishes as a consequence of ${\rm tr}\hat{\sigma}^{\alpha}=0$. Therefore, the pumped spin current to first order in $J$ follows \[\dot{\rho}_{\rm s}^{(1)\alpha}({\bm{x}},t)+{\bm{\nabla}}\cdot{\bm{j}}_{\rm s}^ {(1)\alpha}({\bm{x}},t)=0.\] (46) Hence, the spin of conduction electrons is conserved. <figure><img src="content_image/0912.3607/x7.png"><figcaption>Figure 7: Diagrammatic representations of the spin-relaxation torque of theconduction electrons. (a) This contribution comes from first-order terms in Jbut vanishes. (b) The leading contribution arises from second-order exchangeinteraction terms. This contribution corresponds to local spin damping S×˙S.</figcaption></figure> ### Second-order calculations in $J$ Here we derive the spin current to second order in the exchange coupling as shown in Fig. 2(b). This is calculated in the same manner as the first-order calculations. The pumped spin current reads \[{\bm{j}}_{\rm s}^{(2)\alpha}({\bm{x}},t)= \frac{\hbar^{3}J^{2}}{mV}\sum_{{\bm{k}},{\bm{k}}^{\prime},{\bm{q} },{\bm{q}}^{\prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{\bm {x}}+i\Omega t}\big{(}{\bm{S}}_{{\bm{q}}^{\prime},\Omega^{\prime}}\times{\bm{S }}_{{\bm{q}}-{\bm{q}}^{\prime},\Omega-\Omega^{\prime}}\big{)}^{\alpha}{\bm{k}} \Big{(}\Lambda^{\rm aa(2)}_{{\bm{k}},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega, \Omega^{\prime}}\] \[-\Lambda^{\rm rr(2)}_{{\bm{k}},{\bm{q}},{\bm{q}}^{\prime};\omega, \Omega,\Omega^{\prime}}+\Lambda^{\rm ra(2)}_{{\bm{k}},{\bm{q}},{\bm{q}}^{ \prime};\omega,\Omega,\Omega^{\prime}}+\Gamma^{\rm ra}_{{\bm{k}},{\bm{q}}; \omega,\Omega}\Pi^{\rm ra}_{{\bm{q}};\omega,\Omega}\Lambda^{\rm ra(2)}_{{\bm{k }}^{\prime},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}\Big{)}\] \[\simeq \frac{\hbar^{2}J^{2}}{3mV}\sum_{{\bm{k}},{\bm{k}}^{\prime},{\bm{q }},{\bm{q}}^{\prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{ \bm{x}}+i\Omega t}\big{(}{\bm{S}}_{{\bm{q}}^{\prime},\Omega^{\prime}}\times{ \bm{S}}_{{\bm{q}}-{\bm{q}}^{\prime},\Omega-\Omega^{\prime}}\big{)}^{\alpha}f^{ \prime}_{\omega}\bigg{\{}i{\bm{q}}^{\prime}{\rm Im}(g^{\rm a}_{{\bm{k}},\omega })^{2}\] \[-4\tau\Omega^{\prime}\Big{[}{\rm Im}\varepsilon_{\bm{k}}g^{\rm r} _{{\bm{k}},\omega}(g^{\rm a}_{{\bm{k}},\omega})^{2}\Big{]}\bigg{[}({\bm{q}}+{ \bm{q}}^{\prime})+\frac{n_{\rm i}u_{\rm i}^{2}{\bm{q}}}{V}\Pi^{\rm ra}_{{\bm{q }};\omega,\Omega}g^{\rm r}_{{\bm{k}}^{\prime},\omega}g^{\rm a}_{{\bm{k}}^{ \prime},\omega}\bigg{]}\bigg{\}}\] \[\simeq -\frac{\hbar^{2}\nu J^{2}}{12m\varepsilon_{\rm F}V}\big{[}{\bm{S} }({\bm{x}},t)\times{\bm{\nabla}}{\bm{S}}({\bm{x}},t)\big{]}^{\alpha}-\frac{2 \nu J^{2}\tau^{2}{\rm D}}{V}{\bm{\nabla}}\big{\langle}{\big{[}{\bm{S}}({\bm{x} },t)\times\dot{{\bm{S}}}({\bm{x}},t)\big{]}^{\alpha}}\big{\rangle}.\] (47) In this case, we should pay particular attention to ${\rm tr}\big{(}\hat{\sigma}^{\alpha}\hat{\sigma}^{\beta}\hat{\sigma}^{\gamma} \big{)}=i2\epsilon_{\alpha\beta\gamma}$ and the contribution from the right-hand side of the diagram being given as \[\begin{split}\Lambda^{\rm aa(rr)(2)}_{{\bm{k}},{\bm{q}},{\bm{q}}^ {\prime};\omega,\Omega,\Omega^{\prime}}\equiv& f_{\omega-(+)\frac {\Omega}{2}}g^{\rm a(r)}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}} g^{\rm a(r)}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega-\frac{ \Omega-2\Omega^{\prime}}{2}}g^{\rm a(r)}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+ \frac{\Omega}{2}},\\ \Lambda^{\rm ra(2)}_{{\bm{k}},{\bm{q}}.{\bm{q}}^{\prime};\omega, \Omega,\Omega^{\prime}}\equiv&\Big{(}f_{\omega-\frac{\Omega-2 \Omega^{\prime}}{2}}-f_{\omega-\frac{\Omega}{2}}\Big{)}g^{\rm r}_{{\bm{k}}- \frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}g^{\rm a}_{{\bm{k}}-\frac{{\bm{q}}- 2{\bm{q}}^{\prime}}{2},\omega-\frac{\Omega-2\Omega^{\prime}}{2}}g^{\rm a}_{{ \bm{k}}+\frac{{\bm{q}}}{2},\omega+\frac{\Omega}{2}}\\ &+\Big{(}f_{\omega+\frac{\Omega}{2}}-f_{\omega-\frac{\Omega-2 \Omega^{\prime}}{2}}\Big{)}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac {\Omega}{2}}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega- \frac{\Omega-2\Omega^{\prime}}{2}}g^{\rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2}, \omega+\frac{\Omega}{2}}.\end{split}\] (48) The pumped spin current contains an equilibrium flow given by estimating $(g^{\rm a})^{2}$, \[\sum_{\bm{k}}(g^{\rm a}_{\bm{k}})^{2}\simeq-\frac{i\pi\nu}{2\varepsilon_{\rm F }}.\] (49) The spin density is calculated in a similar manner \[\rho_{\rm s}^{(2)\alpha}({\bm{x}},t)=\] \[\times\Big{(}\Lambda^{\rm aa(2)}_{{\bm{k}},{\bm{q}},{\bm{q}}^{ \prime};\omega,\Omega,\Omega^{\prime}}-\Lambda^{\rm rr(2)}_{{\bm{k}},{\bm{q}}, {\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}+\Pi^{\rm ra}_{{\bm{q}};\omega ,\Omega}\Lambda^{\rm ra(2)}_{{\bm{k}},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega ,\Omega^{\prime}}\Big{)}\] \[\simeq \frac{i2\hbar J^{2}\tau}{V}\sum_{{\bm{k}},{\bm{q}},{\bm{q}}^{ \prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{\bm{x}}+i\Omega t }\big{(}{\bm{S}}_{{\bm{q}}^{\prime},\Omega^{\prime}}\times{\bm{S}}_{{\bm{q}}-{ \bm{q}}^{\prime},\Omega-\Omega^{\prime}}\big{)}^{\alpha}\Omega^{\prime}f^{ \prime}_{\omega}\Pi^{\rm ra}_{{\bm{q}};\omega,\Omega}g^{\rm r}_{{\bm{k}}, \omega}g^{\rm a}_{{\bm{k}},\omega}\] \[\simeq \frac{2\nu J^{2}\tau^{2}}{V}\big{\langle}{\big{[}{\bm{S}}({\bm{x} },t)\times\dot{{\bm{S}}}({\bm{x}},t)\big{]}^{\alpha}}\big{\rangle}.\] (50) To second-order in the exchange coupling, the spin of conduction electrons is not conserved and follows the general spin continuity equation \[\dot{\rho}_{\rm s}^{(2)\alpha}({\bm{x}},t)+{\bm{\nabla}}\cdot{\bm{j}}_{\rm s}^ {(2)\alpha}({\bm{x}},t)=\mathcal{T}_{\rm s}^{(2)\alpha}({\bm{x}},t).\] (51) Here the spin relaxation torque is represented by the diagram in Fig. 7(b) and we obtain \[\mathcal{T}_{\rm s}^{(2)\alpha}({\bm{x}},t)= -\frac{i2\hbar J^{2}}{V}\sum_{{\bm{k}},{\bm{q}}}\sum_{\omega, \Omega}e^{-i{\bm{q}}\cdot{\bm{x}}+i\Omega t}\big{[}{\bm{S}}({\bm{x}},t)\times{ \bm{S}}_{{\bm{q}},\Omega}\big{]}^{\alpha}\Big{(}\Lambda^{\rm aa(1)}_{{\bm{k}}, {\bm{q}};\omega,\Omega}-\Lambda^{\rm rr(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}+ \Lambda^{\rm ra(1)}_{{\bm{k}},{\bm{q}};\omega,\Omega}\Big{)}\] \[\simeq -\frac{i2\hbar J^{2}}{V}\sum_{{\bm{k}},{\bm{q}}}\sum_{\omega, \Omega}e^{-i{\bm{q}}\cdot{\bm{x}}+i\Omega t}\big{[}{\bm{S}}({\bm{x}},t)\times{ \bm{S}}_{{\bm{q}},\Omega}\big{]}^{\alpha}f^{\prime}_{\omega}\bigg{[}\frac{i \hbar{\bm{q}}^{2}}{6m}{\rm Im}(g^{\rm a}_{{\bm{k}},\omega})^{2}+\Omega g^{\rm r }_{{\bm{k}},\omega}g^{\rm a}_{{\bm{k}},\omega}\bigg{]}\] \[\simeq -\frac{\hbar^{2}\nu J^{2}}{12m\varepsilon_{\rm F}V}\big{[}{\bm{S} }({\bm{x}},t)\times{\bm{\nabla}}^{2}{\bm{S}}({\bm{x}},t)\big{]}^{\alpha}+\frac {2\nu J^{2}\tau}{V}\big{[}{\bm{S}}({\bm{x}},t)\times\dot{{\bm{S}}}({\bm{x}},t) \big{]}^{\alpha}.\] (52) ## Appendix B Magnetization-pumped charge current in a Rashba system Here, we calculate the charge current stemming from magnetization pumping in the presence of Rashba spin-orbit interactions. In the Rashba system, the Dyson equation is modified as \[G_{{\bm{k}}\alpha,{\bm{k}}^{\prime}\alpha^{\prime}}(t,t^{\prime})= \delta_{{\bm{k}}{\bm{k}}^{\prime}}\delta_{\alpha\alpha^{\prime}}g _{{\bm{k}}\alpha}(t,t^{\prime})\] \[+\frac{u_{\rm i}}{V}\int_{\rm C_{K}}{dT}\sum_{n}\sum_{\bm{p}}e^{i {\bm{p}}\cdot{\bm{r}}_{n}}g_{{\bm{k}}\alpha}(t,T)G_{{\bm{k}}+{\bm{p}}\alpha,{ \bm{k}}^{\prime}\alpha^{\prime}}(T,t^{\prime})\] \[-J\int_{\rm C_{K}}{dT}\sum_{\beta}\sum_{\bm{q}}g_{{\bm{k}}\alpha} (t,T)\big{[}{\bm{\sigma}}_{\alpha\beta}\cdot{\bm{S}}_{\bm{q}}(T)\big{]}G_{{\bm {k}}+{\bm{q}}\beta,{\bm{k}}^{\prime}\alpha^{\prime}}(T,t^{\prime}).\] \[-\int_{\rm C_{K}}{dT}\sum_{\beta}\sum_{\bm{q}}g_{{\bm{k}}\alpha}( t,T)\big{[}{\bm{E}}_{\rm so}\cdot({\bm{k}}\times{\bm{\sigma}}_{\alpha\beta}) \big{]}G_{{\bm{k}}\beta,{\bm{k}}^{\prime}\alpha^{\prime}}(T,t^{\prime}).\] (53) Since the spin-orbit interactions are generally weak, subject to $E_{\rm so}k_{\rm F}\ll\hbar/\tau$, we perform the perturbation expansion with respect to both the exchange interaction and the Rashba spin-orbit interaction. By iteration, we obtain the leading contribution shown in Fig. 4, \[j_{{\rm c}i}({\bm{x}},t)= -\frac{2eJ^{2}}{V}\sum_{{\bm{k}},{\bm{k}}^{\prime},{\bm{q}},{\bm{ q}}^{\prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{\bm{x}}+i \Omega t}\Big{[}{\bm{E}}_{\rm so}\times\big{(}{\bm{S}}_{{\bm{q}}^{\prime}, \Omega^{\prime}}\times{\bm{S}}_{{\bm{q}}-{\bm{q}}^{\prime},\Omega-\Omega^{ \prime}}\big{)}\Big{]}_{j}\] \[\times\bigg{[}\frac{\hbar^{2}k_{i}}{m}\Big{(}\tilde{\Lambda}_{j;{ \bm{k}},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}^{\rm aa(2)}- \tilde{\Lambda}_{j;{\bm{k}},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{ \prime}}^{\rm rr(2)}+\tilde{\Lambda}_{j;{\bm{k}},{\bm{q}},{\bm{q}}^{\prime}; \omega,\Omega,\Omega^{\prime}}^{\rm ra(2)}+\Gamma^{\rm ra}_{{\bm{k}},{\bm{q}}; \omega,\Omega}\Pi^{\rm ra}_{{\bm{q}};\omega,\Omega}\tilde{\Lambda}_{j;{\bm{k}} ^{\prime},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}^{\rm ra(2) }\Big{)}\] \[+\delta_{ij}\Big{(}\Lambda^{\rm aa(2)}_{{\bm{k}},{\bm{q}},{\bm{q} }^{\prime};\omega,\Omega,\Omega^{\prime}}-\Lambda^{\rm rr(2)}_{{\bm{k}},{\bm{q }},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}+\Lambda^{\rm ra(2)}_{{\bm{ k}},{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}\Big{)}\bigg{]}\] \[\simeq \frac{ieJ^{2}\tau}{mV}\sum_{{\bm{k}},{\bm{k}}^{\prime},{\bm{q}},{ \bm{q}}^{\prime}}\sum_{\omega,\Omega,\Omega^{\prime}}e^{-i{\bm{q}}\cdot{\bm{x} }+i\Omega t}\Big{[}{\bm{E}}_{\rm so}\times\big{(}{\bm{S}}_{{\bm{q}}^{\prime}, \Omega^{\prime}}\times{\bm{S}}_{{\bm{q}}-{\bm{q}}^{\prime},\Omega-\Omega^{ \prime}}\big{)}\Big{]}_{j}\Omega^{\prime}f^{\prime}_{\omega}g^{\rm r}_{{\bm{k} },\omega}g^{\rm a}_{{\bm{k}},\omega}\] \[\times\bigg{\{}-\frac{m}{\hbar}\delta_{ij}+\frac{\hbar q_{i}q_{j} }{3\pi\nu}\Big{[}{\rm Im}\varepsilon_{{\bm{k}}^{\prime}}g^{\rm r}_{{\bm{k}}^{ \prime},\omega}(g^{\rm a}_{{\bm{k}}^{\prime},\omega})^{2}\Big{]}\Pi^{\rm ra}_{ {\bm{q}};\omega,\Omega}\bigg{\}}\] \[\simeq\] (54) Here we have defined contributions from the right-hand side of the diagrams as \[\begin{split}\tilde{\Lambda}^{\rm aa(rr)(2)}_{i;{\bm{k}},{\bm{q}} ,{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}\equiv& f_{ \omega-(+)\frac{\Omega}{2}}\bigg{[}\frac{m}{\hbar^{2}}\frac{\partial}{\partial k _{i}}\Big{(}g^{\rm a(r)}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}} g^{\rm a(r)}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega-\frac{ \Omega-2\Omega^{\prime}}{2}}g^{\rm a(r)}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+ \frac{\Omega}{2}}\Big{)}\\ &-2\bigg{(}{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2}\bigg{)} _{i}g^{\rm a(r)}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}\Big{(}g ^{\rm a(r)}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega-\frac{ \Omega-2\Omega^{\prime}}{2}}\Big{)}^{2}g^{\rm a(r)}_{{\bm{k}}+\frac{{\bm{q}}}{ 2},\omega+\frac{\Omega}{2}}\bigg{]},\\ \tilde{\Lambda}^{\rm ra(2)}_{i;{\bm{k}},{\bm{q}},{\bm{q}}^{\prime };\omega,\Omega,\Omega^{\prime}}\equiv&\Big{(}f_{\omega-\frac{ \Omega-2\Omega^{\prime}}{2}}-f_{\omega-\frac{\Omega}{2}}\Big{)}\bigg{[}\frac{m }{\hbar^{2}}\frac{\partial}{\partial k_{i}}\Big{(}g^{\rm r}_{{\bm{k}}-\frac{{ \bm{q}}}{2},\omega-\frac{\Omega}{2}}g^{\rm a}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q }}^{\prime}}{2},\omega-\frac{\Omega-2\Omega^{\prime}}{2}}g^{\rm a}_{{\bm{k}}+ \frac{{\bm{q}}}{2},\omega+\frac{\Omega}{2}}\Big{)}\\ &-2\bigg{(}{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2}\bigg{)} _{i}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}\Big{(}g^{ \rm a}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega-\frac{\Omega-2 \Omega^{\prime}}{2}}\Big{)}^{2}g^{\rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+ \frac{\Omega}{2}}\bigg{]}\\ &+\Big{(}f_{\omega+\frac{\Omega}{2}}-f_{\omega-\frac{\Omega-2 \Omega^{\prime}}{2}}\Big{)}\bigg{[}\frac{m}{\hbar^{2}}\frac{\partial}{\partial k _{i}}\Big{(}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}g^{ \rm r}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega-\frac{\Omega-2 \Omega^{\prime}}{2}}g^{\rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+\frac{\Omega }{2}}\Big{)}\\ &-2\bigg{(}{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2}\bigg{)} _{i}g^{\rm r}_{{\bm{k}}-\frac{{\bm{q}}}{2},\omega-\frac{\Omega}{2}}\Big{(}g^{ \rm r}_{{\bm{k}}-\frac{{\bm{q}}-2{\bm{q}}^{\prime}}{2},\omega-\frac{\Omega-2 \Omega^{\prime}}{2}}\Big{)}^{2}g^{\rm a}_{{\bm{k}}+\frac{{\bm{q}}}{2},\omega+ \frac{\Omega}{2}}\bigg{]}.\end{split}\] (55) The charge density is written in terms of the lesser component of the nonequilibrium Green’s function \[\rho_{\rm c}({\bm{x}},t)=\frac{ie\hbar}{V}\sum_{{\bm{k}},{\bm{q}}}e^{-i{\bm{q} }\cdot{\bm{x}}}{\rm tr}\hat{G}^{<}_{{\bm{k}}-\frac{{\bm{q}}}{2},{\bm{k}}+\frac {{\bm{q}}}{2}}(t,t).\] (56) In a similar manner to the charge current, the charge density is calculated as \[\rho_{\rm c}({\bm{x}},t)=\] \[\times\Big{(}\tilde{\Lambda}_{i;{\bm{k}},{\bm{q}},{\bm{q}}^{ \prime};\omega,\Omega,\Omega^{\prime}}^{\rm aa(2)}-\tilde{\Lambda}_{i;{\bm{k}} ,{\bm{q}},{\bm{q}}^{\prime};\omega,\Omega,\Omega^{\prime}}^{\rm rr(2)}+\Pi^{ \rm ra}_{{\bm{q}};\omega,\Omega}\tilde{\Lambda}_{i;{\bm{k}},{\bm{q}},{\bm{q}}^ {\prime};\omega,\Omega,\Omega^{\prime}}^{\rm ra(2)}\Big{)}\] \[\simeq\] \[\simeq\] (57) The charge and its current densities that we have obtained satisfy the following charge conservation law: \[\dot{\rho}_{\rm c}({\bm{x}},t)+{\bm{\nabla}}\cdot{\bm{j}}_{\rm c}({\bm{x}},t)=0,\] (58) indicating that our calculation has been performed correctly. ###### Acknowledgements. 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1712.03066	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 43275, "num_imgs": 8, "llama3_tokens_count": 13046 }	[ "content_image/1712.03066/x1.png", "content_image/1712.03066/x2.png", "content_image/1712.03066/x3.png", "content_image/1712.03066/x4.png", "content_image/1712.03066/x5.png", "content_image/1712.03066/x6.png", "content_image/1712.03066/x7.png", "content_image/1712.03066/x8.png" ]	# Cosmic anisotropy with Reduced Relativistic Gas Simpliciano Castardelli dos Reis _E-mail address:_ simplim15@hotmail.com1Departamento de Física, ICE, Universidade Federal de Juiz de Fora Campus Universitário - Juiz de Fora, 36036-330, MG, Brazil 1 Ilya L. Shapiro _E-mail address:_ shapiro@fisica.ufjf.br1Departamento de Física, ICE, Universidade Federal de Juiz de Fora Campus Universitário - Juiz de Fora, 36036-330, MG, Brazil 12 Tomsk State Pedagogical University, Tomsk, 634041, Russia 23 Tomsk State University, Tomsk, 634050, Russia3 Received: date / Revised version: date ###### Abstract The dynamics of cosmological anisotropies is investigated for Bianchi type I universe filled by a relativistic matter represented by the reduced relativistic gas model (RRG), with equation of state interpolating between radiation and matter. Previously it was shown that the interpolation is observed in the background cosmological solutions for homogeneous and isotropic universe and also for the linear cosmological perturbations. We extend the application of RRG to the Bianchi type I anisotropic model and find that the solutions evolve to the isotropic universe with the pressureless matter contents. pacs: _MSC:_81T16, 81T17, 81T20 and PACS:04.62.+v, 11.10.Hi, 11.15.Tk ## 1 Introduction The standard cosmological model describes a universe with homogeneous and isotropic geometry. The matter contents is described by a set of cosmic fluids satisfying some equations of state (EoS). The inhomogeneities are allowed only in the form of small perturbations, which define most of the observables which are used to define most of relevant observables. We know that the spectrum of CMB, Large Scale Structure, BAO and other observations demonstrate the correctness of this description, that is the dynamics of perturbations proves that the expanding universe is very close to homogeneity and isotropy of expanding universe at the sufficiently large scale. The question is whether universe was “born” isotropic and homogeneous or it became such due to some internal mechanism at the early stage of its evolution. The complete analytical description of the possible anisotropies and non-homogeneities of the early universe is impossible. Therefore the standard approach is to assume certain symmetry of the metric tensor. For instance, homogeneity and isotropy are possible symmetries. In formulating a more general metrics one of the possibilities is to consider an anisotropic but homogeneous space-time. The pioneer work [1] explored the case of the metric anisotropic in two space directions with a given group of symmetries, in the universe filled by dust, while [2] dealt with the special cases of locally rotationally symmetric and shear-free dust. The homogeneous models can be grouped by the possible space symmetries given by the Bianchi classification, which is based on the Lie algebras satisfied by the Killing vectors or, equivalently, the structure constants of the hypersurface’s tetrad system [3]. The first possible space of this classification, called Bianchi type I, has three Killing vectors corresponding to the three spatial translations. Due to the simplicity of the Bianchi-I metric, it was extensively used in anisotropic cosmological models, including the Kasner vacuum solution [4]. More sophisticated cosmological models based on other types of Bianchi classification, are possible. One can mention, for instance, the renowed works on the Bianchi type IX models by Belinskii, Khalatnikov and Lifshitz [5; 6], and the Mixmaster universe model by Misner [7; 8], which shows a chaotic behaviour. One of the main questions concerning anisotropic cosmological models is whether the universe could be anisotropic in the early epoch and evolve to be isotropic? It is certainly interesting to identify a mechanism which could be responsible by such an isotropization. It is highly desirable to have a maximally simple description of such a universe, such that further analysis of the perturbations could provide an observational evidence of isotropization. Some of mechanisms of this kind consist from the analysis of the asymptotic behaviour of solutions with isotropic classical fluids [9; 10] (see also [11]), viscous and anisotropic stress tensor [12], primordial magnetic field [13; 14] and quantum effects in primordial universe [15; 16]. In what follows we concentrate on the Bianchi-I model. From the mentioned references we know that the speed of isotropization of the metric may depend of the EoS of the contents of the universe and, in particular, is different for matter or radiation. The situation is qualitatively similar to the dynamics of metric and density perturbations on the isotropic background, but we do not need to treat anisotropies as small perturbations. Usually, the EoS is assumed to be a linear relation between pressure and energy density, $p=\omega\rho$, with a constant $\omega$. The value of $\omega$ corresponds to the type of a fluid. For example, $\omega=-1$ means cosmological constant, $\omega=0$ dust and $\omega=1/3$ radiation. According to the recent data (see, e.g., [17] and [18]) the present-day universe is dominated by non-luminous sources, such as Dark Matter (DM) and Dark Energy. It is most likely that the DM is a gas of weakly interacting massive particles, while the main candidate to be Dark Energy is the cosmological constant. The observational data show that most of its history universe was very isotropic, and therefore the isotropization should occur very early. Since in the past the universe was much hotter than now, the contribution of the cosmological constant to the overall energy density balance at the epoch of isotropization was very small [19]. At the same time, regardless the mass and warmness of the DM particles are unknown, the DM is supposed to be very hot in the early universe and then to became relatively cold at the later stage. Therefore it makes sense to explore the isotropization mechanism for the case of a universe filled by baryonic and dark matter, which are hot in the early universe and dust-like in the present epoch. The simplest appropriate description for the particles in a very early universe is the ideal relativistic gas of massive particles. Perhaps the most useful representation of such a gas is through the Reduced Relativistic Gas model (RRG), which provides a simplified approximation to the Maxwell distribution. The EoS of RRG was originally invented by A.D. Sakharov in the famous 1966 paper [20], to interpolate between radiation and dust regimes. In this work the interpolating EoS has been used for the first derivation of the CMB spectrum, but the details of how to obtain the EoS of the model were not given. More recently RRG model was reinvented by our group in Refs. [21; 22]. The main advantage of this model includes the fact that the solutions for the background cosmology can be obtained in a closed, analytic form for a wide class of models including RRG and other fluids [23], while the EoS is very close to the one of the relativistic gas of ideal particles [21]. Consequently RRG has been used for a simplified evaluation of the bounds of warmness of DM [22; 24], description of energy exchange between matter and radiation and for an overall rough estimate for the cosmological observables in the model with the running cosmological constant [25]. In the present work we apply RRG to describe the isotropization of the universe in the transition period when the matter contents of the universe is in the transition from the radiation to the dust EoS. We will follow the classical works [9; 13], but instead of dealing with radiation and dust cases separately, consider the RRG fluid which interpolates smoothly between the two regimes. The paper is organized as follows. In Sec. 2 we present a new derivation of the EoS of the RRG [20]. This new derivation is instructive and more formal than the previous one in [21]. In Sec. 3 we formulate the equations describing the dynamics of Bianchi type I model in the universe filled by RRG. Sec. 4 describes the simplest approximation for solving these equations. In particular it is shown that the previously known radiation and dust cases represent the limiting cases of the new system of equations. The solution in the general case of RRG can be possible only by means of numerical methods, as described in Sec. 5. Finally, in Sec. 6 we draw our conclusions and describe the perspectives for the further work. ## 2 Reduced relativistic gas: equation of state Let us consider the EoS for the RRG model in a way different from [21]. The model describes ideal relativistic gas of massive identical particles. The main simplification compared to the J$\ddot{\rm u}$ttner model [26] (see also the book [27]) is that within RRG particles have identical kinetic energies. This assumption make the EoS very simple and, in particular, provides great simplification in cosmology, both at the background and perturbations level [20]. At the same time, the difference with the EoS of the J$\ddot{\rm u}$ttner model, derived on the basis of Maxwell distribution does not exceed $2.5\%$[21]. For the cosmological applications, since J$\ddot{\rm u}$ttner model and, in general, an ideal gas of identical particles, is certainly just an approximation, the RRG is perfectly justified and useful model. The derivation of EoS in [21] is very simple, one can say it is at at the high-school level. Let us present a little bit more formal scheme of deriving this equation in the flat Minkowski metric, which enables one, in principle, to evaluate the difference with the J$\ddot{\rm u}$ttner model analytically. The number of particles $N$ is evaluated on a three-dimensional space-like hypersurface with the normal vector $n^{\mu}$, with the hypersurface element area $d\sigma$. The general expression for a non-degenerate gas composted of identical particles is [28] \[N=\int d\sigma\,d^{4}p\,\,n_{\mu}p^{\mu}\,f(x,p)\,\delta(p^{2}-m ^{2}),\] (1) where $p^{2}=(p^{0})^{2}-\delta_{ij}\,p^{i}\,p^{j}$. The distribution function $f(x,p)$ depends of space-time coordinates and momenta, denoted by $x$ and $p$. Taking the integral over $dp_{0}$ and using the properties of the delta function, we get \[N=\int d\sigma\,\frac{d^{3}p}{p^{0}}\,n_{\mu}\,p^{\mu}\,f(x,p).\] (2) For the constant time hypersurface $n^{\mu}=\delta^{\mu}_{0}$ and $d\sigma=d^{3}x$ we arrive at the expression \[N=\int d^{3}x\,d^{3}p\,\,f(x,p).\] (3) The RRG corresponds to the _ansatz_ for for distribution function, \[f(x,p)\,=\,C\,\delta(E-E_{0}),\] (4) where $C$ is a normalization constant, $E=p^{0}=\sqrt{{{\bf p}}^{2}+m^{2}}$ and $E_{0}$ is a constant energy of a gas particle. Using the expression for distribution function in (3), one can easily obtain \[N\,=C\int d^{3}x\,d\Omega\,dE\,\,E\,\sqrt{E^{2}-m^{2}}\,\delta(E -E_{0}),\] (5) where $d\Omega$ is the solid angle element. From the last expression, one can determine the constant C, leading to the final form of the distribution function, \[f\,=\,\frac{n}{4\pi E_{0}\,\sqrt{E_{0}^{2}-m^{2}}}\,\delta(E-E_{ 0}),\] (6) Here $n=N/V$ is the concentration (number of particles per volume) of the gas. The expression for the energy-momentum tensor is [3; 28] (see also brief derivation in the Appendix) \[T^{\mu\nu}\,=\,\int\,d^{3}p\,\,\frac{p^{\mu}p^{\nu}}{p^{0}}\,f(x ,p).\] (7) In the reference frame of an observer with four-velocity $u^{\mu}$ the projection of the energy-momentum tensor onto the hypersurface with normal vector $u^{\mu}$ leads to the energy density $\rho$ and pressure $p$. According to Ref. [11], \[\rho = u_{\mu}u_{\nu}T^{\mu\,\nu},\qquad p\,=\,-\frac{1}{3}\,h_{\mu\nu} \,T^{\mu\,\nu},\] (8) where $\,h_{\mu\nu}=\eta_{\mu\nu}-u_{\mu}u_{\nu}$. In case of a comoving reference frame, in which observer has the four-velocity $u^{\mu}=\delta^{\mu}_{0}$ with the distribution function (6), the expressions (8) become \[\rho=nE_{0}\quad\mbox{and}\quad p=\frac{n(E_{0}^{2}-m^{2})}{3E_{0 }}.\] (9) Defining the rest energy density $\,\rho_{d}=nm$, the pressure and energy density are related by the expression \[p = \frac{\rho}{3}\,\Big{(}1-\frac{\rho_{d}^{2}}{\rho^{2}}\Big{)},\] (10) which is nothing else but the EoS of the RRG model [20; 21]. It is easy to see that this EoS interpolates between radiation, $p\sim\rho/3$, at high energies, when $\rho^{2}\gg\rho_{d}^{2}$, and dust $p\sim 0$, at low energies, when $\rho^{2}\approx\rho_{d}^{2}$. ## 3 Bianchi-I type cosmology with RRG Consider the anisotropic cosmology with the RRG fluid. Our starting point will be the space of Bianchi-I type, with the metric of the form [3], \[ds^{2}=dt^{2}-a^{2}_{1}(t)\,dx^{2}-a_{2}^{2}(t)\,dy^{2}-a_{3}^{2 }(t)\,dz^{2}.\] (11) A useful parametrization of anisotropic metric was introduced by Misner in [7; 8], \[a_{1/2}(t)=a(t)\,e^{\beta_{+}(t)\pm\sqrt{3}\,\beta_{-}(t)}\,, \qquad a_{3}(t)=a(t)\,e^{-2\,\beta_{+}(t)},\] (12) where $a(t)$, $\beta_{+}(t)$ and $\beta_{-}(t)$ are unknown functions of time. In this parametrization $\sqrt{-g}=a_{1}a_{2}a_{3}=a^{3}\,$ and the relation between $\,\beta_{\pm}$ and $a_{i}$ is (13) In Ref. [11], within the _1+3_ covariant formalism of a system of time-like geodesic congruence, the change of a connecting vector between geodesics, expressing the relative distance, is split in the irreducible parts called _shear_, _vorticity_ and _expansion_. In particular, the functions $\beta_{\pm}$ are the independent components of a traceless tensor, which represents the shear. In what follows, we analyse the dynamics of gravitational field for the metric (12), generated by Einstein equations. The matter contents of the universe is modelled by an isotropic RRG, where pressure is assumed to be the same in all spatial directions. The energy-momentum tensor is \[T_{\mu\nu}=(\,\rho+p\,)\,u_{\mu}\,u_{\nu}+p\,g_{\mu\nu}.\] (14) The conservation equation $\nabla_{\mu}\,T^{\mu\nu}=0$ leads to \[\dot{\rho}+3\,H\,(\,\rho+p\,)=0,\qquad H=\frac{\dot{a}}{a}\,,\] (15) where the dot means derivative with respect to the physical time. Let us note that the anisotropy of the metric does not affect the last equation because of the isotropic pressure. Eq. (15) can be integrated by using the EoS of the RRG, yielding the same result as in the isotropic case [21], \[\rho=\sqrt{\rho^{2}_{1}\,\Big{(}\,\frac{a_{0}}{a}\,\Big{)}^{6}+ \rho^{2}_{2}\,\Big{(}\,\frac{a_{0}}{a}\,\Big{)}^{8}},\] (16) where $\rho_{1}$, $\rho_{2}$ and $a_{0}$ are integration constants. As usual, one can distinguish two extreme regimes in the solution (16). In the case $\rho_{1}\ll\rho_{2}$, one meets the ultra-relativistic case, that is RRG demonstrates radiation-like behaviour. On the other hand, for $\rho_{1}\gg\rho_{2}$, RRG behaves like a dust. Now we are in a position to consider the Einstein equations for the Bianchi - I metric. According to Ref. [12], the Einstein tensor, $G_{\mu\nu}$ for the metric (12) assumes the form \[G_{0\,0} = 3\,H^{2}\,-\,3\big{(}\,\dot{\beta}_{+}^{2}+\dot{\beta}_{-}^{2}\, \big{)}\,,\] \[G_{1\,1} = -3H^{2}-2\dot{H}\,-\,3\big{(}\,\dot{\beta}_{+}^{2}+\dot{\beta}_{- }^{2}\,\big{)}\,+\] \[+\Big{(}\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\Big{)}\,\big{(}\, \beta_{+}+\sqrt{3}\,\beta_{-}\,\big{)},\] \[G_{2\,2} = -3H^{2}-2\dot{H}\,-\,3\big{(}\,\dot{\beta}_{+}^{2}+\dot{\beta}_{- }^{2}\,\big{)}\,+\] \[+\Big{(}\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\Big{)}\,\big{(}\, \beta_{+}-\sqrt{3}\,\beta_{-}\,\big{)},\] \[G_{3\,3} = -3H^{2}-2\dot{H}\,-\,3\big{(}\,\dot{\beta}_{+}^{2}+\dot{\beta}_{- }^{2}\,\big{)}\,-\] (17) \[-2\,\Big{(}\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\Big{)}\,\beta_{+}.\] The Einstein equations are given by $G_{\mu\nu}=8\pi GT_{\mu\nu}$. For an isotropic $T_{\mu\nu}$ tensor, Einstein equations can be rewritten such that the pressure of matter does not enter the equations. Following [12], we define the new quantities \[G_{+} = \frac{1}{6}\,\big{(}\,G_{11}+G_{22}-2\,G_{33}\,\big{)},\] \[G_{-} = \frac{1}{2\,\sqrt{3}}\,\big{(}\,G_{11}-G_{22}\,\big{)},\] (18) yielding significant simplifications compared to (17), \[G_{\pm} = \ddot{\beta}_{\pm}+3\,H\dot{\beta}_{\pm}.\] (19) Einstein equations boil down to¹ [FOOTNOTE:1][ENDFOOTNOTE] \[G_{+} = \frac{8\,\pi\,G}{6}\,\big{(}T_{11}+T_{22}-2\,T_{33}\big{)},\] \[G_{-} = \frac{8\,\pi\,G}{2\sqrt{3}}\,\big{(}T_{11}-T_{22}\,\big{)}.\] (20) Furthermore, since we assume isotropic energy-momentum tensor, $T_{11}=T_{22}=T_{33}$ and \[G_{\pm}=0.\] (21) Finally, $00$-component of Einstein equations, together with Eqs. (21) and (19), yield \[H^{2}\,-\,\big{(}\dot{\beta}_{+}^{2}+\dot{\beta}_{-}^{2}\big{)} = \frac{8\pi G}{3}\,\rho\] (22) \[\mbox{and }\qquad 3H\,\dot{\beta_{\pm}}+\ddot{\beta_{\pm}} = 0.\] (23) A first integral of (23) can be easily found in the form \[\dot{\beta_{\pm}}=\gamma_{\pm}\,a^{-3},\] (24) where $\gamma_{\pm}$ are integration constants. The last result transforms (22) into an equation for the conformal factor of isotropic expansion $a(t)$. Defining useful constants $\Gamma$ and $\phi$, \[\gamma_{+} = \Gamma\cos\phi\,,\qquad\gamma_{-}\,=\,\Gamma\sin\phi\,,\] (25) we arrive at the generalized form of Friedmann equation for anisotropic Bianchi-I metric with RRG matter contents, \[H^{2}\,=\,\frac{\dot{a}^{2}}{a^{2}} = \Gamma^{2}\,\Big{(}\,\frac{a_{0}}{a}\,\Big{)}^{6}\,+\] (26) \[+\frac{8\pi G}{3}\,\rho_{1}\Big{(}\,\frac{a_{0}}{a}\,\Big{)}^{4} \,\sqrt{\Big{(}\,\frac{a_{0}}{a}\,\Big{)}^{-2}+b^{2}}\,,\] where $b=\rho_{2}/\rho_{1}$ is the warmness parameter [22]. The specific new element compared to isotropic cosmological model is the first term in the _r.h.s._. This term has the ultrarelativistic $a^{-6}$ scaling feature and hence it is irrelevant for the late cosmology. At the same time, it may be quite relevant in the early universe. Due the new term, caused by anisotropy, the very early universe behaves according to \[a\,\sim t^{1/3}\,,\] (27) different from the radiation-dominated universe. It is well-known that the same dynamics of the conformal factor can be achieved in the isotropic plane universe with an ideal fluid with EoS $p=\rho$. In order to see this consider the EoS $p=w\rho$, with constant $w$. Using the conservation law results in $\rho=\rho_{0}a^{-3\,(w+1)}$. By comparing this result to the first term of the _r.h.s_ of (26), we arrive at $w=1$. A fluid of this kind was called _stiff matter_, when first introduced by Zel’dovich [29]. We have seen that this EoS results from integrating anisotropies at the early stage of the evolution of the universe. The solution of Eqs. (24) can be expressed as \[\beta_{\pm}(t)-\beta^{0}_{\pm}=\gamma_{\pm}W(t),\qquad W(t)=\int_ {t_{0}}^{t}\frac{dt^{\prime}}{a^{3}(t^{\prime})}.\] (28) In this expression $t_{0}$ correspond to the initial moment of time and $\beta^{0}_{\pm}$ are integration constants. One can notice that both $\beta_{\pm}$, with exception of the integration constants, will lead to a same functional form. After $W(t)$ is found, the parameters $\,\gamma_{\pm}\,$ determine $\beta_{\pm}$ and consequently the metric components by Eqs. (12). ## 4 Approximations It is easy to present the solution for (26) and (28) in the form of quadratures, however the integrals are not elementary functions and the qualitative analysis becomes cumbersome. Therefore, in order to have better idea about the physical output of these equations, we split the derivation of the scale factor dependence into two different considerations. In the present section we consider three approximations, namely, vacuum, radiation and dust. In the next section we present the results of a numerical solution in the general case. The radiation and dust approximations come from the limits of the RRG EoS depending on the value of parameter $b$. The approximation for vacuum will be explained bellow. The considerations in this section are almost completely non-original and are presented as to serve as reference for the consequent numerical solutions. When $a(t)$ is very small, one can keep only the first (stiff matter of anisotropic origin) term on the _r.h.s_ of (26). This procedure is equivalent to taking a vacuum solution, because in this regime we are disregarding the terms coming from the matter contents. Indeed, it is known that for the evolution of homogeneous and anisotropic models in the vicinity of the singularity the matter contents has no much relevance [30] (see also [31]). The vacuum metric of Bianchi type I is called Kasner solution. Following [12], we arrive at \[\frac{\dot{a}^{2}}{a^{2}}=\Gamma^{2}\Big{(}\frac{a_{0}}{a}\Big{)} ^{6},\] (29) which can be solved in the form \[\Big{(}\frac{a}{a_{0}}\Big{)}^{3}=3\Gamma(t-t_{0}).\] (30) Setting $a_{0}=1$, the equations (28) can be integrated, yielding \[\beta_{\pm}(t)\,=\,\beta^{(0)}_{\pm}\,+\,\frac{\gamma_{\pm}}{3\, \Gamma}\,\,\mbox{ln}\,\Big{(}\,\frac{t}{t_{0}}\,\Big{)}.\] (31) Here $\beta^{(0)}_{\pm}$ and $t_{0}$ are integration constants. From the angular relations (25) the functions $a_{k}(t)$ can be presented as \[a_{k}(t) = (3\Gamma)^{1/3}\,t^{p_{k}}\,,\qquad k=1,2,3\,.\] (32) Parameters $p_{k}$ can be written down using notations (25), \[p_{1/2} = \frac{1}{3}\,\big{(}1+\cos\,\phi\pm\sqrt{3}\,\sin\,\phi\big{)}\,,\] \[p_{3} = \frac{1}{3}\,\big{(}1-2\,\cos\,\phi\big{)}\,.\] (33) and the line element as \[ds^{2}=dt^{2}-\big{(}3\Gamma\big{)}^{2/3}\,\big{[}t^{2p_{1}}\,dx ^{2}-t^{2p_{2}}\,dy^{2}-t^{2p_{3}}\,dz^{2}\,\big{]},\] (34) The multiplicative constant can be absorbed into the spatial coordinates, providing the standard form [3], \[ds^{2}=dt^{2}-t^{2p_{1}}\,dx^{2}-t^{2p_{2}}\,dy^{2}-t^{2p_{3}}\, dz^{2},\] (35) where the parameters $p_{k}$, $p_{2}$ and $p_{3}$ satisfy the algebraic constraints \[p_{1}^{2}+p_{2}^{2}+p_{3}^{2}=1,\quad p_{1}+p_{2}+p_{3}=1.\] (36) Finally, in the Kasner solution \[a(t)=\big{[}\,a_{1}(t)\,a_{2}(t)\,a_{3}(t)\,\big{]}^{1/3}=t^{ \frac{1}{3}\,(p_{1}+p_{2}+p_{3})}=t^{1/3}.\] (37) The approximations for which the analytic solution can be easily obtained correspond to the ultra-relativistic, $b^{-1}\to 0$ or dust, $b\to 0$ regimes. In what follows we consider these two cases separately. Let us note that the general form of solutions (28) remains the same independent on the approximations for the isotropic energy-momentum tensor. In the ultra-relativistic case one can perform the expansion up to the first order in $b^{-1}$ in (26). Taking $a_{0}=1$, we arrive at \[\dot{a}^{2}\,=\,\frac{\Gamma^{2}}{a^{4}}+\frac{8\pi G\rho_{1}b}{3 a^{2}}\,\Big{(}1+\frac{a^{2}}{2b^{2}}\Big{)}.\] (38) Taking into account the $a^{2}/b^{2}$-term in the parenthesis, this is the Bianchi type I model with radiation, which has initial density expressed by $\rho_{2}$. This is exactly the classical result of [9] for the radiation, but we obtained it as a limit of the RRG solution. It proves useful to make a change of variables \[a = \frac{1}{\chi_{rad}}\,\sinh\,\xi,\quad\kappa_{\rm rad}^{2}\,=\, \frac{8\pi G\,\rho_{2}}{3\Gamma^{2}},\quad 0\leq\,\xi\,<\infty\,,\] (39) in Eq. (26). This results in the relation \[dt\,=\,\frac{1}{\Gamma\kappa_{\rm rad}^{3}}\,\sinh^{2}\xi\,d\xi\,.\] (40) Then Eqs. (26) and (28) become the parametric relations \[\Gamma t = \frac{1}{4\kappa_{\rm rad}^{3}}\,(\sinh 2\xi\,-\,2\xi)\,,\quad \beta_{\pm}\,=\,\frac{\Gamma_{\pm}}{\Gamma}\,\,\mbox{ln}\,(\tanh\xi)\,.\] (41) From the relation between $a$ and $\xi$ in (39), one can obtain \[\tanh\xi = \Big{(}1+\frac{1}{a^{2}\,\kappa_{\rm rad}^{2}}\,\Big{)}^{-\,\frac {1}{2}}.\] (42) In case $(a^{2}\,\kappa_{\rm rad}^{2})^{-1}$ is very small, one gets the relation \[\tanh\xi = 1-\frac{1}{2}\,\frac{1}{a^{2}\,\kappa_{\rm rad}^{2}}\,+\,\dots\,.\] (43) Consequently, due to the (41), \[\beta_{\pm}\,=\,-\,\frac{\gamma_{\pm}}{\Gamma}\,\Big{[}\,\frac{1} {a^{2}\,\kappa_{\rm rad}^{2}}\,+\,\dots\Big{]}.\] (44) In the radiation approximation, if we disregard the term $\big{(}a^{2}\kappa_{\rm rad}^{2}\big{)}^{-1}$, then $\beta_{\pm}$ tend to zero for great values of $a$, and effectively there is isotropization. Another way to arrive at the same conclusion is by observing that when $\xi\to\infty$, we have $\beta_{\pm}\to 0$. In the same limit \[\sinh\xi\sim\cosh\xi\sim\frac{1}{2}\,e^{\xi}\] (45) and dominate in the Eq. (41). Using (39), \[t\,\approx\,\frac{2a^{2}}{\kappa_{\rm rad}},\] (46) which yields the standard expression for the isotropic radiation dominated universe, \[a=\Big{(}\frac{2\pi G\,\rho_{2}}{3\Gamma^{2}}\,\Big{)}^{1/4}\, \sqrt{t}.\] (47) This expression means that the role of anisotropy is negligible for the evolution of the scale factor and hence we have isotropization. Let us now consider the limit $a\gg b$, which means a dust-dominated universe. The solution of the dynamical equations (26) and (28) for dust is simpler than for the radiation-dominated case [12] and was originally obtained in [32]. Here we will try to arrive at the same result by taking the corresponding limit in the general solution for RRG, which interpolates between radiation and dust. The solutions of Eqs. (26,28) for $a(t)$ and $\beta_{\pm}(t)$ are given by \[a^{3}=\frac{3\Gamma}{t_{I}}\,t\,\big{(}\,t+t_{I}\,\big{)},\qquad \beta_{\pm}=\frac{\gamma_{\pm}}{3\Gamma}\,\,\mbox{ln}\,\Big{(}\frac{t}{t+t_{I} }\Big{)}.\] (48) Here \[t_{I}=\frac{4}{3\Gamma\kappa_{\rm dust}^{2}}\qquad\mbox{and} \qquad\kappa_{\rm dust}^{2}=\frac{8\pi G\rho_{1}}{3\Gamma^{2}}\] (49) are constants. The solutions (48) in the dust-dominated approximation can be considered in two different asymptotic situations. The first one is $t\ll t_{I}$, which implicates in the expansions \[a^{3} = 3\Gamma\,\Big{[}\,\Big{(}\frac{t}{t_{I}}\Big{)}^{2}+t\,\Big{]},\] \[\beta_{\pm} = \frac{\gamma_{\pm}}{3\Gamma}\,\,\mbox{ln}\,\Big{[}\frac{t}{t_{I}} \,\Big{(}\,1+\frac{t}{t_{I}}+\dots\,\Big{)}\Big{]}.\] (50) Disregarding terms with powers greater than two, the solutions tend to the Kasner expressions (30) and (31). In the second case $t\gg t_{I}$ one can use the same scheme as before, but now making expansion in the powers of $t_{I}/t$. In this way we obtain the standard solution for the dust, with $a\sim t^{2/3}$ and $\beta_{\pm}\to 0$. Following the same logic as in the radiation case, we conclude that the behaviour in the late times demonstrates isotropization. ## 5 Numerical Solution Let us consider numerical solution of the dynamical system of Eqs. (26) and (28) without assuming high- or low-energy approximations. Exactly as it was done in the previous section, we consider a simplified model with one fluid described by RRG and the anisotropy which enters the general energy balance by means of the stiff matter energy density. It proves useful to express the solution in terms of initial values of the relative energy densities parameters $\Omega_{an}^{(i)}$ and $\Omega_{RRG}^{(i)}$, defined by \[\Omega_{an}=\frac{\Gamma^{2}}{H^{2}},\quad\Omega_{RRG}=\frac{8\, \pi\,G\,\rho_{1}}{3\,H^{2}}\,\sqrt{1+b^{2}}=1-\Omega_{an}.\] (51) The subscript $(i)$ denotes the values of the parameters in the initial moment of time. Our purpose is to evaluate the isotropization of the universe starting from the initial moment of time $t=0$, when $a(0)=a_{i}=1$ and $H(0)=H_{i}$. Therefore, in the initial instant of time the values are $\,\Omega_{an}^{(i)}\,$ and $\,\Omega_{RRG}^{(i)}$, corresponding to $H=H_{i}$ in (51). It proves useful to define the dimensionless time variable $\tau=H_{i}t$. In this way we arrive in the equations \[\frac{\dot{a}^{2}}{a^{2}} = \frac{\Omega_{an}^{(i)}}{a^{6}}+\frac{\Omega_{RRG}^{(i)}}{a^{4}\, \sqrt{1+b^{2}}}\,\sqrt{a^{2}+b^{2}},\] \[\dot{\beta}_{\pm} = \frac{\sqrt{\Omega_{an}^{(i)}}\,\gamma_{\pm}}{\Gamma a^{3}},\] (52) where the dots mean derivatives with respect to $\tau$, The value of $\Omega_{an}(t)$ measures the amount of anisotropy, such that greater values correspond to higher degree of anisotropy. As before, $b$ is the warmness parameter of the RRG matter. In the nowadays universe the value of $\Gamma$ is very small implying in a very small value of $\Omega^{0}_{an}$. The warmness $b$ today is bounded from above by approximately $0.001$ for the dominating fluid, namely for the Dark Matter [25]. Indeed, in the early universe when $\Omega^{0}_{an}$ was significant, the warmness could have a large value. The framework of RRG enables one to see how the warmness affects the time of isotropization, that is the typical time of transition from large value of $\Omega_{an}^{(i)}$ to a small value at the later period. The second equation in (52) can be expressed via the angular parameter in (25). This angle becomes relevant only in the vicinity of the singularity, when the metric can be approximated by the Kasner solution, and in the subsequent numerical analysis it will not play much role. Let us present the numerical solutions for different values of the warmness parameter $b$ using Mathematica software [33]. We used the initial conditions $a=a_{i}=1$, $\beta_{+}=10$, $\beta_{-}=15$, such that $\Omega_{an}^{(i)}=0.99$ and $\Omega_{RRG}^{(i)}=0.01$ at $\tau=0$. In all plots the scale factor and anisotropy measure $\Omega_{an}$ are compared with the plots for the cases of vacuum, radiation and dust, by assuming the same initial values of $\Omega^{(i)}$’s. The Figs. 1 and 2 clearly shows that RRG behaviour tends to Kasner at the early stage of evolution, and is very close of radiation during some time for both scale factor and $\Omega_{an}(\tau)$. In the Figs. 3 and 4, the isotropisation can be observed, because $\beta_{+}$ and $\beta_{-}$ tend to constants. It is easy to see that the isotropization for for RRG occurs faster than for the dust-like contents, close to the rate in the radiation case. <figure><img src="content_image/1712.03066/x1.png"><figcaption>Figure 1: Plots of scale factors for the initial values of b=10 andΩ(i)ani=0.99.</figcaption></figure> <figure><img src="content_image/1712.03066/x2.png"><figcaption>Figure 2: Plots of Ωani(τ) for the initial values of b=10 and Ω(i)ani=0.99.</figcaption></figure> <figure><img src="content_image/1712.03066/x3.png"><figcaption>Figure 3: Plots of β+ corresponding to the parameters b=10 and Ω(i)ani=0.99,while the initial condition β+=10.</figcaption></figure> <figure><img src="content_image/1712.03066/x4.png"><figcaption>Figure 4: Plots of β− corresponding to the parameters b=10 and Ω(i)ani=0.99,while the initial condition β−=15.</figcaption></figure> For smaller warmness, $b=0.5$, one can observe in Figs. 5-8 another behaviour, when RRG plot is (quite naturally) close to dust. <figure><img src="content_image/1712.03066/x5.png"><figcaption>Figure 5: Plots of scale factors for the moderate warmness. Parameters are asfollows: b=0.5, Ω(i)ani=0.99.</figcaption></figure> <figure><img src="content_image/1712.03066/x6.png"><figcaption>Figure 6: Plots of Ωani(τ) for the moderate warmness. Parameters are asfollows: b=0.5, Ω(i)ani=0.99.</figcaption></figure> <figure><img src="content_image/1712.03066/x7.png"><figcaption>Figure 7: Plots of β+ for the moderate warmness. Parameters are as follows:b=0.5, Ω(i)ani=0.99 with the initial condition β+=10.</figcaption></figure> <figure><img src="content_image/1712.03066/x8.png"><figcaption>Figure 8: Plots of β− for the moderate warmness. Parameters are as follows:b=0.5, Ω(i)ani=0.99 while the initial condition is β−=15.</figcaption></figure> The plots presented above show that the RRG is perfectly well interpolating between radiation and dust regimes, as it should be expected. The asymptotic behavior of $\beta_{\pm}(\tau)$ is constant, which means an effective isotropization of the solutions. Concerning the time of isotropization, depending on warmness the RRG model can be closer to dust or radiation. ## 6 Conclusions We formulated the framework of RRG model applied to the dynamics of anisotropy in the early epoch, where the universe was filled by radiation and matter (baryonic and dark), which was so hot that has the EoS which interpolates between the radiation and pressureless matter. For the Bianchi-I universe away from the singularity region the gravitational theory based on the Einstein-Hilbert action provides an isotropization mechanism for RRG, exactly like for both radiation and dust matter contents with isotropic EoS. This physical situation is a subject of current interest, see, e.g., [34]. More complicated spaces may require a more complicated gravitational theories to explain isotropization mechanism. We believe that the simple and efficient RRG model can be useful for describing the realistic matter contents in these complicated cases, as it was for the rather simples Bianchi-I universe described above. Another potentially interesting application of our results is related to the cosmic perturbations in the anisotropic universe, which is not sufficiently well explored. Since the problem is technically complicated, it maybe very useful to have a simple albeit realistic description of the matter contents in the early universe in the epoch when the isotropization occurs. In this respect the framework RRG looks perfect, since it is extremely simple and enables one to quantify the transition from radiation to matter epochs, exactly as it was used in the pioneer work of Sakharov [20]. As it was expected from the previous works on this model [21; 22; 25], the RRG shows the behaviour which is intermediate between radiation and dust and approaches one or another depending on the value of warmness parameter. We have shown that this feature can be extended to the simplest anisotropic Bianchi-I model. One of the natural further developments can be related to the derivation and analysis of density and metric perturbations in the universe filled by RRG, including the case with interaction between RRG and radiation [35]. One can expect that RRG would be eventually useful as a model which helps to explore the observables which can tell us about the dynamics of anisotropies in the early universe. The formalism which was developed in the present work can be useful for the description of Bianchi I phase between the two FLRW phases of the history of universe in the models proposed in [36; 37]. In this case the matter contents of the universe is supposed to be hot and therefore the RRG can be helpful in its efficient description. One can also use the same description of the hot or warm matter in other approaches to anisotropy, like the recent consideration of gravity with $R^{2}$ term [38] or with the sigma-model like scalar field [39], or even in loop quantum gravity [40]. ## Appendix. Brief derivation of Eq. (7) Let us present a very brief derivation of the main expression for the energy-momentum tensor which was used in the main text to arrive at the EoS of the RRG model. More details can be found in [28] and also in [3]. Consider a gas of free massive relativistic particles with equal masses in the equilibrium state. Once in the comoving frame for each particle $T^{00}$ is the energy density, the standard arguments show that the energy-momentum tensor of the gas can be expressed as a sum over particles which are labeled by the subscript $a$, \[T^{\mu\nu}(x)\,=\,\sum_{a}\,\int\,ds\,\,\delta^{4}\big{(}x-x_{a} (s)\big{)}\,\,\frac{p_{a}^{\mu}(s)\,p_{a}^{\nu}(s)}{m_{a}},\] (53) and $s$ is an integration over the proper time for individual particles. Using the definition of Dirac’s delta function, one can rewrite this expression in the form \[T^{\mu\nu}(x) = \int d^{4}p\,\,\,p^{\mu}p^{\nu}\,f(x,p),\] (54) where \[f(x,p)\,=\,\sum_{a}\int ds\,\frac{\delta^{4}\big{(}p-p_{a}(s) \big{)}\,\delta^{4}\big{(}x-x_{a}(s)\big{)}}{m_{a}}.\] (55) The expression (54) includes an integral over four-momenta. As far as each of the free particles satisfies a dispersion relation $p^{2}=m^{2}$ with $p^{0}\geq 0$, one can replace $d^{4}p$ by the expression (56) Taking the integral over $\,p^{0}\,$ one has to replace the invariant element of integration in four dimensions $d^{4}p$ to the invariant element of integration in the space sector, $(m/p^{0})d^{3}p$, because the normal vector to the $p^{2}=m^{2}$ has the same direction as $p^{\mu}$[3]. Finally, using the properties of the delta function leads us to Eq. (7), where the distribution function $f$ depends only on ${{\bf p}}$. Let us stress that the distribution function $f(x,p)$ is defined to be dependent on the motion of all particles. For the many-body system the use of the methods of Statistical Mechanics, in the case of a thermal equilibrium in Minkowski space leads to the distribution function of the J$\ddot{\rm u}$ttner model. 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1908.04456	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 27469, "num_imgs": 3, "llama3_tokens_count": 7242 }	[ "content_image/1908.04456/x1.png", "content_image/1908.04456/x2.png", "content_image/1908.04456/x3.png" ]	# Synthetic gauge field in a single optomechanical resonator Yuan Chen${}^{1,2,\dagger}$, Yan-Lei Zhang${}^{1,2,\dagger}$, Zhen Shen${}^{1,2,\dagger}$, Chang-Ling Zou${}^{1,2}$, Guang-Can Guo${}^{1,2}$, and Chun-Hua Dong${}^{1,2}$ ${}^{1}$CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, P. R. China. ${}^{2}$CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China. February 20, 2024 † [FOOTNOTE:†][ENDFOOTNOTE] Synthetic gauge fields have recently emerged Aidelsburger _et al._(2018); Hey and Li (2018), arising in the context of quantum simulations, topological matter, and the protected transportation of excitations against defects. For example, an ultracold atom experiences a light-induced effective magnetic field when tunnelling in an optical lattice Goldman _et al._(2014, 2016), and offering a platform to simulate the quantum Hall effect and topological insulators. Similarly, the magnetic field associated with photon transport between sites has been demonstrated in a coupled resonator array Hafezi _et al._(2013); Mittal _et al._(2018). Here, we report the first experimental demonstration of a synthetic gauge field in the virtual dimension of bosonic modes in a single optomechanical resonator. By employing degenerate clockwise (CW) and counter-clockwise (CCW) optical modes and a mechanical mode, a controllable synthetic gauge field is realized by tuning the phase of the driving lasers. The non-reciprocal conversion between the three modes is realized for different synthetic magnetic fluxes. As a proof-of-principle demonstration, we also show the dynamics of the system under a fast-varying synthetic gauge field. Our demonstration not only provides a versatile and controllable platform for studying synthetic gauge fields in high dimensions but also enables an exploration of ultra-fast gauge field tuning with a large dynamic range, which is restricted for a magnetic field. <figure><img src="content_image/1908.04456/x1.png"><figcaption>Figure 1: Schematic illustration of a synthetic gauge field in anoptomechanical resonator. a Two phase-related strong opposite propagatingdriving fields enhance the optomechanical coupling between a mechanical modeand the coupled optical modes inside a microcavity. Ecw(ccw) are theamplitudes of the clockwise (CW) and counter-clockwise (CCW) drives, and θ isthe phase difference between them. The CW and CCW modes are coupled throughbackscattering in the microresonator with the modal coupling strength of J. bSchematic of the CW and CCW optical modes coupling with the radial breathingmode. The synthetic magnetic flux ΦB arises from the phase-related drivingfields. acw, accw, and m are the bosonic operators for the optical andmechanical modes, respectively. c Energy diagram for the mode conversionbetween the CW , and CCW optical modes and the breathing mode, where the probeis transferred from the CW (c) or CCW (d) direction. The accumulated phase inthe CW to CCW optical mode conversion is reversed for the opposite propagatingprobe direction, which results in the non-reciprocal power transmission of theoptical probe field.</figcaption></figure> When charged particles move around in a closed loop, the enclosed magnetic flux $\Phi_{B}$ induces a phase that leads to interesting topological phenomena in condensed matter physics. Non-trivial topological phases and topologically protected quantum states have been extensively explored for fundamental studies and quantum information processing. For uncharged particles or bosonic excitations, the essential effect of a gauge field can also be realized through a geometric phase Berry (1984), where a particle or excitation acquires a path-dependent phase by a carefully engineered Hamiltonian. Such a synthetic gauge field enables the simulation of quantum many-body physics with unprecedented precision and unconventional control of bosons Aidelsburger _et al._ (2018); Hey and Li (2018). Thus, synthetic gauge fields have aroused tremendous research interest recently and have been realized in various systems, including ultracold atoms Goldman _et al._ (2014, 2016), optical photons Hafezi _et al._ (2013); Mittal _et al._ (2018); Ozawa _et al._ (2019), phonons Ma _et al._ (2019) and other bosonic quasi-particles Anderson _et al._ (2016); Roushan _et al._ (2017). A synthetic gauge field is commonly realized in real space, where the excitation hops between sites and the phase accumulated for a closed loop path is not trivial. For photons, it has been realized in an artificial photonic microcavity array with engineered spatially dependent couplings between sites Hafezi _et al._ (2013); Mittal _et al._ (2018); Ozawa _et al._ (2019), while the hopping phase determined by the structural parameters is fixed and requires nanofabrication. Recently, a reconfigurable synthetic gauge field was proposed and realized in coupled resonators by employing parametric interactions Estep _et al._ (2014); Schmidt _et al._ (2015); Fang _et al._ (2017); Bernier _et al._ (2017); Hey and Li (2018). For example, a closed loop of sites with a synthetic gauge field was realized by two coupled optomechanical resonators Fang _et al._ (2017), where the appropriate spatially dependent hopping phases between sites were realized through external drives. <figure><img src="content_image/1908.04456/x2.png"><figcaption>Figure 2: The typical photon-photon conversion and photon-phonon conversionmodulated by the synthetic magnetic flux. a Schematic of the mode conversionsof the CW optical mode, CCW optical mode, and mechanical mode modulated by thesynthetic magnetic flux. b,c The spectra of the CW emission power (T) and theCW to CCW optical mode conversion (Ro) with the synthetic magnetic flux atθ=0.06π,0.52π,1.11π,1.43π,1.84π. The incident driving powers are 3.7 mW (CW)and 1.6 mw (CCW). The dots are the experimental results. The solid lines arethe results of calculations using the parameters ωm/2π=98.72 MHz, κ0/2π=5 MHz,κe/2π=36 MHz, γm/2π=92 kHz, J/2π=5.3 MHz, Gcw/2π=0.6 MHz, and Gccw/2π=0.4 MHz,respectively. The relative optical emission power spectra of T and Ro arecalculated at 10th μs. d The photon-phonon conversion (Rm) with the syntheticmagnetic flux at θ=0.13π,0.62π,1.05π,1.55π,1.89π. The relative phonon powerdensity is measured at 12th μs. The corresponding parameters are Gcw/2π=0.53MHz, and Gccw/2π=0.38 MHz, respectively. The optomechanical coupling rate ofthe read laser is GReadcw/2π=0.21 MHz. The other parameters are the same as inb. e Ro and Rm at δ=0 are related to the synthetic magnetic flux. Ro is nearlymaximum when Rm is minimum. The red dots and grey dots are the experimentalresults of the CW to CCW optical mode conversion and photon-phonon conversion,respectively. The solid red line and solid grey line are the simulationresults of these conversions, respectively. The probe propagates in the CWdirection.</figcaption></figure> In this Letter, a synthetic gauge field is realized in a virtual dimension by exploring the natural optical and mechanical modes in a single spherical microresonator without the requirement of fabricating an array of identical microstructures. The effective magnetic flux can be precisely controlled by changing the phases of the external driving lasers. Additionally, such a synthetic gauge field can be arbitrarily tuned, offering the opportunity to study time-dependent gauge field dynamics. Benefiting from the great progress achieved in the coherent optomechanical interaction in microcavities and also the coherent nonlinear optical effects, our demonstration of the synthetic gauge field can be scaled up to larger dimensions but with full controllability, which is not available in real-space lattices. Therefore, exotic topological photonics and non-reciprocal quantum frequency conversion can be readily realized in a single microresonator. <figure><img src="content_image/1908.04456/x3.png"><figcaption>Figure 3: a Schematic of the non-reciprocal emission power spectra of T and Rounder a dynamic synthetic magnetic flux. b Frequency detuning of T and Ro witherror bars under a dynamic synthetic magnetic flux. c-f The non-reciprocalemission power spectra of T and Ro with a dynamic synthetic magnetic flux∂θ(t)/∂t from +100kHz to +600kHz. The experimental results are shown in c ande, and the corresponding simulation results are shown in d and f,respectively. The corresponding parameters are Gcw/2π=0.69 MHz andGccw/2π=0.31 MHz. The other parameters are the same as in 2b.</figcaption></figure> The synthetic gauge field can be realized in a single optomechanical system, as illustrated in Fig.$\,$1a. A single silica microsphere supports massive high-quality optical whispering gallery modes (WGMs) and mechanical modes Kippenberg and Vahala (2007); Park and Wang (2009), by which the coherent conversion between the modes has been extensively studied in the context of optomechanics Weis _et al._ (2010); Safavi-Naeini _et al._ (2011); Dong _et al._ (2012); Zhang _et al._ (2017). Therefore, by exploiting those modes as virtual dimensions, we can study an interesting model of bosonic lattices, as depicted in Fig.$\,$1b, with each vertex node for the bosonic mode serving as a site and the edges corresponding to coherent hops between sites. For modes with non-degenerate frequencies, the coherent conversion between the modes can be stimulated by an external driving laser, which compensates for the energy difference between the modes. In the lattice, we can find the simplest closed loop of a triangle plaquette (blue surface in Fig.$\,$1b), which consists of two optical modes and one mechanical mode. In a microsphere made of a non-magnetic material, the clockwise (CW) and counter-clockwise (CCW) optical modes are degenerate. However, the surface roughness and material defects induce backscattering of the photons Kippenberg _et al._ (2002), as described by an optical modal coupling strength $J$ (Fig.$\,$1a). Both the CW and CCW optical modes can optomechanically couple with the mechanical mode, while the CW (or CCW) photons can only couple with the phonon when there is an off-resonant intracavity field along the CW (or CCW) direction due to the conservation of momentum Shen _et al._ (2016). By introducing driving lasers in both the CW and CCW directions, coherent coupling between the optical modes and mechanical mode can be realized, with the hopping strength being proportional to the drives $E_{\mathrm{cw}}e^{i\left(\theta+\theta_{0}\right)}$ and $E_{\mathrm{ccw}}e^{i\theta_{0}}$. Here, $E_{\mathrm{cw(ccw)}}$ is the amplitude of the drive, and $\theta$ is the phase difference between them. We know that the synthetic gauge field is from the phase difference, therefore we assume the common phase factor $\theta_{0}=0$ for convenience. Figures$\,$1c-d provide a detailed model of the triangle plaquette, which is described by the Hamiltonian (see Supplementary Material for details) \[H=Ja_{\mathrm{cw}}a_{\mathrm{ccw}}^{\dagger}+G_{\mathrm{cw}}e^{i\theta}ma_{ \mathrm{cw}}^{\dagger}+G_{\mathrm{ccw}}ma_{\mathrm{ccw}}^{\dagger}+\mathrm{H.c .},\] (1) where $m$, $a_{\mathrm{cw}}$ and $a_{\mathrm{ccw}}$ are the bosonic operators for the mechanical and optical modes, respectively. $G_{\mathrm{cw(ccw)}}\propto E_{\mathrm{cw(ccw)}}$ is the stimulated photon-phonon coupling strength produced by red-sideband-detuned drives Weis _et al._ (2010); Safavi-Naeini _et al._ (2011). The remaining phase $\theta$ is gauge-independent and actually represents the phase gain by the bosonic excitations when circulating the plaquette. Such a phase $\theta$ for bosonic excitations produces the Aharonov-Bohm effect for electrons, where electrons travel along a closed loop and gain a phase $\theta=\frac{e}{\hbar c}\Phi_{B}$. Here, $\Phi_{B}$ is the magnetic flux enclosed by the loop, and $e$, $\hbar$ and $c$ are the electron charge, reduced Plank’s constant and the light velocity, respectively. Therefore, non-trivial effective gauge fields are synthesized for photons and phonons, while the flux $\Phi_{B}$ can be controlled optically. In our experiments, we fabricate a microsphere with a diameter of approximately $32\mu\mathrm{m}$, which supports degenerated high-quality-factor WGMs near $780\,\mathrm{nm}$ (intrinsic optical linewidth of $\kappa_{0}/2\pi=5\:\mathrm{MHz}$, and a backscattering-induced modal coupling rate of $J/2\pi=5.3\mathrm{\:MHz}$). In the same resonator, the radial breathing mechanical mode has a frequency of $\omega_{m}/2\pi=98.72\,\mathrm{MHz}$ and an optomechanical mode linewidth of $\gamma_{m}/2\pi=92\,\mathrm{kHz}$. In the experiments, both drives are generated from a laser passing through acoustic-optic modulators (AOMs), and the probe photons are generated by an electro-optic modulator (EOM). The phase $\theta$ of the drives is precisely controlled through the relative phase delay between the radio-frequency (RF) signals driving the AOMs. To verify the synthetic gauge field in our optomechanical resonator, we send probe laser to the optical CW mode. Comparing Figs.$\,$1c and d, the model is symmetric under parity and time-reversal symmetry, i.e., $\theta\rightarrow-\theta$ and CW$\rightarrow$CCW. Therefore, the system behaviour for a probe input coupled to the optical CCW mode (Fig.$\,$1d)is equivalent to the scheme in Fig.$\,$1c by flipping the phase of the drives from $\theta$ to $-\theta$. Therefore, we can prove non-reciprocal transmission by studying the probe field from the CW port with various $\theta$ in the following experiments. Two drives and probe pulses (duration of $\tau_{p}=10\mu s$) are sent into the system; thus, the transient bosonic excitation transportation under the synthetic gauge field can be experimentally investigated. As depicted in Fig.$\,$2a, the amplitudes of $a_{\mathrm{cw}}$, $a_{\mathrm{ccw}}$ and $m$ are separately detected. The optical outputs are instantly and simultaneously measured since their lifetimes are relatively low compared to mechanical mode, and the CW and CCW outputs are separated in the forward and backward directions. To probe the mechanical excitation, we introduce another read pulse with a duration of $5\mu s$ at the same frequency of the drives, which arrives $2\mu s$ after the drives Dong _et al._ (2012). Because of the longer lifetime of the phonons, the excited phonons during the first pulses can be converted into photons and measured with a time gate detection to obtain the displacement power spectral density of the mechanical mode (see Supplementary Materials). Figures$\,$2b-e summarize the experimental results of the system with a static synthetic gauge field ($\theta$). In Figs.$\,$2b-d, the intensities of the CW and CCW photons and the phonons for different input probe detunings are presented. For all of the spectra, we observe a significant change in the spectral shape due to the variation of $\theta$, with all the other experimental conditions being fixed. Such a phase-dependent response of the coupled three-mode system manifests the synthetic gauge field, since the conversions from the CW probe to CCW photon and phonon modes are effectively manipulated, which resembles the dynamics of charged particles in a magnetic field. The CW population contains both the direct emission of the probe field and the weak indirect conversion from other modes of the closed loop, only the latter part of which is related to the phase. Therefore, the spectra are slightly modified by the synthetic gauge field $\theta$. We further summarize the derived photon and phonon populations from the spectra in Fig.$\,$2e. By varying $\theta$ over a range of $7\pi$, we find the mode populations of a periodic oscillation with a period of $2\pi$, which means the phase dependence is repeatable and further confirms the nature of the gauge field. Since the CW photons and phonons exhibit a complementary oscillation, the conversion efficiency between the nodes in the loop-path of the three modes can be fine-tuned. The breaking of time-reversal symmetry is also an important consequence of the synthetic gauge field. Therefore, the system should show different responses when changing $\theta$ to $2n\pi-\theta$, with $n$ being an arbitrary integer. Such breaking of the time-reversal symmetry is obvious in Fig.$\,$2c, and the most remarkable difference can be found at $\theta=0.5\pi$ and $2\pi-\theta$. Additionally, according to the symmetry under $\theta\rightarrow-\theta$ and CW$\rightarrow$CCW, for a non-zero fixed $\theta$ that corresponds to broken time-reversal symmetry and broken parity symmetry, non-reciprocal bosonic transportation is most significant when $\theta=0.5\pi+2n\pi$ with the probe laser conversion from the CW direction to the CCW direction. The observed phase dependence and broken time-reversal symmetry of the bosonic excitation conversion in our system verify the static synthetic gauge field. Compared with a magnetic field, the synthetic gauge field has the advantages of a large dynamic range and ultra-fast tuning; thus, it provides a unique route to study the dynamics under a fast-varying gauge field Singleton and Vagenas (2013); Jing _et al._ (2017). As an example, we study the system response with a linearly varying gauge field $\Phi_{B}\left(t\right)$, as illustrated in Fig.$\,$3a. For various synthetic gauge fields $\partial\theta\left(t\right)/\partial t$ at a few hundred kHz, we observe a peak in the spectra of $T$, which is off-resonant with the mechanical mode. Similarly, an extra dip can be observed in the spectra of $R_{o}$ when compared with Fig.$\,$2c, and the detuning approximately $\Delta\approx\partial\theta\left(t\right)/\partial t$. Figures.$\,$3d and f show the theoretically calculated spectra, which agree with the experimental results.**Such dynamics under a temporally varying $\Phi_{B}\left(t\right)$ can be interpreted in the frequency domain, as the frequency of the CW drive is detuned with respect to the CCW drive by $\partial\theta\left(t\right)/\partial t$. Therefore, the induced CW photon-phonon conversion is slightly shifted from the resonance by $\partial\theta\left(t\right)/\partial t$ and introduces a modification to the spectra at the detuning. In Fig.$\,$3b, the extracted frequency of the varying gauge-field-induced peak or dip is plotted, which shows excellent agreement with $\partial\theta\left(t\right)/\partial t$. The results demonstrate the potential of our synthetic gauge field to study more complex time-dependent gauge fields Singleton and Vagenas (2013); Jing _et al._ (2017), as an arbitrary adjustable $\Phi_{B}\left(t\right)$ can be realized in our system by employing arbitrary wave generators for the RF AOM inputs. By exploiting the virtual photon and phonon degrees of freedom (DOFs) in an optomechanical resonator, coherent coupling between the modes can be achieved by external driving lasers, without requiring photonic and phononic device fabrication. Beyond the simplest three-mode model, our demonstration can be scaled to more modes in such a single resonator. Benefiting from the enhanced nonlinear optical effects due to the high quality factor and small mode volume, such as four-wave mixing Gaeta _et al._ (2019) and Brillouin scattering Dong _et al._ (2015), we could generalize the synthetic gauge field to higher virtual dimensions. Therefore, a single resonator is sufficient for studying interesting topological physics in high dimensions and is feasible for manipulating photons with unprecedented abilities. Additionally, nonlinear optical and optomechanical effects enable the generation of entangled bosonic excitations, which has potential for realizing topologically protected quantum squeezing Peano _et al._ (2016), entanglement Blanco-Redondo _et al._ (2018) and one-way quantum steering Cavalcanti and Skrzypczyk (2017). Acknowledgments The authors thank Xiang Xi for discussions. This work was supported by the National Key Research and Development (R&D) Program of China (grant 2016YFA0301303), the National Natural Science Foundation of China (grants 11722436 and 11704370, 11874342, 61805229), and Anhui Initiative in Quantum Information Technologies (grant AHY130200). This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication. Author contributions Y. C., Y.-L. Z. and Z.S. contributed equally to this work. Y.-L. Z., Z.S. C.-L.Z. and C.-H. D. conceived and designed the experiment. Y.C., Z.S. and C.H.D. prepared the samples, built the experimental setup and carried out experiment measurements. Y.-L.Z. and C.-L.Z. provided theoretical support and analysed the data. C.-H.D. and C.-L.Z. wrote the manuscript with inputs from all authors. C.-H.D, C.-L.Z. and G.-C.G. supervised the project. 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1709.10410	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 66617, "num_imgs": 14, "llama3_tokens_count": 21671 }	[ "content_image/1709.10410/fig1a.jpg", "content_image/1709.10410/fig2.jpg", "content_image/1709.10410/fig3.jpg", "content_image/1709.10410/fig4a.jpg", "content_image/1709.10410/fig5.jpg", "content_image/1709.10410/fig6.jpg", "content_image/1709.10410/fig7a.jpg", "content_image/1709.10410/fig8.jpg", "content_image/1709.10410/fig9a.jpg", "content_image/1709.10410/fig10a.jpg", "content_image/1709.10410/fig11.jpg", "content_image/1709.10410/fig12.jpg", "content_image/1709.10410/fig13.jpg", "content_image/1709.10410/fig14.jpg" ]	# Generalization of nonlinear control for nonlinear discrete systems D. Dmitrishin, A. Stokolos, I. Skrynnik, E. Franzheva ###### Abstract. The problem of stabilization of unstable periodic orbits of discrete nonlinear systems is considered in the article. A new generalization of the delayed feedback, which solves the stabilization problem, is proposed. The feedback is represented as a convex combination of nonlinear feedback and semilinear feedback introduced by O. Morgul. In this article, the O. Morgul method was transferred from the scalar case to the vector one. It is shown that the additional introduction of the semilinear feedback into the equation makes it possible to significantly reduce the length of the prehistory used in the control and to increase the rate of convergence of the perturbed solutions to periodic ones. As an application of the proposed stabilization scheme, a possible computational algorithm for finding solutions of systems of algebraic equations is given. The numerical simulation results are presented. ## 1. Introduction By control of chaos we mean a small external influence on the system or a small change in the structure of the system in order to transform the chaotic behavior of the system into regular (or chaotic, but with different properties) [1]. The problem of optimal influence on the chaotic regime is one of the fundamental problems in nonlinear dynamics [2, 3]. It is assumed that the dynamic system has a chaotic attractor, which contains a countable set of unstable cycles of different periods. If the control action locally stabilizes a cycle, then the trajectory of the system remains in its neighborhood, i.e. regular movements will be observed in the system. Hence, one of the ways to control chaos is the local stabilization of certain orbits from a chaotic attractor. To solve the stabilization problem, various control schemes were proposed [4], among which controls based on the Delayed Feedback Control (DFC) principle are quite popular [5]. Such controls, under certain conditions, allow local stabilization of equilibrium positions or cycles, which, generally speaking, are not known in advance. Among the DFC schemes, linear schemes are the simplest for physical implementation. However, they have significant limitations: they can be used only for a narrow area of the parameter space that enter the original nonlinear system. The necessary conditions for the applicability of the linear feedback are formulated more precisely in Section 2.1. To extend the class of systems to which the DFC scheme applies, it is necessary to introduce non-linear elements into the control. For the first time, a nonlinear DFC with one delay was considered in [6], where the advantages of such a modification are also noted, in particular, the fact that the control becomes robust. In [7, 8] the concept of nonlinear control with one delay from [6] was extended: to the vector case; to manage with several delays; to the case of an arbitrary period $T$. It is shown that the control allows to stabilize cycles of arbitrary lengths, unless the multipliers are real and greater than one. A relationship is established between the size of the localization set of multipliers and the amount of delay in the nonlinear feedback. In [9, 10], a semilinear DFC scheme with linear and nonlinear elements was investigated. In spite of the fact that this scheme contains only one difference in control, nevertheless, it is possible to stabilize cycles with length $T=1,2$ under sufficiently general assumptions about cycle multipliers. For $T\geq 3$, the situation changes critically, and the stabilization of cycles is possible only if the rigid constraints on multipliers are met. The scheme of O. Morgul is considered in detail in Section 2.3. It will be generalized to the case of several differences in control, and transferred from scalar to vector case. The purpose of the presented work is to improve the algorithms of M.Vieira de Souza, A.J. Lichtenberg, O. Morgul, and D. Dmitrishin: suppression of chaos in nonlinear discrete systems by local stabilization of cycles of a given length. Accordingly, the problem consists of choosing the structure and parameters of the control system, in which beforehand unknown cycles of a given length would be locally asymptotically stable. The paper considers delayed feedback in the form of a convex combination of nonlinear control and generalized control by O. Morgul. The characteristic polynomial of a closed system for a cycle of length $T$ is deduced and its structure has turned out to be quite simple. As a particular case, this polynomial contains the characteristic polynomials for non-linear control and generalized control of O. Morgul. The solution of the problem for stabilizing cycles of length one is given, i.e. equilibrium positions, and a theoretical basis is prepared for solving the problem in a general formulation for cycles of arbitrary length. The special structure of the characteristic polynomial allows the use of methods of complex analysis. That is why the main method of constructing controls and investigating the conditions for their applicability is the geometric theory of functions of a complex variable. From the perspective of this theory, O. Morgul’s approach to stabilizing cycles and the conditions for its applicability are analyzed. The analysis of the influence of the control parameters on the quality of control is carried out and it is indicated why the combined control is better than the nonlinear or semi-linear control separately. Finally, applications of the proposed scheme of combined control to the improvement of iterative methods for solving algebraic equations are considered. ## 2. Review and preliminary results We consider a nonlinear discrete system, which in the absence of control has the form (1) \[{x_{n+1}}=f\left({x_{n}}\right),\quad{x_{n}}\in{\mathbb{R}^{m}},\quad n=1,2,\ldots,\] where $f\left(x\right)$ is a differentiable vector function of the corresponding dimension. It is assumed that the system (1) has an invariant convex set $A$, that is, if $\xi\in A$, then $f\left(\xi\right)\in A$. It is also assumed that in this system there is one or more unstable $T$-cycles $\left({{\eta_{1}},\ldots,{\eta_{T}}}\right)$, where all the vectors ${\eta_{1}},\ldots,{\eta_{T}}$ are distinct and belong to the invariant set $A$, i.e. ${\eta_{j+1}}=f\left({{\eta_{j}}}\right),j=1,\ldots,T-1,{\eta_{1}}=f\left({{ \eta_{T}}}\right)$. The multipliers of the unstable cycles under consideration are defined as the eigenvalues of the products of the Jacobian matrices $\prod_{j=1}^{T}{{{f^{\prime}}}\left({{\eta_{j}}}\right)}$ of dimensions $m\times m$. As a rule, the cycles $\left({{\eta_{1}},\ldots,{\eta_{T}}}\right)$ of the system (1) are not a priori known as well as the spectrum of the matrix $\left\{{{\mu_{1}},\ldots,{\mu_{m}}}\right\}$ of the matrix $\prod_{j=1}^{T}{f^{\prime}\left({{\eta_{j}}}\right)}$. It is required to describe the set $M$ in which it is possible to locally stabilize the $T$-cycle of the system (1) by one control from the admissible control class for all multipliers localized in $M$, $M\subset\bar{C}$ ($\bar{C}$ is the extended complex plane). I.e., so that the system \[{x_{n+1}}=f\left({x_{n}}\right)+{u_{n}}\] would have a locally asymptotically stable $T$-cycle with multipliers in $M$, and on this cycle the control ${u_{n}}$ would vanish. In other words, we assume that for a given cycle length $T$, we know the estimate of the localization set of the multipliers $M$. In other words, we believe that the dynamic system is characterized not so much by the function $f$ (or a family of functions) as by the set of localization of multipliers of a cycle (or cycles) of known length. ### Linear control As a control, let us consider a law based on linear feedback (2) \[{u_{n}}=-\sum_{j=1}^{N-1}{{\varepsilon_{j}}\left({{x_{n-jT+T}}-{x_{n-jT}}} \right)},\] where the gain should be limited: $\left\|{{\varepsilon_{j}}}\right\|<1$, $j=1,\ldots,N-1$, $T=1,2,\ldots$. Accordingly, the system closed by such a control has the form (3) \[{x_{n+1}}=f\left({x_{n}}\right)-\sum_{j=1}^{N-1}{{\varepsilon_{j}}\left({{x_{n -jT+T}}-{x_{n-jT}}}\right)}.\] Note that when the state ${x_{k+T}}={x_{k}}$, $k=1,2,\ldots$ is synchronized, the control (2) vanishes, i.e. the closed system (3) takes the form as in the absence of control. This means that the $T$-cycles of the system (1) are $T$-cycles of the system (3). Consider the case $T=1$. It is required to find the necessary conditions in terms of the localization set of the multipliers $M$ for which the equilibrium position of the system (3) is locally asymptotically stable (or sufficient conditions under which this equilibrium position is unstable). It is shown in [11] that the set $M$ of localization of multipliers of system (1) can not be arbitrarily large for any linear control of the form (2), more precisely, its diameter can not exceed sixteen, and the diameter of its each connected component is at most four and not depending on the dimension $m$ of the system neither on the number $N$ in the control (2). This conclusion imposes significant limitations on the practical application of linear control. We also note one more drawback of linear control (2): the invariant convex set $A$ of system (1) will not be invariant for system (3). ### Nonlinear control Another type of feedback – nonlinear – has the form (4) \[{u_{n}}=-\sum_{j=1}^{N-1}{{\varepsilon_{j}}\left({f({x_{n-jT+T}})-f({x_{n-jT}} )}\right)},\] and the corresponding closed system (5) \[{x_{n+1}}=\sum_{j=1}^{N}{{a_{j}}f\left({{x_{n-jT+T}}}\right)},\] where ${a_{1}}=1-{\varepsilon_{1}}$, ${a_{j}}={\varepsilon_{j-1}}-{\varepsilon_{j}}$, $j=2$, $\ldots$, $N-1$, ${a_{N}}={\varepsilon_{N-1}}$. It is clear that $\sum_{j=1}^{N}{{a_{j}}=1}$. Only those controls of the form (4) for which $0\leq{a_{j}}\leq 1$, $j=1,\ldots,N$ are considered admissible. When the state ${x_{k+T}}={x_{k}}$, $k=1,2,\ldots$ is synchronized, the control (4) vanishes, and the closed system (5) takes the form as in the absence of control. Therefore, the $T$-cycles of the system (1) are $T$-cycles of the system (5). In addition, the invariant convex set $A$ of system (1) remains invariant for system (5). As shown in [15], for any set $M$ of localization of the multipliers of $T$-cycles of system (1) that does not contain real numbers greater than one, there exists a control of the form (4) for which in system (5) these $T$-cycles will be locally asymptotically stable. Thus, the specified control will have the property of robustness. We give a solution of the problem of choosing the coefficients ${a_{j}}$, $j=1,\ldots,N$, for special cases of the localization sets of multipliers, case $\mathbf{A}$: $M=\left\{{\mu\in\mathbb{R}:\mu\in\left({-\hat{\mu},1}\right)}\right\}$, $\hat{\mu}>1$, case $\mathbf{B}$: $M=\left\{{\mu\in\mathbb{C}:\left\|{\mu+R}\right\|<R}\right\}$, $R>{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$. The algorithm for finding the minimal $N$ and the coefficients $\left\{{{a_{1}},\ldots,{a_{N}}}\right\}$ consists of the following steps [12]: a) nodes are calculated: \[{\psi_{j}}=\pi\frac{{\sigma+T(2j-1)}}{{\sigma+T(N-1)}},\] $j=1,2,\ldots,\frac{{N-2}}{2}$, if $N$ is even; $j=1,2,\ldots,\frac{{N-1}}{2}$, if $N$ is odd; in case $\mathbf{A}$, we should assume $\sigma=2$, and in case $\mathbf{B}$, assume $\sigma=1$; b) the following polynomials are constructed \[{\eta_{N}}\left(z\right)=z\left({z+1}\right)\prod_{j=1}^{\frac{{N-2}}{2}}{ \left({z-{e^{i{\psi_{j}}}}}\right)\left({z-{e^{-i{\psi_{j}}}}}\right)},\] if $N$ is even; \[{\eta_{N}}\left(z\right)=z\prod_{j=1}^{\frac{{N-1}}{2}}{\left({z-{e^{i{\psi_{j }}}}}\right)\left({z-{e^{-i{\psi_{j}}}}}\right)},\] if $N$ is odd; c) the coefficients of the polynomial ${\eta_{N}}\left(z\right)=\sum_{j=1}^{N}{{c_{j}}{z^{j}}}$ are calculated (for example, by Vieta’s formulas); d) the coefficients $a_{j}$ are computed: \[{a_{j}}=\frac{{\left({1-\frac{{1+(j-1)T}}{{2+(N-1)T}}}\right){c_{j}}}}{{\sum_{ k=1}^{N}{\left({1-\frac{{1+(j-1)T}}{{2+(N-1)T}}}\right){c_{k}}}}},j=1,\ldots,N;\] e) in case $\mathbf{A}$, we introduce the quantities \[J_{N}^{(T)}=-{\left[{\frac{T}{{2+(N-1)T}}\prod_{k=1}^{\frac{{N-2}}{2}}{ct{g^{2 }}\frac{{\pi(2+T(2k-1))}}{{2(2+(N-1)T)}}}}\right]^{T}}\] if $N$ is even; \[J_{N}^{(T)}=-{\left[{\prod_{k=1}^{\frac{{N-1}}{2}}{ct{g^{2}}\frac{{\pi(2+T(2k- 1))}}{{2(2+(N-1)T)}}}}\right]^{T}}\] if $N$ is odd; the optimal value of $N$ is computed as a minimal natural number satisfying the inequality \[{\mu^{}}\leq\frac{1}{{\left\|{J_{{}_{N}}^{(T)}}\right\|}};\] f) in case $\mathbf{B}$ we introduce the quantities \[\hat{J}_{N}^{(T)}=-{\left[{\frac{T}{{1+(N-1)T}}\prod_{k=1}^{\frac{{N-2}}{2}}{ ct{g^{2}}\frac{{\pi(1+T(2k-1))}}{{2(1+(N-1)T)}}}}\right]^{T}}\] if $N$ is even; \[\hat{J}_{N}^{(T)}=-{\left[{\prod_{k=1}^{\frac{{N-1}}{2}}{ct{g^{2}}\frac{{\pi(1 +T(2k-1))}}{{2(1+(N-1)T)}}}}\right]^{T}}\] if $N$ is odd; the optimal value of $N$ is computed as a minimal natural number satisfying the inequality \[R\leq\frac{1}{{2\left\|{\hat{J}_{{}_{N}}^{(T)}}\right\|}}.\] We note that for $\sigma\in\left\{{1,2}\right\}$ and $T=1,2$, the polynomials ${F_{T}}\left(z\right)=z{\left({{a_{1}}+\ldots+{a_{N}}{z^{N-1}}}\right)^{T}}$ are univalent in the central unit disk $D=\left\{{z\in\mathbb{C}:\left\|z\right\|<1}\right\}$. Apparently, the univalence property of polynomials is true for $\sigma\in\left[{0,2}\right]$ and for all $T.$ For different $\sigma$ the set $M$ of localization of the multipliers of $T$-cycles of the system (1) must lie in the half-plane $\left\{{z\in\mathbb{C}:{\mathop{\rm Re}\nolimits}z<1}\right\}$ (Figure 1-a, 1-b). <figure><img src="content_image/1709.10410/fig1a.jpg"><figcaption>Figure 1. Coverings of the set M of localization of multipliers under a) T=3,N=7, σ∈[0,1]: σ=1 – black, σ=0.66 – blue, σ=0.33 – green, σ=0 – red; b) T=3,N=7, σ∈[1,2]: σ=1 – black, σ=1.33 – blue, σ=1.66 – green, σ=2 – red</figcaption></figure> ### Semilinear control To stabilize the cycle of the length $T=3$ O. Morgul [9, 10] proposed a feedback control that includes linear and nonlinear elements, i.e., a semilinear feedback control of the form (6) \[{u_{n}}=-\varepsilon\left({f({x_{n}})-{x_{n-T+1}}}\right),\] for which a corresponding closed-loop system is (7) \[x_{n+1}=(1-\varepsilon)f(x)+\varepsilon x_{n-T+1},\] where $\varepsilon\in\left[{0,1}\right)$. On the cycle, the conditions $f({x_{n}})={x_{n+1}}={x_{n-T+1}}$ are fulfilled, therefore, on the cycle ${u_{n}}\equiv 0$. We note that in [9] only the scalar case $f:\mathbb{R}\to\mathbb{R}$ was considered. However, Morgul’s scheme can be generalized to the vector case, as will be shown below. In setting the invariant convex set $A$ of system (1) remains invariant for the system (7) too. If we assume that $\varepsilon\in\left[{0,\infty}\right)$ [9], then the convex invariant set can not be preserved, although in this case it is possible to stabilize the equilibrium positions with multipliers from the half-plane $\left\{z\in\mathbb{C}:{\mathop{\rm Re}\nolimits}z>1\right\}$. The characteristic equation for the $T$-cycle, in the scalar case, has the form [10] (8) \[\left(\lambda-\varepsilon\right)^{T}-\mu{\left(1-\varepsilon\right)^{T}}{ \lambda^{T-1}}=0,\] where $\mu$ is the multiplier of the cycle. Accordingly, in the vector case the characteristic equation takes the form (9) \[\prod_{j=1}^{m}{\left[{\left({\lambda-\varepsilon}\right){{}^{T}}-{\mu_{j}}{{ \left({1-\varepsilon}\right)}^{T}}{\lambda^{T-1}}}\right]}=0,\] where $\mu_{j}$ are multipliers of the cycle ($j=1,\ldots,m$), in general, complex. Equation (9) is obtained as a special case of a more general characteristic equation, which we derive in Section 3. If all the roots of equation (9) lie in an open central unit disc $D$, then the $T$-cycle is locally asymptotically stable [10, 13]. If the multipliers $\mu{{}_{j}}$, $j=1,\ldots,m$ are known exactly, then one can check whether the roots belong to the central unit circle by known criteria such as Schur-Cohn, Clark, Jury [14]. However, cycles are not known, hence, multipliers are not known. In this case, the geometric criterion of A. Solyanik proved to be effective for the stability of cycles of discrete systems [15]. Let us apply this criterion. Making the change $\lambda=\frac{1}{z}$, we write equation (9) as a set of equations \[\left[\begin{array}[]{{20}{c}}\frac{1}{\mu_{j}}=\Phi\left(z\right),\\ j=1,\ldots,m,\end{array}\right.\] where $\Phi\left(z\right)={\left({1-\varepsilon}\right)^{T}}\frac{z}{{{{\left({1- \varepsilon z}\right)}^{T}}}}$. The following observation is extremely useful in our settings. Lemma 1.: _All the roots of equation (9) lie in the central unit circle if and only if_ (10) \[\mu_{j}\in\left(\overline{\mathbb{C}}\backslash\Phi(\overline{D})\right)^{}, \quad j=1,\ldots,m,\] _where $\overline{D}=\left\{z\in\mathbb{C}:\left\|z\right\|\leq 1\right\}$ is a closed central unit disk, $\overline{\mathbb{C}}$ is an extended complex plane, the asterisk denotes the inversion ${\left(z\right)^{}}=\frac{1}{{\bar{z}}}$. Here $\bar{z}$ denotes the complex conjugated of $z$._ Note that the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ is the inverse of the set of exceptional values of the image of the disk under the mapping $\Phi\left(z\right)$. According to Lemma 1, the $T$-cycle will be locally asymptotically stable if the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ covers the set $M$ of localization of multipliers. The condition (10) can be rewritten as $M\subseteq{\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$, or in the equivalent form $\Phi(D)\subseteq\bar{C}\backslash{\left({\bar{M}}\right)^{}}$. This means that the set ${\left({\bar{M}}\right)^{}}$ must be exceptional for the image of the disk $D$ under the mapping $\Phi\left(z\right)$. Let us consider some examples. <figure><img src="content_image/1709.10410/fig2.jpg"><figcaption>Figure 2. Множество (¯C∖Φ(¯D))∗ for T=1, γ=0.8.</figcaption></figure> Example 1. Let $T=1$. In this case the set \[{\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}=\left\{{w\in\mathbb{C}:w=- \frac{\varepsilon}{{1-\varepsilon}}+\frac{1}{{1-\varepsilon}}z,z\in D}\right\},\] that is, this set is an open circle with center at the point $\left(-\frac{\varepsilon}{1-\varepsilon},0\right)$ and of radius $\frac{1}{\left\|1-\varepsilon\right\|}$ (Fig 2). If $\varepsilon\to 1^{-}$, the disk converges to the half-plane $\left\{{w\in\mathbb{C}:w<1}\right\}$. If $\varepsilon\to 1^{+}$ the disc converges to the half-plane $\left\{{w\in\mathbb{C}:w>1}\right\}$. Therefore, if the set $M$ lies in the half-plane $\left\{{w\in\mathbb{C}:w<1}\right\}$ or $\left\{{w\in\mathbb{C}:w>1}\right\}$, then the equilibrium position of the system (1) can be stabilized by the control of the form (6). Example 2. Let $T=2$. Then \[{\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}=\left\{{w\in\mathbb{C}:w= \frac{{2\varepsilon}}{{{{\left({1-\varepsilon}\right)}^{2}}}}\left({\frac{1}{2 }(\frac{1}{{\varepsilon z}}+\varepsilon z)-1}\right),z\in D}\right\},\] that is, it is the interior of an ellipse with semiaxes $\left\{{\frac{{1+{\varepsilon^{2}}}}{{{{(1-\varepsilon)}^{2}}}},\frac{{1+ \varepsilon}}{{1-\varepsilon}}}\right\}$ and centered at the point $\left({-\frac{{2\varepsilon}}{{{{(1-\varepsilon)}^{2}}}},0}\right)$ (Fig 3). The ellipse foci are at the points $\left({-\frac{{{{(1+\varepsilon)}^{2}}}}{{{{(1-\varepsilon)}^{2}}}},0}\right)$ and $\left({1,0}\right)$. Therefore, if the set $M$ lies in the half-plane $\left\{{w\in\mathbb{C}:w<1}\right\}$, then the 2-cycle of the system (1) can be stabilized by the control of the form (6 ). <figure><img src="content_image/1709.10410/fig3.jpg"><figcaption>Figure 3. The set (¯C∖Φ(¯D))∗ for T=2, γ=0.8</figcaption></figure> O. Morgul considered only the scalar case, and in the scalar case the set $M$ can consist only of real numbers. Then the condition of stabilization of the equilibrium is the following: $\mu\in\left({-\frac{{1+\varepsilon}}{{1-\varepsilon}},1}\right)$ or $\mu\in\left({1,\frac{{1+\varepsilon}}{{1-\varepsilon}}}\right)$. Accordingly, for the 2-cycle, the stabilizability condition has the form: $\mu\in\left({-{{\left({\frac{{1+\varepsilon}}{{1-\varepsilon}}}\right)}^{2}},1 }\right)$. In the case $T=1$, the function $\Phi\left(z\right)=(1-\varepsilon)\frac{z}{{1-\varepsilon z}}$ is univalent for all $\varepsilon\in\left({-\infty,\infty}\right)$ in the entire complex plane, with the exception of the point ${z_{0}}=\frac{1}{\varepsilon}$. For $T=2$ and $\varepsilon\neq 0$, the function $\Phi\left(z\right)={(1-\varepsilon)^{2}}\frac{z}{{{{(1-\varepsilon z)}^{2}}}}$ is univalent in the open central disk $\left\{{z\in\mathbb{C}:\left\|z\right\|<\frac{1}{{\left\|\varepsilon\right\|}}}\right\}$. In these cases, the functions $\Phi\left(z\right)$ for $\varepsilon\in\left[{0,1}\right)$ are univalent in the open central unit disc $D$. For $T\geq 3$ the situation becomes different. The function $\Phi\left(z\right)={(1-\varepsilon)^{T}}\frac{z}{{{{(1-\varepsilon z)}^{T}}}}$ will not be univalent in the disk $D$ for all $\varepsilon\in\left[{0,1}\right)$, but only for $\varepsilon\in\left[{0,\frac{1}{{T-1}}}\right)$ [16]. Since $\Phi\left({-1}\right)=-{\left({\frac{{1-\varepsilon}}{{1+\varepsilon}}}\right) ^{T}}$, then for $\varepsilon\in\left[{0,\frac{1}{{T-1}}}\right)$, the condition for the stabilizability of the $T$ cycle in the scalar case takes the form $\mu\in\left({-{{\left({\frac{{1-\varepsilon}}{{1+\varepsilon}}}\right)}^{T}},1 }\right)$. The function ${\left({\frac{{1-\varepsilon}}{{1+\varepsilon}}}\right)^{T}}$ increases by $\varepsilon$, hence the maximum size for the multiplier localization set will be $\varepsilon=\frac{1}{{T-1}}$, i.e. $\mu\in\left({-{{\left({\frac{T}{{T-2}}}\right)}^{T}},1}\right)$. For $\varepsilon>\frac{1}{{T-1}}$, the function $\Phi\left(z\right)$ fails to be univalent, and the interval for the multiplier will decrease (Figure 4). The function ${\left({\frac{T}{{T-2}}}\right)^{T}}$ decreases as $T\geq 3$ asymptotically tending to ${e^{2}}\approx 7.389$. <figure><img src="content_image/1709.10410/fig4a.jpg"><figcaption>Figure 4. The inverse image of the boundary of the disk D (red) and the set(¯C∖Φ(¯D))∗ (gray) for T=5, a) ε=0.25, b) ε=0.3.</figcaption></figure> ## 3. Generalized semilinear control ### Formulation of the problem A natural generalization of delayed feedback control is the joint use of the linear, nonlinear, and semilinear feedback considered in Section 2. Let us take into account that the linear feedback is ineffective, and we choose the control in the form of a convex combination of nonlinear and generalized semilinear feedbacks, specifically, (11) \[{u_{n}}=-\left({1-\gamma}\right)\sum_{j=1}^{N-1}{\varepsilon_{j}^{(1)}\left({f ({x_{n-jT+T}})-f({x_{n-jT}})}\right)}-\gamma\sum_{j=1}^{N}{\varepsilon_{j}^{(2 )}\left({f({x_{n-jT+T}})-{x_{n-jT+1}})}\right)},\] where $\gamma\in\left[{0,1}\right)$. Note that the control (11) disappears on a cycle of length $T$. The motivation for using a control of the form (11) for $T\geq 3$ is completely obvious. In the general case, the semilinear control does not allow stabilizing cycles of system (1) of length three or more. However, the combined use of semilinear and non-linear controls can allow reducing the necessary length of history used in the feedback. For $T=1,2$, we can also expect qualitatively new effects when the equilibrium position is stabilized due to a larger number of control parameters: an increase in the rate of convergence of the perturbed solutions to periodic ones, expansion of the basin of attraction of a locally stable periodic solution, and so on. In other words, combined control should improve the properties of both nonlinear and semilinear control. Let us close the system (1) by the control (11), then we get (12) \[{x_{n+1}}=\left({1-\gamma}\right)\sum_{j=1}^{N}{{a_{j}}f({x_{n-jT+T}})}+\gamma \sum_{j=1}^{N}{{b_{j}}{x_{n-jT+1}}},\] where the coefficients ${a_{1}},\ldots,{a_{N}}$, ${b_{1}},\ldots,{b_{N}}$ are associated with the parameters $\varepsilon_{1}^{(1)},\ldots,\varepsilon_{N-1}^{(1)}$, $\varepsilon_{1}^{(2)},\ldots,\varepsilon_{N}^{(2)}$ by a linear bijection \[\left\{{\begin{array}[]{{20}{c}}{{a_{1}}=\frac{1}{{1-\gamma}}-\varepsilon_{1} ^{(1)}-\frac{\gamma}{{1-\gamma}}\varepsilon_{1}^{(2)},}\\ {{a_{j}}=-(\varepsilon_{j}^{(1)}-\varepsilon_{j-1}^{(1)})-\frac{\gamma}{{1- \gamma}}\varepsilon_{j}^{(2)},j=2,\ldots,N-1,}\\ {{a_{N}}=\varepsilon_{N-1}^{(1)}-\frac{\gamma}{{1-\gamma}}\varepsilon_{N}^{(2) },}\\ {{b_{j}}=\varepsilon_{j}^{(2)},j=1,\ldots,N.}\end{array}}\right.\] We request the invariant convex set $A$ of system (1) to be invariant for system (12). Therefore, we must require the following relations: ${a_{j}}\in\left[{0,1}\right]$, ${b_{j}}\in\left[{0,1}\right]$, $j=1,\ldots,N$, $\sum_{j=1}^{N}{{a_{j}}}=1$, $\sum_{j=1}^{N}{{b_{j}}}=1$. To this end, additional restrictions must be imposed on the control (11). Namely, $\sum_{j=1}^{N}{\varepsilon_{j}^{(2)}}=1$; $\frac{1}{{1-\gamma}}-\varepsilon_{1}^{(1)}\geq\frac{\gamma}{{1-\gamma}} \varepsilon_{1}^{(2)}\geq 0$, $\varepsilon_{j-1}^{(1)}-\varepsilon_{j}^{(1)}\geq\frac{\gamma}{{1-\gamma}} \varepsilon_{j}^{(2)}\geq 0$, $j=2,\ldots,N-1$, $\varepsilon_{N-1}^{(1)}\geq\frac{\gamma}{{1-\gamma}}\varepsilon_{N}^{(2)}\geq 0$. It is required to select the parameters ${a_{1}},\ldots,{a_{N}}$, ${b_{1}},\ldots,{b_{N}}$, satisfying the given constraints, so that the $T$-cycle of the system (12) is locally asymptotically stable, and $N$ would be the smallest. If we let $\gamma=0$ in (12), then we obtain the system (5), i.e. the system (1), closed by nonlinear feedback. If we let $N=1$ in (12), then ${a_{1}}={b_{1}}=1$, therefore, we get the system (7), that is, as in the case of closure by the semilinear feedback. Thus, the system (12) contains the systems (5) and (7) as particular cases. ### Construction of the characteristic polynomial We begin the investigation of the stability of the $T$ -cycle of the system (12) with the derivation of the characteristic equation for this cycle. The classical way is the construction of the Jacobi matrix of a special mapping in the neighborhood of the cycle [10], and finding the characteristic polynomial of this matrix. As a result, this characteristic polynomial will have a cumbersome form, and the path of its simplification is not at all obvious [10]. The same polynomial can be constructed from a different mapping, and the polynomial is obtained in a very convenient form for further investigations [17, 18]. In [18] the equivalence of the classical O. Morgul method and the alternative method is proved, which will be applied below. The solution of system (12) can be represented in the form (13) \[\left\{{\begin{array}[]{{20}{c}}{{x_{Ts}}={\eta_{1}}+u_{s}^{1}}\\ {{x_{Ts+1}}={\eta_{2}}+u_{s}^{2}}\\ \ldots\\ {{x_{Ts+T-1}}={\eta_{T}}+u_{s}^{T}}\end{array}}\right.,\] $s=0,1,\ldots$. We substitute solution (13) into (12), assuming that in the neighborhood of the cycle the norms of the vectors $u_{s}^{1},\ldots,u_{s}^{T}$ are small. Let $n=Ts$. Then \[{x_{n+1}}={x_{Ts+1}}={\eta_{2}}+u_{s}^{2},{x_{n+2}}={x_{Ts+2}}={\eta_{3}}+u_{s }^{3},\ldots,{x_{n+T}}={x_{T(s+1)}}={\eta_{1}}+u_{s+1}^{1}.\] Selecting the linear part and taking into account that ${\eta_{1}}=f\left({{\eta_{2}}}\right),\ldots,{\eta_{T}}=f\left({{\eta_{1}}}\right)$, we get (14) \[\left\{{\begin{array}[]{{20}{c}}{u_{s}^{2}=(1-\gamma)f^{\prime}\left({{\eta_{ 1}}}\right)({a_{1}}u_{s}^{1}+\ldots+{a_{N}}u_{s-N+1}^{1})+\gamma({b_{1}}u_{s-1 }^{2}+\ldots+{b_{N}}u_{s-N}^{2})}\\ \ldots\\ {u_{s}^{T}=(1-\gamma)f^{\prime}\left({{\eta_{T-1}}}\right)({a_{1}}u_{s}^{T-1}+ \ldots+{a_{N}}u_{s-N+1}^{T-1})+\gamma({b_{1}}u_{s-1}^{T}+\ldots+{b_{N}}u_{s-N} ^{T})}\\ {u_{s+1}^{1}=(1-\gamma)f^{\prime}\left({{\eta_{T}}}\right)({a_{1}}u_{s}^{T}+ \ldots+{a_{N}}u_{s-N+1}^{T})+\gamma({b_{1}}u_{s}^{1}+\ldots+{b_{N}}u_{s-N+1}^{ 1})}\end{array}}\right.,\] where $f^{\prime}\left({{\eta_{j}}}\right)$, $j=1,\ldots,T$, are Jacobian matrices of dimension $m\times m$. The system (14) is linear, so its solutions are represented in the form (15) \[\left({\begin{array}[]{{20}{c}}{u_{s}^{1}}\\ {\ldots}\\ {u_{s}^{T}}\end{array}}\right)=\left({\begin{array}[]{{20}{c}}{{c_{1}}}\\ {\ldots}\\ {{c_{T}}}\end{array}}\right){\lambda^{s}},\] Where $\lambda$ is a complex number to be determined. Substituting (15) into (14), we obtain (16) Denote by $z=\frac{1}{\lambda}$, $q\left(z\right)={a_{1}}+{a_{2}}z+\ldots+{a_{N}}{z^{N-1}}$, $p\left(z\right)={b_{1}}z+{b_{2}}{z^{2}}+\ldots+{b_{N}}{z^{N}}$. The system (16) considered with respect to the vectors ${c_{1}},\ldots,{c_{T}}$, will have a nontrivial solution if and only if the determinant of the matrix \[\left({\begin{array}[]{{20}{c}}{-(1-\gamma)q(z)f^{\prime}\left({{\eta_{1}}} \right)}&{(1-\gamma p(z))I}&{\rm O}&\ldots&{\rm O}&{\rm O}\\ {\rm O}&{-(1-\gamma)q(z)f^{\prime}\left({{\eta_{2}}}\right)}&{(1-\gamma p(z))I }&\ldots&{\rm O}&{\rm O}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ {\rm O}&{\rm O}&{\rm O}&\ldots&{-(1-\gamma)q(z)f^{\prime}\left({{\eta_{T-1}}} \right)}&{(1-\gamma p(z))I}\\ {{z^{-1}}(1-\gamma p(z))I}&{\rm O}&{\rm O}&\ldots&{\rm O}&{-(1-\gamma)q(z)f^{ \prime}\left({{\eta_{T}}}\right)}\end{array}}\right),\] is non-zero, where ${\rm O}$ is zero matrix of dimension $m\times m$, $I$ is the identity matrix of dimension $m\times m$. That is \[\det\left(z^{-1}(1-\gamma p(z))^{T}I-\left((1-\gamma)q\left(z\right)\right)^{T }\prod_{j=1}^{T}f^{\prime}\left(\eta_{j}\right)\right)=0.\] Let the eigenvalues of the product of the Jacobi matrices $\prod_{j=1}^{T}{f^{\prime}\left({{\eta_{j}}}\right)}$ be equal to ${\mu_{1}},\ldots,{\mu_{m}}$. Then, replacing this product by Jordan’s canonical form, we obtain the final form of the characteristic equation (17) \[\prod_{j=1}^{m}{\left({\frac{{{{(1-\gamma p(z))}^{T}}}}{{z{{\left({(1-\gamma)q \left(z\right)}\right)}^{T}}}}-{\mu_{j}}}\right)}=0.\] Hence, the desired characteristic polynomial has the form (18) \[\tilde{f}\left(\lambda\right)=\prod_{j=1}^{m}{\left({{{\left[{{\lambda^{N}}- \gamma{\lambda^{N}}p({\lambda^{-1}})}\right]}^{T}}-{{(1-\gamma)}^{T}}{\mu_{j}} {\lambda^{T-1}}{{\left[{{\lambda^{N-1}}q({\lambda^{-1}})}\right]}^{T}}}\right)}.\] The polynomial (18) contains, as a special case for $N=1$, the polynomial (9). ### Geometric criterion for local asymptotic stability of a cycle The next step in the study of the stability of cycles is to analyze the location of the zeros of the characteristic polynomial (18) on the complex plane. Or, equivalently, the roots of equation (17). The local stability of cycles of difference systems is equivalent to the Schur stability of the characteristic polynomial corresponding to this cycle [cf. Ex. 13]. We present this fact in the following two Lemmas. Lemma 2.: _The $T$-cycle of system (12) is locally asymptotically stable if and only if all zeros of polynomial (18) lie in the open central unit disk $D$._ As noted in Section 2.3, it is not possible to apply the known criteria for testing the Schur stability of the polynomial (18), since the quantities ${\mu_{1}},\ldots,{\mu_{m}}$ are not known. Therefore, to verify the local stability of cycles of system (12), we apply the geometric criterion of stability suggested by A. Solyanik. Denote by $\Phi\left(z\right)=\left(1-\gamma\right)^{T}\frac{z\left(q(z)\right)^{T}}{ \left(1-\gamma p(z)\right)^{T}}$. Lemma 3.: _All roots of the polynomial (18) lie in the open central unit disc $D$ if and only if_ (19) \[{\mu_{j}}\in{\left({\bar{C}\backslash\Phi(\overline{D})}\right)^{}},j=1, \ldots,m,\] _where $\overline{D}$ is a closed central unit disk, $\overline{\mathbb{C}}$ is an extended complex plane, the asterisk denotes the inversion operation: $\left(z\right)^{}=\frac{1}{\bar{z}}$._ Proof.: The polynomial (18) is Shur-stable if and only if $\tilde{f}(\lambda)\neq 0$ for all $\lambda\in\bar{C}\backslash D$. This is equivalent to $\frac{1}{{{\mu_{j}}}}\neq\Phi(z)$, $z\in\bar{D}$, $j=1,\ldots,m$. Consequently, the necessary and sufficient conditions for the stability of the Schur polynomial (18) are the inclusions: $\frac{1}{{{\mu_{j}}}}\notin\Phi(\bar{D})$, or $\frac{1}{{{\mu_{j}}}}\in\bar{C}\backslash\Phi(\bar{D})$, or ${\mu_{j}}\in{\left({\bar{C}\backslash\Phi(\overline{D})}\right)^{}}$, $j=1,\ldots,m$. ∎ In the general case, cycle multipliers are not known; hence, the $T$ -cycle is locally asymptotically stable if the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ covers the set $M$ of localization of multipliers. This means that the set ${\left({\bar{M}}\right)^{}}$ must be exceptional for the image of the disk $D$ under the mapping $\Phi\left(z\right)$. This property will be the main one for constructing the control coefficients ${a_{1}},\ldots,{a_{N}},{b_{1}},\ldots,{b_{N}}$. ### The design of controls that stabilize cycles The next step is to construct a function $\Phi\left(z\right)$ so that the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ covers the set $M$ of localization of multipliers. In this case, it is necessary to estimate the size of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ as a function of $N$ and $\gamma$. The function $\Phi\left(z\right)={\left(1-\gamma\right)^{T}}\frac{z\left(q(z)\right)^{T}}{ \left(1-\gamma p(z)\right)^{T}}$ depends on the polynomials $q(z)$, $p(z)$, and the parameter $\gamma\in\left[{0,1}\right)$, and $q(1)=1$, $p(0)=0$, $p(1)=1$, hence $\Phi(0)=0$, $\Phi(1)=1$. To further advance the formulation of the problem, we impose an essential restriction on the function $\Phi\left(z\right)$, namely, we know that the polynomial $q(z)$ is calculated by the formulas indicated in Section 2.2 (for some $\sigma\in\left[{1,2}\right]$). That ensures that $M\subseteq\left\{{\mu\in\mathbb{R}:\mu\in\left({-\hat{\mu},1}\right)}\right\}$ ($\hat{\mu}>1$) or $M\subseteq\left\{{\mu\in\mathbb{C}:\left\|{\mu+R}\right\|<R}\right\}$ ($R>{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$) for any admissible $\hat{\mu}$ and $R$, at least for a sufficiently large $N$ and $\gamma=0$. We want to choose the polynomial $p(z)$ and the parameter $\gamma$ so that a certain desired linear dimension of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ is maximal (or the set $\Phi\left(D\right)$ is minimal). The linear dimensions depend on the value $\Phi\left({-1}\right)=-{\left({1-\gamma}\right)^{T}}\frac{{{{\left({q(-1)} \right)}^{T}}}}{{{{\left({1-\gamma p(-1)}\right)}^{T}}}}$. Since all ${b_{j}}$, $j=1,\ldots,N$ are not negative, then $\left\|{p(-1)}\right\|<1$. Consequently, it is not possible to make $\Phi\left({-1}\right)$ small at the expense of the polynomial $p(z)$. The polynomial $p(z)$ has a very specific role: this polynomial should be chosen so that the parameter $\gamma$ can be varied within the widest limits. We formulate this requirement. We consider the family of functions (20) \[\left\{\Phi\left(z\right)=\left(1-\gamma\right)^{T}\frac{z\left(q(z)\right)^{T }}{\left({1-\gamma p(z)}\right)^{T}}:\gamma\in\left[0,\gamma^{}\right]\right\},\] where the polynomial $q(z)$ is defined as above. It is required to find the polynomial $p(z)$ (with a given degree and given normalization conditions) so that the family of functions (20) is univalent in the disc $D$, and $\gamma^{}$ is maximal. If it is necessary to maximize the linear dimension of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ in the direction of the negative real axis, then the requirement of univalence of the family (20) can be replaced by a weaker requirement to be typically real. We recall that an analytic function in $D$ is said to be typically real in the sense of Rogozinsky if real preimages correspond to real values of the function [19]. In other words, a function that is typically real in $D$ must map an open upper semicircle to an open upper (or lower) half-plane. We give a solution of this problem for $T=1$. The function $\Phi\left(z\right)=\left({1-\gamma}\right)\frac{{z}}{{1-\gamma z}}$ is univalent in $D$ for $\gamma\in\left[{0,1}\right)$. The polynomial $zq(z)$ is also univalent for $z\in D$. Therefore, the function $\Phi\left(z\right)=\left({1-\gamma}\right)\frac{{zq(z)}}{{1-\gamma zq(z)}}$ is univalent for $\gamma\in\left[{0,1}\right)$ and $z\in D$ as a superposition of univalent functions. In this case, the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}=\left\{{\frac{1}{{1-\gamma }}\left({\frac{1}{{zq(z)}}-\gamma}\right):z\in D}\right\}$. This set is obtained as a result of shifting the set $\left\{\frac{1}{zq(z)}:z\in D\right\}$ by $\gamma$ and the subsequent extension in $\frac{1}{1-\gamma}$ times (Fig 5). <figure><img src="content_image/1709.10410/fig5.jpg"><figcaption>Figure 5. The sets (¯C∖Φ(¯D))∗ for N=9, σ=1.8, γ=0 – blue, γ=0.6 – black.</figcaption></figure> The system (12) takes the form (21) \[{x_{n+1}}=\left({1-\gamma}\right)\sum_{j=1}^{N}{{a_{j}}f({x_{n-j+1}})}+\gamma \sum_{j=1}^{N}{{a_{j}}{x_{n-j+1}}}.\] The role of the parameter $\gamma$ is clearly visible in Fig 5. The value of $N$ plays a dual role with respect to the parameter $\gamma$. Increasing $N$ can reduce $\gamma$, leaving the linear dimensions of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ almost unchanged. We consider the system (12) for $N=1$: (22) \[{x_{n+1}}=\left({1-\gamma}\right)f({x_{n}})+\gamma{x_{n}}.\] The boundary of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ is a circle passing through the point $\left({-\frac{{1+\gamma}}{{1-\gamma}},0}\right)$. Consider system (12) with other averaging parameters ${\gamma_{1}}$, ${a_{1}},\ldots,{a_{N}}$ (23) \[{x_{n+1}}=\left({1-{\gamma_{1}}}\right)\sum_{j=1}^{N}{{a_{j}}f({x_{n-j+1}})}+{ \gamma_{1}}\sum_{j=1}^{N}{{a_{j}}{x_{n-j+1}}}.\] For this system, the boundary of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ passes through the point $\left({-\frac{{q_{N}^{-1}+{\gamma_{1}}}}{{1-{\gamma_{1}}}},0}\right)$, where ${q_{N}}=-\sum_{j=1}^{N}{{{(-1)}^{j}}{a_{j}}}$, and the coefficients ${a_{1}},\ldots,{a_{N}}$ are calculated using the formulas of Section 2.2 for some $\sigma\in\left[{1,2}\right]$. Let $q_{N}^{-1}<\frac{{1+{\gamma_{1}}}}{{1-{\gamma_{1}}}}$. This means that the linear dimension of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ for the system (22) is greater than for the system (23). In order for the linear dimensions of these sets to be almost equal, the second set must be stretched. The stretching coefficient is determined by the parameter ${\gamma_{1}}$. It is not difficult to establish a connection between the parameters $\gamma$ and ${\gamma_{1}}$: \[{\gamma_{1}}=\frac{{1-q_{N}^{-1}}}{2}+\frac{{1+q_{N}^{-1}}}{2}\gamma.\] Let, for example, $\gamma=0.9$, $N=5$, $\sigma=1.0$. Calculate $q_{N}^{-1}\approx 5.0$. Then ${\gamma_{1}}\approx 0.7$. This example shows how much the ${\gamma_{1}}$ parameter can be made smaller compared to $\gamma$. The sets ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ for systems (22), (23) are shown in Fig. 6. <figure><img src="content_image/1709.10410/fig6.jpg"><figcaption>Figure 6. The sets (¯C∖Φ(¯D))∗ for the systems (22) (blue) and (23) (black)for N=5, σ=1.0, γ=0.9, γ1≈0.7.</figcaption></figure> Let us give some more examples for $\sigma=1.4$, $\sigma=1.8$, $\sigma=2.0$ and $N=5$. If $\sigma=1.4$, then $q_{N}^{-1}\approx 7.856$ and ${\gamma_{1}}\approx 0.557$. If $\sigma=1.8$, then $q_{N}^{-1}\approx 11.640$ and ${\gamma_{1}}\approx 0.368$. If $\sigma=2.0$, then $q_{N}^{-1}\approx 13.928$ and ${\gamma_{1}}\approx 0.254$. <figure><img src="content_image/1709.10410/fig7a.jpg"><figcaption>Figure 7. The sets (¯C∖Φ(¯D))∗ for the systems (22) (blue) and (23) (black)for a) N=5, σ=1.4, γ=0.9, γ1≈0.557; b) N=5, σ=1.8, γ=0.9, γ1≈0.368 c) N=5,σ=2.0, γ=0.9, γ1≈0.254.</figcaption></figure> Thus, the introduction of retardation in the feedback allows us to reduce the value of the averaging parameter $\gamma$. The advantages of this approach are discussed below. ### On the rate of convergence of perturbed solutions to the equilibrium position Consider again the system (22), and let $M$ be the localization set of the multipliers of the equilibrium position of this system. Let the control parameters $\gamma,{a_{1}},\ldots,{a_{N}}$ be chosen so that the domain ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ covers the set $M$, where $\Phi\left(z\right)=\left({1-\gamma}\right)\frac{{zq(z)}}{{1-\gamma zq(z)}}$, and $q(z)={a_{1}}+{a_{2}}z+\ldots+{a_{N}}{z^{N-1}}$. In this case, the equilibrium position will be locally asymptotically stable. This means that for the initial vectors ${x_{1}},\ldots,{x_{N}}$ lying in a sufficiently small neighborhood of the equilibrium position, the solution of the system (22) determined by these initial vectors tends to the equilibrium position. Such a neighborhood is called the basin of attraction of the equilibrium position of the system (22) in the space of the initial vectors. Evaluation of the basin of attraction is, in general, a very complicated task, and it is not the subject of this article. We note, however, that even when all the conditions for attraction of the perturbed solution to the equilibrium position are satisfied, the behavior of the perturbed solution may turn out to be complicated, and approach the equilibrium very slowly. This is the case when the multiplier of the system is near the boundary of the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$. The rate at which the perturbed solution approaches the equilibrium is determined by the maximum ${\lambda^{}}$ among the moduli of zeros of the characteristic polynomial (18) (with $q(z)={a_{1}}+{a_{2}}z+\ldots+{a_{N}}{z^{N-1}}$, $p(z)=zq(z)$, $z=\lambda^{-1}$). Constructing the $\Phi\left({\frac{1}{\rho}{e^{it}}}\right)$ maps as $\rho\leq 1$, one can obtain the level lines ${\lambda^{}}=\rho$. Fig 8 and 9 show these level lines for the polynomial (9) for $\gamma=0.9$ and the polynomial (18) for $\gamma=0.9$, $N=5$, $\sigma\in\left\{{1,1.4,1.8,2}\right\}$. A darker color shows the level lines corresponding to a larger value of $\rho$. <figure><img src="content_image/1709.10410/fig8.jpg"><figcaption>Figure 8. The set (¯C∖Φ(¯D))∗, where Φ(z)=(1−γ)z1−γz, γ=0.9.</figcaption></figure> <figure><img src="content_image/1709.10410/fig9a.jpg"><figcaption>Figure 9. The set (¯C∖Φ(¯D))∗, where Φ(z)=(1−γ)zq(z)1−γzq(z), N=5, for a)γ=0.7, σ=1, b) γ≈0.557, σ=1.4, c) γ≈0.368, σ=1.8, d) γ≈0.254, σ=2.</figcaption></figure> In Fig. 8, 9, the darker regions correspond to those values of the multipliers $\mu$ for which the maximum ${\lambda^{}}$ among the moduli of zeros of the characteristic polynomial is closer to unity. Let us consider in more detail the diagrams shown in Fig. 8 and 9. The light regions determine the effective coverage of the localization set of the multipliers of the equilibrium position of the system (1). If we use the semi-linear control by O. Morgul, then the set ${\left({\bar{C}\backslash\Phi(\bar{D})}\right)^{}}$ can theoretically be made arbitrarily large, letting $\gamma$ to unity , covering arbitrarily large regions of localization of multipliers. However, in this case, the effective coverage region shifts to a neighborhood of the point $\left({-\frac{\gamma}{{1-\gamma}},0}\right)$ i.e. it is significantly shifted from zero. Thus, if it is necessary to cover multipliers, one of which in the unit circle and the other on the negative real axis and at a considerable distance from zero, the first of them will appear in the “dark” region, and therefore the eigenvalue that corresponds to it, lies close to the boundary of the unit circle. This, in turn, means a very slow aspiration of the perturbed solution to the equilibrium position. When generalized semilinear control is used, the effective coverage area of the localization set of the multipliers is sufficiently close to zero and extends to the negative real axis at $\gamma\to 1$ or $N\to\infty$. Also from Fig. 9, the role of the parameter $\sigma$ is seen. Thus, the use of generalized semilinear control makes it possible to accelerate the convergence of perturbed solutions to the equilibrium position in comparison with the control by O. Morgul. Especially in the case of a large spread of multipliers, the equilibrium positions of the system (1). ## 4. Applications to computational methods for solving systems of equations In this section, we consider several examples of the application of the method of stabilizing the equilibrium position of the system (1) by control (11) to the possibility of generalizing the known iterative processes for solving systems of linear and nonlinear equations [20]. ### Nonlinear equations Consider the computational scheme of the method of simple iterations (or the Richardson method) of solving a system of algebraic equations, generally speaking with complex coefficients (24) \[F\left(x\right)=0,\] where the differentiable function $F:{C^{m}}\to{C^{m}}$. To solve the system (24) an auxiliary difference system is constructed (25) \[{x_{n+1}}={x_{n}}+G\left({x_{n}}\right)F\left({x_{n}}\right),\] where $G\left({x_{n}}\right)$ is a matrix to be chosen. The equilibrium positions of system (25) coincide with the solutions of system (24). In the classical scheme of simple iterations, the matrix $G\left({x_{n}}\right)$ is chosen from the condition that the multipliers of the equilibrium position of the system (25) belong to the interval $\left({-1,1}\right)$. This condition can be weakened: the matrix $G\left({x_{n}}\right)$ should be chosen so that the multipliers of the equilibrium position of the system (25) are real and less than unity. For example, we can take $G\left({x_{n}}\right)=-{\left[{F^{\prime}({x_{n}})}\right]^{}}$, where $F\left(x\right)$ is the Jacobian matrix, the sign $$ means Hermitian transposition. Then the system (25) takes the form (26) \[{x_{n+1}}={x_{n}}-{\left[{F^{\prime}\left({x_{n}}\right)}\right]^{}}F\left({x _{n}}\right).\] Let $F\left(\xi\right)=0$. If the matrix $F^{\prime}(\xi)$ is not degenerated, then the matrix ${\left[{F^{\prime}(\xi)}\right]^{}}F^{\prime}(\xi)$ is positive definite, i.e all its eigenvalues are greater than zero. Consequently, all the eigenvalues of the matrix $\left({I-{{\left[{F^{\prime}(\xi)}\right]}^{}}F^{\prime}(\xi)}\right)$, where $I$ is unit matrix, are real and less than one. Let these eigenvalues lie in the interval $\left({-\hat{\mu},1}\right)$. We organize the iteration process for system (26) according to the scheme (12): (27) \[{x_{n+1}}=\sum_{j=1}^{N}{{a_{j}}{x_{n-j+1}}}-(1-\gamma)\sum_{j=1}^{N}{{a_{j}}{ {\left[{F^{\prime}\left({{x_{n-j+1}}}\right)}\right]}^{}}F\left({{x_{n-j+1}}} \right)},\] where $0<\gamma<1$, the coefficients ${a_{1}},\ldots,{a_{N}}$ are calculated using the formulas of Section 2.2 (for some $\sigma\in\left[{1,2}\right]$). Denote by ${q_{N}}=-\sum_{j=1}^{N}{{{(-1)}^{j}}{a_{j}}}$. Then $\gamma$ and $N$ should be chosen from the conditions: $\frac{{q_{N}^{-1}+\gamma}}{{1-\gamma}}>\hat{\mu}$, $0<\gamma<1$. For example, if $\sigma=2$, then ${a_{j}}=2\tan\frac{\pi}{{2(N+1)}}\left({1-\frac{j}{{N+1}}}\right)\sin\frac{{ \pi j}}{{N+1}},j=1,\ldots,N$, ${q_{N}}={\tan^{2}}\frac{\pi}{{2(N+1)}}$, and the inequality $\frac{{{{\cot}^{2}}\frac{\pi}{{2(N+1)}}+\gamma}}{{1-\gamma}}>\hat{\mu}$ must hold. If $\sigma=1$, then ${a_{j}}=\frac{2}{N}\left({1-\frac{j}{{N+1}}}\right),j=1,\ldots,N$, ${q_{N}}=\frac{1}{N}$, and the inequality $\frac{{N+\gamma}}{{1-\gamma}}>\hat{\mu}$ must hold. The iterative process will converge to the equilibrium position, provided that the initial vectors lie in the region of attraction of this equilibrium position. We note that the scheme (27) can be replaced by a similar, more economical from the computational point of view \[\left\{{\begin{array}[]{{20}{c}}{{{\hat{x}}_{n}}=\sum_{j=1}^{N}{{a_{j}}{x_{n- j+1}}},}\\ {{x_{n+1}}={{\hat{x}}_{n}}-(1-\gamma){{\left[{F^{\prime}\left({{{\hat{x}}_{n}} }\right)}\right]}^{}}F\left({{{\hat{x}}_{n}}}\right),n>N}\end{array}}\right..\] ### A generalized method for a simple iteration of the solution of systems of linear equations If the system (24) is linear, i.e. $Ax-b=0$, then the system (26) takes the form \[{x_{n+1}}=\left({I-{A^{}}A}\right){x_{n}}+{A^{}}b,\] then, accordingly, the control system (27) becomes (28) \[\left\{{\begin{array}[]{{20}{c}}{{{\hat{x}}_{n}}=\sum_{j=1}^{N}{{a_{j}}{x_{n- j+1}}},}\\ {{x_{n+1}}=\left({I-(1-\gamma){A^{}}A}\right){{\hat{x}}_{n}}+(1-\gamma){A^{} }b,n>N}\end{array}}\right..\] In the case when the matrix $A$ is symmetric positive definite, the iteration scheme is simplified (29) A similar scheme is also suitable for inversion of matrices (30) \[\left\{{\begin{array}[]{{20}{c}}{{{\hat{X}}_{n}}=\sum_{j=1}^{N}{{a_{j}}{X_{n- j+1}}},}\\ {{X_{n+1}}=\left({I-(1-\gamma){A^{}}A}\right){{\hat{X}}_{n}}+(1-\gamma){A^{} },n>N}\end{array}}\right.\] or for a symmetric positive definite matrix $A$ (31) where ${X_{n}}$ is a matrix. Theoretically, the iterative processes (28), (29), (30), (31) converge for any initial values, unlike the usual simple iteration schemes. One advantage of these schemes over other methods of solving linear equations is the absence of division operations in computational processes, which makes it possible to carry out calculations with ill posed matrices. We note that for $\gamma=0$, $N=1$, the generalized simple iteration method coincides with the classical simple iteration method. ### The generalized Seidel method for solving systems of linear equations Suppose that the diagonal elements of $A$ are nonzero. We represent the matrix $A$ in the form \[A=L+\hat{D}+U,\] where $\hat{D}$ is a diagonal matrix, the matrices $L$ and $U$ are lower and upper triangular matrices with zero diagonals. The classical Seidel method consists of assigning the initial vector ${x_{0}}$ and sequentially computing the vectors ${x_{n}}$: $(L+\hat{D}){x_{n+1}}=-U{x_{n}}+b$, and then ${x_{n+1}}=-{(L+\hat{D})^{-1}}U{x_{n}}+{(L+\hat{D})^{-1}}b$. Of course, this method does not need to build the matrix ${(L+\hat{D})^{-1}}$. The Seidel method converges if all the eigenvalues of the matrix ${(L+\hat{D})^{-1}}U$ lie in the central unit disc of the complex plane. This condition is satisfied, for example, if the matrix $A$ is symmetric positive definite. Let us generalize the Seidel method. We apply to the system ${x_{n+1}}=-{(L+\hat{D})^{-1}}U{x_{n}}+{(L+\hat{D})^{-1}}b$ the computational scheme (21). We get \[\left\{\begin{array}[]{{20}{c}}\hat{x}_{n}=\sum_{j=1}^{N}{{a_{j}}{x_{n-j+1}}} ,\\ {x_{n+1}}=\left({\gamma I-(1-\gamma){{(L+\hat{D})}^{-1}}U}\right){{\hat{x}}_{n }}+(1-\gamma){{(L+\hat{D})}^{-1}}b,\quad n>N.\end{array}\right.\] After simple transformations, the generalized method of P.L. Seidel is reduced to an iterative scheme (32) \[\left\{\begin{array}[]{{20}{c}}\hat{x}_{n}=\sum_{j=1}^{N}a_{j}x_{n-j+1},\\ (L+\hat{D}){x_{n+1}}=\left({-U+\gamma A}\right){{\hat{x}}_{n}}+(1-\gamma)b, \quad n>N.\end{array}\right.\] For $\gamma=0$, $N=1$, the generalized Seidel method coincides with the classical one. If the matrix $A$ is symmetric positive definite, then it is sufficient to take $N=1$ in (32). Let us study the question of the convergence of the iterative scheme (32). Let ${\mu_{1}},\ldots,{\mu_{m}}$ be the eigenvalues of the matrix $-{(L+\hat{D})^{-1}}U$. We consider the polynomial (18) for $p({\lambda^{-1}})={a_{1}}{\lambda^{-1}}+\ldots+{a_{N}}{\lambda^{-N}}$, $q({\lambda^{-1}})={a_{1}}+\ldots+{a_{N}}{\lambda^{-N+1}}$. If all zeros of this polynomial lie in the central unit circle, then the iterative scheme (32) converges. For a suitable choice of $N$, ${a_{1}},\ldots,{a_{N}}$, the scheme (32) converges if the eigenvalues of the matrix $-{(L+\hat{D})^{-1}}U$ lie, for example, in the set $M\subseteq\left\{{\mu\in\mathbb{C}:\left\|\mu\right\|<1}\right\}\cup\left\{{\mu \in\mathbb{R}:\mu\in\left({-\hat{\mu},1}\right)}\right\}$, $\hat{\mu}>1$, or in the set $M\subseteq\left\{{\mu\in\mathbb{C}:\left\|\mu\right\|<1}\right\}\cup\left\{{\mu \in\mathbb{C}:\left\|{\mu+R}\right\|<R}\right\}$, $R>{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}$. For the inversion of the matrix $A=L+\hat{D}+U$, we can apply the scheme (33) \[\left\{{\begin{array}[]{{20}{c}}{{{\hat{X}}_{n}}=\sum_{j=1}^{N}{{a_{j}}{X_{n- j+1}}},}\\ (L+\hat{D}){X_{n+1}}=\left({-U+\gamma A}\right){{\hat{X}}_{n}}+(1-\gamma)I, \quad n>N.\end{array}}\right.\] We note that the complexity of generalized methods increases quite insignificantly compared with the classical ones, at each iteration it is additionally necessary to perform several addition and multiplication operations. Similarly, we can generalize other stationary, and even nonstationary, iterative methods for solving systems of algebraic equations ## 5. Numerical simulation Example 1. Consider a system of nonlinear equations (34) \[\left\{{\begin{array}[]{{20}{c}}{{f_{i}}\left({x,y,z}\right)=0}\\ {i=1,2,3,}\end{array}}\right.\] where ${f_{1}}\left({x,y,z}\right)=-x+{x^{3}}+{y^{2}}+7{z^{4}}-1$, ${f_{2}}\left({x,y,z}\right)=x-y+2z$, ${f_{3}}\left({x,y,z}\right)={(x-y-8z)^{4}}-z$. This system was studied in [21]. For its solution we used simple iteration and Newton methods. These methods were used to find the solutions $\left({1,1,0}\right)$ and $\left({-1,-1,0}\right)$, where it was noted that for the convergence, the initial approximation $\left({{x_{0}},{y_{0}},{z_{0}}}\right)$ must be close to the solution. This is especially true for ${z_{0}}$. To solve system (34) we apply the iterative process (27). We calculate \[{\left[{F^{\prime}\left({x,y,z}\right)}\right]^{}}=\left({\begin{array}[]{{2 0}{c}}{-1+3{x^{2}}}&1&{4{{(x-y-8z)}^{3}}}\\ {2y}&{-1}&{-4{{(x-y-8z)}^{3}}}\\ {28{z^{3}}}&2&{-32{{(x-y-8z)}^{3}}-1}\end{array}}\right)\] and set $N=3$, $\gamma=0.91$, $\sigma=1.4$ in (27). Then ${a_{1}}\approx 0.46798$, ${a_{2}}\approx 0.37603$, ${a_{3}}\approx 0.15600$. For the initial approximations we take three points \[\left({{x_{0}},{y_{0}},{z_{0}}}\right)=\left({1.55,0.74,0.12}\right),\] \[\left({{x_{0}},{y_{0}},{z_{0}}}\right)=\left({0.84,0.8,-0.01}\right),\] \[\left({{x_{0}},{y_{0}},{z_{0}}}\right)=\left({-0.91,-1.1,-0.005}\right).\] Next, for each of these points, put $\left({{x_{i}},{y_{i}},{z_{i}}}\right)=\left({{x_{0}},{y_{0}},{z_{0}}}\right)$, $i=1,2$. Then the iterative process (27) converges, and for each initial point to a different solution: for the first one to $\left({0.95134,1.04417,0.04642}\right)$, for the second one to $\left({1,1,0}\right)$ and for the third one to $\left({-1,-1,0}\right)$. The graphs $\left({n,{x_{n}}}\right)$, $\left({n,{y_{n}}}\right)$, $\left({n,{z_{n}}}\right)$ for different the initial points are shown in Fig. 10 (for the first – black, for the second – green, for the third – blue). <figure><img src="content_image/1709.10410/fig10a.jpg"><figcaption>Figure 10. The graphs a) (n,xn), b) (n,yn), c) (n,zn) of the iterative process(27) of the solution of the system (34) for different initial points</figcaption></figure> Fig. 11 shows the graphs of the discrepancy $\left({n,{\varepsilon_{n}}}\right)$, where \[{\varepsilon_{n}}=\left\|{{f_{1}}({x_{n}},{y_{n}},{z_{n}})}\right\|+\left\|{{f_{2 }}({x_{n}},{y_{n}},{z_{n}})}\right\|+\left\|{{f_{3}}({x_{n}},{y_{n}},{z_{n}})} \right\|.\] <figure><img src="content_image/1709.10410/fig11.jpg"><figcaption>Figure 11. Graphs of the residual (n,εn) of the iterative process (27) of thesolution of system (34) for different initial points</figcaption></figure> We also give the values of the first iterations and the discrepancy graph for the iterative process (26) and the initial vector $\left({{x_{0}},{y_{0}},{z_{0}}}\right)=\left({1.00001,0.99999,0}\right)$: $\left({{x_{7}},{y_{7}},{z_{7}}}\right)=\left({1.086,0.910,0.246}\right)$, $\left({{x_{8}},{y_{8}},{z_{8}}}\right)=\left({234.865,-233.087,-1867.571}\right)$. <figure><img src="content_image/1709.10410/fig12.jpg"><figcaption>Figure 12. The discrepancy graph of the (n,εn) iterative process (26) of thesystem solution (34)</figcaption></figure> Compared with the simple iteration method and the Newton method, the proposed method turned out to be more efficient, allowing us to find one more solution, and the basin of attraction of the equilibrium turns out to be much larger. Example 2. Consider the matrix (35) \[A=\left({\begin{array}[]{{20}{c}}1&2&3\\ 2&{-2}&{-10}\\ 3&{-10}&1\end{array}}\right)\] and let us apply the iterative process (33) for its inversion. Since the eigenvalues of the matrix $A$ are $\left\{{-11.58,1.85,9.73}\right\}$, the method of simple iterations of the inversion of this matrix will diverge. We expand the matrix $A$ as \[A=L+\hat{D}+U=\left({\begin{array}[]{{20}{c}}0&0&0\\ 2&0&0\\ 3&{-10}&0\end{array}}\right)+\left({\begin{array}[]{{20}{c}}1&0&0\\ 0&{-2}&0\\ 0&0&1\end{array}}\right)+\left({\begin{array}[]{{20}{c}}0&2&3\\ 0&0&{-10}\\ 0&0&0\end{array}}\right),\] and find the eigenvalues of the matrix $-{\left({L+\hat{D}}\right)^{-1}}U$: $\left\{{0,-0.41,-72.59}\right\}$. Since these eigenvalues do not lie in the central unit circle, the Seidel method is not applicable for inversion of the matrix (35). But these eigenvalues are less than unity, therefore, we apply the generalized Seidel method. In the formula (33) we take $N=7$, $\gamma=0.743$, $\sigma=1.8$. Then ${a_{1}}\approx 0.14722$, ${a_{2}}\approx 0.21348$, ${a_{3}}\approx 0.22286$, ${a_{4}}\approx 0.19052$, ${a_{5}}\approx 0.13372$, ${a_{6}}\approx 0.07116$. As initial approximations, we take matrices: $X[1]$ is a unit matrix, $X[2],\ldots,X[7]$ is zero. Then \[X[250]=\left({\begin{array}[]{{20}{c}}{0.490}&{0.154}&{0.067}\\ {0.154}&{0.038}&{-0.077}\\ {0.067}&{-0.077}&{0.029}\end{array}}\right).\] Denote by ${\varepsilon_{n}}={\left\\|{X[n]A-I}\right\\|_{1}}$, where the norm ${\varepsilon_{n}}={\left\\|\bullet\right\\|_{1}}$ is defined as the sum of the absolute values of the matrix components. We calculate $\varepsilon_{250}\approx 3\cdot 10^{-9}$. To visualize the convergence of the matrix inversion process, we plot the discrepancy graph <figure><img src="content_image/1709.10410/fig13.jpg"><figcaption>Figure 13. The discrepancy graph (n,εn) of the iterative process (33) of theinverse of the matrix (35)</figcaption></figure> Because of poor initial approximation, the discrepancy at the first few steps increased sharply, however, after ten steps this discrepancy began to decrease rapidly. This confirms the practical effectiveness of the proposed iterative scheme. We note that, both with increasing $\gamma$, and with decreasing $N$, the rate of convergence will decrease. For comparison, we give numerical calculations using Morgul’s scheme, i.e. In the formula (33) we take $N=1$. For this case, the best value for $\gamma$ will be $0.974$. The required accuracy is achieved at 800 step. We give the graphs of the discrepancy of the previous scheme and Morgul’s scheme for the first 80 iterations. <figure><img src="content_image/1709.10410/fig14.jpg"><figcaption>Figure 14. Graphs of the discrepancy (n,εn) of the iterative process (33) ofthe inverse of the matrix (35) for N=7 (black) and N=1 (blue)</figcaption></figure> It can be seen that the discrepancy of Morgul’s method decreases monotonically, but is much slower than the discrepancy of the generalized semilinear control. ## 6. Conclusion In the article the problem of stabilization of unstable and a priori unknown periodic orbits of nonlinear systems with discrete time is considered. A new approach to constructing delayed feedback, which solves the stabilization problem, is proposed. The feedback is represented as a convex combination of nonlinear feedback and semilinear feedback introduced by O. Morgul. This preserves the advantages of both types of feedback. The methods of geometric complex analysis were used to construct the nonlinear feedback gain factors and to obtain the conditions for the applicability of such control. These methods are used to analyze the possibility of using Morgul’s scheme. The necessary and sufficient conditions for stabilization in the form of a geometric criterion for local asymptotic stability are obtained. Morgul’s method was also transferred from the scalar case to the vector one. It is important to note that the characteristic polynomials for periodic orbits in the nonlinear and semilinear cases have a very simple structure, although, naturally, different. It was this circumstance that stimulated the integration of the two approaches mentioned above. The resulting characteristic polynomial also has a rather simple structure and contains, as special cases, polynomials of nonlinear and semilinear control schemes. The geometric criterion of stability in the nonlinear and semilinear cases consists of the analysis of images of the central unit circle under a special polynomial mapping. In a combined nonlinear-semilinear control method, instead of polynomial mappings, one has to study rational mappings. In this paper, we give a solution to the construction of quasioptimal fractional-rational maps for the case $T=1$, that is, to stabilize the equilibrium positions. An additional introduction to the control of semilinear feedback allows us to significantly reduce the length of the used prehistory in the delayed feedback and to increase the rate of convergence of the perturbed solutions to the periodic ones. As an application of the proposed stabilization scheme, a possible computational algorithm for finding solutions of systems of algebraic equations is presented, based on the modification of known iterative schemes. In these schemes, the values of the variables computed in the previous steps are used. At the same time, the complexity of the new iterative schemes practically does not increase. The above results of numerical solutions of systems of linear and nonlinear equations confirm our solution as an improvement of previous work and the effectiveness of the proposed equilibrium stabilization schemes. ## 7. acknowledgment The authors are deeply grateful to Alexei Solyanik and Emil Iacob for their valuable comments and help in preparation of manuscript. ## References [1] Jackson E.A. Perspectives of Nonlinear Dinamics. Vol. I, II, - Cambridge Univ. Press, Cambridge, 1980, 1990 Chaos II, ed. Hao Bai-Lin. – World Sci., (1990) * [2] Ott E., Grebodgi C., Yorke J.A. Controlling chaos. Phys. Rev. Lett. 64, 1196-1199 (1990) * [3] Chen G., Dong X. From chaos to order: Methodologies, Perspectives and Application. World Scientific, Singapore (1999) * [4] Andrievsky B. R., Fradkov A. L. Control of Chaos: Methods and Applications. I. Methods, Avtomat. i Telemekh., (2003), no. 5, 3–45 * [5] Pyragas K. Continuous control of chaos by self controlling feedback. Phys. Rev. Lett. 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Oxford University press, London (1939) * [17] Khamitova A., Characteristic polynomials for a cycle of non-linear discrete systems with time delays, Vestnik of St. Petersburg State University, Series 10, Applied Mathematics, Computer Science, Control Processes (2016), Issue 4., 104-115 * [18] Dmitrishin D., Hagelstein P., Khamitova A., and Stokolos A., On the stability of cycles by delayed feedback control, Linear and Multilinear Algebra 64 (2016), 1538-1549 * [19] Rogosinski W.W. Uber positive harmonische sinusentwicklungen. Jber.Deutsch. Math. Ver., 40 (1931), 2., Abt. 33 – 35 * [20] Kelley C.T. Iterative methods for linear and nonlinear equations. SIAM, Philadelfia (1995) * [21]https://www.youtube.com/watch?v=S5hevRtjMI8 D. Dmitrishin, I. Skrynnik and E. Franzheva, Odessa National Polytechnic University, 1 Shevchenko Ave, Odessa 65044, Ukraine. e-mail: dmitrishin@opu.ua A. Stokolos, Georgia Southern University, Statesboro, GA 30458, USA. e-mail: astokolos@georgiasouthern.edu
1111.6583	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 62021, "num_imgs": 9, "llama3_tokens_count": 13869 }	[ "content_image/1111.6583/x1.png", "content_image/1111.6583/x1.png", "content_image/1111.6583/x3.png", "content_image/1111.6583/x3.png", "content_image/1111.6583/SW400x200-0days.png", "content_image/1111.6583/initial_tracer.png", "content_image/1111.6583/shockbubble_classic.png", "content_image/1111.6583/checker.png", "content_image/1111.6583/stress_checker.png" ]	# PyClaw: Accessible, Extensible, Scalable Tools for Wave Propagation Problems David I. Ketcheson King Abdullah University of Science and Technology, Box 4700, Thuwal, Saudi Arabia, 23955-6900 (david.ketcheson@kaust.edu.sa) Kyle T. Mandli University of Texas at Austin, 1 University Station C0200 Austin, TX 78712-0027 (kyle@ices.utexas.edu) Aron J. Ahmadia King Abdullah University of Science and Technology, Box 4700, Thuwal, Saudi Arabia, 23955-6900 (aron.ahmadia@kaust.edu.sa) Amal Alghamdi King Abdullah University of Science and Technology, Box 4700, Thuwal, Saudi Arabia, 23955-6900 (amal.alghamdi@kaust.edu.sa) Manuel Quezada de Luna Texas A&M University, College Station, TX 77843 (mquezada@math.tamu.edu) Matteo Parsani King Abdullah University of Science and Technology, Box 4700, Thuwal, Saudi Arabia, 23955-6900 (matteo.parsani@kaust.edu.sa) Matthew G. Knepley University of Chicago, 5735 S. Ellis Ave. Chicago, IL 60637 (knepley@ci.uchicago.edu) Matthew Emmett University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 (memmett@unc.edu) ###### Abstract Development of scientific software involves tradeoffs between ease of use, generality, and performance. We describe the design of a general hyperbolic PDE solver that can be operated with the convenience of MATLAB yet achieves efficiency near that of hand-coded Fortran and scales to the largest supercomputers. This is achieved by using Python for most of the code while employing automatically-wrapped Fortran kernels for computationally intensive routines, and using Python bindings to interface with a parallel computing library and other numerical packages. The software described here is PyClaw, a Python-based structured grid solver for general systems of hyperbolic PDEs [18]. PyClaw provides a powerful and intuitive interface to the algorithms of the existing Fortran codes Clawpack and SharpClaw, simplifying code development and use while providing massive parallelism and scalable solvers via the PETSc library. The package is further augmented by use of PyWENO for generation of efficient high-order weighted essentially non-oscillatory reconstruction code. The simplicity, capability, and performance of this approach are demonstrated through application to example problems in shallow water flow, compressible flow and elasticity. ## 1 Introduction Traditionally, scientific codes have been developed in compiled languages like Fortran or C. There exists an abundance of well-tested, efficient, often serial implementations of numerical algorithms in those languages. It is often desirable to parallelize and extended such codes with new algorithmic components in order to apply them to increasingly challenging problems. More recently, high-level scientific programming languages, such as MATLAB, IDL, and R, have also become an important platform for numerical codes. These languages offer powerful advantages: they allow code to be written in a language more familiar to scientists and they permit development to occur in an evolutionary fashion. Problem parameters can be specified and plotting can be performed interactively, bypassing the comparatively slow edit/compile/run/plot cycle of development in Fortran or C [23]. However, programs written in such high-level languages are not portable to high performance computing platforms and may be very slow compared to compiled code. We present one approach to leveraging the advantages of both kinds of languages. Our starting point is Clawpack [16]: a widely used, state-of-the-art package for solving hyperbolic systems of partial differential equations, such as those arising in fluid mechanics, astrophysics, geodynamics, magnetohydrodynamics, oceanography, porous media flow, and numerous other fields. In this paper we present PyClaw, a package that brings greater accessibility, flexibility, and parallelization to Clawpack and related packages. PyClaw is used as an illustration to describe, demonstrate, and provide support for a particular approach to construction of scientific software. The approach we advocate consists of three steps: 1. use Python to create a convenient interface to serial legacy code; 2. use Python to interface the resulting code with software tools for parallelization of the code, with minimal modification of the serial legacy code; 3. use Python to interface the resulting parallel code to other packages that provide extended functionality. In the case of PyClaw, (i) consists of a Python interface to the Clawpack and SharpClaw Fortran-based packages for numerical solution of systems of hyperbolic PDEs. This interface allows the code to be operated in the same convenient and interactive way that one works with MATLAB. In step (ii), PyClaw was parallelized by interfacing with PETSc, a state-of-the-art library for parallel scientific computing. This enables parallel computation that scales to large supercomputers and achieves on-core performance close to that of the legacy code. Finally, step (iii) is illustrated in that PyClaw was interfaced with PyWENO, to increase the available order of accuracy of numerical approximation. A key in all three steps is the use of the numerical Python package numpy [22]. The idea of using a layer of numpy-based Python code on top of Fortran, C, or C++ kernels to solve PDEs has become increasingly popular over the past several years; see for instance [19, 6]. We consider PyClaw to be a very convincing case study. PyClaw is one of the most highly scalable Python-based codes available. Perhaps the first well-known scientific project to provide a parallel solver in Python is GPAW, which extends the Python interpreter itself with parallel data structures and algorithms for electronic structure calculations [20]. Python has also previously been used as a tool for parallelizing Fortran and C codes by introducing parallel code in a Python layer that also calls the Fortran/C kernels [21]. In the FiPy package, parallelism is achieved by using an existing parallel library (Trilinos) through its Python bindings [6]. PyClaw takes an approach similar to that of FiPy, in which all parallel operations over distributed-memory processes are handled by the PETSc library through the petsc4py Python package (http://code.google.com/p/petsc4py/). This approach offers the advantages of utilizing a documented, well-designed abstract parallel interface for developers that is already known to achieve excellent scaling on many architectures. The algorithms of Clawpack and its high-order extension, SharpClaw, are described in Section 2. The PyClaw framework is described in Section 3. Section 4 describes the parallelization of PyClaw. As demonstrated in Section 5, PyClaw maintains both the serial performance of Clawpack and the parallel scalability of PETSc. In Section 6, we briefly highlight some of the software development practices that have contributed to the success of PyClaw. The combination of wave propagation algorithms and scalable parallelization enables efficient solution of interesting scientific problems, as demonstrated through three examples in Section 7. A repository containing the data, software, and hardware environments for reproducing all experimental results and figures in this paper is available online (http://bitbucket.org/ahmadia/pyclaw-sisc-rr). The most recent release of the PyClaw code is hosted at http://github.com/clawpack/pyclaw and can be installed alongside its dependencies in the clawpack distribution in a few minutes with the following pip commands (pip is a freely available Python package installer and manager): ``` 1pip install numpy 2pip install clawpack ``` Listing 1: Installing the most recent release of PyClaw and its dependencies The petsc4py package is not an explicit dependency of PyClaw, but if it has been installed, the PetClaw parallel extension described in Section 4 is seamlessly enabled. It is our hope that readers will download and try the code. To whet the reader’s appetite, an example of a complete PyClaw program is shown in Listing 2. This example sets up, runs, and plots the solution of a two-dimensional inviscid fluid dynamics problem (specifically, test case 6 of [13]). <figure><img src="content_image/1111.6583/x1.png"><figcaption>(a) Structure of a PyClaw Solution object, which may contain multiple Gridobjects, each of which may have multiple associated State objects. Each Stateobject has a associated fields of conserved quantities (e.g., density,momentum) and optionally, associated auxiliary property fields.</figcaption></figure> ## 2 Finite Volume Hyperbolic PDE solvers The numerical methods in PyClaw compute approximate solutions of systems of hyperbolic conservation laws: \[\kappa(\mathbf{x})\mathbf{q}_{t}+\nabla\cdot\mathbf{f}(\mathbf{q} ,\mathbf{x})_{x} =\mathbf{s}(\mathbf{q},\mathbf{x}).\] (1) Here $\mathbf{q}(\mathbf{x},t)$ is a vector of conserved quantities (e.g., density, momentum, energy) and $\mathbf{f}(\mathbf{q},\mathbf{x})$ represents the flux (modeling wave-like phenomena), while $\mathbf{s}(\mathbf{q},\mathbf{x})$ represents additional non-hyperbolic _source_ terms, such as diffusion or chemical reactions. The _capacity function_$\kappa(\mathbf{x})$ is frequently useful for taking into account variations in material properties or in the use of non-uniform grids (see [15, Chapter 6]). Here we describe high-resolution shock capturing methods, which is one of the most successful classes of numerical methods for solving (1). Computing solutions to nonlinear hyperbolic equations is often costly. Solutions of (1) generically develop singularities (shocks) in finite time, even if the initial data are smooth. Accurate modeling of solutions with shocks or strong convective character requires computationally expensive techniques, such as Riemann solvers and nonlinear limiters. In a finite volume method, the unknowns at time level $t^{n}$ are taken to be the averages of $q$ over each cell: \[Q^{n}_{i}=\frac{1}{\Delta x}\int_{x_{i-\frac{1}{2}}}^{x_{i+\frac {1}{2}}}q(x,t^{n})\,dx,\] (2) where $\Delta x=x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}$ and $i$ are the local grid spacing and the cell’s index, respectively. A simple update of the cell averages based on the resulting waves gives the classic Godunov method, a robust but only first-order accurate numerical scheme: \[Q^{n+1}_{i} =Q^{n}_{i}-\frac{\Delta t}{\Delta x}\left(F^{n}_{i+\frac{1}{2}}-F ^{n}_{i-\frac{1}{2}}\right),\] (3) where $F$ and $\Delta t$ are the _numerical flux_ function and the time step. Godunov’s method results from taking a particular choice of $F$ referred to as the _upwind flux_. The first-order method just described is very dissipative. Higher-order extensions require the computation of higher derivatives of the solution or flux. Near a solution discontinuity, or shock, spurious oscillations tend to arise due to dispersion and numerical differencing across the discontinuity. In order to combat this, shock-capturing methods use special nonlinear algorithms to compute numerical derivatives in a non-oscillatory way by limiting the the value of the computed derivative in the vicinity of a discontinuity [15]. The classic Clawpack algorithm is based on the second-order Lax-Wendroff difference scheme that was later extended by LeVeque [14, 15]. This scheme can be written in the flux-differencing form (3) by an appropriate choice of numerical flux, which has the form \[F^{n}_{i-\frac{1}{2}} =F_{\textup{upwind}}+F_{\textup{correction}},\] (4) where $F_{\textup{upwind}}$ is the Godunov flux. The classic Clawpack algorithm is based on modifying (4) by applying a limiter to $F_{\textup{correction}}$. However, in LeVeque’s extension, rather than applying the limiter to the flux variables, the limiter is applied directly to the waves computed by the Riemann solver. This allows for better accuracy by limiting only the characteristic families that are discontinuous in a given neighborhood. Furthermore, the first-order contribution is written in terms of _fluctuations_ (which approximate the quasilinear term $Aq_{x}$) rather than fluxes (which approximate $f(q)$). This allows the algorithm to be applied to hyperbolic systems not in conservation form. While the Lax-Wendroff approach can be extended to even higher order, this is cumbersome because of the large number of high order terms appearing in the Taylor series. A simpler alternative is the method of lines, in which the spatial derivatives are discretized first, leading to a system of ODEs that can be solved by traditional methods. This is the approach taken in SharpClaw [10, 9, 11]. First, a non-oscillatory approximation of the solution is reconstructed from the cell averages to give high order accurate point values just to the left and right of each cell interface. This reconstruction is performed using weighted essentially non-oscillatory (WENO) reconstruction in order to avoid spurious oscillations near discontinuities. As in the classic Clawpack algorithm, the scheme is written in terms of fluctuations rather than fluxes, so that it can be applied to non-conservative problems. ### Clawpack and SharpClaw We now describe the "legacy" Fortran codes on which PyClaw is built. The classic algorithm implemented in Clawpack (“Conservation Laws Package”) includes extensions to two and three dimensions, adaptive mesh refinement, and other enhancements [16]. The unsplit multidimensional Clawpack algorithms include additional correction terms, computed by a secondary “transverse” Riemann solver, which approximates corner transport terms. The Clawpack software (http://www.clawpack.org) and its extensions, consisting of open source Fortran code, have been freely available since 1994. More than 7,000 users have registered to download Clawpack. Clawpack is a very general tool in the sense that it is easily adapted to solve any hyperbolic system of conservation laws. The only specialized code required in order to solve a particular hyperbolic system is the Riemann solver routine. A wide range of Riemann solvers, including several for the most widely studied hyperbolic systems, have been developed by Clawpack users and are also freely available. Clawpack handles not only simple Cartesian grids but any logically quadrilateral grid, provided that there is a map from a uniform Cartesian domain to the desired physical domain (see Section 7.1). Non-hyperbolic source terms ($\mathbf{s}(\mathbf{q},\mathbf{x})$) can be easily included via operator splitting. For more examples and details regarding Clawpack, see [14] and [15, Chapter 23]. The high-order WENO-based wave propagation method is implemented in SharpClaw, another Fortran package designed similarly to Clawpack and which makes use of the same Riemann solver routines. The default options in SharpClaw employ fifth-order WENO reconstruction in space and the fourth-order strong stability preserving (SSP) Runge–Kutta method of [8] in time. In multi-dimensions SharpClaw requires propagation of waves only in the normal direction to each edge. ## 3 PyClaw PyClaw is an object-oriented framework that incorporates the functionality of Clawpack and SharpClaw. This functionality may be provided via either pure Python or calls made to the underlying Fortran routines included in Clawpack and SharpClaw. PyClaw is designed such that the user provides appropriate call-back functions, such as a Riemann solver, that describe and implement the problem in question, PyClaw than manages the “main” routine including appropriate time stepping and output. It also avoids the need to deal with strictly formatted data files and reduces the need to write custom Fortran routines for new problems. Instead, problems can be set up interactively or in a simple scripting language. PyClaw also allows for simulation and visualization to be done in a single, interactive environment. Users may engage with this framework at different levels, depending on their expertise and requirements. These interfaces are described in Section 3.1. PyClaw wraps the full functionality of the “classic” 1D, 2D and 3D Clawpack code including the use of capacity functions, mapped grids, and both dimensionally-split and fully-multidimensional algorithms. It also provides the full functionality of SharpClaw, and adds to this with higher order WENO reconstruction. It does not presently include adaptive mesh refinement, which is part of the AMRClaw and GeoClaw extensions of Clawpack. ### Interfaces The PyClaw distribution includes pre-written application scripts that solve problems in acoustics, elasticity, compressible flow, shallow water flow, and other application domains. These application scripts represent the typical “main” routines that lower-level language developers are used to, and are written with ease of understanding as their most important goal. These scripts run both on serial workstations and from batch processing queues for, e.g., 8,000-node parallel jobs without modification. Novice users can solve a wide range of problems by modifying the example scripts to deal with different domains, initial conditions, boundary conditions, and so forth. This requires only simple understanding of high-level scripting code, but allows users to compute solutions of complex problems on large supercomputers. The scripts for the applications in this paper have all been added to the distributed examples, and we plan to continue this practice with respect to future publications that use PyClaw. Novice or advanced users may also run problems and analyze results in an interactive shell. When Python is invoked with standard input connected to a tty device, it reads and executes commands interactively. This feature simplifies serial development, debugging, and visualization, and is familiar to users of commercial software such as MATLAB and Mathematica. PyClaw’s top level classes present the same API whether used in serial or parallel, allowing users to develop interactively, then run production jobs in batch environments. Advanced users may want to solve a hyperbolic system that is not included among the example applications. In Clawpack a number of Riemann solvers have been written for specific systems and are included with Clawpack. Since PyClaw and Clawpack Riemann solvers are interoperable, many systems of interest have already been implemented and can be used immediately in PyClaw. A user also has the option of writing his/her own Riemann solver in Fortran which can then be utilized in Clawpack as well as PyClaw. Clawpack users who wish to run an existing serial Clawpack application in parallel can do so easily by wrapping any problem-specific Fortran routines (such as those that set initial conditions or compute source terms) automatically with f2py and using the resulting Python function handles in PyClaw. Numerical analysts are often interested in comparing solutions of a problem obtained with different methods or different values of method parameters. The sample application scripts provide a common functional interface that can be accessed from the command line for selecting between solvers, choosing serial or parallel computation, and other options. Users are free to extend this interface to allow more programmatic flexibility at the command line or from within a batch environment. PyClaw also fully exposes the full range of command line options available from the underlying PETSc library, allowing advanced users to tweak low-level settings such as message-passing communication strategies. Frequently, research scientists are interested in comparing the performance of numerical methods. PyClaw enables this comparison by allowing scientific developers to extend the software with a new Solver class. The Solver class, described in the next section, is responsible for prescribing a single time step of the numerical algorithm. In PyClaw this is accomplished by implementing a homogenous_step routine which evolves the solution of the PDE from time $t$ to $t+\Delta t$. Since most existing codes have such a routine already, it is often straightforward to include legacy code in the PyClaw framework by simply wrapping this function. Non-hyperbolic terms $\mathbf{s}(\mathbf{q},\mathbf{x})$ can also be incorporated via operator splitting. This is the most common way to extend the Solver class and allows for easy comparison between different numerical methods. ### Classes The primary abstractions used in PyClaw are the Solver and the Solution. The Solver class provides an abstract interface to evolve a Solution object forward in time. The Solution class is a data abstraction containing information about the domain of the PDE and the state of the solution at a particular time inside the domain. Here we will discuss these classes and how they interact. The role of the Solver is illustrated in Figure 1(b). The Solver class prescribes how a State $Q^{n}$ is evolved forward in time to obtain $Q^{n+1}$; in general this consists of three parts: 1. set ghost cell values based on the prescribed boundary conditions; 2. advance the solution based on the hyperbolic terms (i.e., $\mathbf{q}_{t}+\nabla\cdot\mathbf{f}(\mathbf{q},\mathbf{x})=0$); 3. advance the solution based on the source term $\mathbf{s}(\mathbf{q},\mathbf{x})$ (if present) by a fractional-step approach [15, Chapter 17]. The base Solver class implements the basic interface to each of these functions and a subclass of the Solver class is expected to implement the appropriate functions depending on the numerical method being implemented. The Solver class is sufficiently abstract to accommodate algorithms that are based on the method of lines, such as SharpClaw, as well as algorithms that are not, such as Clawpack. The Solution class, depicted in Figure 1(a), has two purposes: * Describe the problem domain * Keep track of the values of the state variables $Q^{n}$ and PDE coefficients Like Clawpack, PyClaw simulations are always based on a computational domain composed of tensor products of one-dimensional equispaced discretizations of space. More general physical domains may be used as long as it is possible to map them to that computational domain. PyClaw includes a set of geometry classes that implement these abstractions. The solution values $Q^{n}$ and the PDE coefficients are contained in numpy arrays stored in the Solution object. Thus the full Solution class represents a snapshot of the gridded data. The class acts as a container object with one or more Grid and State objects such as in the case of adaptive mesh refinement or nested grids, both of which are possible with Clawpack algorithms, though not yet available in PyClaw. This hierarchical class structure allows the underlying data and algorithms to be modified without the knowledge of the interacting objects and without changing the interface presented to the user. An example of this is the PetClaw State object, which reimplements State functionality over distributed memory. In the future, we intend to provide State objects that implement other strategies for accelerating performance. <figure><img src="content_image/1111.6583/x1.png"><figcaption>(a) Structure of a PyClaw Solution object, which may contain multiple Gridobjects, each of which may have multiple associated State objects. Each Stateobject has a associated fields of conserved quantities (e.g., density,momentum) and optionally, associated auxiliary property fields.</figcaption></figure> ### Extension using PyWENO One of the principal achievements of PyClaw has been to facilitate the extension of Clawpack and SharpClaw by interfacing with other packages. For example, PyWENO [3] has been used to add much higher-order functionality to the existing SharpClaw code, within PyClaw. The Fortran code SharpClaw contains only fifth-order WENO routines for evaluation at cell interfaces. New WENO routines for PyClaw were generated by PyWENO, which is a standalone package for building custom WENO codes. For a given (arbitrarily high) order of reconstruction, PyWENO symbolically computes the smoothness coefficients, reconstruction coefficients, and optimal (linear) weights of the WENO method. From these symbolic results, PyWENO generates Fortran kernels that perform the WENO reconstructions (it can also generate C and OpenCL kernels). The generated kernels can optionally perform individual steps of the reconstruction process (i.e., computing smoothness indicators, nonlinear weights, or the final reconstruction) or combinations thereof. This affords authors some flexibility in avoiding redundant computations or minimizing memory allocations and accesses. Furthermore, the points within each cell at which the WENO reconstruction is performed are also arbitrary. Negative weights are automatically split into positive and negative parts [26], allowing PyWENO to generate kernels for routines for arbitrary sets of points (such as arbitrary order Gauss-Legendre quadrature points). For PyClaw, odd-order WENO routines to approximate the solution at the left and right edges of each cell were generated from fifth to seventeenth order. All aspects of the WENO reconstruction are wrapped into standalone subroutines and no temporary work arrays are allocated. Using these routines is trivially easy; for instance, to use the 9th-order WENO method instead of the classic algorithm in Listing 2 one needs only to replace line 9 by the two lines ``` 1 solver = pyclaw.SharpClawSolver1D() 2 solver.weno_order = 9 ``` Listing 3: Using SharpClaw (with PyWENO) ## 4 Parallelization Like many finite difference and finite volume codes, Clawpack and SharpClaw implement boundary conditions through the use of _ghost cells_. In this approach, fictitious layers of cells are added around the edge of the problem domain; the number of layers depends on the width of the stencil of the numerical scheme. At the beginning of each step, the ghost cell values are set to satisfy the specified boundary conditions. Then the numerical scheme is applied on all the interior (non-ghost) cells. Many types of boundary conditions can be handled well in this manner, including periodicity, reflection, and outflow (non-reflecting). Custom boundary conditions may also be specified, for instance to model time-dependent inflow. This approach is highly amenable to parallelization, since it is based on the idea that information at the edge of a domain is filled in by a routine that is independent of the rest of the numerical scheme. Therefore, the serial kernels can be applied on each processor of a distributed parallel machine as long as some routine first fills the ghost cells on the processor either by appeal to boundary conditions or through communication with neighboring processors, as appropriate. Only this ghost cell routine needs to know the global topology of the problem; the serial kernels operate based entirely on local information. This orthogonality allows independent development of serial numerical schemes and parallel communication patterns, and is a key strategy in combining the work of computational mathematicians and computer scientists. The same global-local decomposition is employed in PETSc. The PETSc library includes a DMDA object that implements parallelization through the use of ghost cells. The DMDA is a highly scalable class for data layout across parallel, structured grids. All storage allocation and communication of ghost values is handled by the DMDA, but storage is returned to the PyClaw program as numpy arrays so that no code changes are necessary and the user experience is identical. In Figure 2, we show three different representations of data over a simple $5\times 6$ structured grid. The global ordering is used as input to PETSc linear and nonlinear solvers, whereas the natural ordering is used for output since it is independent of the particular parallel partition. Local ordering is used to extract data over a “halo” region, including ghost unknowns shared with other processes. This is, in fact, how PyClaw makes use of the DMDA structure. Local vectors are extracted with a given number of overlap unknowns, and computations are performed using the same serial routines. These local vectors are then used to update a global vector, and PETSc performs the appropriate accumulation for shared unknowns. This simple mechanism in PETSc for integrating local and global data (which works also for unstructured grids) allows easy parallelization. Thus PyClaw relies on Clawpack and SharpClaw to provide computational kernels for time-dependent nonlinear wave propagation and on PETSc (through petsc4py) to manage distributed data arrays and the communication between them. The data structures in PETSc and Clawpack/SharpClaw are directly interfaced through the Python package numpy [22]. The parallel extension of PyClaw consists of only about 300 lines of Python code. Any PyClaw script can be run in parallel simply by replacing the statement from clawpack import pyclaw with ``` 1from clawpack import petclaw as pyclaw ``` Listing 4: Running in parallel and invoking the Python script with mpirun. <figure><img src="content_image/1111.6583/x3.png"><figcaption>Figure 3: Weak scaling performance profile of the shock bubble problem with160,000 grid cells per core</figcaption></figure> The serial PyClaw routines handle discretization, Riemann solves, limiting and reconstruction, since they only depend on local data. PETSc handles parallel layout and communication, but has no information about the local computations. PETSc allows fine-grained control of the ghost value communication patterns so that parallel performance can be tuned to different supercomputing architectures, but by default a user does not need to manage parallelism or see PETSc code. In fact, the PetClaw user is shielded from PETSc in much the same way that a PETSc user is shielded from MPI. This separation can enable future development. For instance, an unstructured mesh topology of hexahedral elements could be managed by PETSc, using a Riemann solver which could accommodate deformed elements, without changes to PyClaw. In addition to communication of ghost cell values, parallel hyperbolic solvers require communication of the maximum wave speed occurring on each processor in order to check whether a prescribed stability condition (generally phrased in terms of the Courant number) has been satisfied and choose the size of the next time step appropriately. This is also handled by a single PETSc call. Although the basic Clawpack and SharpClaw algorithms are explicit and require no algebraic solver, a powerful advantage gained by using PETSc for parallelization is the possibility of employing PETSc’s solvers for implicit integration of hyperbolic problems that are stiff due to source terms or fast waves that do not need to be accurately resolved. This is the subject of ongoing work. A particular challenge of using Python is that most parallel debuggers support only C or Fortran, making parallel debugging of Python codes difficult [4]. This is yet another motivation for using a tested parallel library like PETSc. ## 5 Performance A few previous works have considered efficiency of scientific Python codes in serial as well as in parallel; see for instance [1, 12, 21, 4]. Those studies consisted mainly of simple code snippets run in serial or on up to a few dozen processors until the recent work [4], which includes scalability studies up to 16,384 cores. In this section, we investigate the efficiency of a full object-oriented Python framework (PyClaw) compared with hand-coded Fortran (Clawpack). We also consider the scaling of PetClaw on all 65,536 cores of the Shaheen supercomputer at KAUST. We consider only the second-order classic Clawpack algorithm here, as we are mainly interested in the effect of using a Python framework (in the serial case) and the cost of communication (in the parallel case). In terms of these factors, roughly similar results may be expected for the performance of the higher order algorithms, and preliminary tests (not described here) indicate good scaling of those also. ### Serial performance For a detailed serial performance comparison of an explicit stencil-based PDE code in Python, see [12]. In that work, vectorized numpy code was found to be fast enough for some operations, while wrapped Fortran loops performed identically to a pure Fortran code. In contrast to the simple kernel code considered there, we present tests of a full object-oriented solver framework. Our results thus extend those of [12], providing an indication of the efficiency that can be expected for a sophisticated Python-based PDE solver framework. Table 1 shows an on-core serial comparison between the Fortran-only Clawpack code and the corresponding hybrid PyClaw implementation for two systems of equations on two different platforms. The hyperbolic systems considered are the 2D linear acoustics equation and the 2D shallow water (SW) equations [15]. The acoustics test involves a very simple Riemann solver (amounting to a $3\times 3$ matrix-vector multiply) and is intended to provide an upper bound on the performance loss arising from the Python code overhead. The shallow water test involves a more typical, costly Riemann solver (specifically, a Roe solver with an entropy fix) and should be considered as more representative of realistic nonlinear application problems. Clawpack and PyClaw rely on similar Fortran kernels that differ only in the array layout. Because most of the computational cost is in executing the low-level Fortran kernels, the difference in performance is relatively small – though not negligible. The results for the Shallow water equations are in rough agreement with the 10% overhead reported in [4]. A 10-30% increase in computational time (for realistic applications) seems well worth the advantages provided by the use of Python (in particular, easy parallelism). The overhead is expected to be even smaller for more complex systems or in three dimensions. Application \| Processor \| Clawpack \| PyClaw \| Ratio ---\|---\|---\|---\|--- Acoustics \| Intel Xeon \| 28s \| 41s \| 1.5 PowerPC 450 \| 192s \| 316s \| 1.6 Shallow Water \| Intel Xeon \| 79s \| 99s \| 1.3 PowerPC 450 \| 714s \| 800s \| 1.1 Table 1: Timing results (in seconds) for on-core serial experiments solving acoustics and shallow water problems implemented in both Clawpack and PyClaw on Intel Xeon and the IBM BlueGene/P PowerPC 450 processors ### Parallel performance We now investigate the parallel performance of PetClaw on the Shaheen supercomputer at KAUST, an IBM BlueGene/P system consisting of 65,536 cores. When characterizing the performance of scientific codes on supercomputers, a commonly used characterization is that of _weak scalability_, which is assessed by studying how the run time of the code is affected when the resolution of the simulation and the number of processors is increased commensurately to maintain a fixed amount of work per processor. The parallel efficiency is given by dividing the run time of the single processor job by the run time of the parallel job. The problem used for the comparisons is of a compressible, inviscid flow that consists of a shock impacting a low-density bubble (examined in detail in Section 7.2). We investigate weak scaling by running the problem for a fixed number of time steps and with a fixed number of grid cells ($400\times 400=160,000$) per core, while increasing the number of cores from one up to the whole machine. Figure 3 shows the results, with parallel efficiency provided in the last row. It is important to note that the time required to load the necessary Python packages and shared objects, which occurs only once at the beginning of a simulation (or series of batch simulations) has been excluded from the results presented here. This load time is discussed in the next section. Observe that in all parallel runs, more than 90% of the time is spent in the computational kernels. The parallel operations scale extremely well: the the CFL condition-related reduction takes essentially the same amount of time for all runs from 16 processors up, as does the communication of ghost cell values in localToGlobal. Together these parallel operations consume about 6% of the total run time. Parallel initialization consists of PETSc parallel object construction, including memory allocation and MPI communicator initialization. Note that the parallel initialization, while significant in these artificial test runs, will not contribute significantly to the cost of real simulations because it is a one-time cost. <figure><img src="content_image/1111.6583/x3.png"><figcaption>Figure 3: Weak scaling performance profile of the shock bubble problem with160,000 grid cells per core</figcaption></figure> ### Dynamic Loading As alluded to already, the work of loading Python libraries and objects dynamically at run-time does not currently scale well on the Shaheen system. Large-scale supercomputers such as Shaheen rely on parallel file systems that are designed to support large distributed loads, with each process independently accessing data. Dynamic loading does not follow this pattern because every process is attempting to access the same data simultaneously. This issue was partially addressed in [4], but an implementation capable of supporting dynamic library loading is still lacking. The dynamic loading time for the PetClaw runs in Section 7 is less than 5% of the total simulation time, and this will generally be the case for 2D wave propagation problems because the CFL condition means that large simulations of hyperbolic problems necessarily require long run times in order for waves to propagate across the full domain. ## 6 Software Development Practices The majority of software development practices utilized in PyClaw are inherited from the open source software community. The community’s atmosphere of free and open sharing complements the tradition of scientific inquiry. In fact, growing movements within the scientific community seek to embrace _scientific reproducibility_ for software tools used in conducting mathematical and scientific research [5, 27]. In addition to making our results reproducible, we also intend that our software be useful as a reference for understanding numerical methods involved in solving hyperbolic PDEs and as a platform for extending and applying these techniques. As such, we also seek to provide a friendly and inviting context for scientists working in this cross-disciplinary environment to conduct their research. ### Application Tests The goal of reproducibility in research is to improve not only confidence in results, but their extensibility as well. A series of regression tests have been devised for every major application where PyClaw has been used. The script and parameters for generating the test are stored in the repository (typically in a single file), along with verified output from a successful run. Where possible, the output is verified against a different solver implementation or by analysis. These “application tests” produce the same output regardless of the choice of solver type, kernel implementation, or computer architecture. The python-nose (http://code.google.com/p/python-nose/) unit testing framework simplifies development and selective execution of the tests. Programmatic test code generation is used to exercise the full range of solver and kernel options for each test. Scientific results are archived as application tests within the unit testing framework, ensuring that our published results are reproducible in current and future versions of the PyClaw solver. In our experience, the application tests are the single greatest factor in facilitating adoption, refactoring, and extension of the code. New users are confident that they have a working installation (or can tell us what doesn’t work on their architectures) and are capable of reproducing our published scientific results. Developers refactoring the code for improved organization, performance, or readability can rely on the application tests for regression testing, to ensure that their changes have not incidentally broken anything. Perhaps most importantly, new solver methods and their implementations can be verified against known solutions with the application tests. This facilitates and encourages the development of new ideas within the PyClaw framework. ### Hosted Distributed Version Control Our use of git (http://git-scm.com/), a modern, distributed version control system, provides many benefits. Development need not be synchronized through a master server, which makes it easier to incorporate subprojects from developers loosely attached to the core team. Management of releases and bugfix updates has been greatly simplified. However, perhaps the greatest beneficiary is the user. Users do not have to wait for PyClaw releases in order to retrieve bugfixes for particular machines or improvements which are under active development, they need only update to a given changeset. Moreover, a user can easily switch between an approved release and experimental code for comparison with a single version control command. This allows the user a much finer-grained manipulation of versioning than was previously possible. There are many excellent open source distributed version control hosting sites, including Bitbucket (http://www.bitbucket.org) and GitHub (http://www.github.org), which provide a range a services to both developers and community users. PyClaw leverages the services provided at GitHub, which includes wiki webpages for user communication, as well as a bug reporting and tracking infrastructure integrated with the hosted version control repository. We have separately engaged the use of Google Groups to provide a mailing list for the PyClaw user and developer community. ### Documentation PyClaw is provided with a range of documentation suitable for the variety of users interacting with the software. While this paper provides a high-level overview of the capabilities of the code and its application, it is our experience from using other projects that the best software documentation includes a separate tutorial and user’s guide with a class reference section. The tutorial and user’s guide are maintained in the ReStructured Text format, from which they can be translated into HTML, PDF, and several other output formats using for instance Sphinx (http://sphinx.pocoo.org/). The PyClaw code itself is documented inline using Python’s docstring conventions, allowing us to automatically generate class and module reference sections for our documentation. ## 7 Applications The numerical algorithms made accessible in PyClaw, empowered by parallelization, are capable of modeling challenging wave propagation phenomena. In this section, we provide three example applications. They are not intended to break new scientific ground, but rather to demonstrate the versatility of the algorithms accessible through PyClaw, and (in the third application) the power of PetClaw as a scalable parallel solver. ### Shallow Water Flow on the Sphere Classical shallow water equations on a sphere are an approximation of the flow on the earth’s surface. They are of interest because they capture most of the flow’s features of a thin layer of fluid tangent to the surface of the sphere. Therefore, they are often used by meteorologists, climatologists and geophysicists to model both atmosphere and oceans. In three-dimensional Cartesian coordinates, using $h$ and $\mathbf{u}=\left(u,v,w\right)^{T}$ to define the height and the fluid velocity vector, the shallow water equations are of the form (1), with $\mathbf{q}=\left(h,hu,hv,hw\right)^{T}$ and \[\mathbf{f}\left(\mathbf{q}\right)=\left(\begin{array}[]{ccc}hu&hv&hw\\ hu^{2}+\frac{1}{2}gh&huv&huw\\ huv&hv^{2}+\frac{1}{2}gh&hvw\\ huw&hvw&hw^{2}+\frac{1}{2}gh\end{array}\right),\] (5) where $g$ is the gravitational acceleration. The source term $\mathbf{s}\left(\mathbf{q},\mathbf{x}\right)$ includes the Coriolis force and an additional term that ensures that the velocity is tangent to the sphere: \[\mathbf{s}\left(\mathbf{q},\mathbf{x}\right)=-\frac{2\Omega}{a}z\left(\mathbf{ x}\times h\mathbf{u}\right)+\left(\mathbf{x}\cdot\left(\mathbf{\nabla}\cdot \tilde{\mathbf{f}}\right)\right)\mathbf{x}.\] (6) Here $\Omega$ and $a$ are the rotation rate and the radius of the earth, respectively. In (6) $\tilde{\mathbf{f}}$ is the part of the flux matrix associated with the momentum equations [2]. In this framework, we consider the numerical solution of the zonal wave number 4 Rossby-Haurwitz problem [30]. Rossby-Haurwitz waves are steadily propagating initial conditions of the nonlinear non-divergent barotropic vorticity equation on a rotating sphere [7]. Although they do not represent exact steady solutions of the shallow water equations, they are expected to evolve in a similar way [29]. For this reason Rossby-Haurwitz waves have also been used to test shallow water numerical models and are among the standard shallow water model test cases proposed by Williamson et al. [30]. The problem consists of a divergence-free initial velocity field that rotates about the $z-$axis without significantly changing form on short time scales (dynamical weak instabilities). On longer time scales (50-100 days) the instabilities effects lead to the breakdown of the wave structure. Previous studies have shown that the time at which the solution symmetry breaks depends strongly on the numerical truncation error of the scheme [29]. Because of this the Rossby-Haurwitz problem is frequently used to assess the accuracy of a numerical algorithm for the solution of the shallow water equations on a rotating sphere. The three-dimensional shallow water equations are solved on the logically rectangular grid introduced in [2], using the same approach employed there, namely the classic Clawpack method with Strang splitting for the source term [28]. Simulations have been performed on four grids with $100\times 50$, $200\times 100$, $400\times 200$ and $800\times 400$ cells. Table 2 lists the breakdown time. With the coarsest mesh, the diffusive part of the numerical error suppresses the instability completely (cfr. [29]). The finer grid results confirm that the time at which the initial velocity loses its symmetry is sensitive to the numerical error. Grid \| Breakdown ---\|--- 100×50 \| - 200×100 \| ≈ 34 d 400×200 \| ≈ 45 d 800×400 \| ≈ 46 d Table 2: Time at which the symmetry of the Rossby-Haurwitz wave breaks down. Figure 4 shows the contour line of the water height at day 0 (initial solution), 38, 45 and 48, for the grid with $400\times 200$ cells. These plots show qualitatively the evolution of the instability. <figure><img src="content_image/1111.6583/SW400x200-0days.png"><figcaption>(a) 0 days.</figcaption></figure> ### Shock-Bubble Interaction The Euler equations for a compressible, inviscid fluid with cylindrical symmetry can be written as \[\rho_{t}+(\rho u)_{z}+(\rho v)_{r} =-\frac{\rho v}{r},\] (7) \[(\rho u)_{t}+(\rho u^{2}+p)_{z}+(\rho uv)_{r} =-\frac{\rho uv}{r},\] \[(\rho v)_{t}+(\rho uv)_{z}+(\rho v^{2}+p)_{r} =-\frac{\rho v^{2}}{r},\] \[=-\frac{(\rho E+p)v}{r}.\] Here the $z$-coordinate represents distance along the axis of symmetry while the $r$-coordinate measures distance away from the axis of symmetry. The quantities $\rho,E,p$ represent density, total energy, and pressure, respectively, while $u$ and $v$ are the $z$- and $r$-components of velocity. We consider an ideal gas with $\gamma=1.4$ in the cylindrical domain $[0,2]\times[0,0.5]$. The problem consists of a planar shock traveling in the $z$-direction that impacts a spherical bubble of lower-density fluid. In front of the shock $u=v=0$ and $\rho=p=1$ except inside the bubble, where $p=1,\rho=0.1$. Behind the shock, $p=5,\rho\approx 2.82,v\approx 1.61$, and these conditions are also imposed at the left edge of the domain. In addition to (7), we solve a simple advection equation for a tracer that is initially set to unity in the bubble and zero elsewhere in order to visualize where the fluid interior to the bubble is transported. Reflecting boundary conditions are imposed at the bottom of the domain while outflow conditions are imposed at the top and right boundaries. Since the edge of the bubble is curved, it does not align with the Cartesian grid. Thus, in the cells that are partly inside and partly outside the bubble, the initial condition used is a weighted average of the different solution values, based on the fraction of the cell that is inside. This fraction is computed by adaptive quadrature using the scipy.integrate.quad package. Figure 5 shows the initial condition and results of this problem, using a $1280\times 320$ grid and the unsplit classic Clawpack algorithm with full transverse corrections. The bubble is observed to transform into a “smoke ring”. Considerably more detailed structure is evident in this simulation compared to lower-resolution adaptively refined results from AMRClaw that are published at http://depts.washington.edu/clawpack/clawpack-4.3/applications/euler/2d/shockbubble/amr/www/index.html. Figure 6 shows a close-up of the smoke ring solution obtained with the classic Clawpack algorithm, as well as solutions obtained using the SharpClaw algorithm with fifth- (WENO5) and seventh-order (WENO7) reconstructions. All runs were performed on a $1280\times 320$ grid with a maximum CFL number of 0.8. Although the overall features of the solutions are similar, more fine structure is apparent in the SharpClaw results. For instance, several vortices can be seen to the left of the smoke ring in the WENO7 run that are not resolved in the classic run. <figure><img src="content_image/1111.6583/initial_tracer.png"><figcaption>(a) Initial tracer showing location of low-density bubble.</figcaption></figure> <figure><img src="content_image/1111.6583/shockbubble_classic.png"><figcaption>(a) Classic Clawpack solution.</figcaption></figure> ### Cylindrical Solitary Waves in a Periodic Medium The problem considered in this section is taken from [24]. It involves the propagation of nonlinear waves in a two-dimensional crystal, leading to the formation of solitary radial waves or “rings”. For this purpose, we consider the 2D $p$-system with spatially-varying coefficients as a model for elastic waves: \[\epsilon_{t}-u_{x}-v_{y} =0,\] (8) \[\rho(x,y)u_{t}-\sigma(\epsilon,x,y)_{x} =0,\] \[\rho(x,y)v_{t}-\sigma(\epsilon,x,y)_{y} =0.\] Here $\epsilon$ represents the strain, $u$ and $v$ are the velocities in $x$ and $y$ respectively, $\rho(x,y)$ is the spatially-varying material density and $\sigma(\epsilon,x,y)$ is the stress. The system (8) is closed by introducing a nonlinear constitutive relation $\sigma(\epsilon,x,y)$. Similar to [17], we take \[\sigma(\epsilon,x,y)=\exp(K(x,y)\epsilon)+1,\] (9) where $K(x,y)$ is the spatially-varying bulk modulus. The medium, a small part of which is shown in Figure 7(a), is in a checkerboard pattern with alternating squares of two different materials: (10) The problem is quite challenging, for multiple reasons. First, the flux is spatially varying and even discontinuous – meaning that the solution variables (strain and momentum) are also discontinuous. Furthermore, these discontinuities include corner singularities. Finally, and owing in part to these discontinuities, it is necessary to use a large number of cells per material period ($\gtrsim 100$) in order to get even qualitatively accurate results. As we are interested in a phenomenon that arises only after waves pass through many ($>100$) material periods, this leads to a very high computational cost. The problem is solved using the SharpClaw algorithm with fifth-order WENO reconstruction. As explained in Section 2.1, the implementation in SharpClaw is based on solving normal Riemann problems at the grid interfaces; see [25] for a detailed explanation of the approximate Riemann solvers employed and for a much more detailed study of this problem. A close-up view of the initial stress is shown in Figure 7(b). The stress is a Gaussian pulse with an amplitude of $5$ and a variance in $x$ and $y$ of $5$, centered at the origin. The velocity is initially zero. The problem is symmetric with respect to reflection about the $x$- and $y$- axes, so the computational domain is restricted to the positive quadrant and reflecting boundary conditions are imposed at the left and bottom boundaries. Outflow (zero-order extrapolation) conditions are imposed at the top and right boundaries. In units of the medium periodi, the domain considered is $200\times 200$, and the grid spacing is $\Delta x=\Delta y=1/240$. Hence the full simulation involves $6.8\times 10^{9}$ unknowns. It was run on 16,384 cores of the Shaheen supercomputer at KAUST, over a period of about 3.2 days. <figure><img src="content_image/1111.6583/checker.png"><figcaption>(a) Detail of checkerboard medium.</figcaption></figure> The formation of solitary wave rings is seen clearly in Figure 8(a), which depicts the stress at $t=200$. The structure of these waves is shown in Figure 8(b), which displays slices of the stress at $45^{\circ}$ (solid line) and $0^{\circ}$ (dashed line) with respect to the $x$-axis. <figure><img src="content_image/1111.6583/stress_checker.png"><figcaption>(a) Solitary wave train (stress shown).</figcaption></figure> ## 8 Discussion and future plans This work demonstrates that the use of Python in combination with existing Fortran and C codes allows the production of scientific software that is accessible, extensible, and efficient. The serial performance loss is relatively small, and is more than compensated for even on a typical laptop by the ability to run in parallel without any additional effort. Combining scalable parallelism with the algorithms of Clawpack and SharpClaw yields a potent tool for exploring novel wave phenomena. We are in the process of extending PyClaw to include three-dimensional wave propagation and implicit time stepping. Both of these are straightforward steps, since the PyClaw framework was written with such developments in mind and the related software packages (Clawpack and PETSc) already support these features. Preliminary implementations are under testing. The use of Python in scientific computing has many potential benefits [23]. The Python packages numpy, scipy, and matplotlib offer essential numerical tools with interfaces familiar to users of MATLAB (the _lingua franca_ of numerical methods) in a general-purpose programming language. An increasing number of important libraries (like PETSc and Trilinos) now have Python bindings, making it relatively easy to add powerful capabilities like massively scalable parallelism to Python codes. As discussed in Section 6, the Python community promotes a range of positive code development practices that are not common in scientific teams but are often picked up by those who begin to work in Python [6]. While the existence of multiple scientific codes for solving the same problems is healthy, it is widely recognized that increased code sharing and reuse would benefit the numerical analysis and scientific computing communities. Closer integration of code developed by different groups would not only allow researchers to be more productive (by reducing duplication of effort), but would also allow useful algorithmic improvements to be more rapidly distinguished from insignificant ones by simplifying the task of comparing them. In our experience, the adoption of Python as a high-level scientific coding language dramatically increases opportunities for code-sharing and reuse. Indeed, the results described in this paper consist largely of combining a few powerful existing pieces of scientific software. Acknowledgments. We are very grateful to Randall LeVeque, and all the Clawpack developers, without whom PyClaw would not exist. 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1108.1055	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 24862, "num_imgs": 2, "llama3_tokens_count": 7556 }	[ "content_image/1108.1055/x1.png", "content_image/1108.1055/x2.png" ]	# Wireless Capacity With Arbitrary Gain Matrix Magnús M. Halldórsson School of Computer Science Reykjavik University Reykjavik 101, Iceland mmh@ru.is Pradipta Mitra School of Computer Science Reykjavik University Reykjavik 101, Iceland ppmitra@gmail.com ###### Abstract. Given a set of wireless links, a fundamental problem is to find the largest subset that can transmit simultaneously, within the SINR model of interference. Significant progress on this problem has been made in recent years. In this note, we study the problem in the setting where we are given a fixed set of arbitrary powers each sender must use, and an arbitrary gain matrix defining how signals fade. This variation of the problem appears immune to most algorithmic approaches studied in the literature. Indeed it is very hard to approximate since it generalizes the max independent set problem. Here, we propose a simple semi-definite programming approach to the problem that yields constant factor approximation, if the optimal solution is strictly larger than half of the input size. Key words and phrases:Wireless Networks, Capacity, SINR Model, Semidefinite programming. ## 1. Introduction We consider the fundamental problem of wireless network capacity. Given is a set $L=\{\ell_{1},\ell_{2},\ldots,\ell_{n}\}$ of links, where each link $\ell_{v}$ represents a communication request from a sender $s_{v}$ to a receiver $r_{v}$. We are also given, for every $\ell_{v}\in L$, a transmission power $P_{v}>0$. The powers received from senders to receivers are defined by an $n\times n$ dimensional gain matrix $G$ with positive entries. Specifically, the signal received from $s_{v}$ at $r_{w}$ is $G_{wv}\cdot P_{v}$. Thus an instance in this model can be described by the tuple $(L,P,G)$ where $P$ is the vector of the power assignments $P_{v}$ for all $\ell_{v}$. Simultaneously communicating links interfere with each other, following the physical model or “SINR model” of interference. Due to its higher fidelity to reality [10, 22, 26], this model of interference has recently gained substantial attention in the analysis of wireless networks. In this model, a receiver $r_{v}$ successfully receives a message from a sender $s_{v}$ if and only if the following condition holds: (1) \[\frac{G_{vv}\cdot P_{v}}{\sum_{\ell_{w}\in S\setminus\{\ell_{v}\}}G_{vw}\cdot P _{w}+N}\geq\beta,\] where $N$ is a universal constant denoting the ambient noise, $\beta\geq 1$ denotes the minimum SINR (signal-to-interference-noise-ratio) required for a message to be successfully received, and $S$ is the set of concurrently scheduled links in the same _slot_. We say that a link $\ell_{v}$ is feasible in $S$ if Eqn. 1 is satisfied for $\ell_{v}$. A set $S$ is feasible if each of its link is feasible. Note that what we described above is the abstract SINR model. In the more commonly studied geometric SINR model, $G_{vw}$ is a polynomial function of $d(s_{w},r_{v})$, where $d(x,y)$ is the distance between two points $x$ and $y$. Our results naturally apply to that model as well. Given that the geometric SINR model does not capture obstacles, reflections and other real life distortions, it is interesting to see what can be proven in the abstract model. Our setting where the powers are given as part of the input is often called the _fixed_ power case, as opposed to the _power control_ case where the algorithm can choose the power assignment. So far, research on fixed power has focused on _oblivious_ power assignments, where the power of a link is a (usually simple) function of the length of the link [13, 5, 20, 12]. Recently, a constant factor approximation algorithm to find the capacity in the power control case has also been achieved [19]. Unfortunately, none of these techniques appear to extend to the case of arbitrary fixed powers (for either arbitrary or geometric gain matrices). Yet, the problem of arbitrary fixed powers is not only natural, but has practical relevance, as commercial hardware often do not have the capacity of choosing precise powers to implement either an arbitrary assignment _à la_[19], or to implement many of the oblivious power assignments found in literature. In this paper, we prove the following theorem. Assume $(L,P,G)$ is an instance of the capacity problem in the abstract SINR model, such that $\|OPT\|>\frac{1}{2}(1+\epsilon)\|L\|$ for some $\epsilon>0$, where $OPT$ is the maximum feasible subset of $L$ using $P$. Then there is a polynomial time randomized algorithm to find a feasible set of size $\Omega(\epsilon\|L\|)$, with probability $1-o(1)$. We do this by means of a semi-definite programming relaxation, which we show how to successfully round if the condition $\|OPT\|>\frac{1}{2}(1+\epsilon)\|L\|$ holds. In addition, we discuss numerical experiments we have performed. These experiments show that the algorithm appears to work quite well on random instances, even better than the guarantees of Thm. 1. Semi-definite programming has been a staple in designing approximation algorithms for NP-hard problems ever since the seminal work of Goemans and Williams on the Max-CUT problem [7]. It is interesting to note that the discrete “classical” problems closest to wireless capacity, namely the independent set problem and the graph coloring problem, have been fruitfully studied using semi-definite programming [15, 18]. The vertex cover problem, also relevant via its connection to the independent set problem, also has SDP-based approximation algorithms [14, 17]. Given this background, one may expect some of the techniques to easily carry over to the capacity problem. Yet that does not appear to be the case, at least not in a straightforward manner. A study of the aforementioned papers reveal that the discreteness of the problem plays an important role in the bounds. For example, in [18], the analysis proceeds by bounding the probability of vectors representing edges not being cut by a random hyperplane. Given the additive nature of the SINR model, it is not obvious how to extend that analysis to this case. There have also been a number of results for these problems on hypergraphs [21, 2, 3]. Though hypergraphs appear to be closer in spirit to the additive wireless model, they are still different, because the effect of each node on any other node doesn’t change in the SINR model (as opposed to in a hypergraph, where it can be different based on which edge they are in). Thus, the (sophisticated) methods on hypergraphs do not appear to translate immediately to the SINR model either. Our SDP relaxation and rounding algorithms are quite simple in contrast to some of the previously mentioned work. Whether or not advanced techniques can be extended to the SINR model remains to be seen. ### Related Work. Moscibroda and Wattenhofer [24] were the first to study of the _scheduling complexity_ of arbitrary set of wireless links. Early work on approximation algorithms produced approximation factors that grew with structural properties of the network [27, 25, 1]. The first constant factor approximation algorithm was obtained for capacity problem for uniform power in [8] (see also [13]) in $\mathbf{R^{2}}$ with $\alpha>2$. Fanghänel, Kesselheim and Vöcking [6] gave an algorithm that uses at most $O(OPT+\log^{2}n)$ slots for the scheduling problem with _linear_ power assignment $P_{v}=d(s_{v},r_{v})^{\alpha}$, that holds in general distance metrics. Kesselheim obtained a $O(1)$-approximation algorithm for the capacity problem with power control for doubling metrics [19]. Around the same time, the first constant factor algorithm for all sub-linear, length monotone power assignments was achieved on general metrics [12]. Other recent studies in the SINR model include work on topological maps [16], distributed algorithms for scheduling [11], distributed power control [4] and auction based spectrum allocation [23]. ## 2. SDP-based algorithm. First, some notation. Vectors are denoted by $\vec{x},\vec{s_{w}}$ etc. The standard $2$-norm of the vector $\vec{x}$ is $\\|\vec{x}\\|$. The $i^{th}$ entry of $\vec{x}$ is $\vec{x}(i)$. The inner product of vectors $\vec{x}$ and $\vec{y}$ is denoted $(\vec{x}\cdot\vec{y})$. Define $g_{vv}=P_{v}G_{vv}-\beta N$ and $g_{vw}=P_{w}G_{vw}$ for $v\neq w$. Note that we can assume without loss of generality that $g_{vv}\geq 0,\forall v$. Let $OPT$ be a feasible subset of $L$ of maximum size. Note that $n=\|L\|$. Consider the following program. \[\max\sum_{v}(\vec{s_{v}}\cdot\vec{s}),\text{ subject to}\] \[(\vec{s_{v}}\cdot\vec{s})g_{vv}\geq\beta\left(\sum_{w\neq v}(\vec {s_{v}}\cdot\vec{s_{w}})g_{vw}\right),\forall v\] \[(\vec{s_{v}}\cdot\vec{s})\geq 0,\forall v\] \[(\vec{s_{v}}\cdot\vec{s_{w}})\geq 0,\forall v,w\] \[(\vec{s_{v}}\cdot\vec{s_{w}})\geq(\vec{s_{v}}\cdot\vec{s})+(\vec{ s_{w}}\cdot\vec{s})-1,\forall v,w\] \[\\|\vec{s_{v}}\\|^{2}=1,\forall v\text{ and }\\|\vec{s}\\|^{2}=1\ .\] where $\vec{s_{v}},\vec{s}\in\mathbb{R}^{n+1}$. Each link $\ell_{v}$ has a vector variable $\vec{s_{v}}$ associated with it. The dot product of $\vec{s_{v}}$ with a vector $\vec{s}$ denotes the (fractional) extent to which $\ell_{v}$ is selected in the solution. Since the objective function and constraints are all linear functions of vector inner products, this problem is a SDP. Thus the program can be solved up to an additive error of $\varepsilon>0$ in time that is polynomial in $n$ and $\log\varepsilon$[28]. Since $\varepsilon$ can be made small enough to not matter, we will simply assume that the problem can be solved exactly. We can rotate the vectors to fix $\vec{s}=\{1,0\ldots 0\}$, thus the above program is equivalent to: \[\max\sum_{v}\vec{s_{v}}(1),s.t.\] (2) \[\vec{s_{v}}(1)g_{vv}\geq\beta\left(\sum_{w\neq v}(\vec{s_{v}} \cdot\vec{s_{w}})g_{vw}\right),\forall v\] (3) \[\vec{s_{v}}(1)\geq 0,\forall v\] (4) \[(\vec{s_{v}}\cdot\vec{s_{w}})\geq 0,\forall v,w\] (5) \[(\vec{s_{v}}\cdot\vec{s_{w}})\geq\vec{s_{v}}(1)+\vec{s_{w}}(1)-1, \forall v,w\] (6) \[\\|\vec{s_{v}}\\|^{2}=1,\forall v\ .\] Let us verify that this program is a relaxation of the maximum capacity problem. The SDP is a relaxation of the original problem. Proof.: Consider any optimal solution $OPT$ to the capacity problem. For all $\ell_{v}\in OPT$, set $\vec{s_{v}}=\vec{s}=\{1,0,0,0\ldots 0\}$. If $\ell_{v}\in L\setminus OPT$ set \[\vec{s_{v}}(i)=\left\{\begin{array}[]{rl}1&\text{if }i=v+1\\ 0&\text{otherwise}\\ \end{array}\right.\] In other words, we make sure that each unselected link chooses a different position for the single $1$ in the vector. Given these assignments, Equations 3, 4 and 6 can easily seen to hold. To show that Eqn. 2 is satisfied, first assume $\ell_{v}\in OPT$. The following observation is immediate: If $\ell_{v},\ell_{w}\in OPT$ then $\vec{s_{v}}(1)=\vec{s_{w}}(1)=(\vec{s_{v}}\cdot\vec{s_{w}})=1$. If $\ell_{v}\in L\setminus OPT$ then $\vec{s_{v}}(1)=0$ and $(\vec{s_{v}}\cdot\vec{s_{w}})=0$ for any $\ell_{w}\neq\ell_{v}$. Since $\ell_{v}\in OPT$, \[\vec{s_{v}}(1)g_{vv}=g_{vv}\] And, \[\beta\left(\sum_{w\neq v}(\vec{s_{v}}\cdot\vec{s_{w}})g_{vw}\right)\] \[= \beta\left(\sum_{w\in OPT\setminus\{v\}}(\vec{s_{v}}\cdot\vec{s_{ w}})g_{vw}\right)+\beta\left(\sum_{w\in L\setminus(OPT\cup\{v\})}(\vec{s_{v}} \cdot\vec{s_{w}})g_{vw}\right)\] \[= \beta\left(\sum_{w\in OPT\setminus\{v\}}g_{vw}\right)\] where the second equality follows from the claim above. Now, since $\ell_{v}\in OPT$, $g_{vv}\geq\beta\left(\sum_{w\in OPT\setminus\{v\}}g_{vw}\right)$ (by Eqn 1). Thus, the above two equations show that Eqn. 2 is satisfied when $\ell_{v}\in OPT$. The case where $\ell_{v}\not\in OPT$ is similar. For Eqn. 5, the following observations suffice: * If $\ell_{v},\ell_{w}\in OPT$, $(\vec{s_{v}}\cdot\vec{s_{w}})=1=\vec{s_{v}}(1)+\vec{s_{w}}(1)-1$ * If $\ell_{v},\ell_{w}\not\in OPT$, they have $1$s in different positions and $(\vec{s_{v}}\cdot\vec{s_{w}})=0\geq 0+0-1$ * If $\ell_{v}\in OPT,\ell_{w}\not\in OPT$, they have 1s in different positions and $(\vec{s_{v}}\cdot\vec{s_{w}})=0=1+0-1$ ∎∎ Now we present our algorithm and the proof of Thm. 1. We need two related definitions. Let $\delta_{v}=\max\{\vec{s_{v}}(1)-\frac{1}{2},0\}$ for all $\ell_{v}\in L$. Further, define $L^{+}=\{\ell_{v}\in L:\delta_{v}>0\}$. The algorithm is as follows. ``` 1: Solve the SDP 2: Select each link $\ell_{v}\in L^{+}$ with probability $\frac{\delta_{v}}{2}$ in to a set $R$ 3: Output $\{\ell_{v}\in R:\ell_{v}\text{ is feasible in }R\}$ ``` Algorithm 1 Capacity1 If $\|OPT\|\geq(1+\epsilon)n/2$, then $\sum_{\ell_{v}\in L^{+}}\delta_{v}\geq\frac{n\epsilon}{2}$. Proof.: Since $\|OPT\|\geq(1+\epsilon)n/2$, it follows that $\sum_{v}\vec{s_{v}}(1)\geq(1+\epsilon)n/2$ (since the SDP is a relaxation of the original problem). Now by definition of $\delta_{v}$, $\delta_{v}+\frac{1}{2}\geq\vec{s_{v}}(1)$. Thus, \[\sum_{\ell_{v}\in L}\left(\frac{1}{2}+\delta_{v}\right)\geq(1+ \epsilon)n/2\] \[\Rightarrow \sum_{\ell_{v}\in L}\delta_{v}\geq(1+\epsilon)n/2-\|L\|/2=(1+ \epsilon)n/2-n/2=\frac{\epsilon n}{2}\] Observing that $\delta_{v}=0$ for $\ell_{v}\not\in L^{+}$ completes the proof. ∎∎ We can now prove the main Theorem. Proof.: of Thm. 1 Assume that the random binary variable $X_{v}$ describes whether or not $\ell_{v}\in L^{+}$ is chosen into $R$. We observe that $\Ex(X_{v})=\frac{\delta_{v}}{2}$, according to the algorithm. Then for any $\ell_{v}$, (7) \[\Ex\left(\beta\left(\sum_{w\in R\setminus\{v\}}g_{vw}\right) \right)=\Ex\left(\beta\left(\sum_{w\in L^{+}\setminus\{v\}}g_{vw}X_{w}\right)\right)\] \[= \beta\left(\sum_{w\in L^{+}\setminus\{v\}}g_{vw}\Ex(X_{w})\right) =\beta\left(\sum_{w\in L^{+}\setminus\{v\}}g_{vw}\frac{\delta_{w}}{2}\right)\] \[\Rightarrow \Ex\left(\beta\left(\sum_{w\in R\setminus\{v\}}g_{vw}\right) \right)=\frac{1}{2}\beta\left(\sum_{w\in L^{+}\setminus\{v\}}g_{vw}\delta_{w}\right)\] Now, by Eqn. 2, \[\vec{s_{v}}(1)g_{vv}\geq\beta\left(\sum_{w\neq v}(\vec{s_{v}}\cdot\vec{s_{w}}) g_{vw}\right),\forall v\in L^{+}\] Since $\vec{s_{v}}(1)\geq\frac{1}{2}$ for $v\in L^{+}$ and $(\vec{s_{v}}\cdot\vec{s_{w}})g_{vw}$ is always non-negative, we get for $\ell_{v}\in L^{+}$, (8) \[g_{vv} \geq \beta\left(\sum_{w\in L^{+}\setminus\{v\}}(\vec{s_{v}}\cdot\vec{s _{w}})g_{vw}\right)\] \[\geq \beta\left(\sum_{w\in L^{+}\setminus\{v\}}(\vec{s_{v}}(1)+\vec{s_ {w}}(1)-1)g_{vw}\right)=\beta\left(\sum_{w\in L^{+}\setminus\{v\}}(\delta_{v}+ \delta_{w})g_{vw}\right)\] \[\geq \beta\left(\sum_{w\in L^{+}\setminus\{v\}}\delta_{w}g_{vw}\right)\] where the second inequality follows from Eqn. 5, and the first equality follows from observing that $\delta_{v}=\vec{s_{v}}(1)-\frac{1}{2}$ for $\ell_{v}\in L^{+}$. Then, for $\ell_{v}\in L^{+}$, (9) \[\Pro(\ell_{v}\text{ is infeasible in $R$}) =\Pro\left(\beta\left(\sum_{w\in R\setminus\{v\}}g_{vw}\right)>g_ {vv}\right)\] \[\leq\frac{\Ex(\beta(\sum_{w\in R\setminus\{v\}}g_{vw}))}{g_{vv}} \leq\frac{1}{2}\] The first equality is the definition of infeasiblity. The first inequality is Markov’s inequality. The last inequality follows from Equations 7 and 8. Now the expected size of the output is \[\Ex\left(\|\{\ell_{v}\in R:\ell_{v}\text{ is feasible in }R\}\|\right) =\sum_{\ell_{v}\in L^{+}}\Pro(\ell_{v}\in R\text{ and }\ell_{v} \text{ is feasible in }R)\] \[=\sum_{\ell_{v}\in L^{+}}\Pro(\ell_{v}\text{ is feasible in }R) \Pro(\ell_{v}\in R)\] \[\geq\sum_{\ell_{v}\in L^{+}}\frac{1}{2}\frac{\delta_{v}}{2}=\frac {1}{4}\sum_{\ell_{v}\in L^{+}}\delta_{v}\geq\frac{n\epsilon}{8}\] The second equality follows from the independence of the events concerned. The first inequality follows from Eqn. 2. The last inequality follows from Lemma 1. Thus the expected size of the feasible output is $\Omega(\epsilon n)$. It is not difficult to boost the probability of getting a $\Omega(\epsilon n)$ size subset to complete the proof of the theorem. ∎∎ ## 3. Numerical Experiments We ran simulations to test how well the algorithm does in practice. We used CVX, a package for specifying and solving convex programs using MATLAB [9]. We ran it on version 7.8 of MATLAB running on a Macbook with a 2 GHz Intel Core 2 Duo Processor and 2 GB of RAM. We generated a number of problem instances where $n=61$ and $\|OPT\|=21,26,31,36$ and $41$. The instances were generated as follows. To generate the feasible subset a large random instance $M$ of links on the 2d plane was generated. Each sender $s_{v}=(s_{v}(x),s_{v}(y))$ is a random point in a $450\times 450$ box. The receiver $r_{v}$ is defined by $(s_{v}(x)+\text{random}_{v}(x),s_{v}(y)+\text{random}_{v}(y))$ where $\text{random}_{v}(x)$ and $\text{random}_{v}(y)$ are sampled uniformly at random from $[-20,20]$. We generated corresponding gain matrices using the geometric SINR model setting $\alpha=2.5$ (thus $G_{vw}=\frac{1}{\\|s_{w}-r_{v}\\|^{\alpha}}$). We used both uniform ($P_{v}$ is a constant) and mean power assignments ($P_{v}=\\|s_{v}-r_{v}\\|^{\alpha/2}$) to generate the gain matrix. We set the noise $N=0$ throughout the experiments. To generate the input instance $G$ (which is a $n\times n$ matrix), we combined a subset of $M$ with random entries. More specifically, first we retrieved a random feasible subset $R$ of $M$ (found greedily). This defined a $R\times R$ submatrix of $G$. The remaining entries were chosen iid randomly from $[0,\kappa]$, where $\kappa$ was chosen large enough so that the remaining $n-\|R\|$ links would not contain a large feasible subset, thus $R$ would be $OPT$ for the instance. Though computationally slow (for $n=60$ the SDP took a few minutes to be solved), the algorithm performed extremely well. Indeed, it took some time to come up with instances where the algorithm didn’t have a perfectly integral solution. If the random entries of $G$ corresponding to $L\setminus OPT$ were too large (corresponding to a large $\kappa$, meaning that $L\setminus OPT$ contained only very small subsets that were feasible) or if $OPT$ was too _loosely_ feasible (ie, Eqn. 2 was far from being tight for most of the links), the algorithm did exceedingly well. <figure><img src="content_image/1108.1055/x1.png"><figcaption>Figure 1. OPT vs the average size of the set found by the SDP algorithm. Ineach case n=61.</figcaption></figure> <figure><img src="content_image/1108.1055/x2.png"><figcaption>Figure 2. OPT vs the average size of the set found by the SDP algorithm. Ineach case n=61 and the links in OPT were generated using mean power. Thelabels in the x-axis describe the configuration of the instances. Thus, in thefirst case, the instance is an union of 3 feasible sets of size 21, 20 and 20,respectively, where the latter two are copies of subsets of the first one.</figcaption></figure> Even after trying to make the problem more difficult, the algorithm did quite well, only degrading when $OPT<n/2$, for which we claim no theoretical guarantee anyway, though even in these cases the output was not unsatisfactory. Indeed, in all these cases, using the simple filtering $(\vec{s_{v}}\cdot\vec{s})>0.51$ identified $OPT$ almost exactly. Our sampling algorithm, by design cannot achieve better than a factor $2$ approximation in general, and that is almost what we achieved in all cases, as illustrated in Figure 1 for uniform power (the results for mean power were essentially identical). As we have mentioned, in the above experiments, the algorithm sharply identified $OPT$. To create more ambiguous instances, we also tried the following. In this setting, we took a feasible set, and added copies of subsets of it. Thus the instance would be of the form $L_{1}\cup L_{2}$ or $L_{1}\cup L_{2}\cup L_{3}$ where $L_{1}$ is feasible, and $L_{2},L_{3}$ are copies of subsets of $L_{1}$. One expects the solution to be more “spread out” in this case, and that is exactly what we found. The algorithm still performed rather well, even below theoretically guaranteed levels, though the behavior is somewhat different. Figure 2 demonstrates the case for mean power. ## 4. Conclusion We have shown how to use semi-definite programming to approximate the wireless capacity problem in cases where the capacity is known to be large. It is an interesting question whether or not these results can be further improved, potentially using the power of geometric SINR model. Questions about the integrality gap and hardness of the problem (apart from what is known via the fact that the problem generalizes max independent set) also deserve attention. Though we have performed some preliminary numerical experiments, the efficacy of this method both in terms of accuracy and computational efficiency also is an interesting avenue of further investigation. ## References * [1] Chafekar, D., Kumar, V., Marathe, M., Parthasarathy, S., Srinivasan, A.: Cross-layer Latency Minimization for Wireless Networks using SINR Constraints. In: Mobihoc (2007) * [2] Chlamtac, E.: Approximation algorithms using hierarchies of semidefinite programming relaxations. In: FOCS. pp. 691–701. 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1106.6299	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 27309, "num_imgs": 0, "llama3_tokens_count": 8915 }	[]	# Holograms of Conformal Chern–Simons Gravity Hamid Afshar afshar@ipm.ir Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria, Europe School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran Department of Physics, Sharif University of Technology, P. O. Box 11365-9161, Tehran, Iran Branislav Cvetković cbranislav@ipb.ac.rs University of Belgrade, Institute of Physics, P. O. Box 57, 11001 Belgrade, Serbia Sabine Ertl sertl@hep.itp.tuwien.ac.at Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria, Europe Daniel Grumiller grumil@hep.itp.tuwien.ac.at Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria, Europe Niklas Johansson niklasj@hep.itp.tuwien.ac.at Institute for Theoretical Physics, Vienna University of Technology, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria, Europe February 25, 2024 ###### Abstract We show that conformal Chern–Simons gravity in three dimensions has various holographic descriptions. They depend on the boundary conditions on the conformal equivalence class and the Weyl factor, even when the former is restricted to asymptotic Anti-deSitter behavior. For constant or fixed Weyl factor our results agree with a suitable scaling limit of topologically massive gravity results. For varying Weyl factor we find an enhancement of the asymptotic symmetry group, the details of which depend on certain choices. We focus on a particular example where an affine $\hat{u}(1)$ algebra related to holomorphic Weyl rescalings shifts one of the central charges by 1. The Weyl factor then behaves as a free chiral boson in the dual conformal field theory. pacs: 04.60.Rt,04.20.Ha,11.25.Tq,11.15.Wx,11.15.Yc † [FOOTNOTE:†][ENDFOOTNOTE] Conformal Chern–Simons gravity (CSG) Deser et al. (1); Horne and Witten (1989) is a 3-dimensional third-derivative theory of gravity that has a parity-odd non-covariant action \[S_{\textrm{CSG}}=\frac{k}{4\pi}\,\int\!\operatorname{d}\!^{3}x\,\epsilon^{ \lambda\mu\nu}\,\Gamma^{\sigma}_{\lambda\rho}\,\Big{(}\partial_{\mu}\Gamma^{ \rho}_{\nu\sigma}+\tfrac{2}{3}\,\Gamma^{\rho}_{\mu\tau}\Gamma^{\tau}_{\nu \sigma}\Big{)}\] (1) but covariant equations of motion \[C_{\mu\nu}:=\varepsilon_{\mu}{}^{\lambda\sigma}\nabla_{\lambda}\big{(}R_{\nu \sigma}-\frac{1}{4}\,g_{\nu\sigma}R\big{)}=0\,.\] (2) Here $k$ is the Chern–Simons level, which we assume to be some positive integer, $\Gamma$ are the Christoffel symbols and $C_{\mu\nu}$ is the Cotton tensor, the vanishing of which is equivalent to conformal flatness in three dimensions. In the bulk the theory (1) is not only diffeomorphism invariant but also invariant under local Weyl rescalings \[g_{\mu\nu}\to e^{2\Omega}g_{\mu\nu}\,.\] (3) Consequently, the theory defined by the action (1) is topological in the sense that it has zero physical bulk degrees of freedom. Interesting physical properties emerge if a boundary is introduced Brown and Henneaux (1986); Witten (1988) — for instance an asymptotic boundary, like in the holographic AdS/CFT correspondence Maldacena (1998); Aharony et al. (2000). In this paper we show that CSG allows for various qualitatively different holographic conformal field theory (CFT) duals, depending on the boundary conditions imposed on the Weyl factor. We focus on presenting and discussing the main results, and we shall provide a more detailed account of the calculations elsewhere [7]. We assume that the manifold $M$ has a (connected) boundary $\partial M$ with cylindrical or toric topology; $\partial M$ may but need not be an asymptotic boundary. It is convenient to parametrize the boundary such that one of the coordinates, $y$, is constant on it. With no loss of generality we assume $y=0$ at $\partial M$. In the vicinity of the boundary we write the metric $g_{\mu\nu}$ as \[g_{\mu\nu}=e^{2\phi}\,\bar{g}_{\mu\nu}=e^{2\phi}\big{(}g_{\mu\nu}^{\rm AAdS}+h _{\mu\nu}\big{)}\] (4) and impose the condition that the metric $\bar{g}_{\mu\nu}$ be asymptotically AdS. More specifically, with the leading metric \[g_{\mu\nu}^{\rm AAdS}\,\operatorname{d}\!x^{\mu}\operatorname{d}\!x^{\nu}= \frac{\operatorname{d}\!x^{+}\operatorname{d}\!x^{-}+\operatorname{d}\!y^{2}}{ y^{2}}\] (5) we require that the subleading state-dependent part $h_{\mu\nu}$ take the form \[\left(\begin{array}[]{lll}h_{++}={\cal O}(1/y)&h_{+-}={\cal O}(1)&h_{+y}={\cal O }(1)\\ &h_{--}={\cal O}(1)&h_{-y}={\cal O}(1)\\ &&h_{yy}={\cal O}(1)\end{array}\right)\,.\] (6) The boundary conditions (5) with (6) restrict the conformal equivalence class of the metric. The Weyl factor $\phi$ has to be considered separately. Depending on its properties we distinguish three cases: 1. Trivial Weyl factor $\phi=\rm const.$ ($=0$) 2. Fixed Weyl factor $\phi\neq\rm const.$ 3. Free Weyl factor $\phi$ not fixed completely The reason for our split into boundary conditions on the conformal class and on the Weyl factor is the enhanced gauge invariance of CSG: if $g$ is a solution to the equations of motion (2) then also $e^{2\phi}\,g$ is a solution. The boundary conditions on the conformal class of the metric, (5)-(6), are chosen such that AdS is allowed as a background and that most of the linearized excitations around AdS are admissible. However, we cannot consistently allow all such excitations. The rationale behind the precise choices above will be explained elsewhere [7]. For the present context it is sufficient to point out that the boundary conditions (4)-(6) should not be made stronger, since this would eliminate interesting solutions, and cannot be made looser, since this would lead to inconsistencies, like infinite charges. ## I Trivial Weyl factor In this section we focus on case I. Since all the results in this section turn out to coincide with suitable scaling limits of topologically massive gravity (TMG) the presentation will be condensed, and we refer to [7] for a more detailed analysis. The boundary conditions (6) are preserved by diffeomorphisms generated by a vector field $\xi$ with the properties \[\xi^{\pm} =\varepsilon^{\pm}(x^{\pm})-\frac{1}{2}\,y^{2}\,\partial_{\mp}^{2 }\varepsilon^{\mp}(x^{\mp})+{\cal O}(y^{3})\,,\] (7a) \[\xi^{y} =\frac{y}{2}\,\big{(}\partial_{+}\varepsilon^{+}+\partial_{-} \varepsilon^{-}\big{)}+{\cal O}(y^{3})\,.\] (7b) They also allow asymptotic Weyl rescalings (3) with \[\Omega={\cal O}(y^{2})\,.\] (8) These Weyl rescalings are trivial symmetries, which are modded out in the asymptotic symmetry algebra. We calculate now the response functions using the standard AdS/CFT dictionary Aharony et al. (2000). To this end we need the first variation of the on-shell action $\delta S=\delta S_{\rm CSG}\|_{\rm EOM}+\delta S_{b}\|_{\rm EOM}$, including appropriate boundary terms $S_{b}$, which we take from Guica et al. (2011). The result is \[\delta S=\frac{1}{2}\,\int_{\partial M}\!\!\!\operatorname{d}\!^{2}x\sqrt{- \gamma^{(0)}}\,\Big{(}T^{\alpha\beta}\,\delta\gamma^{(0)}_{\alpha\beta}+J^{ \alpha\beta}\,\delta\gamma^{(1)}_{\alpha\beta}\Big{)}.\] (9) For convenience we use Gaussian normal coordinates in the asymptotic expansion ($e^{\rho}\propto 1/y$) \[\operatorname{d}\!s^{2}=\operatorname{d}\!\rho^{2}+\big{(}\gamma^{(0)}_{\alpha \beta}\,e^{2\rho}+\gamma^{(1)}_{\alpha\beta}\,e^{\rho}+\gamma^{(2)}_{\alpha \beta}+\dots\big{)}\,\operatorname{d}\!x^{\alpha}\operatorname{d}\!x^{\beta}\,,\] (10) where $\gamma^{(0)}$ is the boundary metric, $\gamma^{(1)}$ describes (partially massless) Weyl gravitons and their sources, and $\gamma^{(2)}$ contains information about the left- and right-moving massless boundary gravitons. The appearance of $\gamma^{(1)}$ is the only difference to the situation studied by Brown and Henneaux in their seminal paper Brown and Henneaux (1986). The response functions $T^{\alpha\beta}$ and $J^{\alpha\beta}$ are Brown–York stress tensor and partially massless response, respectively. They correspond to operators in the dual CFT and are given by [7] \[T^{\alpha\beta} =\frac{k}{\pi}\,\varepsilon^{(\alpha}{}_{\gamma}\,\gamma^{\beta) \gamma}_{(2)}=\frac{k}{2\pi}\,\varepsilon^{\alpha\gamma}\,\gamma^{\beta\,(2)}_ {\;\,\gamma}+(\alpha\leftrightarrow\beta)\,,\] (11) \[J^{\alpha\beta} =\frac{k}{4\pi}\,\big{(}\delta^{(\alpha}_{\gamma}-\varepsilon^{( \alpha}{}_{\gamma}\big{)}\,\big{(}\gamma^{\beta)\gamma}_{(1)}-\frac{1}{2}\, \gamma^{\beta)\gamma}_{(0)}\gamma_{\sigma\delta}^{(1)}\gamma^{\sigma\delta}_{( 0)}\big{)}\,.\] (12) The results (11), (12) for the 1-point functions agree with corresponding results in TMG Kraus and Larsen (2006); Skenderis et al. (2009). The same applies to 2-point functions Skenderis et al. (2009) and 3-point functions Grumiller and Sachs (2010). The non-vanishing 2-point functions are given by [$T^{R/L}$ are the (anti-)holomorphic flux components of the stress-energy tensor, with $x^{\pm}=\varphi\pm t$, $z=\varphi+it$]: \[\langle J(z,\bar{z})J(0,0)\rangle=\frac{2k\,\bar{z}}{z^{3}}\] (13) \[\langle T^{R}(z)T^{R}(0)\rangle=\frac{6k}{z^{4}}=-\overline{ \langle T^{L}(\bar{z})T^{L}(0)\rangle}\] (14) The result (13) shows that one of the conformal weights of the partially massless Weyl gravitons is negative, $\bar{h}=-1/2$, in agreement with the classical Grumiller et al. (2011) and 1-loop analysis Bertin et al. (2011). From the 2-point functions (14) we can read off the central charges of the dual CFT. \[c_{R/L}=\pm 12k\] (15) The result (15) agrees with a suitable limit of the TMG results for the central charges Kraus and Larsen (2006); Solodukhin (2006). ## II Fixed Weyl factor Let us now address case II where the Weyl factor $\phi$ in (4) is arbitrary but fixed. In order to recover the desired asymptotic diffeomorphisms (7) it turns out that we need to restrict its asymptotic behavior to ¹ [FOOTNOTE:1][ENDFOOTNOTE] \[\phi(x^{+},\,x^{-},\,y)=f(x^{+},\,x^{-})+\ldots\] (16) We then find that the asymptotic diffeomorphisms (7) have to be accompanied by a compensating infinitesimal Weyl rescaling (3) with \[\Omega=-\big{(}\varepsilon^{+}\partial_{+}+\varepsilon^{-}\partial_{-}\big{)} \,f+{\cal O}(y^{2})\,.\] (17) Thus, the boundary condition preserving gauge transformations are precisely as for case I, but we have to simultaneously rescale the metric with a Weyl factor (17). The only essential difference to case I is the result for the Brown–York stress tensor, which no longer is conserved. \[\nabla_{\alpha}T^{\alpha\beta}\propto\varepsilon^{\beta\gamma}\partial^{\alpha }\partial_{\alpha}\partial_{\gamma}f\] (18) This effect was explained in Kraus and Larsen (2006). On the gravity side, the reason for the anomalous conservation (18) is that the action is not diffeomorphism invariant: it transforms by a boundary term. This anomaly vanishes only for flat boundary metrics. Indeed, requiring the stress tensor to be conserved implies [we drop a term proportional to $x^{+}x^{-}$ since it is not periodic in $\varphi=(x^{+}+x^{-})/2$]. \[f(x^{+},x^{-})=f_{+}(x^{+})+f_{-}(x^{-})\,.\] (19) Restricting to this class of Weyl factors $\phi$ produces a class of diffeomorphism anomaly free CFTs. Interestingly, and not unexpectedly, the potential non-conservation of the charges implied by the Brown–York analysis above is invisible in a canonical analysis. The difference between the Brown–York and the canonical result comes about because the former is based upon the variation of the action (1), which is not diffeomorphism invariant at the boundary, while the latter is based upon the first order action ($T_{i}:=de_{i}+\varepsilon_{ijk}\omega^{j}e^{k}$) \[S^{(1)}=\frac{k}{2\pi}\,\int\Big{[}\omega^{i}\operatorname{d}\!\omega_{i}+ \tfrac{1}{3}\varepsilon_{ijk}\,\omega^{i}\omega^{j}\omega^{k}+\lambda^{i}T_{i} \Big{]}\] (20) which is manifestly diffeomorphism invariant. See again Kraus and Larsen (2006). A key result of the canonical analysis [7] (using the same methods as in Blagojevic and Cvetkovic (2011)) is an expression for the diffeomorphism charges $Q^{P}[\xi^{\rho}]$ ($\delta Q$ denotes the difference in charge between two states in the theory): \[\delta Q_{P}[\xi^{\rho}]=-\frac{k}{2\pi}\,\int\limits_{0}^{2\pi} \operatorname{d}\!\varphi\,\Big{[}\xi^{\rho}\big{(}e^{i}{}_{\rho}\,\delta \lambda_{i\varphi}+\lambda^{i}{}_{\rho}\,\delta e_{i\varphi}\\ +2\omega^{i}{}_{\rho}\,\delta\omega_{i\varphi}\big{)}+2\theta^{i} [\xi^{\rho}]\,\delta\omega_{i\varphi}\Big{]}\,.\] (21) The last term proportional to the Lorentz parameter $\theta^{i}[\xi^{\rho}]$ vanishes asymptotically for cases I and II. The result (21) can be derived from requiring functional differentiability of the canonical Poincaré generator $\tilde{G}_{P}[\xi^{\rho}]$Blagojevic and Cvetkovic (2011); 7. Functional differentiability of the canonical Weyl generator $\tilde{G}_{W}[\Omega]$ in general also leads to Weyl charges $Q_{W}[\Omega]$, which, however, are trivial for cases I and II. ## III Varying Weyl factor Case III has several similarities to case II, but also some essential differences. For the same reason as before we restrict the Weyl factor $\phi$ as in (16), but with $f$ now being free rather than fixed. The gauge transformations preserving the boundary conditions still include the asymptotic diffeomorphisms (7), while the allowed Weyl rescalings now include all $\Omega$ of the form \[\Omega=\Omega(x^{+},\,x^{-})+{\cal O}(y^{2})\] (22) with an arbitrary function $\Omega(x^{+},\,x^{-})$. Relatedly, we allow variations of the Weyl factor $\delta\phi$ with an arbitrary function $\delta f(x^{+},\,x^{-})$. \[\delta\phi=\delta f(x^{+},\,x^{-})+\ldots\] (23) Let us start with stating the canonical result for the Weyl charge [7], which for case III becomes non-trivial. \[\delta\,Q_{W}[\Omega]=-\frac{k}{\pi}\int\limits_{0}^{2\pi}\operatorname{d}\! \varphi\,\delta f\,\partial_{\varphi}\Omega\,.\] (24) Clearly, the charge (24) is not conserved for arbitrary functions $f$ and $\Omega$. It is conserved if and only if $\partial_{t}(f\partial_{\varphi}\Omega)$ is a total $\varphi$-derivative. Let us consider explicitly the case when the stress tensor is conserved, i.e., when the flatness condition (19) holds. To preserve this form, we must also have $\Omega=\Omega_{+}(x^{+})+\Omega_{-}(x^{-})$. Fourier expanding these functions as \[f_{\pm} =\frac{f_{0}}{2}+\frac{p_{f}}{2}(t\pm\varphi)+\sum_{n\neq 0}f_{ \pm}^{(n)}e^{-in(t\pm\varphi)}\,,\] (25) \[\Omega_{\pm} =\frac{\Omega_{0}}{2}+\frac{p_{\Omega}}{2}(t\pm\varphi)+\sum_{n \neq 0}\Omega_{\pm}^{(n)}e^{-in(t\pm\varphi)}\,,\] (26) one finds straightforwardly that the conservation of the Weyl charge is equivalent to requiring \[f_{+}^{(n)}\Omega_{-}^{(n)}=f_{-}^{(n)}\Omega_{+}^{(n)}\qquad\forall n\neq 0\,.\] (27) This means that the Weyl rescaling $\Omega$ preserves some functional relation between $f_{+}$ and $f_{-}$ of the form \[f_{+}^{(n)}=C_{n}f_{-}^{(n)}\] (28) for some constants $C_{n}$. Particularly simple choices are \[f_{-}=0\,,\quad f_{+}=0\,,\quad\mbox{or}\quad f_{+}(x)=C\,f_{-}(x)\,.\] (29) Any such choice leads to an infinite tower of conserved charges. For instance, the first choice in Eq. (29) leads to the tower \[Q_{W}[\Omega=-e^{in(t+\varphi)}]=2ik\,nf_{+}^{(n)}\,.\] (30) The generators ${\cal J}_{n}=\tilde{G}_{W}[\Omega=-e^{in(t+\varphi)}]$ obey a simple Dirac bracket algebra. \[i\,\{{\cal J}_{n},{\cal J}_{m}\}^{\ast}=2kn\,\delta_{n+m,0}\] (31) The asterisk denotes Dirac brackets defined in a specific partially reduced phase space [7]. Let us now turn to the Virasoro charges. The asymptotic expansion of the dreibein $\bar{e}^{i}{}_{\mu}:=\exp{(-\phi)}\,e^{i}{}_{\mu}$ reads \[\bar{e}^{\,i}{}_{\mu}=\frac{1}{y}\,\delta^{i}_{\mu}+\begin{pmatrix}0&0&0\\ \bar{e}_{(1)+}^{\,-}&0&0\\ 0&0&0\end{pmatrix}+y\begin{pmatrix}\bar{e}_{(2)+}^{\,+}&\bar{e}_{(2)-}^{\,+}&0 \\ \bar{e}_{(2)+}^{\,-}&\bar{e}_{(2)-}^{\,-}&0\\ 0&0&0\end{pmatrix}+\ldots\] (32) and similarly for $\delta\bar{e}^{i}{}_{\mu}$. The expansions for the dualized spin-connection $\omega^{i}$ and the Lagrange multiplier 1-form $\lambda^{i}$ then follow straightforwardly from the equations of motion descending from the first order action (20). Putting these expressions into the variation (21) produces (33) with \[\delta C_{\pm}=\delta\bar{e}_{(2)\pm}^{\mp}+\partial_{\pm} \partial_{\varphi}\delta\phi-(\partial_{\pm}\phi)\partial_{\varphi}\delta\phi+ {\cal O}(y)\,,\] (34) \[\delta C_{y}=\frac{1}{y}\partial_{\varphi}\delta\phi+{\cal O}(1)\,,\] (35) \[\theta^{\hat{y}}[\xi^{\rho}]\,\delta\omega_{\hat{y}\varphi}=\frac {1}{2}\,\big{(}\partial_{+}\varepsilon^{+}-\partial_{-}\varepsilon^{-}\big{)} \partial_{t}\delta\phi+{\cal O}(y)\,.\] (36) To reduce clutter we consider a Virasoro transformation with only $\varepsilon^{+}$ nonzero. (The formulas are completely analogous for the general case.) The diffeomorphism charges become \[\delta Q_{P}[\xi^{\rho}]=-\frac{k}{\pi}\int\limits_{0}^{2\pi}\operatorname{d} \!\varphi\,\varepsilon^{+}\,\Big{[}\delta\bar{e}_{(2)+}^{\,-}-(\partial_{+} \phi)\partial_{\varphi}\delta\phi\Big{]}\,,\] (37) where we dropped a total $\varphi$-derivative. Note that the second term ($\sim\partial_{+}\phi\partial_{\varphi}\delta\phi$) is not integrable in general. However, considering a compensating Weyl rescaling as in case II we have \[\delta Q_{W}[\xi^{\rho}]=-\frac{k}{\pi}\int\limits_{0}^{2\pi}\operatorname{d} \!\varphi\,\left[-\delta\phi\,\partial_{\varphi}(\varepsilon^{+}\partial_{+} \phi)\right]\,.\] (38) This expression results from combining (24) and (17). Adding these contributions, $\delta Q[\xi^{\rho}]=\delta Q_{P}[\xi^{\rho}]+\delta Q_{W}[\xi^{\rho}]$, turns the terms bilinear in $\phi$ and $\delta\phi$ into a $\varphi$-derivative. Taking into account also the terms proportional to $\varepsilon^{-}$ we obtain the charges \[Q[\xi^{\rho}]=-\frac{k}{\pi}\,\int\limits_{0}^{2\pi}\operatorname{d}\!\varphi \,\Big{[}\varepsilon^{+}\,\bar{e}_{(2)+}^{\,-}+\varepsilon^{-}\,\bar{e}_{(2)-} ^{\,+}\Big{]}\,.\] (39) The result (39) coincides with the one for cases I and II. Thus, the canonical charges (39) are conserved. ## IV CFT interpretation Based upon the results above we conjecture that CSG with boundary conditions (4)-(6), (16) and $f=f(x^{+})$ is dual to a CFT with enhanced symmetries. These symmetries are generated by the Virasoro operators $L_{n}$, $\bar{L}_{n}$ as well as the generators ${\cal J}_{n}$. The Virasoro generators are defined as a combination of diffeomorphisms (7) accompanied by compensating Weyl rescalings (17): $L_{n}=\tilde{G}_{P}[\varepsilon^{+}=e^{inx^{+}}]+\tilde{G}_{W}[\Omega=\Omega( \varepsilon^{+})]$ and $\bar{L}_{n}=\tilde{G}_{P}[\varepsilon^{-}=-e^{-inx^{-}}]+\tilde{G}_{W}[\Omega= \Omega(\varepsilon^{-})]$. Therefore the non-zero commutators are \[[L_{n},\,L_{m}] =(n-m)\,L_{n+m}+\frac{c_{R}}{12}\,(n^{3}-n)\,\delta_{n+m,0}\,,\] (40a) \[=(n-m)\,\bar{L}_{n+m}+\frac{c_{L}}{12}\,(n^{3}-n)\,\delta_{n+m,0}\,,\] (40b) \[=c_{0}\,n\,\delta_{n+m,0}\,.\] (40c) The values of the central charges are determined by the Chern–Simons level $k$ from the action (1): \[c_{R}=-c_{L}=6\,c_{0}=12\,k\] (41) Of course, the value of $c_{0}$ is defined only with respect to a given normalization of the generators ${\cal J}_{n}$. The commutator $[L_{n},\,{\cal J}_{m}]$ vanishes due to the peculiar way the Virasoro generators $L_{n}$, $\bar{L}_{n}$ arise. By construction they act trivially on the 3-dimensional Weyl factor. We define now Sugawara-shifted generators ${\cal L}_{n}$ that generate only holomorphic diffeomorphisms ($::$ denotes normal ordering): \[L_{n}\to{\cal L}_{n}=L_{n}+\frac{1}{4k}\,\sum_{m\in\mathbb{Z}}:{\cal J}_{m}{ \cal J}_{n-m}:\] (42) To show that the generators ${\cal L}_{n}$ produce only diffeomorphisms it is sufficient to check that the compensating Weyl charge (38) by virtue of (30) can be written as \[\delta Q_{W}=\frac{k}{\pi}\,\int\limits_{0}^{2\pi}\operatorname{d }\!\varphi\,\delta f\partial_{\varphi}(e^{inx^{+}}\partial_{+}f)\\ =k\,\delta\Big{(}\sum_{m\in\mathbb{Z}}m(n-m)f_{+}^{(m)}f_{+}^{(n- m)}\Big{)}\Rightarrow\\ \tilde{G}_{W}=-\frac{1}{4k}\,\sum_{m\in\mathbb{Z}}:{\cal J}_{m}{ \cal J}_{n-m}:\] (43) In the last equality we have converted classical expressions into quantum operators, with corresponding ordering ambiguities. We have fixed the latter by requiring normal ordering. Since $L_{n}$ is a sum of a diffeomorphism and a holomorphic Weyl rescaling with Weyl charge (43), the shifted Virasoro operators ${\cal L}_{n}$ in (42) generate by construction solely diffeomorphisms. The definition (42) together with the old algebra (40) establish a new algebra that contains an affine $\hat{u}(1)_{R}$ generated by ${\cal J}_{n}$. \[[{\cal L}_{n},\,{\cal L}_{m}] =(n-m)\,{\cal L}_{n+m}+\big{(}k+\frac{1}{12}\big{)}\,(n^{3}-n)\, \delta_{n+m,0}\] (44a) \[=(n-m)\,\bar{L}_{n+m}-k\,(n^{3}-n)\,\delta_{n+m,0}\] (44b) \[=2k\,n\,\delta_{n+m,0}\] (44c) \[=-m\,{\cal J}_{n+m}\] (44d) Note in particular that the last commutator is now non-vanishing and shows that ${\cal J}_{n}$ behaves as an operator with the appropriate conformal weights $(1,0)$. As compared to cases I and II, in case III the holomorphic Weyl factor constitutes an additional free chiral boson in the theory. The shift of $c_{R}\to c_{R}+1$ (and no shift of $c_{L}$) is precisely what one would expect from a free chiral boson. Note that there is no corresponding shift of the left central charge or the left Virasoro generators, since charge conservation demands that we accompany a diffeomorphism generated by $\xi[\varepsilon^{-}=-e^{inx^{-}}]$ with a compensating anti-holomorphic Weyl rescaling (17) with $\varepsilon^{-}=-e^{inx^{-}}$. It is worthwhile pointing out the peculiar way in which the chiral boson arises on the gravity side. In the bulk there is no scalar field, but only the Weyl factor $\phi$ in (4). The bulk equations of motion (2) do not restrict $\phi$ at all! The whole dynamics of $\phi$ emerges through consistency conditions imposed at the boundary. If the stress-energy tensor is postulated to be conserved then the flatness condition (19) must hold, which is equivalent to demanding that $\phi$ obeys the massless Klein-Gordon equation at $\partial M$. If additionally the Weyl charges are required to be conserved then $\phi$ is restricted further, as we have shown above. Thus, the whole dynamics of the scalar field arises solely through boundary and consistency conditions, and not through an interplay between bulk and boundary dynamics. It will be interesting to relax some of these requirements and study consequences for the dual CFT [7]. For instance, considering a curved boundary metric with Ricci scalar ${\cal R}$ we expect the condition (19) to be replaced by the Liouville equation $\nabla^{2}f\propto{\cal R}$. Another interesting generalization is to allow for Weyl factors $\phi$ that contain a term $b\,\ln{y}$ (see footnote [18]). For the second choice in (29) essentially the same discussion applies, with suitable changes $L\leftrightarrow R$. Other choices of the constants $C_{n}$ in (28) lead to correspondingly different dual CFTs. One particular class of choices exhibits an interesting feature. Assuming \[C_{n}C_{-n}=\|C_{n}\|^{2}=1\qquad\forall n\] (45) we find that all Weyl charges vanish and the generators ${\cal J}_{n}$ commute with each other. A more comprehensive discussion of these and more general choices, as well as their consequences, will be presented elsewhere [7]. Some of the cases II (and case I) can be interpreted as constant Weyl charge superselection sectors of a specific case III CFT. For instance, case II with holomorphic Weyl factor $\Omega(x^{+})$ is recovered from the CFT defined by (40)-(41) by choosing $\Omega$ correspondingly and by setting all Weyl charges to zero. If $\Omega$ is constant then case I is recovered. However, not all cases II can be recovered in this way, because of the restriction (28). The CFTs discussed here cannot be unitary since one of the central charges always is negative (except for some non-integer $k$), e.g. $c_{L}<0$. Redefining $\bar{L}_{n}\to-\bar{L}_{-n}$ makes the central charge positive, but the corresponding CFT still is not unitary, since there are states with negative weights, as evident from the spectrum of Weyl gravitons, see (13). It remains a challenge to construct some unitary dual CFT for CSG. Finally, it would be of interest to generalize our results, where applicable, to 4-dimensional conformal gravity, see Lu and Pope (2011); Maldacena (2011) and references therein. ###### Acknowledgements. We thank Radoslav Rashkov and Thomas Zojer for discussions. HA thanks Hessamaddin Arfaei for his support and encouragement as well as ITP members and secretaries in Vienna for their hospitality. HA is supported by the Ministry of Science, Research and Technology in Iran, and during the final stages also by the START project Y435-N16 of the Austrian Science Fund (FWF). BC is supported by the Serbian Science Foundation, Grant No. 171031. SE, DG and NJ are supported by the FWF projects Y435-N16 and P21927-N16. BC acknowledges travel support from the FWF project Y435-N16. ## References * Deser et al. (1982a) S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. 48, 975 (1982a). Ann. Phys. 140, 372 (1982b). Erratum-ibid. 185, 406 (1988). * Horne and Witten (1989) J. H. Horne and E. Witten, Phys. Rev. Lett. 62, 501 (1989). * Brown and Henneaux (1986) J. D. Brown and M. Henneaux, Commun. Math. Phys. 104, 207 (1986). * Witten (1988) E. Witten, Nucl. Phys. B311, 46 (1988). * Maldacena (1998) J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), eprint hep-th/9711200. * Aharony et al. (2000) O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Phys. Rept. 323, 183 (2000), eprint hep-th/9905111. * (7) H. Afshar, B. Cvetkovic, S. Ertl, D. Grumiller, and N. Johansson, in preparation. * Guica et al. (2011) M. Guica, K. Skenderis, M. Taylor, and B. C. van Rees, JHEP 02, 056 (2011), eprint 1008.1991. * Kraus and Larsen (2006) P. Kraus and F. Larsen, JHEP 01, 022 (2006), eprint hep-th/0508218. * Skenderis et al. (2009) K. Skenderis, M. Taylor, and B. C. van Rees, JHEP 09, 045 (2009), eprint 0906.4926. * Grumiller and Sachs (2010) D. Grumiller and I. Sachs, JHEP 03, 012 (2010), eprint 0910.5241. * Grumiller et al. (2011) D. Grumiller, N. Johansson, and T. Zojer, JHEP 01, 090 (2011), eprint 1010.4449. * Bertin et al. (2011) M. Bertin, D. Grumiller, D. Vassilevich, and T. Zojer JHEP 06, 111 (2011), eprint 1103.5468. * Solodukhin (2006) S. N. Solodukhin, Phys. Rev. D74, 024015 (2006), eprint hep-th/0509148. * Blagojevic and Cvetkovic (2011) M. Blagojevic and B. Cvetkovic, JHEP 01, 082 (2011), eprint 1010.2596. * Lu and Pope (2011) H. Lu and C. Pope, Phys. Rev. Lett. 106, 181302 (2011), eprint 1101.1971. * Maldacena (2011) J. Maldacena, eprint 1105.5632.
1004.5557	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 15714, "num_imgs": 4, "llama3_tokens_count": 4457 }	[ "content_image/1004.5557/Figure1.png", "content_image/1004.5557/Figure2.png", "content_image/1004.5557/Figure3.jpg", "content_image/1004.5557/Figure4.jpg" ]	# Coherent interfacial bonding on the FeAs tetrahedron in Fe/Ba(Fe${}_{1-x}$Co${}_{x}$)${}_{2}$As${}_{2}$ bilayers T. Thersleff T.Thersleff@ifw-dresden.de. IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany K. Iida K.Iida@ifw-dresden.de. IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany S. Haindl IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany M. Kidszun IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany D. Pohl IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany A. Hartmann IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany F. Kurth IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany J. Hänisch IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany R. Hühne IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany B. Rellinghaus IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany L. Schultz IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany B. Holzapfel IFW Dresden, Institute for Metallic Materials, P.O. Box 270116, D-01171 Dresden, Germany February 18, 2024 ###### Abstract We demonstrate the growth of epitaxial Fe/Ba(Fe${}_{1-x}$Co${}_{x}$)${}_{2}$As${}_{2}$ (Fe/Ba–122) bilayers on MgO(001) and LSAT(001) single crystal substrates using Pulsed Laser Deposition (PLD). By exploiting the metallic nature of the FeAs tetrahedron in the Ba-122 crystal structure, we achieve a coherent interfacial bond between bcc iron and Co-doped Ba–122. $T_{\rm c}$ values for both bilayers were close to that of the PLD target. Direct observation of interfacial bonding between Fe and the Ba–122 FeAs sublattice by atomic resolution transmission electron microscopy implies that this bilayer architecture may work for other iron pnictide systems and pave the way for the fabrication of superconducting/ferromagnetic heterostructures. FeAs superconductors, interface analysis, Fe/Ba–122 bilayers pacs: 74.70.Xa, 68.37.-d † [FOOTNOTE:†][ENDFOOTNOTE] Rapidly following the discovery of superconductivity in the AE(Fe${}_{1-x}$Co${}_{x}$)${}_{2}$As${}_{2}$ (AE-122, AE = Alkaline Earth) systemRotter _et al._ (2008), thin films were produced to probe the fundamental properties of and assess potential applications for these unique materialsHiramatsu _et al._ (2008); Choi _et al._ (2009); Hiramatsu _et al._ (2009); Katase _et al._ (2009); Lee _et al._ (2009); Iida _et al._ (2009); Lee _et al._ (2010); Iida _et al._ (2010); Katase _et al._ (2010). However, difficulties overcoming the poor metal/oxide bond at the interface of many substrates has necessitated the need for significant optimization of the deposition parametersIida _et al._ (2010); Katase _et al._ (2010) as well as the use of various intermediate layersLee _et al._ (2010) to produce well-textured films. In spite of these efforts, nearly all of these films contain an unintentional amorphous or iron-containing layer at the interface. While the nature of this interface is not yet fully understood, the disruption of local crystallographic ordering associated with it precludes the use of these films for interface-sensitive applications such as multilayers or heterostructures where coherent and chemically inert phase boundaries are required. Moreover, it may be responsible for the challenging growth of epitaxial Ba–122 films in generalIida _et al._ (2009, 2010) as well as the generation of pinning-active columnar defects observed to originate at this interface in some filmsLee _et al._ (2010). A careful TEM investigation of the interface between epitaxially-grown Co-doped Ba–122 and bare (La,Sr)(Al,Ta)O${}_{3}$ (LSAT) substrates revealed significant amounts of textured body-centered cubic (bcc) ironIida _et al._ (2010). The orientation of this iron layer was rotated 45${}^{\circ}$ in-plane to both the substrate and the Ba–122 phase and is the likely culprit for the Fe (200) reflection in XRD patterns. In this orientation, the (100) surface plane of iron has an approximately 2% lattice mismatch with the square-planar iron sublayer defining the FeAs tetrahedron in the Ba–122 unit cell and thus offers a natural location for metallic bonding (figure 1). Furthermore, since the quality of our films containing these iron regions is consistently very high, it appears to be advantageous to their epitaxial growth and superconducting properties. <figure><img src="content_image/1004.5557/Figure1.png"><figcaption>Figure 1: (Color online) The mismatch between the (001) surface plane of bcciron (left) and the iron sublayer in the Co-doped Ba–122 unit cell (right) isabout 2% . Atomic radii are not to scale.</figcaption></figure> To investigate the nature of bonding at this interface, we deposited Fe/Ba–122 bilayers on MgO(001) and LSAT(001) substrates. An Fe layer of 20 nm was deposited onto MgO and LSAT at 620${}^{\circ}$C using the standard on-axis Pulsed Laser Deposition (PLD) technique in a $10^{-8}$ mbar chamber with a 248 nm KrF laser operating at 10Hz. Subsequently, a Ba–122 layer of around 130 nm was deposited at 700${}^{\circ}$C. We observed island growth of Fe in the resulting films on both substrates; however, considerable research on the optimization of iron thin film growth on MgO already existsJordan _et al._ (1998). Accordingly, iron on the MgO substrate was first deposited at room temperature and then heated to 700${}^{\circ}$C for the deposition of the Ba–122 phase under identical conditions used for the Ba–122 layer on Fe-buffered LSAT. The XRD data acquired using the Bragg-Brentano geometry with Co $K_{\alpha}$ radiation (figure 2a) reveal _c_-axis textured growth for both iron and Ba–122 layers on MgO and LSAT substrates. Neither bilayer shows evidence for secondary phases. The additional Ba–122 texture component on the LSAT substrate leading to the (110) peak in the XRD scan has a distinct epitaxial relationship to the substrate with $(110)[001]$Ba–122 $\parallel$$(001)[110]$ LSAT and $(110)[001]$Ba–122 $\parallel$$(001)[\overline{1}10]$ LSAT. As a result, two additional satellite peaks appear in the (103) pole figure near the peak for the main texture component (figure 2d). On MgO, no additional texture components could be identified suggesting pure epitaxial growth (figure 2c). The Ba–122 (103) reflection on both substrates exhibits four-fold symmetry with $\Delta\phi_{\textup{MgO,LSAT}}=0.95^{\circ},1.17^{\circ}$ and (004) rocking curves reveal $\Delta\omega_{\textup{MgO,LSAT}}=0.64^{\circ},1.01^{\circ}$. In figure 2e,f, the iron layer appears well textured on both substrates with a 45${}^{\circ}$ in-plane rotation to the Ba–122 phase. <figure><img src="content_image/1004.5557/Figure2.png"><figcaption>Figure 2: (Color online) (a) X-Ray diffraction pattern for bilayers on bothLSAT and MgO in the Bragg-Brentano geometry. Neither bilayer shows evidencefor secondary phases and both exhibit a strong c-axis texture. The LSATbilayer additionally contains a second texture component evidenced by theBa–122 (110) reflection. (b) Resistively measured Tc curves for Ba–122deposited upon bare LSAT and MgO as well as the new Fe/Ba–122 bilayers. Tc,90is 24.4 K and 24.8 K for the MgO and LSAT bilayers respectively. (c) Polefigure of the Ba–122 (103) reflection for the MgO bilayer reveals highlytextured growth with a Δϕ value of 0.95∘. (d) While the Ba–122 (103)reflection exhibits excellent texture with a Δϕ value of 1.17∘, the additional(110) component also appears to be textured. (e,f) Iron on both substratesgrows with a 45∘ in-plane rotation and exhibits four-fold symmetry. Theadditional peak in the center of (e) arises from the (002) plane of the MgOsubstrate. All pole figures are plotted on a square root scale.</figcaption></figure> Resistively-measured $T_{\rm c}$ values (figure 2b) for both films are very high showing a $T_{c,90}$ of 24.4 K and 24.8 K on MgO and LSAT respectively. These values are among the highest reported for any Co-doped Ba–122 thin film to date and are nearly equal to that of the PLD target used ($T_{\rm c}=25.5$ K as measured with a vibrating sample magnetometer). To reveal the location of the misaligned Ba–122 on the LSAT bilayer as well as to elucidate the nature of the Fe/Ba–122 interface common to the bilayers on both substrates, a comprehensive Transmission Electron Microscopy (TEM) investigation on this film was initiated. A TEM lamella was prepared using the Focused Ion Beam (FIB) in-situ lift-out techniqueLangford (2006). The bright field TEM overview shown in figure 3a confirms the nucleation of the (001) faceted iron islands discussed previously. Significantly, the (110) oriented Ba–122 component observed in figure 2a,d appears to grow exclusively between these faceted islands whereas the (001) iron surface plane provides an effective interface for the epitaxial growth of the Ba–122 phase, showing no misalignment of the texture over large sample areas. On the MgO substrate, the optimized deposition parameters for the iron layer eliminated any island growth as evidenced by a scanning electron microscopy image of a FIB cross-section provided in figure 3b. Consequently, no misaligned texture is observed in figure 2 for the MgO bilayer. <figure><img src="content_image/1004.5557/Figure3.jpg"><figcaption>Figure 3: (Color online) (a) Bright field TEM overview of two faceted ironislands. Ba–122 between iron islands grows with (110)[001]Ba–122 ∥ (001)[110]LSAT and (110)[001]Ba–122 ∥ (001)[¯¯¯110] LSAT whereas atop the (100) surfacefacet, Ba–122 grows with (001)[100] Fe ∥(001)[110] Ba–122. (b) FIB cut imagedusing a secondary electron in-lens detector on the MgO bilayer reveals noisland growth.</figcaption></figure> The interface between Fe and Ba–122 was studied on the TEM lamella described above using High Resolution Scanning TEM (HRSTEM) on an FEI Titan${}^{3}$ 80–300 microscope with an image $C_{s}$ corrector operating at 300 kV. Figure 4a shows a HRSTEM image obtained with a High Angle Annular Dark Field (HAADF) detector. Directly below, higher resolution data obtained from a different sample region is presented. By selecting a camera length of 363 mm, the atomic columns appear as bright dots and the crystallographic symmetry of the Ba–122 phase becomes evident. In the lower portion of figure 4, the location of atomic columns is denoted using artificial colors and a schematic model. The Ba–122 phase is observed to terminate on the upper As sublayer of the FeAs tetrahedron. <figure><img src="content_image/1004.5557/Figure4.jpg"><figcaption>Figure 4: (Color online) (a) HRSTEM image of the Fe/Ba–122 interface. Directlybelow, a higher resolution image of a neighboring region is shown. (b) HRTEMimage from the Fe/Ba–122 interface with an enlargement below. In both (a) and(b), the atomic columns appear white and are identified through the use ofartificial coloration and a schematic model. The interface is highly coherentand bonding takes place directly on the FeAs sublattice.</figcaption></figure> As an independent confirmation of this analysis, High Resolution TEM (HRTEM) was undertaken on a separate sample region and is presented in figure 4b. This image was acquired such that the atomic columns appear as white dots, significantly easing the image interpretation. Below, the atomic positions are identified in the same manner as previously. Combined with the HRSTEM data in figure 4a, these observations constitute compelling evidence that the square-planar iron sublayer in the Ba–122 unit cell is directly replaced by the (001) surface plane of the bcc iron layer resulting in a coherent interfacial bond on the FeAs sublattice. The results of this study contain some wide-reaching implications. First, the metallic nature of and excellent lattice matching between the Ba–122 iron sublayer and the bcc iron (001) surface plane ensure that this interface will be highly coherent. This suggests that the epitaxial growth of Ba–122 will be favorable on any substrate upon which a planar iron (001) facet can be grown, as directly demonstrated by the well-textured growth of an Fe/Ba–122 bilayer on MgO(001). On bare MgO, full epitaxy was not obtained due to a large lattice misfit of around 6% and $T_{\rm c}$ was significantly reduced. Second, in addition to the excellent texture of these bilayers, their $T_{\rm c}$ values remain close to that of the Ba–122 target material, thus representing a way to retain good superconducting properties in epitaxially-grown thin films. Third, the interfacial bond between the iron layer and the Ba–122 phase is directly observed to take place on the iron sublayer within the FeAs tetrahedron using two independent imaging techniques. Since the FeAs tetrahedron is the one structural feature common to every type of iron pnictide, these results suggest that similar bilayer structures from other iron pnictide systems can be realized. Finally, the clean and coherent nature of the Fe/Ba–122 interface may enable the fabrication of ferromagnetic/superconducting heterostructures thus paving the way for future studies on the interplay between magnetism and superconductivity in the iron pnictides. ###### Acknowledgements. We wish to acknowledge J. Scheiter for help with the TEM lamella preparation as well as S. Fähler, J. Engelmann, and S. Trommler for the scientific discussions. ## References * (1) * Rotter _et al._ (2008)M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett., 101, 107006 (2008). * Hiramatsu _et al._ (2008)H. Hiramatsu, T. Katase, T. Kamiya, M. Hirano, and H. Hosono, Appl. Phys. Express, 1, 101702 (2008). * Choi _et al._ (2009)E.-M. Choi, S.-G. Jung, N. H. Lee, Y.-S. Kwon, W. N. Kang, D. H. Kim, M.-H. Jung, S.-I. Lee, and L. Sun, Appl. Phys. Lett., 95, 062507 (2009). * Hiramatsu _et al._ (2009)H. Hiramatsu, T. Katase, T. Kamiya, M. Hirano, and H. Hosono, Phys. Rev. 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1405.1221	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 54700, "num_imgs": 6, "llama3_tokens_count": 11486 }	[ "content_image/1405.1221/x1.png", "content_image/1405.1221/x2.png", "content_image/1405.1221/x3.png", "content_image/1405.1221/x4.png", "content_image/1405.1221/x5.png", "content_image/1405.1221/x6.png" ]	# KMCLib: A general framework for lattice kinetic Monte Carlo (KMC) simulations Mikael Leetmaa Natalia V. Skorodumova Multiscale Materials Modelling, Materials Science and Engineering, School of Industrial Engineering and Management, KTH - Royal Institute of Technology, Brinellvägen 23, 100 44 Stockholm, Sweden Department of Physics and Astronomy, Uppsala University, Box 516, 751 20 Uppsala, Sweden 4/22/2014 ###### Abstract KMCLib is a general framework for _lattice_ kinetic Monte Carlo (KMC) simulations. The program can handle simulations of the diffusion and reaction of millions of particles in one, two, or three dimensions, and is designed to be easily extended and customized by the user to allow for the development of complex custom KMC models for specific systems without having to modify the core functionality of the program. Analysis modules and on-the-fly elementary step diffusion rate calculations can be implemented as plugins following a well-defined API. The plugin modules are loosely coupled to the core KMCLib program via the Python scripting language. KMCLib is written as a Python module with a backend C++ library. After initial compilation of the backend library KMCLib is used as a Python module; input to the program is given as a Python script executed using a standard Python interpreter. We give a detailed description of the features and implementation of the code and demonstrate its scaling behavior and parallel performance with a simple one-dimensional A-B-C lattice KMC model and a more complex three-dimensional lattice KMC model of oxygen-vacancy diffusion in a fluorite structured metal oxide. KMCLib can keep track of individual particle movements and includes tools for mean square displacement analysis, and is therefore particularly well suited for studying diffusion processes at surfaces and in solids. keywords: KMC; kinetic Monte Carlo; diffusion; simulation framework; Python † [FOOTNOTE:†][ENDFOOTNOTE] PROGRAM SUMMARY _Program Title:_ KMCLib _Journal Reference:_ _Catalogue identifier:_ _Licensing provisions:_ GPLv3 _Programming language:_ Python and C++ _Computer:_ Any computer that can run a C++ compiler and a Python interpreter _Operating system:_ Tested on Ubuntu 12.4 LTS, CentOS release 5.9, Mac OSX 10.5.8 and Mac OSX 10.8.2, but should run on any system that can have a C++ compiler, MPI and a Python interpreter. _RAM:_ From a few megabytes to several gigabytes depending on input parameters and the size of the system to simulate. _Number of processors used:_ From one to hundreds of processors depending on the on the type of input and simulation. _Supplementary material:_ The full documentation of the program is distributed with the code and can also be found at http://www.github.com/leetmaa/KMCLib/manual _Keywords:_ KMC; lattice; kinetic; Monte Carlo; diffusion; simulation; framework; plugin; Python _Classification:_ 4.13, 16.13 _External routines/libraries:_ KMCLib uses and external Mersenne Twister pseudo random number generator that is included in the code. A Python 2.7 interpreter and a standard C++ runtime library is needed to run the serial version of the code. For running the parallel version an MPI implementation is needed, such as _e.g._ MPICH from http://www.mpich.org or OpenMPI from http://www.open-mpi.org. SWIG (obtainable from http://www.swig.org/) and CMake (obtainable from http://www.cmake.org/) are needed for building the backend module, Sphinx (obtainable from http://sphinx-doc.org) for building the documentation and CPPUNIT (obtainable from http://sourceforge.net/projects/cppunit/) for building the C++ unit tests. _Nature of problem:_ Atomic scale simulation of slowly evolving dynamics is a great challenge in many areas of computational materials science and catalysis. When the rare-events dynamics of interest is orders of magnitude slower than the typical atomic vibrational frequencies a straight-forward propagation of the equations of motions for the particles in the simulation can not reach time scales of relevance for modeling the slow dynamics. _Solution method:_ KMCLib provides an implementation of the kinetic Monte Carlo (KMC) method that solves the slow dynamics problem by utilizing the separation of time scales between fast vibrational motion and the slowly evolving rare-events dynamics. Only the latter is treated explicitly and the system is simulated as jumping between fully equilibrated local energy minima on the slow-dynamics potential energy surface. _Restrictions:_ KMCLib implements the _lattice_ KMC method and is as such restricted to geometries that can be expressed on a grid in space. _Unusual features:_ KMCLib has been designed to be easily customized, to allow for user-defined functionality and integration with other codes. The user can define her own on-the-fly rate calculator via a Python API, so that site-specific elementary process rates, or rates depending on long-range interactions or complex geometrical features can easily be included. KMCLib also allows for on-the-fly analysis with user-defined analysis modules. KMCLib can keep track of individual particle movements and includes tools for mean square displacement analysis, and is therefore particularly well suited for studying diffusion processes at surfaces and in solids. _Additional comments:_ The program can be obtained as a git repository from http://www.github.com/leetmaa/KMCLib _Running time:_ From a few seconds to several days depending on the type of simulation and input parameters. ## 1 Introduction Atomic scale simulations of slowly evolving dynamics pose a great challenge in many areas of computational materials science and catalysis. The dynamics of processes such as chemical reactions on surfaces and slow diffusion events in solids often involve time-scales that are impossible to access with conventional molecular dynamics (MD) simulations. When atomic resolution is demanded to properly describe the investigated physical system, but the time-scale of the events of interest renders MD simulations unsuitable, some other method working efficiently with atomic resolution on longer time scales must be employed. There exists today a range of accelerated dynamics methods handling the dynamics in one way or another to allow for the rare events to occur much more frequently in the simulation [1]. Kinetic Monte Carlo (KMC) [2], [3] is one such method. The idea behind KMC is to utilize the separation of time scales between on the one hand the fast molecular vibrations, and on the other hand the rare events of interest evolving the slow dynamics. Only the latter is explicitly simulated, while the former is treated as thermally equilibrated at the time one of the rare events takes place. This is theoretically justified when the separation of the vibrations and rare events time scales is large [4], [5]. For many systems of high scientific and technological interest this condition is known to be well satisfied, paving the ground for the popularity of different flavors of the KMC method [5], [6], [7], [8], [9], with common applications ranging from biological systems [8] and heterogeneous catalysis [10] to the electrochemistry in fuel-cells [11] and atomic transport mechanisms in alloys [12]. We will in this work focus on the _lattice_ KMC method, a particularly efficient form of KMC for simulating ordered – as opposed to amorphous – systems, where the geometry of the simulated system can be conveniently described on a regular grid in space [5]. KMCLib implements the _lattice_ KMC algorithm in the form of the algorithm known as the variable step size method (VSSM) or the _n-fold_ way[2], [3], [5]. Although several other methods are possible [13] the VSSM methods is theoretically well founded [14] and particularly well suited for an efficient implementation for simulations of large systems. Despite its maturity and the popularity of the approach there has been surprisingly few publicly available computer programs implementing the _lattice_ KMC method [13], [15], [16], [17]. Although this situation is improving [18], [19], [20], most available codes are still particularly geared towards surface studies [13], [15], [20], or closed source [15] preventing straightforward customization. Due to this shortage of publicly available open sourced software together with the relative simplicity of the algorithm and the need for significant method customizations in many cases, is has not been uncommon that a whole new KMC code gets written for each new set of physical problems to solve. Many research groups each have their own custom code aimed at simulating a particular type of systems, and it is not uncommon that these codes never make it to the public. We hope to reverse this trend by providing a fast and efficient general framework for _lattice_ KMC simulations that is easy to customize and to integrate with other software. Our hope is that KMCLib will be a useful platform for implementing custom KMC models, meeting the highest expectations for flexibility, robustness and efficiency. KMCLib consists of a highly optimized C++ core program with a well-defined front-end API in Python, and is after compilation of the C++ code used as a Python module. Code extensions can be written in Python with no need to recompile the underlying C++ code. We have introduced the possibility for using custom rate calculations on-the-fly during the simulation, via a Python interface, that facilitates usage of any external code for calculating elementary process rates. This makes it possible to include _e.g._ long-range interactions or site and geometry dependent process-rates in the simulations, without modifying the main program. A plugin API for analysis on-the-fly during the simulation has also been specified to allow for user-defined analysis tools to be integrated with the code. The code is also equipped with an implementation of a recent mean square displacement (MDS) algorithm for non-equidistant time step KMC simulations [21] to facilitate diffusion studies with the code. This paper describes the main simulation algorithms implemented in KMCLib with and without the custom rate calculation modifications. The code is then described in more detail with focus on technical solutions. The scaling behavior of the implementation is demonstrated using a simple one-dimensional A-B-C model as well as with a more complex three-dimensional fluorite structure metal oxide vacancy-diffusion model. Parallel performance is finally discussed and demonstrated. The KMC formalism itself will not be described in detail since it is well-known and has been described elsewhere in the literature (see e.g. [2], [3], [5], [6]). ## 2 Algorithms ### Outline of the main KMC algorithm Figure 1 shows a schematic illustration of the version of the VSSM or _n-fold_ way _lattice_ KMC algorithm implemented in KMCLib. 1. Start with a set of particles at a set of points in space (lattice sites), and a list of all possible elementary processes _e.g._ diffusion or reaction events that can take the system of particles from the present state to the next state. Each elementary process $p_{i}$ is associated with a rate constant $r_{i}$. 2. Match all processes with the geometry at each lattice site to construct a table of the availability of each process and obtain the total rate of each process $R_{i}$ by multiplying the process’ rate $r_{i}$ with the number of sites where the process is available $N_{i}$ \[R_{i}=r_{i}N_{i}\] (1) For the list of processes a vector ${\bf P}$ is constructed with the incremental total rates, such that for the $j$:th process the incremental value is \[P_{j}=\sum_{i=1}^{j}R_{i}\] (2) 3. A process to perform is chosen at random, proportionally to the relative total rate of the process. This is done by drawing a random number $\rho$ between 0 and the last value in ${\bf P}$, $P_{k}$ and a binary search is performed over the elements in ${\bf P}$ to determine which process to take. Starting from the lower end, the last process $p_{j}$ for which $P_{j}<\rho$ is chosen. 4. A site to perform the process at is chosen by drawing from the list of sites where the process is available, with each site being equally likely. 5. The chosen process is performed at the chosen site by updating the geometry according to the specifications of the particular process. Each process holds information about the local geometry around a site before and after the process is performed. The _before_ geometry is used for matching the local geometry of a lattice site with the process, while the _after_ geometry is used to update the local geometry around the lattice site where the process is performed. 6. Time is propagated by drawing a random number between 0 and 1, and the time step $\Delta t$ to add to the simulation time is determined by \[\Delta t=\frac{-\ln(\rho)}{P_{k}}\] (3) with $P_{k}$ being the total available rate in the system. 7. Repeat from step 2 until the maximum allowed number of steps is reached. The simulation is run for a fixed number of steps. Geometry and time step data can either be stored in a trajectory file for post-processing and analysis, or analysis can be performed on-the-fly during the simulation. <figure><img src="content_image/1405.1221/x1.png"><figcaption>Figure 1: Flowchart of the main KMC algorithm implemented in KMCLib. See textfor details.</figcaption></figure> ### Algorithm modifications due to custom rate calculations An important feature of KMCLib is its ability to handle processes where the rates are not given at the start of the simulation, but are instead calculated and updated on-the-fly during the simulation. The two different ways to determine the process rates (at simulation start-up or during the simulation) can furthermore be combined, such that some of the processes gets their rates updated during the simulation while others do not. The way the rate update framework is implemented allows for great flexibility allowing the user to define the desired behavior via Python code. It is _e.g._ possible to couple the update of the process rates to global or local geometrical parameters, such that a temperature or pressure gradient can affect the local rates of diffusion processes differently at different sites, or that the process rates are being modified at sites in the vicinity of certain atom types or geometrical features that may arise during the simulation. The handling of custom rates requires some modifications to the above described algorithm: 1. At start-up, not only are the initial set of particles and their geometry and all possible processes given, but also a custom rate calculator. The custom rate calculator is a Python class that can receive information about a process and the local geometry around a lattice site and return an updated rate constant for the process at that particular site. The custom rate calculator has to follow a well defined API to assure compatibility with the program and is described in more detail in section 3.3.2 below, as well as in the program user manual. 2. All processes are matched with the geometry at each site to construct a table of the availability and total rate of each process. At each site where a process is possible the rate is also updated using the custom rate calculator. The total rate of each process $R_{i}$ is now obtained by summing the updated local rate for each site where the process is available. Note that due to the use of custom rates the same elementary process will now have a different rate at different sites. With $N_{i}$ being the available sites for process $i$, and $r_{j}$ being the locally updated rate at site $j$ the total rate for process $i$ is given by \[R_{i}=\sum_{j=1}^{N_{i}}r_{j}\] (4) The vector ${\bf P}$ with the incremental total rate for all processes is constructed according to (2). A similar vector ${\bf S}$ of incremental rates is also constructed for each process with the available sites and their respective rates \[S_{j}=\sum_{i=1}^{j}r_{i}\] (5) where $r_{i}$ is the rate at site $i$ of the process. 3. A process to perform is chosen at random, proportionally to the relative total rate of the process. This is done in the same way as described previously by drawing a random number between 0 and $P_{k}$ and performing a binary search over the ${\bf P}$ vector. 4. A site to perform the chosen process at, is chosen by drawing from the list of sites where the chosen process is available. Since the rate now may vary over the sites at which the process is available it becomes necessary to draw the site with a similar procedure as when the process is chosen. A random number $\rho$ is generated between 0 and the last value in ${\bf S}$, $S_{j}$, and a binary search is performed over the elements in ${\bf S}$ to determine which site to take. Starting from the lower end, the last site $i$ for which $S_{i}<\rho$ is chosen. The reminder of the algorithm (perform the process, propagate simulation time and collect data) is identical with the procedure described in section 2.1 above. ### Outline of the mean square displacement algorithm To collect mean square displacement (MSD) data on the fly during the simulation a history buffer is used where a fixed number of coordinates and corresponding simulation times are stored for each particle of interest. Squared displacements of the coordinates are calculated and stored in a histogram. 1. Before the simulation starts the history buffer is populated with the initial coordinate and time for each particle. 2. When a particle is moved its new coordinates along with the current time of the simulation is stored at the first position in the history buffer for that particle. Space is made for the new data in the buffer by moving each coordinate and time already in the buffer to the next position in memory. If the buffer is full the last value is thrown away. 3. The squared displacements between the first (latest added) coordinate in the buffer and all subsequent coordinates stored are calculated and binned in a squared-displacements histogram. The number of values added to each bin is collected in a MSD bin count histogram. 4. After the simulation is finished the MSD data is obtained by dividing the value at each bin in the squared-displacements histogram with the corresponding bin value in the MSD bin count histogram and the standard deviation of the final MSD data is estimated. A more detailed account of the algorithm as well as the somewhat involved procedure for deriving the correct error estimate of the final MSD data has been described in detail elsewhere [21]. ## 3 Description of the program KMCLib is a general platform from which custom _lattice_ KMC models can be implemented. The Python programming language is used to define a well tested and well documented user-interface that is flexible enough to easily define custom KMC models. This is combined with a well-structured performance optimized core program written in C++. The potential need for customization of the underlying C++ code has largely been eliminated through the use of a custom rate calculator interface that allows for algorithmic modifications of the rate determination without the need to re-compile the underlying C++ code. Analysis can be performed and data collected during the simulation, with custom analysis modules loosely coupled to KMCLib via an analysis API in Python. <figure><img src="content_image/1405.1221/x2.png"><figcaption>Figure 2: Code organization overview. Deployment code to the left anddocumentation and test code to the right. Boxes represent logical entitieswhile arrows indicate paths of communication.</figcaption></figure> ### Code overview Figure 2 shows a schematic picture of how the code is organized. The code consists of two major parts. One part is the deployment code for actually performing KMC simulations (figure 2 left), while the other part consists of code for testing and includes also the documentation of the Python user-interface (figure 2 right). The deployment code consists of a set of Python classes that defines the user-interface to the KMCLib module (including the custom rate calculator API and the on-the-fly analysis API), as well as a C++ backend library wrapped to Python using the SWIG framework. The python frontend is responsible for handling user-input and all other communication with the outside world, as well as controlling the flow of the main Monte Carlo loop and communicating with the C++ backend via the SWIG interface. All computationally heavy tasks, from step #2 through to step #5 in figure 1, including matching, picking and updating, are implemented in the C++ backend. All software needs testing to ensure its correct functioning. If the tests are performed systematically and at all levels of the code the risk of having a malfunctioning program can be greatly reduced [22]. The automated KMCLib tests are split up in three parts: _functionality tests_, written using the Python unittest module testing the whole program at an aggregate level; _Python unittests_ (also written using the Python unittest module) testing the Python code at a detailed level; and the _C++ unittests_ implementing detailed tests of the C++ code. The C++ unittests are setup using the CPPUNIT framework. The unittests are intended to test each building block of the code at as low level as possible, while the functionality tests set up realistic but small simulations using the KMCLib Python interface much in the same way a user would do. The functionality tests represent good usage examples and can as such be used by new users to get their first simulations up and running. The Python code is marked up with doc-strings using the Sphinx documentation format. A manual for the Python interface can be auto generated from the source code and the doc-strings using Sphinx and the provided makefile. The documentation included with KMCLib also describes the installation procedure and the code usage. All header files in C++ are documented using the doxygen format to aid in any deeper level customization and development work. ### Input structure Input to KMCLib is written as a Python script and executed using a standard Python interpreter. The input script should start with loading the KMCLib module using a from KMCLib import * statement. This will load all API classes that are needed to setup and run a lattice KMC calculation with KMCLib. Input is then built up by specifying the geometry and all processes via the KMCConfiguration and KMCInteractions classes (see the reference manual for details). On-the-fly custom rate calculations can at this stage be introduced simply by providing an instance of a user-given class inheriting from the KMCRateCalculatorPlugin interface when setting up the KMCInteractions object. If provided, the matching procedure later in the algorithm will use this calculator to determine an updated rate for each process at each site where the process is available. The KMCRateCalculatorPlugin Python class in turn inherits from the RateCalculator C++ class. This is achieved by declaring the C++ base class a “director” in SWIG. When a process is matched the matching routine in C++ calls the user-specified Python CustomRateCalculator via its inherited C++ interface. The custom rates interface thus allows the user to provide an altered rate for the process at the specific site in question. The capacity to use custom rates greatly increases the flexibility of the program and opens up the possibilities for performing KMC simulations with site specific rates or to take into account longer-range interactions when determining the rate of each elementary process. Note that the user is only required to write a Python extension to the program and there is no need to re-compile the underlying C++ code. Details of the custom rate calculator interface can be found in the reference manual. After specifying the geometry and all interactions the input script sets up an instance of the main simulation object the KMCModel, and the model is run by calling the run method on the KMCModel object with an instance of the KMCControlParameters class as input holding all information on the number of steps in the simulation and when analysis and trajectory data should be gathered. If on-the-fly analysis is required a list of user-specified analysis classes, each inheriting from the KMCAnalysisPlugin class, is given as an argument to the run method. Details of how to set up the necessary objects with the correct input can be found in the program reference manual. ### Features and implementation details #### 3.3.1 Match and update algorithm Since the matching of local geometries with possible processes plays a central role in the performance of any lattice KMC implementation we will below describe our matching algorithm in some detail. The KMCLib backend uses a type of neighbor lists that we will refer to as _match-lists_. They are used for representing local geometries around lattice sites in the configuration. When the underlying C++ configuration is constructed from the user-given Python input a match-list is generated for each lattice site. Each match-list holds the lattice-site type information (used for labeling different elements or particle types in the simulation), the corresponding lattice site indices and the relative local coordinates for all neighboring lattices sites. Each lattice site match-lists is sorted according to its relative coordinates. The size of the match-lists is determined from the user-defined processes. The process with the largest included neighborhood will determine the cutoff used for setting up all lattice site match-lists. KMCLib has a dual representation of the geometry. The configuration is represented on a lattice and there is information on each lattice-point about the lattice-point type. But apart from the lattice geometry we also have a list of atom-ID tags with their corresponding current types and Cartesian coordinates. The two representations are connected by the lattice that keeps track of which atom that sits on each lattice-point. This might seem trivial at first, but provides a powerful tool for keeping track of the individual particles on the lattice, which in turn is a prerequisite for any efficient diffusion calculation. Each user given process holds information about the local configuration around a lattice site before and after the process is performed, as well as information about how each of the involved particles should be moved from the initial to the final state of the process. The local geometry information is stored in match-lists similar to the ones used at the lattice-sites. We will refer to these match-lists as the _before_ and _after_ process match-lists. The _before_ process match-list will be used for matching against local geometries on the lattice, while the _after_ process match-list will be used to update the lattice when the process is performed. Each process match-list is sorted in the same way as the lattice site match-lists. If the user has allowed match-list wild-cards the coordinates of the process match-list are compared to the corresponding set of local coordinates at the lattice and any missing entry in the process match-list is inserted with a wild-card identifier as the type. The movements of the individual particles are represented as a set of move-vectors, which in turn will be applied to the Cartesian coordinates of the atom-IDs involved in the process. The matching of a process with a lattice site is performed by comparing the site match-list with the _before_ process match-list. Since the lists are sorted in the same way only the type information needs to be compared. If wild-cards are not allowed in the process match-lists the coordinates must also be compared. Performing a process is done by comparing the site match-list and the process _after_ match-list. Any mismatch between the two will result in the corresponding types information being updated on the lattice. The atom-IDs are updated on the lattice sites according to the specifications given from the process move-vectors and the move-vectors are furthermore added to the corresponding atom-ID coordinates. In this way the lattice configuration and the atom-ID coordinates are guaranteed to be synchronized, with the important difference that the atom-ID coordinates are never checked for periodic boundary conditions. Before the simulation enters the Monte Carlo loop the matching must be done between all sites and all processes. This results in an $O(n_{L}\times n_{P}\times m)$ scaling behavior, where $n_{L}$ is the number of lattice sites, $n_{P}$ is the number of processes and $m$ is the average number of match-list entries that is looped over for each match. After a process has been performed, however, there is no need to re-match the processes at all sites. Only sites that are in the vicinity of the site where the process was performed are affected by the move and a number of affected sites $n_{A}\ll n_{L}$, which is constant with the size of the simulation will require a re-match. This results in an $O(n_{A}\times n_{P}\times m)$ scaling behavior of the matching in the Monte Carlo loop, and the matching is thus _constant_$O(1)$ with respect to the size of the simulation. #### 3.3.2 On-the-fly rate calculations To use the custom rate calculation abilities with KMCLib it is enough to define a class that inherits from CustomRateCalculator, and overload its rate method. The custom rate calculator is given to the interactions object using its setRateCalculator method before the interactions object is used to construct the KMC model. When a rate update is required for a process at a specific site the custom rate calculator’s rate method will be called with parameters specifying the global coordinates of the site and its local geometry and elements (types) before and after the process is applied. When the rate method is called with a site and a process it is guaranteed that the process is available at the particular site (i.e. that the site and the process match). The rate constant associated with the process is also given and it is then fully up to the user to specify any modifications to this rate from the given geometry and process input. Details of the custom rate interface can be found in the reference manual. It should be noted that the flexibility of using custom rate updates inevitably comes with a performance penalty. It is therefore not generally recommended to implement models using a custom rate calculator that can easily be implemented with processes with fixed rates. As an illustrating example we can look the Ising model of [2] that is provided with the functionality tests, where it is implemented both using a custom rate calculator in Python and using processes with fixed rates. The equivalent simulation with fixed rates runs roughly four times faster. #### 3.3.3 Trajectory format A simulation trajectory can be produced in two formats using KMCLib, depending on the type of simulation. For pure lattice KMC simulations that are only concerned with lattice sites and their types, without keeping track of the movement of specific atoms, a lattice trajectory format is used where the type at each lattice site is stored. Since the lattice positions are fixed during the simulation it is enough to print them only once to the trajectory file. For simulations where move-vectors have been provided with the processes, so that the program can keep track of the individual atoms during the simulation, it is also possible to use a simple xyz format that prints the type and coordinate for each atom along the simulation. Note that although the lattice sites do not move during the simulation the individual atoms do move according to the provided move vectors. The user can set the interval between steps to save to the trajectory. To avoid unnecessary file IO operations the trajectory data is stored in an internal buffer. The buffer size and time since last file IO is saved and the trajectory buffer is printed to file only when the buffer has reached its size limit or when a long enough time has passed since last print to file. #### 3.3.4 On-the-fly analysis Instead of storing trajectory data to file it can be handy to run analysis directly while running the simulation. KMCLib provides the possibility to perform on-the-fly analysis via an analysis plugin interface. The user defines an object that inherits from the KMCAnalysisPlugin class and overloads the setup, registerStep and finalize methods for desired behavior. Right before the Monte Carlo loop is entered each user-provided analysis object will in turn get access to the initialized system. This is done by calling each analysis object’s setup method with the configuration and simulation start time as arguments, allowing for the analysis objects to set up internal memory and collect initial data. After a step in the Monte Carlo loop has been performed program flow control is returned to Python where all required analysis and trajectory handling is performed (the “collect data” step in figure 1). Analysis is performed by letting each user-provided analysis object in turn get access to all details of the present state of the simulation by calling each analysis object’s registerStep method with the configuration and simulation time as arguments. It is then up to each analysis object to extract relevant information at this stage in the simulation. At the end of the simulation the finalize method is called on each analysis object to allow for post-processing of collected data before the program stops. The mean square displacement analysis described in section 3.3.5 below is implemented using the analysis plugin interface. A detailed account of the analysis plugin interface can be found in the reference manual. #### 3.3.5 On-the-fly mean square displacements To facilitate diffusion studies with KMCLib the algorithm described in [21] for calculating mean square displacements with correct error estimates on-the-fly during a simulation is implemented. To run a simulation with mean square displacement analysis it is required that move vectors are provided with the processes to allow the program to keep track of the individual atoms. An atom type to track must be provided by the user as well as parameters specifying the number of history steps to use and the histogram spacing. It is also important that the primitive unit cell of the configuration is given in correct units, since the transformation to Cartesian coordinates is done using this information before summing contributions for each time-step. The mean square displacement analysis is implemented as an analysis plugin as described in section 3.3.4 above and several examples of its use are included in the functionality tests. ### Parallelization strategy The MPI parallelization scheme in KMCLib is based on the assumption that the matching of a process with a local geometry, including rate evaluation when required, is by far the most time-consuming part of each step in the Monte Carlo loop. This condition is particularly well satisfied for all cases where custom rate calculations involve anything more than a simple table look-up. When the list of sites to match against the list of processes is generated after a process is performed, the process-site pairs to match are distributed over all ranks. The matching is performed in parallel and the matching result is communicated to all ranks. If a custom rate calculator is used the list of process-site pairs that need their rates to be updated is also split up over all ranks and calculated in parallel, and the updated rates are communicated to all ranks. This parallelization strategy is similar to the approach in aKMC [6] where the expensive saddle-point searches are performed in parallel. Contrary to parallelization schemes that rely on a spatial decomposition of the simulation box with several processes performed simultaneously on different ranks, our approach does not affect the overall serial algorithm. Running in serial or parallel is thus guaranteed to generate exactly the same simulation results. To achieve a good work-load using this parallelization scheme it is necessary that the number of process-site pairs to parallelize over is much longer than, or exactly matches, the number of MPI ranks. As will be apparent in section 4.3 below, the parallelization will furthermore, as expected, work more efficiently for simulations with heavier custom rate calculations. ### Random numbers Good quality random numbers are absolutely crucial for results from any Monte Carlo method to be valid [23]. Over the years, the quality of available pseudo random numbers has improved significantly and today the Mersenne-Twister [24] algorithm is considered the gold standard for non-cryptographic applications. A publicly available library Mersenne-Twister implementation [25] is therefore included in KMCLib. To assure that the included random number generator works correctly we tested it against the standard C++11 implementation provided with the gcc (Ubuntu/Linaro 4.6.3-1ubuntu5) 4.6.3 compiler. Both pseudo random number generators generated identical results for several million consecutive numbers. Interchanging the included random number generator with the standard C++11 implementation in KMCLib furthermore produced identical simulation results. An internal wrapper was finally implemented around the pseudo random number generator in KMCLib to facilitate use of any other pseudo random number generator implementation, or the standard C++11 implementation, with minimal modifications to the code. ## 4 Timings and scaling We use two different systems for demonstrating the scaling behavior of KMCLib. The first system is a simple one-dimensional A-B-C model, where particles of type A, B or C sit on a one-dimensional lattice. Only two processes are used for this system, the flipping of A to B and B to A, leaving the C particles inert. This allows us to vary the total size of the system while keeping the number of active A+B sites constant, simply by varying the number of C particles. Both included processes have the same rate and the number of A particles was always the same as the number of B particles at the start of each simulation. The second system is a more complex model of oxygen-vacancy diffusion in a fluorite metal oxide structure with a large fraction of dopants on the metal sites and vacancies in the oxygen sub-lattice. The primitive cell is based on the cubic oxygen sub-lattice. There are two basis points in the cell, (0,0,0) and (0.5,0.5,0.5). Since we use the primitive cell of the oxygen sub-lattice to represent a fluorite metal oxide structure half the (0.5,0.5,0.5) sites will be occupied with metal ions and half will be marked as empty. Oxygen-vacancy diffusion was described with six processes describing nearest-neighbor hops along the Cartesian axes. The system was run with two different settings. In the first setting oxygen diffusion was modeled with equal rates for all six processes. In the second setting the system was run with a custom rate calculator that modified the process rates based on the local distribution of dopants within three shells of primitive cells (258 neighbor sites) from the vacancy site of interest. <figure><img src="content_image/1405.1221/x3.png"><figcaption>Figure 3: Scaling behavior (A-B-C model) when varying the total number oflattice sites while keeping the number of active (A+B) sites constant. Theinitialization time (a) and memory usage (c) grows linearly with the systemsize, while the time per step in the loop (b) stays constant.</figcaption></figure> <figure><img src="content_image/1405.1221/x4.png"><figcaption>Figure 4: The scaling behavior of the fluorite metal oxide model in the firstsetting (with short rate evaluation time, see text). The system size is variedkeeping the ratio between different lattice site types constant. Theinitialization time (a) and memory usage (c) grows linearly with system size.The short time spent in rate evaluation makes updating the lists of possiblesites for each process the dominating term and the time per step in the loop(b) therefore grows linearly with system size.</figcaption></figure> <figure><img src="content_image/1405.1221/x5.png"><figcaption>Figure 5: The scaling behavior of the fluorite metal oxide model in the secondsetting (with long rate evaluation time, see text). The system size is variedwith the ratio between lattice site type occupation kept constant. Theinitialization time (a) and memory usage (c) grows linearly with system size.The rate evaluation is here the dominating term and the time per step in theloop (b) is therefore close to constant.</figcaption></figure> ### Serial performance All serial performance tests were carried out on a laptop computer running Ubuntu-12.04 LTS, with an Intel i7-3517U 1.90 GHz CPU with 4 MB cache, and with 8 GB RAM. Only one of the CPU’s cores was used. In figure 3 we show the scaling with system size for the simple A-B-C model when the number of active (A+B) sites are kept constant at 2000, varying only the number of C sites. The initialization time (figure 3a) is seen to grow linearly with the system size. This is because the initialization loops over all sites to set up their match-lists. Linearity is achieved by dividing the simulation box up in blocks of primitive unit-cells and restricting the calculation of distances for the match-lists within a limited number of such neighboring cells. To make sure all relevant neighbor information is included the number of neighbor cells to consider is determined by the longest process cutoff as defined by the user. The memory usage (figure 3c) is also seen to grow linearly with system size as expected. The time per step in the loop (figure 3b) is constant. This is also expected since no extra work should be required per step as long as the number of active (A+B) sites is kept constant. Figure 4 shows the scaling with system size for the fluorite metal oxide oxygen-vacancy diffusion system in the first setting. Both the initialization time and memory usage (4a and c) scales linearly with system size as expected. Contrary to the constant scaling for the A-B-C model, the time spent for each step in the loop (figure 4b) also grows linearly with system size. This linear behavior arises since the system size is scaled homogeneously so that the number of active sites (sites where the processes can be applied) grows linearly with the total size of the system. When a site is matched against a process the list of available sites for that process must be searched through for the site of interest to determine if the site should be added to (or removed from) this list. When the number of available sites grows with system size the list to search through grows accordingly, which gives rise to the observed linear scaling behavior. The situation is quite different for the fluorite metal oxide oxygen-vacancy diffusion system in the second setting (figure 5). While the initialization time and memory usage (figure 5a and c) still scales linearly, the time per step in the loop (figure 5b) is close to constant, with only a slight increase with system size. This behavior is expected and can be explained by noting the difference in scale on the y-axis in figures 4b and 5b. The linear component from figure 4b is still present in figure 5b, but the time spent in the loop is completely dominated by the rate evaluation resulting in close to constant $O(1)$ scaling. We expect this type of roughly constant $O(1)$ scaling for most realistic applications even when custom rates are not used, since the matching step is typically far more time consuming than the search through the availability vectors of the processes. ### System size and simulation time A peculiarity when it comes to the scaling behavior of any KMC simulation implementing the VSSM algorithm, which at first can be easy to overlook and therefore well worth noting, is how the simulation time (as opposed to the wall-clock time measured in the timings above) varies with the system size. We recall that the simulation time is propagated according to the expression (3) where the sum of the rates in the system appears in the denominator. The length of each time-step in the simulation is thus inversely proportional to the total rate in the system. This means that doubling the size of the system (and the number of available sites) and running the simulation for the same number of elementary steps will only propagate the simulation time half as far. <figure><img src="content_image/1405.1221/x6.png"><figcaption>Figure 6: Scaling with the number of MPI processes for three cases withdifferent time spent in the custom rate calculation (0.001, 0.01 and 0.1seconds additional time per rate evaluation as indicated). Black dotsrepresent measured values and the gray line indicates ideal linear scaling.</figcaption></figure> ### Parallel performance All parallel performance tests were run on a CentOS Linux-based cluster with HP Proliant SL2x170z nodes, each equipped with two quad-core Intel Xeon E5520 CPUs and 36 GB RAM, with infiniband interconnect. One core was used for each MPI process. As discussed in section 3.4 above the parallel performance is expected to depend strongly on the amount of work needed during the custom rate calculations. To run the same test system for different times spend in the rate calculation function a sleep statement was introduced. Three different cases were considered, waiting an additional 1, 10 and 100 milliseconds respectively each time the rate calculation function was called. Figure 6 shows the scaling calculated as the total time running on one MPI process over the total time running in parallel. As expected, the more time spent in the custom rate calculation the better the scaling. When only 1 millisecond is added to the rate calculation the scaling flattens out above 16 processes, but with 100 additional milliseconds in the rate calculation the scaling is excellent all the way up to the 128 processes. Clearly, the heavier the custom rate calculations the more is to gain from running in parallel, but the number of sites to re-match at each step will also influence the parallel performance. The test system used in figure 6 is three-dimensional with two basis sites for each primitive unit cell and a large cutoff which resulted in more than 500 process-site pairs to re-evaluate at each step. For a good load balance it is necessary that the number of process-site pairs to re-match is either much larger than, or an exact multiple of, the number of MPI processes. Running efficiently in parallel will therefore be limited to a few MPI processes for systems with few neighbors (with low dimensionality) and short cutoff. ### Caching of calculated rates In those cases where the on-the-fly rates are kept constant over time, i.e., when no time-evolution is allowed in the custom rate expressions, it is possible to use caching techniques to save calculated rates for later retrieval when needed. The efficiency of such a caching scheme is highly dependent on the details of the simulated system and on the caching algorithm used, but it can in some cases provide a significant speedup. For one of the simpler systems investigated, the two-dimensional Ising spin model (see section 3.3.2), the custom rate implementation can be sped up almost to the level of the fixed rates implementation using a prototype caching scheme where all the calculated custom rates are saved. For systems with larger number of possible lateral interactions saving all calculated rates is not feasible due to memory limitations. A caching scheme with several internal hash tables with fixed maximum size, where the oldest table filled is emptied when the latest table reaches its maximum size limit would resolve the memory issue. No cashing mechanism is implemented in the present version of the code but will be further investigated for future releases, however a prototype implementation is available in a development branch of the KMCLib git repository. The structure of the custom rate framework meanwhile makes it easy for the user to implement her own cashing mechanism. ## 5 Concluding remarks We have presented a general framework for lattice KMC simulations designed to easily be customized, to facilitate implementation of custom KMC models without having to re-do all the work of implementing and testing the simulation engine for each new set of problems to investigate. We have made our code freely available since we believe this will be of great benefit for the larger research community, by saving time and letting researchers focus their effort on the design of new models to solve challenging physical problems, rather than spending time re-implementing a well-known method. This paper has focused mainly on technical details and on the modifications we have made to the standard lattice KMC algorithm. We have demonstrated the scaling behavior in serial and shown how the parallel performance depends on the simulated system. The code itself is separately documented from a users perspective, and includes several usage examples in the functionality tests. We have therefore not given a full usage example here, but trust that the reader will find the code manual and examples enough to get started. To run the tests and set up new calculations require some knowledge of Python programming but following the examples and the documentation this barrier should be easy to overcome. The modular plugin design of the program with a well-defined API is ideal for sharing programming work with the community. Custom rate calculators, analysis modules and scripts for setting up geometries and processes and to visualize results can be written and published separately, and we hope to see many such contributions from the research community, to include in future releases. Acknowledgments We gratefully acknowledge Christian Stigen Larsen for making his Mersenne-Twister implementation publicly available under the GPL license. We acknowledge financial support by the Swedish Energy Agency (Energimyndigheten, STEM), the Swedish Research Council (Vetenskapsrådet) and the Carl Trygger Foundation. Supercomputer time was granted by the Swedish National Infrastructure for Computing (SNIC). ## References * (1) Arthur F. Voter, Francesco Montalenti, Thimothy C. Germann, Extending the Time Scale in Atomistic Simulation of Materials, Annu. Rev. Mater. Res. 32 (2002) 321. * (2) A. B. Bortz, M. H. Kalos, J. L. Lebowitz, A New Algorithm for Monte Carlo Simulations of Ising Spin Systems, J. Comput. Phys. 17 (1975) 10. * (3) Daniel T. Gillespie, A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions, J. Comput. Phys. 22 (1976) 403. * (4) Kristen A. Fichthorn, W. H. Weinberg, Theoretical foundations of dynamical Monte Carlo simulations, J. Chem. Phys. 95 (1991) 1090. * (5) Corbett C. Battaile, The Kinetic Monte Carlo method: Foundation, implementation and application, Comput. Methods Appl. Mech. Engrg. 197 (2008) 3386. * (6) Lijun Xu, Graeme Henkelman, Adaptive kinetic Monte Carlo for first-principles accelerated dynamics, J. Chem. Phys. 129 (2008) 114104. * (7) Abhijit Chatterjee, Arthus F. Voter, Accurate acceleration kinetic Monte Carlo simulations through the modification of rate constants, J. Chem. Phys. 132 (2010) 194101. * (8) Alexander Slepoy, Aidan P. Thompson, Steven J. Pilmpton, A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks, J. Chem. Phys. 128 (2008) 205101. * (9) Daniel T. Gillespie, Andreas Hellander, Linda R. Petzold, Perspective: Stochastic algorithms for chemical kinetics, J. Chem. Phys. 138 (2013) 170901. * (10) Michail Stamatakis, Dionisios G. Vlachos, Unraveling the Complexity of Catalytic Reactions via Kinetic Monte Carlo Simulation: Current Status and Frontiers, ACS Catal. 2 (2012) 2648. * (11) Xiang Wang, Kah Chun Lau, C. Heath Turner, Brett I. Dunlap, Kinetic Monte Carlo Simulations of the elementary electrochemistry in a hydrogen-powered solid oxide fuel cell, J. Power Sources 195 (2010) 4177. * (12) Qingchuan Xu, Anton Van der Ven, Atomic transoirt in ordered compointds mediated by local disorder: Diffusion in B2-Ni${}_{x}$Al${}_{1-x}$, Phys. Rev. B 81 (2010) 064303. * (13) J. J. Lukkien, J. P. L. Segers, P. A. J. Hilbers, R. J. Gelten, A. P. J. Jansen, Efficient Monte Carlo methods for the simulation of catalytic surface reactions, Phys. Rev. E 58 (1998) 2598. * (14) K. A. Fichthorn, W. H. Weinberg, Theoretical foundations of dynamical Monte Carlo simulations, J. Chem. Phys. 95 (1991) 1090. * (15) J. J. Lukkien, CARLOS web page, http://carlos.win.tue.nl/ (Apr. 2014). * (16) S. Plimpton, C. Battaile, M. Chandross, L. Holm, A. Thompson, V. Tikare, G. Wagner, E. Webb, X. Zhou, C. Garcia Cardona, A. Slepoy, Crossing the Mesoscale No-Man’s Land via Parallel Kinetic Monte Carlo, Sandia report SAND2009-6226. * (17) SPPARKS web page, http://spparks.sandia.gov/ (Apr. 2014). * (18) M. J. Hoffmann, S. Matera, K. Reuter, kmos: A lattice kinetic Monte Carlo framework, arXiv:1401.5278. * (19) M. J. Hoffmann, KMOS webpage, http://mhoffman.github.io/kmos/ (Apr. 2014). * (20) J. Nielsen, M. d’Avezac, J. Hetherington, M. Stamatakis, Parallel kinetic Monte Carlo simulation framework incorporating accurate models of adsorbate lateral interactions, J. Chem. Phys. 139 (2013) 224706. * (21) Mikael Leetmaa, Natalia V. Skorodumova, Mean square displacements with error estimates from non-equidistant time-step kinetic Monte Carlo simulations, _Submitted_. * (22) Nachiappan Nagappan, E. Michael Maximilien, Thirumalesh Bhat, Laurie Williams, Realizing quality improvement through test driven development: results and experiences of four industrial teams, Empir. Software Eng. 13 (2008) 289. * (23) M. P. Allen, D. J. Tildesley, Computer Simulations of Liquids, Oxford Science Publicaions, 1987. * (24) M. Matsumoto and T. Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation 8 (1998) 3. * (25) Christian S. Larsen, Mersenne-Twister implementation, https://github.com/cslarsen/mersenne-twister (May 2013).
1512.00634	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 6988, "num_imgs": 0, "llama3_tokens_count": 1758 }	[]	# On the Interaction in a Quartet of Galaxies A. A. Yeghiazaryan, A. A. Hakobyan, and T. A. Nazaryan Byurakan Astrophysical Observatory, 0213 Byurakan, Aragatsotn province, Armenia; hakobyan@bao.sci.am ###### Abstract We performed the Fabry-Perot scanning interferometry of the quartet of galaxies NGC 7769, 7770, 7771 and 7771A in H$\alpha$ line and studied their velocity fields. We found that the rotation curve of NGC 7769 is weakly distorted. The rotation curve of NGC 7771 is strongly distorted with the tidal arms caused by direct flyby of NGC 7769 and flyby of a smaller neighbor NGC 7770. The rotation curve of NGC 7770 is significantly skewed because of the interaction with much massive NGC 7771. The rotation curves and morphological disturbances suggest that the NGC 7769 and NGC 7771 have passed the first pericenter stage, however, probably the second encounter has not happened yet. A. A. Yeghiazaryananahit_y@hotmail.comByurakan Astrophysical ObservatoryByurakanAragatsotn province0213Armenia A. A. Hakobyanhakobyan@bao.sci.am0000-0001-7392-1765Byurakan Astrophysical ObservatoryByurakanAragatsotn province0213Armenia T. A. Nazaryannazaryan@bao.sci.amByurakan Astrophysical ObservatoryByurakanAragatsotn province0213Armenia ## 1 Velocity fields and rotation curves We report the results of the optical interferometry of the interacting system of galaxies NGC 7769, 7770, 7771 and 7771A and analyze their kinematics. A detailed description of the morphological features of the galaxies as well as photometry and color analysis of NGC 7769 are presented in the complete version of the study: Yeghiazaryan et al. (2015). We also discuss the influence of interaction on the kinematics, dynamics and star formation in the system. Known models of galaxy interactions are based mostly on statistical observational data. We try to illustrate how and to what extend these models can be applied to explain the features of the galaxies in this system. In order to study the velocity fields of the galaxies, the observations were carried out at the 2.6m telescope of the Byurakan Astrophysical Observatory (BAO, Armenia) on 8 November 1996, with the ByuFOSC (Byurakan Faint Object Spectral Camera) in the interferometric mode, attached at the prime focus of the telescope. Based on the H$\alpha$ velocity fields (the right-hand panels of Figure 1), we calculated the rotation curves of the galaxies (the left-hand panels of Figure 1) by using data points within sectors along the maximal gradient direction, see isovelocity contours in the right-hand panels of Figure 1. Maximal rotational velocity of NGC 7769 is observed at the radius of around 15 arcsec from the galaxy nucleus. The rotational velocities in Figure 1 are in good agreement with the HI measurements ($316~{}{\rm km\,s^{-1}}$) in Chengalur et al. (1993). Our measurements of velocities, having a better spatial resolution compared with those of the previous studies (Chengalur et al. 1993; Nordgren et al. 1997), reveal weak perturbations of the rotation curve of NGC 7769, which may be caused by interaction with NGC 7771. The same cannot be said about the velocity field of NGC 7771. Figure 1 shows that there are perturbations and large dispersion in radial velocities at the distances larger than about 10-15 arcsec from the nuclei. This distance is about half radius of the bar. Evidently, this scatter of radial velocities can be explained by the fact that part of the arms are included in the sector used to calculate radial velocities (sector angle is $40^{\circ}$). However the asymmetric profile along the major axis suggests that Northern and Southern arms do not have the same radial velocity profiles. The asymmetric tidal forces of NGC 7769 and NGC 7770 affecting on NGC 7771, seem to be a natural cause of that. The rotation curve of NGC 7770 is significantly skewed. This is probably because of the strong harassing interaction with the more massive NGC 7771, see Alonso-Herrero et al. (2012). The rotation curve of NGC 7771A is typical for a late type Sm galaxy. [FIGURE:S1.F1][ENDFIGURE] By analyzing velocity fields, sizes, and shapes of spiral arms of NGC 7771 and NGC 7769, in Nordgren et al. (1997) it has been suggested that NGC 7771 and NGC 7769, which have a 2:1 mass ratio, appear to be having a prograde-retrograde interaction, with NGC 7769 being the retrograde one. Our better data support this conclusion. This conclusion is in agreement with the latest models of galaxy collisions (Di Matteo et al. 2007) showing that during direct collisions tidally induced spiral arms are much longer and brighter than those during retrograde collisions. We can conclude that galaxies NGC 7769 and NGC 7771 already have passed the first pericenter stage, however, probably the second encounter has not happened yet. The first pericenter distance should have been large enough (around few sizes of the galaxies), so that large disturbances in rotation curves have not appeared yet. ## 2 Summary The quartet of galaxies NGC 7769, 7770, 7771 and 7771A is a system of interacting galaxies. Here, we present a Fabry-Perot imaging study of the system in H$\alpha$ line. We came to the following main conclusions: * Close interaction between the component galaxies of the system has produced morphological features that are characteristic of the interactions. We have detected features such as tidal arms, spiral arms induced by close interaction, bars and induced star formation. * From the results of our interferometric observations, we obtained the radial velocity profiles of galaxies. The rotation curve of NGC 7769 is weakly distorted. The rotation curve of NGC 7771 is strongly distorted by the tidal arms caused by direct flyby of NGC 7769 and flyby of a smaller neighbor NGC 7770. The rotation curve of NGC 7770 is significantly skewed because of the interaction with much massive NGC 7771. * The radial velocity profiles and morphological disturbances suggest that the NGC 7769 and NGC 7771 have passed the first pericenter stage, however, probably the second encounter has not happened yet. Study of such systems with methods combining photometric and visual analysis is an effective way to clarify features of star formation in different stages of interaction. Ongoing and future surveys using integral field spectroscopy will allow also to explore the spatial distribution of star formation in interacting systems. ## References * Alonso-Herrero et al. (2012) Alonso-Herrero, A., Rosales-Ortega, F. F., Sánchez, S. F., et al. 2012, MNRAS, 425, L46 * Chengalur et al. (1993) Chengalur, J. N., Salpeter, E. E., & Terzian, Y. 1993, ApJ, 419, 30 * Di Matteo et al. (2007) Di Matteo, P., Combes, F., Melchior, A.-L., & Semelin, B. 2007, A&A, 468, 61 * Nordgren et al. (1997) Nordgren, T. E., Chengalur, J. N., Salpeter, E. E., & Terzian, Y. 1997, AJ, 114, 77 * Yeghiazaryan et al. (2015) Yeghiazaryan, A. A., Nazaryan, T. A., & Hakobyan, A. A. 2015, arXiv:1510.00193
1211.0462	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 74767, "num_imgs": 6, "llama3_tokens_count": 21058 }	[ "content_image/1211.0462/x1.png", "content_image/1211.0462/x2.png", "content_image/1211.0462/x3.png", "content_image/1211.0462/anal-ps.png", "content_image/1211.0462/x4.png", "content_image/1211.0462/x5.png" ]	Stochastic pattern formation and spontaneous polarisation: the linear noise approximation and beyond Alan J. McKane${}^{1}$, Tommaso Biancalani${}^{1}$ and Tim Rogers${}^{1,2}$ ${}^{1}$Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom ${}^{2}$Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom Abstract We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically-inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales. ## 1 Introduction The evidence from the talks delivered at this meeting should leave little doubt that stochastic models are required to understand a wide range of biological phenomena. The models do need to be carefully formulated, with the nature of the interactions between the constituents clearly specified, but when this is carried out in an unambiguous fashion, their analysis can begin. In the great majority of cases this will be numerical, and many of the contributions at this meeting were concerned with the development and implementation of efficient algorithms for carrying this analysis out. In this paper we will take a different path. We will similarly take care in formulating the precise form of the model, stressing the need to begin from a microscopic individual-based model (IBM). However we will be primarily be interested in using mathematical analysis to understand the model. To facilitate this, the interactions between the constituents of the model will be represented by chemical-like reactions occurring at specified rates (Black and McKane, 2012). If the underlying stochastic process is Markov, which will usually be the case, a master equation can be written down which will describe the time evolution of the probability of the system being in a given state (van Kampen, 2007). One analytic procedure that can always be carried out on the master equation is to calculate the equations describing the time evolution of the averages of the state variables. These deterministic equations give a macroscopic description of the dynamics of the system. They are the other major methodology for the theoretical study of biological systems, and their use is exemplified by the book by Murray (2008). This older tradition involved both the study of simple, analytically tractable, models and dynamical systems theory. The former was concerned with the mathematical investigation of specific differential equations of few variables and the latter with general results on stability of attractors, topological notions, bifurcation theory, and so on (Wiggins, 2003). A parallel methodology for the mathematical analysis of the full stochastic model also exists, however it is much less widely appreciated than that for the corresponding deterministic analysis. Stochastic differential equations (SDEs) can be derived when the number of constituents is large (but not infinite); the stochasticity originating from the discreteness of the underlying IBM. Techniques from the theory of stochastic processes, as well as general results from this theory, can be used to understand these equations analytically, just as in the deterministic case. Here we will describe this methodology with reference to two specific models in order to illustrate the basic ideas and techniques. We will begin in section 2 with the definition of the state variables of the model, and the specification of the interactions between them in terms of chemical reactions. This allows us to write down the master equation for the process. In section 3 we derive the macroscopic and mesoscopic equations governing the dynamics of the process from this master equation. We then go on to illustrate the use of this formalism, within the linear noise approximation (LNA), to the particular example of the Brusselator model in section 4. However, the LNA is not always able to capture the details of stochastic ordering phenomena, and in section 5 we show that, in some situations, we are able to go beyond the LNA. This is illustrated on the spontaneous emergence of cell polarity; this example being motivated by the talk given by Linda Petzold at the meeting. Finally we conclude with an overview of the techniques and applications that we have discussed in section 6. ## 2 Definition of the models and the master equation The stochastic effects that will interest us here occur when the number of constituents, $N$, is large but finite. In this case a _mesoscopic_ description is required: the state variables are assumed continuous — unlike in the microscopic IBM description — but the effect of the discreteness is not lost: it is manifested in the form of Gaussian distributed ‘noise’. The mesoscopic form of the model is not obvious; we have to begin from a microscopic IBM and derive it as an approximation which holds when $N$ is large. It cannot be found from a macroscopic description, since this consists of time evolution equations for average quantities, whereas the micro- or mesoscopic formulations describe the time evolution of the entire probability density function (pdf). But many stochastic processes have the same average, so there is in principle no unique way of determining the micro- or mesoscopic model from the macroscopic one. This will be especially clear later, when we derive the mesoscopic form of the equations and compare them to their macroscopic counterparts. To set up the IBM we first need to decide what are the variables which describe the state of the system. For the simplest case of a single type of constituent with no spatial, class, or other structure, it would be a single integer $n=1,2,\ldots,N$ representing the number of constituents in the system. In ecological models this could be the number of individuals in the population. There would then be a transition rate from state $n$ to state $n^{\prime}$ caused by births, deaths, competition, predation, etc. This rate will be denoted by $T(n^{\prime}\|n)$, with the initial state on the left (the other convention with the initial state on the right is also sometimes used). The probability of finding the system in state $n$ at time $t$, $P_{n}(t)$, changes according to the master equation: \[\frac{\mathrm{d}P_{n}(t)}{\mathrm{d}t}=\sum_{n^{\prime}\neq n}\,T(n\|n^{\prime} )\,P_{n^{\prime}}(t)-\sum_{n^{\prime}\neq n}\,T(n^{\prime}\|n)\,P_{n}(t).\] (1) The first term on the right-hand side describes the rate at which $P_{n}(t)$ increases due to transitions into the state $n$ from all other states $n^{\prime}$ while the second term on the right-hand side describes the rate at which $P_{n}(t)$ decreases due to transitions out of the state $n$ to all other states $n^{\prime}$. The net change then gives $dP_{n}(t)/dt$. Although it is intuitively clear, it can also be derived from the Chapman-Kolmogorov equation for Markov processes (Gardiner, 2009). It gives the ‘microscopic’ description of the system and will be the starting point for deriving the meso- and macroscopic descriptions. All this generalises to several types of constituents with numbers $\ell,m,\ldots$ at a given time. Spatial structure can also be introduced with constituents located in a particular small volume $j=1,2\ldots$ at a given time. For notational simplicity we will combine the index $i$ labelling these small volumes with an index $s$ which labels the types or classes of constituent, into one label $J=\{j,s\}$. Later, when carrying out the analysis of the differential equations we will separate the indices, and may also take the continuum limit in which volume labels become continuous. An additional comment is worth making at this stage. There is no agreed nomenclature for the small volumes: describing them as ‘cells’ is potentially confusing in a biochemical context and the term ‘patches’ is usually only used in an ecological context when the constituents are individuals. We will use the neutral term ‘domain’ and talk about their ‘volume’ $V$, even when they are one- or two-dimensional, as will be the case with the models we discuss in this paper. If we now write $n_{I}$ for the number of constituents of a particular type in a particular domain, we can specify the state of the system through the vector of integers $\bm{n}=(n_{1},n_{2},\ldots)$. The master equation is then simply Eq. (1) with $n$ replaced by $\bm{n}$: \[\frac{\mathrm{d}P_{\bm{n}}(t)}{\mathrm{d}t}=\sum_{\bm{n}^{\prime}\neq\bm{n}}\, T(\bm{n}\|\bm{n}^{\prime})P_{\bm{n}^{\prime}}(t)-\sum_{\bm{n}^{\prime}\neq\bm{n }}\,T(\bm{n}^{\prime}\|\bm{n})P_{\bm{n}}(t).\] (2) Having decided what the fundamental constituents of the system are, and so how the states of the system are defined, the next step is to give the transition rates $T(\bm{n}\|\bm{n}^{\prime})$. This will define the model, and is best done through examples. We will naturally choose as examples the models that will be analysed later in the paper. The first is the Brusselator (Glansdorff and Prigogine, 1971) and the second is a model of cell polarity introduced by Altschuler et al. (2008). The notation used to describe the chemical types and the rates in the Brusselator model follows that of Cross and Greenside (2009). In every domain $i$ molecules of two species $X$ and $Y$ interact through the reactions of the Brusselator model (Glansdorff and Prigogine, 1971): \[\emptyset \stackrel{{ a}}{{\rightarrow}} X_{i},\] \[X_{i} \stackrel{{ b}}{{\rightarrow}} Y_{i},\] \[2X_{i}+Y_{i} \stackrel{{ c}}{{\rightarrow}} 3X_{i},\] \[X_{i} \stackrel{{ d}}{{\rightarrow}} \emptyset.\] (3) In order, these reactions describe: (i) the creation of a new $X$ molecule, (ii) an $X$ molecule spontaneously transforming into a $Y$ molecule, (iii) two $X$ molecules reacting with a $Y$, changing it to an $X$, and (iv) $X$ molecules being removed from the system. The rates at which the reactions occur are denoted by $a,b,c$ and $d$, and $X_{i}$ and $Y_{i}$ are the molecules that are in domain $i$ at the time that the reaction occurs. Each of these reactions are assumed to occur independently, and without memory of the previous states of the system. In addition to the reactions given in Eq. (3), migration reactions, which describe molecular diffusion from one domain to another, have to be specified. For every pair of neighbouring domains $i$ and $j$, molecules of the two species $X$ and $Y$ may diffuse from one domain to the other according to \[X_{i}\stackrel{{\alpha}}{{\rightarrow}}X_{j},\ \ \ Y _{i}\stackrel{{\beta}}{{\rightarrow}}Y_{j}\ .\] (4) The second example consists of a two-dimensional ‘cell’ enclosed within a perfectly circular membrane (Altschuler et al., 2008). The centre of the cell is known as the ‘cytoplasmic pool’ and contains signalling molecules which we denote by $C$. In our implementation, the one-dimensional circular membrane is divided up into domains labelled by an index $i$; molecules lying within domains $i$ are denoted by $M_{i}$. The following reactions take place: \[C \stackrel{{ k_{\rm on}}}{{\rightarrow}} M_{i},\] \[M_{i} \stackrel{{ k_{\rm off}}}{{\rightarrow}} C,\] \[M_{i}+C \stackrel{{ k_{fb}}}{{\rightarrow}} 2M_{i}.\] (5) The first two reactions describe a molecule in the cytoplasmic pool attaching itself to a random domain on the membrane and a molecule detaching and returning to the cytoplasmic pool, respectively. The third reaction represents a molecule on the membrane attempting to recruit another molecule from the cytoplasmic pool to the same domain. As for the Brusselator, there is also diffusion, in this case along the membrane: \[M_{i} \stackrel{{\alpha}}{{\rightarrow}} M_{j}\,,\] (6) for any pair of neighbouring membrane domains $i$ and $j$. The reaction scheme is illustrated in Fig. 1 (after Altschuler _et al_). <figure><img src="content_image/1211.0462/x1.png"><figcaption>Figure 1: Illustration of the reactions of the cell polarisation model. Thethree reactions are (a) attachment of signalling molecules to the membrane,(b) the return of a membrane molecule to the cytoplasmic pool, (c) recruitmentof new signalling molecules by those already on the membrane, and (d)diffusion along the membrane, which we implement as migration betweenneighbouring domains.</figcaption></figure> Once the reaction scheme has been picked and laid out as in Eqs. (3)-(6), the transition rates $T(\bm{n}\|\bm{n}^{\prime})$ can be chosen. This effectively specifies the model. When writing the transition rates, we will only list the variables in the domains that are involved in the reaction. For the Brusselator, the number of $X_{i}$ and $Y_{i}$ will be denoted by $\ell_{i}$ and $m_{i}$, respectively. We then invoke mass action, that is, the rate of the transition for a given reaction is proportional to the product of the densities of the reactants, to write the transition rates as \[T_{1}(\ell_{i}+1,m_{i}\|\ell_{i},m_{i}) = a,\] \[T_{2}(\ell_{i}-1,m_{i}+1\|\ell_{i},m_{i}) = b\;\frac{\ell_{i}}{V},\] \[T_{3}(\ell_{i}+1,m_{i}-1\|\ell_{i},m_{i}) = c\;\frac{{\ell_{i}}^{2}m_{i}}{V^{3}}\] \[T_{4}(\ell_{i}-1,m_{i}\|\ell_{i},m_{i}) = d\;\frac{\ell_{i}}{V},\] (7) where the subscripts on the rates refer to the four reactions in Eq. (3) and where $V$ is the volume of the domain. These transition rates are as in the usual Brusselator model (Boland et al., 2009; Biancalani et al., 2011), although it is worth noting that we have not imposed a fixed limit on the number of molecules permitted in a domain (in contrast with some previous work (Lugo and McKane, 2008; Biancalani et al., 2010); this is reflected in the fact we use the inverse volume of a domain, $V^{-1}$, rather than the inverse total number of molecules in a domain, as the expansion parameter). The reaction rates which describe molecular diffusion reactions in Eq. (4) are given by \[T_{5}(\ell_{i}-1,\ell_{j}+1\|\ell_{i},\ell_{j}) = \alpha\;\frac{\ell_{i}}{zV},\] \[T_{6}(m_{i}-1,m_{j}+1\|m_{i},m_{j}) = \beta\;\frac{m_{i}}{zV}.\] (8) Here $z$ is the number of nearest neighbours of a domain and the index $j$ denotes a nearest neighbour of domain $i$. To model spontaneous cell polarisation we denote the number of $C$ and $M_{i}$ molecules by $\ell$ and $m_{i}$, respectively. The transition rates are then taken to be \[T_{1}(\ell-1,m_{i}+1\|\ell,m_{i}) = k_{\rm on}\;\frac{\ell}{V}\] \[T_{2}(\ell+1,m_{i}-1\|\ell,m_{i}) = k_{\rm off}\;\frac{m_{i}}{V},\] \[T_{3}(\ell-1,m_{i}+1\|\ell,m_{i}) = k_{\rm fb}\;\frac{\ell m_{i}}{V^{2}},\] (9) for the reactions (5) and \[T_{4}(m_{i}-1,m_{j}+1\|m_{i},m_{j}) = \alpha\;\frac{m_{i}}{2V},\] (10) for the reactions (6). The transition rates for these two models can be substituted into Eq. (2) which can then, together with suitable initial and boundary conditions, be used to solve for $P_{\bm{n}}(t)$. They can also be used as the basis for setting up a simulation using the Gillespie algorithm (Gillespie, 1976, 1977). Our approach in this paper will be to devise approximation schemes for the master equation and to check the results obtained from such schemes with simulations based on the Gillespie algorithm. We end this section by generalising the above formulation so that it applies to a general biochemical network and by extension to any network of interacting agents. To do this, suppose that there are $L$ different constituents in the system. These could be labelled by their type, the domain they occupy, etc. We will denote them by $Z_{I},I=1,\ldots,L$ and at a given time there will be $n_{I}$ of them, so that the state of the system can be specified by $\bm{n}=(n_{1},\ldots,n_{L})$, as described earlier in this section. We suppose that there are $M$ reactions which interconvert species: \[\sum_{I=1}^{L}r_{I\mu}Z_{I}\ {\longrightarrow}\ \sum_{I=1}^{L}p_{I\mu}Z_{I},\ \ \ \mu=1,2,...M.\] (11) Here the numbers $r_{I\mu}$ and $p_{I\mu}\ (I=1,\ldots,L;\mu=1,\ldots,M)$ describe respectively the population of the reactants and the products involved in the reaction. The reactions Eqs. (3)-(6) are simple examples of this general set of reactions. A quantity which is central to the structure of both the mesoscopic and macroscopic equations is the stoichiometry matrix, $\nu_{I\mu}\equiv r_{I\mu}-p_{I\mu}$, which describes how many molecules of species $Z_{I}$ are transformed due to the reaction $\mu$. In the notation introduced above for the master equation, $\bm{n}^{\prime}=\bm{n}-\bm{\nu}$, where $\bm{\nu}_{\mu}=(\nu_{1\mu},\ldots,\nu_{L\mu})$ is the stoichiometric vector corresponding to reaction $\mu$. Therefore the master equation (2) may be equivalently written as \[\frac{\mathrm{d}P_{\bm{n}}(t)}{\mathrm{d}t}=\sum^{M}_{\mu=1}\left[T_{\mu}(\bm{ n}\|\bm{n}-\bm{\nu}_{\mu})P_{\bm{n}-\bm{\nu}_{\mu}}(t)-T_{\mu}(\bm{n}+\bm{\nu}_ {\mu}\|\bm{n})P_{\bm{n}}(t)\right].\] (12) Many models of interest involve only a handful of different reactions; in this situation, it is often convenient to rewrite the master equation as a sum over reactions as in Eq. (12), rather than over pairs of states $\bm{n}$ and $\bm{n}^{\prime}$ as in Eq. (2). In the next section we will describe how the mesoscopic description of the models can be obtained from the master equation (12). These can then be used as the basis for the calculational schemes which we wish to implement. ## 3 Derivation of the macro- and mesoscopic equations ### Macroscopic equation Before we derive the mesoscopic equations from the master equation, we will carry out the simpler task of deriving the macroscopic equations. This will be done for the case of the general biochemical network described in section 2. This is achieved by multiplying Eq. (12) by $\bm{n}$, and summing over all possible values of $\bm{n}$. After making the change of variable $\bm{n}\rightarrow\bm{n}+\bm{\nu}$ in the first summation, one finds that \[\frac{\mathrm{d}\langle\bm{n}(t)\rangle}{\mathrm{d}t}=\sum^{M}_{\mu=1}\bm{\nu} _{\mu}\big{\langle}T_{\mu}(\bm{n}+\bm{\nu}_{\mu}\|\bm{n})\big{\rangle},\] (13) where the angle brackets define the expectation value: \[\langle\cdots\rangle=\sum_{\bm{n}}(\cdots)P_{\bm{n}}(t)\,.\] (14) In the limit where both the particle numbers and the volume become large, we will take the state variables to be the concentration of the constituents $n_{I}/V$, rather than their number $n_{I}$. These will be assumed to have a finite limit as $V\to\infty$. Specifically, the state of the system will be determined by the new variables \[y_{I}=\lim_{V\to\infty}\frac{\langle n_{I}\rangle}{V}\,,\quad\text{where}\quad I =1,\ldots,L\,.\] From Eq. (13) we have that \[\frac{\mathrm{d}y_{I}}{\mathrm{d}\tau}=\sum_{\mu=1}^{M}\nu_{I\mu}f_{\mu}(\bm{y }),\ \ \ I=1,\ldots,L,\] (15) where $\tau=t/V$ and where \[f_{\mu}(\bm{y}) = \lim_{V\to\infty}\big{\langle}T_{\mu}(\bm{n}+\bm{\nu}_{\mu}\|\bm{n })\big{\rangle}\bigg{.}\] (16) \[= \lim_{V\to\infty}T_{\mu}\big{(}\langle\bm{n}\rangle+\bm{\nu}_{\mu }\|\langle\bm{n}\rangle\big{)}\bigg{.}\] \[= \lim_{V\to\infty}T_{\mu}(V\bm{y}+\bm{\nu}_{\mu}\|V\bm{y})\bigg{.}\,.\] In the above we have used the fact that in the macroscopic limit the probability distribution functions are Dirac delta functions and so, for instance, $\langle n^{m}\rangle=\langle n\rangle^{m}$, for any integer $m$. The equation \[\frac{\mathrm{d}y_{I}}{\mathrm{d}\tau}=A_{I}(\bm{y}),\] (17) where \[A_{I}(\bm{y})\equiv\sum_{\mu=1}^{M}\nu_{I\mu}f_{\mu}(\bm{y}),\ \ \ I=1,\ldots,L,\] (18) is the macroscopic equation corresponding to the microscopic master equation (12). It can be calculated from a knowledge of the stoichiometric matrix $\nu_{I\mu}$ and the transition rates $T_{\mu}(\bm{n}+\bm{\nu}_{\mu}\|\bm{n})$. We scaled time by a factor of $V$ simply because the choice we made for the transition rates (7)-(10) were finite as $V\to\infty$, but we could have easily incorporated an extra factor of $V$ in these rates through a time rescaling. We also chose particularly simple forms for these transition rates in that they were all functions of the species concentration $n_{I}/V$. More generally, they might separately be functions of $n_{I}$ and $V$, which become functions of the species concentration $n_{I}/V$ only when both the particle numbers and the volume become large, so that in the limit $V\to\infty$ they become functions of the macroscopic state variable $\bm{y}$. ### Mesoscopic equation It is perhaps useful at this stage to recall precisely what is meant by the terms ‘microscopic’, ‘mesoscopic’ and ‘macroscopic’. The microscopic description is the one based on the fundamental constituents whose reactions are described by relations such as those in Eqs. (3)-(6). The dynamics of the processes are described by the master equation or by the Gillespie algorithm. The macroscopic description has been derived from this microscopic description above: it only involves average quantities and their time evolution. In between these two levels of description is the ‘meso’-level description, where the continuous variable of the macro-description is used, but where the stochastic effects due to the discrete nature of the individuals is retained. Some other authors include master equations at the meso-level leaving the micro-level for the world of atoms and molecules, but this does not seem such a useful assignment in the biological context in which we are working. The derivation of the mesoscopic equation follows similar lines to the calculation above, with the important difference that we do not take an average, or equivalently, do not take the limit $V\to\infty$. We begin by substituting $y_{I}=n_{I}/V$ directly into the master equation. Since, as discussed above, our transition rates are all functions of $n_{I}/V$ we simply replace $T_{\mu}(\bm{n}+\bm{\nu}_{\mu}\|\bm{n})$ by $f_{\mu}(\bm{y})$ in the notation of Eq. (16). In addition we will denote the pdf $P_{\bm{n}}(t)$ where $\bm{n}$ has been replaced by $V\bm{y}$ as $P(\bm{y},t)$. With these changes we may write the master equation (12) as \[\frac{\partial P(\bm{y},t)}{\partial t}=\sum^{M}_{\mu=1}\bigg{[}f_{\mu}\Big{(} \bm{y}-\frac{\bm{\nu}_{\mu}}{V}\Big{)}P\Big{(}\bm{y}-\frac{\bm{\nu}_{\mu}}{V}, t\Big{)}-f_{\mu}(\bm{y})P(\bm{y},t)\bigg{]}\] For $V$ large, the steps $\bm{\nu}_{\mu}/V$ are likely to be very small, suggesting that we may expand the functions $P$ and $f$ as Taylor series around $\bm{y}$. Truncating at order $\mathcal{O}(V^{-2})$, we arrive at \[\frac{\partial P(\bm{y},\tau)}{\partial\tau} = -\sum^{M}_{\mu=1}\sum_{I}\nu_{I\mu}\frac{\partial}{\partial y_{I} }\left[f_{\mu}(\bm{y})P(\bm{y},\tau)\right]\] \[+ \sum^{M}_{\mu=1}\frac{1}{2V}\sum_{I,J}\nu_{I\mu}\nu_{J\mu}\frac{ \partial^{2}}{\partial y_{I}\partial y_{J}}\left[f_{\mu}(\bm{y})P(\bm{y},\tau) \right]\,,\] where as before we have absorbed a factor of $V$ into the rescaled time variable $\tau=t/V$. This is a Fokker-Planck equation which can be cast into the standard form (Risken, 1989; Gardiner, 2009) \[\frac{\partial P(\bm{y},\tau)}{\partial\tau}=-\sum_{I}\frac{\partial}{\partial y _{I}}\left[A_{I}(\bm{y})P(\bm{y},\tau)\right]+\frac{1}{2V}\sum_{I,J}\frac{ \partial^{2}}{\partial y_{I}\partial y_{J}}\left[B_{IJ}(\bm{y})P(\bm{y},\tau) \right],\] (20) where $A_{I}(\mathbf{y})$ is defined by Eq. (18) and where \[B_{IJ}(\bm{y})=\sum^{M}_{\mu=1}\nu_{I\mu}\nu_{J\mu}f_{\mu}(\bm{y}),\ \ I,J=1, \ldots,L.\] (21) In the Fokker-Planck equation (20), the continuous nature of the state variables indicates that the individual nature of the constituents has been lost. However, the stochasticity due to this discreteness has not: it now manifests itself through the function $B_{IJ}(\bm{y})$. We can see this is the case through the presence of the factor $1/V$. One might ask if this approach is consistent with the previous macroscopic derivation. As $V\to\infty$, the Fokker-Planck equation reduces to the Liouville equation \[\frac{\partial P(\bm{y},\tau)}{\partial\tau}=-\sum_{I}\frac{\partial}{\partial y _{I}}\left[A_{I}(\bm{y})P(\bm{y},\tau)\right].\] (22) One can check by direct substitution that the solution to this equation is $P(\bm{y},\tau)=\delta(\bm{y}(\tau)-\bm{y})$ where $\delta$ is the Dirac delta function and where $\bm{y}(\tau)$ is the solution of the macroscopic system (17); see (Gardiner, 2009) for details. It is also natural to ask if it is useful to include higher order terms in $V^{-1}$. There are sound mathematical reasons for not going to higher order, for instance the pdf may become negative (Risken, 1989). As we will see, for the problems that we are interested in here (and many others) very good agreement with simulations can be found by working with the Fokker-Planck equation (20). The Fokker-Planck equation (20) provides a mesoscopic description of the system but, like the master equation (2) from which it originated, it is an equation for a pdf. It is therefore quite distinct from the macroscopic equation (17), which is an equation for the state variables themselves. There does, however, exist an equation for the state variables which is completely equivalent to the Fokker-Planck equation (20) (Gardiner, 2009). This equation takes the form \[\frac{\mathrm{d}y_{I}}{\mathrm{d}\tau}=A_{I}(\bm{y})+\frac{1}{\sqrt{V}}\sum_{J }g_{IJ}(\bm{y})\eta_{J}(\tau),\] (23) where the $\eta_{J}(\tau)$ are Gaussian white noises with zero mean and correlator \[\langle\eta_{I}(\tau)\eta_{J}(\tau^{\prime})\rangle=\delta_{IJ}\delta(\tau- \tau^{\prime}),\] (24) and where $g_{IJ}(\bm{y})$ is related to $B_{IJ}(\bm{y})$ by \[B_{IJ}(\bm{y})=\sum_{K}g_{IK}(\bm{y})g_{JK}(\bm{y}).\] (25) The mesoscopic equation (23) generalises the macroscopic ordinary differential equation (17) with the addition of noise terms $\bm{\eta}(\tau)$ and so is a stochastic differential equation (SDE). As we will discuss below we need to specify that it is to be interpreted in the sense of Itō (Gardiner, 2009). Notice the direct relationship between this SDE and the macroscopic ODE: sending $V\to\infty$ in Eq. (23) immediately yields equation (17). It is important to point out that the matrices $g_{IJ}(\bm{y})$ which define the behaviour of the noise cannot be found from the macroscopic equations, and a knowledge of the microscopic stochastic dynamics is essential if one is to understand the effects of noise. It is not permissible in this context to simply ‘add noise terms’ to the macroscopic equations to obtain a mesoscopic description, as some authors have done in the past. The only situation in which it is permissible to do this is if the noise is external to the system, that is, it does not originate from the internal dynamics of the system. We end this section with two general comments on the mesoscopic equation (23). The first is that while there are no strong restrictions on the form of $A_{I}(\bm{y})$, there are on $B_{IJ}(\bm{y})$. From Eq. (21) we see that the matrix $B$ is symmetric, but also that for any non-zero vector $\bm{w}$, \[\sum_{I,J}w_{I}B_{IJ}w_{J}=\sum^{M}_{\mu=1}\left(\bm{w}\cdot\bm{\nu}\right)^{2 }f_{\mu}(\bm{y})\geq 0,\] (26) since $f_{\mu}(\bm{y})\geq 0$. Thus $B$ is positive semi-definite (Mehta, 1989). It follows that $B=g\,g^{\rm T}$ for some non-singular matrix $g$, where T denotes transpose (Mehta, 1989). One way of constructing such a matrix is to note that since $B$ is symmetric, it can be diagonalised by an orthogonal transformation defined through a matrix $O_{IJ}$. Then since $B$ is positive semi-definite, its eigenvalues are non-negative, and so \[B=O\Lambda O^{\rm T}=g\,g^{\rm T},\ \ {\rm where\ }g=O\Lambda^{1/2},\] (27) and where $\Lambda$ and $\Lambda^{1/2}$ are the diagonal matrices with respectively the eigenvalues and square root of the eigenvalues of $B$ as entries. We can take the positive roots of the eigenvalues without loss if generality, since the sign can always be absorbed in the $\eta_{J}$ factor in Eq. (23) (its distribution is Gaussian and so invariant under sign changes). It should also be pointed out that we can go further and make an orthogonal transformation on the noise, $\zeta_{J}=\sum_{I}S_{IJ}\eta_{J}$, and leave Eq. (24), and so its distribution, unchanged. The transformation matrix $S$ can then be used to define a new matrix $G_{IJ}=\sum_{K}g_{IK}S_{JK}$, so that the form of the mesoscopic equation (23) is unchanged. So while the procedure outlined above gives us a way of constructing $g_{IJ}(\bm{y})$ from $B_{IJ}(\bm{y})$, it is not unique. The second comment relates to the statement made earlier, that Eq. (23) is to be interpreted in the Itō sense. The singular nature of white noise means that in some cases SDEs are not uniquely defined by simply writing down the equation, but have to be supplemented with the additional information on how the singular nature of the process is to be interpreted (van Kampen, 2007; Gardiner, 2009). This happens when $g_{IJ}$ depends on the state of the system $\bm{y}$; the noise is then said to be multiplicative. As we will see in the next section, this subtlety is not relevant within the LNA, since there the $g_{IJ}$ is evaluated at a fixed point of the dynamics, and so ceases to depend on the state of the system. However, when going beyond the LNA, it is an important consideration. If one wishes to manipulate a multiplicative noise SDE like Eq. (23), then one must employ modified rules of calculus which take into account the contribution from the noise. We refer to Gardiner (2009) for details, and a complete discussion on the relationship between the Itō formulation and the alternatives. This completes our general discussion of the derivation and form of the mesoscopic equation. In the next two sections we will apply it to the two models which we introduced in Section 2. In the first case we will consider use of the LNA is sufficient for our requirements, but in the second case we have to go beyond the LNA. ## 4 Stochastic patterns in population models One of the reasons why deterministic reaction-diffusion systems are interesting is the fact that they may give rise to ordered structures, either in space or time. Many models displaying several different kinds of patterns have been extensively discussed in the literature (Murray, 2008; Cross and Greenside, 2009). However, the mathematical mechanisms which are responsible for the pattern formation are few and universal, and they can be conveniently analysed using the simplest models. Perhaps the most famous of these mechanisms was put forward by Turing in his pioneering study of morphogenesis (Turing, 1952), and it is now referred to as the ‘Turing instability’ (Murray, 2008; Cross and Greenside, 2009). It is invoked as a central paradigm in various areas of science to explain the emergence of steady spatial structures which typically look like ‘stripes’, ‘hexagons’ or ‘spots’ (Murray, 2008; Cross and Greenside, 2009). In this section, we will be interested in reaction-diffusion systems exhibiting a Turing instability which are composed of discrete entities as described in section 2. The intrinsic noise in the system will render it stochastic. As we shall see, by means of the LNA, one is able to make analytical progress and so clarify the role of demographic noise in the pattern formation process. We shall show that systems of this kind display ‘stochastic patterns’ in addition to the conventional ‘Turing patterns’. It has been suggested that stochastic patterns are responsible for the robustness of the patterning observed in population systems (Butler and Goldenfeld, 2009, 2011; Biancalani et al., 2010) and they have been applied in several ecological models (Butler and Goldenfeld, 2009; Bonachela et al., 2012), in the dynamics of hallucination (Butler et al., 2012) and in a biological model with stochastic growth (Woolley et al., 2011). In a similar way, the emergence of stochastic travelling waves has been studied (Biancalani et al., 2011), which has found application in a marine predator-prey system (Datta et al., 2010). There is also an existing literature on stochastic patterning in arid ecosystems (Ridolfi et al., 26) where the origin of the noise is extrinsic rather than intrinsic. The following analysis employs the Brusselator model to exemplify the general theory. This is a reaction scheme introduced by Lefever and Prigogine in the 1960s (Glansdorff and Prigogine, 1971) as a model of biochemical reactions which showed oscillatory behaviour. For our purposes, its interest lies in the fact that its spatial version is one of the simplest models which exhibits a Turing instability. Before we begin the analysis of this model we need to discuss some aspects of the notation we will use. As explained in section 2 the labels that we have been using so far combine a spatial index with an index for the type of constituent, for example $J=\{j,s\}$. This was done in order to reduce the clutter of indices. However in the analysis below we will need to separate them, since we will assume a regular lattice structure for the domains, which will allow us to use Fourier analysis to effectively diagonalise the spatial part of the system. The Fourier components of a given function will be labelled through the argument of that function, leaving only the index (e.g. $s$) labelling the type. Specifically we will choose the spatial structure to be a regular $D$-dimensional hypercubic lattice with periodic boundary conditions and domain length $l$. Following the conventions of (Chaikin and Lubensky, 2000), the discrete spatial Fourier transform is then defined as: \[\tilde{f}_{k}=l^{D}\sum_{j=1}^{\Omega}e^{-ilk\cdot j}f_{j},\ {\rm with\ }\ f_{ j}=l^{-D}\Omega^{-1}\sum_{k=1}^{\Omega}e^{ilk\cdot j}\tilde{f}_{k},\] (28) where $\Omega$ is the number of lattice points, $j$ is a $D$-dimensional spatial vector and $k$ is its Fourier conjugate. Note that $i$ here is imaginary unit and not a spatial index, and although both $j$ and $k$ are $D$-dimensional vectors, we do not explicitly show this, in line with the notation adopted in section 2. We begin the analysis by deriving the well-known deterministic Brusselator equations from this microscopic description using Eqs. (17) and (18): \[\frac{\mathrm{d}u_{i}}{\mathrm{d}\tau} = a-\left(b+d\right)u_{i}+cu^{2}_{i}v_{i}+\alpha\Delta u_{i},\] \[\frac{\mathrm{d}v_{i}}{\mathrm{d}\tau} = bu_{i}-cu^{2}_{i}v_{i}+\beta\Delta v_{i},\] (29) where $u_{i}=\ell_{i}/V$ and $v_{i}=m_{i}/V$, where $u$ is the density of chemical species $X$ and $v$ is the density of chemical species $Y$. The symbol $\Delta$ represents the discrete Laplacian operator $\Delta f_{j}=(2/z)\,\sum_{j^{\prime}\in\partial j}\left(f_{j}-f_{j^{\prime}}\right)$ where $j^{\prime}\in\partial j$ indicates that the domain $j^{\prime}$ is a nearest neighbour of the domain $j$ and $z$ is the co-ordination number of the lattice. The spatial Fourier transform of the discrete Laplacian operator reads (Lugo and McKane, 2008): \[\tilde{\Delta}_{k}=\frac{2}{D}\sum^{D}_{s=1}\left[\cos\left(k_{s}\,l\right)-1 \right].\] (30) It is possible to obtain a continuous spatial description, in the deterministic limit, by taking the limit of small domain length scale, $l\to 0$(Lugo and McKane, 2008). By doing so, one can recover the traditional partial differential equations for reaction-diffusion systems: \[\frac{\partial u}{\partial\tau} = a-\left(b+d\right)u+cu^{2}v+D_{1}\nabla^{2}u,\] \[\frac{\partial v}{\partial\tau} = bu-cu^{2}v+D_{2}\nabla^{2}v,\] (31) where $D_{1}$ and $D_{2}$ are obtained by scaling the diffusivities $\alpha$ and $\beta$ according to: \[\frac{1}{2D}l^{2}\alpha\mapsto D_{1},\ \ \frac{1}{2D}l^{2}\beta\mapsto D_{2}.\] (32) However, we shall keep the space discrete in the following analysis because the theory is simpler to describe and it is most convenient for carrying out stochastic simulations. We shall also set $l=1$, since this simply amounts to a choice of length scale, and this is the simplest choice. The macroscopic equations (4) are Eq. (17) for the particular case of the Brusselator model. To find the corresponding mesoscopic equations we need to find the particular form of Eq. (23) for the Brusselator. This we can do by calculating $B_{IJ}(\bm{y})$, defined in Eq. (21), but we will find that we do not need to utilise the non-linear equation (23) to take the fluctuations into account; it is sufficient to use only a linearised form. This is the LNA, and is implemented by writing \[y_{I}(t)=\langle y_{I}(t)\rangle+\frac{\xi_{I}(t)}{\sqrt{V}},\] (33) where $\langle y_{I}(t)\rangle$ satisfies the macroscopic equation (17). Substituting Eq. (33) into Eq. (23), we expand in powers of $1/\sqrt{V}$. The terms which are proportional to $1/\sqrt{V}$ give an equation for $\xi_{I}$: \[\frac{\mathrm{d}\xi_{I}}{\mathrm{d}\tau}=\sum_{J}{\mathcal{J}}_{IJ}(\langle\bm {y}\rangle)\xi_{J}+\sum_{J}g_{IJ}(\langle\bm{y}\rangle)\eta_{J}(\tau),\] (34) where ${\mathcal{J}}$ is the Jacobian of the system. In many situations, including the one we are describing here, we are only interested in the fixed points of the macroscopic equation, in which case $\langle\bm{y}\rangle=\bm{y}^{}$ and the matrices ${\mathcal{J}}$ and $g$ can be replaced by their values at the fixed point $\bm{y}^{}$. The SDE (34) now involves only constant matrices: \[\frac{\mathrm{d}\xi_{I}}{\mathrm{d}\tau}=\sum_{J}{\mathcal{J}}^{}_{IJ}\xi_{J} +\sum_{J}g^{}_{IJ}\eta_{J}(\tau).\] (35) For the specific case of the Brusselator the index $I$ includes the spatial index $i$ and an index $s=1,2$ which distinguishes between the variables $u$ and $v$. If we take the spatial Fourier transform of Eq. (35), translational invariance implies that the matrices ${\mathcal{J}}^{}$ and $g^{}$ are diagonalised in the spatial variables, and so this equation becomes \[\frac{\partial\tilde{\xi}_{\gamma}(k,\tau)}{\partial\tau}=\sum_{\delta=1}^{2}{ \mathcal{J}}^{}_{\gamma\delta}(k)\,\tilde{\xi}_{\delta}(k,\tau)+\sum_{\delta= 1}^{2}g^{}_{\gamma\delta}(k)\,\tilde{\eta}_{\delta}(k,\tau).\] (36) We are now in a position to discuss both the classical Turing patterns found in deterministic equations such as Eqs. (31) and the stochastic Turing patterns found in the corresponding mesoscopic equations. The homogeneous fixed point of Eqs. (31) is given by \[u^{}=\frac{a}{d},\ \ v^{}=\frac{bd}{ac},\] (37) although from now on we will set $c=d=1$, as is common in the literature (Glansdorff and Prigogine, 1971). In the deterministic case, the linear stability analysis about this fixed point is a special case of that carried out above in the stochastic picture, and corresponds to ignoring the noise term in Eq. (36). Therefore the small, deterministic, spatially inhomogeneous, perturbations $\tilde{\xi}_{\gamma}(k,\tau)$ satisfy the equation \[\frac{\partial\tilde{\xi}_{\gamma}(k,\tau)}{\partial\tau}=\sum_{\delta=1}^{2}{ \mathcal{J}}^{}_{\gamma\delta}(k)\,\tilde{\xi}_{\gamma}(k,\tau),\] (38) where the Jacobian is found to be \[{\mathcal{J}}^{}(k)=\left(\begin{array}[]{cc}b-1+\alpha\tilde{\Delta}_{k}&a^{ 2}\\ -b&-a^{2}+\beta\tilde{\Delta}_{k}\end{array}\right).\] (39) The eigenvalues of the Jacobian, $\lambda_{\gamma}(k)$ ($\gamma=1,2$), give information about the stability of the homogeneous state. In particular, the perturbations $\tilde{\xi}_{\gamma}(k,\tau)$ grow like linear combinations of $e^{\lambda_{\gamma}(k)\,t}$, therefore if $\mbox{Re}[\lambda_{\gamma}(k)]$ is positive for some $k$ and some $\gamma$, then the perturbation will grow with time and the homogeneous state will be unstable for this value of $k$. Turing’s insight was that the pattern eventually formed as a result of the perturbation is characterised by this value of $k$. The overall scenario is complicated by the nature of the boundary conditions, the presence of other attractors and the effect of the non-linearities (Cross and Greenside, 2009), but in the following we shall ignore these, and consider only the simplest case in order to understand the main concepts. In this most straightforward situation, the small perturbation which excites the unstable $k$-th Fourier mode will cause the concentrations $u$ and $v$ to develop a sinusoidal profile about their fixed point values characterised by the wave-number $k$. The pattern is steady or pulsating depending on whether or not the imaginary part $\mbox{Im}[\lambda_{\gamma}(k)]$ is zero. In both cases, the amplitude of sinusoidal profile increases exponentially with a time-scale $1/\mbox{Re}[\lambda_{\gamma}(k)]$, and so clearly the eigenvalue with the largest real part will dominate. By moving away from the homogeneous state the linear approximation will eventually lose its validity and the effect of the non-linear terms will become relevant. If the system admits no solutions which diverge, the growth will be damped by the non-linearities to some non-zero value, which defines the final amplitude of the spatial pattern. Typically, the interesting case occurs when a control parameter triggers the pattern formation by making the real part of one of the eigenvalues positive. For the Brusselator this is illustrated in Fig. 2, where the relevant eigenvalue of $\mathcal{J}$, (i.e. the one which becomes positive) is shown for different values of the parameter $b$. Here $b$ is the control parameter with the other free parameter $a$ fixed and equal to $1.5$. For $b<b_{c}\approx 2.34$, the real part of both eigenvalues is negative and thus the homogeneous state is stable. This corresponds to the situation where there are no patterns. The critical value of $b$, $b_{c}$, occurs when the real part of one of the eigenvalues is tangent to the $k$-axis. For values of $b$ larger than $b_{c}$, a window of unstable modes sets in, delimited by the intersections of $\mbox{Re}[\lambda]$ with the $k$-axis. Each mode contributes to the pattern, although the wavelength which maximises $\mbox{Re}[\lambda]$ is the one with bigger amplitude as it grows faster than the other modes. <figure><img src="content_image/1211.0462/x2.png"><figcaption>Figure 2: Real part (solid lines) and imaginary part (dashed line) of the mostunstable eigenvalue of matrix J. Parameter values are a=1.5, α=2.8 and β=22.4,(Cross and Greenside, 2009). Solid lines correspond to b=1.8 (black),b≡bc=2.34 (blue) and b=2.9 (red). The imaginary part is shown for b=2.34 only,although it looks qualitatively the same for the range of b values displayed.</figcaption></figure> As we have mentioned already no Turing patterns emerge from the deterministic equations if both eigenvalues of ${\mathcal{J}^{}}$ have a negative real part for all $k$, since then the perturbations decay away. However, when the system is described as an IBM, intrinsic noise is present which acts as a continuous perturbation on the homogeneous state, exciting each $k$-mode. Given that the homogeneous state is stable for all $k$, every excitation decays, although each with a different time-scale given by $1/\mbox{Re}[\lambda_{\gamma}(k)]$. Thus, the closer $\mbox{Re}[\lambda_{\gamma}(k)]$ is to zero, the slower the relaxation of the corresponding $k$-mode. The situation is illustrated in Fig. 2, where the value of $k$ at which this occurs for $b=1.8<b_{c}$ is seen to be the maximum of the curve of $\mbox{Re}[\lambda_{\gamma}(k)]$ versus $k$. The net effect is that only a window of modes around the maximum of $\mbox{Re}[\lambda_{\gamma}(k)]$ is visible in the dynamics, the others having died away. This patterns have been called stochastic Turing patterns (Biancalani et al., 2010). Figure 3 shows the result of numerically simulating a two-dimensional Brusselator model using the Gillespie algorithm in a parameter regime where the homogeneous state is stable to perturbations for all $k$. <figure><img src="content_image/1211.0462/x3.png"><figcaption>Figure 3: (Colour online) Snapshot of two-dimensional stochastic Turingpatterns for species X. The system consists of 40×40 domains with periodicboundary conditions. The parameters are the same as in Fig. 2, except that b=2and V=500. Simulations started close to (u∗,v∗) and ran for t/V=15. We haveused warm colours for values of u>u∗ and cold colours for u<u∗. White pixelsindicate the fixed point value, u∗.</figcaption></figure> From the argument given above we would expect that in order to study stochastic Turing patterns it would be sufficient to linearise about the homogeneous state as before, but now to include noise. In addition, we would expect that the detailed nature of the noise would be unimportant, only its strength. This justifies the use of the LNA in the analysis of stochastic Turing patterns and implies that the arguments which we have used can be made quantitative through the use of Eq. (36). To do this we first take the temporal Fourier transform of this equation to obtain \[\tilde{\xi}_{\gamma}(k,\omega)=\sum_{\delta,\sigma}\,\Phi^{-1}_{\gamma\delta}( k,\omega)\,g^{}_{\delta\sigma}(k)\,\tilde{\eta}_{\sigma}(k,\omega),\] (40) where $\Phi_{\gamma\delta}(k,\omega)=-i\omega\delta_{\gamma\delta}-{\mathcal{J}}^{}_ {\gamma\delta}$. From Eq. (40) we can find an expression for the power spectrum of the fluctuations which is the quantity we use to analyse the patterns: \[P_{\gamma}(k,\omega)=\langle\|\tilde{\xi}_{\gamma}(k,\omega)\|^{2}\rangle=\sum_{ \delta,\sigma}\,\Phi^{-1}_{\gamma\delta}(k,\omega)\,\tilde{B}^{}_{\delta \sigma}(k)\,(\Phi^{{\dagger}})^{-1}_{\sigma\gamma}(k,\omega),\] (41) where $\tilde{B}^{}(k)$ is obtained by Fourier transforming in space the matrix $B(\bm{y})$ given by Eq. (25), evaluated at the fixed point. The details of this calculation can be found in Lugo and McKane (2008); here we simply state the final formulae — which holds for any two-species system which has one spatial dimension: \[\tilde{B}^{}_{11}(k)=B^{}_{11}-2u^{}\alpha\tilde{\Delta}_{k},\] \[\tilde{B}^{}_{12}(k)=B^{}_{12},\quad\tilde{B}^{}_{12}(k)=B^{} _{12},\] \[\tilde{B}^{}_{22}(k)=B^{}_{22}-2v^{}\beta\tilde{\Delta}_{k}.\] (42) Here, the matrix $B^{}$ indicates the correlation matrix of Eq. (25) calculated at the fixed point for the corresponding non-spatial system. For instance, in the case of the Brusselator this is obtained by considering only the reactions (3) without those of Eqs. (4), which yields: $B^{}_{11}=2a(1+b)$, $B^{}_{12}=B^{}_{21}=-2ab$ and $B^{}_{22}=2ab$. <figure><img src="content_image/1211.0462/anal-ps.png"><figcaption>Figure 4: (Colour online) Analytical power spectrum of species X, obtainedwith the same parameter values as in Fig. 3, and from the analyticalexpression (41).</figcaption></figure> The expression (41) for the power is plotted in Fig. 4. We have also measured the numerical power spectrum via the Gillespie algorithm, and found good agreement with the analytical expression, confirming that the dynamics is captured within the approximation scheme we have used. The power spectrum shows a peak at $k\neq 0$ and $\omega=0$ which indicates the presence of stochastic Turing patterns of length scale characterised by $k$. As shown in previous studies (Butler and Goldenfeld, 2009; Biancalani et al., 2010), one can compute the region of parameters for which stochastic Turing patterns arise by looking at when $P_{\gamma}(k,0)$ has a maximum for some non-zero $k$. It has been found that those regions are greatly enlarged, making the pattern formation a much more robust mechanism. Specifically, stochastic patterns may appear even for equal diffusivities, a condition for which deterministic patterns cannot occur (Butler and Goldenfeld, 2009; Biancalani et al., 2010; Butler and Goldenfeld, 2011). Notice that unlike their deterministic analogue, stochastic patterns are not steady but they continuously decay whilst they are re-created by the effect of the noise (Scott et al., 2011; Biancalani et al., 2011). In Fig. 5 this effect is shown by means of the dynamics of a one-dimensional Brusselator. The noisy nature of patterns makes them hard to detect by the naked eye, and the emergence of a length scale only becomes clear by means of a Fourier analysis. <figure><img src="content_image/1211.0462/x4.png"><figcaption>Figure 5: (Colour online) Dynamics of a one-dimensional system of 50 domainsrun for 2∗103 time (τ=t/V) units. Parameter values are the same as in Fig. 3.</figcaption></figure> The theory we have presented here is rather general and also applies to other types of pattern instabilities. For instance, if the power spectrum showed a peak at $k\neq 0$ and $\omega\neq 0$ the overall pattern would consist of stochastic travelling waves (Biancalani et al., 2011). Finally, it should be mentioned that the amplitude of stochastic patterns scales as $1/\sqrt{V}$ and it therefore vanishes in the limit $V\rightarrow\infty$, in which the deterministic picture is recovered. Stochastic patterns arise because of the noise related to the discreteness of the populations and they are therefore less relevant for populations in which the number of individuals is macroscopic. ## 5 Spontaneous Emergence of Cell Polarity Many important biological functions require cells to break their rotational symmetry and form a distinguished ‘nose’ and ‘tail’. The emergence of this symmetry-breaking is known as polarisation, and the precise mechanisms responsible are not yet fully understood. A few years ago Altschuler et al. (2008) proposed a very simple model of polarisation in which signalling molecules self-recruit to the cell membrane, before diffusing away. They showed through simulations that, depending on the choice of model parameters, the membrane molecules may spontaneously aggregate. Further investigations have followed several different lines, including more detailed simulations (Petzold et al, 2012) and mathematically rigorous studies (Gupta, 2012). In this section, we will show how the mesoscopic framework developed in previous sections may be used to analytically describe the spontaneous emergence of cell polarity in this model. This effect is stronger than those described by the LNA, and it will require a different theoretical approach. Before beginning the analysis, it is worth noticing that in this model the total number of molecules does not change; there is thus no need to distinguish between this and the volume, so we write $N=V$. Moreover, the variables $\ell$ (giving the number of molecules in the cytoplasmic pool) and $m_{i}$ (the number of molecules in membrane domain $i$) are related by: $\ell+\sum_{i}m_{i}=V\,.$ As usual, we first explore the behaviour of the macroscopic equations. Substituting the transition rates (5) and (6) into equations (17) and (16), we arrive at \[\frac{du}{d\tau} = (k_{\text{off}}-u\,k_{\text{fb}})\sum_{i}v_{i}-u\,k_{\text{on}}\] \[\frac{dv_{i}}{d\tau} = u\,k_{\text{on}}+(u\,k_{\text{fb}}-k_{\text{off}})\,v_{i}+\alpha \Delta v_{i}\,.\] Conservation of the total number of molecules implies that $u+\sum_{j}v_{j}=1$, so we may eliminate $u$ from the above system to obtain \[\frac{dv_{i}}{d\tau}=\Big{(}1-\sum_{j}v_{j}\Big{)}\big{(}k_{\text{on}}+v_{i}\, k_{\text{fb}}\big{)}-v_{i}k_{\text{off}}+\alpha\Delta v_{i}\,.\] (43) In the appropriate continuum limit, this equation agrees with that of Altschuler et al. (2008). As with the Brusselator, there is a homogeneous fixed point. Putting $v_{i}\equiv v$ and setting $dv_{i}/d\tau=0$, we find the quadratic equation \[(1-\Omega\,v)(k_{\text{on}}+v\,k_{\text{fb}})-v\,k_{\text{off}}=0\,.\] (44) For simplicity, we consider the case in which $k_{\text{on}}\approx 0$ (that is, almost all membrane molecules exist as result of the feedback mechanism). In this limit, the homogeneous fixed point is given by $v=v^{}/\Omega$, where $v^{}$ is the mean fraction of molecules on the membrane: \[v^{}=\begin{cases}\left(1-\frac{k_{\text{off}}}{k_{\text{fb}}} \right)&\text{if}\quad k_{\text{fb}}>k_{\text{off}}\\ 0&\text{otherwise.}\end{cases}\] (45) To gain further insight, we pass to Fourier space, where Eq. (43) with $k_{\text{on}}=0$ becomes \[\frac{d\tilde{v}_{k}}{dt}=\tilde{v}_{k}\Big{[}k_{\text{fb}}(1-l^{-1}\tilde{v}_ {0})-k_{\text{off}}+\alpha\big{(}\cos(lk)-1\big{)}\Big{]}\,.\] (46) The Jacobian for this system is diagonal, and it is straightforward to read off the eigenvalues at the fixed point $\tilde{v}_{k}=\delta_{w,0}\,l\,v^{}$ as \[\lambda_{k}=\begin{cases}k_{\text{off}}-k_{\text{fb}}\Big{.}\,,&\text{if}\quad k =0\\ \Big{.}\alpha\big{(}\cos(lk)-1\big{)}&\text{if}\quad k\neq 0\,.\end{cases}\] We can conclude from this analysis that provided $k_{\text{fb}}>k_{\text{off}}$ the homogeneous fixed point is non-zero and stable. This is a puzzle: the homogeneous state corresponds to the signalling molecules being spread uniformly around the membrane, if this state is stable, then how can polarisation occur? We postulate that the answer lies in the following observation. Notice that if the diffusion coefficient $\alpha$ is small then the modes with wave number $k\neq 0$ are only marginally stable; their associated eigenvalues are close to zero. In this regime, a small random perturbation (resulting from intrinsic noise, for example) may be enough to push the system very far from its equilibrium state. Moreover, the stochastic dynamics in this regime cannot be understood within the framework of the LNA, for the simple reason that when the system has been pushed far from its steady state, linearisation around that state is no longer representative of the true dynamics. To make analytical progress, we will need to deal with the non-linearity of the model some other way. We begin by writing down the mesoscopic equations. For our purposes, the Fokker-Planck equation (20) is the most useful, with $A$ and $B$ given by \[A_{i}(\bm{v}) = \Big{(}1-\sum_{j}v_{j}\Big{)}\big{(}k_{\text{on}}+v_{i}\,k_{\text {fb}}\big{)}-v_{i}k_{\text{off}}+\alpha\Delta v_{i}\,,\] \[B_{ij}(\bm{v}) = \delta_{ij}\bigg{[}\Big{(}1-\sum_{k}v_{k}\Big{)}\big{(}k_{\text{ on}}+v_{i}\,k_{\text{fb}}\big{)}+v_{i}k_{\text{off}}\bigg{]}\,.\] Note that we have neglected terms of order $\alpha/V$ from the noise, as we are interested in behaviour when $\alpha$ is small and $V$ large, so $\alpha/V$ is negligible. As with the macroscopic equations, this system is easier to analyse in Fourier space. We introduce the distribution $P(\bm{\tilde{v}},\tau)$ of Fourier variables $\tilde{v}_{k}$, which satisfies the Fokker-Planck equation \[\frac{\partial P(\bm{\tilde{v}},\tau)}{\partial\tau} = -\sum_{k}\frac{\partial}{\partial\tilde{v}_{k}}\left[\tilde{A}_{k }(\bm{\tilde{v}})P(\bm{\tilde{v}},\tau)\right]\] \[+ \frac{1}{2V}\sum_{k,k^{\prime}}\frac{\partial}{\partial\tilde{v}_ {k}}\frac{\partial}{\partial\overline{\tilde{v}_{k^{\prime}}}}\left[\tilde{B}_ {k,k^{\prime}}(\bm{\tilde{v}})P(\bm{\tilde{v}},\tau)\right]\,,\] where \[\tilde{A}_{k}(\bm{\tilde{v}}) = \tilde{v}_{k}\,\Big{[}k_{\text{fb}}(1-l^{-1}\tilde{v}_{0})-k_{ \text{off}}+\alpha\big{(}\cos(lk)-1\big{)}\Big{]}\] \[\tilde{B}_{k,k^{\prime}}(\bm{\tilde{v}}) = \tilde{v}_{k+k^{\prime}}\,l\,\Big{[}k_{\text{fb}}(1-l^{-1}\tilde{ v}_{0})+k_{\text{off}}\Big{]}\,.\] (48) Note that the Fourier modes may take complex values, and we use $\partial/\partial\overline{\tilde{v}_{k^{\prime}}}$ to denote differentiating with respect to the complex conjugate. For later convenience, we assume $\Omega$ is odd and number the modes by $k\in\{-(\Omega-1)/2,\ldots,(\Omega-1)/2\}$, so that $\tilde{v}_{-k}=\overline{\tilde{v}_{k}}$. Our remaining analysis is informed by two observations. First, we note that the non-linearity in equation (48) arises only from terms involving $\tilde{v}_{0}$. Second, in the interesting regime $k_{\text{fb}}-k_{\text{off}}\gg\alpha$, we have that the eigenvalues of the macroscopic system satisfy $\lambda_{0}\ll\lambda_{k}<0$ and thus $\tilde{v}_{0}$ is (comparatively) very stable near $v^{}$. This implies a separation of time-scales in the problem: we expect $\tilde{v}_{0}$ to relax very quickly to a value near its equilibrium, whilst the other modes fluctuate stochastically on much slower time-scales. Combining these facts suggests the following strategy: we restrict our attention to only those trajectories in which $\tilde{v}_{0}$ is held constant at $lv^{}$. Conditioning on the value of $\tilde{v}_{0}$ alters the structure of the noise correlation matrix for the other modes; details of the general formulation are given in Appendix B of Rogers et al. (29). The result is a Fokker-Planck equation for the distribution $P^{(c)}(\bm{\tilde{v}},\tau)$ of the remaining Fourier modes $\tilde{v}_{k}$ with $k\neq 0$, conditioned on $\tilde{v}_{0}$ taking the value $lv^{}$: \[\frac{\partial P^{(c)}(\bm{\tilde{v}},\tau)}{\partial\tau} = -\sum_{ki\neq 0}\frac{\partial}{\partial\tilde{v}_{i}}\left[ \tilde{A}^{(c)}_{k}(\bm{\tilde{v}})P(\bm{\tilde{v}},\tau)\right]\] \[+ \frac{1}{2V}\sum_{k,k^{\prime}\neq 0}\frac{\partial}{\partial \tilde{v}_{k}}\frac{\partial}{\partial\overline{\tilde{v}_{k^{\prime}}}}\left[ \tilde{B}^{(c)}_{k,k^{\prime}}(\bm{\tilde{v}})P(\bm{\tilde{v}},\tau)\right]\,,\] where \[\tilde{A}^{(c)}_{k}(\bm{\tilde{v}}) = \alpha\,\tilde{v}_{k}\,\big{(}\cos(lk)-1\big{)}\bigg{.}\] \[\tilde{B}^{(c)}_{k,k^{\prime}}(\bm{\tilde{v}}) = 2k_{\text{off}}\bigg{[}l\,\tilde{v}_{k+k^{\prime}}-\frac{\tilde{ v}_{k}\tilde{v}_{k^{\prime}}}{v^{}}\bigg{]}\,.\] (50) Multiplying (5) by $\tilde{v}_{k}$ and integrating over all $\bm{\tilde{v}}$ we obtain a differential equation for the mode averages: \[\frac{d}{d\tau}\langle\tilde{v}_{k}\rangle=\alpha\langle\tilde{v}_{k}\rangle \big{(}\cos(lk)-1\big{)}\,.\] (51) Thus, for all $\alpha>0$ we have \[\langle\tilde{v}_{k}\rangle\to\delta_{k,0}\,lv^{}\,.\] (52) For the second-order moments the behaviour is not so trivial. Multiplying (5) this time by $\tilde{v}_{k}\tilde{v}_{k^{\prime}}$ and integrating yields \[\frac{d}{d\tau}\langle\tilde{v}_{k}\tilde{v}_{k^{\prime}}\rangle=\frac{lk_{ \text{off}}}{V}\langle\tilde{v}_{k+k^{\prime}}\rangle+\langle\tilde{v}_{k} \tilde{v}_{k^{\prime}}\rangle\Big{[}\alpha\big{(}\cos(lk)+\cos(lk^{\prime})-2 \big{)}-\frac{1}{V}\frac{k_{\text{off}}}{v^{}}\Big{]}\,.\] (53) The equilibrium values are thus $\langle\tilde{v}_{k}\tilde{v}_{k^{\prime}}\rangle\to 0$ if $k+k^{\prime}\neq 0\mod\Omega$, and \[\langle\|\tilde{v}_{k}\|^{2}\rangle\to\frac{(lv^{})^{2}}{1+\alpha V(1-\cos(lk)) v^{}/k_{\text{off}}}\,.\] (54) This is our main result, although further analysis is required to interpret the implications for the behaviour of the model. In Altschuler et al. (2008) it was observed that simulations of a continuum version of this model exhibited the curious phenomenon of the membrane molecules grouping together, despite the macroscopic equations suggesting they should be spread uniformly around the membrane. We introduce a summary statistic to measure this effect. Suppose the membrane molecules are distributed according to an angular density field $v(x)$, and let $\Lambda$ denote the mean angular separation of two molecules: \[\Lambda=\left(\frac{1}{v^{}}\right)^{2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}d(x, y)\big{\langle}v(x)v(y)\big{\rangle}\,dx\,dy\,,\] (55) where \[d(x,y)=\begin{cases}\|x-y\|&\text{if}\quad\|x-y\|<\pi\\ 2\pi-\|x-y\|&\text{otherwise.}\end{cases}\] (56) To compute $\Lambda$ from our result (54) we pass to the continuum limit, taking $\Omega\to\infty$ and using the modes $\tilde{v}_{k}$ as the coefficients of the Fourier series of the membrane angular density field $v(x)$. Taking an angular prescription for the membrane so that $l=2\pi/\Omega$, we renormalise by a factor of $1/2\pi l$ and reverse the Fourier transform to obtain \[v(x)=\lim_{\Omega\to\infty}\frac{1}{2\pi l}\sum_{k=1}^{\Omega}e^{ikx}\tilde{v} _{k}\,,\quad\text{for }x\in[-\pi,\pi)\,.\] (57) The calculation proceeds thus: \[\Lambda =\] (58) \[= \frac{1}{2\pi}\left(\frac{1}{lv^{}}\right)^{2}\sum_{k,k^{\prime} }\langle\tilde{v}_{k}\tilde{v}_{k^{\prime}}\rangle\int_{-\pi}^{\pi}\|x\|e^{ikx} \,dx\] \[= \frac{\pi}{2}+\sum_{k\neq 0}\frac{1}{1+\varphi^{2}k^{2}}\int_{- \pi}^{\pi}\|x\|e^{ikx}\,dx\,,\] where $\varphi=(\alpha Vl^{2}v^{}/k_{\text{off}})^{1/2}$, which we assume to have a finite value in the limit $l\to 0$. The first equality above comes from the rotational invariance of the model meaning that we may fix $y=0$, after which we employ Eq. (57) and Eq. (54) in turn. Now, for $k\neq 0$ \[\int_{-\pi}^{\pi}\|x\|e^{ikx}\,dx=\begin{cases}-4k^{-2}&\text{for odd $k$}\\ 0&\text{for even $k$.}\end{cases}\] Also, the following infinite series (Bromwich, 1926) will be useful: \[\sum_{k\,\text{odd}}\frac{1}{x^{2}+k^{2}}=\frac{\pi}{2x}\tanh\left(\frac{\pi x }{2}\right)\,.\] Altogether, we have \[\Lambda(\varphi) = \frac{\pi}{2}-\frac{2}{\pi}\sum_{k\,\text{odd}}\left(\frac{1}{1+ \varphi^{2}k^{2}}\right)\left(\frac{1}{k^{2}}\right)\] (59) \[= \frac{\pi}{2}-\frac{2}{\pi}\left(\,\sum_{k\,\text{odd}}\frac{1}{k ^{2}}-\sum_{k\,\text{odd}}\frac{1}{\varphi^{-2}+k^{2}}\right)\] \[= \Bigg{.}\varphi\tanh\left(\frac{\pi}{2\varphi}\right)\,.\] The limits are $\Lambda(\infty)=\pi/2$, which corresponds to molecules spread uniformly around the membrane, and $\Lambda(0)=0$, which corresponds to complete localisation. In figure 6 we compare this theoretical prediction with the result of simulations for $\varphi$ varying over several orders of magnitude. For small values of $\varphi$ the membrane molecules cluster into a tight group, meaning that the cell has become polarised. As $\varphi$ is increased (caused by an over-abundance of signalling molecules in relation to their rate of diffusion around the membrane) this effect is weakened, and the cell loses polarity. <figure><img src="content_image/1211.0462/x5.png"><figcaption>Figure 6: (Colour online) Mean angular distance between membrane molecules, asφ is varied over several orders of magnitude. Red circles are the results ofsimulations, the blue line shows the result from Eq. (59). On the right aretwo snapshots from simulations, with φ corresponding to the points (a) and (b)in the main figure. Clearly (a) is polarised and (b) is not, as predicted bythe theory.</figcaption></figure> Finally, we discuss the physical meaning of the parameter $\varphi$. The average number of molecules on the membrane at equilibrium is $Vv^{}$, the typical lifetime of a membrane molecule is $1/k_{\text{off}}$, and the continuum diffusion coefficient is $\alpha l^{2}$. The quantity $\varphi$ can thus be interpreted as the mean total distance travelled by all the membrane molecules during their lifetime: if this quantity is small, they must remain localised. ## 6 Discussion and Conclusion The main aim of this paper has been to show that the analysis of stochastic models in biology need not only be numerical: a range of analytical techniques are available, just as for deterministic models. In fact the treatment of stochastic models may be simpler, since in many cases the noise can be considered to be a linear perturbation (the LNA) to the deterministic form of the dynamical equations. Linear equations such as these are easier to solve, especially when the fluctuations are around a stationary state. The deterministic, or macroscopic, description of the system is valid when the individual nature of the constituents is unimportant, for example when they are so numerous as to effectively be infinite in number. The ensemble average of the stochastic variables will also obey the same deterministic equation. The general form of this equation is Eq. (17), that is, $\dot{y}_{I}=A_{I}(\bm{y})$, where the dot denotes a time derivative and the index $I$ includes both a position label and another identifying the constituent type. The function $A_{I}$ can be calculated from the elementary reactions of the process using Eq. (18). The mesoscopic, or stochastic, description of the system which uses the same variables as the macroscopic description, is in principle no different. The general form of this equation is (23), that is, $\dot{y}_{I}=A_{I}(\bm{y})+V^{-1/2}\sum_{J}g_{IJ}(\bm{y})\eta_{J}$, where $\eta_{J}$ is a Gaussian white noise with zero mean and unit strength. The only additional function which appears over and above that in the deterministic equation is $g_{IJ}$, which can, like $A_{I}$, be calculated from the elementary reactions of the process using Eqs. (21) and (25). Although Eq. (23) is a straightforward generalisation of Eq. (17), it is much less well-known. There are several reasons for the perceived difficulty of using Eq. (23), probably the most important being the unfamiliarity of many biologists with the general theory of stochastic processes. We have tried to show in this paper that the stochastic theory which is required need not be more complicated than that of dynamical systems theory, which is applicable to equations such as (17). This is especially true if the LNA is a valid approximation for the particular system under study. If the multiplicative nature of the noise cannot be neglected, as in section 5, then care is required because of the singular nature of white noise. However, even in this case, a systematic theory has been developed that may be applicable in situations in which there is a separation of time-scales (Rogers et al., 28; Biancalani et al., 2012; Rogers et al., 29). We applied Eq. (23) to two sets of processes, one for which the LNA was applicable and one for which it was not. The former situation was discussed in section 4 where we revisited the problem of the emergence of spatial structures for systems of populations, in the paradigmatic example of the Brusselator model. Intrinsic fluctuations, which are intuitively thought of as a disturbing source, appear instead to be critical for the emergence of spatial order. More specifically, we showed how Turing patterns can arise for parameter values for which the macroscopic equations predict relaxation to a homogeneous state. We called these patterns ‘stochastic patterns’, as they are generated by the continuous action of noise present in the system. However, it can be argued that the amplitude of stochastic patterns might be so small that they can hardly be observed in a real population, given that the amplitude of the noise is small as well. Whilst this might be true for some systems, a recent study (Ridolfi et al., 25) has suggested that the response to a small perturbation in a pattern-forming system can be unexpectedly large, if the system displays a sufficient degree of ‘non-normality’. The connection between non-normality and stochastic patterns is so far largely unexplored, and constitutes a possible further investigation in this line of research. In section 5 we discussed an example of a stochastic phenomenon which goes beyond what can be understood within the LNA. The stochastic patterns appearing in the Brusselator are noise-driven perturbations around the homogeneous state, having characteristic magnitude $1/\sqrt{V}$ (and thus disappearing in the limit $V\to\infty$). By contrast, the spontaneous emergence of cell polarity in the model of Altschuler _et al_ requires the noise to have a more complex structure, which can lead the system to a state very far removed from the homogeneous fixed point of the deterministic equations. To characterise this process, it was necessary to study the full effect of the non-linear terms in the mesoscopic equations. To achieve this, we exploited the natural separation of time-scales occurring between the dynamics of the zeroth Fourier mode (which relaxes quickly to its equilibrium value) and the remaining degrees of freedom. This is a non-standard technique, however, it can be made relatively systematic and general, as will be outlined in a forthcoming paper. The LNA has played an important role in boosting the recognition of the importance of stochastic effects in the literature; we hope that methods employing the separation of time-scales may provide the next theoretical advance. Acknowledgements. 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1612.02015	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 72449, "num_imgs": 7, "llama3_tokens_count": 21887 }	[ "content_image/1612.02015/x1.png", "content_image/1612.02015/x2.png", "content_image/1612.02015/x3.png", "content_image/1612.02015/x5.png", "content_image/1612.02015/x7.png", "content_image/1612.02015/x8.png", "content_image/1612.02015/x11.png" ]	# A Black Hole Mass Determination for the Compact Galaxy Mrk 1216 Jonelle L. Walsh${}^{1}$, Remco C. E. van den Bosch${}^{2}$, Karl Gebhardt${}^{3}$, Akın Yıldırım${}^{2,4}$, Kayhan Gültekin${}^{5}$, Bernd Husemann${}^{2,6}$, and Douglas O. Richstone${}^{5}$ ${}^{1}$ George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843, USA; walsh@physics.tamu.edu ${}^{2}$ Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany ${}^{3}$ Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA ${}^{4}$ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany ${}^{5}$ Department of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI 48109, USA ${}^{6}$ European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany ###### Abstract Mrk 1216 is a nearby, early-type galaxy with a small effective radius of 2.8 kpc and a large stellar velocity dispersion of 308 km s${}^{-1}$ for its $K$-band luminosity of $1.4\times 10^{11}\ L_{\odot}$. Using integral-field spectroscopy assisted by adaptive optics from Gemini North, we measure spatially resolved stellar kinematics within $\sim$450 pc of the galaxy nucleus. The galaxy exhibits regular rotation with velocities of $\pm 180$ km s${}^{-1}$ and a sharply peaked velocity dispersion profile that reaches 425 km s${}^{-1}$ at the center. We fit axisymmetric, orbit-based dynamical models to the combination of these high angular resolution kinematics, large-scale kinematics extending to roughly three effective radii, and _Hubble Space Telescope_ imaging, resulting in a constraint of the mass of the central black hole in Mrk 1216. After exploring several possible sources of systematics that commonly affect stellar-dynamical black hole mass measurements, we find a black hole mass of $M_{\mathrm{BH}}=(4.9\pm 1.7)\times 10^{9}\ M_{\odot}$ and a $H$-band stellar mass-to-light ratio of $\Upsilon_{H}=1.3\pm 0.4\ \Upsilon_{\odot}$ (1$\sigma$ uncertainties). Mrk 1216 is consistent with the local black hole mass – stellar velocity dispersion relation, but is a factor of $\sim$$5-10$ larger than expectations from the black hole mass – bulge luminosity and black hole mass – bulge mass correlations when conservatively using the galaxy’s total luminosity or stellar mass. This behavior is quite similar to the extensively studied compact galaxy NGC 1277. Resembling the $z\sim 2$ quiescent galaxies, Mrk 1216 may be a passively evolved descendant, and perhaps reflects a previous era when galaxies contained over-massive black holes relative to their bulge luminosities/masses, and the growth of host galaxies had yet to catch up. Subject headings:galaxies: elliptical and lenticular, cD – galaxies: individual (Mrk 1216) – galaxies: kinematics and dynamics – galaxies: nuclei – black hole physics ## 1. Introduction Our understanding of the connection between supermassive black holes and their host galaxies is anchored by $\sim$100 dynamical black hole mass ($M_{\mathrm{BH}}$) measurements that have been made over the past two decades (e.g., Kormendy & Ho, 2013; van den Bosch, 2016, and references therein). Strong correlations have emerged between $M_{\mathrm{BH}}$ and large-scale galaxy properties, like the bulge luminosity ($L_{\mathrm{bul}}$; e.g., Kormendy & Richstone 1995; Marconi & Hunt 2003; Kormendy & Ho 2013) or mass ($M_{\mathrm{bul}}$; e.g., Häring & Rix 2004; Sani et al. 2011; McConnell & Ma 2013), and the stellar velocity dispersion ($\sigma_{\star}$; e.g., Ferrarese & Merritt 2000; Gebhardt et al. 2000; Gültekin et al. 2009). These relations are connected to each other, and the search for the most fundamental one is still ongoing (Beifiori et al., 2012; Saglia et al., 2016; van den Bosch, 2016). The empirical relationships imply that black holes are key components of galaxies and regulate galaxy properties via feedback mechanisms (Silk & Rees, 1998; Fabian, 1999), although a non-causal origin in which black holes do not actively shape their host galaxies is also possible (Peng, 2007; Jahnke & Macciò, 2011). Establishing the exact role of black holes in galaxy evolution and accurately inferring the black hole mass function requires increasing the number of $M_{\mathrm{BH}}$ measurements, specifically targeting galaxies with diverse properties that have experienced varied growth channels. As we begin to examine a broader range of black hole masses and hosts, galaxies with different structural properties show surprises in the scaling relations. Recent progress detecting high-mass black holes in Brightest Cluster Galaxies (BCGs) and other large early-type galaxies hint that these objects may be positive outliers from $M_{\mathrm{BH}}-\sigma_{\star}$ and $M_{\mathrm{BH}}-L_{\mathrm{bul}}$(e.g., McConnell et al., 2012; Rusli et al., 2013; Thomas et al., 2016), but there are still too few measurements to firmly characterize the scaling relations at $M_{\mathrm{BH}}\gtrsim 10^{9}\ M_{\odot}$. The uncertainties are equally severe at the opposite end, where spiral galaxies with low-mass black holes ($M_{\mathrm{BH}}\lesssim 10^{7}\ M_{\odot}$) measured from water megamaser disks exhibit substantial scatter below the global black hole – host galaxy relations (e.g., Greene et al., 2010; Läsker et al., 2016; Greene et al., 2016). There are also new observations of compact galaxies, whose black holes are a remarkably large fraction of the galaxy’s stellar mass (e.g., Seth et al., 2014; Walsh et al., 2015, 2016; Saglia et al., 2016). NGC 1277 and NGC 1271 are two such compact galaxies, with NGC 1277 being widely studied over the last few years (e.g., van den Bosch et al., 2012; Emsellem, 2013; Walsh et al., 2016; Scharwächter et al., 2016; Graham et al., 26). Both were originally discovered by the HET Massive Galaxy Survey (van den Bosch et al., 2015) and share considerable similarities with the $z\sim 2$ quiescent galaxies (e.g., Trujillo et al. 2014; Ferré-Mateu et al. 2015; Yıldırım et al. 2015; Yıldırım et al. 2016b, in prep). Like the $z\sim 2$ red nuggets, NGC 1277 and NGC 1271 have small effective radii of $R_{e}\sim 1-2$ kpc, stellar masses of $M_{\star}\sim 10^{11}\ M_{\odot}$, and stellar mass surface density profiles that are elevated at the center and drop off steeply at larger radii compared to low-redshift early-type galaxies. In addition, NGC 1277 and NGC 1271 are rotating, consistent with the disk-like flattened structures of the $z\sim 2$ red nuggets (van der Wel et al., 2011), and have uniformly old stellar populations (ages of $\sim$10 Gyr) extending out to several $R_{e}$. The red nuggets are thought to grow in size and moderately in mass through mergers to produce the present-day massive galaxies (e.g., van Dokkum et al., 2010), with a small fraction experiencing passive evolution since $z\sim 2$(e.g., Trujillo et al., 2009; Wellons et al., 2016). NGC 1277 and NGC 1271 appear to be such relics of the red nuggets, and could provide the unique opportunity to gain insight into the black holes at earlier epochs. Of the current galaxies with dynamical $M_{\mathrm{BH}}$ measurements, NGC 1277 and NGC 1271 are most similar to NGC 4342 (Cretton & van den Bosch, 1999) and NGC 1332. All are nearby galaxies that are flattened and rotating, with small effective radii and large stellar velocity dispersions for their luminosities. They contain black holes that are more massive than the predictions from $M_{\mathrm{BH}}-L_{\mathrm{bul}}$, yet are consistent with $M_{\mathrm{BH}}-\sigma_{\star}$. The magnitude of the offset from $M_{\mathrm{BH}}-L_{\mathrm{bul}}$ depends on the adopted bulge luminosity (e.g., Walsh et al., 2015, 2016; Graham et al., 26, 27), although we note that NGC 1332 may very well be consistent with the black hole relations given the uncertainties associated with both $M_{\mathrm{BH}}$(Rusli et al., 2011; Barth et al., 3, 4) and the bulge component (Rusli et al., 2011; Kormendy & Ho, 2013; Savorgnan & Graham, 2016; Saglia et al., 2016). Clearly, additional mass measurements of black holes in NGC 1277-like galaxies are needed. A significant sample of local analogs to the $z\sim 2$ red nuggets that are also positive outliers from $M_{\mathrm{BH}}-L_{\mathrm{bul}}$ would suggest that this black hole scaling relation did not apply at earlier times, galaxies instead harbored over-massive black holes, and subsequent galaxy growth had yet to occur. Beyond the connections to the $z\sim 2$ red nuggets, the compact galaxies are interesting because they occupy the sparsely populated upper-end of the $M_{\mathrm{BH}}$ relationships, and their properties are quite distinct from the BCGs and giant ellipticals that are expected to house the most massive black holes in the Universe. Currently, at the high-mass end, the differing behaviors of $M_{\mathrm{BH}}-\sigma_{\star}$ and $M_{\mathrm{BH}}-L_{\mathrm{bul}}$, and the poorly characterized scatter in $M_{\mathrm{BH}}$ for fixed $\sigma_{\star}$ or $L_{\mathrm{bul}}$, lead to strongly divergent predictions of the black hole mass function (Lauer et al., 43). This in turn affects inferences about black hole growth histories and constraints of the mean radiative efficiency of black hole accretion, the duty cycle of active galactic nuclei, and the redshift evolution of the scaling relations (e.g., Marconi et al., 2004; Lauer et al., 44; Shankar et al., 2009; Robertson et al., 2006). In this paper, we examine Mrk 1216, an early-type, compact ($R_{e}=2.8$ kpc, $M_{\star}=1.6\times 10^{11}\ M_{\odot}$), high-dispersion ($\sigma_{\star}=308$ km s${}^{-1}$) galaxy found through the HET Massive Galaxy Survey. Yıldırım et al. (2015) presented wide-field integral-field spectroscopy of Mrk 1216, and constructed axisymmetric Schwarschild models in order to learn about the galaxy’s dynamical stellar mass-to-light ratio and dark matter halo. The spatial resolution, however, was insufficient to pin down the black hole mass, and Yıldırım et al. (2015) set an upper-limit of $M_{\mathrm{BH}}<1\times 10^{10}\ M_{\odot}$. Here, we use Gemini North observations assisted by adaptive optics (AO) to resolve the region where the black hole dominates the gravitational potential (the black hole sphere of influence; $r_{\mathrm{sphere}}=GM_{\mathrm{BH}}/\sigma_{\star}^{2}$), thereby obtaining a secure stellar-dynamical $M_{\mathrm{BH}}$ measurement. We assume a distance of 94 Mpc to Mrk 1216. This is the same distance adopted by Yıldırım et al. (2015) and is the Virgo + Great Attractor + Shapley Supercluster Infall value (Mould et al., 2000) for $H_{0}=70.5$ km s${}^{-1}$ Mpc${}^{-1}$, $\Omega_{M}=0.27$ and $\Omega_{\Lambda}=0.73$. At this distance, 1″ corresponds to 456 pc. The paper is structured as follows. We review the imaging observations and the luminous mass model in Section 2. In Sections 3 and 4, we describe the high angular resolution and the large-scale spectroscopic observations, the measured stellar kinematics, and the point-spread function (PSF) characterizations. An overview of the stellar-dynamical models and the results from those models, including an examination of the black hole mass error budget, is provided in Section 5. We study the galaxy’s orbital structure in 6, and discuss the location of Mrk 1216 on the black hole mass – host galaxy relationships, as well as the implications, in Sections 7 and 8. Concluding remarks are provided in Section 9. ## 2. HST Imaging We obtained an _HST_ Wide-Field Camera 3 (WFC3) $F160W$ image of Mrk 1216. The WFC3/IR observations were executed under program GO-13050, and included dithered full array images and brief subarray exposures with a total integration time of 1354 s. The data were reduced, and the flattened, calibrated images were corrected for geometric distortions, cleaned, and combined using AstroDrizzle (Gonzaga et al., 2012) to produce a super-sampled image with a scale of 0$\farcs$06 pixel${}^{-1}$. After masking the foreground stars, we described the galaxy’s stellar surface brightness distribution as the sum of two-dimensional (2D) Gaussians. Such a Multi-Gaussian Expansion (MGE; Monnet et al., 1992; Emsellem et al., 1994) is able to recover the surface brightness profiles of realistic multi-component galaxies while also allowing for the intrinsic luminosity density to be determined through an analytical deprojection. During the MGE fit, we took into account the WFC3 PSF from van der Wel et al. (2012). The PSF was generated with TinyTim (Krist & Hook, 2004) for the $F160W$ filter and a G2 V star at the center of the WFC3 detector, then drizzled to produce the same scale as our final Mrk 1216 image. The Mrk 1216 MGE is composed of 10 Gaussians with dispersions, measured along the major axis, of 0$\farcs$09$-$29$\farcs$76, and projected axis ratios between 0.52 and 0.99. The components have the same center, and a position angle of 70.2${}^{\circ}$ east of north. The final parameter values, after correction for Galactic extinction using the Schlafly & Finkbeiner (2011) WFC3 $F160W$ value of 0.017 mag and assuming an $H$-band absolute solar magnitude of 3.32 (Binney & Merrifield, 1998), are given in Yıldırım et al. (2015). The MGE fits the _HST_ image very well within the central $\sim$40″, and allows us to accurately infer the stellar gravitational potential. We refer the reader to Yıldırım et al. (2015) for additional details regarding the imaging observations, data reduction, and construction of the MGE model. ## 3. NIFS Observations and Measurements In addition to the luminous mass model, stellar kinematics on scales comparable to the black hole sphere of influence are crucial inputs into the dynamical models. We therefore observed Mrk 1216 with the Near-infrared Integral Field Spectrometer (NIFS; McGregor et al., 2003) aided by the ALTtitude conjugate Adaptive optics for the InfraRed (Herriot et al., 2000; Boccas et al., 2006) system on Gemini North. The observations were taken on 21 Dec 2013 under program GN-2013A-Q-1 in laser guide star (LGS) mode using an $R=13.7$ mag star located 22″ from the galaxy nucleus as the tip-tilt reference. Four blocks of consecutive Object-Sky-Object sequences with 600 s exposures were recorded. The observations were acquired with the $H+K$ filter and the $K$ grating centered on $2.2$$\mu$m. We observed the tip-tilt star once during the night for an estimate of the PSF and an A0 V star for telluric correction. The normal baseline calibrations consisting of dark frames, flat fields, Argon/Xeon arc lamp exposures, and a Ronchi mask (to establish the spatial rectification) were taken as well. We processed the NIFS data using PyRAF ¹ and the Gemini data reduction package version 1.11, following the steps in the NIFS example scripts² for calibration, telluric, and science exposures. For the galaxy, the basic procedure consisted of preparing the raw images for processing within the NIFS data reduction package, subtracting sky frames from adjacent object exposures, flat fielding, removing bad pixels and cosmic rays, wavelength calibration, and spatial rectification. We corrected for telluric features using an A0 V star, whose spectrum had been divided by a blackbody with a temperature of 9480 K after interpolating over the Br$\gamma$ absorption line. We then assembled data cubes having $x$ and $y$ spatial dimensions, each with a scale of 0$\farcs$05 pixel${}^{-1}$, and a wavelength axis. We determined the spatial offsets between individual galaxy cubes by summing over the wavelength dimension and cross-correlating the images. These eight cubes, corresponding to a total of 1.3 hours on-source, were aligned and combined to produce the final Mrk 1216 data cube. We followed similar steps to reduce the NIFS observation of the PSF star. [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] ### Stellar Kinematics From the reduced Mrk 1216 data cube, we measured the stellar kinematics as a function of spatial location. Specifically, we extracted the line-of-sight velocity distribution (LOSVD), parameterized by the first four Gauss-Hermite moments, in 67 Voronoi spatial bins (Cappellari & Copin, 2003) using the penalized pixel fitting (pPXF) code of Cappellari & Emsellem (2004). The spatial bins were chosen so that the galaxy spectra had a signal-to-noise ratio (S/N) $\gtrsim$ 40, where the S/N was measured as the median flux divided by the standard deviation of the pPXF model residuals. Such a high S/N spectrum is required in order to measure the LOSVD’s deviation from a Gaussian (e.g., van der Marel & Franx, 1993; Bender et al., 1994). We provided pPXF with a velocity template library composed of 12 stars (K0$-$M5 giant stars and two late-type supergiants), which were observed with NIFS in the $K$ band. The stars are a subset of those presented in Winge et al. (2009), but we have reduced the data ourselves (Walsh et al., 2016), starting with the raw frames and their calibration files retrieved from the Gemini Science Archive. During the fit with pPXF, we corrected for slight differences in the continuum shape and equivalent width between the LOSVD-convolved template stars and the observed galaxy spectra via an additive constant and a multiplicative Legendre polynomial of degree 1. The LOSVD was largely constrained by the strong ${}^{12}$CO(2$-$0) and ${}^{12}$CO(3$-$1) bandheads, which were contained within the $2.26-2.39$$\mu$m fitting region. We masked the Ca i absorption line because our template library does not include cool dwarf stars (e.g., Krajnović et al., 2009), and further excluded a few artificial features, likely the result of imperfect sky subtraction or telluric correction. Example fits with pPXF to the observed galaxy spectra located at the nucleus, in an intermediate region, and in one of the outermost spatial bins are given in Figure 1. <figure><img src="content_image/1612.02015/x1.png"><figcaption>Figure 1.— Example fits with pPXF to the observed Mrk 1216 spectra located atthe nucleus (top), at an intermediate location (middle), and in one of theoutermost spatial bins (bottom). The red line shows the optimal stellartemplate convolved with the LOSVD, whose shape is further adjusted with anadditive constant and a first-degree multiplicative Legendre polynomial.Several wavelength regions shown in gray were masked during fit, including theCa I absorption line and artifacts from imperfect data reduction. The greendots are the model residuals that have been shifted upward by a constant,arbitrary amount.</figcaption></figure> After an initial fit to the galaxy spectrum in each spatial bin, we ran a Monte Carlo simulation with 100 iterations. During each realization, we generated a synthetic spectrum by taking the best-fit model and adding random Gaussian noise based upon the standard deviation of the model residuals. We re-fit the spectrum using pPXF with the penalization turned off. From the resulting distribution for each Gauss-Hermite moment, we took the mean to be the kinematic value and the standard deviation to be the 1$\sigma$ uncertainty. Finally, we point-symmetrize the kinematics using the method described in van den Bosch & de Zeeuw (2010), which also removes the systematic offsets in the odd Gauss-Hermite moments. The resulting radial velocity ($V$) map shows that Mrk 1216 is rotating, such that the southwest side of the galaxy is blueshifted and the northeast side is redshifted with values of $\pm$180 km s${}^{-1}$. The velocity dispersion ($\sigma$) rises from 230 km s${}^{-1}$ at a projected radius of $\sim$1″ to 425 km s${}^{-1}$ at the nucleus. The third Gauss-Hermite moment ($h_{3}$), or skewness, falls between $\pm$0.07, and we observe a $h_{3}-V$ anti-correlation, which is a common for rotating, axisymmetric systems (e.g., Fisher, 1997). The map of the fourth Gauss-Hermite moment ($h_{4}$), or the kurtosis, has a slight peak at the nucleus to a value of 0.08. The kinematics have median errors of 15 km s${}^{-1}$, 18 km s${}^{-1}$, 0.04, and 0.04 for $V$, $\sigma$, $h_{3}$, and $h_{4}$, respectively. Table 1 provides the extracted Gauss-Hermite moments for each NIFS spatial bin, and the 2D velocity fields are shown in Figure 2. x \| y \| V \| ΔV \| σ \| Δσ \| h3 \| Δh3 \| h4 \| Δh4 ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- (″) \| (″) \| (km s−1) \| (km s−1) \| (km s−1) \| (km s−1) \| (7) \| (8) \| (9) \| (10) -0.055 \| 0.003 \| -1.205 \| 12.708 \| 425.376 \| 17.927 \| -0.010 \| 0.024 \| 0.075 \| 0.031 0.012 \| -0.069 \| 54.401 \| 14.221 \| 424.049 \| 20.599 \| -0.023 \| 0.028 \| 0.082 \| 0.034 0.013 \| 0.079 \| -52.747 \| 14.410 \| 420.777 \| 20.519 \| 0.024 \| 0.028 \| 0.083 \| 0.033 0.087 \| -0.012 \| 17.161 \| 13.271 \| 415.750 \| 18.381 \| -0.000 \| 0.026 \| 0.072 \| 0.034 0.107 \| 0.092 \| -46.839 \| 13.620 \| 369.604 \| 17.681 \| 0.031 \| 0.027 \| 0.063 \| 0.033 Note. – The first two columns provide the x and y locations of the Voronoi bin generators, while the remaining columns present the point-symmetrized NIFS kinematics and their uncertainties. The position angle is 283.94∘, defined counterclockwise from the galaxy’s major axis to x. This table is available in its entirety in machine-readable form. Table 1NIFS Kinematics <figure><img src="content_image/1612.02015/x2.png"><figcaption>Figure 2.— The Mrk 1216 stellar kinematics from NIFS assisted by LGS AO. Themeasurements are shown on the scale given by the color bar to the right, andthe range of values is provided at the top of the V, σ, h3, and h4 maps. Mrk1216 exhibits regular rotation, a central rise in the stellar velocitydispersion, a h3−V anti-correlation, and a slight peak in h4 at the nucleus.The blueshifted velocities correspond to the southwest side of the galaxy.</figcaption></figure> We also examined the robustness of the stellar kinematics by changing how the measurements were made with pPXF. We modified the fitting region to be longer ($2.26-2.43$$\mu$m) and shorter ($2.26-2.37$$\mu$m), included the Ca I absorption line in the fit, required that the relative mix of template stars remain the same between spatial bins, and tested different combinations of degree $0-2$ multiplicative polynomials along with no additive component, an additive constant, and first-degree additive Legendre polynomial. We found the kinematics were consistent within 2$\sigma$ compared to our default fitting approach. We note that there were a number of bins whose kinematics were inconsistent at the 1$\sigma$ level, with the number of discrepant bins ranging from $0-15$ depending on the fitting method being adopted. In Section 5.2, we test the effect on $M_{\mathrm{BH}}$ if the kinematics from an alternative fitting approach in which a multiplicative degree 2 polynomial and no additive term is used instead. ### PSF Model We described the NIFS PSF as the sum of two concentric, circular Gaussians. Following past work (e.g., Krajnović et al., 2009; Seth et al., 2014), the MGE in Section 2 was convolved with the NIFS PSF and compared to the Mrk 1216 data cube, after collapsing along the wavelength axis. This resulted in best-fit values of 0$\farcs$07 and 0$\farcs$36 for the dispersions, and 0.56 and 0.44 for the weights of the inner and and outer Gaussian components, respectively. The method further allowed us to determine the center of the NIFS aperture. Attempts to fit a three-Gaussian PSF model produced a negligible component that contributed 1% to the total flux. Our characterization of the NIFS PSF is consistent with expectations of the Gemini ALTAIR system (e.g., Gebhardt et al., 2011; Onken et al., 2014; Drehmer et al., 2015). We also measured the PSF using an LGS AO NIFS observation of the galaxy’s tip-tilt star. With the 2D image decomposition package Galfit (Peng et al., 2010), we found that the sum of three circular Gaussians fit the collapsed NIFS data cube of the star well. The Gaussians have dispersions of 0$\farcs$07, 0$\farcs$14, and 0$\farcs$37 with weights of 0.39, 0.21, and 0.40. Due to the temporal variability of the AO correction, and because the observations of the star were conducted on-axis in contrast to the off-axis observations of the galaxy, we view this second PSF determination as a rough estimate. Nevertheless, we use this result to test how sensitive the inferred $M_{\mathrm{BH}}$ is to the assumed PSF in Section 5.1. ## 4. Large-Scale Spectroscopy The NIFS kinematics are complemented by large-scale spectroscopic observations that provide important constraints on the stellar mass-to-light ratio and the orbital distribution (e.g., Shapiro et al., 2006). The large-scale spectra were obtained with the Potsdam Multi Aperture Spectrograph (PMAS; Roth et al., 2005) in the Pmas fiber PAcK (PPAK; Verheijen et al., 2004; Kelz et al., 2006) mode from the 3.5 m telescope at Calar Alto Observatory, and from the Marcario Low-Resolution Spectrograph (LRS; Hill et al., 1998) on the Hobby-Eberly Telescope at McDonald Observatory. Yıldırım et al. (2015) and van den Bosch et al. (2015) presented the PPAK and HET data, but we provide a brief summary below. The PPAK integration time was 1.5 hours on-source, with two 900 s exposures taken at three dither positions to fully sample the 331 2$\farcs$7-wide science fibers. We acquired the data on 5 Dec 2011 with the medium resolution V1200 grating, covering $3650-4620$ Å with a spectral resolving power of $R$$\sim$1650 at 4000 Å. Data reduction followed the approach of the Calar Alto Legacy Integral Field Area Survey (Sánchez et al., 2012; Husemann et al., 2013). We extracted $V$, $\sigma$, $h_{3}$, and $h_{4}$ in 41 Voronoi spatial bins using pPXF, the Indo-U.S. Library of Coudé Feed Stellar Spectra (Valdes et al., 2004), and an additive Legendre polynomial of degree 15. As a final step, we point-symmetrized the stellar kinematics. The PSF was reconstructed by comparing the collapsed PPAK data cube to the Mrk 1216 MGE. The PSF has an inner Gaussian component with a dispersion of 1$\farcs$24 that contributes 77% to the total flux, while the second Gaussian component has a weight of 23% and a dispersion of 3$\farcs$72. Moreover, we have a single 900 s exposure of Mrk 1216 taken with the HET/LRS 2″-wide slit aligned with the galaxy major axis. The g2 grating and $2\times 2$ binning provided coverage of $4200-7400$ Å and an instrumental dispersion of 180 km s${}^{-1}$. After the initial data processing, we constructed 21 spatial bins and measured the stellar kinematics with pPXF and the MILES template library (Sánchez-Blázquez et al., 2006; Falcón-Barroso et al., 2011). Measurements of $V$ and $\sigma$ were made in each of the spatial bins, with $h_{3}$ and $h_{4}$ being extracted from the inner 10 bins. The HET data have slightly better spatial resolution than the PPAK observations, and the PSF is given by the sum of two Gaussians with dispersions of 1$\farcs$19 and 3$\farcs$39, each weighted by 0.55 and 0.45, respectively. The PPAK and HET kinematics extend out to $\sim$3 $R_{e}$ and show features that are very similar to those seen from NIFS. The large-scale kinematics reveal that the galaxy is rotating with redshifted velocities of $\sim$160 km s${}^{-1}$ to the northeast, a peak in the central velocity dispersion to $\sim$350 km s${}^{-1}$, and a clear anti-correlation between $h_{3}$ and $V$. The PPAK and HET kinematics are consistent with the NIFS kinematics over the radial extent they share in common, after accounting for differences in spatial resolution and binning. ## 5. Stellar-Dynamical Models In order to constrain the mass of the central black hole in Mrk 1216, we calculated three-integral, orbit-based dynamical models using the triaxial Schwarzschild code of van den Bosch et al. (2008). We ran the code in the axisymmetric limit, meaning that triaxial orbit families (e.g., box orbits) are included in the orbital libraries but we adopt a nearly oblate axisymmetric shape with an intermediate to long axis ratio of 0.99. The assumption of axisymmetry is justified by the galaxy’s rotation and the absence of isophotal and kinematic twists in the _HST_ image and the NIFS/PPAK data. We deprojected the MGE in Section 2 using an inclination angle of 70${}^{\circ}$. The same inclination was used by Yıldırım et al. (2015), and is mid-way between the range of angles for which the MGE can be deprojected given the apparent axis ratio of the flattest Gaussian component. Since Mrk 1216 does not contain a nuclear dust disk, we are unable to derive an independent estimate of the inclination angle, as has been possible for a few nearby galaxies (e.g., van den Bosch et al., 2012; Yıldırım et al., 102). The stellar potential is combined with the gravitational potential due to a black hole and a Navarro-Frenk-White (NFW; Navarro et al., 1996) dark matter halo. We created an orbit library that samples 32 equipotential shells with logarithmically spaced radii beginning at 0$\farcs$003, with 9 angular and 9 radial values at each energy. We ensure a smooth distribution function by bundling together 125 orbits with similar initial conditions. The 972,000 orbits were then integrated in the galaxy’s potential. We used a non-negative least squares solver to assign weights to the orbits such that the superposition matches the stellar kinematics, as well as the intrinsic and projected stellar masses to an accuracy of 1%, while accounting for PSF effects and aperture binning. The free parameters in the model are the black hole mass, the $H$-band stellar mass-to-light ratio ($\Upsilon_{H}$), the concentration ($c$) of the NFW halo, and the fraction of dark matter relative to the stellar mass ($f_{\mathrm{DM}}$). The stellar mass-to-light ratio is assumed to be constant with radius, which is supported by the lack of color gradient in _HST_ WFC3 F814W and F160W images (Yıldırım et al., 2015). We generated model grids that sampled 41 values of $M_{\mathrm{BH}}$ with $8.5\leq\log(M_{\mathrm{BH}}/M_{\odot})\leq 10.5$, 28 values of $\Upsilon_{H}$ between 0.3 and $3.0\ \Upsilon_{\odot}$, and 24 NFW halos with $c=5,10,15$ and $\log(f_{\mathrm{DM}})=0.5-4.0$. Models without a dark halo were run as well. While the results of our fiducial model grid presented in Section 5.1 were obtained by fitting to the NIFS$+$PPAK kinematics, in Section 5.2 we also test fitting NIFS-only and NIFS+HET kinematics. With four Gauss-Hermite moments in 108 NIFS$+$PPAK spatial bins, there are 432 observables. Ultimately, the best-fit model is the one with the lowest $\chi^{2}$ ($\chi^{2}_{\mathrm{min}}$), and the 1$\sigma$ statistical uncertainty for a given parameter is set by marginalizing over the other free parameters and searching for where the change in $\chi^{2}$ ($\Delta\chi^{2}\equiv\chi^{2}-\chi^{2}_{\mathrm{min}}$) is 1.0. The 3$\sigma$ model fitting error is taken to be where $\Delta\chi^{2}=9.0$. ### Modeling Results We present the results of our fiducial model grid in the left panel of Figure 3, and provide a comparison between the best-fit model and NIFS/PPAK kinematics in Figure 4. We find that $M_{\mathrm{BH}}=4.9\times 10^{9}\ M_{\odot}$, $\Upsilon_{H}=1.3\ \Upsilon_{\odot}$, $c=10$, and $\log(f_{\mathrm{DM}})=3.5$, which translates to a halo virial mass of $5\times 10^{14}\ M_{\odot}$. This model reproduces the observed kinematics well, and has a reduced $\chi^{2}$ of 0.6. The 1$\sigma$ statistical uncertainties on $M_{\mathrm{BH}}$ and $\Upsilon_{H}$ correspond to $(4.9^{+0.8}_{-0.7})\times 10^{9}\ M_{\odot}$ and $1.3\pm 0.1\ \Upsilon_{\odot}$, respectively, whereas the 3$\sigma$ statistical uncertainties translate to $M_{\mathrm{BH}}=(4.9^{+1.8}_{-1.9})\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.3\pm 0.3\ \Upsilon_{\odot}$. In contrast, the dark halo parameters are not well constrained. All three values of $c$ are allowed within 3$\sigma$ and $\log(f_{\mathrm{DM}})>2.5$. Models without a dark halo are clearly ruled out, as $\chi^{2}_{\mathrm{min}}=320$ for the models without a dark halo, which corresponds an increase of 65 relative to the best-fit model incorporating an NFW dark matter halo. <figure><img src="content_image/1612.02015/x3.png"><figcaption>Figure 3.— Contours of χ2 for various stellar-dynamical models (gray dots)with different combinations of black hole mass and H-band stellar mass-to-light ratio after marginalizing over the dark halo parameters. The red squareis the best-fit model, the red contour indicates where Δχ2=2.3, and thesubsequent black contours correspond to Δχ2=6.2 and 11.8, respectively. TheseΔχ2 values correspond to 1σ, 2σ, and 3σ confidence regions for two parameters.The results are shown for dynamical models fit to the combination of NIFS andPPAK data sets (left) and for models fit to only the NIFS kinematics (right).The two grids produce consistent results.</figcaption></figure> <figure><img src="content_image/1612.02015/x5.png"><figcaption>Figure 4.— The observed NIFS (left) and PPAK (right) kinematics, plotted as afunction of projected radial distance from the nucleus, are compared to thebest-fit stellar dynamical model (red) with MBH=4.9×109 M⊙ and ΥH=1.3 Υ⊙. Thedata are folded and multiple position angles are depicted. The best-fit modelreproduces the kinematic features well, and has a reduced χ2 of 0.6.</figcaption></figure> Moreover, we build into the $M_{\mathrm{BH}}$ and $\Upsilon_{H}$ error budgets the effect of adopting a different NIFS PSF model, using unsymmetrized kinematics, and assuming a different inclination angle. Due to the difficulties in measuring the AO PSF, it is important to test how other reasonable PSF characterizations might affect $M_{\mathrm{BH}}$. Our fiducial model above utilizes a two-Gaussian description determined by comparing the galaxy’s MGE to the Mrk 1216 collapsed data cube. If instead the NIFS PSF is taken to be the sum of three concentric, circular Gaussians measured from the NIFS observation of the galaxy’s tip-tilt star, we find $M_{\mathrm{BH}}=5.5\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.3\ \Upsilon_{\odot}$. A similar change occurs when the observed stellar kinematics are not forced to be point-symmetric. The fiducial model was fit to NIFS and PPAK kinematics that were averaged in a two-fold manner over the major and minor axes in order to reduce noise in the kinematic measurements. When only the systematic offsets in the odd Gauss-Hermite moments are subtracted off, but no other adjustments are made, we find that $M_{\mathrm{BH}}=5.8\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.1\ \Upsilon_{\odot}$. In addition to fitting to the point-symmetric NIFS$+$PPAK kinematics, the fiducial model was run for an inclination angle of 70${}^{\circ}$. Often Schwarzschild models are calculated for a single viewing orientation, as it is computationally expensive to sample over $M_{\mathrm{BH}}$, $\Upsilon_{H}$, two dark halo parameters, and the inclination (or three angles in the case of triaxiality). In the few cases where inclination was allowed to vary, the parameter was not well constrained by the 2D line-of-sight kinematics (Krajnović et al., 2005; van den Bosch & van de Ven, 2009; Walsh et al., 2012). Therefore, we determined the effect on $M_{\mathrm{BH}}$ and $\Upsilon_{H}$ if a near edge-on angle of $85^{\circ}$ is used instead. We find that $M_{\mathrm{BH}}$ decreases by 22% to $3.8\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}$ changes by 23% to $1.6\ \Upsilon_{\odot}$. By adding in quadrature the 1$\sigma$ formal model fitting uncertainty and the percent change in the best-fit values relative to the fiducial model above, we ultimately find that $M_{\mathrm{BH}}=(4.9\pm 1.7)\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.3\pm 0.4\ \Upsilon_{\odot}$ for Mrk 1216. These values are consistent with models fit to only the (bi-symmetrized) PPAK kinematics – Yıldırım et al. (2015) found a black hole mass upper-limit of $1.0\times 10^{10}\ M_{\odot}$ and $\Upsilon_{H}=1.8^{+0.5}_{-0.8}\ \Upsilon_{\odot}$, along with an NFW halo parameterized by $c=10$ and $\log(f_{\mathrm{DM}})=2.9^{+1.1}_{-2.2}$ (3$\sigma$ statistical uncertainties). In addition, our dynamical $H$-band mass-to-light ratio is in agreement with expectations from stellar population synthesis models for both a Kroupa ($1.2\ \Upsilon_{\odot}$) and a Salpeter ($1.7\ \Upsilon_{\odot}$) initial mass function, assuming solar metallicity and a $\sim$13 Gyr age (Vazdekis et al., 1996). Given the black hole mass of $4.9\times 10^{9}\ M_{\odot}$ and the bulge stellar velocity dispersion of 308 km s${}^{-1}$ (see Section 7), the NIFS data, with a central 0$\farcs$1 spatial bin, have easily resolved the 0$\farcs$49 black hole sphere of influence. ### Additional Models The PPAK data cube provides an increase in 2D spatial coverage, more spatial bins, smaller uncertainties on the extracted kinematics, and better spectral resolution than the HET long-slit spectroscopy. Thus, we use the PPAK kinematics in place of the HET kinematics when constructing the dynamical models. If instead Schwarzschild models are fit to the NIFS$+$HET kinematics, we recover the same results, with $M_{\mathrm{BH}}=(4.9^{+0.8}_{-0.7})\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.3\pm 0.1\ \Upsilon_{\odot}$ (1$\sigma$). Likewise, we don’t see significant changes in $M_{\mathrm{BH}}$ and $\Upsilon_{H}$ if models are constrained by only the NIFS kinematics. By fitting to the NIFS data alone, the black hole mass is less susceptible to systematic effects that commonly plague stellar-dynamical models, such as assumptions about the dark matter halo (e.g., Gebhardt & Thomas, 2009; Schulze & Gebhardt, 2011; Rusli et al., 2013) and the radial form of the stellar mass-to-light ratio (e.g., McConnell et al., 2013). This comes at the expense of larger $M_{\mathrm{BH}}$ statistical uncertainties due to the poor constraint on $\Upsilon_{H}$. Results of fitting orbit-based models to the NIFS kinematics alone are shown in the right panel of Figure 3, and we find that $M_{\mathrm{BH}}=(5.9^{+1.0}_{-1.7})\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.4^{+0.5}_{-0.2}\ \Upsilon_{\odot}$ (1$\sigma$). Finally, the NIFS kinematics were measured using an additive constant and a degree 1 multiplicative polynomial to account for differences in continuum shape between the velocity template library and the observed galaxy spectra. We consider these NIFS kinematics robust, as changes to how pPXF is run (see Section 3.1) produce similar kinematics. However, slight adjustments to the degree of the additive/multiplicative polynomials can cause inconsistent kinematics at the 1$\sigma$ level (all are consistent within 2$\sigma$). In particular, one of the largest differences is seen when running pPXF with a multiplicative degree 2 polynomial (with no additive component), which produces 11 spatial bins in which the kinematics differ by more than 1$\sigma$ relative to our adopted set of NIFS kinematics. If instead we use the NIFS kinematics extracted with pPXF and a multiplicative degree 2 polynomial, we infer $M_{\mathrm{BH}}=(3.5^{+0.4}_{-0.5})\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.3\pm 0.1\ \Upsilon_{\odot}$ (1$\sigma$). We note that this second set of NIFS kinematics show systematically smaller dispersions compared to the central PPAK kinematics, hence we perform this test as a sanity check but do not incorporate the results into our $M_{\mathrm{BH}}$ and $\Upsilon_{H}$ error budgets. The results from each of the three model grids above are in agreement with our final black hole mass and mass-to-light ratio measurements for Mrk 1216 of $M_{\mathrm{BH}}=(4.9\pm 1.7)\times 10^{9}\ M_{\odot}$ and $\Upsilon_{H}=1.3\pm 0.4\ \Upsilon_{\odot}$. ## 6. The Orbital Structure of Mrk 1216 Not only do the Schwarzschild models provide us with constraints on the black hole mass and the stellar mass-to-light ratio, but they also allow for an examination of the galaxy’s orbital structure. Figure 5 illustrates the amount of anisotropy and the orbit type as a function of radius. Using the best-fit model from Section 5.1, we plot the ratio $\sigma_{r}$/$\sigma_{t}$, where the tangential velocity dispersion is given by $\sigma_{t}^{2}=(\sigma_{\phi}^{2}+\sigma_{\theta}^{2})/2$ and $(r,\theta,\phi)$ are spherical coordinates. In order to gain an idea of the uncertainties in $\sigma_{r}$/$\sigma_{t}$, we show all the models within $\Delta\chi^{2}=1$ from our fiducial grid, and the best-fit models from the grid searches in which we assumed a different NIFS PSF, unsymmetrized kinematics, and a near edge-on inclination angle. We find that Mrk 1216 is roughly isotropic within the black hole sphere of influence, but becomes radially anisotropic with $\sigma_{r}$/$\sigma_{t}\sim 1.5$ at the radial extent of the PPAK kinematics. The galaxy was modeled using a triaxial Schwarzschild code (van den Bosch et al., 2008) that was run in the axisymmetric limit. Thus, the best-fit model includes contributions from box orbits, albeit a small one, making up 10% of the orbits near the nucleus and 20% at a radius of 10″. Instead, short-axis tube orbits dominate our best-fit model, contributing $\sim$$60-90$% at all radii. Long-axis tube orbits, which are important for prolate and triaxial systems, are negligible. <figure><img src="content_image/1612.02015/x7.png"><figcaption>Figure 5.— Mrk 1216’s orbital structure over the radial extent of the NIFS andPPAK kinematics. The anisotropy (top) and orbit type (bottom) are determinedfrom the best-fit Schwarzschild model (solid lines) and a rough estimate ofthe uncertainties is depicted by the shaded regions. The galaxy is essentiallyisotropic (gray dashed horizontal line) within the black hole sphere ofinfluence (gray dot-dashed vertical line), and becomes radially anisotropic atlarger distances from the nucleus. The orbits are composed mainly of short-axis tubes (red), although a small fraction of box orbits (blue) are alsopresent.</figcaption></figure> ## 7. The Black Hole – Host Galaxy Relations Placing Mrk 1216 on the black hole–host galaxy relations further requires identifying the bulge component, and currently there is a broad range of measurements in the literature. From Galfit models of the _HST_ F160W image, Yıldırım et al. (2015) find an upper limit on the $H$-band bulge luminosity and effective radius of $L_{H,\mathrm{bul}}=8.3\times 10^{10}\ L_{\odot}$ and $R_{\mathrm{e,bul}}=$ 3$\farcs$42, and a lower limit of $L_{H,\mathrm{bul}}=1.5\times 10^{10}\ L_{\odot}$ and $R_{\mathrm{e,bul}}=$1$\farcs$34. These numbers were derived from 2-component and 4-component 2D Sersic fits for the upper and lower limits, respectively, and correspond to a bulge-to-total ratio of $B/T=0.69$ and $0.13$. The former characterization includes a centrally concentrated component with a Sersic index of 3.61 and a projected axis ratio of 0.56. Since this double Sersic model produced pronounced residuals, Yıldırım et al. (2015) increased the number of Sersic functions to four to obtain a good fit to the galaxy’s complex structure. Each of the four components, however, have rather low Sersic indices between 0.99 and 1.61, which complicates a morphological classification. A dynamical decomposition using orbital weights from the Yıldırım et al. (2015) best-fit Schwarzschild model support the picture depicted by the 4-component photometric decomposition. In contrast, Savorgnan & Graham (2016) argue that Mrk 1216’s bulge component has an $H$-band luminosity of $\sim$$1.2\times 10^{11}\ L_{\odot}$ based on a one-dimensional (1D) multi-component Sersic fit to the _HST_ F160W surface brightness brightness profile. Their fit includes an intermediate-scale disk, in addition to a nuclear exponential disk and a spheroidal component. Both Yıldırım et al. (2015) and Savorgnan & Graham (2016) find that more than one Sersic component is required to match the surface brightness despite the elliptical galaxy classification in the NASA/IPAC Extragalactic Database and Hyperleda. For Mrk 1216, we conservatively adopt limits that extend from the total quantities down to the smallest bulge measurements in the literature, with values set to the result from a 2-component 2D Sersic fit (Yıldırım et al., 2015). The galaxy’s total luminosity was determined from the MGE model in Section 2, which also agrees with a single Sersic Galfit model of the _HST_ F160W image. After assuming $H-K=0.2$ and a $K$-band absolute solar magnitude of $3.28$(Binney & Merrifield, 1998), we establish a bulge luminosity for Mrk 1216 of $L_{K,\mathrm{bul}}=(9.6^{+4.4}_{-7.9})\times 10^{10}\ L_{\odot}$. For comparison, the upper bound on $L_{K,\mathrm{bul}}$ derived in this manner is 36% smaller than the growth curve analysis of Two Micron All Sky Survey (Skrutskie et al., 2006) images by van den Bosch (2016). We determine the bulge mass using $\Upsilon_{H}=1.3\ \Upsilon_{\odot}$ from the best-fit model in Section 5.1. Although a similar compact, high-dispersion galaxy, NGC 1277, showed evidence for a radially varying $V$-band stellar mass-to-light ratio (e.g., Martín-Navarro et al., 2015), there is no _HST_ WFC3 F814W$-$F160W color gradient observed for Mrk 1216 (Yıldırım et al., 2015) and a similar $\Upsilon_{H}$ from dynamical models fit to only the small-scale NIFS data in Section 5.2 and models fit to only the large-scale PPAK data (Yıldırım et al., 2015) are found. Hence, adopting $\Upsilon_{H}=1.3\ \Upsilon_{\odot}$ is justified and we find that $M_{\mathrm{bul}}=(1.1^{+0.5}_{-0.9})\times 10^{11}\ M_{\odot}$ for Mrk 1216. Finally, we calculate $\sigma_{\star}$ for bulge effective radii of $R_{\mathrm{e,bul}}=$1$\farcs$34, 3$\farcs$42, and 6$\farcs$34, which correspond to the measurements from a 4-component, a 2-component, and a 1-component Sersic Galfit model (Yıldırım et al., 2015). We use our best-fit stellar-dynamical model to predict the luminosity-weighted second moment within a circular aperture whose radius equals $R_{\mathrm{e,bul}}$, while also excluding the region within the black hole sphere of influence (e.g., Gebhardt et al., 2011; McConnell & Ma, 2013). Thus, we determine that $\sigma_{\star}=308^{+16}_{-6}$ km s${}^{-1}$ for Mrk 1216, which agrees well with the previous measurement of the PPAK velocity dispersion within a circular aperture that contains half of the light by Yıldırım et al. (2015). As can be seen in Figure 6, Mrk 1216 is an outlier from the $M_{\mathrm{BH}}-L_{K,\mathrm{bul}}$ and $M_{\mathrm{BH}}-M_{\mathrm{bul}}$ relations, but is consistent with $M_{\mathrm{BH}}-\sigma_{\star}$. Even when using the galaxy’s total luminosity or total stellar mass, Mrk 1216 falls a factor of $\sim$6 and $\sim$10 above the Kormendy & Ho (2013) and Läsker et al. (2014)$M_{\mathrm{BH}}-L_{K,\mathrm{bul}}$ correlations, as well as a factor of $\sim$$5-10$ above $M_{\mathrm{BH}}-M_{\mathrm{bul}}$ depending on whether the relation from McConnell & Ma (2013), Kormendy & Ho (2013), Savorgnan et al. (2016), Saglia et al. (2016) is assumed. Thus, using the total luminosity (stellar mass) makes Mrk 1216 a $2.2-2.5\sigma$ outlier ($1.4-3.0\sigma$ outlier) from the $M_{\mathrm{BH}}-L_{K,\mathrm{bul}}$ ($M_{\mathrm{BH}}-M_{\mathrm{bul}}$) relation given the various calibrations and scatter of the correlations. In Figure 7, we show predictions of a model with a $5.8\times 10^{8}\ M_{\odot}$ black hole, which is the mass expected from $M_{\mathrm{BH}}-M_{\mathrm{bul}}$(Saglia et al., 2016) for Mrk 1216’s total stellar mass of $1.6\times 10^{11}\ M_{\odot}$. These $\sigma$ and $h_{4}$ predictions are compared to the NIFS observations and the best-fit model from Section 5.1 with $M_{\mathrm{BH}}=4.9\times 10^{9}\ M_{\odot}$. Our best-fit model exhibits a similar central velocity dispersion peak and elevated central $h_{4}$ values as the observed kinematics, whereas the model based on $M_{\mathrm{BH}}-M_{\mathrm{bul}}$ cannot reproduce these features. For comparison, the two other compact, high-dispersion galaxies from the HET Massive Galaxy Survey are shown in Figure 6. We follow the same conventions for characterizing the NGC 1277 and NGC 1271 bulge quantities as was used for Mrk 1216. In particular, from the MGE descriptions of _HST_ images (van den Bosch et al., 2012; Walsh et al., 2015) and the best-fit stellar mass-to-light ratios from dynamical models fit to AO observations (Walsh et al., 2015, 2016), we measure total luminosities of $L_{V}=1.7\times 10^{10}\ L_{\odot}$ and $L_{H}=7.2\times 10^{10}\ L_{\odot}$, and total stellar masses of $1.6\times 10^{11}\ M_{\odot}$ and $1.0\times 10^{11}\ M_{\odot}$ for NGC 1277 and NGC 1271, respectively. These values are similar to the total luminosities and masses reported by Emsellem (2013) for NGC 1277 and Graham et al. (27) for NGC 1271. Graham et al. (26) calculate a larger total luminosity for NGC 1277 based on modeling the 1D light profile, however their MGE model suggests a smaller total luminosity than the one adopted here (their MGE model has 43% less light than their 1D component analysis). Despite using total properties, NGC 1277 and NGC 1271 remain outliers from $M_{\mathrm{BH}}-L_{K,\mathrm{bul}}$ by $2.1-2.8\sigma$(Kormendy & Ho, 2013; Läsker et al., 2014) and $M_{\mathrm{BH}}-M_{\mathrm{bul}}$ by $1.4-3.0\sigma$(McConnell & Ma, 2013; Kormendy & Ho, 2013; Savorgnan et al., 2016; Saglia et al., 2016). The two galaxies are in good agreement with the expectations from $M_{\mathrm{BH}}-\sigma_{\star}$(McConnell & Ma, 2013; Kormendy & Ho, 2013; Saglia et al., 2016; van den Bosch, 2016). van den Bosch (2016) conclude that $M_{\mathrm{BH}}-\sigma_{\star}$ is the best empirical relationship available and that a multi-variate scaling relation between $M_{\mathrm{BH}}$, $L_{K}$ and $R_{e}$ is a projection of $M_{\mathrm{BH}}-\sigma_{\star}$ with an equal amount of intrinsic scatter. Previous attempts to explore whether the inclusion of an additional parameter leads to tighter scaling relations (e.g., Beifiori et al., 2012; Saglia et al., 2016) have also generally found no significant decreases in the amount of intrinsic scatter compared to the single-parameter $M_{\mathrm{BH}}-\sigma_{\star}$ relation. Given the arguments of van den Bosch (2016), it is not surprising that these compact galaxies are consistent with $M_{\mathrm{BH}}-\sigma_{\star}$ but with their small sizes are outliers from $M_{\mathrm{BH}}-L_{\mathrm{bul}}$. The three HET compact galaxies are in the compilation of van den Bosch (2016). With this new black hole mass and luminosities derived from the _HST_ images, we find that Mrk 1216 and NGC 1271, are within $\sim$$1.7\sigma$ of the black hole mass – galaxy luminosity – galaxy size relation correlation, and that NGC 1277 is consistent with the relation, given the intrinsic scatter. Future work on multi-variate scaling relations requires overcoming a sampling bias in the known $M_{\mathrm{BH}}$ hosts; currently there is a very limited spread in effective radii at a given $K$-band galaxy luminosity (van den Bosch et al., 2015). <figure><img src="content_image/1612.02015/x8.png"><figcaption>Figure 6.— Location of Mrk 1216 (red square) on the black hole – host galaxyrelations. We show multiple calibrations of the correlations, but for clarityonly display the intrinsic scatter (gray) measured by Saglia et al. (2016) forthe black hole mass – stellar velocity dispersion (top) and the black holemass – bulge mass (bottom) relationships. The intrinsic scatter from bothKormendy & Ho (2013) and Läsker et al. (2014) are shown for the black holemass – K-band bulge luminosity (middle) relationship. NGC 1277 and NGC 1271(blue asterisks) are two compact galaxies similar to Mrk 1216 from the HETMassive Galaxy Survey with MBH measurements. Due to uncertainties in the bulgecomponents, we show limits that extend from the total quantities (upper boundof the horizontal solid line) to the smallest bulge estimates (lower bound ofthe horizontal dotted line) in the bottom two panels. Even when using thetotal luminosity/stellar mass, Mrk 1216, NGC 1277, and NGC 1271 are outliersfrom MBH−LK,bul and MBH−Mbul, yet are consistent with MBH−σ⋆.</figcaption></figure> <figure><img src="content_image/1612.02015/x11.png"><figcaption>Figure 7.— Comparison of the stellar kinematics measured from NIFS (left) tothe best-fit model with a 4.9×109 M⊙ black hole (middle) and a model with a5.8×108 M⊙ black hole (right), which is the mass expected from MBH−Mbul(Saglia et al., 2016) when conservatively using Mrk 1216’s total stellar massof 1.6×1011 M⊙. When generating models for the 5.8×108 M⊙ black hole, wesampled over a range stellar mass-to-light ratios and dark matter halos andpresent the model with the lowest χ2. The best-fit model is a good match tothe data, while the model with a smaller black hole cannot reproduce the sharprise in the velocity dispersion (top) or the elevated h4 values (bottom) atthe nucleus.</figcaption></figure> ## 8. Discussion Mrk 1216 hosts one of the largest black holes dynamically detected to date, naturally leading to the question of how such a massive black hole ended up in a relatively modest galaxy. One interesting explanation is that Mrk 1216 is a relic of the $z\sim 2$ quiescent galaxies, and avoided the same series of mergers that produced the typical massive early-type galaxies of today. Mrk 1216 perhaps reflects an earlier period when galaxies contained over-massive black holes for their bulge luminosities/masses, and galaxy growth had yet to follow. Such a scenario has been proposed to explain the locations of NGC 1277 and NGC 1271 on the black hole scaling relations based upon having small effective radii for the stellar masses, stellar mass surface density profiles comparable to the $z\sim 2$ red nuggets, and a uniformly old stellar population out to several $R_{e}$ (e.g., Trujillo et al. 2014; Ferré-Mateu et al. 2015; Martín-Navarro et al. 2015; Yıldırım et al. 2016b, in prep). Mrk 1216 is akin to NGC 1277 and NGC 1271, and is similar to the $z\sim 2$ red nuggets. The galaxy is a clear outlier from the local galaxy mass–size relation, and has an elevated central stellar mass surface density (Yıldırım et al. 2016b, in prep). However, there are hints that Mrk 1216 has experienced some growth and begun the process of becoming a normal early-type galaxy. Mrk 1216 has one of the largest effective radii ($R_{e}=2.8$ kpc) in our sample of compact galaxies found through the HET Massive Galaxy Survey, and the stellar mass surface density profile is more extended in the outer regions (Yıldırım et al. 2016b, in prep). With future work, it would be informative to study Mrk 1216’s stellar population over the extent of the galaxy. Another possible explanation for the location of Mrk 1216 on the $M_{\mathrm{BH}}-L_{\mathrm{bul}}$ and $M_{\mathrm{BH}}-M_{\mathrm{bul}}$ relations is that we simply do not have enough measurements at the upper-end of the correlations. With the limited number of objects in this high-mass regime, neither the form of the correlations nor the magnitude and distribution of the scatter are well determined (e.g., McConnell & Ma, 2013). Therefore, Mrk 1216 could fall in the tails of a distribution between black hole mass and galaxy properties that still needs to be fully flushed out. We note that many of the compact galaxies in the HET Massive Galaxy Survey have nuclear dust disks, indicating the presence of cleanly rotating gas (Ho et al., 2002; Alatalo et al., 2013), from which independent black hole mass measurements can be derived for comparison to the stellar-dynamical determinations. Such cross-checks between mass measurement methods is essential for establishing the amount of intrinsic scatter in the black hole correlations, and eventually for assessing how strongly Mrk 1216, NGC 1277, and NGC 1271 deviate from the relations. Presently, a majority of the meaningful comparisons between the stellar and gas-dynamical methods have led to discrepancies where the stellar-dynamical $M_{\mathrm{BH}}$ exceeds the gas-dynamical mass by factors of $2-3$(de Francesco et al., 2006; Rusli et al., 2011; Gebhardt et al., 2011; Walsh et al., 2012, 2013; Barth et al., 4), although there are a very small number of direct comparison studies. Finally, over-massive black holes can result from tidal stripping events. This idea was used to explain the low-mass galaxy M60-UCD1, which lies a mere 6.6 kpc away from the giant elliptical galaxy M60 (Seth et al., 2014). Indeed, the EAGLE cosmological, hydrodynamical simulation (Schaye et al., 2015; Crain et al., 2015) shows that tidal stripping is the dominant process responsible for the extreme outliers from the $M_{\mathrm{BH}}-M_{\star}$ correlation, but the simulation is most sensitive to galaxies with $M_{\mathrm{BH}}\sim 10^{8}\ M_{\odot}$ and $M_{\star}\sim 10^{10}\ M_{\odot}$. Due to the limited box size, predictions cannot be made for more massive NGC 1277-like galaxies (Barber et al., 2016). Contrary to NGC 1277 and NGC 1271, which are members of the Perseus cluster, Mrk 1216 is an isolated galaxy in the field, with only two other galaxies within 1 Mpc at its distance (Yıldırım et al., 2015). Combined with the regular isophotes and lack of tidal signatures in the _HST_ image, the idea that Mrk 1216 was once the center of a more massive galaxy seems unlikely. ## 9. Conclusion In summary, we measured the 2D stellar kinematics of the compact, high-dispersion galaxy Mrk 1216 using newly acquired AO NIFS observations that probe within the black hole sphere of influence. Mrk 1216 is rotating, has a distinct rise in the stellar velocity dispersion at the nucleus, exhibits the expected anti-correlation between $V$ and $h_{3}$, and has elevated central $h_{4}$ values. The high angular resolution NIFS kinematics, along with large-scale kinematic measurements and the luminous mass model from an _HST_ image, are fit with stellar-dynamical models based upon the Schwarzschild superposition method. We constrain the mass of the central black hole in Mrk 1216 to be $(4.9\pm 1.7)\times 10^{9}\ M_{\odot}$ and the $H$-band stellar mass-to-light ratio to be $1.3\pm 0.4\ \Upsilon_{\odot}$. The error budget incorporates some possible systematic effects and the formal 1$\sigma$ model fitting uncertainties. With $\sigma_{\star}=308$ km s${}^{-1}$, Mrk 1216 is consistent with the $M_{\mathrm{BH}}-\sigma_{\star}$ relationship, but is a surprising positive outlier from $M_{\mathrm{BH}}-L_{\mathrm{bul}}$ and $M_{\mathrm{BH}}-M_{\mathrm{bul}}$, even when conservatively using the galaxy’s total luminosity and stellar mass of $L_{K}=1.4\times 10^{11}\ L_{\odot}$ and $M_{\star}=1.6\times 10^{11}\ M_{\odot}$. Mrk 1216 is similar to NGC 1277 and NGC 1271 – the two compact, high-dispersion galaxies from the HET Massive Galaxy Survey that have prior stellar-dynamical $M_{\mathrm{BH}}$ measurements from AO observations. All three galaxies resemble the quiescent galaxies at $z\sim 2$ given their small but massive nature, their stellar mass surface density profiles, their strong rotation, and (in the cases of NGC 1277 and NGC 1271) their uniformly old stellar populations. Therefore, Mrk 1216, as well as NGC 1277 and NGC 1271, appear to be relics of the $z\sim 2$ red nuggets, and their black holes may imply that the normalization of the $M_{\mathrm{BH}}-L_{\mathrm{bul}}$ and $M_{\mathrm{BH}}-M_{\mathrm{bul}}$ relations were higher at earlier times. In other words, perhaps black hole growth precedes that of its host galaxy. Another possibility is that the galaxies are simply unusual and are in the tail of a distribution between $M_{\mathrm{BH}}$ and galaxy properties that still needs to be firmly established. Distinguishing between the two scenarios requires obtaining a more complete census of local black holes in a wide range of galaxies with diverse evolutionary histories. Based on observations obtained at the Gemini Observatory acquired through the Gemini Science Archive and processed using the Gemini IRAF package, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina), under program GN-2013A-Q-1. Also based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program #13050. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC). This work is further based on observations obtained with the Hobby-Eberly Telescope (HET). The HET is a joint project of the University of Texas at Austin, the Pennsylvania State University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen. The HET is named in honor of its principal benefactors, William P. Hobby and Robert E. Eberly. This material is based in part upon work J. L. W. conducted as an NSF Astronomy and Astrophysics Postdoctoral Fellow under Award No. 1102845. The authors further acknowledge the Texas Advanced Computing Center (TACC; http://www.tacc.utexas. edu) at the University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper. 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1201.6683	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 49590, "num_imgs": 1, "llama3_tokens_count": 17187 }	[ "content_image/1201.6683/x1.png" ]	# Homogenization of the boundary value for the Dirichlet Problem Ki-ahm Lee Seoul National University, Seoul, 151-747, Korea & Korea Institute for Advanced Study, Seoul,130-722, Korea kiahm@snu.ac.kr Henrik Shahgholian Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden henriksh@kth.se ###### Abstract. The Dirichlet problem with oscillating boundary data is the subject of study in this paper. It turns out that due to integral representation of such problems we can reduce the study to the case of surface integrals of rapidly oscillating functions, and their limit behavior: \[\lim_{\varepsilon\to 0}\int_{\Gamma}g\left(y,\frac{y}{\varepsilon}\right)d \sigma_{y},\] where $g(x,y)$, represents the boundary value in the Dirichlet problem. The lower dimensional character of the surface $\Gamma$ produces unexpected and surprising effective limits. In general, the limit of the integral depends strongly on the sequence $\varepsilon=\varepsilon_{j}$ chosen. Notwithstanding this, when the surface does not have flat portions with _rational directions_ a full averaging takes place and we obtain a unique effective limit in the above integral. The results here are connected to recent works of D. Gérard-Varet and N. Masmoudi, where they study this problem in combination with homogenization of the operator, in convex domains. H. Shahgholian has been supported in part by Swedish Research Council. K. Lee has been supported by Korea-Sweden Research Cooperation Program. This project is part of an STINT (Sweden)-NRF (Korea) research cooperation program. ###### Contents Contents * 1 Introduction * 1.1 Background * 1.2 Heuristics * 1.3 Plan of the paper * 2 Definitions and Notation * 2.1 Averaging and Ergodic theory * 3 Technical preliminaries * 4 Surface integrals of oscillating functions * 5 The case of Layered densities, almost periodicity, and ergodicity * 5.1 Layered Materials * 5.2 Almost Periodic Case * 5.3 Ergodic Case * 6 Applications to Partial Differential Equations * 6.1 Dirichlet problem: Elliptic Case * 6.2 Dirichlet problem: Parabolic Case * 6.3 Neumann problem * 7 Examples and illustrations * 7.1 Example 1 * 7.2 Example 2 ## 1. Introduction ### Background In this paper we consider Dirichlet and related problems, with rapidly oscillating data. Although we treat the case of Laplace equation, our analysis straightforwardly extends to general cases for equations that admit a Poisson/Green representation. Such integral representation, in turn, reduces the study of the problem to the corresponding integral equation, and hence the analysis of the integral of rapidly oscillating functions becomes central. To fix ideas, let $\Gamma$ be a $C^{1}$ surface in ${\mathbb{R}}^{n}$$(n\geq 2)$, not necessarily bounded. We assume $g(x,y)$ is integrable in both variables over $\Gamma$, and $1$-periodic in $y$-variable i.e., $g(x,y+k)=g(x,y)$ for $k\in{\mathbb{Z}}^{n}$. In this paper, we shall study possible limit behaviors of the integral (1) \[\lim_{\varepsilon\to 0}\int_{\Gamma}g\left(y,\frac{y}{\varepsilon}\right)d \sigma_{y},\] and we shall prove that under mild conditions on the surface $\Gamma$ there is an effective limit as $\varepsilon$ tends to zero. In general we show that the above integrals stay within the bounds of the interval $\left[\int_{\Gamma}\overline{g}_{}(y,\nu_{y}),\int_{\Gamma}\overline{g}^{}(y ,\nu_{y})\right]$ where $\overline{g}_{},\overline{g}^{}$ are defined below, as the infimum respectively supremum of the average-integrals of $g(y,\cdot)$ over closed loops of the plane $\{x:x\cdot\nu_{y}=0\}$ on the torus; here $\nu_{y}$ denotes the normal vector of $\Gamma$ at $y$. To illuminate the application of this to the Dirichlet problem, let us consider a bounded $C^{1}$-domain $\Omega\subset{\mathbb{R}}^{n}$$(n\geq 2)$. Let further $g(x,y)$ and $f(x,y)$ be continuous in both variables, and periodic in $y$. Let $u_{\varepsilon}$ be a solution to the Dirichlet problem ( \[P_{\varepsilon}\] ) \[\triangle u_{\varepsilon}=-\mu_{\varepsilon}\quad\text{ in }\Omega,\qquad u_{ \epsilon}=g(x,x/\varepsilon)\quad\text{ on }\partial\Omega,\] where $\mu_{\varepsilon}=f(x,x/\varepsilon)\chi_{\Gamma_{0}}d\sigma_{x}$, and $\Gamma_{0}$ is a $C^{1}$ surface compactly in $\Omega$. Since solutions to this problem can be represented by surface integrals of $g$ and $f$ (through Poisson and Green functions), we may directly apply our results. Hence, a by-product of our surface integral homogenization, is nontrivial limit scenarios, as $\varepsilon$ tends to zero, of equation ($P_{\varepsilon}$) and its solution. It is noteworthy that, through integral equations of Fredholm type, or just standard functional minimization, our technique applies to homogenizations with oscillating Neumann data (see Section 6.3). For fully nonlinear operator the Neumann problem has been treated in [4], see also [3]. Another issue that occurs along such analysis is the study of the speed of convergence. In problems, where governing partial differential equations have oscillating coefficients, one uses the standard method of expansion \[u_{\epsilon}(x)=u_{0}+\epsilon u_{1}(x,x/\epsilon)+\epsilon u_{bl,\epsilon}+O( \epsilon),\] in an appropriate space. Here $u_{bl,\epsilon}$ is the so-called boundary layer term, and finding it is part of the problem. The function $u_{bl,\epsilon}$ will then solve a Dirichlet problem with oscillating boundary data. We refer to [1] for some backgrounds and to [5] for recent developments in the topic. Remark 1.1.: _It is noteworthy that our approach applies to general equations of the type_ \[\hbox{div}(A(x/\varepsilon)\nabla u)=f(x/\varepsilon)\] _with oscillating Dirichlet/Neumann data. This, even though straightforward, becomes quite technical and is therefore outside the scope of this paper. Thus for clarity of the exposition we shall treat the Laplacian case, only._ ### Heuristics Let us set $x=y\text{(mod $1$)}$ if $x\in Q^{+}_{1}=\{0\leq x_{i}<1\}$, and $y-x\in{\mathbb{Z}}^{n}$, and adopt the standard notation from homogenization, as well as that of ergodic theory. For clarity of the ideas, let us also deal with problem ($P_{\varepsilon}$) in the case $f\equiv 0$, and see how it reduces to the integral averaging, and then how the limit integral is obtained. Since harmonic functions can be represented using Poisson kernel $P(x,y)$, \[u_{\epsilon}(x)=\int_{\partial\Omega}P(x,y)g(y,y/\epsilon)\,d\sigma_{y},\] we must then analyze the behavior of this integrals as $\varepsilon$ tends to zero. For $x\not\in\partial\Omega$, $P(x,y)$ is continuous and we can rewrite the integral (2) \[u_{\epsilon}(x)\approx\sum_{j}P(x,y^{j})\|\partial\Omega\cap Q_{r_{j}}(y^{j})\| \fint_{\partial\Omega\cap Q_{r_{j}}(y^{j})}g(y^{j},y/\epsilon)\,d\sigma_{y},\] where $r_{j}$ is small enough, independent of $\varepsilon$, and $Q_{r_{j}}(y^{j})$ is a cube with size $r_{j}$ and center $y^{j}$. This obviously brings us to integral (1). Observe that at this stage we cannot replace the second variable of $g$ with $y^{j}$, due to rapid oscillation for $\epsilon\approx 0$. Further assume that $\partial\Omega\cap Q_{r_{j}}(y^{j})$ is so flat that at $\epsilon$-scale it is approximately $\epsilon^{2}$-away from its tangent plane; this requires a $C^{2}$-graph locally, but our techniques/proofs work for $C^{1}$-surfaces. The idea now is to cover the boundary $\partial\Omega\cap Q_{r_{j}}(y^{j})$ with finite many, and small enough cubes, so that each part of the surfaces is as flat as we want to. In particular we have \[\fint_{\partial\Omega\cap Q_{r_{j}}(y^{j})}g(y^{j},y/\epsilon)\,d\sigma_{y}= \sum_{i}\fint_{\partial\Omega\cap Q_{\varepsilon_{i}r_{j}}(y^{i,j})}g(y^{j},y/ \epsilon)\,d\sigma_{y},\] where $\varepsilon_{i}$ is to be chosen small enough. This part is slightly more delicate, and needs extra care. We would also prefer to take $\varepsilon_{i}=\sqrt{\varepsilon}$. The trouble is that this might not work, as the points are changing, and we may loose lots of information about the behavior of the surface $\partial\Omega\cap Q_{\varepsilon_{i}r_{j}}(y^{i,j})$. Indeed, an scaling of the integral gives \[\fint_{\partial\Omega\cap Q_{(\varepsilon_{i}/\varepsilon)r_{j}}(y^{i,j})}g(y^ {j},y)\,d\sigma_{y},\] and the question is whether this integral will converge to the mean $\fint_{Q_{1}^{+}}g(y^{j},y)dy$. This would happen exactly when the normal vector of the surface $\partial\Omega$ at $y^{i,j}$ is irrational. In other words the surface $\partial\Omega\cap Q_{(\varepsilon_{i}/\varepsilon)r_{j}}(y^{i,j})$ (which is almost a plane) will foliate the $n$-dimensional torus, as $\varepsilon$ tends to zero, provided we have chosen $\varepsilon_{i}/\varepsilon\to\infty$, and the normal vector of the surface $\partial\Omega$ at $y^{i,j}$ is irrational. In particular (3) \[\lim_{\varepsilon\to 0}\fint_{\partial\Omega\cap Q_{(\varepsilon_{i}/ \varepsilon)r_{j}}(y^{i,j})}g(y^{j},y)\,d\sigma_{y}=\overline{g}(y^{j}).\] If the set of points on $\partial\Omega$ with rational direction has zero surface measure (this may fail if there are so-called flat spots with rational directions) then we could cover the boundary with small cubes centered at points with irrational normals. From here combining (2)-(3) we obtain \[\lim_{\varepsilon\to 0}u_{\varepsilon}(x)=\int_{\partial\Omega}P(x,y)\overline {g}(y)\,d\sigma_{y}.\] It is apparent that if there is a flat portion of the boundary with rational normal, then a full foliation cannot take place at such portions. Consequently, the resulting (mod 1) surface will be a close simple curve over the $n$-dimensional torus. Hence different sequence of $\epsilon$ may give different shifts of this close loop, and hence the possibility of a parameter family of values, in between $[\overline{g}_{}(y,\nu_{y}),\overline{g}^{}(y,\nu_{y})]$. Remark 1.2.: _A word of caution: As $\varepsilon$ tends to zero, one may obviously rescale the integral, by a change of variables, as we did in our heuristic explanations above. This scaling makes the surface to be scaled (we assume the origin is NOT on the surface) so as to disappear in the limit. This naturally would make it impossible to compute the limit integral. However, the periodicity of the function $g$ in its second variable, implies that we can bring back the surface so as it passes through the fundamental cube $Q_{1}^{+}$ using (mod1) argument. In particular this means that despite the integration is on a fixed surface, the surface itself will start jumping forth and back due to the variable $\varepsilon$._ _It would be a good idea for the reader to consider simple examples such as integration over a line segment in the plane, by varying the normal direction of the plan from rational to irrational._ ### Plan of the paper In the next section we shall introduce all definitions and notations. We take care of some technicalities in Section 3. We shall formulate our main results concerning surface integration in Section 4, and its generalization to other type of functions, such as layered-densities, almost periodic functions will appear in Section 5. Several interesting applications to PDE are mentioned in Section 6, and several Examples are also given in Section 7. ## 2. Definitions and Notation ### Averaging and Ergodic theory Definition 2.1.: _Let $\nu$ be a vector in ${\mathbb{R}}^{n}$, $z$ a fixed given point, and $Q_{r}(z)=\{x:\,\|x_{i}-z_{i}\|<r\}$. Let $g(x,y)$, be integrable in both variables over the plain $\{(y-z)\cdot\nu=0\}$, and periodic in $y$-variable. We define_ \[\overline{g}^{}(z,\nu):=\varlimsup_{\varepsilon\to 0}\varlimsup_{r \rightarrow\infty}\fint_{\{(y-z)\cdot\nu=0\}{\cap Q_{r}(z)}}g\left(z,\frac{y}{ \varepsilon}\right)d\sigma_{y},\] _and_ \[\overline{g}_{}(z,\nu):=\varliminf_{\varepsilon\to 0}\varliminf_{r \rightarrow\infty}\fint_{\{(y-z)\cdot\nu=0\}\cap Q_{r}(z)}g\left(z,\frac{y}{ \varepsilon}\right)d\sigma_{y}.\] _Later we shall consider cases where $\nu=\nu_{z}$, is the normal vector at $z$ on a given surface $\Gamma$. We also define the average of $g$ as_ \[\overline{g}(z)=\fint_{Q^{+}_{1}}g\left(z,y\right)\ dy,\] _for $Q^{+}_{1}=\{x:\,0\leq x_{i}\leq 1\}$._ Henceforth we shall assume all vectors have length one, unless otherwise stated. We shall also without loss of generality assume that the surface is orientable, and fix a consistent choice of normal in clockwise direction (this choice is obvious if $\Gamma$ is the boundary of a domain). It should be noted that if the set $\{(y-z)\cdot\nu=0\}(mod~{}1)$ foliates the cube $Q^{+}_{1}$ then the integral \[\lim_{r\rightarrow\infty}\fint_{\{(y-z)\cdot\nu=0\}{\cap Q_{r}(z)}}g\left(z, \frac{y}{\varepsilon}\right)d\sigma_{y}\] converges to the average $\overline{g}(z)$, and this happens exactly when $\nu$ is irrational direction (see Lemma 3.2). When the set $\{(y-z)\cdot\nu=0\}(mod~{}1)$ does not foliate the cube $Q^{+}_{1}$ (or the $n$-dimensional torus) then we shall get the limit as an integral over a closed loop flow on the torus. In particular the value of the integral exists, and depends on $\nu$ and $\epsilon$. For different values of $\epsilon$ this loop translates over the torus and will give rise to supremum respectively infimum values of the integrals, as defined above by $\overline{g}^{},\overline{g}_{}$, respectively. Remark 2.1.: _It should be remarked that in the definitions of the above averages $\overline{g}^{},\overline{g}_{}$ we could have replaced_ \[\fint_{\{(y-z)\cdot\nu=0\}\cap Q_{r}(z)}g\left(z,\frac{y}{\varepsilon}\right)d \sigma_{y},\] _by_ \[\fint_{\{(y-z)\cdot\nu=0\}\cap D}g\left(z,\frac{y}{\varepsilon}\right)d\sigma_ {y},\] _for any domain $D$ containing the point $z$. The reason for this is that the set $T=\{(y-z)\cdot\nu=0\}(mod~{}1)$ either foliates the whole $n$-dimensional torus, or it is a closed simple curve on the torus. In either case, when $\epsilon$ tends to zero, the piece of plane $\{(y-z)\cdot\nu=0\}\cap D$ will have the same effect as $T$._ Since the direction of the normal of the plane $\{(y-z)\cdot\nu=0\}$ will play a crucial role in our analysis, we introduce proper definitions well-known in Ergodic theory. For readers’ convenience we also prove some of these well-known results here. Definition 2.2.: _(IDDC)_ 1. _The vector_ $\nu\in{\mathbb{S}}^{n-1}$ _is a rational direction if there is_ $m\in{\mathbb{Z}}^{n}$ _such that_ $\nu=\frac{m}{\|m\|}$_._ 2. _The vector_ $\nu\in{\mathbb{S}}^{n-1}$ _is an irrational direction if_ $\nu$ _is not a rational direction._ 3. _A smooth surface_ $\Gamma$ _satisfies Irrational Direction Dense Condition (IDDC) if_ \[\sigma_{\Gamma}\left(\Gamma\backslash\{x\in\Gamma:\text{$\nu_{x}$ is a irrational direction. }\}\right)=0\] _for the surface measure_ $\sigma_{\Gamma}$_. Here_ $\nu_{x}$ _denotes the normal to_ $\Gamma$ _at_ $x$_._ ## 3. Technical preliminaries In this section we shall recall some standard facts from Ergodic theory, and also state and prove some averaging results that will be needed for the proof of the main theorems. The first lemma is a version of Weyl’s Lemma and is well-known fact about uniform distribution. Lemma 3.1 (Weyl’s Lemma).: _Let $\nu=(\nu_{1},\cdots,\nu_{d})\in{\mathbb{R}}^{d}$ ($d\geq 1$), be such that $(1,\nu_{1},\cdots,\nu_{d})$ is an irrational direction. Then, for $h\in L^{2}(0,1)$, and $a_{k}=\sum_{i=1}^{d}k_{i}\nu_{i}(mod~{}1)$ we have_ \[\frac{1}{\|I_{N}\|}\sum_{k\in I_{N}}h(a_{k})\rightarrow\int_{0}^{1}h(t)dt,\qquad k =(k_{1},\cdots,k_{d})\] _as $N\to\infty$. Here $I_{N}=\{k\in{\mathbb{Z}}^{d}:\,\|k_{i}\|\leq N,i=1,\cdots,d\}$._ That $(1,\nu_{1},\cdots,\nu_{d})$ is an irrational direction, means that at least one of the numbers $\nu_{j}$ is irrational. In this paper $d=n-1$ is the only case that is used. Proof.: Set \[\mu_{\nu,N}=\frac{1}{\|I_{N}\|}\sum_{k\in I_{N}}\delta_{k\cdot\nu}\] to be the sum of Dirac delta functions. First, let us consider the case $h=h_{m}=e^{2\mathbf{i}\pi mt}$ and $m\neq 0$. By the assumption there is $\nu_{l}$, which is an irrational number, and hence (4) \[\left\|\mu_{\nu,N}(h_{m})\right\|=\frac{1}{\|I_{N}\|}\Pi_{j=1}^{d}\left\|\sum_{k_{j }=-N}^{N}e^{2\mathbf{i}\pi mk_{j}\nu_{j}}\right\|\leq\frac{C}{2N\left\|1-e^{2 \mathbf{i}\pi m\nu_{l}}\right\|}\quad\rightarrow_{N\to\infty}0,\] since $m\nu_{l}$ cannot be an integer i.e. $1-e^{2\mathbf{i}\pi m\nu_{l}}\neq 0$. Now by Fourier expansion (extending $h$ as 1-periodic function), for some $b_{m}\in{\mathbb{R}}$, \[h(s)=\int_{0}^{1}h(t)dt+\sum_{m=1}b_{m}e^{2\mathbf{i}\pi ms},\] which in combination with (4) results in \[\mu_{\nu,N}(h)=\int_{0}^{1}h(t)dt+\sum_{m=1}^{\infty}b_{m}\mu_{\nu,N}(e^{ \mathbf{i}2\pi ms})\rightarrow\int_{0}^{1}h(t)dt.\] ∎ Lemma 3.2.: _Let $g$, $\overline{g}^{}$, $\overline{g}_{}$, and $\overline{g}$ be as in Definition 2.1. Then the following hold:_ 1. _If_ $\nu_{x}$ _is an irrational direction, then_ \[\overline{g}^{}(x,\nu_{x})=\overline{g}_{}(x,\nu_{x})=\overline{g}(x).\] 2. _The following inequalities always hold_ \[\overline{g}^{}(x,\nu_{x})\geq\overline{g}(x)\geq\overline{g}_{}(x,\nu_{x}).\] Proof.: First note that we can replace the integral in the definition of $\overline{g}_{}$, and $\overline{g}^{}$ by \[\fint_{\{(y-x)\cdot\nu=0\}\cap Q_{N}}g\left(x,\frac{y}{\varepsilon}\right)d \sigma_{y},\] with $N$ positive integers, and $N\to\infty$. This simplifies the matter slightly. Set $\Pi=\Pi(\nu,x)=\{(y-x)\cdot\nu=0\}$. We may without loss of generality assume $\nu=(1,\nu^{\prime})$. Since $\nu$ is irrational direction, $\Pi(mod~{}1)$ will foliate the unit cell, according to standard foliation theory and results in integral average over $Q_{1}^{+}$. To see this we assume (by using periodicity) that $x\in Q_{1}^{+}$ and that the plane $\Pi$ cuts the $x_{1}$-axis (or some other axis). let $a_{0}$ be the point of intersection between the $x_{1}$-axis and this plane, so that $\Pi(\nu,x)=\{x:\,x_{1}=a_{0}-\nu^{\prime}\cdot x^{\prime}\}$, and $(1,\nu^{\prime})$ is irrational (by the assumption), and $0\leq a_{0}<1$ due to periodicity of $g(x,y)$ in $y$. Having fixed $\nu$, we shall now use the notation $\Pi_{t}=\Pi(\nu,te_{1})$, the plane with normal $\nu$ through the point $(t,0^{\prime})$. For $k^{\prime}\in{\mathbb{Z}}^{n-1}$ define $S_{k^{\prime}}$ to be the cylinder generated by the $x_{1}$-axis and the $(n-1)$-dimensional cube $k^{\prime}+\{y:\,0<y_{i}<1,\,i=2,\cdots,n\}$ (these are cubes in ${\mathbb{R}}^{n-1}$ translated by the vector $k^{\prime}$). Set further $a_{k^{\prime}}=a_{0}-k^{\prime}\cdot\nu^{\prime}(mod~{}1)$ ($k^{\prime}=(k_{2},\cdots,k_{n})$), then $\tilde{\Pi}_{a_{k^{\prime}}}=(\Pi\cap S_{k^{\prime}})~{}(mod~{}1)$. Let $N>0$ be a large number, and $I^{\prime}_{N}=\{k^{\prime}\in{\mathbb{Z}}^{n-1}:\ \|k_{i}\|\leq N\}$. Then by periodicity of $g(x,\cdot)$ \[\int_{Q_{N}(x)\cap\Pi}g\left(x,y\right)d\sigma_{y}=\sum_{k^{\prime}\in I^{ \prime}_{N}}\int_{\tilde{\Pi}_{a_{k^{\prime}}}}g\left(x,y\right)d\sigma_{y}.\] Set now $w(a_{k^{\prime}})=w(x,a_{k^{\prime}}):=\int_{\tilde{\Pi}_{a_{k^{\prime}}}}g \left(x,y\right)d\sigma_{y}$. Then \[\fint_{Q_{N}\cap\Pi}g\left(x,y\right)d\sigma_{y}=\frac{1}{\|\nu\|}\frac{1}{\|I_{N }^{\prime}\|}\sum_{k^{\prime}\in I_{N}^{\prime}}w(a_{k^{\prime}}),\] where we have used $\|Q_{N}\cap\Pi\|=(2N)^{n-1}\|\nu\|=\|I^{\prime}_{N}\|\|\nu\|$. Since $\nu$ is irrational, from Lemma 3.1 we conclude (5) \[\lim_{N\to\infty}\fint_{Q_{N}\cap\Pi}g\left(x,y\right)d\sigma_{y}=\frac{1}{\| \nu\|}\int_{0}^{1}w(t)dt=\frac{1}{\|\nu\|}\int_{0}^{1}\int_{\tilde{\Pi}_{t}}g(x,y )d\sigma_{y}dt=\overline{g}(x),\] where $\tilde{\Pi}_{t}=\Pi_{t}\cap S_{0^{\prime}}~{}(mod~{}1)$. Therefore $\lim_{r\rightarrow\infty}\fint_{\{(y-x)\cdot\nu=0\}\cap Q_{r}}g\left(x,\frac{y }{\varepsilon}\right)d\sigma_{y}=\overline{g}(x)$ which is independent of $\varepsilon$. Now we have conclusion (i) from the definition of $\overline{g}^{}(x,\nu)$ and $\overline{g}_{}(x,\nu)$. To prove (ii) let us suppose that $\nu$ is a rational direction, otherwise the conclusion follows by the equality in (i). Since $\nu$ is rational, it is not hard to see that the restriction of $g(x,y)$ on this hyperplane will be periodic with period $T$, say; for $g(x,y/\varepsilon)$ the period will be $\varepsilon T$. In particular the integral can be seen as integration over a spiral-like plane on the torus (a closed loop). The two dimensional case can be illustrated by a curve on the torus that loops over itself. In particular the limit integral (w.r.t. $r$) in the definition of $\overline{g}_{},\overline{g}^{}$ can be replaced by \[\overline{g}_{}(x,\nu_{x})=\varliminf_{\varepsilon}\fint_{\{(y-{x})\cdot\nu=0 \}\cap Q_{\varepsilon T}}g\left(x,\frac{y}{\varepsilon}\right)d\sigma_{y}= \varliminf_{\varepsilon}\fint_{\{(y-{x_{\varepsilon}})\cdot\nu=0\}\cap Q_{T}}g \left(x,y\right)d\sigma_{y},\] \[\overline{g}^{}(x,\nu_{x})=\varlimsup_{\varepsilon}\fint_{\{(y-{x})\cdot\nu=0 \}\cap Q_{\varepsilon T}}g\left(x,\frac{y}{\varepsilon}\right)d\sigma_{y}= \varlimsup_{\varepsilon}\fint_{\{(y-{x_{\varepsilon}})\cdot\nu=0\}\cap Q_{T}}g \left(x,y\right)d\sigma_{y}.\] An observation here is that the supremum value of the mean integral w.r.t. $\epsilon$ actually is taken for some value $\epsilon_{0}$, and naturally for many other values, due to periodicity in $\epsilon$. One can think of the situation as parallel planes in ${\mathbb{R}}^{n}$ with fixed distance $\epsilon_{1}$ from each other are moving, simultaneously by keeping a fixed distance between them, in the orthogonal direction of the plane ($mod~{}1$), when $\epsilon$ changes. Obviously, these planes foliate $Q^{+}_{1}$ once $\epsilon$ ranges $[0,1)$. From here one deduces \[\fint_{Q_{1}^{+}}g(x,y)dy=\fint_{0}^{\epsilon_{1}}d\epsilon\fint_{\Pi(\nu,x_{ \epsilon})(mod~{}1)}g(x,y)d\sigma_{y}\] \[\leq\fint_{\Pi(\nu,x_{\epsilon_{0}})(mod~{}1)}g(x,y)d\sigma_{y}\leq\overline{g }^{}(x,\nu).\] Similarly \[\fint_{Q_{1}^{+}}g(x,y)dy\geq\overline{g}_{}(x,\nu).\] ∎ In our analysis of the limit behavior of the integral (1) we will use Definition 2.1 in a slightly different way. Next lemma will give us a hint in that direction. Lemma 3.3.: _Let $g$ be as before, $\Pi(\nu,z)=\{(y-z)\cdot\nu=0\}$, and $z\in\Gamma$. Then for any $R_{\varepsilon}\nearrow\infty$ (as $\varepsilon\searrow 0$) we have_ \[\limsup_{\varepsilon\to 0}\fint_{\Pi(\nu,z)\cap Q_{\varepsilon R_{ \varepsilon}}(z)}g(z,y/\varepsilon)d\sigma_{y}\leq\overline{g}^{}(z,\nu),\] _and_ \[\liminf_{\varepsilon\to 0}\fint_{\Pi(\nu,z)\cap Q_{\varepsilon R_{ \varepsilon}}(z)}g(z,y/\varepsilon)d\sigma_{y}\geq\overline{g}_{}(z,\nu).\] _In case $\nu$ is irrational we obtain_ \[\lim_{\varepsilon\to 0}\fint_{\Pi(\nu,z)\cap Q_{\varepsilon R_{ \varepsilon}}(z)}g(z,y/\varepsilon)d\sigma_{y}=\overline{g}(z).\] We leave the the reader to verify this simple fact. We shall next make the previous lemma even more general by letting the plane be replaced by very smooth surfaces. Let $\Gamma$ be a smooth surface, with module of continuity $\tau=\tau_{\Gamma}$ for its $C^{1}$ norm. Define (6) \[M_{\epsilon}=\sqrt{\min(\frac{1}{\tau(\varepsilon)},\frac{1}{\varepsilon})} \nearrow\infty,\qquad\hbox{as }\epsilon\searrow 0.\] Then (7) \[M_{\epsilon}\tau(\epsilon M_{\epsilon})\leq\sqrt{\tau(\sqrt{\epsilon})}\to 0, \qquad\hbox{as }\epsilon\to 0.\] Lemma 3.4.: _Let $g$ be as before, and $\Gamma$ a smooth $C^{1}$ surface, with module of continuity $\tau=\tau_{\Gamma}$ for its $C^{1}$ norm. Let further $\rho_{\varepsilon}=\epsilon M_{\epsilon}$ where $M_{\epsilon}$ is as in (6). Then, for $z\in\Gamma$, and $\eta>0$ there exists $\varepsilon_{\nu_{z},\eta}$ such that for all $\varepsilon\leq\varepsilon_{\nu_{z},\eta}$ we have_ \[\overline{g}_{}(z,\nu_{z})-\eta\leq\fint_{\Gamma\cap Q_{\rho_{\epsilon}}(z)}g \left(z,\frac{y}{\varepsilon}\right)d\sigma_{y}\leq\overline{g}^{}(z,\nu_{z}) +\eta.\] Proof.: Set $z_{\varepsilon}=\frac{z}{\varepsilon}\,(mod~{}1)$, and $\Gamma_{z}^{\epsilon}:=\{x:\epsilon(x+z)\in\Gamma\}$. Then we have \[\fint_{\Gamma\cap Q_{\rho_{\epsilon}}(z)}g\left(z,\frac{y}{\varepsilon}\right) d\sigma_{y}=\fint_{\Gamma^{\varepsilon}\cap Q_{M_{\epsilon}}(0)}g\left(z,z_{ \varepsilon}+y\right)d\sigma_{y},\] where \[\Gamma_{z}^{\varepsilon}\cap Q_{M_{\varepsilon}}(0)\subset\left\{x\in Q_{M_{ \varepsilon}}(0):\|x\cdot\nu_{z}\|<\frac{\rho_{\epsilon}\tau(\rho_{\varepsilon}) }{\varepsilon}=M_{\varepsilon}\tau(\varepsilon M_{\varepsilon})\right\}.\] If we set $\Pi_{\nu_{z}}=\{x:\,x\cdot\nu_{z}=0\}$, then by continuity of $g(z,.)$, and that for $\varepsilon$ small enough the surface $\Gamma_{z}^{\varepsilon}$ is $M_{\varepsilon}\tau(\varepsilon M_{\varepsilon})(\approx 0)$ close to $\Pi_{\nu_{z}}$ in $B_{M_{\varepsilon}}(0)$ we will have \[\fint_{\Gamma^{\varepsilon}_{z}\cap Q_{M_{\epsilon}}(0)}g\left(z,z_{ \varepsilon}+y\right)d\sigma_{y}=\fint_{\Pi_{\nu_{z}}\cap Q_{M_{\epsilon}}(0)} g\left(z,z_{\varepsilon}+y\right)d\sigma_{y}+I,\] where \[I=\fint_{\Gamma^{\varepsilon}_{z}\cap Q_{M_{\epsilon}}(0)}\left(g\left(z,z_{ \varepsilon}+\tilde{y}\right)-g\left(z,z_{\varepsilon}+y\right)\right)d\sigma_ {y}.\] Here $\tilde{y}=y+\nu_{z}\|y\|s_{\varepsilon}$ with $s_{\varepsilon}=\tau(\varepsilon M_{\varepsilon})$. Estimating $I$ we obtain $\|I\|\leq C\tau_{g}(\|y\|s_{\varepsilon})\leq C\tau_{g}(M_{\varepsilon}s_{ \varepsilon})<\eta/2$, for $\varepsilon$ small enough. Here $\tau_{g}$ is the module of continuity for $g$ From here, and Lemma 3.3 the statements in the lemma follows, provided we have taken $\varepsilon$ small enough depending on $\nu_{z}$. ∎ ## 4. Surface integrals of oscillating functions Theorem 4.1.: _Let $\Gamma$ be a $C^{1}$ surface in ${\mathbb{R}}^{n}$, and $g(x,y)$ be integrable in $x$-variable over $\Gamma$, and continuos and 1-periodic in $y$ in ${\mathbb{R}}^{n}$. Then_ \[\limsup_{\varepsilon\to 0}\int_{\Gamma}g\left(y,\frac{y}{\varepsilon} \right)d\sigma_{y}\leq\int_{\Gamma}\overline{g}^{}(y,\nu_{y})d\sigma_{y},\] _and_ \[\liminf_{\varepsilon\to 0}\int_{\Gamma}g\left(y,\frac{y}{\varepsilon} \right)d\sigma_{y}\geq\int_{\Gamma}\overline{g}_{}(y,\nu_{y})d\sigma_{y}.\] _Moreover if $\Gamma$ satisfies IDDC then an effective limit exists and we have_ \[\lim_{\varepsilon\to 0}\int_{\Gamma}g\left(y,\frac{y}{\varepsilon} \right)d\sigma_{y}=\int_{\Gamma}\overline{g}(y)d\sigma_{y}.\] Proof.: We shall prove the limit superior estimate only. The limit inferior estimates follows in a similar way. The last statement follows in an obvious manner. Let us fix a small positive constant $\eta>0$, to be decided later. Without loss of generality, we may assume $\Gamma$ is bounded, and $g(x,y)$ is uniformly continuous function on $\Gamma\times{\mathbb{R}}^{n}$. Next from Lemma 3.4, for each $z\in\Gamma$ we may take $\varepsilon_{\nu_{z},\eta}$ such that the estimate (8) \[\overline{g}_{}(z,\nu_{z})-\eta\leq\fint_{\Gamma\cap Q_{\rho_{\epsilon}}(z)}g \left(z,\frac{y}{\varepsilon}\right)d\sigma_{y}\leq\overline{g}^{}(z,\nu_{z}) +\eta.\] holds for all $\varepsilon<\varepsilon_{\nu_{z},\eta}$. Next we cover $\Gamma$ by cubes $Q_{\rho_{\varepsilon}}(z)$ for all $z\in\Gamma$ and all $\rho_{\varepsilon}=\varepsilon M_{\varepsilon}$ with $\varepsilon<\varepsilon_{\nu_{z},\eta}$. This is a fine cover of $\Gamma$, and therefore there is a _finite_ disjoint sub-cover such that $\Gamma\setminus\cup_{j=1}^{N}Q_{\rho_{\varepsilon_{j}}}(z^{j})$ has surface measure less than $\eta$. Here $z^{j}\in\Gamma$, and $\varepsilon_{j}=\varepsilon_{\nu_{z^{j}}}$. Next \[\int_{\Gamma}g(y,y/\varepsilon)d\sigma_{y}\leq\sum_{j=1}^{N}\int_{\Gamma\cap Q _{\rho_{\varepsilon_{j}}}(z^{j})}g(y,y/\varepsilon)d\sigma_{y}+\eta\|\|g\|\|_{ \infty},\] and by (8) we obtain \[\int_{\Gamma}g(y,y/\varepsilon)d\sigma_{y}\leq\sum_{j=1}^{N}\left(\overline{g} ^{}(z^{j},\nu_{z^{j}})+\eta\right)\|\Gamma\cap Q_{\rho_{\varepsilon_{j}}}(z^{j })\|+\eta\|\|g\|\|_{\infty}\] \[\leq\int_{\Gamma}\overline{g}^{}(y,\nu_{y})d\sigma_{y}+(1+\|\|g\|\|_{\infty}+\| \Gamma\|)\eta<\int_{\Gamma}\overline{g}^{}(y,\nu_{y})d\sigma_{y}+\eta_{0},\] for any $\eta_{0}$ as small as we wish, provided $\eta$ is small enough. In a similar way we obtain the estimate from below \[\int_{\Gamma}g(y,y/\varepsilon)d\sigma_{y}\geq\int_{\Gamma}\overline{g}_{}(y, \nu_{y})d\sigma_{y}-\eta_{0}.\] As $\eta_{0}$ was arbitrary we have the two main estimates in the statement of the theorem. Putting these together along with the IDDC we shall have the third statement. Indeed, due to IDDC we have that the set $\{x\in\Gamma:\ \nu_{x}\hbox{ rational }\}$ has zero surface measure, and the integral over this set is zero. For the rest of $\Gamma$ we have irrational normals only, and hence the full averaging (Lemma 3.2) takes place and we obtain $\overline{g}_{}=\overline{g}$. ∎ ## 5. The case of Layered densities, almost periodicity, and ergodicity In this section we shall deduce, similar results of that in Theorem 4.1 while replacing the periodicity assumption with layered materials/densities, almost-periodic and ergodic case. Nevertheless we shall only mention the results without deepening much into the analysis. The reader may easily verify the statements. ### Layered Materials If we assume the function $g(x,y)$ is independent of $(y_{k+1},\cdots,y_{n})$ and is $1$-periodic in $(y_{1},\cdots,y_{k})$ then one may naturally obtain results reminiscent of that of layered materials in homogenizations for PDE. Indeed one can obtain the following obvious result: _If the surface $\Gamma$ does not have any flat parts in directions $e_{i}$ ($i=1,\cdots,k$), then the averaging takes place._ ### Almost Periodic Case In the case of almost periodic functions (see [2]) one obtains similar results by replacing the average integral $\overline{g}(x)$ with \[\hat{g}(y)=\lim_{r\to\infty}\fint_{Q_{r}^{+}}g(x,y)dy.\] The obvious details are left to the reader. ### Ergodic Case The results of our main theorem can be generalized further to the case of functions with ergodic properties (see [6]). Indeed, if we assume $g(x,y,\omega)$ is defined on $\Gamma\times{\mathbb{R}}^{n}\times D$, for some $D\subset{\mathbb{R}}^{n}$ and that $g(x,y,\omega)$ is statistically homogeneous field in $(y,\omega)$-variable, i.e., $g(x,y,\omega)=h(x,T_{y}\omega)$ for some random variable $h(x,z)$ (random w.r.t. second variable) with underlying probability space and an ergodic $n$-dimensional dynamical system $T_{y}$. Hence $h(x,T_{y}\omega)$ admits an averaging \[\lim_{\varepsilon\to 0}\fint_{K}h(x,T_{y/\varepsilon}\omega)dx=\hbox{Exp}(h(x, \cdot)).\] One may now deduce similar results as that in Theorem 4.1 with \[\overline{g}(z)=\hbox{Exp}(g(z,\cdot,\cdot)).\] ## 6. Applications to Partial Differential Equations ### Dirichlet problem: Elliptic Case Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{n}$, with piecewise smooth boundary. Let $g(x,y)$ be as before, and $f(x,y)$ have the same property as $g$. For simplicity, we shall assume $f,g$ are continuous in both variables (actually $L^{2}(\partial\Omega)$ would suffice). Let further $\Gamma_{0}\subset\Omega$ be a piecewise $C^{1}$-curve, and define \[\overline{\mu}^{}=\overline{f}^{}\chi_{\Gamma_{0}}d\sigma,\qquad\overline{ \mu}_{}=\overline{f}_{}\chi_{\Gamma_{0}}d\sigma\qquad\overline{\mu}= \overline{f}\chi_{\Gamma_{0}}d\sigma.\] Then the following result hold. Theorem 6.1.: _For a solution $u_{\varepsilon}$ of ($P_{\varepsilon}$), we define (in $\Omega$)_ \[u^{}(x):=\limsup_{\varepsilon\to 0}u_{\varepsilon}(x)\qquad\hbox{ and }u_{}(x):=\liminf_{\varepsilon\to 0}u_{\varepsilon}(x)\ .\] _Then the following hold (in the weak sense):_ \[\Delta\overline{u}^{}\geq-\overline{\mu}^{},\qquad\Delta\overline{u}^{}\leq -\overline{\mu}_{}\qquad\hbox{in }\Omega,\] \[u^{}(x)\leq\overline{g}^{}(x,\nu_{x})\quad\text{ on }\partial\Omega,\qquad u _{}(x)\geq\overline{g}_{}(x,\nu_{x})\quad\text{ on }\partial\Omega.\] _Moreover for any sequence of $u_{\varepsilon}$, there is a subsequence converging to a function $u$ in $\Omega$, satisfying_ \[-\overline{\mu}^{}\leq\Delta u\leq-\overline{\mu}_{}\qquad\hbox{in the weak sense in }\Omega,\] \[\overline{g}_{}(x,\nu_{x})\leq\varliminf_{z\to x}u(z)\leq\varlimsup_{ z\to x}u(z)\leq\overline{g}^{}(x,\nu_{x}),\qquad x\in\partial\Omega.\] Proof.: By Green’s and Poisson’s representations we have where $P$ and $G$ are Poisson respectively the Green functions of the domain. By Theorem 4.1 \[u^{}(x)\leq\int_{\partial\Omega}P(x,y)\overline{g}^{}(y,\nu_{y})d\sigma_{y}+ \int_{\Gamma_{0}}G(x,y)\overline{f}^{}(y,\nu_{y})d\sigma_{y},\] and \[u_{}(x)\geq\int_{\partial\Omega}P(x,y)\overline{g}_{}(y,\nu_{y})d\sigma_{y}+ \int_{\Gamma_{0}}G(x,y)\overline{f}_{}(y,\nu_{y})d\sigma_{y},\] which implies the first two statements in the theorem. Now the last statement follows from the above inequalities, in an obvious way. ∎ To obtain an effective limit, in the above theorem, one needs to consider domains satisfying IDDC. In particular, using the above theorem and Lemma 3.2 (ii) we have the following result. Corollary 6.2.: _Let $\partial\Omega$ and $\Gamma_{0}$ satisfy IDDC, and $u_{\epsilon}$ be a solution to problem ($P_{\varepsilon}$). Then, $u_{\varepsilon}$ converges to a function $u$ in $\Omega$, satisfying_ \[\Delta u=-\overline{\mu}(x),\quad\text{ in }\Omega,\qquad u=\overline{g}(x) \quad\text{ on }\partial\Omega.\] The general nature of the method employed here suggests that we can apply this to situations where integral representations are possible. This can naturally go beyond the Dirichlet problem, or the Laplace operator, and as general as to systems, and equations of higher orders. We state explicitly that once one has an integral representation of any function, through kernel functions, then one may conclude similar statement as that of Theorem 6.1 and Corollary 6.2. We leave it to the reader to apply this to their favorite scenarios. ### Dirichlet problem: Parabolic Case Let us now consider the case of parabolic equations (9) \[\Delta u_{\varepsilon}-\partial_{t}u_{\varepsilon}=-\mu_{\varepsilon}(x,t) \quad\hbox{in }\Omega,\qquad u(x,t)=g_{\varepsilon}(x,t),\] where $d\mu_{\varepsilon}(x,t)=f_{\varepsilon}(x,t)\chi_{\Gamma_{0}}d\sigma$, $f_{\varepsilon}(x,t)=f(x,t,x/\varepsilon,t/\varepsilon)$, $g_{\varepsilon}(x,t)=g(x,t,x/\varepsilon,t/\varepsilon)$, and $f(x,y,s,t),g(x,y,s,t)$ are integrable in $(x,s)$-variables over $\Gamma_{0}$ and $\partial\Omega$ respectively, and $1$-periodic in $(y,t)$-variable. Here one may consider different cases, such as $\Omega=D\times(0,T)$, with $D$ a domain in ${\mathbb{R}}^{n}$, or $\Omega\subset{\mathbb{R}}^{n}\times{\mathbb{R}}$ and time varying. Also $\Gamma_{0}$ can be take to be either time independent or varying in time. Now a similar argument, using Poisson and Green representations, as in Section 6.1 can give us various type of results. In the case $\Omega=D\times(0,T)$ one obtains same type of results as that of Theorem 6.1. To obtain results of the nature of Corollary 6.2 one needs * either to assume $\Omega=D\times(0,T)$, and $D$ has IDDC condition along with $f$, and $g$ being independent of their fourth variable, i.e. $f_{\varepsilon}(x,t)=f(x,x/\varepsilon,t)$, $g_{\varepsilon}(x,t)=g(x,x/\varepsilon,t)$ * or $\Omega$ and $\Gamma_{0}$, both have IDDC condition in ${\mathbb{R}}^{n+1}$. In addition, Corollary 6.2 can be obtained similarly for $f_{\varepsilon}(x,t)=f(x,x/\varepsilon,t/\varepsilon^{2})$, $g_{\varepsilon}(x,t)=g(x,x/\varepsilon,t/\varepsilon^{2})$ with $\Omega=D\times(0,T)$, and $D$ has IDDC condition. ### Neumann problem As mentioned at the end of the previous section, one can apply the technique of this paper to far reaching scenarios and problems, involving integral representations. The Neumann problem naturally fits into this category through Fredholm’s alternative, and an integral representation. Indeed, let $u_{\varepsilon}$ be a solution to the problem \[\Delta u_{\varepsilon}=0,\quad\text{ in }\Omega,\qquad\partial_{\nu}u_{ \varepsilon}=g(x,x/\varepsilon)\quad\text{ on }\partial\Omega,\] with $g$ satisfying compatibility condition $\int_{\partial\Omega}g(y,y/\varepsilon)d\sigma_{y}=0$ for all $\varepsilon>0$. Then it is well-known that \[u_{\varepsilon}(x)=\int_{\partial\Omega}F(x,y)\phi_{\varepsilon}(y)d\sigma_{y},\] where $F$ is the Fundamental solution for Laplace operator (or the corresponding operator), and $\phi$ solves the Voltera integral equation of second kind, i.e., \[\phi_{\varepsilon}(x)=\int_{\partial\Omega}\partial_{\nu}F(x,y)\phi_{ \varepsilon}(y)d\sigma_{y}+g(x,x/\varepsilon).\] As $\varepsilon$ tends to zero, $\phi_{\varepsilon}$ tends (weakly in $L^{1}(\partial\Omega)$) to a limit $\phi_{0}$ solving \[\phi_{0}(x)=\int_{\partial\Omega}\partial_{\nu}F(x,y)\phi_{0}(y)d\sigma_{y}+ \overline{g}(x),\] in a weak sense over the boundary of $\Omega$. This happens exactly when the boundary of $\Omega$ has IDDC. More accurately, the kernel of the bounded operator \[T^{\varepsilon}(v)=v(x)-\int_{\partial\Omega}\partial_{\nu}F(x,y)v(y)d\sigma_{ y}-g(x,x/\varepsilon),\] acting on $L^{1}(\partial\Omega)$ space, is upper semi-continuous and has a unique element. In particular $\varlimsup_{\varepsilon}ker(T^{\varepsilon})\subset ker(T^{0})$, where \[T^{0}(v)=v(x)-\int_{\partial\Omega}\partial_{\nu}F(x,y)v(y)d\sigma_{y}- \overline{g}(x).\] By uniqueness of the solutions to the Fredholm operator this kernel must have only one element, and hence $\phi_{\varepsilon}\to\phi_{0}$, with $\phi_{0}\in Ker(T^{0})$. In other words $\phi_{0}$ solves the Voltera equation above for the function $\overline{g}(x)$. From here it follows that $u_{\varepsilon}$ converges to $u_{0}=\int_{\partial\Omega}F(x,y)\phi_{0}(y)d\sigma_{y}$, with $\phi_{0}$ solving \[\phi_{0}(x)=\int_{\partial\Omega}\partial_{\nu}F(x,y)\phi_{0}(y)d\sigma_{y}+ \overline{g}(x).\] Hence $u_{0}$ solves the averaged/effective Neumann problem \[\Delta u_{0}=0,\quad\text{ in }\Omega,\qquad\partial_{\nu}u_{\varepsilon}= \overline{g}(x)\quad\text{ on }\partial\Omega.\] A different way of analyzing this is to consider the solution of the Neumann problem in the weak form \[\int_{\Omega}\nabla u_{\varepsilon}(y)\cdot\nabla\phi(y)dy=\int_{\partial \Omega}g(y,y/\varepsilon)\phi(y)d\sigma_{y},\] where $\phi$ is a test function in a reasonable class. Letting $\varepsilon\to 0$ we see that the integrals converge to \[\int_{\Omega}\nabla u_{0}(y)\cdot\nabla\phi(y)dy=\int_{\partial\Omega} \overline{g}(y)\phi(y)d\sigma_{y}.\] The latter in turn solves the Neuman problem with $\overline{g}(y)$ as the amount of flux at each boundary point. ## 7. Examples and illustrations The behavior of the limit integrals in 2.1 are directly related to foliation of the fundamental cell $Q_{1}^{+}$. To illustrate this (in ${\mathbb{R}}^{2}$) consider a sequence $p_{j}=(-\sqrt{2}/j,1)$. For $a\in[0,1)$, let $l^{a}_{j}$ be the line through the point $(0,a)$ and orthogonal to $p_{j}$. Then, due to the fact that $p_{j}$ is rationally independent Lemma 3.2 implies \[\lim_{\varepsilon\to 0}\fint_{l_{j}\cap Q_{\rho_{\varepsilon}}}g(x,y/ \varepsilon)d\sigma_{y}=\overline{g}(x).\] On the other hand $l_{j}\to l_{0}$ which is a line through $(0,a)$ and parallel to the $x_{1}$-axis. For the limit of the average for this line we then have \[\lim_{\varepsilon\to 0}\fint_{l_{0}\cap Q_{\rho_{\varepsilon}}}g(x,y/ \varepsilon)d\sigma_{y}=\overline{g}(x,\cdot,a).\] where averaging takes place only on $y_{1}$-variable. In general $\overline{g}(x,\cdot,a)\neq\overline{g}(x).$ More importantly, we can in general not expect to take a very small $\varepsilon_{0}$ and expect \[\fint_{l_{0}\cap Q_{\rho_{\varepsilon}}}g(x,y/\varepsilon_{0})d\sigma_{y}\leq \overline{g}^{}(x),\] for estimates from above, or $\geq\overline{g}_{}(x)$, when considering estimates from below. The reader may easily verify this by taking $g(x,y)=g(y)=\|\sin\pi y_{1}\sin\pi y_{2}\|$, and make computations to arrive at \[\lim_{\varepsilon\to 0}\fint_{l_{j}\cap Q_{\rho_{\varepsilon}}}\|\sin(\pi y_{1} /\varepsilon)\sin(\pi y_{2}/\varepsilon)\|d\sigma_{y}=\frac{4}{\pi^{2}},\] and \[\lim_{\varepsilon\to 0}\fint_{l_{0}\cap Q_{\rho_{\varepsilon}}}\|\sin(\pi y_{1} /\varepsilon)\sin(\pi y_{2}/\varepsilon)\|d\sigma_{y}=\frac{2\sin(\pi a)}{\pi}.\] Let us set \[L^{a}_{j}=l_{j}^{a}\cap Q_{\rho_{\varepsilon_{0}}/\varepsilon_{0}}(mod~{}1).\] Then one readily verifies that for $\varepsilon_{0}$ fixed, and very small \[L^{a}_{j}\subset Q_{1}^{+}\cap\left\{x:a-\frac{M_{\varepsilon_{0}}\sqrt{2}}{j} \leq x_{2}\leq a+\frac{M_{\varepsilon_{0}}\sqrt{2}}{j}\right\},\] where $M_{\varepsilon_{0}}=\rho_{\varepsilon_{0}}/\varepsilon_{0}$. In particular, for $j$ large $L_{j}$ will never foliate the unit cell, and hence it is impossible to approximate the integral over $\partial\Omega$ by any covering, however small. ### Example 1 We consider the case when $g$ is periodic only in $x_{1}$-direction and when the domain is a slab with a unit normal direction $\nu$. For a $\nu\in{\mathbb{S}}^{n-1}$, set $\Omega=\{x:-R_{1}<x\cdot\nu<R_{2}\}$. Let $g(x)=g(x_{1})$ be independent of $(x_{2},\cdots,x_{n})$ and $1$-periodic, i.e. $g(x+k)=g(x_{1}+k_{1})=g(x_{1})=g(x)$ for $k\in{\mathbb{Z}}^{n}$. Now let $u_{\varepsilon}$ be a solution of the following equation (10) \[\begin{cases}\triangle u_{\varepsilon}=0&\text{ in }\Omega,\\ u_{\varepsilon}(x)=M&\text{ on }x\cdot\nu=-R_{1},\\ u_{\varepsilon}(x)=g\left(\frac{x}{\varepsilon}\right)&\text{ on }x\cdot\nu=R_ {2}.\end{cases}\] We discuss three possible limits of $u_{\varepsilon}$ whose homogenized equation can be found. Namely, (11) \[u^{}=\limsup_{\varepsilon\to 0}u_{\varepsilon},\quad u_{}=\liminf_{ \varepsilon\to 0}u_{\varepsilon},\quad\text{and}\quad\overline{u}=\lim _{j\rightarrow\infty}u_{\varepsilon_{i}},\] where the subsequence $u_{\varepsilon_{i}}$ is such that $E(u_{\varepsilon_{i}})=\frac{1}{2}\int_{\Omega}\|\nabla u_{\varepsilon_{i}}\|^{2 }dx\rightarrow\liminf_{\varepsilon\to 0}E(u_{\varepsilon})$ as $i\rightarrow\infty$ (See Figure 2). It turns out that $u^{}$, $u_{}$, and $\overline{u}$ don’t follow simple homogenization whose boundary data is a simple average $\overline{g}$. It means there are nontrivial homogenization processes for each different limits. For $R\neq 0$, set \[\overline{g}^{}(R,e_{1})=\max g,\quad\overline{g}_{}(R,e_{1})=\min g,\] and (12) \[\overline{g}(R,M)=\begin{cases}\overline{g}^{}(R,e_{1})\quad\text{for $M>\sup g $},\\ \overline{g}_{}(R,e_{1})\quad\text{for $M<\inf g$},\\ M\quad\text{for $\inf g\leq M\leq\sup g$}.\end{cases}\] If $R=0$, let \[\overline{g}^{}(0,e_{1})=\overline{g}_{}(0,e_{1})=\overline{g}(0,e_{1})=g(0).\] Proposition 7.1.: _For the particular choice $\nu=e_{1}$, in equation (10), the limit functions $u^{}$, $u_{}$, and $\overline{u}$ will satisfy_ (13) \[\begin{cases}\triangle u=0&\text{ in }\Omega,\\ u(x)=M&\text{ on }x\cdot e_{1}=-R_{1},\\ u(x)=A&\text{ on }x\cdot e_{1}=R_{2},\end{cases}\] _where_ (14) \[A=\begin{cases}\overline{g}^{}(R_{2},e_{1})\quad\text{ if $u=u^{}$},\\ \overline{g}_{}(R_{2},e_{1})\quad\text{ if $u=u_{}$},\\ \overline{g}(R_{2},e_{1})\quad\text{ if $u=\overline{u}$}.\end{cases}\] Proof.: Select $\varepsilon_{i}$ such that $g\left(\frac{x_{1}}{\varepsilon_{i}}\right)=\max g$ and then $u_{\varepsilon_{i}}=u^{}$ since all $u_{\varepsilon_{i}}$ have the same boundary values. Hence it is clear that $\limsup_{\varepsilon\to 0}u_{\varepsilon}=u^{}$. Similar argument can be applied to $u_{}$ and $\overline{u}$ to have the conclusion. ∎ Proposition 7.2.: _In equation (10), when $\nu\neq e_{1}$, $u_{\varepsilon}$ converges to $\overline{u}$ which is a solution to_ (15) \[\begin{cases}\triangle u=0&\text{ in }\Omega,\\ u(x)=M&\text{ on }x\cdot\nu=-R_{1},\\ u(x)=\overline{g}&\text{ on }x\cdot\nu=R_{2}.\end{cases}\] Proof.: Choose any point $x$ such that $x\cdot\nu=R_{2}$ and $\frac{x}{\varepsilon}=\tilde{x}\text{ (mod $1$)}$ for some $\tilde{x}\in Q_{1}$. Then, for $\nu\neq e_{1}$, (16) \[\begin{split}\fint_{\{(y-\tilde{x})\cdot\nu=0\}\cap Q_{r}}& g\left(\frac{y}{\varepsilon}\right)d\sigma_{y}=e_{1}\cdot\nu \fint_{\{\|y_{1}-\tilde{x}_{1}\|\leq r\}}g\left(\frac{y_{1}}{\varepsilon}\right) \frac{1}{e_{1}\cdot\nu}dy_{1}\\ &=\fint_{\{\|y_{1}-\tilde{x}_{1}\|\leq r\}}g\left(\frac{y_{1}}{ \varepsilon}\right)dy_{1}\rightarrow\overline{g},\qquad(\text{as}\ \epsilon\to 0 ),\end{split}\] which is independent of the choice of $\tilde{x}$. It implies $\overline{g}^{}(x,\nu)=\overline{g}_{}(x,\nu)=\overline{g}(x)$. Let $P(x-y)$ be a Poisson Kernel of the the domain $\Omega$. Then \[u_{\varepsilon}(x)=M\int_{x\cdot\nu=-R_{1}}P(x-y)d\sigma_{y}+\int_{x\cdot\nu=R _{2}}P(x-y)g\left(\frac{y_{1}}{\varepsilon}\right)d\sigma_{y}.\] Letting $\epsilon$ tend to zero and using first Lemma 3.3, and then Lemma 3.2 we shall have \[\lim_{\epsilon\to 0}\int_{x\cdot\nu=R_{2}}P(x-y)g\left(\frac{y_{1}}{ \varepsilon}\right)d\sigma_{y}=\int_{x\cdot\nu=R_{2}}P(x-y)\overline{g}d\sigma _{y}\ ,\] which implies the conclusion. ∎ <figure><img src="content_image/1201.6683/x1.png"><figcaption>(a) fig 2.a</figcaption></figure> The next interesting question is to find the limit equation for the general converging sequence. In the following lemma, we will show there is a converging subsequence whose limit takes any value between the supremum of $g$ and its infimum on $x\cdot e_{1}=R_{2}$. Proposition 7.3.: _For any $A$ such that $\min g\leq A\leq\max g$, there is a sequence $\{u_{\varepsilon_{i}}\}$ converging uniformly to $u$ such that its limit $u$ satisfies_ (17) \[\begin{cases}\triangle u=0&\text{ in }\Omega,\\ u(x)=M&\text{ on }x\cdot e_{1}=-R_{1},\\ u(x)=A&\text{ on }x\cdot e_{1}=R_{2}.\end{cases}\] Proof.: There are $\varepsilon_{i}\to 0$ such that $g\left(\frac{x_{1}}{\varepsilon}\right)=A$ since $g$ is 1-dimensional. Therefore all $u_{\varepsilon_{i}}$ have the same boundary values, implying $u_{\varepsilon_{i}}=u(x)$. ∎ ### Example 2 In this example we confine ourselves to ${\mathbb{R}}^{2}$. We consider the case when $g$ is periodic only in $x_{1}$-direction and the domain $\Omega$ is convex with two parallel flat parts of boundaries, orthogonal to $\nu=e_{1}$. For exactness we consider the following stadium like domain \[\Omega=\left\{x:\|x_{1}\|<R,\|x_{2}\|\leq 1+\sqrt{R^{2}-x_{1}^{2}}\right\}.\] Let $g(x)=g(x_{1})$ be $1$-periodic, and independent of $x_{2}$-direction. Now let $u_{\varepsilon}$ be a solution of the following equation (18) \[\begin{cases}\triangle u_{\varepsilon}=0&\text{ in }\Omega,\\ u_{\varepsilon}(x)=g\left(\frac{x}{\varepsilon}\right)&\text{ on }\partial \Omega,\end{cases}\] then $u^{}$ and $u_{}$ (see (11)) are sub- and super-solutions of (18) respectively. In general, they are not solutions. In the next result we state that the homogenized boundary data may not be continuous even though $g(x)$ is smooth. Proposition 7.4.: _There is a a smooth $1$-periodic function $g(x_{1})$ of one variable $x_{1}$ and a subsequence of solutions $\{u_{\varepsilon_{i}}\}$ to equation (18), converging to $u$ such that $u$ is not continuous on $\partial\Omega$._ Proof.: If $x_{1}\neq\pm R$, then $x=(x_{1},x_{2})\in\partial\Omega$ satisfies IDD condition and then any limit will satisfy the boundary condition $u(x)=\overline{g}$. Now we select $\varepsilon_{i}$ so that $g\left(\frac{x_{1}}{\varepsilon_{i}}\right)=\max g\neq\overline{g}$. Then $u_{\varepsilon_{i}}$ has a converging subsequence to $u$ which will be discontinuous at $x=(R,1)$. ∎ ## References [AA] G. Allaire, and M. Amar; Boundary layer tails in periodic homogenization. ESAIM Control Optim. Calc. Var., 4:209–243 (electronic), 1999. * [Bohr] H. Bohr, _Almost Periodic Functions_, Chelsea Publishing Company, New York, N.Y., 1947. ii+114 pp, * [BDLS]Barles, G., Da Lio, F.; Lions, P.-L.; Souganidis, P. E. Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions. Indiana Univ. Math. J. 57 (2008), no. 5, 2355–2375. * [CKL] S. Choi, I. Kim, K. Lee; Homogenization of Neumann boundary data with fully nonlinear operator (preprint). * [GM] Gérard-Varet, David; Masmoudi, Nader; Homogenization and boundary layer, preprint. * [JKO] V. V. Jikov, S.M. Kozlov, O.A. Oleĭnik, Homogenization of differential operators and integral functionals. (English summary) Translated from the Russian by G. A. Yosifian [G. A. Iosifʹyan]. Springer-Verlag, Berlin, 1994. xii+570 pp. ISBN: 3-540-54809-2.
1607.07365	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 34800, "num_imgs": 5, "llama3_tokens_count": 9582 }	[ "content_image/1607.07365/x1.png", "content_image/1607.07365/x2.png", "content_image/1607.07365/x3.png", "content_image/1607.07365/x4.png", "content_image/1607.07365/x5.png" ]	# Scheduling of Dynamic Electric Loads Using Energy Storage and Short Term Power Forecasting Raymond A. de Callafon, Abdulelah H. Habib and Jan Kleissl R.A. de Callafon, A.H. Habib and J. Kleissl are with the Department of Mechanical and Aerospace Engineering, University of California, San Diego. Email: callafon,ahhabib,jkleissl@ucsd.edu ###### Abstract In this paper we formulate an optimization approach to schedule electrical loads given a short term prediction of time-varying power production and the ability to store only a limited amount of electrical energy. The proposed approach is unique and versatile as it allows scheduling of electrical loads that each have their own dynamic power demand during on/off switching, while also allowing the specification of minimum on/off times for each loads separately. The optimization approach is formulated as a parallel enumeration of all possible on/off times of the electrical loads using a moving time approach in which only a short term power production forecast is needed, while at the same time taking into account constraints on electrical energy storage and power delivery of a battery system. It is shown that the complexity of the optimization (number of enumerations) is limited by the number of data points in the short term power production forecast and the minimum on/off time of the electrical loads. The limited complexity along with parallel enumeration allows real-time operational scheduling of a large number of loads. The simulation results shown in this paper illustrate that relatively short term power forecast profiles can be used to effectively schedule dynamic loads with various dynamic load profiles. ## I Introduction One of main challenges in optimal sizing of battery systems for standalone or islanding microgrid applications is the power volatility of distributed renewable energy resources within that microgrid. It is well understood that a combination of battery systems for local energy storage and scheduling of loads for modifying power demands would allow an islanding microgrid to perform reliably, despite power production fluctuations [1, 2] and [3]. Even a non-islanding (grid connected) microgrid would benefit from local energy storage and scheduling of loads to allow for power demand following at the point of common coupling to mitigate costly power surge demands and facilitate demand flexibility [4]. Load scheduling and load shedding plays an important role in tracking power production, where loads are turned on/off at optimal times to follow the (predicted) power production as closely as possible [5]. Load scheduling and shedding applications can also be used as ancillary services to curtail power surge demands and provide voltage stability. Demonstrations and commercial applications can be seen in EV charging [6, 7], building load scheduling [8], household appliance scheduling [9], HVAC system scheduling [10] and several use cases motivated by environmental and financial incentives [11, 12, 13]. An energy storage unit in the form of a battery system can alleviate the effect of power volatility and power demand surges [14], but still requires financial decisions with respect to optimal sizing of the battery in combination with load scheduling and load shedding. Most approaches to optimal load or demand scheduling use some form of Model Predictive Control (MPC) [15, 16] to compute optimal control or scheduling signals. In MPC typically a constrained optimization problem is solved over a (short term) moving time horizon to compute an optimal control signal for a dynamical system in real time. Countless examples of innovative MPC based approaches for either load scheduling, grid tied storage systems or maintaining voltage stability can be found in [17], [18], [19] or [20]. Although MPC approaches are extremely powerful, models of dynamic systems are typically linear and control signal are allowed to attain any real value during the optimization [21, 22, 23]. For proper load scheduling and shedding along with a battery, the optimal control may be non-linear and must switch loads with different on/off switching dynamics, while maintaining constraints on battery power and state of charge. In this paper we define a load scheduling algorithm as the optimal on/off timing of a set of distinct electric loads using an MPC approach in which only a short term power production forecast is needed, while at the same time taking into account constraints on electrical energy storage and power delivery of a battery system via a barrier function. In our problem formulation, each electric load may have a different dynamic power response during on/off switching and a different minimum on/off time. The optimal scheduling is solved by a parallel enumeration of all possible on/off timing combinations that is limited by the number of data points in the short term power production forecast and the minimum on/off time of the electrical loads. The limited complexity along with parallel enumeration of load switching combinations allows real-time operational scheduling of a large number of loads as will be illustrated in the simulation results of this paper. ## II Electric Loads With Switching Dynamics ### _Switch Signal and Minimum On/Off Time_ To formulate the scheduling algorithm, first the (binary) load switching signal and the transient dynamics describing the (real) power demand of each load is defined here, similar to work presented in [24]. The dynamic properties of a load is characterized by the time-varying power demand during on/off switching and abbreviated to the “switching dynamics” of a load. The switching dynamics of each load is important in order to track power production, while maintaining constraints on battery power and state of charge during load switching. For notational purposes, consider a fixed number of $n$ loads where the time-varying power demand $p_{i}(t)$, $i=1,2,\ldots,n$ as a function of time $t$ for each load $i$ is determined by a binary switching signal $w_{i}(t)=\{0,1\}$ used to switch loads “on” or “off”. To maintain generality, the switching dynamics used to describe the power demand $p_{i}(t)$ during on/off switching may be different for each load, whereas the switching dynamics for a particular load $i$ may also depend on the transition of the binary switching signal $w_{i}(t)=\{0,1\}$. As a result, the switching dynamics for the time dependent power demand $p_{i}(t)$ for each load $i$ is different when the binary switching signal $w_{i}(t)=\{0,1\}$ transitions from 0 to 1 (rising edge, turning load $i$ on) or from 1 to 0 (falling edge, turning load $i$ off). Each load $i$ is also assumed to have a known minimum duration $T_{i}^{\mbox{\tiny on}}>0$ for the “on” time of the load when $w_{i}(t)=1$ and a minimum duration $T_{i}^{\mbox{\tiny off}}>0$ for the “off” time of the load when $w_{i}(t)=0$. The minimum on/off times $T_{i}^{\mbox{\tiny off}}$ and $T_{i}^{\mbox{\tiny on}}$ avoid undesirable chattering of the switch signal $w_{i}(t)$ during load scheduling and directly reduce the number of possible switching combinations. For the computation of the time varying binary switch signal $w_{i}(t)$, a Model Predictive Control (MPC) approach is used over a moving time optimization horizon $T<\infty$. The finite optimization horizon $T$ is required to satisfy \[\max_{i}(T_{i}^{\mbox{\tiny on}},T_{i}^{\mbox{\tiny off}})<T<\infty\] (1) to ensure the effect of the load switching signal $w_{i}(\tau)$ at $\tau=t$ for loads with minimum on/off times $T_{i}^{\mbox{\tiny on}}$ and $T_{i}^{\mbox{\tiny off}}$ can be predicted into the future over the time interval $T$ when $t\leq\tau\leq t+T$. Finally, it is assumed that all loads are initially switched “off”, e.g., $w_{i}(0)=0$ for $i=1,2,\ldots,n$. ### _Finite Set of Admissible Switching Signals_ With the minimum on/off duration times $T_{i}^{\mbox{\tiny on}},T_{i}^{\mbox{\tiny off}}$ and the finite optimization horizon $T$ for load switching, on/off switching of a load at time $t$ can now be formalized. The load switching signal $w_{i}(t),~{}i=1,2,\ldots,n$ will be a Zero Order Hold (ZOH) binary signal, where $w_{i}(t)\in\{0,1\}$ is allowed to change its binary state only over a finite number $N$ of switching time opportunities within the prediction horizon of length $T$. For binary load switching, an MPC optimization problem quickly becomes intractable due to a combinatorial problem where the number of switching combinations grows exponentially in the number of switching opportunities $N$ and the number $n$ of loads. However, it was shown in the [24] that the requirement of the minimum on/off duration times $T_{i}^{\mbox{\tiny on}},T_{i}^{\mbox{\tiny off}}$ over the finite time prediction horizon $T$ constrained by (1) significantly reduces the number of switching combinations and alleviates the combinatorial problem. In fact, it is shown in [24] that the number of possible combinations is much smaller than a trivial exponential growth of $(2^{n})^{N-1}$. With the ZOH approximation, the switching signal $w_{i}(t)\in\{0,1\}$ is kept constant in between the finite number $N$ of switching time opportunities within the prediction horizon of length $T$. As a result, the admissible on/off transition signal $w_{i}(t)=\{w_{i}^{\mbox{\tiny on}}(t),w_{i}^{\mbox{\tiny off}}(t)\}$ of a load at time $t=\tau_{i}$ can now be formalized by the ZOH switching signal \[w_{i}^{\mbox{\tiny on}}(t)=\left\{\begin{array}[]{rclcl}0&\mbox{for}&t<\tau_{i }&\mbox{and}&\tau_{i}\geq T_{i,last}^{\mbox{\tiny off}}+T_{i}^{\mbox{\tiny off }}\\ 1&\mbox{for}&t\geq\tau_{i}&\mbox{and}&\tau_{i}\leq T-T_{i}^{\mbox{\tiny on}} \end{array}\right.\] (2) where $T_{i,last}^{\mbox{\tiny off}}$ denotes the most recent (last) time stamp at which the load $i$ was switched “off”, and \[w_{i}^{\mbox{\tiny off}}(t)=\left\{\begin{array}[]{rclcl}1&\mbox{for}&t<\tau_{ i}&\mbox{and}&\tau_{i}\geq T_{i,last}^{\mbox{\tiny on}}+T_{i}^{\mbox{\tiny on} }\\ 0&\mbox{for}&t\geq\tau_{i}\end{array}\right.\] (3) where $T_{i,last}^{\mbox{\tiny on}}$ denotes the most recent (last) time stamp at which the load $i$ was switched “on”. ### _Dynamic Load Models_ It is clear that the switching time(s) $\tau_{i}$ for the signal $w_{i}(t)=\{0,1\}$ depends on the time-varying power demand $p_{i}(t)$ that is different for each load. For the computational results presented in this paper, continuous-time (linear) dynamic models will be used to model the power switching dynamics of a load. It should be pointed out that the computational analysis is not limited to the use of a linear dynamic model, as long as the dynamic model allow the numerical computation of a time-varying power demand $p_{i}(t)$ as a function of the switching signal $w_{i}(t)$ over a finite time interval $T$. To allow different dynamics for the time dependent power demands $p_{i}(t)$ when the binary switching signal $w_{i}(t)=\{0,1\}$ transitions from 0 to 1 (”on”) or from 1 to 0 (”off”), different dynamics is used for each of the load models. This allows power demands $p_{i}(t)$ to be modeled at different rates when switching loads. Using the Laplace transform ${\cal L}\{\cdot\}$ and referring back to the admissible on/off transition signals $w_{i}^{\mbox{\tiny on}}(t)$ and $w_{i}^{\mbox{\tiny off}}(t)$ respectively in (2) and (3), the switched linear order continuous-time dynamic models for the loads are assumed to be of the form \[p_{i}(s)=G_{i}^{\mbox{\tiny on}}(s)x_{i}w_{i}(s)~{}\mbox{and}~{}w_{i}(s)={\cal L }\{w_{i}^{\mbox{\tiny on}}(t)\}\] (4) and \[p_{i}(s)=G_{i}^{\mbox{\tiny off}}(s)x_{i}w_{i}(s)~{}\mbox{and}~{}w_{i}(s)={ \cal L}\{w_{i}^{\mbox{\tiny off}}(t)\}\] (5) where $G_{i}^{\mbox{\tiny on}}(s)$ and $G_{i}^{\mbox{\tiny on}}(s)$ represent the dynamics of the power demand for turning the load $i$ ”on” or ”off”. Both models satisfy $G_{i}^{\mbox{\tiny on}}(0)=1$ and $G_{i}^{\mbox{\tiny on}}(0)=1$ and a steady-state load demand parameter $x_{i}$ is used to model the relative size of the load, but different dynamics is used to model respectively the on/off dynamic switching of the load [24]. ### _Discretization of Switching Dynamics_ In order to be able to compute the time dependent power demand $p_{i}(t)$ for each load $i$, the time response of the switched dynamic models given in (4) and (5) needs to be computed over the prediction horizon $T$. The switching signal $w_{i}(t)$ and the power demand $p_{i}(t)$ for each load $i$ is time discretized at $t_{k}=k\Delta_{t}$ where $\Delta_{t}$ is the sampling time $k=0,1,\ldots$ is an integer index. The discretized load switching signal $w_{i}(t_{k}),~{}i=1,2,\ldots,n$ is a Zero Order Hold (ZOH) binary signal. Since $w_{i}(t_{k})\in\{0,1\}$ is allowed to change its binary state only over a finite number $N$ of switching times $\tau_{i}$ within the prediction horizon of length $T$, we assume that both the switching times \[\tau_{i}=N_{i}\Delta_{t}\] (6) and the minimum on/off duration times \[\begin{array}[]{rcl}T_{i}^{\mbox{\tiny on}}&=&N_{i}^{\mbox{\tiny on}}\Delta_{t }\\ T_{i}^{\mbox{\tiny off}}&=&N_{i}^{\mbox{\tiny off}}\Delta_{t}\end{array}\] (7) are all multiple of the sampling time $\Delta_{t}$. As the switching signal $w_{i}(t_{k})$ is _also_ held constant between subsequent time samples $t_{k}$ and $t_{k+1}$, the computation of the time discretized power demand $p_{i}(t_{k})$ for each load can be achieved using a Zero Order Hold (ZOH) discrete-time equivalent of the continuous-time models given earlier in (4) and (5). Using the z-transform ${\cal Z}\{\cdot\}$, the ZOH discrete-time equivalent dynamic models are given by \[p_{i}(z)=G_{i}^{\mbox{\tiny on}}(z)x_{i}w_{i}(z)~{}\mbox{and}~{}w_{i}(z)={\cal Z }\{w_{i}^{\mbox{\tiny on}}(t_{k})\}\] for “on” switching of the load and \[p_{i}(z)=G_{i}^{\mbox{\tiny off}}(z)x_{i}w_{i}(z)~{}\mbox{and}~{}w_{i}(z)={ \cal Z}\{w_{i}^{\mbox{\tiny off}}(t_{k})\}\] for “off” switching of the load, where $G_{i}^{\mbox{\tiny on}}(z)$ and $G_{i}^{\mbox{\tiny off}}(z)$ are the ZOH discrete-time equivalents of $G_{i}^{\mbox{\tiny on}}(s)$ and $G_{i}^{\mbox{\tiny off}}(s)$ using a sampling time $\Delta_{t}$. As a result, the power demand dynamics of each load is fully determined by $G_{i}^{\mbox{\tiny on}}(z)$, $G_{i}^{\mbox{\tiny off}}(z)$, static load demand $x_{i}$ and the chosen sampling time $\Delta_{t}$. ## III Load Scheduling Algorithm ### _Battery System Constraints_ The discrete-time switching signals $w_{i}(t_{k})$ for the loads $i=1,2,\ldots,n$ lead to a total (real) power demand \[p(t_{k})=\sum_{i=1}^{n}p_{i}(t_{k})\] for the $n$ schedulable loads. To formulate a dynamic load scheduling algorithm, first the power tracking error \[e(t_{k})=P(t_{k})-\sum_{i=1}^{n}p_{i}(t_{k})\] (8) is defined as the error between the anticipated or predicted power generation $P(t_{k})$ and total real power demand $p(t_{k})$ is considered. The time-dependent variable $P(t_{k})$ may refer to any result obtained from net power generation and prediction, e.g. solar power prediction. Since power generation prediction is not the main objective or contribution of this paper, $P(t_{k})$ is left as general as possible here. Due to the limited number $n$ of loads and possible errors in power prediction $P(t_{k})$, any (predicted) power tracking error $e(t_{k})$ will be absorbed/delivered by an energy storage system to match net power flow. Using a battery system for energy storage and assuming power tracking errors $e(t_{k})$ can be absorbed or delivered by the battery, the signal $e(t_{k})$ is subjected to several constraints imposed by the battery system. For a typical battery system, the first constraint involves the maximum power delivery and absorption capability \[P\cdot\|e(t_{k})\|<1\] (9) of the battery normalized to Power Units (PU) by $P$. The second constraint involves the maximum energy storage capability \[0.1\leq S\cdot\sum_{m=0}^{k}e(t_{k})<0.9\] (10) of the battery normalized to State of Charge (SOC) Units by $S$. The lower bound of 10% and upper of 90% is chosen as a safeguard to protect the battery against under- and overcharging, but can be chosen closer to 0 or 1 if so desired. ### _Admissible Discrete-Time Switching Combinations_ With the imposed time discretization given in (6), (7) and a finite number $N$ of switching times $\tau_{i}$ within the prediction horizon of length $T$, the admissible on/off transition signal in (2) reduces to (11) where $N_{i,last}^{\mbox{\tiny off}}$ now denotes the most recent discrete-time index at which the load $i$ was switched “off”. Similarly, (3) reduces to \[w_{i}^{\mbox{\tiny off}}(t_{k})=\left\{\begin{array}[]{rclcl}1&\mbox{for}&k<N_ {i}&\mbox{and}&N_{i}\geq N_{i,last}^{\mbox{\tiny on}}+N_{i}^{\mbox{\tiny on}} \\ 0&\mbox{for}&k\geq N_{i}\end{array}\right.\] (12) where $N_{i,last}^{\mbox{\tiny on}}$ denotes the most recent discrete-time index at which the load $i$ was switched “on”. Collectively, the signals $w_{i}^{\mbox{\tiny on}}(t_{k})$ in (11) and $w_{i}^{\mbox{\tiny off}}(t_{k})$ (12) define a set ${\cal W}$ of binary values for admissible discrete-time switching signals defined by \[{\cal W}=\left\{\begin{array}[]{c}w_{i}(t_{k})\in\{w_{i}^{\mbox{\tiny on}}(t_{ k}),w_{i}^{\mbox{\tiny off}}(t_{k})\},\\ i=1,2,\ldots,n,~{}k=1,2,\ldots,N\\ \mbox{where}~{}\begin{array}[]{c}w_{i}^{\mbox{\tiny on}}(t_{k})\in\{0,1\}~{} \mbox{given in}~{}\mbox{(\ref{eq:wiondiscrete})}\\ w_{i}^{\mbox{\tiny off}}(t_{k})\in\{0,1\}~{}\mbox{given in}~{}\mbox{(\ref{eq: wioffdiscrete})}\end{array}\end{array}\right\}\] (13) It is worthwhile to note that the number of binary elements in the set ${\cal W}$ in (13) is always (much) smaller than the trivial exponential number of $(2^{n})^{N-1}$[24]. This due to required minimum number of on/off samples $N_{i}^{\mbox{\tiny on}},N_{i}^{\mbox{\tiny off}}$ for the loads given in (11) and (12). ### _Moving Horizon Optimization_ Following the power tracking error $e(t_{k})$ defined in (8), the dynamic load scheduling optimization problem is formulated as a moving horizon optimization problem \[\begin{array}[]{c}w_{i}(t_{m})\\ i=1,2,\ldots,n\\ m=k,\ldots,k+N-1\end{array}=\mbox{arg}\min_{w_{i}(t_{m})\in{\cal W}}f(e(t_{m})),\] (14) where $f(e(t_{m}))\geq 0$, $i=1,2,\ldots,n$ refers to the $n$ loads and $m=k,\ldots,k+N-1$ refers to the $N$ switching time combinations within a prediction horizon of length $T$ over the admissible set ${\cal W}$ defined in (13). Similar to the ideas in Model Predictive Control (MPC), the $N\times n$ dimensional optimal switching signal $w_{i}(t_{m})$ is computed over the prediction horizon $m=k,\ldots,k+N-1$. Once the optimal switching signal $w_{i}(t_{m})\in{\cal W}$, $m=k,\ldots,k+N-1$ is computed, the optimal signal is applied to the loads _only_ at the time instant $t_{k}$, after which the time index $k$ is incremented and the optimization in (14) is recomputed over the moving time horizon. It should be noted that the admissible set ${\cal W}$ defined in (13) has a finite and countable number of binary combinations for the switching signal that is (much) smaller than the trivial exponential number of $(2^{n})^{N-1}$[24]. Therefore, the $N\times n$ dimensional optimal switching signal $w_{i}(t_{m})\in{\cal W}$ is computed simply by a finite number of evaluation of the criterion function $f(e(t_{l}))>0$. Furthermore, evaluation of the power tracking error $e(t_{m})$ in (8) for each possible switching combination $w_{i}(t_{m})\in{\cal W}$ can be done with a full parallel computation, as different $w_{i}(t_{m})$ for $i=1,2,\ldots,n$ and $m=k,2,\ldots,k+N-1$ are independent of each other. Instead of formulating a gradient based optimization or Mixed Integer Linear Programming (MILP) for loads with different switching dynamics, this approach allows extremely fast (parallel) numerical evaluation of the power tracking error $e(t_{m})$ in (8) and the battery constraints (9) and (10) for the finite number of switching signal combinations $w_{i}(t_{m})$ within the set ${\cal W}$ for real-time operation dynamic scheduling of loads. To incorporate the battery constraints given earlier in (9) and (10) and to be able to track the predicted power $P(t_{k})$, the optimization function $f(e(t_{m}))>0$ is defined as the sum of a least squares criterion $\\|e(t_{m})\\|_{2}$ and (smooth) boundary functions $B_{j}(e(e(t_{m}))$. In particular, the optimization function $f(e(t_{m}))$ is defined as \[f(e(t_{m}))=\\|e(t_{m})\\|_{2}+\sum_{j=1}^{4}B_{j}(e(t_{m}))\] (15) where \[\\|e(t_{m})\\|_{2}=\sum_{m=k+1}^{k+N-1}tr\{e(t_{m})e(t_{m})^{T}\] and the barrier functions $B_{j}(e(t_{m}))$ are defined as follows * $ B_{1}(e(t_{m}))=C_{1}\cdot(P\max_{m}\|e(t_{m})\|-1)$ if $\max_{m}P\|e(t_{m})\|\geq 1$, else $B_{1}(e(t_{m}))=0$. * $ B_{2}(e(t_{m}))=-C_{2}\cdot\Delta_{t}\sum_{m}e(t_{m})$ if $\sum_{m}e(t_{m})\leq 0$, else $B_{2}(e(t_{m}))=0$. * $ B_{3}(e(t_{m}))=C_{3}\cdot(S\Delta_{t}\sum_{m}e(t_{m})-0.9)$ if $\sum_{m}e(t_{m})\geq 0.9$, else $B_{3}(e(t_{m}))=0$. * $ B_{4}(e(t_{m}))=-C_{4}\cdot(S\Delta_{t}\sum_{m}e(t_{m})-0.1)$ if $\sum_{m}e(t_{m})\leq 0.1$, else $B_{4}(e(t_{m}))=0$. The elaborate definition of $f(e(t_{m}))$ in (15) ensures that $f(e(t_{m}))\geq 0$ and the constraints (9) and (10) are taken into account with a linear weighting scaled by the constants $C_{j}$. Although the barrier functions $B_{j}(e(t_{m}))$ are not “true” barrier functions that approach $\infty$ at the constraint, the additional linear weighting ensures that solutions are found that are forced away from the constraints. Increasing the value of $C_{j}$ will make this enforcement stronger and typically $C_{2}>>1$ to enforce that the normalized SOC \[S\Delta_{t}\sum_{m}e(t_{m})\] always remains positive. In the application example used in this paper, the coefficients $C_{j}$ were set to $C_{1}=C_{3}=C_{4}=10$ whereas $C_{2}=1000$. The reason why not a true barrier function is used is that the optimization of the function $f(e(t_{m}))\geq 0$ in (14) still allows for a solution in case the constraints are (temporarily) violated instead of giving no possible solution for load scheduling. Temporarily violation of constraints can be used during battery storage design to indicate that a larger battery is required, while in operation it may be used to allow for a (temporary) solution for load scheduling instead of providing simply “no” solution due unanticipated constrain violation. ## IV Application Example ### _Simulated Load Switching Dynamics_ To illustrate the results of the scheduling algorithm for loads with distinct load switching dynamics, three loads are selected with different switching dynamics for on/off switching. Load sizes were elected using optimal static load size selection [24]. Furthermore, load no. 2 exhibits a resonant power dynamics behavior requiring a temporary power surge to power on the load. The dynamic characteristics of the loads with their minimum on/off time used in this simulation study are summarized in Table I. \| Loads --- char. Size (%) \| Poles\tiny oni \| Poles\tiny offi \| T\tiny oni \| T\tiny offi x1 \| 60.00 \| -0.01 \| -0.04 \| 180 \| 180 x2 \| 25.86 \| \- 0.05 ± j0.06 \| -0.05 \| 240 \| 240 x3 \| 12.22 \| -0.02 \| -0.02 \| 300 \| 300 TABLE I: Loads characteristics: relative size in PU, poles of denominator dynamics, and minimum on/off time in seconds. To illustrate the variability in the dynamics of the loads summarized in Table I, the dynamic response of switching dynamics of the three loads in our case study are depicted in Figure 1. Although the same switching signal $w_{i}(t_{k})$ is used, the loads exhibit different power demand transitions $p_{i}(t_{k})$. Load 2 shows the typical behavior of a second order dynamics model with an initial larger peak load, typically seen in AC motors used in HVAC systems. <figure><img src="content_image/1607.07365/x1.png"><figcaption>Fig. 1: Dynamics of power demand pi(tk) (colored lines) of the three loadsi=1,2,3 defined in Table I as a function of the same binary switch signalwi(tk). Note: time scale is in seconds.</figcaption></figure> ### _Dynamic Load Switching Results at 50% SOC_ For the simulation results in this paper, the power curve to be tracked is a power production curve produced by a solar power unit with an irregular bell shaped curve due to solar variability during a 4 hour (240 minute) period. It is worth mentioning that a single optimization for $n=3$ loads over a prediction horizon of $N=6$ switching time opportunities (every minute) with the load dynamics summarized in Table I (discretized at every second) takes less then 0.3 second to compute in Matlab on a standard 4 core CPU system. The optimization results displayed in each of the figures that follow was therefore computed in less than 70 seconds over the 240 switching opportunities along the 4 hour power curve. As loads are scheduled to switch each minute, the 0.3 second optimization time clearly poses no problem for real-time operation. For the first results depicted in this paper, the scheduling algorithm is initialized for a battery with an initial SOC of 50% and leads to the final result depicted in Figure 2. It can be seen that the load scheduling manages to track the irregular power curve (green line) by scheduling of the loads at the appropriate times, while keeping the battery energy in SOH between the boundaries of 10% and 90%. Power demand on the battery also remained within 10% of the total power demand in PU. <figure><img src="content_image/1607.07365/x2.png"><figcaption>Fig. 2: Load scheduling with a battery initialized at 50% SOC. Top figure:Power curve (PU) to be tracked (green line) with total real power demand dueto load switching (black line) and individual dynamic load demands (coloredlines). Middle and bottom figure are battery energy (100% SOC) and batterypower demand (PU) where a positive value indicates charging.</figcaption></figure> One may be tempted to concluded that the load switching results are relatively easily obtained due to 50% SOC level of the battery. However, if the constraints on battery energy weighted by the barrier function $B_{j}(e(t_{m}))$ are ignored (e.g. coefficients $C_{2}=C_{3}=C_{4}=0$), no constraints on battery energy are taken into account. In that case, the battery may be overcharged as indicated in the results summarized in Figure 3. <figure><img src="content_image/1607.07365/x3.png"><figcaption>Fig. 3: Load scheduling with a battery initialized at 50% SOC, but withoutconstraints on the battery energy. Middle figure, showing the battery energy(100% SOC), clearly indicates overcharging, despite power curve tracking andconstraints on battery power.</figcaption></figure> ### _Dynamic Load Switching Results with Extreme SOC_ Starting the battery at a nearly discharged state (SOC of 10%) or fully charged state (100% SOC) would require a careful switching regime of loads at the beginning of the power curve to be tracked. Running the scheduling algorithm for a battery using a moving time prediction horizon where the load scheduling is computed according to the MPC approach (14) with the optimization function given in (15) automatically decides on carefully switched loads at the beginning of the power curve to ensure the power curve is tracked, while bringing the battery back into its allowed operating regime. The results of the scheduling algorithm for a battery with a SOC of 10% and 100% are summarized in Figure 4 and Figure 5 respectively. <figure><img src="content_image/1607.07365/x4.png"><figcaption>Fig. 4: Load scheduling with a battery initialized at 10% SOC. Top figure:Power curve (PU) to be tracked (green line) with total real power demand dueto load switching (black line) and individual dynamic load demands (coloredlines). Middle and bottom figure are battery energy (100% SOC) and batterypower demand (PU) where a positive value indicates charging.</figcaption></figure> <figure><img src="content_image/1607.07365/x5.png"><figcaption>Fig. 5: Load scheduling with a battery initialized at 100% SOC. Top figure:Power curve (PU) to be tracked (green line) with total real power demand dueto load switching (black line) and individual dynamic load demands (coloredlines). Middle and bottom figure are battery energy (100% SOC) and batterypower demand (PU) where a positive value indicates charging.</figcaption></figure> The subtle differences between the load scheduling summarized in Figure 4 and Figure 5 show how the proposed load scheduling algorithm can handle different SOC conditions of the battery, while still tracking the power curve. All this is done, despite the difference in switching dynamics between the loads summarized earlier in Figure 1. ## V Conclusions The optimal scheduling of electrical loads with known and distinct time dependent power demand profiles is solved by formulating a model predictive approach in which only a short term forecast of power production is needed. The length of the short term forecast is determined by the minimum on/off time of the electrical loads. The optimal scheduling is solved by computing the finite number of possible on/off load switching combinations over the short term power forecast and formulating an objective function that minimizes the difference between (real) power production and (real) power demand, while at the same time taking into account constraints on electrical energy storage and power delivery of a battery system. 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1905.00893	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 47816, "num_imgs": 3, "llama3_tokens_count": 14340 }	[ "content_image/1905.00893/x1.png", "content_image/1905.00893/x2.png", "content_image/1905.00893/x4.png" ]	# Dark matter and bubble nucleation in old neutron stars A. Herrero${}^{1}$ M. A. Pérez-García${}^{1}$ mperezga@usal.es J. Silk${}^{2,3,4}$ silk@iap.fr C. Albertus${}^{1}$ albertus@usal.es ${}^{1}$Department of Fundamental Physics, University of Salamanca, Plaza de la Merced S/N E-37008, Salamanca, Spain ${}^{2}$Institut d’Astrophysique, UMR 7095 CNRS, Sorbonne Université, 98bis Blvd Arago, 75014 Paris, France ${}^{3}$Department of Physics and Astronomy, The Johns Hopkins University, Baltimore MD 21218, USA ${}^{4}$Beecroft Institute of Particle Astrophysics and Cosmology, Department of Physics, University of Oxford, Oxford OX1 3RH, UK February 28, 2024 ###### Abstract We study the probability for nucleation of quark matter droplets in the dense cold cores of old neutron stars induced by the presence of a self-annihilating dark matter component, $\chi$. Using a parameterized form of the equation of state for hadronic and quark phases of ordinary matter, we explore the thermodynamical conditions under which droplet formation is facilitated by the energy injection from $\chi$ self-annihilations. We obtain the droplet nucleation time as a function of the dark matter candidate mass, $m_{\chi}$. We discuss further observational consequences. ## I Introduction Neutron stars (NSs) are compact astrophysical stellar objects where the low temperature and high density regions of ordinary matter phase space can be explored [1]. Typically, their measured masses do not exceed a maximum value $M_{}\sim 2M_{\odot}$ and radii $R_{}\sim 11-13$ km. They are thought to be composed out of nucleons, mainly neutrons, with a little fraction of protons and possibly other heavier baryons or even more exotic components [2] besides a leptonic fraction to keep electrical charge neutrality. Central nucleon number densities are thought to be several times that of nuclear saturation density, $n_{0}\sim 0.17\,\rm fm^{-3}$ and effective measured temperatures are in the range $T^{\infty}\sim 10^{5.3}-10^{6}$ K for old NSs with lifetimes $\tau_{\rm NS}\gtrsim 10^{4}$ yr [3]. Under these conditions ordinary matter is typically degenerate since baryonic Fermi energies are of the order of $E_{\rm F,B}\sim 30$ MeV, whereas the internal temperature drops below $T\sim 1$ MeV $(k_{\rm B}=1)$ within $\sim 100$ seconds after the birth of the NS [4]. Although the usual description of the interior of these objects is based on effective nuclear degrees of freedom i.e. nucleons and mesons, other realizations based on quark constituents could indeed happen in nature. Early since the pioneering work by Bodmer and Witten [5; 6] the conjecture of the existence of a most fundamental, quark deconfined, state of matter in the NS core has remained an intriguing possibility. This has been explored in the literature, see for example [7; 8; 10; 9; 11] and references therein. There are two principal mechanisms capable of triggering the deconfinement transition. One involves the increase of the central pressure [15] due to either the accretion of a small amount of hadronic matter or slowed rotation, while the other relies on temperature effects [16; 17; 18; 19; 20]. In [21], the concept of a limiting conversion temperature in the proto-hadronic star is introduced as an indicator of the thermal energy $\Theta\sim 10-30$ MeV that can induce nucleation, provided central densities (or stellar masses) are large enough. In brief, both rely on overcoming the hadronic potential barrier that confines quarks and tunneling out of the nucleon bag that is a few fm in size. Microscopically, the locally deconfined two-flavoured $ud$ quark phase $(Q^{})$ first forms and later proceeds to a $\beta$-equilibrated three-flavoured $uds$ quark matter (QM). Note, however, that whether this former $ud$ phase truly decays has been questioned by recent works relying on arguments of energetic stability [25]. Matter at such high densities can only be partially tested by terrestrial experiments. Sites such as the GSI with the FAiR accelerator, BNL with RHIC, and CERN with the LHC, can use heavy ion collisions (HIC) to produce the so-called quark-gluon plasma [26] consisting of a highly excited hadronic system several fm in size and a lifetime of approximately 20 fm/c. Although temperatures in the fire-ball that is produced are initially high $T\gtrsim 80$ MeV, HIC can test supra-saturation densities and provide tighter constraints on magnitudes such as the high-density behavior of the symmetry energy, the tidal deformability and the equation of state (EoS) of nuclear matter itself, thus linking different areas of interest ranging from astrophysics to nuclear and particle physics [27]. In this work, we are interested in the study of nucleation of quark droplets in a hadronic (nucleon) medium inside the NS core with a novel mechanism mediated by dark matter (DM). This type of matter is one of the key ingredients in our presently accepted cosmological model that remains as yet undetected. It is found to constitute $\sim 26\%$ of our Universe. There are nowadays plenty of candidates from extensions beyond the Standard Model (SM) of particle physics that have been proposed to populate the dark sector [28]. Experimental searches with different strategies try to put constraints on the mass and scattering cross-section phase space $(m_{\chi},\sigma_{\chi})$. For example, while the weakly interacting candidates (WIMPs) have been thoroughly searched for in the last decade with null results [29], other candidates have lately attracted much attention, see [30]. Within standard cosmology, the present relic density can be calculated reliably if the WIMPs were in thermal and chemical equilibrium with the hot SM particles after inflation. In this same context, scanning the $m_{\chi}\sim 10-1000$ GeV range (we use $c=1$) has yielded some results. Direct detection generation II experiments based on nuclear recoils are currently approaching the $\sigma_{\chi}\sim 10^{-47}$$\rm cm^{2}$[31] close to the atmospheric/solar coherent neutrino interaction floor. In addition, they exclude some of the preferred regions arising from scans of SUSY models, see fig 26.1 in [29] in the range $m_{\chi}\sim 30-60$ GeV and $m_{\chi}\sim 10^{2}-10^{3}$ GeV. In indirect searches [32], products of DM annihilation including neutrinos, gamma rays, positrons, antiprotons, and antinuclei can be detected. There are additional sources with constraining power such as those arising from large-scale structure of the Universe that rule out a $m_{\chi}\lesssim 400$ eV WIMP under the Tremaine-Gunn bound [33] along with others. It is generally believed that massive DM particles interact gravitationally and that any non-gravitational couplings are expected to be weakly or strongly interacting in order to at least maintain equilibrium with luminous matter in our Universe. The structure of this contribution is as follows. In Sec. II we present the effective field theory approaches to describe the hadronic content of the interior of the NS with a possible quark phase. We introduce the relativistic Lifshitz-Kagan theory used to describe the induced bubble nucleation due to the presence of a component of DM inside the NS. Later, in Sec. III, we present our results discussing the thermodynamical conditions along with model details more favorable for nucleation of QM bubbles in hadronic matter. We evaluate the nucleation time taking into account the mass of the DM candidate. We discuss the different sources of uncertainty in our modelling and possible further astrophysical observable consequences. Finally, in Sec. IV, we give our conclusions. ## II Modeling the NS interior with a DM component We assume that a non-vanishing component of DM is present in the NS. In an evolved NS, this may be the result of various processes taking place during the lifetime of the star, including its progenitor phase as well. In dense DM environments such as the galactic centre, where there is a high density of neutron stars, clumps of DM of typical mass corresponding to the free streaming mass, of order $10^{-6}\rm M_{\odot}$ but possibly larger Profumo et al. (2006), can occasionally be accreted by NSs. This can provide a rare but substantial enhancement of the DM component of the NS. As we explain below, gravitational capture and depletion processes (typically self-annihilation or decay [35; 36]) modulate the DM population inside the star [37; 38]. Complementary constraints on DM annihilation processes from additional isotropic gamma-ray or reionization and heating of the intergalactic gas backgrounds have been summarized in recent contributions [39]. We will consider DM that self-annihilates through reactions involving quark pairs $\chi{\bar{\chi}}\to q{\bar{q}}\to N\gamma$ producing photon final states. Note that although other channels are indeed available, for simplicity we will stick to this case in what follows but we will later discuss other possibilities. DM candidates ($\chi$) of interest to us will be those with cross sections $\sigma_{\chi N}$ scattering off nucleons (N) and masses $m_{\chi}$ in non-excluded regions of the currently available phase space [29]. We first consider the cold NS core as a system described by the well-known relativistic lagrangian model from [40] consisting of a baryon sector $B=n,p$ (neutrons and protons) interacting through mesonic fields $\mathcal{M}=\sigma,\omega,\rho$ and a minimal leptonic sector $l=e$ (electrons). Besides, the usual non-linear self-interacting potential is included under the form \[\mathcal{U}(\sigma)=\frac{1}{3}a_{1}m_{B}(m_{B}-m^{}_{B})^{3}+\frac{1}{4}a_{2 }(m_{B}-m^{}_{B})^{4},\] (1) with $m_{B}$ being the bare baryon mass and $m^{}_{B}=m_{B}-g_{\sigma B}\sigma$ the effective baryon mass in the medium. Specific values for the couplings of the baryon and meson fields as well as particle masses used can be found in [41]. In the cold system, baryonic density can be expressed in terms of the Fermi momentum $k_{F,B}$ for each particle component as $n_{b}=\sum_{B}\frac{k^{3}_{F,B}}{3\pi^{2}}$. Similarly, for the leptons we have $n_{l}=n_{e}=\frac{k^{3}_{F,l}}{3\pi^{2}}$. The equations for the mesonic fields in the extended system are obtained using a mean field approach so that we replace a generic field $\phi(x)\rightarrow\langle\phi(x)\rangle=\phi$. By doing this the equations for the non-vanishing mesonic field components are obtained. For the $\omega_{0}\equiv\omega$ field one obtains \[m_{\omega}^{2}\omega=\sum\limits_{B}g_{\omega B}n_{B},\] (2) while for the $\rho_{03}\equiv\rho$ field \[m_{\rho}^{2}\rho=\sum\limits_{B}g_{\rho B}\tau_{3B}n_{B},\] (3) being $\tau_{3B}={\rm diag}(1/2,-1/2)$ the isospin 3-rd component matrix operator. Finally for the $\sigma$ field, \[m_{\sigma}^{2}\sigma=-\frac{d\mathcal{U}(\sigma)}{d\sigma}+\frac {1}{2\pi^{2}}\sum\limits_{B}g_{\sigma B}{m^{}_{B}}^{3}\left[t_{B}\sqrt{1+t_{B }^{2}}\right.-\] (4) \[{\rm ln}\left.\left(t_{B}+\sqrt{1+t_{B}^{2}}\right)\right].\] where we have defined $t_{B}=k_{F,B}/m^{}_{B}$. We can also write the chemical potential for baryons as $\mu_{B}=\sqrt{k_{F,B}^{2}+{m^{}_{B}}^{2}}+g_{\omega B}\omega+g_{\rho B}\tau_{ 3B}\rho$ while for leptons $\mu_{l}=\sqrt{k_{F,l}^{2}+m_{l}^{2}}$. Additionally, imposing conditions for electrical charge neutrality $n_{e}=n_{p}$ and $\beta$-equilibrium $\mu_{n}=\mu_{p}+\mu_{e}$ (no neutrinos are trapped) we can obtain the solution for the mesonic fields, provided a $n_{b}$ value is set. This allows to obtain the EoS from contribution of all particle species. From this, the total energy density and pressure to describe the interior of the old NS can be written as a sum of hadronic (Had) and leptonic (l) terms, $\varepsilon=\varepsilon_{\rm Had}+\varepsilon_{l}$ and $P=P_{\rm Had}+P_{l}$, respectively. More explicitly, \[\varepsilon= \frac{1}{8\pi^{2}}\sum\limits_{i=B,l}{m^{}}_{i}^{4}\left[(2t_{i} ^{2}+1)t_{i}\sqrt{1+t_{i}^{2}}-ln\left(t_{i}+\sqrt{1+t_{i}^{2}}\right)\right]\] (5) \[+\mathcal{U}(\sigma)+\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1} {2}m_{\omega}^{2}\omega^{2}+\frac{1}{2}m_{\rho}^{2}\rho^{2},\] and \[P=-\sum\limits_{i=B,l}\varepsilon_{i}+\sum\limits_{i=B,l}n_{i}\mu_{i}.\] (6) We use $t_{l}=k_{F,l}/m_{l}$ and ${m^{}}_{l}=m_{l}=m_{e}$. Let us note that although a proper treatment of the hadronic (quark) system would require using the framework of Quantum Chromodynamics (QCD) at intermediate energies, this calculation is, in practice, technically infeasible. It is for this reason that phenomenologically different approaches using effective field theories with baryonic (mesonic) and more fundamental quark (gluon) degrees of freedom have been exploited in order to determine the EoS of both forms of matter consistently [42] and explore their thermodynamical conditions of stability. In this spirit, to describe the deconfined quark matter phase we use the MIT Bag model [43] where gluon fields are effectively considered through a vacuum pressure, $B$, including interactions through the strong coupling constant $\alpha_{S}$[44]. The perturbative QCD parameter $\alpha_{S}$ characterizes the degree of the quark interaction correction, with $\alpha_{S}=0$ corresponding to no QCD corrections (Fermi gas approximation). For our purposes we will explore values $\alpha_{S}=0.4,0.5$ in line with previous works [10]. In addition, these selected values are also included in the allowed range arising from gravitational wave and astrophysical constraints quoted in [45; 46]. We use this model as it provides a tractable and meaningful way to describe a hypothetical more fundamental configuration of matter although other more refined approaches exist [47]. Regarding this aspect we expect no dramatic modification of the results we find. <figure><img src="content_image/1905.00893/x1.png"><figcaption>Figure 1: Pressure as a function of the baryonic chemical potential forhadronic matter (Had) and deconfined ud matter. We use αs=0.4, 0.5,B=70,100,150 MeVfm−3.</figcaption></figure> For a cold $uds$ quark system, the thermodynamical potential $\Omega_{q}$ for each light flavour with mass $m_{q}$ and chemical potential $\mu_{q}$ ($q=u,d,s$), has a general form \[\Omega_{q}= -\frac{1}{4\pi^{2}}\left[\mu_{q}p_{\rm F,q}\left(\mu_{q}^{2}- \frac{5}{2}m_{q}^{2}\right)+\frac{3}{2}m_{q}^{4}\,\rm{ln}\left(\frac{\mu_{q}+p _{\rm F,q}}{m_{q}}\right)\right]\] (7) \[+\frac{\alpha_{S}}{2\pi^{3}}\Bigg{\{}3\left[\mu_{q}p_{\rm F,q}-m_ {q}^{2}\,\rm{ln}\left(\frac{\mu_{q}+p_{\rm F,q}}{m_{q}}\right)\right]^{2}-2p^{ 4}_{\rm F,q})\Bigg{\}},\] where $p_{\rm F,q}=(\mu_{q}^{2}-m_{q}^{2})^{1/2}$. Here we use the approximation $m_{u}=m_{d}\equiv 0$ and $m_{s}=150$ MeV. Thus for massless quarks $\Omega_{q}$ adopts the simplified form $\Omega_{q}=-\frac{\mu_{q}^{4}}{4\pi^{2}}\left(1-\frac{2\alpha_{S}}{\pi}\right)$. The quark number density is obtained as $n_{q}=-\frac{\partial\Omega_{q}}{\partial\mu_{q}}$. Finally, the energy density for $uds$ matter includes the contribution from the effective bag constant $B$ under the form $\varepsilon_{\rm Quark}=\sum\limits_{q}\left(\Omega_{q}+\mu_{q}n_{q}\right)+B$ or, more explicitly, \[\varepsilon_{\rm Quark}= \frac{3}{4\pi^{2}}\left(1-\frac{2\alpha_{s}}{\pi}\right)(\mu_{u}^ {4}+\mu_{d}^{4})+\frac{3}{8\pi^{2}}m_{s}^{4}\left[x_{s}\eta_{s}(2x_{s}^{2}+1)\right.\] (8) \[-\left.{\rm ln}(x_{s}+\eta_{s})\right]-\frac{\alpha_{s}}{2\pi^{3} }m_{s}^{4}\left\{2x_{s}^{2}(x_{s}^{2}+2\eta_{s}^{2})-3\left[x_{s}\eta_{s} \right.\right.\] \[+\left.\left.{\rm ln}(x_{s}+\eta_{s})\right]^{2}\right\}+B,\] where $x_{s}=\sqrt{\mu_{s}^{2}-m_{s}^{2}}/m_{s}$, $\eta_{s}=\sqrt{1+x_{s}^{2}}$. Note that an additional lepton component $\varepsilon_{l}$ must be present in the total $\varepsilon$ contribution, analogous to that in Eq. (5). For the quark-gluon pressure we have \[P_{\rm Quark}=-\sum\limits_{q}\Omega_{q}-B\] (9) and the total pressure is $P=P_{\rm Quark}+P_{l}$. As before, the conservation of baryonic charge, weak equilibrium and the additional constraint of electric charge neutrality in NS matter can be expressed for the quark phases as $n_{b}=\frac{1}{3}\sum_{q}n_{q}$, $\mu_{d}=\mu_{u}+\mu_{e}$, $\mu_{s}=\mu_{d}$ and $\frac{2}{3}n_{u}=\frac{1}{3}(n_{d}+n_{s})+n_{e}$. In this model the actual first order phase transition comes determined by the thermodynamical conditions of the cold NS interior [21]. For the multicomponent system under scrutiny the Gibbs criteria imposes the equality of baryonic chemical potentials at a given pressure $P_{0}$ where equilibrium holds, \[\mu_{b}(P_{0})\Big{\|}_{\textrm{Had}}=\mu_{b}(P_{0})\Big{\|}_{ \textrm{Quark}},\] (10) and $\mu_{b}=\frac{\varepsilon+P}{n_{b}}$ in the cold system. Let us remark here that, on general grounds, the nature of the transition between the Hadron-Resonance Gas (HRG) and Quark-Gluon Plasma (QGP) phases of nuclear matter can be presented in terms of the phase diagram of QCD as a function of temperature T and baryon chemical potential $\mu_{b}$. At large T and small $\mu_{b}$, the transition is expected on the basis of lattice calculations to be a rapid crossover, whereas it is naturally expected that the phase transition along the $\mu_{b}$-axis (with $T=0$) is an actual first-order phase transition [22; 23]. Somewhere along the phase transition line the point at which this occurs is commonly known as the QCD critical point that has not yet been experimentally confirmed nor predicted with theoretical certainty [24]. In our stellar scenario DM with $m_{\chi}\gtrsim 10$ GeV and cross sections $\sigma_{\chi N}$ in the currently allowed phase space can thermalize and display a radial distribution inside the NS, $\rho_{\chi,}(r)$, built on top of that of ordinary baryonic matter. The ratio of gravitational to thermal energy makes DM to follow a Gaussian distribution $\rho_{\chi,}(r)\sim e^{-(r/r_{\rm th})^{2}}$ with a radius $r_{\rm th}=\sqrt{\frac{3k_{B}T}{2\pi G\rho_{B}m_{\chi}}}$. This central region is where most DM annihilations will take place. We consider that the inner NS core has a baryonic mass density around three times that of nuclear saturation density, $\rho_{B}\sim 7\times 10^{14}$$\rm g/cm^{3}$, and an internal temperature $T\sim 10^{5}$ K at a given galactic location with a corresponding ambient DM density, $\rho_{\chi}$. Typical values in the solar neighbourhood are $\rho_{\chi,\rm local}=(0.39\pm 0.03)$$\rm GeV/cm^{3}$[48] although NSs are found closer to the galactic centre where they can be higher by a factor of $\sim$100. The NS captures DM, up to factors of order unity, at a rate $C_{\chi}$[(49; 50)] \[C_{\chi}\simeq 1.8\times 10^{23}\left(\frac{100\,\rm GeV}{m_{\chi}}\right) \left(\frac{\rho_{\chi}}{\rho_{\chi,\rm local}}\right)f_{\chi,N}\,\,\rm s^{-1}.\] (11) $f_{\chi,N}$ denotes a phenomenological factor dealing with the opacity of stellar matter as it depends on the ratio of the leading contribution of $\chi N$ scattering cross section $\sigma_{\chi N}$ to the minimum geometrical cross section defined as $\sigma_{0}\sim\frac{m_{B}}{M_{}}R_{}^{2}\sim 10^{-45}\rm cm^{2}$. Thus this factor saturates to unity $f_{\chi,N}\sim 1$ if $\sigma_{\chi N}\gtrsim{\sigma_{0}}$, whereas it decreases the capture rate otherwise. As current experimental efforts foresee sensitivities below $\sigma_{0}$ for some $m_{\chi}$ windows, this would imply lower amounts of DM inside the NS by this same factor. However this aspect is not critical as a tiny content of DM can induce important changes in the NS EoS, as we will explain. The DM particle population number inside the star, $N_{\chi}$, will not only depend on the capture rate $C_{\chi}$ but also on the self-annihilation rate, $C_{a}$, in our scenario. This latter is model-dependent but we can estimate it to be $C_{a}\sim\langle\sigma_{a}v\rangle/r_{\rm th}^{3}$. Numerically, \[C_{a}=1.1\times 10^{-30}\left(\frac{\langle\sigma_{a}v\rangle}{3\times 10^{-26 }\,\rm cm^{3}s^{-1}}\right)\left(\frac{10^{5}\,\rm K}{T}\frac{m_{\chi}}{100\, \rm GeV}\right)^{3/2}\,\rm s^{-1}.\] (12) Therefore $N_{\chi}$ can be obtained as a function of time $t$ by solving the differential equation for the NS core $\frac{dN_{\chi}}{dt}=C_{\chi}-C_{a}N^{2}_{\chi}$ considering the two competing processes of capture and self-annihilation. The solution can be written as \[N_{\chi}(t)=\sqrt{\frac{C_{\chi}}{C_{a}}}\rm\,tanh\left[\frac{t}{\tau}+\gamma( N_{\chi,0})\right],\] (13) where $N_{\chi,0}$ is the DM population in the final stage of its progenitor phase and $\gamma(N_{\chi,0})={\rm tanh}^{-1}\left(\sqrt{\frac{C_{\chi}}{C_{a}}}N_{\chi,0 }\right)$ with $\tau^{-1}=\sqrt{C_{\chi}C_{a}}$. For times large enough so that the system has reached the steady state, $t\gg\tau$, the asymptotic population is given by $N_{\chi}(t_{\infty})\simeq\sqrt{\frac{C_{\chi}}{C_{a}}}$. Under such conditions, we model the average energy release from annihilation processes as obtained from the spectrum of the reaction $\chi{\bar{\chi}}\to q{\bar{q}}\to N\gamma$. Although other channels are indeed possible [51], we restrict our modeling to this one as the mechanism presented here will not be dramatically altered. The spectrum $\frac{dN_{\gamma}}{dE}$ provides the average energy release as $\langle E\rangle=\int_{0}^{m_{\chi}}E\frac{dN_{\gamma}}{dE}\,dE$. Note that the upper limit in the integral is due to the fact that the center of mass energy is $E_{\rm CM}=\sqrt{s}=2m_{\chi}$ being $s$ the Mandelstam variable and each quark (antiquark) carries a maximum value $E_{\rm q}=\sqrt{s}/2=m_{\chi}$. We have used the PYTHIA package [52] to obtain that as an indication, for light quarks approximately $67\%$ of the DM mass $m_{\chi}$ is deposited in the medium in the form of photons. Heavier quarks, alternative channels or even corrections due to energy loss and final state kinematics from the most energetic photons would indeed change the size of the energy injection. Since the energetics of the microscopic particle physics event in the very dense medium inside the NS core with $\sim 10^{34}$$\rm nucleons/cm^{3}$ is rather complex this treatment constitutes just an approximate description. However, we do not expect it dramatically alters the nucleation mechanism itself as exposed here. Typically, the photons produced will be obtained from different highly energetic hadronic as well as electromagnetic showers involving inelastic interactions with the dense medium. In this way, the hadrons with a cross-section $\sigma_{\rm Had}$, which is largely uncertain but can be roughly estimated to be $\sigma_{\rm Had}\sim 1$$\rm fm^{2}$ will produce showers with typical sizes depending on the baryonic density in the NS core. Taking a density $n_{b}\sim(3-5)n_{0}$ and typical energies $E\sim 1-10^{5}$ GeV for each event we obtain a size $X_{\rm Had}\sim\frac{1}{n_{b}\sigma_{\rm Had}}\rm Log\left(\frac{E}{10\,\rm MeV }\right)\lesssim 10$ fm for the hadronic showers while $X_{\rm EM}\sim 1$ fm for the electromagnetic showers [53]. The injected energy is mainly contained into the bubble region with radius $R$ as long as $X_{\rm Had}\lesssim 2R$. <figure><img src="content_image/1905.00893/x2.png"><figcaption>Figure 2: Potential barrier energy as a function of radius for differentcombinations of (αS,B,σ). Solid, dashed and dotted lines depict P=P0+32 MeV,P=P0+42 MeV, P=P0+52 MeV for B=120 MeV/fm3 (upper panel) and B=100 MeV/fm3(lower panel). Red curves on the upper panel have been scaled a factor 1/20 toimprove readability. See text for details.</figcaption></figure> The global picture is thus that of a central DM annihilation volume, where both types of matter coexist, and baryonic matter is subject to steady state energy injection from the quoted DM reactions. To explore whether this scenario could lead to induced quark bubble nucleation we use the relativistic Lifshitz-Kagan theory [54]. We aim to describe the microscopics of a locally induced phase transition from an effective (metastable) hadronic phase to a more fundamental deconfined state. In this formalism the relativistic lagrangian that describes the formation of a fluctuation i.e. a spherical bubble of mass $M$ and radius $R$ is given by \[\small\mathcal{L}(R,\dot{R})=-M(R)\sqrt{1-\dot{R}^{2}}+M(R)-U(R),\] (14) where $\dot{R}=dR/dt$ is the radial growth rate and $U(R)$ is an effective potential depending on the thermodynamical conditions of the medium \[U(R)=\frac{4}{3}\pi R^{3}n_{\rm Quark}(\mu_{\rm Quark}-\mu_{\rm Had})+4\pi \sigma R^{2}.\] (15) We label $\mu_{\rm Had}$ ( $\mu_{\rm Quark}$) as the chemical potentials of the hadronic (quark) phases of matter at fixed pressure value, $P$. $n_{\rm Quark}$ is the number density of the quark phase and $\sigma$ is the surface tension among the two phases. Contributions from volume as well as surface terms are the most energetically relevant although more corrections can be included [55]. As written, this expression may be familiar to the reader as it also used in terrestrial experiments for DM searches such as PICO [56] at SNOLAB, with an analogous strategy based on detecting $\chi$ scattering events in bubble chambers filled with overheated liquids at much smaller densities. When nucleated, QM bubbles can be characterized by a critical radius for stability, \[R_{c}=\frac{3\sigma}{n_{\rm Quark}(\mu_{\rm Quark}-\mu_{\rm Had})},\] (16) fulfilling the relation $U(R_{c})=0$ such that for $R>R_{c}$, the bubble is energetically stable. The potential barrier maximum height to be tunneled is \[U_{\rm max}=\frac{16}{27}\pi\sigma R_{c}^{2}.\] (17) In order to drive further changes at the macroscopic level QM bubbles must have the capability to last sufficiently and, in any case, longer than the time scale for stellar dynamical collapse $\tau_{D}\sim\sqrt{2R_{}^{3}/2GM_{}}\sim 3\times 10^{-5}$ s. Making a transformation to the phase space canonical variables $q,p$ one can obtain the Hamiltonian associated to the lagrangian in Eq.(14) as $\mathcal{H}(q,p)=p\,\dot{q}-\mathcal{L}$. As explained in [10] by taking the _WKB_ approximation [66], one can compute the energy of the ground state, $E_{0}$. The associated oscillation frequency $\nu^{-1}_{0}=\frac{dI}{dE}\|_{E=E_{0}}$ is obtained [10] using the expression \[I(E)={2}\int_{0}^{R_{-}}\sqrt{\left[2M(R)+E-U(R)\right]\left[U(R)-E\right]}\,dR,\] (18) being $R_{-}$ the smaller turn around radius. The action under the potential barrier \[A(E)={2}\int_{R_{-}}^{R_{+}}\sqrt{\left[2M(R)+E-U(R)\right]\left[U(R)-E\right] }\,dR.\] (19) determines the tunneling probability $p_{0}=\rm exp\left(-\frac{A(E)}{\hbar}\right)$. In this system, the average energy per quark injected into the nucleon bag from DM self-annihilation events necessary to create a stable spherical droplet of size $\sim R_{c}$ can be approximated by $E_{\rm inj}\sim\frac{\langle E\rangle}{n_{b}4\pi R^{3}_{c}}$. We assume that the energetic shower is contained into the bubble, however, if this is not the case a decreasing factor $\xi\sim(2\rm R/X_{\rm Had})$ must be included. Thus the photon yields from DM self-annihilation act effectively to raise the energy level of otherwise confined quarks. The bubble formation time is obtained as $\tau^{-1}=\nu_{0}p_{0}$. It is important to notice that the nucleation process may most probably happen over the whole DM thermal volume $\sim r^{3}_{\rm th}$. Therefore a number of nucleation centers are available for this process and can be estimated as $N_{C}\sim(\frac{r_{\rm th}}{R_{c}})^{3}$. In such a case the corrected nucleation time is given by $\tau_{N}=\tau/N_{C}$. As explained in the introduction section of this contribution, the nucleation of quark droplets relying on standard astrophysical mechanisms is highly suppressed. This means, in practice, that nucleation times may be much larger than stellar lifetimes. As we will explain below, as a result of an extra energy injection from DM self-annihilation the hadronic system can be driven into an excited configuration allowing the formation of stable quark droplets more easily. In our description of the cold nuclear system we have not considered the possibility of the existence of a fraction or paired nucleons. Typically for the high densities and low temperatures in the NS core the paired nucleon fraction is less than $\sim 10\%$. At supranuclear densities the pairing gaps, $\Delta$, for neutrons (${}^{3}P_{2}$–${}^{3}F_{2}$ channels) and for protons (${}^{1}S_{0}$ channel) have been estimated yielding sizes up to a few MeV. Instead, for quarks they may be as large as a hundred MeV, see for example [57] for a review. In brief, the density of paired nucleons depends on this gap [58] as well as the quantity $\sqrt{(E_{N}(k,m^{}_{N})-\mu^{}_{N})^{2}+\Delta^{2}(k)}$ involving the nucleon single-particle energy $E_{N}$, effective mass $m^{}_{N}$ and effective chemical potential $\mu^{}_{N}$ for a fixed momentum $k<k_{F,N}$. The existence of such a gapped fraction adds an extra amount of energy ($E_{\rm BCS\,pair\,break}>2\Delta(k_{F,N}$)) that must be overcome in order to break the correlated pair as dictated by the BCS theory [59]. In this sense, the possible existence of a paired fraction tends to suppress the nucleation process. However, it is worth noting that, safely, for nucleons $\Delta(k_{F,N})/m_{\chi}\ll 1$ for typical NS core densities and most of the $m_{\chi}$ range explored in our study and similar applies for quarks. Thus we expect that the possible corrections from kinematically blocked states in the final phase space or paired components will not have a dramatic impact and the scattered fermions will have energies above the Fermi level for DM masses larger than a few GeV. However, this fact must be carefrully reconsidered for $m_{\chi}\lesssim 1$ GeV as this could be a competing effect quenching the efficiency of nucleation. For simplicity we have neglected this possibility leaving it for a future contribution. ## III Results In this section we present our results. We have used arbitrary values $B\in[70,150]$$\rm MeV\,fm^{-3}$ as usually done in the literature, however recent calculations [60] restrict stable solutions of $T=0$$uds$ QM to a window $60\lesssim B\lesssim 80$$\rm MeV\,fm^{-3}$. In addition, if confirmed, quark stars having masses beyond $\sim 2M_{\odot}$ limit would imply even smaller $B$ values [45]. In addition values of the surface tension are poorly known. This quantity is relevant for bubble nucleation in different already studied environments, for example quark matter in supernovae [61] and neutron star mergers [62]. It is also relevant for a possible quark-hadron mixed phase in the interior of neutron stars [63]. Some available calculations have estimated its value using different complementary approaches such as Nambu-Jona-Lasinio models, quark-meson models [64] or chiral models [65] where in the latter maximum values point towards $\sigma\sim 15$$\rm MeV\,fm^{-2}$. For our purposes we will consider $\sigma\in[10,30]$$\rm MeV\,fm^{-2}$. In Fig.(1) we plot pressure as a function of the baryonic chemical potential $\mu_{b}-m_{B}$ for hadronic matter (Had) and deconfined $ud$ matter. We use $\alpha_{S}=0.4$, $0.5$, $B=70,100,150$$\rm MeV\,fm^{-3}$ and $\sigma=30$$\rm MeV\,fm^{-2}$. As the baryonic chemical potential (density) grows the softer quark matter EoS overpasses that of hadronic matter at a transition point i.e. the crossing of both curves. The higher the $B$ ($\alpha_{S}$) the higher the chemical potential associated to the transition density. The transition pressures and baryonic densities for $\alpha_{S}=0.4$ are $P_{0}=588,483,401$$\rm MeV\,fm^{-3}$ and $n_{b}=0.92,0.87,\,0.78$$\rm fm^{-3}$ for $B=150,100,70$$\rm MeVfm^{-3}$, respectively. These results do not depend on $\sigma$. If instead we use $\alpha_{S}=0.5$ and $B=100$$\rm MeV\,fm^{-3}$, we find $P_{0}=600$$\rm MeV\,fm^{-3}$, $n_{\rm Had}=0.95$$\rm fm^{-3}$, $n_{\rm Quark}=1.80$$\rm fm^{-3}$. In Fig.(2) we plot the potential barrier $U(R)$ as a function of radius for different parameter sets. Solid, dashed and dotted lines depict $P=P_{0}+32$ MeV, $P=P_{0}+42$ MeV, $P=P_{0}+52$ MeV for each case. In the upper panel we fix $\alpha_{S}=0.4$, $B=120$$\rm MeV\,fm^{-3}$, $\sigma=10$$\rm MeV\,fm^{-2}$ (black curves) and $\sigma=30$$\rm MeV\,fm^{-2}$ (red curves). The latter case (red lines) has been scaled a factor $1/20$ in order to numerically compare on the same axis. The pressure at the transition point is $P_{0}=525$$\rm MeV\,fm^{-3}$ while the density is $n_{\rm Had}=0.88$$\rm fm^{-3}$, $n_{\rm Quark}=1.58$$\rm fm^{-3}$. We can see that for higher $\sigma$ values, larger bubbles are required for stability and, at the same time, more energetic barriers form. Instead, for increasing pressure, smaller bubbles can survive. In the lower panel we fix $B=100$$\rm MeV\,fm^{-3}$, $\sigma=30$$\rm MeV\,fm^{-2}$, $\alpha_{S}=0.4$ (blue curves) and $\alpha_{s}=0.5$ (green curves). We can clearly see that for larger $\alpha_{S}$ values i.e. including less strong corrections, smaller bubbles are predicted. <figure><img src="content_image/1905.00893/x4.png"><figcaption>Figure 3: Nucleation time as a function of the DM particle mass for differentcases. Coloured bands depict old NS ages (blue horizontal) and SUSY favouredregions (pink vertical). The Universe lifetime τU∼1017.6 s is also shown. Seetext for details.</figcaption></figure> In Fig.(3) we plot the logarithm (base 10) of the nucleation time as a function of the DM particle mass for different parameter sets used. The amount of DM for each $m_{\chi}$ value is obtained from the asymptotic population $N_{\chi}(t_{\infty})$ ranging from $N_{\chi}\sim 10^{30}$ for $m_{\chi}=1$ GeV to $N_{\chi}\sim 10^{23}$ for $m_{\chi}=10^{6}$ GeV. These values are in all cases below the critical value given by their fermionic or bosonic nature, see [36; 67]. In addition, given the spread of the hadronic/quark parameters scanned, the pressure values considered for evaluation are selected above their corresponding $P_{0}$ for each case. For the first set we fix $\alpha_{S}=0.4$, $B=70$$\rm MeV\,fm^{-3}$, $\sigma=10$$\rm MeV\,fm^{-2}$ and select $P=406$$\rm MeV\,fm^{-3}$ (long dashed). For the second set (same $\alpha_{S}$ as before) $B=120$$\rm MeV\,fm^{-3}$, $\sigma=30$$\rm MeV\,fm^{-2}$ with $P=535$$\rm MeV\,fm^{-3}$ (short dashed), and third set $\sigma=10$$\rm MeV\,fm^{-2}$ (solid). For the fourth one $B=100$$\rm MeV\,fm^{-3}$, $\sigma=30$$\rm MeV\,fm^{-2}$ with $P=515$$\rm MeV\,fm^{-3}$ (dotted) and, finally, the fith set is the same as the latter case but using $\alpha_{S}=0.5$ (dotdashed). We plot additional coloured regions with constraints coming from NS ages typically measured (blue horizontal), see Table I in [3] with $\tau_{NS}\sim 10^{9.5}-10^{13.5}$ s. They can even reach $\sim 4.9$ Gyr as in PSR J0437–4715 [68], close to the upper limit provided by the Universe lifetime (yellow doble dash-dotted line) with $\tau_{U}\sim 10^{17.6}$ s. We also plot the regions (pink vertical) denoting the preferred four typical SUSY models, CMSSM, NUHM1, NUHM2, pMSSM10 which integrates constraints set by ATLAS Run 1 [29]. We can see that nucleation times are critically dependent on the $B,\alpha_{S}$ and $\sigma$ values of the quark phase. The thermodynamical conditions (central pressure) produce a moderate impact on the results. Increasing the bubble critical radius i.e. $\sigma$, produces a high energetic cost. The energy injection obtained from the self-annihilating $\chi$ pair signals the kink where probability of nucleation $p_{0}\sim 1$. The results obtained for the $B=70$$\rm MeV\,fm^{-3}$ set properly belong to the quoted stability window of QM as obtained in earlier works [60]. This case requires a $m_{\chi}\gtrsim 4\times 10^{5}$ GeV to induce nucleation in the central thermal volume inside the NS. In this scenario, light DM candidates ($m_{\chi}<10$ GeV) would less favour such a conversion in ordinary NSs. Baryonic densities at the transition point found for each case are indeed in the previously estimated interval $\sim(3-4)n_{0}$. Note that additional corrections due to final state limitations of the energy injected as well as hadronic cascade sizes versus bubble radius will correct the energy efficiency by factors $\mathcal{O}(1)$. Thermal energy loses are found to be negligible in a cold system. In addition, other basic channels into which DM particles may annihilate could happen, including heavy quarks, leptons, weak or Higgs bosons [39]. Besides, a direct neutrino/antineutrino annihilation channel or neutrinos originating from the decays of the particles produced in the annihilation could take place. This could modify effectively the energy injected into the central hadronic region altering the efficiency of the process [69]. The hierarchy problem, also framed as requiring naturalness or the absence of fine tuning, prefers WIMP masses below about a few TeV, which is considerably more constraining than the WIMP coincidence which essentially allows particles from a few GeV to about 100 TeV [12]. Nevertheless in our analysis we have considered maximum values of $m_{\chi}\lesssim 10^{3}$ TeV in consistency with the wimp-like scenario depicted here and with capture rates in accordance with a typical weakly interacting DM candidate being accreted by the NS, see results from CTA [13] or MAGIC [14]. If more massive DM candidates were considered additional corrections to several quantities e. g. capture rates, effective hadronic potentials, corrections from final phase space multiple channels or excited states of quark degrees of freedom should be considered. A remark is due at this point regarding the feasibility of the percolation transition in the macroscopic stellar object. The mechanism quoted for nucleation in the relativistic Lifshitz-Kagan theory involves matter undergoing this process is at a metastable state i.e. at pressures larger than the transition pressure $P_{0}$. Given an object with mass $M_{}$ and radius $R_{}$ it may happen that displays either pure hadronic or nuclear-quark hybrid mixed nature. As we mentioned in the introduction section we have focused on the nucleation mechanism for matter in NS cores under the hadronic state in consistency with the DM-nucleon scattering cross sections used in the capture rate in Eq. (11). Additional corrections would arise from a hybrid NS or even a quark star and will be treated elsewere. In the hadronic stellar object the existence of such a macroscopic percolated region is expected to convert the full NS as already explained in [15] producing an energetic $10^{51}-10^{53}$ erg gamma ray burst (GRB). On the other hand for some of the parameter sets explored in our work (see dot-dashed line curve in Fig. (3)) this mechanism would induce a rapid nucleation that, if progressing to macroscopic, seems to be hardly reconcilable with the frequency of very short GRBs observed [70]. The correspondance of the $m_{\chi}$ to the probability of nucleation of quark droplets seems remarkably sensitive under this mechanism. At a microscopic level, a dynamical study of the droplet boundary once formed has not been studied in detail yet. If the full NS converts to a more compact QM star this exotic type of matter would be ejected leaving an imprint on the cosmic rays or scintillation patterns [71; 72; 73]. Boundary conditions for isolated clusters have been somewhat explored in [74] where, arising from the strange quark content, the name of _strangelet_ is coined. If a strangelet is not in flavour equilibrium, it can decay via weak semileptonic decays, weak radiative decays and electron capture. Other modes of decay reduce the baryon number instead. Further work on this is under progress and will be reported elsewhere. ## IV Conclusions We have studied the nucleation of quark droplets in the NS core facilitated by the energy injection due to a component of self-annihilating DM present inside neutron stars. We find that under the effective field theoretical description of the hadronic and quark phases, the latter using a MIT bag model, the dark matter candidate mass highly influences the nucleation time $\tau_{N}$. Depending on the central pressure (density) conditions inside the star nucleation times may span 40 orders of magnitude. Within this scenario light dark matter is less favoured to produce such a percolation phase transition inside the core central region. However, for parameter sets within a window of energetic stability for stellar mass-radius values of $m_{\chi}\gtrsim 4\times 10^{5}$ GeV are found capable of nucleating the macroscopic thermal DM core during the NS lifetime and drive a conversion into a more compact star. Emission of radiation (GRB) or chunks of matter (cosmic rays) is to be expected along with gravitational waves. To determine the temporal sequence of the multi-messenger signal and magnitude of the effects presented here, detailed calculations of the central EoS instability are needed and are left for future work. ## V Acknowledgments We would like to thank S. Heinemeyer, C. Kouvaris for useful discussions. This work has been supported by Junta de Castilla y León SA083P17. 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1703.03520	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 28793, "num_imgs": 4, "llama3_tokens_count": 8460 }	[ "content_image/1703.03520/x1.png", "content_image/1703.03520/x2.png", "content_image/1703.03520/x3.png", "content_image/1703.03520/Fig04_aniket.jpg" ]	# Resistively-detected NMR lineshapes in a quasi-one dimensional electron system M. H. Fauzi Department of Physics, Tohoku University, Sendai 980-8578, Japan A. Singha Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India M. F. Sahdan Department of Physics, Tohoku University, Sendai 980-8578, Japan M. Takahashi Department of Physics, Tohoku University, Sendai 980-8578, Japan K. Sato Department of Physics, Tohoku University, Sendai 980-8578, Japan K. Nagase Department of Physics, Tohoku University, Sendai 980-8578, Japan B. Muralidharan Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India Y. Hirayama Department of Physics, Tohoku University, Sendai 980-8578, Japan Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan February 26, 2024 ###### Abstract We observe variation in the resistively-detected nuclear magnetic resonance (RDNMR) lineshapes in quantum Hall breakdown. The breakdown is locally occurred in a gate-defined quantum point contact (QPC) region. Of particular interest is the observation of a dispersive lineshape occured when the bulk 2D electron gas (2DEG) is set to $\nu_{\rm{b}}=2$ and the QPC filling factor to the vicinity of $\nu_{\rm{QPC}}=1$, strikingly resemble the dispersive lineshape observed on a 2D quantum Hall state. This previously unobserved lineshape in a QPC points to simultaneous occurrence of two hyperfine-mediated spin flip-flop processes within the QPC. Those events give rise to two different sets of nuclei polarized in the opposite direction and positioned at a separate region with different degree of electronic spin polarization. pacs: 73.43.Fj, 76.60.$-$k Recent advent in NMR technique through a resistive detection (RDNMR) has made it possible to study various spin physics in a 2D quantum Hall systemKumada _et al._ (2007); Zhang _et al._ (2007); Stern _et al._ (2012); Tiemann _et al._ (2012); Friess _et al._ (2014); Tiemann _et al._ (2014); Piot _et al._ (2016), and a quasi-1D channelKou _et al._ (2010); Kawamura _et al._ (2015). Despite the success achieved, a certain aspect related to the origin of the RDNMR lineshape variations noted experimentally in continuous wave (cw) mode is still poorly understood. One of them involved the puzzling observation of a dispersive lineshape in the quantum Hall state, a resistance dip followed by a resistance peak resonance line with increasing radio frequencyGervais (2009). It is first reported by Desrat et alDesrat _et al._ (2002) in the vicinity of $\nu_{\rm{b}}=1$ and has been confirmed in a number of follow-up papersPiot _et al._ (2016); Tracy _et al._ (2006); Kodera _et al._ (2006); Dean _et al._ (2009); Bowers _et al._ (2010); Desrat _et al._ (2013, 2015). Similar dispersive like lineshape has been observed as well in the vicinity of $\nu_{\rm{b}}=2/9$Gervais _et al._ (2005), $\nu_{\rm{b}}=2/3$, $\nu_{\rm{b}}=1/3$Stern _et al._ (2004), and at $\nu_{\rm{b}}=2$ Landau level crossingYang _et al._ (2011). A number of appealing explanations has been put forward, but none of them provides a comprehensive explanation. Part of the reason why it still is difficult to unravel its physical origin is that we do not have a mature level of understanding about many-body 2D electronic states at the first Landau level yet, let alone their coupling to the nuclear spin. Thus, it would be highly desirable to study the lineshape variations in a platform where one can avoid such complexity. <figure><img src="content_image/1703.03520/x1.png"><figcaption>Figure 1: (a)-(b) Schematic of potential barrier seen by up-spin and down-spinelectrons without (solid line) and with (dashed line) the presence of positiveand negative nuclear polarization, respectively. The chemical potential windowsits at νQPC<1, so that only the up-spin channel affects the transport. (c)Differential diagonal resistance Rd≡dVd/dIAC curve versus split gate biasvoltage (VSG) at a field of 4.5 T (black) and 4.25 T (red). The left and rightsplit gate are biased equally. The center gate voltage VCG is fixed to −0.425V and −0.4 V, respectively. Upper inset displays a schematic drawing ofdevice. Cross marks represent Ohmic contact pads. Triple Schottky gatesdeposited on top of the Hall bar defined a quantum point contact (see SEMimage). The lithographic gap(width) between(of) a pair of split gate is600(500) nm. An extra gate (center gate) with lithographic width of 200 nm isdeposited in between the split gates. An excitation current IAC=1 nA withf=13.7 Hz is applied to the device for transport measurement. Lower insetshows typical Rd time trace during current-induced dynamic nuclearpolarization with IAC=10 nA.</figcaption></figure> <figure><img src="content_image/1703.03520/x2.png"><figcaption>Figure 2: (a) Lower plot shows a 2D color map of 75As RDNMR traces at theupper flank of the νQPC=1 plateau, −0.70≤VSG≤−0.41 V, measured at 4.5 T. Thebackground resistance has been subtracted from the spectrum. Upper plot showsthe blown-up spectra in between −0.46≤VSG≤−0.41 V to accentuate the dispersivestructure. (b) The RDNMR amplitude percentage vs split gate normalized to theoff-resonance resistance, \absΔRd/Rd, extracted from panel (a). (c)-(d) RawRDNMR data sliced at the VSG=−0.7 and VSG=−0.41 V, respectively. RDNMR in reddots superimposed in panel (d) measured very close to the bulk 2DEG ν=2plateau, served as a reference signal with almost zero Knight shift. Thesignal is obtained by applying IAC=100 nA. The red line is a Gaussian fit tothe spectrum with the FWHM of 8.8 kHz. (e) The position of peak resonancefrequency (black dots) and dip resonance frequency (red dots) extracted frompanel (a) for −0.50≤VSG≤−0.41 V. (f) The peak-to-dip resonance frequencyseparation Δf extracted from panel (e). All the spectra measured with IAC=10nA (except the ref. signal) and RF power is −30 dBm.</figcaption></figure> In this Rapid Communication, we resort to a quasi-one dimensional system in a gate-defined quantum point contact (QPC) to study various possible lineshapes including the dispersive lineshape noted experimentally in cw mode. Unlike on the 2D system, the mechanism for generation and resistive detection of nuclear spin polarization is tractable, allowing conveniently a direct interpretation of the observed lineshapes. Generation and detection of nuclear spin polarization are achieved by setting the filling factor in the bulk 2DEG to $\nu_{\rm{b}}=2$ and $\nu_{\rm{QPC}}=1$ in the QPCWald _et al._ (1994); Dixon _et al._ (1997); Machida _et al._ (2002); Würtz _et al._ (2002); Deviatov _et al._ (2004); Würtz _et al._ (2005); Masubuchi _et al._ (2006); Córcoles _et al._ (2009); Keane _et al._ (2011); Chida _et al._ (2012). Fig. 1(a)-(b) schematically displays how the nuclear polarization affects the transmission probability through the potential barrier of the QPC. For $\nu_{\rm{QPC}}<1$ (the down-spin channel $T_{\downarrow}$ does not affect the transport), the up-spin channel $T_{\uparrow}$ sees an increase(decrease) in the barrier potential in the presence of positive(negative) nuclear polarization, where positive (negative) means nuclear polarization is parallel (opposite) to the external magnetic field. Consequently, the transmission probability of the up-spin channel is reduced(enhanced). Therefore, the transmission is modified by a dynamic nuclear polarization (DNP) under a steady state where nuclear spins diffuse from the polarized regions to the center of the QPC. At sufficiently high current densities, there are two possible tractable DNPs by hyperfine-mediated inter edge spin-flip scattering within the lowest Landau level, namely forward and backward spin-flip scatteringsWald _et al._ (1994); Dixon _et al._ (1997); Singha _et al._ (2017). The first (second) one involves a spin-flip scattering from the forward propagating up-spin (down-spin) channel to the forward (backward) propagating down-spin (up-spin) channel, which in turn produces the positive (negative) nuclear polarization through the spin flip-flop process. On sweeping the rf field after the polarization reaches a steady state, those two different sets of nuclear polarization would leave a different trace in the RDNMR signal; with the positive (negative) one resulting in a resistance dip (peak). Here we demonstrate that under certain electronic state in the QPC, those two sets of nuclei can be generated simultaneously in a separate region within the QPC. Since they experience different degree of electron spin polarization, one can observe a combination of a resistance dip and peak resonance line in the RDNMR spectrum, namely dispersive lineshape. Our studies are carried out on a 20-nm-wide doped GaAs quantum well with the 2DEG located $165$ nm beneath the surface. The wafer is photo-lithographically carved into a $30$-$\mu$m-wide and $100$-$\mu$m-long Hall bar geometry. The low temperature electron mobility is $84.5$ m${}^{2}$/Vs at an electron density of $1.0\times 10^{15}$ m${}^{-2}$. A single QPC defined by triple Schottky gates is patterned on top of the Hall bar by Ti/Au evaporation. The bulk 2DEG density $n$ can be tuned by applying back gate voltage ($V_{\rm{BG}}$) to Si-doped GaAs substrate. It enables us to control the filling factor of interest in the bulk 2DEG $\nu=\frac{h}{eB}n$ with back gate $V_{\rm{BG}}$ and magnetic field $B$. The samples are mounted inside a single-shot cryogenic-free ${}^{3}$He refrigerator with a temperature of 300 mK. A six-turn rf coil wrapped the sample to be able to apply an oscillating magnetic field in the plane of the 2DEG. Throughout this study, the amplitude of rf power delivered to the top of the cryostat is fixed to $-30$ dBm (unless specified otherwise). Fig. 1(c) displays two sets of diagonal resistance traces as a function of split gate bias voltage across the QPC measured at a field of $4.5$ (black line) and $4.25$ (red line) T. The center gate is fixed to $V_{\rm{CG}}=-0.425$ V and $V_{\rm{CG}}=-0.4$ V, respectively[32]. We start with fully filled first Landau level in the bulk 2DEG ($\nu_{\rm{b}}=2$), where both the up-spin and down-spin electrons are available for transmission. Applying negative voltage on the split gates allows us to selectively transmit the up-spin channel through the constriction and reflect the down-spin channel. The nuclear spins is dynamically polarized by applying $I_{\rm{AC}}=10$ nA at a selected operating point along the diagonal resistance trace on both sides of the $\nu_{\rm{QPC}}=1$ plateau. Typically, the resistance increases exponentially and reaches a point of saturation on the time scale of a few hundred seconds with the characteristic exponential rise time of about $150$ seconds (see the lower inset of Fig. 1), similar time scale characteristic is reported previously on other QPC structuresCórcoles _et al._ (2009). Once the resistance saturated, the rf is swept across the Larmor frequency of ${}^{75}$As nuclei while measuring its resistance. The rf sweep rate is set to $100$ Hz/s[33]. We observe variation in the RDNMR lineshape spectra on both flank of the $\nu_{\rm{QPC}}=1$ plateau as displayed in Fig. 2 and 3. Let us start with the RDNMR spectra for $\nu_{\rm{QPC}}<1$ case observed at a field of $4.5$ T shown in Fig. 2(a), measured from $V_{\rm{SG}}=-0.41$ up to $V_{\rm{SG}}=-0.7$ V. For ease of comparison, we plot the resistance variation $\Delta R_{\rm{d}}$ with respect to the off-resonance resistance at $f=33$ MHz. The salient feature appears in a narrow portion of the split gate bias voltage region, $-0.50\leq V_{\rm{SG}}\leq-0.41$ V, very close to the $\nu_{\rm{QPC}}=1$ plateau. The spectra have a curious dispersive lineshape, strikingly resemble the dispersive lineshape previously observed in a number of reports on a 2D quantum Hall system in the vicinity of $\nu_{\rm{b}}=1$Desrat _et al._ (2002); Piot _et al._ (2016); Tracy _et al._ (2006); Kodera _et al._ (2006); Dean _et al._ (2009); Bowers _et al._ (2010); Desrat _et al._ (2013, 2015). The lineshape we observe in our system is found to be highly sensitive to the rf power such that the resistance peak resonance line vanishes at a relatively high rf power of -15 dBm[34]. The corresponding signal amplitude normalized to the off resonance resistance $\abs{\Delta R_{\rm{d}}}/R_{\rm{d}}$ is displayed in Fig. 2(b). All the signal amplitude observed here falls below $1\%$, similar to the previous reports in Ref. Córcoles _et al._ (2009); Keane _et al._ (2011). Starting from the observable signal closest to the plateau $V_{\rm{SG}}=-0.41$ V, the dip amplitude shows a sharp upturn and reaches a maximum value at $V_{\rm{SG}}=-0.44$ V. It is then followed by a downturn and takes on a minimum value at $V_{\rm{SG}}=-0.50$ V, precisely at the transition between dispersive-to-single lineshape. The peak amplitude has a smaller amplitude than the dip amplitude and shows a monotonically decrease from $V_{\rm{SG}}=-0.42$ V and eventually vanishes at $V_{\rm{SG}}=-0.51$ V. The spectrum evolves into an expected single dip lineshape for $V_{\rm{SG}}\leq-0.51$ V with the signal amplitude gradually increases. It can be partially explained by an increase in the current density locally in the constriction. Altogether, the facts that the lineshapes, signal amplitudes, as well as resonance point variations with the split gate bias voltage constitute firm evidence that the nuclei is polarized locally in the QPC. We plot in Fig. 2(c)-(d) the raw RDNMR spectra at the two most extreme cases $V_{\rm{SG}}=-0.70$ and $V_{\rm{SG}}=-0.41$, respectively. In order to extract the Knight shift for each spectrum, here we plot in Fig. 2(d) (red dots) the reference signal taken close to $\nu_{\rm{b}}=2$ with nearly zero Knight shift. The spectrum is fitted with a Gaussian functionMasubuchi _et al._ (2006), centered at $33.057$ MHz and FWHM of $8.8$ kHz (red line). Note that the long tail in the higher radio frequency side in the reference spectrum is nothing but reflects a long T1 timeHashimoto _et al._ (2002). Comparing with the reference signal, the observed spectrum at $V_{\rm{SG}}=-0.70$ V is only Knight shifted by about $8$ kHz, reasonable value for the spectrum very far from the plateau at a field of $4.5$ T. The dip frequency in the dispersive lineshape at $V_{\rm{SG}}=-0.41$ V gives the largest observable shift by about $18$ kHz. Interestingly, its peak frequency appears to be substantially unshifted as it is aligned reasonably well with the reference resonance point. RDNMR measurement performed at a smaller field of $4.25$ T reveals similar lineshape patterns[36]. Fig. 2(e) displays the dip and peak resonance line points extracted from the split gate bias voltage segment between $-0.41$ to $-0.50$ V, where the dispersive lineshape is observed. The peak resonance line lies at the resonance reference point with very small variation throughout the range, substantially not Knight shifted. On the other hand, the dip resonance line is upshifted in a linear fashion up to $V_{\rm{SG}}=-0.46$ V and then followed by a slight downshift. The resulting $\Delta f$ values extracted from panel (e) is plotted in Fig. 2(f). The $\Delta f$ value continuously drops down to $12$ kHz in an obviously linear fashion up until $V_{\rm{SG}}=-0.46$ V from its initial value of $18.3$ kHz at $V_{\rm{SG}}=-0.41$ V, bearing a similarity to $\Delta f-B$ plot around $\nu_{\rm{b}}=1$ observed on the 2D systemDesrat _et al._ (2013). The value remains constant at about $12$ kHz throughout the remaining split gate values, an indication that the electronic state in the QPC does not change significantly. Similar trend is observed as well for a field of $4.25$ T[37]. We now move on to discuss the RDNMR taken at the opposite side of the plateau ($\nu_{\rm{QPC}}>1$) as shown in Fig. 3. The data show similar lineshape trend, but with inverted signal and much smaller amplitude than its counterpart. At a field of $4.5$ T displayed in Fig. 3(a), the RDNMR signal is visible only in a confined split gate bias range, $-0.32\leq V_{\rm{SG}}\leq-0.30$ V. The spectra measured very close to the plateau are hindered by a large resistance fluctuation in particular at the point where the diagonal resistance abruptly changes. Nevertheless, one can verify the existence of the inverted dispersive lineshape for $\nu_{\rm{QPC}}>1$ (see the line-cuts at $V_{\rm{SG}}=-0.313$ and $V_{\rm{SG}}=-0.302$ V in Fig. 3(b) for better visual). The RDNMR signal measured at a field of $4.25$ T displayed in Fig. 3(c) has less resistance fluctuation and hence offers better signal to noise ratio. The inverted dispersive lineshape appears at $V_{\rm{SG}}=-0.29$ V (upper Fig. 3d) and turns into a resistance peak lineshape at $V_{\rm{SG}}=-0.285$ V (lower Fig. 3d). In contrast to the case for $\nu_{\rm{QPC}}<1$ where the RDNMR signal is observed in a wide range of split gate bias voltages, the signal observed here vanishes very quickly far from the $\nu_{\rm{QPC}}=1$ plateau region. Recall that the hyperfine-mediated spin flip-flop process relies on the spatial overlap between the up-spin and down-spin channelsWald _et al._ (1994). Thus, the absence of RDNMR signal indicates the critical current for breakdown is higher than for $\nu_{\rm{QPC}}<1$ since the channel is opened widerHwang _et al._ (1993). <figure><img src="content_image/1703.03520/x3.png"><figcaption>Figure 3: (a) 2D color map of 75As RDNMR traces at the lower flank of theνQPC=1 plateau (νQPC>1) measured at a field of 4.5 T. (b) Raw RDNMR tracessliced at VSG=−0.313 (upper) and VSG=−0.302 (lower) V, respectively. (c) 2Dcolor map of 75As RDNMR traces at the lower flank of the νQPC=1 plateau(νQPC>1) measured at a field of 4.25 T. (d) Raw RDNMR traces sliced atVSG=−0.29 (upper) and VSG=−0.285 (lower) V, respectively.</figcaption></figure> The results presented in Fig. 2$-$3 provide important insights onto mechanisms leading to the dispersive lineshape observed in the vicinity of $\nu_{\rm{QPC}}=1$ plateau. Fig. 4 displays all possible hyperfine-mediated spin-flip scattering events where the QPC filling factor is tuned slightly less than 1 for two different alternating current cycles. The forward and backward spin-flip scattering could occur simultaneously within the QPC. The forward scattering occurs at the central region of the QPC where the degree of electron spin polarization is finite, not zero. On the other hand, the backward spin-flip scattering occurs slightly outside the central region where the electron spin polarization is zero. Those scattering events polarize the nuclei in opposite direction and spatially separated. On sweeping of rf with increasing frequency, the positive nuclear polarization is destroyed first due to Knight shift. It results in an increase in the transmissivity of the up-spin channel. On further sweeping the rf, the positive nuclear polarization starts to build up and negative nuclear polarization is destroyed. This results in a decrease in the transmissivity of the up-spin channel. The backward spin-flip scattering is highly suppressed when the QPC filling factor is further tuned to $\nu_{\rm{QPC}}<1$, leaving only positive nuclear polarization build-up at the central region of the QPC. The RDNMR spectrum switches from dispersive-like to dip resonance lineshape. In this scenario, the Knight shift at the central region is determined by $K_{S}\propto\left(n_{\uparrow}-n_{\downarrow}\right)\propto\left(T_{\uparrow}- T_{\downarrow}\right)$, where $n_{\uparrow}(n_{\downarrow})$ and $T_{\uparrow}(T_{\downarrow})$ are up(down)-spin electron density and up(down)-spin transmission probability, respectively. The Knight shift reaches a maximum value when the up-spin channel is completely transmitted ($T_{\uparrow}=1$) while the down spin channel is completely reflected ($T_{\downarrow}=0$). It decreases with reduction of $T_{\uparrow}$, agreeing well with the experimental data shown in Fig. 2(e). <figure><img src="content_image/1703.03520/Fig04_aniket.jpg"><figcaption>Figure 4: (a) Schematics of Landauer-Büttiker edge channel with forward (up-to-down spin flip) and backward (down-to-up spin flip) hyperfine-mediatedspin-flip scatterings occurred at filling factor slightly smaller than νQPC≈1during the first half-clock alternating current cycle (μS>μD) and (b) duringthe second half-clock alternating current cycle (μS<μD). Lighter edgesindicate an empty channel while darker edges indicate a filled channel. Thedrain is held at ground (μD=0) while the source chemical potential μS=0oscillates at a frequency of 13.7 Hz.</figcaption></figure> For $\nu_{\rm{QPC}}>1$ case, similar scenario happens. However, the Overhauser field from the polarized nuclei now affects the transmission of the down-spin channel while the fully transmitted up-spin channel is left unaffected. The nuclear polarization influences the transmissivity of the down-spin channel in an opposite way than that of the up-spin channel. This is the reason why the RDNMR spectrum gets inverted as experimentally confirmed in Fig. 3 and noted in Ref. Keane _et al._ (2011). To summarize, here we observe four variation of the RDNMR lineshapes in a gate-defined QPC. Of particular interest is the emergence of the dispersive lineshape in the RDNMR signal when the bulk filling factor is set to $\nu_{\rm{b}}=2$ and the QPC filling factor to the vicinity of the $\nu_{\rm{QPC}}=1$ plateau. It can be accounted by considering simultaneous occurrence of two hyperfine-mediated spin-flip scattering events due to current-induced dynamic nuclear polarization. These phenomena give rise to localized regions with opposite nuclear polarization in the QPC. Although both of them are in contact with electrons in the QPC, they polarize in a region with different degree of electron spin polarization. Our experimental results further cemented the idea that the observation of the dispersive lineshapes on the 2D system, in particular around $\nu_{\rm{b}}=1$, should reflect the nuclear spin interaction with two electronic sub-systems as suggested by the authors in Ref.Piot _et al._ (2016); Desrat _et al._ (2013). We would like to thank K. Muraki of NTT Basic Research Laboratories for supplying high quality wafers for this study. We thank K. Hashimoto, K. Akiba, T. Aono, T. Tomimatsu, B. Friess, A. Micholic, and D. G. Austing for helpful discussions and/or technical assistance. M.H.F. and Y.H. acknowledge support from Multi-Dimensional program, Tohoku University. A. S., M.T., K.N., and Y.H. acknowledge support from Graduate Program in Spintronics, Tohoku University. B. M. and Y. H. acknowledge support from WPI-AIMR, Tohoku University. Y.H. acknowledges financial support from KAKENHI Grants Nos. $26287059$ and $15\rm{H}05867$. B. M. and A. 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1812.05273	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35099, "num_imgs": 8, "llama3_tokens_count": 9996 }	[ "content_image/1812.05273/Fig1.jpg", "content_image/1812.05273/Fig2.jpg", "content_image/1812.05273/Fig3.jpg", "content_image/1812.05273/FigS1.jpg", "content_image/1812.05273/FigS2.jpg", "content_image/1812.05273/FigS3.jpg", "content_image/1812.05273/FigS4.jpg", "content_image/1812.05273/FigS5.jpg" ]	# Noncentrosymmetric compensated half-metal hosting pure spin Weyl nodes, triple nodal points, nodal loops, and nexus fermions Hyo-Sun Jin${}^{1}$ Young-Joon Song${}^{1}$ Warren E. Pickett${}^{2}$ pickett@physics.ucdavis.edu Kwan-Woo Lee${}^{1,3,4}$ mckwan@korea.ac.kr ${}^{1}$Department of Applied Physics, Graduate School, Korea University, Sejong 30019, Korea ${}^{2}$Department of Physics, University of California, Davis, California 95616, USA ${}^{3}$IFW Dresden, Helmholtzstr. 20, D-01069, Dresden, Germany ${}^{4}$Division of Display and Semiconductor Physics, Korea University, Sejong 30019, Korea February 19, 2024 ###### Abstract Materials containing multiple topological characteristics become more exotic when combined with noncentrosymmetric crystal structures and unusual magnetic phases such as the compensated half-metal state, which is gapped in one spin direction and conducting in the other. First principles calculations reveal these multiple topological features in the compensated half-metal Cr${}_{2}$CoAl having neither time-reversal nor inversion symmetries. In the absence of (minor) spin-orbit coupling (SOC), there are (1) a total of twelve pairs of magnetic Weyl points, (2) three distinct sets of triple nodal points (TNPs) near the Fermi level that are (3) interconnected with six symmetry related nodal lines. This combination gives rise to _fully spin polarized nexus fermions_, in a system with broken time-reversal symmetry but negligible macroscopic magnetic field. The observed high Curie temperature of 750 K and calculated SOC hybridization mixing of several meV should make these nexus fermions readily measurable. Unlike topological features discussed for other Heuslers which emphasize their strong ferromagnetism, this compensated half-metal is impervious to typical magnetic fields, thus providing a complementary set of experimental phenomena. Making use of the soft calculated magnetic state, large magnetic fields can be used to rotate the direction of magnetism, during which certain topological features will evolve. Our results suggest that these features may be common in inverse-Heusler systems, particularly the isostructural and isovalent Ga and In analogs. <figure><img src="content_image/1812.05273/Fig1.jpg"><figcaption>Figure 1: (a) Structure of the inverse-Heusler Cr2CoAl with the sequence ofCr1-Cr2-Co-Al along the diagonal direction. (b) Bulk and (001) surfaceBrillouin zones (BZs) with high symmetry points of the inverse Heuslersystems. The dots in the bulk BZ indicate Weyl points (WPs) with positive(red) and negative (blue) chiralities. There are twelve pairs of WPs protectedby the three twofold rotational axes along the ⟨100⟩ directions, and sixmirrors (M±xy, M±yz, M±zx). (c) Total and atom-resolved densities of states(DOSs) of Cr2CoAl, showing the half-metallic character. The spin down DOS,with a gap at the Fermi level EF, is plotted downward.</figcaption></figure> ## I Introduction and Background Over the past decade various topological phases in insulating, semimetallic, and even metallic materials have been proposed and intensively investigated due to the variety of exotic properties that emerge, some of which have been experimentally realized.[1; 2] More recently three-dimensional (3D) topological features mixing with zero-dimensional (0D) band-crossings (Dirac, Weyl, multi-Weyl, and triple-nodal points [TNPs]) have stimulated further interest,[3] since Dirac and Weyl fermions have been sought among elementary particles, and conversely TNP fermions have no counterpart within the standard model. Nexus fermions are a yet more intricate excitation that have been proposed. Breaking either parity ${\cal P}$ or time-reversal ${\cal T}$ symmetries in 3D systems, or both, allows Weyl points (WPs)[4] with topological character to appear. Weyl semimetal (WS) phases were initially suggested, then observed, in transition metal monophosphides lacking ${\cal P}$ symmetry,[5; 6; 7; 8; 9] viz. TaAs. This class shows an unconventional fermiology, with nodal loop Fermi surfaces[10] in the bulk resulting in surface Fermi arcs connecting WPs of opposite chirality,[6; 7; 8; 9] leading to unconventional transport properties such as large magnetoresistance and chiral anomaly effects. In Weyl semimetals, the impact of breaking of ${\cal P}$ symmetry depends on the strength of spin-orbit coupling (SOC)[11] which often is small. Partially for this reason, the magnetic Weyl (semi-)metals, which break ${\cal T}$, have begun to attract more interest. In such magnetic materials WPs can appear even in the absence of SOC, and often show a much larger separation due to large spin polarization. In addition to the exotic properties of ${\cal P}$ breaking WSs, large anomalous Hall effects are expected in ${\cal T}$-broken cases[12; 13] where low carrier densities exist. Since the number of pairs of WPs in a ${\cal T}$-broken semimetal is odd,[4; 15; 16] fewer WPs are likely than with the ${\cal P}$-broken cases.[12; 13] For example, a large anomalous Hall effect and angle, and a strong anomalous Nernst effect are proposed in the inverse-Heusler compensated half-semimetal Ti${}_{2}$MnAl,[13; 14] which has magnetic WPs just below the Fermi level and the same structure as Cr${}_{2}$CoAl. Such a large anomalous Hall effect is also observed recently in a ferromagnetic van der Waals nodal line semimetal Fe${}_{3}$GeTe${}_{2}$.[17] So far, only a few candidates have been predicted in the Co-based full Heuslers[15; 12], half-metallic CrO${}_{2}$,[18] and the tetragonal $\beta$-V${}_{2}$OPO${}_{4}$,[16] all of which do possess ${\cal P}$ symmetry. No magnetic Weyl phase has yet been observed, whereas a ${\cal P}$-broken WS was observed just a few months after the predictions.[5; 9] Another anomalous 0D band-crossing degeneracy is the TNP. Initially, TNPs were predicted along the symmetry line with $C_{3v}$ symmetry in both symmorphic and nonsymmorphic space groups,[19; 20] and extended to lines of $C_{6v}$ and $C_{4v}$ symmetries. In the $C_{4v}$ case, SOC removes the 6-fold degeneracy (considering spin) and results in reverting to 4-fold Dirac or 2-fold Weyl nodes depending on existence of the ${\cal T}$ symmetry.[22; 21] An exotic phase has been expected when a 0D TNP coincides with a 1D nodal line. This interconnection was dubbed nexus fermions by Chang _et al._ who proposed it along the $C_{3v}$ symmetry line in tungsten carbide WC.[23] However, that nexus point lies far above the Fermi level $E_{F}$ in WC. Dispersion around a nexus point has similarities to the low energy excitations of the chiral superfluid ${}^{3}$He-A, as noted by Heikkilä and Volovik.[24] Chang _et al._ derived an unusual Landau level spectrum quite distinct from that of Weyl semimetals, suggesting novel magnetotransport response.[23] No realistic system has been proposed for such an exotic phase. In this Rapid Communication we describe a unique material, Cr${}_{2}$CoAl, that displays _all four of the non-trivial degeneracies_ mentioned above, as well as additional rare properties. Cr${}_{2}$CoAl, with the non-centrosymmetric inverse Heusler structure, is a ferrimagnetic metal displaying simultaneous WPs, TNPs, nodal loops, and nexus fermions in the absence of SOC (which is minor due partially to cancellations). It is furthermore a half metal, so the various fermionic degeneracies mentioned above are pure spin. Finally, Cr${}_{2}$CoAl is a rare compensated half-metal, producing no macroscopic magnetic field. In the metallic spin-up channel the set of topologically features lie close to $E_{F}$. Due to a combination of space group symmetries in this system the TNPs appear unexpectedly along the $C_{2v}$ symmetry line. The Cr${}_{2}$Co${\cal Z}$ systems (${\cal Z}$=Al, Ga, In) show Curie temperatures T${}_{C}$ near 750 K, far above room temperature, and a minute ordered moment (at most a few hundredths $\mu_{B}$),[25] confirming both strong magnetic coupling and magnetic compensation. The Cr1 local moment is compensated by antialigned Co and Cr2 local moments;[26; 27] the Cr1 and Co sites form edge-sharing tetrahedra and Cr2 sites form octahedra, as displayed in Fig. 1(a). We will focus on Cr${}_{2}$CoAl where correlation corrections reveal a half-metallic electronic structure.[27] Our results suggest that several of these features may be common in magnetic inverse-Heusler systems. ## II Calculational Methods Our _ab initio_ calculations were based on the generalized gradient approximation[28] (GGA) implemented in the all-electron full-potential code wien2k.[29] Some topological aspects were confirmed by another full-potential local-orbital code fplo-18.[30] The same detailed conditions were used in these calculations as in our previous study.[27] For these intermetallic compounds, correlation beyond GGA was excluded as its primary effect is solely to widen the down-spin gap,[31] producing a compensated half-metal state without affecting the spin-up bands with the topological character. These inverse-Heusler systems have the symmorphic but noncentrosymmetric $F\bar{4}3m$ (No. 216) space group, comprised of all cubic point group operations except inversion. Its bulk Brillouin zone (BZ) of $fcc$ shape is displayed in Fig. 1(b). The topological characters are investigated by the hybrid Wannier function charge center approach.[32] From the band structure obtained from wien2k, a tight-binding representation was generated in terms of maximally localized Wannier functions as implemented in the wannier90[33] and wien2wannier[34] programs, with an initial guess for the orbitals as the $3d$ orbitals of Cr and Co, and $3s$ and $3p$ orbitals of Al. The surface spectral functions were calculated by the Green’s function approach,[35] implemented in the wanniertools package.[36] <figure><img src="content_image/1812.05273/Fig2.jpg"><figcaption>Figure 2: (a) GGA spin-up band structure of Cr2CoAl near EF=0 in the absenceof SOC. The spin-down bands (not shown) are gapped at EF. The blue squareindicates nodal points. (b,c) Enlarged spin-up plots around the Weyl pointslying around –0.07 eV along the X−W line, and along a line parallel to the[¯110] direction, respectively. W1 (W2) denotes a WP with positive (negative)chirality at (0.4132,0,0.887)2πa [(0,0.4132,0.887)2πa]. (d) Bands enlargedaround the TNPs, lying just 10 meV above EF along the [001] direction (SOCneglected). The red dots denote a pair of symmetry related TNPs, with the Δ1band having essentially zero velocity at the crossing. (e) Effect of SOC onthe TNP, showing a hybridization splitting of 10 meV, for spin along the (001)direction. (f) Plot of the π-Berry phase for the nodal loops, indicatingtopologically nontrivial loops. For the trivial nodal lines, the Berry phaseis zero. (g) Plot of the nodal lines: a trivial along the Γ−X rotational axis,meeting with topologically nontrivial nodal loops centered on X. Only theloops oriented along the ^z direction are shown. Nexus fermions appear midwayalong Γ−X, marked by blue dots. TNPs (blue dots) occur at crossing points oftwo nodal lines lying on two perpendicular mirror planes.</figcaption></figure> ## III Topological character First we address topological properties of the compensated ferrimagnetic half-metal Cr${}_{2}$CoAl, in the absence of SOC. The weak effects of SOC will be discussed below. Compared with the metallic spin-up bands,[27] features of interest in the down bands (not shown) lie relatively far away from $E_{F}$ (lower than about –0.4 eV). Thus only the spin-up bands will be discussed in more detail. Figure 2(a) displays the band structure near $E_{F}$. In this region, the spin-up bands contain triplet $t_{1u}$ and doublet $e_{1u}$ manifolds driven by the hybridization among the $3d$ orbitals of Cr1, Co, and Cr2 ions, as previously discussed by some of the current authors.[27] Along each of the twelve $X-W$ lines in the BZ, a linear crossing occurs below E${}_{F}$ at $E_{WP}=-74$ meV. To reveal the origin of the nodal points, their chiralities are calculated by the Wannier charge center approach implemented in the wanniertools package.[36] Figure 2(c) shows two WPs with opposite chirality along a line parallel to the [$\bar{1}10$] direction. In total, there are twelve pairs of WPs protected by the six mirror planes and three two-fold rotation symmetries, as displayed on the bulk BZ in Fig. 1(b). The positions and chiralities are given in the caption of Fig. 2. As mentioned above, magnetic Weyl semimetals with ${\cal P}$ symmetry have an odd number of pairs of WPs,[4] but this system has an even number due to lack of both ${\cal T}$ and ${\cal P}$ symmetries. A second feature is the band crossing between doublet and singlet bands just above E${}_{F}$ at E${}_{TNP}$=10 meV along the $\Gamma-X$ line having $C_{2v}$ symmetry, enlarged in Fig. 2(d). These Weyl touchings do not fit well with the types of Ghang _et al.[23]_ who classified crossings as of the same sign, or opposite signs. This band touching accidentally occurs when one band has zero velocity: static fermions crossing and mixing with Weyl fermions. In this cubic system, there are three pairs of TNPs. (For the origin of the TNPs, see below.) Each TNP lies midway along the $\Gamma-X$ line. The doublet bands along (0,0,$k_{z}$) have mostly $d_{xz}$, $d_{yz}$ character of the Co and Cr1 ions, whereas the singlet band has mainly $d_{z^{2}}$ character of the Co and Cr2 ions, with some mixing of Cr2 $d_{xy}$. This crossing, however, leads to a third unexpected feature. Around –0.1 eV midway along the $U-L$ line two bands cross, one again with essentially zero velocity. Unusually, in one direction perpendicular to this line, the band touching persists. Nodal line calculations, using the wanniertools code,[36] establish that the crossing leads to two intersecting nodal lines on the two perpendicular mirror planes about the $C_{2}$ rotational axis, as given in Fig. 2(g). Analysis of the Berry phase[38; 39; 37] or the behavior of hybrid Wannier charge centers[40] can clarify the occurrence and topological nature of a nodal line. One way is by calculating the Berry phase by integrating around a closed loop in the BZ,[37; 41] as shown in Fig. 2(f). The integral vanishes (modulo 2$\pi$) unless it encircles a nontrivial nodal line. This method even allows, for topological loops with Zeeman (magnetic) band splitting, the detection of the nodal lines after splitting by SOC.[38; 39] Alternatively, the Berry phase resulting from the sum of the hybrid Wannier charge centers $z(\vec{k}_{\parallel})$ shows a jump when $z(\vec{k}_{\parallel})$ crosses the projection of a topological nodal loop.[40] For this noncentrosymmetric space group the Berry phase is no longer quantized, but the topology-revealing jump still occurs. (See the Supplementary Material.)[42] The TNPs lie at the crossing points of these $X$-centered nodal loops and a trivial nodal line along the rotational axis imposed by crystal symmetry, resulting in the curious nexus fermionic region. With quadratically touching bands, the 1D nodal line along $\Gamma-X$ is topologically trivial.[20; 23] <figure><img src="content_image/1812.05273/Fig3.jpg"><figcaption>Figure 3: Several views of surface spectra neglecting SOC, with bright yellowindicating high surface intensity. Top and bottom rows of panels are for theCr1-Co and Cr2-Al surface terminations, respectively. (a)–(b): The (001)surface spectral functions of the spin-up channel along symmetry lines. Theseare followed by isoenergy spectral densities of the surface states at energylying at the WPs (middle column) and TNPs (right column). The WPs in (c) and(d) are denoted as green and red circles, whereas the light-blue squares of(e) and (f) indicate TNPs. The ¯X and ¯Y points in (a) and (b) lie at eachmidpoint of adjacent faces of the surface BZ, outlined by white lines inpanels (c)–(f). The high-symmetry points relative to bulk are provided in Fig.1(b).</figcaption></figure> ## IV Origin of triple nodal points TNPs occur most commonly along symmetry lines when a nondegenerate band crosses a doubly degenerate band. Since the band eigenfunctions belong to different irreducible representations of the little group, there is no matrix element of the Hamiltonian (which has the full symmetry of the crystal) to mix the bands, and they cross. Nevertheless, the crossing causes the eigenset at the TNP to become non-analytic, leading to possible topological character. (See the Supplementary Materials.)[42] In this magnetic and noncentrosymmetric inverse-Heusler material, TNPs can arise along the $\Gamma-X$ lines (in the absence of SOC). Such TNPs along the $\Gamma-X$ line of the $C_{2v}$ point group have not been considered before. The TNPs along the $\Gamma-X$ line result from the conventional little group symmetries along the $\Delta$ line in the $F{\bar{4}}3m$ space group. ## V Surface states Figures 3(a) and 3(b) show the (001) surface spectral functions along symmetry lines for the two terminations Cr1-Co and Cr2-Al, respectively. The mapping from the bulk BZ onto the surface BZ is presented in Fig. 1(b). Note that the spectra along the $\bar{\Gamma}-\bar{X}$ and $\bar{\Gamma}-\bar{Y}$ lines are asymmetric due to surface termination breaking of square symmetry. This asymmetry is also reflected in the isoenergy spectral densities shown throughout the surface BZ Figs. 3(c)–3(f), where their structures are identical only along the diagonal directions. Near $E_{F}$, several surface states are visible for both terminations. Topological nodal lines lead to drumhead surface states within the projection of the loop, with (usually) low dispersive surface states,[43; 37] suspected to support instabilities toward surface superconductivity or magnetism when they lie near $E_{F}$. (Drumhead states in isoenergy spectra appear as closed contours within projection of the nodal loop, or as lines terminating at the edge of the projection.) Along the $\bar{\Gamma}-\bar{Y}$ line, the weak drumhead related states lying near $E_{F}$ appear around –75 meV (+75 meV) for the Cr1-Co (Cr2-Al) termination, as shown in Figs. 3(a) and 3(b). However, the main nodal loop projection is along the $\bar{\Gamma}-\bar{M}$ (see below). Figures 3(c) and 3(d) show the isoenergy spectral densities of the two terminations at the WP energy $E_{WP}=$–74 meV. Fermi arcs can be seen connecting WPs with opposite chirality. We also show the spectral densities at the TNP energy $E_{TNP}=10$ meV in Figs. 3(e) and 3(f). In addition to the one projecting onto $\bar{\Gamma}$, TNPs are projected near the midpoint of $\bar{\Gamma}-\bar{M}$ lines. In the Cr1-Co termination, Fermi arcs connecting pairs of TNPs are visible, losing intensity as they merge into the bulk spectrum. At the WP energy, Fig. 3(c), arcs extend between TNPs without merging into bulk bands. The nearness of the TNPs to $E_{F}$ make them amenable to measurement. We are not aware of any topological invariant involving TNPs having been identified in a real material.[3] ### Effects of spin-orbit coupling Lowering of symmetry and lifting of degeneracies by SOC depend on the direction of magnetization, thus the (still near) topological character can be anisotropic. Effects of SOC on WPs in inverse-Heusler compounds have been analyzed by Shi _et al._,[13] where they establish the minor effect for most purposes. We calculate the magnetic anisotropy energy of Cr${}_{2}$CoAl to be at most 4 meV/f.u., with the (111) direction favored slightly. Just as the spin moments exactly cancel due to antialigned atomic moments, the SOC-driven orbital moments and change in spin moments, already small, also cancel, with the result $\mu_{s}=0.01,\mu_{orb}=-0.02,\mu_{net}=-0.01,$ in $\mu_{B}$. Figure 2(e) shows a closeup view of the GGA+SOC band structure near the TNPs along the $\Gamma-X_{001}$ line with spin along (001). SOC leads to a hybridization splitting of about 10 meV at the TNPs, but affects the Fermi surface very little. This mixing strength is minor compared with the observed high Curie temperature of $k_{B}$T${}_{C}\approx$ 65 meV. Kim _et al._ have shown that a tiny gap, comparable with the size of thermal fluctuation $E_{TNP}/k_{B}$, results in surface spectra and transport properties similar to those without SOC.[22] ## VI Summary We have also studied the isovalent and isostructural Ga and In analogs. In addition to WPs, these systems also show nexus fermions very near $E_{F}$ midway along the $\Gamma-X$ line in the spin-up channel. The energies are –6 (–190) meV for Cr${}_{2}$CoGa (Cr${}_{2}$CoIn). (See the Supplementary Material.)[42] In summary, using first principles calculations we have uncovered a unique combination of topological character and compensated half-metallic magnetic order in the noncentrosymmetric, time-reversal symmetry breaking inverse Heusler compound Cr${}_{2}$CoAl. Directly associated with the lack of ${\cal P}$ and ${\cal T}$ symmetries, Cr${}_{2}$CoAl displays a combination of four unusual degeneracies: magnetic Weyl points, triple nodal points, both topological and trivial nodal loops that interconnect, and nexus fermions. All of these occur in a half metal with compensating magnetic moments, which provide no macroscopic magnetic field that would complicate some probes. Specifically, the gapped spin-down electrons will not interfere with the spin-up topological features within the gap. The weak SOC in $3d$ metals leads to tiny orbital moments and band shifts that are negligible for most purposes. Unprecedented TNPs emerge along the $C_{2v}$ (_i.e._, $\Gamma-X)$ line due to a combination of the rotation and mirror point group symmetries. These TNPs along the $\Gamma-X$ line interconnect with nodal links on the mirror planes, leading to nexus fermions lying right above the Fermi energy in the spin-up channel. The combination of compensated half-metallicity and nexus points very near the Fermi energy, with high Curie temperature and minor SOC effects, makes Cr${}_{2}$CoAl a promising candidate to realize an observable nexus fermion phase using modern spectroscopies and transport studies. 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B 95, 235116 (2017). * (42) See Supplemental Material at [URL]. for the enlarged band disperions around the nodal points, the sum of the Wannier charge centers for the nodal link, and the band structures and surface spectra of the Ga and In analogs. * (43) A. A. Burkov, M. D. Hook, and L. Balents, Topological nodal semimetals, Phys. Rev. B 84, 235126 (2011). * (44) E. Pachoud, J. Cumby, C. T. Lithgow, and J. P. Attfield, Charge Order and Negative Thermal Expansion in V${}_{2}$OPO${}_{4}$, J. Am. Chem. Soc. 140, 636 (2018). ## IX Characters of triple nodal points and lines Figure 4 provides more detail about the nodal points and lines discussed in the main text; see the caption for explanation. The unusual variations in band dispersion at emerge from the topological TNP are shown in Fig. 5. <figure><img src="content_image/1812.05273/FigS1.jpg"><figcaption>Figure 4: (a),(b) Band crossings around nodal points at →ka=(0,0.1153,0.1153)and →kb=(0,0.3458,0.3458) in units of 2πa. The (blue) filled circles denotequadratic crossings of the type-α (trivial) nodal lines, whereas the (red)circles mark linear crossing of the type-β (nontrivial) nodal loops on themirror planes. The paths are described by dashed lines in (c), which is (001)projection of the bulk Brillouin zone. The (red) solid lines indicate some ofnodal lines and loops. (d) Plot of the hybrid Wannier charge center WCC z(→k∥)of the nodal loops of (c) for →k∥ along the indicated path, showing the jumpin Berry phase as the WCC passes through the projection of the topologicalnodal loop.</figcaption></figure> <figure><img src="content_image/1812.05273/FigS2.jpg"><figcaption>Figure 5: Band dispersions around the non-analytic TNP in Cr2CoAl,illustrating the various types of dispersion that eminate from the TNP at(0.2527,0.2527,0)2πa.</figcaption></figure> ## X Band structures and surface spectra of the Ga and In analogs Figure 6 provides the bands near E${}_{F}$ for the isostructural and isovalent compounds Cr${}_{2}$CoGa and Cr${}_{2}$CoIn, showing how the different energy positions (“different chemistry”) affect the various band crossings. <figure><img src="content_image/1812.05273/FigS3.jpg"><figcaption>Figure 6: GGA blowup spin-up band structures near EF of (a) Cr2CoGa and (b)Cr2CoIn, which are similar to that of Cr2CoAl, in the absence of SOC. The red(blue) dots denote triple (Weyl) nodal points, while the (blue) box indicatesnodal points leading to nodal line on each mirror planes. In (b), thepositions of W1 and W2 points are given by (12,14,−14) and (14,34,12),respectively, in units of 2πa.</figcaption></figure> Figures 7 and 8 show the surface spectral functions for Cr${}_{2}$CoGa and Cr${}_{2}$CoIn, respectively, presented as for Cr${}_{2}$CoAl in Fig. 3 of the main text. See the captions for descriptions of the various panels. <figure><img src="content_image/1812.05273/FigS4.jpg"><figcaption>Figure 7: Surface spectra of Cr2CoGa, neglecting SOC, with bright yellowindicating high surface intensity. Top and bottom rows of panels are for theCr1-Co and Cr2-Ga surface terminations, respectively. (a), (b): the (001)surface spectral functions of the spin-up channel along symmetry lines. Theseare followed by isoenergy spectral densities of the surface states lying atthe WP energy EWP=–48 meV (middle column) and at the TNP energy ETNP=–6 meV(right column). The WPs in (c) and (d) are denoted green and red circles,whereas the light-blue squares of (e) and (f) indicate TNPs. The ¯X and ¯Ypoints in (a),(b) lie at each midpoint of adjacent faces of the surface BZ,outlined by white lines in panels (c)–(f).</figcaption></figure> <figure><img src="content_image/1812.05273/FigS5.jpg"><figcaption>Figure 8: Correspondence of Fig. 7 for Cr2CoIn. Top and bottom rows of panelsare for the Cr1-Co and Cr2-In surface terminations, respectively. Thecorresponding topological points lie at EWP=–130 meV, ETNP=–190 meV.</figcaption></figure>
1707.04518	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 70428, "num_imgs": 6, "llama3_tokens_count": 21571 }	[ "content_image/1707.04518/x1.png", "content_image/1707.04518/x2.png", "content_image/1707.04518/x4.png", "content_image/1707.04518/x7.png", "content_image/1707.04518/x10.png", "content_image/1707.04518/x13.png" ]	# Insights into the orbital magnetism of noncollinear magnetic systems Manuel dos Santos Dias m.dos.santos.dias@fz-juelich.de Samir Lounis s.lounis@fz-juelich.de Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich & JARA, D-52425 Jülich, Germany February 17, 2024 ###### Abstract The orbital magnetic moment is usually associated with the relativistic spin-orbit interaction, but recently it has been shown that noncollinear magnetic structures can also be its driving force. This is important not only for magnetic skyrmions, but also for other noncollinear structures, either bulk-like or at the nanoscale, with consequences regarding their experimental detection. In this work we present a minimal model that contains the effects of both the relativistic spin-orbit interaction and of magnetic noncollinearity on the orbital magnetism. A hierarchy of models is discussed in a step-by-step fashion, highlighting the role of time-reversal symmetry breaking for translational and spin and orbital angular motions. Couplings of spin-orbit and orbit-orbit type are identified as arising from the magnetic noncollinearity. We recover the atomic contribution to the orbital magnetic moment, and a nonlocal one due to the presence of circulating bound currents, exploring different balances between the kinetic energy, the spin exchange interaction, and the relativistic spin-orbit interaction. The connection to the scalar spin chirality is examined. The orbital magnetism driven by magnetic noncollinearity is mostly unexplored, and the presented model contributes to laying its groundwork. Orbital magnetism, spin texture, skyrmion, spin-orbit interaction, electronic structure, tight-binding model, x-ray magnetic circular dichroism ## I Introduction Magnetic skyrmionsBogdanov and Yablonskii (1989) are a kind of topological twist in a ferromagnetic structure, with unusual properties.Nagaosa and Tokura (2013) They have been found in bulk samples and in thin films, also at room temperatureMühlbauer _et al._ (2009); Yu _et al._ (2010); Heinze _et al._ (2011); Shibata _et al._ (2013); Boulle _et al._ (2016); Moreau-Luchaire _et al._ (2016); Woo _et al._ (2016). When electrons travel throught the noncollinear magnetic structure of the skyrmion, they experience emergent electromagnetic fields.Xiao _et al._ (2010); Franz _et al._ (2014); Everschor-Sitte and Sitte (2014) This strong coupling between the electronic and magnetic degrees of freedom leads to very efficient motion of skyrmions with electric currents. Jonietz _et al._ (2010) It also generates a topological contribution to the Hall effect, a transport signature of a skyrmion-hosting sample,Neubauer _et al._ (2009); Schulz _et al._ (2012) and was shown to enable the electrical detection of an isolated skyrmion. Crum _et al._ (2015); Hanneken _et al._ (2015) The link between the magnetic structure and orbital electronic properties was explored for other kinds of magnetic systems before,Shindou and Nagaosa (2001); Tatara and Kawamura (2002); Tatara and Garcia (2003); Tatara and Kohno (2003); Nakamura _et al._ (2003); Bulaevskii _et al._ (2008) with renewed interest since the experimental discovery of skyrmions. Hamamoto _et al._ (2015); Hoffmann _et al._ (2015); Yin _et al._ (2015); dos Santos Dias _et al._ (2016); Hanke _et al._ (2016); Göbel _et al._ (2017); Lux _et al._ (2017) Inspired by the investigations of nanosized skyrmions in the PdFe/Ir(111) system, Romming _et al._ (2013, 2015) we uncovered another manifestation of their topological nature: a new kind of orbital magnetism. In Ref. dos Santos Dias _et al._, 2016, magnetic trimers and skyrmion lattices were compared, and the orbital magnetic moment was shown to have two contributions: a spin-orbit-driven one and a scalar-chirality-driven one. The minimum number of magnetic atoms needed for a non-vanishing scalar chirality is three, $\mathbf{n}_{1}\cdot(\mathbf{n}_{2}\times\mathbf{n}_{3})\neq 0$, with $\mathbf{n}_{i}$ the orientation of their respective spin magnetic moments. This means that the magnetic structure is noncoplanar, a requirement for the appearance of this new kind of orbital magnetism. Magnetic trimers were analyzed in detail within density functional theory (DFT), before considering a skyrmion lattice meant to mimic PdFe/Ir(111). Although some calculations were feasible with DFT, to address larger skyrmion sizes a minimal tight-binding model was constructed from the DFT data. This model reproduced both contributions to the local orbital moment, and showed that the sum of all scalar-chirality-driven contributions leads to a topological orbital magnetic moment for a skyrmion lattice. The goal of the present paper is to get more insight into the physical mechanisms driving the orbital magnetism of systems in which both the relativistic spin-orbit interaction (RSOI) and a noncollinear magnetic structure coexist. To this end, we reprise our tight-binding model of Ref. dos Santos Dias _et al._, 2016 but now applied to magnetic trimers, and present a walkthrough of the different sources of orbital magnetism in this model. In classical physics, the orbital magnetic moment arises from the presence of bound currents in a given material. This is a consequence of the continuity equation for the electronic charge density in equilibrium: \[0=\frac{\partial\rho}{\partial t}=-\nabla\cdot\mathbf{j}\;\;\Longrightarrow\; \;\mathbf{j}(\mathbf{r})=\nabla\times\mathbf{m}(\mathbf{r})\quad.\] (1) From the quantum-mechanical point of view, if the ground state supports such a finite bound current, time-reversal symmetry must be broken. Magnetic materials naturally break time-reversal symmetry, due to the existence of ordered spin magnetic moments stabilized by exchange interactions. As long as correlation effects are only moderately important, the orbital magnetic moment is usually assumed to be due to the RSOI, which can be introduced either in atomic or in Rashba-like form, Manchon _et al._ (2015) \[\mathcal{H}_{\mathrm{SOI}}\propto\mathbf{L}\cdot\bm{\upsigma}\quad\mathrm{or} \quad(\mathbf{E}\times\mathbf{p})\cdot\bm{\upsigma}\quad.\] (2) Here $\mathbf{L}=\mathbf{r}\times\mathbf{p}$ is the atomic orbital angular momentum operator, $\bm{\upsigma}$ the vector of Pauli matrices, $\mathbf{E}$ the electric field and $\mathbf{p}=-\mathrm{i}\,\hbar\,\bm{\nabla}$ the linear momentum operator. For an atom both forms are equivalent. It is the coupling between the finite spin moment in a magnetic material and the orbital degrees of freedom depending on $\mathbf{p}$ or $\mathbf{L}$ that leads to the finite orbital moment. Skomski (2008) Recently another way of coupling spin and orbital degrees of freedom has been identified and explored. Xiao _et al._ (2010); Everschor-Sitte and Sitte (2014) Consider the non-interacting electron hamiltonian consisting of a kinetic term and a spin exchange coupling to an underlying magnetic structure: \[\mathcal{H}=\frac{\mathbf{p}^{2}}{2m}\,\sigma_{0}+J\,\mathbf{m}(\mathbf{r}) \cdot\bm{\upsigma}\quad.\] (3) Here $\sigma_{0}$ is the unit spin matrix and $J$ is the strength of the exchange coupling. The magnetization field is $\mathbf{m}(\mathbf{r})=m(\mathbf{r})\,\mathbf{n}(\mathbf{r})$, with its spatially varying magnitude $m(\mathbf{r})$ and direction $\mathbf{n}(\mathbf{r})$. For collinear magnetic systems, such as ferromagnets or simple antiferromagnets, $\mathbf{m}(\mathbf{r})=m(\mathbf{r})\,\mathbf{n}_{z}$, where $\mathbf{n}_{z}$ is the direction of the ferromagnetic or staggered magnetization, respectively (accordingly, we allow $m(\mathbf{r})$ to be negative). Then the eigenstates of the system can be labelled with the usual ‘up’ and ‘down’ eigenspinors of $\sigma_{z}$, and we get two decoupled hamiltonians, one for spin-up and another for spin-down. In this way the magnetic order decouples from the orbital degrees of freedom, if the RSOI is not considered. If a common spin quantization axis cannot be chosen, i.e. the system has a noncollinear magnetic structure, such a coupling is indeed present. To see how this arises, consider the unitary transformation that at every point in space diagonalizes the exchange term, \[U^{\dagger}(\mathbf{r})\,\mathbf{m}(\mathbf{r})\cdot\bm{\upsigma}\;U(\mathbf{r })=m(\mathbf{r})\,\sigma_{z}\;\;\Longrightarrow\;\;U(\mathbf{r})=e^{\mathrm{i} \mathbf{w}(\mathbf{r})\cdot\bm{\upsigma}}\quad.\] (4) The vector $\mathbf{w}(\mathbf{r})$ describes the spin rotation in the axis-angle representation, but its explicit form is not required for the present argument, only the fact that it must have a spatial dependence for a noncollinear magnetic structure. If this unitary transformation is applied to the whole hamiltonian, it effects a SU(2) gauge transformation Tokatly (2008); Berche _et al._ (2012); Berche and Medina (2013) with the result \[\mathcal{H}^{\prime}=U^{\dagger}\mathcal{H}\,U=\frac{\bm{\Pi}^{2}}{2m}+J\,m( \mathbf{r})\,\sigma_{z}\quad,\qquad\Pi_{\mu}=p_{\mu}\,\sigma_{0}+\sum_{\nu} \left(\hbar\,\partial_{\mu}w_{\nu}\right)\sigma_{\nu}=p_{\mu}\,\sigma_{0}+\sum _{\nu}A_{\mu\nu}\,\sigma_{\nu}\quad.\] (5) The kinetic momentum $\Pi_{\mu}$ consists of the canonical momentum $p_{\mu}$ and a vector potential $A_{\mu\nu}$ that couples to the spin, with $\mu,\nu=x,y,z$. The kinetic energy now has four contributions: \[\sum_{\mu}\Pi_{\mu}^{2}=\sum_{\mu}p_{\mu}^{2}\,\sigma_{0}+2\sum_{\mu\nu}A_{\mu \nu}(\mathbf{r})\,p_{\mu}\,\sigma_{\nu}-\mathrm{i}\,\hbar\sum_{\mu\nu}\left( \partial_{\mu}A_{\mu\nu}(\mathbf{r})\right)\sigma_{\nu}+\sum_{\mu\nu}\big{(}A_ {\mu\nu}(\mathbf{r})\big{)}^{2}\sigma_{0}\quad.\] (6) The first term is the usual spin-independent contribution, the second term is a spin-orbit interaction (coupling spin to linear momentum), the third is a Zeeman-like contribution, and the fourth is a spin-independent potential-like contribution. We thus see that noncollinear magnetic structures lead to emergent fields that couple the spin and orbital degrees of freedom. This line of reasoning has been very successful in explaining the emergent electrodynamics of slowly-varying magnetic textures. Nagaosa and Tokura (2013); Everschor-Sitte and Sitte (2014) When both kinds of spin-orbit interaction are at play, the one of relativistic origin and the one arising from a noncollinear magnetic structure, they compete with each other, and a unified picture can only be given for special limiting cases. First-principles electronic structure calculations provide both qualitative and quantitative insights. Here we adopt the tight-binding model of Ref. dos Santos Dias _et al._, 2016 to analyze the smallest system for which both contributions to the orbital moment are present: a magnetic trimer. This paper is organized as follows. Section II introduces the model and its ingredients, and then a step-by-step construction of the eigenstates with broken time-reversal symmetry is provided. First, in Sec. III we explore how a magnetic field breaks the translational symmetry of the trimer. Then we show in Sec. IV how a noncollinear magnetic structure produces orbital effects analogous to those of an external magnetic field. Finally, we combine noncollinear magnetism and the atomic spin-orbit interaction in Sec. V, drawing parallels between the relativistic and the noncollinear sources of orbital magnetism. We discuss our results and present our conclusions in Sec. VI. ## II Tight-binding model for noncollinear magnetic structures In Ref. dos Santos Dias _et al._, 2016 the following minimal tight-binding model was introducted, to describe two magnetic $d$-bands experiencing the effects of the relativistic spin-orbit interaction and of a noncollinear magnetic structure: \[\mathcal{H}=\mathcal{H}_{\mathrm{kin}}+\mathcal{H}_{\mathrm{mag}}+\mathcal{H}_ {\mathrm{soi}}\quad.\] (7) The kinetic energy is given by \[\mathcal{H}_{\mathrm{kin}}=\sum_{i,j\neq i}\sum_{mm^{\prime}s}c_{ims}^{\dagger }\,t_{im,jm^{\prime}}\,c_{jm^{\prime}s}\quad.\] (8) Here $c_{ims}^{\dagger}$ creates an electron on atomic site $i$ and on the $d$-orbital labelled $m$, with spin projection $s$. The detailed form of the hopping matrices $t_{im,jm^{\prime}}$ is presented later. The coupling to a background magnetic structure is described by \[\mathcal{H}_{\mathrm{mag}}=J\sum_{i}\sum_{mss^{\prime}}c_{ims}^{\dagger}\, \mathbf{n}_{i}\cdot\bm{\upsigma}_{ss^{\prime}}\,c_{ims^{\prime}}\quad,\] (9) with $J$ the strength of the coupling, and $\mathbf{n}_{i}$ the unit vector describing the direction of the background magnetic structure on every atomic site. We see that $\mathcal{H}_{\mathrm{kin}}+\mathcal{H}_{\mathrm{mag}}$ is the tight-binding equivalent of the model of Eq. 3 that was discussed in the introduction. The RSOI is considered in atomic form, \[\mathcal{H}_{\mathrm{soi}}=\xi\sum_{i}\sum_{mss^{\prime}}c_{ims}^{\dagger}\, \mathbf{L}_{mm^{\prime}}\cdot\bm{\upsigma}_{ss^{\prime}}\,c_{ims^{\prime}}\quad,\] (10) with $\xi$ its coupling strength and $\mathbf{L}_{mm^{\prime}}$ the matrix elements of the atomic orbital angular momentum operator for the two $d$-orbitals in the model. Time reversal symmetry is usually described by the antiunitary operator $\mathcal{T}=\mathrm{i}\,\sigma_{y}\,\mathcal{K}$. Sakurai (1994) Here $\mathcal{K}$ takes the complex conjugate of the spinor wavefunction it is applied to, while $\mathrm{i}\,\sigma_{y}$ ensures that the spin is also reversed. The hamiltonian is time-reversal invariant if it commutes with it, $\left[\mathcal{T},\mathcal{H}\right]=0$. The action of $\mathcal{T}$ is illustrated in the following example: \[\Psi_{\mathbf{k}\uparrow}(\mathbf{r})=e^{\mathrm{i}\mathbf{k}\cdot\mathbf{r}} \begin{pmatrix}1\\ 0\end{pmatrix}\;\;\Longrightarrow\;\;\mathcal{T}\,\Psi_{\mathbf{k}\uparrow}( \mathbf{r})=e^{-\mathrm{i}\mathbf{k}\cdot\mathbf{r}}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}\begin{pmatrix}1\\ 0\end{pmatrix}=-e^{-\mathrm{i}\mathbf{k}\cdot\mathbf{r}}\begin{pmatrix}0\\ 1\end{pmatrix}=-\Psi_{-\mathbf{k}\downarrow}(\mathbf{r})\quad.\] (11) It is more helpful to think of $\mathcal{T}$ as reversing the state of motion. We could then write this operator as $\mathcal{T}=\mathcal{T}_{\mathrm{P}}\,\mathcal{T}_{\mathrm{S}}$, where $\mathcal{T}_{\mathrm{P}}$ reverses the orbital part of the motion ($\mathbf{k}\Rightarrow-\mathbf{k}$ in the example) and $\mathcal{T}_{\mathrm{S}}$ reverses the spin angular momentum ($\uparrow\;\Rightarrow\;\downarrow$ in the example). Haake (2010) Hamiltonians describing magnetic systems are typically not time-reversal invariant, either due to the presence of external magnetic fields or to the exchange interactions that stabilize the magnetic ground state. Whether they might still be invariant under reversal of the orbital motion, $\mathcal{T}_{\mathrm{P}}$, is only clear in the momentum representation, as it amounts to $\mathcal{H}(-\mathbf{p},\bm{\upsigma})=\mathcal{H}(\mathbf{p},\bm{\upsigma})$. An alternative is to choose basis functions that are not invariant under either $\mathcal{T}_{\mathrm{P}}$ or $\mathcal{T}_{\mathrm{S}}$, as shown by the combination of a plane-wave with a spinor in the example. Haake (2010) This is the strategy that will be employed in the following. ## III The trimer with simple hopping Consider three identical atomic sites forming an equilateral triangle with sides taken as the unit of length, $a=1$, as shown in Fig. 1(a). To uncover the role of the electronic motion around the trimer, we first consider a simplified model with one orbital per site and no spin dependence. Let $\ket{i}$ be a basis state for one electron being on atom $i$. In this basis, the hamiltonian is given by \[\begin{array}[]{r\|lll}\mathcal{H}_{\mathrm{kin}}&\ket{1}&\ket{2}&\ket{3}\\ \hline\bra{1}&0&\bar{t}&t\\ \bra{2}&t&0&\bar{t}\\ \bra{3}&\bar{t}&t&0\end{array}\quad\Longrightarrow\quad\mathcal{H}_{\mathrm{ kin}}=t\,\mathcal{R}_{0}+\bar{t}\,\mathcal{R}_{0}^{\dagger}\quad.\] (12) $t=\lvert t\rvert\,e^{\mathrm{i}\alpha}$ is the hopping amplitude for counterclockwise hops around the triangle, and its complex conjugate $\bar{t}$ is the one for clockwise hops. The complex hopping breaks the symmetry of translational motion, and could be due to the presence of a magnetic flux threading the triangle. <figure><img src="content_image/1707.04518/x1.png"><figcaption>Figure 1: The magnetic trimer. (a) Atomic structure and choice of coordinateaxes, with golden spheres representing the atomic sites. (b) Some magneticstructures described by Eq. (23), with the choice of angles discussed in thetext. The red arrows the local orientation of the magnetic structure. Thestructures with cosθ=±1 are ferromagnetic, cosθ=0 is the antiferromagneticNéel structure, and the others are noncollinear structures.</figcaption></figure> The operator $\mathcal{R}_{0}$ generates the counterclockwise hops, \[\mathcal{R}_{0}=\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix}\quad,\qquad\mathcal{R}_{0}\ket{i}=\ket{i{+}1}\quad,\qquad \mathcal{R}_{0}^{\dagger}=\mathcal{R}_{0}^{-1}\quad,\qquad\mathcal{R}_{0}^{ \dagger}\ket{i}=\ket{i{-}1}\quad,\] (13) and its spectral representation is \[\mathcal{R}_{0}=\sum_{k}e^{-\mathrm{i}\frac{2\pi k}{3}}\ket{k}\!\bra{k}\quad, \qquad\ket{k}=\frac{1}{\sqrt{3}}\left(\ket{1}+e^{\mathrm{i}\frac{2\pi k}{3}} \ket{2}+e^{-\mathrm{i}\frac{2\pi k}{3}}\ket{3}\right)\quad,\qquad k\in\{0,\pm 1 \}\quad.\] (14) Note that these basis states are not invariant under reversal of the translational motion, $\mathcal{T}\ket{k}=\mathcal{T}_{\mathrm{P}}\ket{k}=\ket{-k}$, as the clockwise motion is the time-reversed form of the counterclockwise motion. The state $k=0$ corresponds to no overall translational motion, so it equals itself under $\mathcal{T}_{\mathrm{P}}$. The hamiltonian commutes with $\mathcal{R}_{0}$, so from $\mathcal{H}_{\mathrm{kin}}\ket{k}=E_{k}\ket{k}$ we find the eigenenergies \[E_{k}=2\,\lvert t\rvert\cos\left(\frac{2\pi k}{3}-\alpha\right)\quad.\] (15) As this model is equivalent to a linear chain of three atoms with periodic boundary conditions, we shall call $k$ the ring momentum, which characterizes the translational motion of each eigenstate. We can also define momentum raising and lowering operators, which will be very useful in the following sections: \[\mathcal{R}_{\pm}=\begin{pmatrix}1&0&0\\ 0&e^{\pm\mathrm{i}\frac{2\pi}{3}}&0\\ 0&0&e^{\mp\mathrm{i}\frac{2\pi}{3}}\end{pmatrix}\quad,\qquad\mathcal{R}_{\pm} \ket{k}=\ket{k{\pm}1}\quad,\qquad(\mathcal{R}_{\pm})^{3}\ket{k}=\ket{k}\quad.\] (16) The last equality is due to the periodicity of the phase, $k\pm 3=k$. A uniform magnetic field perpendicular to the plane of the trimer is a simple example of broken time-reversal symmetry. In the symmetric gauge, with the origin at the center of the triangle, the vector potential is given by \[\mathbf{A}(\mathbf{r})=-\frac{B}{2}\,\mathbf{r}\times\mathbf{n}_{z}\quad.\] (17) The Peierls substitution Peierls (1933); Boykin _et al._ (2001); Ibañez Azpiroz _et al._ (2014) provides the phase acquired by an electron hopping from site $j$ to site $i$: (18) with the path integral evaluated on the straight line connecting the sites. $S=\sqrt{3}\,a^{2}/4\sim$0.1\,\mathrm{n}\mathrm{m}^{2}$$ is the area of the triangle, for typical bond lengths. The sign of the result is $\chi_{ij}=1$ if $i$ is a neighbor of $j$ in the counterclockwise sense, and $\chi_{ij}=-1$ for the clockwise sense. $\Phi_{0}=h/e\approx 4\times 10^{3}\,$\mathrm{T}\,\mathrm{n}\mathrm{m}^{2}$$ is the magnetic flux quantum, and $\Phi_{\mathrm{B}}$ the actual magnetic flux threading the triangle. Due to the magnetic field, hopping in a clockwise sense is no longer equivalent to hopping in a counterclockwise sense, and this leads to $E_{-k}\neq E_{k}$, as already derived above. As $\alpha$ is proportional to the magnetic field $B$, we define the orbital magnetic moment operator as \[\mathcal{M}=-\frac{\partial\mathcal{H}}{\partial\alpha}=-\mathrm{i}\,\lvert t \rvert\left(e^{\mathrm{i}\alpha}\,\mathcal{R}_{0}-e^{-\mathrm{i}\alpha}\, \mathcal{R}_{0}^{\dagger}\right)\quad.\] (19) It represents the net current flowing around the triangle, and can be evaluated for each eigenstate using the eigenfunctions given in Eq. (14), \[M_{k}=-2\,\lvert t\rvert\sin\left(\frac{2\pi k}{3}-\alpha\right)=-\frac{ \partial E_{k}}{\partial\alpha}\quad.\] (20) The last equality is a nice illustration of the Hellmann-Feynman theorem. Hellmann (1937); Feynman (1939) The following properties will be useful to simplify certain matrix elements appearing in the next sections: \[E_{k{-}1}+E_{k}=-E_{k{+}1}\quad,\qquad M_{k{-}1}+M_{k}=-M_{k{+}1}\quad,\] (21a) \[E_{k{-}1}-E_{k}=\sqrt{3}\,M_{k{+}1}\quad,\qquad M_{k{-}1}-M_{k}=-\sqrt{3}\,E_{ k{+}1}\quad.\] (21b) <figure><img src="content_image/1707.04518/x2.png"><figcaption>Figure 2: Properties of the trimer model with one orbital per site and nospin dependence, Eq. (12), as a function of the relative magnetic flux. Here\|t\|=1. (a) Eigenenergies, Eq. (15). (b) Orbital magnetic moment of eacheigenstate, Eq. (20). The dotted lines indicate the values ±√3, the orbitalmoment for k=∓1 when B=0.</figcaption></figure> The eigenvalues and the corresponding orbital moments are plotted in Fig. 2, as a function of the relative magnetic flux. Both quantities are simple periodic functions of the magnetic flux. Whenever two eigenstates are degenerate in energy, see Fig. 2(a), their orbital moments are equal in magnitude and opposite in sign, cancelling each other, while the third eigenstate (which is non-degenerate) has zero orbital moment, as seen in Fig. 2(b). Thus, for an arbitrary electron filling in thermal equilibrium ($0<N_{\mathrm{e}}<3$), a finite net orbital moment requires lifting of the energy degeneracy. For nanosized trimers and for realistic laboratory magnetic fields, $\Phi_{\mathrm{B}}/\Phi_{0}\ll 1$, so the previous discussion might seem fanciful. However, once spin exchange to a noncollinear magnetic structure is considered, such seemingly unrealistic effective magnetic fields do emerge. We analyze this case in the next section. ## IV Noncollinear magnetism We now extend the model from the previous section by including the spin exchange coupling, \[\mathcal{H}=\mathcal{H}_{\mathrm{kin}}+\mathcal{H}_{\mathrm{mag}}=\sum_{i,j \neq i}\sum_{s}c_{is}^{\dagger}\,t_{ij}\,c_{js}+J\sum_{i}\sum_{ss^{\prime}}c_{ is}^{\dagger}\,\mathbf{n}_{i}\cdot\bm{\upsigma}_{ss^{\prime}}\,c_{is^{\prime}}\quad,\] (22) with $s=\pm 1$ the spin projection on an arbitrary quantization axis. We refer to spin-up and spin-down by the associations $\uparrow\;=+1$ and $\downarrow\;=-1$. In tandem with the orbital impact of the magnetic field, accounted for by the complex hopping parameters $t_{ij}$, the standard spin Zeeman coupling will also be considered. The atomic structure is invariant under $2\pi/3$ rotations in real space around the central axis of symmetry of the triangle. We similarly require the atomic plus magnetic structure to be invariant under the combination of a spatial rotation and a spin rotation, both by an angle of $2\pi/3$, around the respective rotation axes. The local direction of the magnetic structure is thus chosen to be \[\mathbf{n}_{i}=\sin\theta\left(\cos\varphi_{i}\,\mathbf{n}_{x}+\sin\varphi_{i} \,\mathbf{n}_{y}\right)+\cos\theta\,\mathbf{n}_{z}\quad,\] (23) in spherical coordinates with respect to the spin quantization axis $\mathbf{n}_{z}$, and the azimuthal angles are $\varphi_{1}=0$, $\varphi_{2}=2\pi/3$, and $\varphi_{3}=-2\pi/3$, see Fig. 1(b). We single out the following magnetic structures: ferromagnetic pointing along $+z$ (F$\uparrow$); planar triangular Néel structure (N); and ferromagnetic pointing along $-z$ (F$\downarrow$). For these magnetic structures the scalar spin chirality takes the form \[\mathbf{n}_{1}\cdot(\mathbf{n}_{2}\times\mathbf{n}_{3})=\frac{3\sqrt{3}}{2} \sin^{2}\theta\cos\theta=\frac{3\sqrt{3}}{2}\,C(\theta)\quad.\] (24) This quantity is expected to play an important role as a driver of orbital magnetism. Shindou and Nagaosa (2001); Tatara and Kawamura (2002); Tatara and Garcia (2003); Tatara and Kohno (2003); Nakamura _et al._ (2003); Bulaevskii _et al._ (2008); Hoffmann _et al._ (2015); dos Santos Dias _et al._ (2016) We take as basis states the tensor product of the ring states from Eq. (14) with the spin-up and spin-down eigenstates of $\sigma_{z}$: (25) The basis states are not invariant under either reversal of translational motion, $\mathcal{T}_{\mathrm{P}}\ket{k\,s}=\ket{-k\,s}$, or of spin angular motion, $\mathcal{T}_{\mathrm{S}}\ket{k\,s}=-s\ket{k\,{-s}}$. Defining $J_{\perp}=J\sin\theta$ and $J_{z}=J\cos\theta$, the spin exchange coupling is then expressed in this basis as \[\mathcal{H}_{\mathrm{mag}}=J\sum_{i}\mathbf{n}_{i}\cdot\bm{\upsigma}=J_{\perp} \big{(}\mathcal{R}_{-}\,\sigma_{+}+\mathcal{R}_{+}\,\sigma_{-}\big{)}+J_{z}\, \sigma_{z}\quad,\] (26) with the spin raising and lowering operators $\sigma_{\pm}=(\sigma_{x}\pm\mathrm{i}\,\sigma_{y})/2$, and the ring momentum raising and lowering operators defined in Eq. (16). Eq. (26) shows that the spin-flip part of the magnetic coupling exchanges spin angular momentum with ring momentum. When the spin on every site is decreased ($\sigma_{-}$), the ring momentum $k$ increases by one unit ($\mathcal{R}_{+}$), and vice-versa. This is the spin-orbit interaction driven by the noncollinear structure, in the present model. The basis states couple pairwise, forming the hamiltonian blocks: $\mathcal{H}_{\mathrm{a}}$ pairs $\ket{{-}1\,{\uparrow}}$ with $\ket{0\,{\downarrow}}$; $\mathcal{H}_{\mathrm{b}}$ pairs $\ket{0\,{\uparrow}}$ with $\ket{{+}1\,{\downarrow}}$; and $\mathcal{H}_{\mathrm{c}}$ pairs $\ket{{+}1\,{\uparrow}}$ pairs with $\ket{{-}1\,{\downarrow}}$. Their matrix elements are \[\begin{array}[]{r \| c c}\mathcal{H}_{\xi}&\ket{k{-}1\,{\uparrow}}&\ket{k\,{ \downarrow}}\\ \hline\bra{k{-}1\,{\uparrow}}&E_{k{-}1}+J_{z}+B&J_{\perp}\\ \bra{k\,{\downarrow}}&J_{\perp}&E_{k}-J_{z}-B\end{array}\quad,\quad\xi\in\{ \mathrm{a},\mathrm{b},\mathrm{c}\}\quad,\] (27) where $E_{k}$ are the eigenenergies defined in Eq. (15), and the spin Zeeman coupling to the external magnetic field was included, $B\,\sigma_{z}$. Each block has then the eigenenergies (see Appendix A and Eq. (21)) \[E_{\xi}=-\frac{E_{k{+}1}}{2}\pm\sqrt{\frac{3}{4}\,\big{(}M_{k{+}1}\big{)}^{2}+ \sqrt{3}\,\big{(}J_{z}+B\big{)}\,M_{k{+}1}+\big{(}J_{z}+B\big{)}^{2}+J_{\perp} ^{2}}\quad.\] (28) For $\alpha=B=0$ and introducing $x=\frac{3\lvert t\rvert}{2J}$, this yields \[E_{\mathrm{a}\pm}=\frac{\lvert t\rvert}{2}\pm J\sqrt{1-2\,x\cos\theta+x^{2}} \quad,\qquad E_{\mathrm{b}\pm}=\frac{\lvert t\rvert}{2}\pm J\sqrt{1+2\,x\cos \theta+x^{2}}\quad,\qquad E_{\mathrm{c}\pm}=-\lvert t\rvert\pm J\quad.\] (29) The competition between kinetic and magnetic energies is encoded in the parameter $x$. The magnetic moment associated with each of the eigenstates can be calculated directly from the eigenenergies. It has two contributions (treating $B$ and $\alpha$ as independent): \[M_{\xi}=\frac{\partial E_{\xi}}{\partial B}-\frac{\partial E_{\xi}}{\partial \alpha}=M_{\mathrm{S}\xi}+M_{\mathrm{P}\xi}\quad.\] (30) $M_{\mathrm{S}\xi}$ is the spin magnetic moment, arising from the Zeeman interaction, and signals the broken symmetry under reversal of spin angular momentum ($\mathcal{T}_{\mathrm{S}}$). $M_{\mathrm{P}\xi}$ arises from the broken symmetry under reversal of the translational motion ($\mathcal{T}_{\mathrm{P}}$), due to the currents flowing around the trimer. This is the orbital magnetic moment already encountered in the previous section. For $\alpha=B=0$ we find the spin moments: \[M_{\mathrm{Sa}\pm}\] (31a) \[M_{\mathrm{Sb}\pm}\] (31b) \[M_{\mathrm{Sc}\pm} =\pm\cos\theta\quad.\] (31c) $M_{\mathrm{S}}$ tells us about the spin character of an eigenstate. A positive sign indicates $\uparrow$, a negative one $\downarrow$, and if it vanishes it has an equal amount of each character. The adiabatic approximation, valid for $J\gg\lvert t\rvert$, makes the electron spin collinear with the direction of the magnetic structure. This would lead to a $\cos\theta$ dependence, which is the first term in the $x\ll 1$ expansion. The expansions were carried out up to the first term with the angular dependence of the scalar chirality, Eq. (24). The orbital moments can then be shown to be simply related to the spin moments: \[M_{\mathrm{Pa}}=M_{\mathrm{max}}\,\frac{M_{\mathrm{Sa}}+1}{2}\quad,\qquad M_{ \mathrm{Pb}}=M_{\mathrm{max}}\,\frac{M_{\mathrm{Sb}}-1}{2}\quad,\qquad M_{ \mathrm{Pc}}=-M_{\mathrm{max}}\,M_{\mathrm{Sc}}\quad,\] (32) with $M_{\mathrm{max}}=\sqrt{3}\,\lvert t\rvert$ the maximum value of the orbital magnetic moment for this model. <figure><img src="content_image/1707.04518/x4.png"><figcaption>Figure 3: Trimer with one orbital per site and a noncollinear magneticstructure, Eq. (22), in the strong exchange coupling regime, t=1 and J=3(x=1/2). (a) Eigenenergies, Eq. (29). (b) Spin magnetic moment of eacheigenstate, Eq. (31). (c) Orbital magnetic moment of each eigenstate, Eq.(32). The magnetic structures are defined by Eq. (23) and illustrated in Fig.1(b). The curves are labelled with the states for F↑ taken as reference: thecolor labels the value of MP; solid lines and dashed lines indicate the signof MS. The following combinations of eigenstates are also plotted:(λ)=(aλ)+(bλ)+(cλ), with λ=±.</figcaption></figure> First let us consider the case of the magnetic exchange dominating the kinetic energy, i.e. $J\gg\lvert t\rvert$, allowing a comparison with the adiabatic approximation. Fig. 3 displays the results for $t=1$ and $J=3$ ($x=1/2$). The eigenenergies form two groups, separated by the exchange splitting $2J$, as seen in Fig. 3(a). The magnetic noncollinearity effectively reduces the kinetic energy, evidenced by the shrinking ‘bandwidth’ of each group when going from the F$\uparrow$ structure to the N structure. Fig. 3(b) shows the spin magnetic moments. The spin moments for the eigenstates labelled (c) follow perfectly the adiabatic approximation, seen as the linear behavior, while those for the eigenstates labelled (a) and (b) show deviations from the linear behavior. Fig. 3(c) shows the orbital magnetic moments. For the F$\uparrow$ structure, the eigenstates for each spin projection are decoupled and are just the ring states previously discussed, with the same orbital moments. The variation of the orbital moments with the magnetic structure reveals the presence of the emergent magnetic field that it generates. Going from F$\uparrow$ to N, we arrive at a new energy degeneracy. Comparison of the evolution of the curves with those in Fig. 2 lets us assign $\Phi_{\mathrm{B}}/\Phi_{0}=-1/2$ to the eigenstates evolving from spin-up, and $\Phi_{\mathrm{B}}/\Phi_{0}=+1/2$ for those evolving from spin-down. The net orbital moment is zero for the ferromagnetic structures, but not for the noncollinear ones. To illustrate this, we sum all the contributions corresponding to the $+$ and $-$ bands, which corresponds to placing three electrons in the three upper or lower eigenstates, see Fig. 3(a). The average spin for these combinations follows the magnetic structure almost linearly, see Fig. 3(b), the behavior expected in the adiabatic limit. From Fig. 3(c) we observe that the average orbital moments are indeed zero for the F endpoints and for the N structure, but are finite for the noncollinear structures. A comparison with the $x\ll 1$ expansion in Eq. 31 shows that $M_{\mathrm{P}\pm}\propto C(\theta)$, the scalar spin chirality, to leading order. <figure><img src="content_image/1707.04518/x7.png"><figcaption>Figure 4: Trimer with one orbital per site and a noncollinear magneticstructure, Eq. (22), in the weak exchange coupling regime, t=1 and J=1(x=3/2). (a) Eigenenergies, Eq. (29). (b) Spin magnetic moment of eacheigenstate, Eq. (31). (c) Orbital magnetic moment of each eigenstate, Eq.(32). The magnetic structures are defined by Eq. (23) and illustrated in Fig.1(b). The curves are labelled with the states for F↑ taken as reference: thecolor labels the value of MP; solid lines and dashed lines indicate the signof MS. The following combinations of eigenstates are also plotted:(+)=(c+)+(c−) and (−)=(a+)+(a−)+(b+)+(b−).</figcaption></figure> Next consider the case of the magnetic exchange being comparable to the kinetic energy, i.e. $J\sim\lvert t\rvert$. Fig. 4 displays the results for $t=1$ and $J=1$ ($x=3/2$). As $2J<3t$, the eigenergies for spin-up and spin-down overlap, see Fig. 4(a). Comparing with the previous case, we see that the ordering of the states for F$\uparrow$ has changed, as indicated by the sequence of colors in the figure. This has a dramatic impact on the behavior of the magnetic moments, Fig. 4(b,c). On the one hand, the eigenstates labelled (c) still follow the linear behavior. On the other hand, both the spin and the orbital moments for the eigenstates labelled (a) and (b) are only weakly modified by the magnetic structure. This has a simple explanation: the ring states coupled in $\mathcal{H}_{\mathrm{c}}$ are degenerate in energy for $J=0$, and so the coupling to the magnetic structure is always non-perturbative, while $\mathcal{H}_{\mathrm{a}}$ and $\mathcal{H}_{\mathrm{b}}$ each couple states split by $3t$ for $J=0$, and so the exchange coupling has only a perturbative effect. To visualize whether there is a net orbital moment also in this case, we sum all the contributions corresponding either to the four lower or to the two higher eigenstates, see Fig. 4(a). Although the net spin moment is zero for the F and N structures, surprisingly it acquires a finite value for the noncollinear structures, see Fig. 4(b). Fig. 4(c) shows that the net orbital moment has a similar behavior. A comparison with the $x\gg 1$ expansion in Eq. 31 shows that $M_{\mathrm{P}\pm}\propto M_{\mathrm{S}\pm}\propto C(\theta)$, the scalar spin chirality, to leading order. We have thus seen how a noncollinear magnetic structure can lead to orbital magnetic effects in the absence of the RSOI. The impact on the electronic structure depends crucially on whether the states which become coupled by the magnetic exchange are initially degenerate in energy or not. For the former the adiabatic approximation is always valid, while for the latter the exchange coupling must overcome the difference in kinetic energy between the states for it to have a strong influence. The picture is also very difference if each eigenstate is considered by itself or if a group of eigenstates is considered together. In the next section the RSOI is introduced in the model, and its consequences analyzed. ## V Interplay between noncollinear magnetism and the relativistic spin-orbit interaction We extend our model one final time, by taking the orbital character of the electrons on every site into account. Following Ref. dos Santos Dias _et al._, 2016, we consider two $d$-orbitals to be present on every site, namely $\ket{xy}$ and $\ket{x^{2}{-}y^{2}}$, assumed to be initially degenerate in energy. We shall work with their complex counterparts, which are eigenstates of $L_{z}$: \[\ket{\pm}=\frac{1}{\sqrt{2}}\left(\ket{x^{2}{-}y^{2}}\pm\mathrm{i}\ket{xy} \right)\quad,\qquad L_{z}\ket{\pm 2}=\pm 2\ket{\pm 2}\quad.\] (33) The RSOI in atomic form reduces to $\mathbf{L}\cdot\bm{\upsigma}=L_{z}\,\sigma_{z}$, as the other angular momentum operators vanish when restriced to these two orbitals. To label the states, we make the identifications $+2={\circlearrowleft}$ and $-2={\circlearrowright}$. However, the kinetic hamiltonian now has to describe the directionality of the orbitals. For zero magnetic field, the hopping matrix in either the real or complex basis is given by \[\begin{array}[]{r\|cc}t_{ij}&\ket{x^{2}{-}y^{2}}&\ket{xy}\\ \hline\bra{x^{2}{-}y^{2}}&t\left(1+\cos\gamma_{ij}\right)&t\sin\gamma_{ij}\\ \bra{xy}&t\sin\gamma_{ij}&t\left(1-\cos\gamma_{ij}\right)\end{array}\qquad \text{or}\qquad\begin{array}[]{r\|cc}t_{ij}&\ket{\circlearrowleft}&\ket{ \circlearrowright}\\ \hline\bra{\circlearrowleft}&t&t\,e^{-\mathrm{i}\gamma_{ij}}\\ \bra{\circlearrowright}&t\,e^{\mathrm{i}\gamma_{ij}}&t\end{array}\quad,\] (34) where $\gamma_{ij}/4$ is the angle between the bond and the $x$-axis. We have $\gamma_{ij}=\gamma_{ji}$ (mod $2\pi$), and $\gamma_{12}=0$, $\gamma_{23}=2\pi/3$ and $\gamma_{31}=-2\pi/3$. This encompasses the fourfold symmetry of the orbitals, and their directionality. For example, if two orbitals are along the $x$-axis, hopping can only occur if they are both of $\ket{x^{2}{-}y^{2}}$ type. We can encode the action of the hopping matrix on the orbitals using a new set of Pauli matrices $\tau_{\mu}$ (to be distinguished from the ones used for spin), \[t_{ij}=\lvert t\rvert\,e^{\mathrm{i}\chi_{ij}\alpha}\big{(}\tau_{0}+e^{- \mathrm{i}\gamma_{ij}}\tau_{+}+e^{\mathrm{i}\gamma_{ij}}\tau_{-}\big{)}=t_{ji} ^{\dagger}\quad,\] (35) where the magnetic field was restored via the Peierls phase, see Eq. (18). Our basis states are the tensor product of the ring states $k=0,\pm 1$, of the two orbitals $m=\pm 2$, and of the spinors $s=\pm 1$: \[\ket{k\,m\,s}=\frac{1}{\sqrt{3}}\left(\ket{1\,m\,s}+e^{\mathrm{i}\frac{2\pi k} {3}}\ket{2\,m\,s}+e^{-\mathrm{i}\frac{2\pi k}{3}}\ket{3\,m\,s}\right)\quad.\] (36) These basis functions are ideal to describe time-reversal symmetry breaking: the time-reversed counterpart of each basis state is $\mathcal{T}\ket{k\,m\,s}=-s\ket{-k\,{-m}\,{-s}}$, which corresponds to reversing translational, orbital and spin motions, i.e., reversing each of the variables describing the state of motion. The action of the hamiltonian can then be separated into a diagonal part (that leaves the basis state unchanged), and different kinds of off-diagonal terms, according to what change they effect on the basis state. The diagonal part of the hamiltonian is \[\mathcal{H}_{0}=t\,\mathcal{R}_{0}+\bar{t}\,\mathcal{R}_{0}^{\dagger}+J_{z}\, \sigma_{z}+\xi\,\tau_{z}\,\sigma_{z}+B\,\tau_{z}\quad.\] (37) The new terms are the RSOI ($\xi$ term), and the orbital Zeeman coupling ($B$ term). This part of the hamiltonian acts on a basis state as, recalling Eq. (15), \[\mathcal{H}_{0}\ket{k\,m\,s}=\big{(}E_{k}+s\,J_{z}+\mathrm{sgn}(m)\,(s\,\xi+B) \big{)}\ket{k\,m\,s}\quad.\] (38) We have already seen from the previous section that there are two terms that exchange spin angular momentum and ring momentum, \[\mathcal{H}_{\mathrm{S}\pm}=J_{\perp}\mathcal{R}_{\mp}\,\sigma_{\pm}\quad, \qquad\mathcal{H}_{\mathrm{S}\pm}\ket{k\,m\,s}=J_{\perp}\ket{k{\mp}1\,m\,s{\pm }1}\quad.\] (39) These terms led to the spin-orbit interaction generated by the noncollinear magnetic structure. The remaining piece of the kinetic term generates two terms that exchange orbital angular momentum and ring momentum, \[\mathcal{H}_{\mathrm{L}\pm}=\big{(}t\,\mathcal{R}_{0}\,\mathcal{R}_{\mp}+\bar{ t}\,\mathcal{R}_{\mp}\,\mathcal{R}_{0}^{\dagger}\big{)}\,\tau_{\pm}\quad,\] (40) recall Eq. (16), with the result \[\mathcal{H}_{\mathrm{L}\pm}\ket{k\,m\,s}=\Big{(}t\,e^{-\mathrm{i} \frac{2\pi(k\mp 1)}{3}}+\bar{t}\,e^{\mathrm{i}\frac{2\pi k}{3}}\Big{)}\ket{k{ \mp}1\,m{\pm}2\,s}=E_{k{\pm}1}\,e^{\mp\mathrm{i}\frac{2\pi}{3}}\ket{k{\mp}1\,m {\pm}2\,s}\quad,\] (41) according to the definition in Eq. (15). This might be called an orbit-orbit interaction, as the translational motion and the local orbital motion are coupled. Our hamiltonian can now be written as initially presented in Eq. (7), \[\mathcal{H}=\mathcal{H}_{\mathrm{kin}}+\mathcal{H}_{\mathrm{mag}}+\mathcal{H}_ {\mathrm{soi}}=\mathcal{H}_{0}+\mathcal{H}_{\mathrm{S}+}+\mathcal{H}_{\mathrm{ S}-}+\mathcal{H}_{\mathrm{L}+}+\mathcal{H}_{\mathrm{L}-}\quad,\] (42) and represents a $12\times 12$ matrix, composed of three $4\times 4$ blocks. The basis states that can be coupled by the hamiltonian are limited by $(\mathcal{H}_{\mathrm{S}\pm})^{2}\ket{k\,m\,s}=0$ and $(\mathcal{H}_{\mathrm{L}\pm})^{2}\ket{k\,m\,s}=0$, as the spin and the atomic angular momentum cannot be raised or lowered more than once. We then have the following chain of coupled states: \[\ket{k{-}1\,{\circlearrowleft}\,{\uparrow}}\;\underset{\mathcal{H}_{\mathrm{L} -}}{\longrightarrow}\;\ket{k\,{\circlearrowright}\,{\uparrow}}\;\underset{ \mathcal{H}_{\mathrm{S}-}}{\longrightarrow}\;\ket{k{+}1\,{\circlearrowright}\, {\downarrow}}\;\underset{\mathcal{H}_{\mathrm{L}+}}{\longrightarrow}\;\ket{k\, {\circlearrowleft}\,{\downarrow}}\;\underset{\mathcal{H}_{\mathrm{S}+}}{ \longrightarrow}\;\ket{k{-}1\,{\circlearrowleft}\,{\uparrow}}\quad.\] (43) The three blocks are generated by the three possible starting values of $k$, and can be organized as follows: $\mathcal{H}_{\mathrm{a}}$ couples $\ket{0\,{\circlearrowleft}\,{\uparrow}}$, $\ket{{+}1\,{\circlearrowright}\,{\uparrow}}$, $\ket{{+}1\,{\circlearrowleft}\,{\downarrow}}$, and $\ket{{-}1\,{\circlearrowright}\,{\downarrow}}$; $\mathcal{H}_{\mathrm{b}}$ couples $\ket{{+}1\,{\circlearrowleft}\,{\uparrow}}$, $\ket{{-}1\,{\circlearrowright}\,{\uparrow}}$, $\ket{{-}1\,{\circlearrowleft}\,{\downarrow}}$, and $\ket{0\,{\circlearrowright}\,{\downarrow}}$; and $\mathcal{H}_{\mathrm{c}}$ couples $\ket{{-}1\,{\circlearrowleft}\,{\uparrow}}$, $\ket{0\,{\circlearrowright}\,{\uparrow}}$, $\ket{0\,{\circlearrowleft}\,{\downarrow}}$, and $\ket{{+}1\,{\circlearrowright}\,{\downarrow}}$. The matrix elements for these blocks have the form \[\begin{array}[]{r \| c c \| c c}\mathcal{H}_{\xi}&\ket{k{-}1\,{\circlearrowleft} \,{\uparrow}}&\ket{k\,{\circlearrowright}\,{\uparrow}}&\ket{k\,{ \circlearrowleft}\,{\downarrow}}&\ket{k{+}1\,{\circlearrowright}\,{\downarrow} }\\ \hline\bra{k{-}1\,{\circlearrowleft}\,{\uparrow}}&E_{k-1}+J_{z}+\xi+B&E_{k{+}1 }\,e^{-\mathrm{i}\frac{2\pi}{3}}&J_{\perp}&0\\ \bra{k\,{\circlearrowright}\,{\uparrow}}&E_{k{+}1}\,e^{\mathrm{i}\frac{2\pi}{3 }}&E_{k}+J_{z}-\xi-B&0&J_{\perp}\\ \hline\bra{k\,{\circlearrowleft}\,{\downarrow}}&J_{\perp}&0&E_{k}-J_{z}-\xi+B& E_{k{-}1}\,e^{-\mathrm{i}\frac{2\pi}{3}}\\ \bra{k{+}1\,{\circlearrowright}\,{\downarrow}}&0&J_{\perp}&E_{k{-}1}\,e^{ \mathrm{i}\frac{2\pi}{3}}&E_{k+1}-J_{z}+\xi-B\end{array}\quad.\] (44) The case of a noncollinear magnetic structure is analytically cumbersome, as the hamiltonian blocks are $4\times 4$ matrices. If the magnetic exchange is much stronger than all the other terms, we can adopt the frequently used adiabatic approximation. Everschor-Sitte and Sitte (2014) The spin projectors that diagonalize the magnetic exchange interaction are (see Appendix A) \[P_{\pm}=\frac{1}{2}\,\big{(}\sigma_{0}\pm(\cos\theta\,\sigma_{z}+\sin\theta\, \sigma_{x})\big{)}\quad.\] (45) As the results for $s=-1$ can be obtained from those for $s=+1$ by the replacements $J\rightarrow-J$ and $\cos\theta\rightarrow-\cos\theta$, we set $s=+1$ and drop the spin label in the following. Tracing over the spin components, we define an effective hamiltonian by (46) which can be written using the orbital Pauli matrices as \[\widetilde{\mathcal{H}}_{\xi}=J\,\tau_{0}\] (47) The only role played by $J$ is to define the energy zero, so we will also set $J=0$ from now on. The eigenergies for the effective hamiltonian blocks $\widetilde{\mathcal{H}}_{\xi}$ are \[E_{\xi}=\frac{1}{4}\,\big{(}E_{k}-\sqrt{3}\,M_{k}\cos\theta\pm \lvert t\rvert\sqrt{D_{\xi}}\,\big{)}\quad,\] (48) with the discriminants \[\lvert t\rvert^{2}D_{\xi}=\big{(}\sqrt{3}\,M_{k}+3\,E_{k}\cos\theta-4\,(\xi \cos\theta+B)\big{)}^{2}+4\,\big{(}E_{k}-\sqrt{3}\,M_{k}\cos\theta\big{)}^{2}\quad.\] (49) The orbital moments can be decomposed into two contributions (taking $B$ and $\alpha$ to be independent), \[M_{\xi}=\frac{\partial E_{\xi}}{\partial B}-\frac{\partial E_{\xi}}{\partial \alpha}=M_{\mathrm{L}\xi}+M_{\mathrm{P}\xi}\quad.\] (50) $M_{\mathrm{L}\xi}$ is the atomic orbital moment, stemming from the orbital Zeeman interaction, and signals the broken symmetry under reversal of the local orbital motion ($\mathcal{T}_{\mathrm{L}}$). In the previous section we already encountered $M_{\mathrm{P}\xi}$, the contribution to the orbital motion from the currents circulating around the trimer. In the adiabatic approximation $M_{\mathrm{S}\xi}=\cos\theta$ by construction, so it does not merit further consideration. For $\alpha=B=0$ and setting $y=4\,\xi/\lvert t\rvert$, we obtain the eigenergies \[E_{\mathrm{a}\pm}=\frac{\lvert t\rvert}{4}\,\Big{(}{-}1+3\cos\theta\pm\sqrt{13 -6\,\big{(}1-y\big{)}\cos\theta+\big{(}45+6\,y+y^{2}\big{)}\cos^{2}\theta}\, \Big{)}\quad,\] (51a) \[E_{\mathrm{b}\pm}=\frac{\lvert t\rvert}{4}\,\Big{(}{-}1-3\cos\theta\pm\sqrt{13 +6\,\big{(}1-y\big{)}\cos\theta+\big{(}45+6\,y+y^{2}\big{)}\cos^{2}\theta}\, \Big{)}\quad,\] (51b) (51c) The atomic orbital moments are \[M_{\mathrm{L}\mathrm{a}\pm}=\pm\frac{(y+3)\cos\theta+3}{\sqrt{D_{\mathrm{a}}}} \quad,\qquad M_{\mathrm{L}\mathrm{b}\pm}=\pm\frac{(y+3)\cos\theta-3}{\sqrt{D_{ \mathrm{b}}}}\quad,\qquad M_{\mathrm{L}\mathrm{c}\pm}=\pm\frac{(y-6)\cos\theta }{\sqrt{D_{\mathrm{c}}}}\quad.\] (52) $M_{\mathrm{L}}$ tells us about the atomic orbital character of an eigenstate. A positive sign indicates $\circlearrowleft$, a negative one $\circlearrowright$, and if it vanishes it has an equal amount of each character. The orbital moments arising from the circulating currents are (recall $M_{\mathrm{max}}=\sqrt{3}\,\lvert t\rvert$) \[M_{\mathrm{P}\mathrm{a}\pm}=-\frac{M_{\mathrm{max}}}{4}\left(\cos\theta+1\pm \frac{-1+(2+y)\cos\theta+3\,(1-y)\cos^{2}\theta}{\sqrt{D_{\mathrm{a}}}}\right)\quad,\] (53a) \[M_{\mathrm{P}\mathrm{b}\pm}=-\frac{M_{\mathrm{max}}}{4}\left(\cos\theta-1\pm \frac{+1+(2+y)\cos\theta-3\,(1-y)\cos^{2}\theta}{\sqrt{D_{\mathrm{b}}}}\right)\quad,\] (53b) \[M_{\mathrm{P}\mathrm{c}\pm}=\frac{M_{\mathrm{max}}}{2}\left(1\pm\frac{2+y}{ \sqrt{D_{\mathrm{c}}}}\right)\cos\theta\quad.\] (53c) They can be used to characterize the translational motion, as in Sec. III. For a ferromagnetic structure, $M_{\mathrm{P}}/M_{\mathrm{max}}\approx\pm 1$ can be associated with $k=\mp 1$, and $M_{\mathrm{P}}/M_{\mathrm{max}}\approx 0$ with $k=0$. Note that there is no simple relation between $M_{\mathrm{L}}$ and $M_{\mathrm{P}}$, in contrast to the results of the previous section. <figure><img src="content_image/1707.04518/x10.png"><figcaption>Figure 5: Trimer with two orbitals per site and a noncollinear magneticstructure in the adiabatic approximation, Eq. (V), and with no relativisticspin-orbit interaction (t=1 and ξ=0). (a) Eigenenergies, Eq. (51). (b) Atomicorbital magnetic moment, Eq. (52). (c) Orbital magnetic moment arising fromthe circulating currents, Eq. (53). The magnetic structures are defined by Eq.(23) and illustrated in Fig. 1(b). The curves are labelled with the states forF↑ taken as reference: the color labels MP, similarly to Mk in Fig. 2(b);solid lines and dashed lines indicate the sign of ML. The followingcombinations of eigenstates are also plotted: (λ)=(aλ)+(bλ)+(cλ), with λ=±.</figcaption></figure> Now that the analytical expressions have been derived, let us explore the physics. We begin by examining what happens when the RSOI is turned off ($\xi=0$), with the results gathered in Fig. 5. Consider first the ferromagnetic structures. There are two pairs of degenerate eigenenergies, and another is non-degenerate, see Fig. 5(a). They can be characterized by their atomic orbital moments, Fig. 5(b), and by their translational motion $M_{\mathrm{P}}$, Fig. 5(c). The degenerate eigenstates have strong $\circlearrowleft$ or $\circlearrowright$ orbital character ($M_{\mathrm{L}}\approx 1$ or $-1$, respectively), and $k=\pm 1$ character ($M_{\mathrm{P}}/M_{\mathrm{max}}\approx\mp 0.5$). The non-degenerate eigenstates are orbitally mixed ($M_{\mathrm{L}}=0$), with $k=0$ character ($M_{\mathrm{P}}=0$). There is overall no net orbital magnetic moment, as non-degenerate eigenstates have zero orbital moment, and degenerate ones have orbital moments with opposite values, thus cancelling out. As already seen in the simpler model of Sec. IV, the noncollinear magnetic structures lift the energy degeneracies and modify the orbital moments of each eigenstate, enabling a net orbital moment without the RSOI being present. To illustrate this, we sum all the contributions corresponding to the $+$ and $-$ bands, which corresponds to placing three electrons in the three upper or lower eigenstates. There is a net atomic orbital moment, see Fig. 5(b), with a $C(\theta)$-like angular dependence (Eq. (24)), but no net current, see Fig. 5(c). <figure><img src="content_image/1707.04518/x13.png"><figcaption>Figure 6: Trimer with two orbitals per site and a noncollinear magneticstructure in the adiabatic approximation, Eq. (V), and with strongrelativistic spin-orbit interaction (t=1 and ξ=5). (a) Eigenenergies, Eq.(51). (b) Atomic orbital magnetic moment, Eq. (52). (c) Orbital magneticmoment arising from the circulating currents, Eq. (53). The magneticstructures are defined by Eq. (23) and illustrated in Fig. 1(b). The curvesare labelled with the states for F↑ taken as reference: the color labels MP,similarly to Mk in Fig. 2(b); solid lines and dashed lines indicate the signof ML. The following combinations of eigenstates are also plotted:(λ)=(aλ)+(bλ)+(cλ), with λ=±.</figcaption></figure> We finally bring the RSOI into play. If it is weak when comparing to the kinetic hopping, $\xi\ll\lvert t\rvert$, the picture is qualitatively similar to the previous one. One major difference is that it lifts the energy degeneracies in the ferromagnetic structures, thus allowing net orbital moments. This is the well-known role of the RSOI in ferromagnetic systems. We focus on the opposite limit, $\xi\gg\lvert t\rvert$, to see how it counteracts the kinetic term. The results for $\xi=5$ and $t=1$ are shown in Fig. 6. In the adiabatic approximation, the RSOI is projected onto the local magnetization direction. Combined with our choice of orbitals, this results in a simple $\cos\theta$ dependence, as seen in Eq. (V). All the results show the same behavior, except for a small window around the Néel magnetic structure, where $\cos\theta=0$, and the kinetic term becomes important. The eigenenergies are thus linear in $\cos\theta$, Fig. 6(a), and the atomic orbital moments are almost saturated to the atomic limit, see Fig. 6(b). The $\xi\gg t$ limit also modifies how the electrons move around the trimer, revealed in the behavior of $M_{\mathrm{P}}$, Fig. 6(b). For the ferromagnetic structures we find values close to those of the model without orbital dependence, $M_{\mathrm{P}}/M_{\mathrm{max}}\approx 0,\pm 1$, compare with Fig. 3(b), and a linear departure from those values when the magnetic structure departs from the ferromagnetic ones. In this limit the trimer approximately decouples into two separate orbital channels, each behaving as described in Sec. IV. When the magnetic structure is close to the Néel structure, there is some subtle behavior. To illustrate this, we sum all the contributions corresponding to the $+$ and $-$ bands, which corresponds to placing three electrons in the three upper or lower eigenstates. The average atomic orbital moment is featureless, see Fig. 6(b), but the net current changes sign before vanishing at the N structure, see Fig. 6(c). ## VI Discussion and conclusions In this paper we discussed a sequence of related models for a trimer, to ascertain how magnetic noncollinearity leads to orbital magnetism, even in the absence of the usual RSOI. The simplest model was introduced in Sec. III, and an external magnetic field was used to define the orbital magnetic moment arising from currents circulating around the trimer. It was augmented with the spin degrees of freedom in Sec. IV, and a family of noncollinear magnetic structures was found to lead to the same kind of orbital moment, even without an external magnetic field. The model was finally endowed with orbital degrees of freedom in Sec. IV, enabling the appearance of the RSOI. The adiabatic approximation was adopted, and the competition between the bond-forming tendencies of the orbital-dependent hopping, and the favoring of current-carrying states by the magnetic noncollinearity and the RSOI was analyzed. Trimer-like structures have been considered in the seminal work of Ref. Shindou and Nagaosa, 2001 ($J\gg\lvert t\rvert$) and of Refs. Tatara and Kawamura, 2002; Tatara and Garcia, 2003; Tatara and Kohno, 2003 ($J\ll\lvert t\rvert$), where the appropriate limits of our model are indicated. Those works established the scalar spin chirality $C(\theta)$, see Eq. 24, as the smoking gun of the non-RSOI-driven orbital effects. It vanishes for ferromagnetic structures and for the triangular antiferromagnetic Néel structure. Our results show that the orbital magnetism of an individual eigenstate is not proportional to $C(\theta)$ (for instance, some have a pure $\cos\theta$ dependence), but that considering a full ‘shell’ or ‘band’ does yield this angular dependence, both in the $J\ll\lvert t\rvert$ and in the $J\gg\lvert t\rvert$ limits. We thus expect partial electron fillings to lead to non-$C(\theta)$ angular behavior, as we already found in DFT calculations for magnetic trimers. dos Santos Dias _et al._ (2016) We also analyzed separately the behavior of the two contributions to the orbital magnetic moment, the atomic one and the one due to circulating (bound) currents. The former is derived from the atomic orbital Zeeman interaction, while the latter follows from the Peierls phase acquired by the hopping amplitudes. In general such a separation is also possible, as established by the modern theory of orbital magnetization. Souza and Vanderbilt (2008); Thonhauser (2011) They give access to two aspects of the persistent (bound) current flowing around the trimer: whether it swirls locally around each atomic site (the local orbital moment), and whether there is a net current circulating around the trimer (the nonlocal orbital moment). Our previous work in Ref. dos Santos Dias _et al._, 2016 focused on the atomic orbital moment in trimers but also in a skyrmion lattice, where a topological contribution was identified, and found to be separable from the RSOI-driven one. As this arose from the magnetic noncollinearity being of a special type for a skyrmion, as encoded in its topological charge, Nagaosa and Tokura (2013) we expect that also the nonlocal orbital moment of skyrmions should also contain such a topological contribution. Lux _et al._ (2017) The orbital magnetic moment can be measured independently of the spin magnetic moment with x-ray magnetic circular dichroism, Thole _et al._ (1992); Carra _et al._ (1993); Chen _et al._ (1995) and there is a theoretical proposal for how to separate the local and nonlocal contributions to the orbital moment. Souza and Vanderbilt (2008) Recent advances in atomic scale manipulation with the tip of a scanning tunneling microscope have enabled à la carte assembly of magnetic nanostructures, including trimers. Hirjibehedin _et al._ (2006); Loth _et al._ (2012); Hermenau _et al._ (2017) The several physical regimes explored in our model can be realized experimentally: the interplay between $J$ and $\lvert t\rvert$ can be tuned by changing the separating between the magnetic atoms, or by assembling them on metallic or (semi-)insulating surfaces, while the strength of the RSOI, $\xi$, can be manipulated by choosing a surface with strong RSOI, or by working with heavy magnetic atoms.Zhang _et al._ (2012) Detection of the orbital magnetism at the atomic scale remains challenging, but recent progress in very sensitive magnetometers utilizing nitrogen vacancies in nanodiamonds might open a way forward. Rondin _et al._ (2014) For a long time the experimental and theoretical study of the orbital magnetic moment has been neglected in favor of its spin counterpart. This is natural, as the spin moment in most cases determines most of the total magnetic moment in a solid, and the magnetic structures and dynamics are governed by the interatomic spin exchange interactions. Orbital interactions are well-known to be important for transport measurements, as can be seen from the large family of Hall effects. The recent focus on the coupling between the itinerant electrons and the spin moments, described by emergent electromagnetic fields, is part of that. Xiao _et al._ (2010); Nagaosa and Tokura (2013) We hope that our work helps bringing the humble orbital magnetic moment back to the limelight. ## Appendix A Eigenvectors for the two-dimensional problem We wish to diagonalize the following matrix with real parameters $w$, $x$, $y$ and $z$, \[A=\begin{pmatrix}a+b_{z}&b_{x}-\mathrm{i}\,b_{y}\\ b_{x}+\mathrm{i}\,b_{y}&a-b_{z}\end{pmatrix}=a\,\sigma_{0}+\mathbf{b}\cdot\bm{ \upsigma}\quad.\] (54) The eigenvalues and the associated eigenspace projectors are then \[\lambda_{\pm}=a\pm\sqrt{\mathbf{b}\cdot\mathbf{b}}\quad,\qquad P_{\pm}=\frac{1 }{2}\left(\sigma_{0}\pm\frac{\mathbf{b}\cdot\bm{\upsigma}}{\sqrt{\mathbf{b} \cdot\mathbf{b}}}\right)\quad.\] (55) The corresponding eigenvectors can be parametrized as \[\ket{+}=\begin{pmatrix}c\\ e^{\mathrm{i}\varphi}s\end{pmatrix}\quad,\qquad\ket{-}=\begin{pmatrix}-e^{- \mathrm{i}\varphi}s\\ c\end{pmatrix}\quad,\] (56) with \[c=\sqrt{\frac{1+\cos\theta}{2}}\quad,\qquad s=\sqrt{\frac{1-\cos\theta}{2}}\quad,\] (57) and the angles \[\cos\theta=\frac{b_{z}}{\sqrt{\mathbf{b}\cdot\mathbf{b}}}\quad,\qquad\theta\in [0,\pi]\quad,\qquad\tan\varphi=\frac{b_{y}}{b_{x}}\in[0,2\pi]\quad.\] (58) ###### Acknowledgements. 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1707.05840	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31472, "num_imgs": 5, "llama3_tokens_count": 9687 }	[ "content_image/1707.05840/time_mnist.png", "content_image/1707.05840/error_bound.png", "content_image/1707.05840/laroue.png", "content_image/1707.05840/error_energy.png", "content_image/1707.05840/rec_example.png" ]	# Multiscale Residual Mixture of PCA: Dynamic Dictionaries for Optimal Basis Learning Randall BALESTRIERO ECE Department Rice University HOUSTON, TX randallbalestriero@gmail.com ###### Abstract In this paper we are interested in the problem of learning an over-complete basis and a methodology such that the reconstruction or inverse problem does not need optimization. We analyze the optimality of the presented approaches, their link to popular already known techniques s.a. Artificial Neural Networks,k-means or Oja’s learning rule. Finally, we will see that one approach to reach the optimal dictionary is a factorial and hierarchical approach. The derived approach lead to a formulation of a Deep Oja Network. We present results on different tasks and present the resulting very efficient learning algorithm which brings a new vision on the training of deep nets. Finally, the theoretical work shows that deep frameworks are one way to efficiently have over-complete (combinatorially large) dictionary yet allowing easy reconstruction. We thus present the Deep Residual Oja Network (DRON). We demonstrate that a recursive deep approach working on the residuals allow exponential decrease of the error w.r.t. the depth. ## 1 Introduction We first discuss briefly the problem of dictionary learning and the different choices one has to make to perform this task depending on the problem at hand. We also introduce notations and standard operators. ### Why a Dictionary ? In order to analyze a given finite dataset $X:=\{x_{n}\in\mathbb{R}^{D},n=1,..,N\}$ different approaches are possible. One of them lies on the assumption that those observations actually come from some latent representation that are mixed together, different mixing leading to different observations but with fixed dictionary $\Phi_{K}:=\{\phi_{k}\in\mathbb{R}^{D},k=1,...,K\}$ with usually $K\ll N$. One can thus rewrite \[x_{n}=f(\Phi,x_{n}).\] (1) In practice, a linear assumption is used for $f$ and we write the prediction as \[\hat{x}_{n}=\hat{f}(\hat{\Phi}_{K},x_{n}),\] (2) where one estimate the functional and the filter-bank through a reconstruction loss defined as \[E_{n}=\|\|x_{n}-\hat{x}_{n}\|\|^{2},\] (3) where we assume here a squared error but any loss function can be used in general. The linear assumption imposed on $f$ leads to the estimation weighting coefficients we denote by $\alpha_{n,k}$ weighting for observation $n$ the atom $k$, and those attributes are the new features used to represent the observations. This new representation of $x_{n}$ via its corresponding feature vector $\bm{\alpha_{n}}$ can be used for many tasks s.a. clustering, denoising, data generation, anomaly detection, compression and much more. Depending on the constraints one imposes on the feature vectors $\bm{\alpha_{n}}$ and the dictionary $\Phi_{K}$, one can come back to standard known frameworks s.a. Principal Component Analysis (PCA)[17], Independent Component Analysis (ICA)[13], Sparse Coding (SC)[20], (Semi) Nonegative Matrix Factorization (sNMF)[18, 10], Gaussian Mixture Model (GMM)[4], and many more, but all those approaches can be categorized into two main categories: Complete Dictionary Learning and Over-Complete Dictionary Learning. The use of a dictionary also extends to standard template matching, the mot popular technique and optimal in the GLRT sense, where new examples are to be mapped to one of the template to be clustered or else. ### (Over)complete Dictionary Learning As we saw, the dictionary learning problem finds many formulations but the main difference resides in the properties of the learned dictionary, namely if it is complete or over-complete. The general case of complete or orthogonal basis imposes the following constraint on the filter-bank: \[<\phi_{j},\phi_{k}>=0,\forall j\not=k,\;K=D,\] (4) and with sometimes the complementary constraints that $\|\|\phi_{k}\|\|=1,\forall k$ leads to an orthonormal basis. The orthogonality of the atoms allow to have exact reconstruction leading to $E_{n}=0,\forall n$ as by definition one has \[\hat{x}_{n}=\sum_{i}\frac{<x_{n},\hat{\phi}_{k}>}{\|\|\hat{\phi}_{k}\|\|^{2}}\hat{ \phi}_{k},\forall n.\] (5) Given a dictionary $\Phi$ this decomposition is unique and thus we have $(\hat{\Phi}_{K},x_{n})\Rightarrow\hat{\bm{\alpha}}_{n},\forall n$. However, while through the Gram-Schmidt process existence of such a basis is guaranteed, it is not unique. On the other hand, $\Phi_{K}$ can be an over-complete basis with the main difference coming that now $K>D$ and with the extreme case of $K=N$. Containing more atoms than the dimension of the space leads to interesting properties in practice such as sparse and independent features (ICA), clustering properties (K-means, NMF), biological understanding as is has been empirically shown that visual and auditory cortex of many mammals contains over-complete dictionaries. Yet, all those benefits coming from the redundancy of the atoms also lead to the non-unique features $\bm{\alpha}_{n}$ even when the dictionary is kept fixed, thus we have that $(\hat{\Phi}_{K},x_{n})\not\Rightarrow\hat{\bm{\alpha}}_{n}$. As a result, one has to solve the additional optimization problem of \[\bm{\hat{\alpha}_{n}}=\operatorname{arg\,min}_{\bm{\alpha}\in\Omega\subset \mathbb{R}^{K}}\|\|x_{n}-\sum_{k}\alpha_{k}\hat{\phi}_{k}\|\|.\] (6) As a result, whether one is in a complete or over-complete dictionary setting, the optimization problems are always ill-posed by the non-uniqueness of the solutions forcing to impose additional structures or constraints in order to reach well posed problems. For the complete case, the main approach consist in imposing that as few atoms as possible are used leading to PCA, a very powerful approach for dimensionality reduction and compression. For over-complete cases, different sparsity criteria are imposed on the features $\bm{\alpha}_{n}$ such as norm-(0,1,2). For a norm-0 we are in the Matching pursuit case, norm1 is sparse coding and norm2 is ridge regression. For each of those cases many very efficient exact or iterative optimization algorithms have been developed to estimate $\hat{\Phi}_{k}$ and $\hat{\bm{\alpha}}_{n}$ yet there still exist a conceptual gap between the two concepts and the two approaches are often seen as orthogonal. As we have seen, both settings lead to different beneficial aspects, compression, easy of projection and reconstruction or sparsity/clustering but more complex optimization problems, but, at a higher level, the signal processing community has always put a gap between those frameworks. As well put by Meyer in [CITE] one has to choose between encoding and representation. We thus propose in this paper a novel approach allowing to have an over-complete dictionary yet given an input, a dynamic basis selection reduces it to an optimal complete dictionary without need for optimization in order to reconstruct. The selection is done without optimization in a forward manner leading to an efficient algorithm. This thus allows to inherit from all the properties induced by orthonormal basis while allowing adaptivity for better allowing to learn an over-complete basis with a nonlinear basis selection leading to an orthonormal basis when conditioned on the input. We also provide results from a low-dimensional manifold perspective and show that our approach perform nonparametric orbit estimation. We validate on compression, dictionary learning tasks and clustering. ## 2 Deep Residual Oja Network ### Shallow Model Is it possible to learn a dictionary inheriting the benefits or complete and over-complete dictionaries ? We present one solution here. We first motivate the need for such a framework as well as present the general approach and notations. Throughout the next sections, the choice of the Oja name for the algorithm will become blatant for the reader. Keeping the previously defined notation, we aim at learning an over-complete basis with the number of atoms defined by $FK$ with $F>1$ called the increase factor, note that $F=1$ lead to a complete basis, $K=D$ unless otherwise defined. By definition, the following projection-reconstruction scheme \[\hat{x}_{n}=\sum_{k}\frac{<x_{,}\hat{\phi}_{k}>}{\|\|\hat{\phi}_{k}\|\|^{2}}\hat{ \phi}_{k},\] (7) can not reach an error $E_{n}<\epsilon$ and in fact $E_{n}$ increases with $F$. One way to resolve this issue comes from an optimal basis selection point of view leading to \[E_{n}:=\|\|x_{n}-\sum_{k}\delta_{n,k}\frac{<x_{,}\hat{\phi}_{k}>}{\|\|\hat{\phi}_{ k}\|\|^{2}}\hat{\phi}_{k}\|\|<\epsilon,\] (8) with $\delta{n,k}\in\{0,1\},\forall n,k$ representing a mask allowing to use a subset of the dictionary $\hat{\Phi}_{FK}$ we denote by $\rho_{\delta_{n,.}}[\hat{\Phi}_{FK}]$. #### 2.1.1 Error Bounds, Learning and Link with Oja Rule We first provide an error upper-bound for the proposed scheme $(S1)_{\kappa=1}$. To simplify notations be also define by $\phi_{\kappa}(x_{n})$ the atom defined by \[\phi_{\kappa}(x_{n})=\phi_{k^{\prime}},k^{\prime}=\operatorname{arg\,max}_{k} \frac{\|<x_{n},\phi_{k}>\|^{2}}{\|\|\phi_{k}\|\|^{2}}.\] (9) Theorem 1.: _The error induced by $(S1)_{\kappa=1}$ is $\|\|x_{n}\|\|^{2}-\frac{\|<x,\phi_{\kappa}(x_{n})>\|^{2}}{\|\|\phi_{\kappa}(x_{n})\|\|^{ 2}}$ which is simply the reconstruction error from the best atom as since only one filter is used, nothing else is present._ Proof.: By the incomplete basis theorem, there exists a basis s.t. it contains the atom $\phi_{\kappa}$, we denote by $\phi_{k}$ such a basis, with $k=1,...,D$, we thus have \[E_{n}= \|\|x_{n}-\frac{<x,\phi_{\kappa}(x_{n})>}{\|\|\phi_{\kappa}(x_{n})\|\|^ {2}}\phi_{\kappa}(x_{n})\|\|^{2}\] \[= \|\|\sum_{k}\frac{<x_{n},\phi_{k}>}{\|\|\phi_{k}\|\|^{2}}\phi_{k}-\frac {<x,\phi_{\kappa}(x_{n})>}{\|\|\phi_{\kappa}(x_{n})\|\|^{2}}\phi_{\kappa}(x_{n})\|\| ^{2}\] incomplete basis theorem \[= \|\|\sum_{k\not=\kappa}\frac{<x_{n},\phi_{k}>}{\|\|\phi_{k}\|\|^{2}} \phi_{k}\|\|^{2}\] \[= \sum_{k\not=\kappa}\frac{\|<x_{n},\phi_{k}>\|^{2}}{\|\|\phi_{k}\|\|^{2}}\] \[= \|\|x\|\|^{2}-\frac{\|<x_{n},\phi_{\kappa}(x_{n})>\|^{2}}{\|\|\phi_{ \kappa}(x_{n})\|\|^{2}}\] Parseval’s Theorem \[=\] (10) And we have by definition $E_{n}\geq 0,E_{n}=0\iff x_{n}=\phi_{\kappa}(x_{n})$ ∎ There is first a few comments on the loss and updates. First of all, the loss is closely related to a k-mean objective with cosine similarity and specifically spherical k-means which is the case where the centers and the data points are re-normalized to unit norm and has the objective to minimize \[\sum_{n}(1-cos(x_{n},p_{c(n)})).\] (11) #### 2.1.2 Learning and Oja Rule In order to learn the filter-bank $\Phi_{K}$, a common approach is to use an alternating scheme between finding the cluster belongings and optimizing the atoms w.r.t. this estimate. We first derive a gradient descent scheme to update the atoms and study some of its characteristics. If we now denote by $n(k):=\{n:n=1,...,N\|\kappa(x_{n})=k\}$ be the collection of the sample index in cluster $k$, he resulting loss $E_{n(k)}:=\frac{\sum_{n\in n(k)}E_{n}}{Card(n(k))}$ we can now derive a gradient descent step as \[\phi_{k}^{(t+1)}(\lambda)=\phi_{k}^{(t)}-\lambda\frac{dE_{n(k)}}{d\phi_{k}},\] (12) with \[\frac{dE_{n(k)}}{d\phi_{k}} =\frac{1}{Card(n(k))}\sum_{n\in n(k)}\frac{2\|<x_{n},\phi_{k}>\|}{\| \|\phi_{k}\|\|^{2}}\Big{(}\frac{\|<x_{n},\phi_{k}>\|\phi_{k}}{\|\|\phi_{k}\|\|^{2}}-(-1 )^{1_{<x_{n},\phi_{k}><0}}x_{n}\Big{)},\] \[=\frac{1}{Card(n(k))}\sum_{n\in n(k)}\frac{2<x_{n},\phi_{k}>}{\|\| \phi_{k}\|\|^{2}}\Big{(}\frac{<x_{n},\phi_{k}>\phi_{k}}{\|\|\phi_{k}\|\|^{2}}-x_{n} \Big{)}.\] (13) On the other hand, if one adopts an adaptive gradient step $\lambda$ per atom and point with one of the two strategies $\lambda_{1},\lambda_{2}$ defined as \[\lambda_{1} =\frac{<x_{n},\phi_{k}>}{2\|\|x_{n}\|\|^{2}}\] (14) \[\lambda_{2} =\frac{\|\|\phi_{k}\|\|^{4}}{2<x_{n},\phi_{k}>^{2}}\] (15) then we have the \[\phi_{k}^{(t+1)}(\lambda_{1}) =\phi_{k}^{(t)}-\frac{1}{\sum_{n}\cos(\theta(x_{n},\phi_{k}))^{2} }\sum_{n\in n(k)}\cos(\theta(x_{n},\phi_{k}))^{2}\Big{(}\frac{<x_{n},\phi_{k}> \phi_{k}}{\|\|\phi_{k}\|\|^{2}}-x_{n}\Big{)},\] (16) \[\phi_{k}^{(t+1)}(\lambda_{2}) =\frac{1}{Card(n(k))}\sum_{n\in n(k)}\frac{\|\|\phi_{k}\|\|^{2}}{<x_{ n},\phi_{k}>}x_{n}\] (17) we thus end up with in the $\lambda_{1}$ case to a simple update rule depending on a weighted average of the points in the cluster based on their cosine similarity squared whereas for $\lambda_{2}$ we obtain a rule a la convex NMF with is a plain without update combination of the points available. On the other hand, it is clear that minimizing $E_{n}$ from Eq. 10 is equivalent to maximizing $E^{+}_{n}=\frac{<x_{n},\phi_{\kappa}(x_{n})>}{\|\|\phi_{\kappa}(x_{n})\|\|^{2}}$. As a result, one can seize in Eq. 13 the Oja rule as we can rewrite a GD update of $\phi_{k}$ as \[\phi_{k}^{(t+1)} =\phi^{(t)}_{k}+\gamma\frac{dE^{+}_{n}}{d\phi_{k}}(\phi^{(t)}_{k})\] (18) \[\phi_{k}^{(t+1)} =\phi^{(t)}_{k}+\gamma\Big{(}x_{n}\frac{<x_{n},\phi_{k}>}{\|\|\phi_ {k}\|\|^{2}}-(\frac{<x_{n},\phi_{k}>}{\|\|\phi_{k}\|\|^{2}})^{2}\phi_{k}\Big{)}\] (19) known as the Oja rule. SPEAK ABOUT OJA RULE. And in fact, the convergence of Oja rule toward the first eigenvector-eigenvalue is not surprising as $E^{+}_{n(k)}$ leads explicitly to \[\phi_{k}=\operatorname{arg\,max}_{\phi}\frac{1}{Card(n(k))}\frac{\phi^{T}X(k) ^{T}X(k)\phi}{\phi^{T}\phi},\] (20) which is known as the Rayleigh quotient and is a formulation of PCA leading to a one step global optimum being the greatest eigenvector-eigenvalue. <figure><img src="content_image/1707.05840/time_mnist.png"><figcaption></figcaption></figure> [doublebox]Filter-Bank Learning strategyX,K Initialize Φ_K not converged k 1 K Compute n(k) with current Φ_K Update _ϕ__k with n(k) and X(k) according to Eq. 20 Φ_k [doublebox]Online Filter-Bank Learning strategyX,K Initialize Φ_K not converged n 1 N _κ_= arg max_k \|<xn,_ϕ_k>\|2\|\|_ϕ_k\|\|2\|\|xn\|\|2 Update _ϕ___κ according to Eq. 16 or Eq.17 or Eq.19_ Φ_k Theorem 2.: _If the distribution of the $x_{n}$ in the space is uniformly distributed, the optimal over-complete basis for $(S1)_{kappa=1}$ is thus the one corresponding of a quantization of the sphere, it is unique up to a change of sign and global rotations (same applied to each atom). For the $2$-dimensional case it is easy to see that the maximum error for any given point $x_{n}$ is exactly upper-bounded by $\|\|x\|\|^{2}\Big{(}1-\cos(\frac{\pi}{2FK})^{2}\Big{)}$ if $FK$ growths exponentially .(?? CHECK POWER OF HIGH DIMENION COSINE decompose as union of 2D spaces)_ Proof.: For 2D we now the upper bound is $\|\|x\|\|^{2}\Big{(}1-\cos(\frac{\pi}{2FK})^{2}\Big{)}$ with $FK$ atoms, we thus rewrite \[\|\|x-\hat{x}\|\|^{2}=\sum_{d=1}^{D/2}\|\|x_{d}-\hat{x}_{d}\|\|^{2}\] and see that in order to have the upper bound for each subspace we need the cartesian product of all the subspace basis $\Phi_{FK}$ leading to $FK$ atoms. Thus one need to grow exponentially the number of atom w.r.t the dimension to have a loss increasing linearly. ∎ <figure><img src="content_image/1707.05840/error_bound.png"><figcaption></figcaption></figure> However this pessimistic upper-bound assumes the worst possible scenario: uniform distribution of the data point in the space $\mathbb{R}^{D}$ which in general is false. In fact, many dataset have inherent structures and at least lye only in a small subset sometime regular of $\mathbb{R}^{D}$. In general, data might be living in unions of subsets and thus providing a general strategy or optimal basis is a priori more complex thus pushing the need to learn the atoms as it is done in general for k-mean applications. Theorem 3.: _A sufficient condition for $(S1)_{kappa=1}$ to be optimal is that the data are already clustered along $FD$ lines, or orbits ??? :)_ We now present one way to tackle the curse of dimensionality in the next section ### Multiple Atoms \[E_{n}=\|\|x_{n}-\sum_{k=1}^{K}\frac{<x,\phi^{k}_{\kappa}(x_{n})>}{\|\|\phi^{k}_{ \kappa}(x_{n})\|\|^{2}}\phi^{k}_{\kappa}(x_{n})\|\|^{2}\] (21) For learning atom after atom a la coordinate ascent we have that \[\hat{\phi}^{k^{\prime}}_{j}= \operatorname{arg\,min}_{\phi^{k^{\prime}}_{j}}\sum_{n}E_{n}\] \[= \operatorname{arg\,min}_{\phi^{k^{\prime}}_{j}}\sum_{n\in n(k,j) }\|\|\Big{(}x_{n}-\sum_{k=1,\not=k^{\prime}}^{K}\frac{<x,\phi^{k}_{\kappa}(x_{n} )>}{\|\|\phi^{k}_{\kappa}(x_{n})\|\|^{2}}\Big{)}-\frac{<x,\phi^{k^{\prime}}_{j}>}{ \|\|\phi^{k^{\prime}}_{j}\|\|^{2}}\phi^{k^{\prime}}_{\kappa}(x_{n})\|\|^{2}\] as we showed in the last section we end up with the same update rule but with the input being substracted by the other used atoms. Thus we still perform PCA but on the input minus the other atoms. Note that this ensures orthogonality between the atoms. ### From Shallow to Deep Residual for better Generalization Error Bounds and Combinatorially Large Dictionaries <figure><img src="content_image/1707.05840/laroue.png"><figcaption></figcaption></figure> We now consider the analysis of the generalization performance on out of bag observations as well as the problem of having really large dataset. Theorem 4.: _If we suppose an finite training set and an Oja Network with sufficient filters in order to reach a training error of $0$ then we know that the generalization error is directly lower-bounded by how close the testing and training examples are. In fact_ \[E_{new}\propto\cos(\theta(x_{\kappa},x_{new})),\kappa=\operatorname{arg\,min} _{n}\cos(\theta(x_{n},x_{new})).\] (22) The proof is straightforward as we know the network is able to perfectly reconstruct the training set and only the training set. As a result, if the training set is well and uniformly sampled among the space of possible observations, a shallow Oja Network can be considered as optimal also for the testing set. However, and especially for computer vision task, it is well known that every observation is very far form each other in term of distance when dealt with in the pixel domain, also, requiring a proper sampling of the space of images is clearly outrageous. We thus now present the result motivating deep architectures in general including the Oja Network. \[R^{(l)}_{n}= R^{(l-1)}_{n}-\frac{<R^{(l-1)}_{n},\phi_{\kappa}^{(l)}>}{\|\|\phi_ {\kappa}^{(l)}\|\|^{2}}\phi_{\kappa}^{(l)},\kappa=\operatorname{arg\,max}_{k} \frac{\|<R^{(l-1)}_{n},\phi_{k}^{(l)}>\|}{\|\|R^{(l-1)}_{n}\|\|^{2}\|\|\phi_{k}^{(l)}\| \|^{2}}\] \[R^{(0)}_{n}= x_{n}\] (23) as a result as soon as the input and the template are not orthogonal there is convergence. Theorem 5.: _Since we have by definition that the selected atom is the one with smaller angle, if it is $0$ it means that the input $R^{(l-1)}$ is orthogonal to all the learned dictionary $\Phi^{(l)}$_ \[\cos\Big{(}\theta(R^{(l-1)}_{n},\phi^{(l)}_{\kappa})\Big{)}^{2}=0\iff R^{(l-1) }indep\phi^{(l)}_{k}\forall k,\] (24) _and thus they live in two orthogonal spaces. TO PROVE : ALL THE NEXT ONES ARE ALSO 0_ Theorem 6.: _The residual decreases exponentially w.r.t. the depth of the model._ Proof.: \[\|\|R^{(l)}_{n}\|\|^{2}= \|\|R^{(l-1)}_{n}-\frac{<R^{(l-1)}_{n},\phi_{\kappa}^{(l)}>}{\|\|\phi _{\kappa}^{(l)}\|\|^{2}}\phi_{\kappa}^{(l)}\|\|^{2}\] \[= \|\|R^{(l-1)}_{n}\|\|^{2}-\frac{<R^{(l-1)}_{n},\phi_{\kappa}^{(l)}>^{ 2}}{\|\|\phi_{\kappa}^{(l)}\|\|^{2}}\] \[= \|\|R^{(l-1)}_{n}\|\|^{2}\Big{(}1-\cos\Big{(}\theta(R^{(l-1)}_{n}, \phi_{\kappa}^{(l)})\Big{)}^{2}\Big{)}\] (25) \[= \|\|x_{n}\|\|^{2}\prod_{l=1}^{l}\Big{(}1-\cos\Big{(}\theta(R^{(l-1)}_ {n},\phi_{\kappa}^{(l)})\Big{)}^{2}\Big{)}\] (26) ∎ The final template can be flattened via \[T_{n}= \sum_{l}\frac{<R^{(l-1)}_{n},\phi_{\kappa}^{(l)}>}{\|\|\phi_{\kappa }^{(l)}\|\|^{2}}\phi_{\kappa}^{(l)}\] (27) \[= \sum_{l}P^{(l)}_{n}\] (28) #### 2.3.1 Learning Computing the gradient finds a great recursion formula we define as follows: \[A_{i,j}=\left\{\begin{matrix}0\iff j<i\\ \frac{<R^{(i-1)},\phi^{(i)}_{\kappa}>I_{d}+R^{(i-1)}\phi^{(i)}_{\kappa}}{\|\| \phi^{(i)}_{\kappa}\|\|^{2}}+\frac{2<R^{(i-1)},\phi^{(i)}_{\kappa}>\phi^{(i)}_{ \kappa}\phi^{(i)T}_{\kappa}}{\|\|\phi^{(i)}_{\kappa}\|\|^{4}}\iff i=j\\ A_{i,j-1}-\frac{\phi^{(j)}_{\kappa}\phi^{(j)T}_{\kappa}A_{i,j-1}}{\|\|\phi^{(j)} _{\kappa}\|\|^{2}}\iff j>i\end{matrix}\right.\] (29) thus $A_{i,j}\in\mathbb{R}^{D\times D}$ then we have \[\mathcal{L}_{n}= \|\|R^{(L)}_{n}\|\|,\] (30) \[\frac{\textbf{d}\mathcal{L}^{2}}{\textbf{d}\phi^{(l)}_{\kappa}}= 2R^{(L)}_{n}\frac{\textbf{d}R^{(L)}_{n}}{\textbf{d}\phi^{(l)}_{ \kappa}}\] (31) However as we will see below we have a nice recursive definition to compute all those derivatives, in fact \[\text{Init. }\begin{cases}\frac{\textbf{d}P^{(l)}_{n}}{\textbf{d}\phi^{(l)}_{ \kappa}}&=\frac{<R^{(l-1)}_{n},\phi^{(l)}_{\kappa}>I_{d}+R^{(l-1)_{n}}\phi^{(l )}_{\kappa}}{\|\|\phi^{(l)}_{\kappa}\|\|^{2}}+\frac{2<R^{(i-1)}_{n},\phi^{(l)}_{ \kappa}>\phi^{(l)}_{\kappa}\phi^{(l)T}_{\kappa}}{\|\|\phi^{(l)}_{\kappa}\|\|^{4}}, \\ \frac{\textbf{d}R^{(l)}_{n}}{\textbf{d}\phi^{(l)}_{\kappa}}&=-\frac{\textbf{d} P^{(l)}_{n}}{\textbf{d}\phi^{(l)}_{\kappa}}\end{cases}\] (33) \[\text{Recursion }\begin{cases}\frac{\textbf{d}P^{(l+1)}_{n}}{\textbf{d}\phi^{( l)}_{\kappa}}&=\frac{\phi^{(l+1)}_{\kappa}\phi^{(l+1)^{T}}_{\kappa}}{\|\|\phi^{( l+1)}_{\kappa}\|\|^{2}}\frac{\textbf{d}R^{(l)}_{n}}{\textbf{d}\phi_{\kappa}^{(l) }},\\ \frac{\textbf{d}R^{(l+1)}_{n}}{\textbf{d}\phi^{(l)}_{\kappa}}&=\frac{\textbf{d }R^{(l)}_{n}}{\textbf{d}\phi^{(l)}_{\kappa}}-\frac{\textbf{d}P^{(l+1)}_{n}}{ \textbf{d}\phi^{(l)}_{\kappa}}\end{cases}\] (34) [doublebox]Residual Oja NetworkX,K R_n X_n, ∀n l 1 L Initialize Φ^(l)_K from R not converged k 1 K Compute n(k) with current Φ^(l)_K Update _ϕ_^(l)_k with n(k) and R(k) according to Eq. 20 R_n = (R_n-<Rn,_ϕ_(l)_κ_>\|\|_ϕ_(l)_κ_\|\|2_ϕ_^(l)__κ_) Φ^(l)_k, ∀l [doublebox]Online Residual Oja NetworkX,K R_n X_n, ∀n l 1 L Initialize Φ^(l)_K from R not converged n 1 N _κ_= arg max_k \|<Rn,_ϕ_(l)k>\|2\|\|_ϕ_(l)k\|\|2\|\|Rn\|\|2 Update _ϕ_^(l)__κ according to Eq. 16 or Eq.17 or Eq.19_ R_n = (R_n-<Rn,_ϕ_(l)_κ_>\|\|_ϕ_(l)_κ_\|\|2_ϕ_^(l)__κ_) Φ^(l)_k, ∀l Theorem 7.: _With a Deep (Oja) Network, the previously presented lower-bound of the generalization error becomes an upper-bound._ In addition of guaranteeing better generalization errors through depth, we also benefit from another gain. The depth as we will see allows for an exponential amount of possible templates to be constructed perfectly with only a linear increase in the number of learned parameters. ``` #################### # INPUT: X(N,channels,Ix,Jx),w(n_filters,channels,Iw,Jw) #################### k = T.nnet.conv.conv2d(x,w,stride=stride, border_mode=’valid’,flip_filters=False,input_shape=(N,channels,Ix,Jx), filters_shape=(n_filters,channels,Iw,Jw))#(N,n_filters,(Ix-Iw)/stride+1,(Jx-Jw)/stride+1) output = ((k>0)2-1)T.signal.pool.max_pool_2d_same_size( theano.tensor.abs_(k).dimshuffle([0,2,3,1]), (1,n_filters)).dimshuffle([0,3,1,2])#(N,n_filters,(Ix-Iw)/stride+1,(Jx-Jw)/stride+1) mask = T.switch(T.eq(output,0),0,1)#(N,n_filters,(Ix-Iw)/stride+1,(Jx-Jw)/stride+1) ``` Code 1: Input to Mask ``` #################### # INPUTS: Z(N,n_filters,Iz,Jz),w(n_filters,channels,Iw,Jw),stride #################### dilated_output = T.set_subtensor(T.zeros((N,n_filters,(Iz-1)stride),(Iz-1)stride), dtype=’float32’)[:,:,::stride,::stride],Z)#(N,n_filters,Ix-Iw+1,Jx-Jw+1) rec = T.nnet.conv.conv2d(dilated_Z,w.dimshuffle([1,0,2,3]),stride=1, border_mode=’full’,flip_filters=False)#(N,channels,Ix,Jx) ``` Code 2: Reconstruction ``` ################### # INPUT : rec(N,C,Ix,Jx),mask(N,n_filters,Iz,Jz),Iw,Jw ################### d_W,outputs=theano.scan(fn=lambda acc,i,X,mask: acc+conv2d(rec[i].dimshuffle([0,’x’,1,2]),mask[i].dimshuffle([0,’x’,1,2]), input_shape=(C,1,Ix,Jx), filter_shape=(n_filters,1,Iz,Jz)).dimshuffle([1,0,2,3]), sequences=[theano.tensor.arange(N,dtype=’int32’)], non_sequences=[rec,mask],outputs_info = T.zeros((n_filters,C,Iw,Jw),dtype=’float32’)) d_W = d_W[-1] \caption{algo} ``` Code 3: Mask to Grad <figure><img src="content_image/1707.05840/error_energy.png"><figcaption>Figure 1: Top : evolution of the reconstruciton error w.r.t. epochs. Bottom:evolution of the energy captured per level during training. At first stage thelast levels capture everything since random initialization makes the globalfilters almost orthogonal to images, during training global filters learn tocapture the low frequencies. Since it is known that natural images have a 1/fdecay of energy over frequencies f we can see that the final energyrepartition is indeed bigger for low-frequency/global filters and go down forsmaller filters.</figcaption></figure> <figure><img src="content_image/1707.05840/rec_example.png"><figcaption>Figure 2: Example of decomposition and reconstruction of some CIFAR10 images.From right to left is the final residual (reconstruction minus original), theoriginal image, the reconstructed images and then all the decompositions, onthe left is the global/large one. Summing elementwise columns 1 to 8 leads tocolumn 9 the reconstrued input.</figcaption></figure> ### Previous Work The proposedm ethod can be seen as bilinear sparse coding with one-hot latent vector $y$[12] for the case of only one filter used. There is also a direct link with the probabilistic version of this work, namely mixture of PPCA [24, 23], as here we are in a ”hard clustering” case similarly to k-means versus GMM. By the selection of the best matching atom, we find some links with matching pursuit [26, 21] and also locality sensitive hashing [15, 16] especially in the cosine similarity distance. This problem can also be seen from a best basis selection point of view coupled with dictionary learning. Popular examples with varying degrees of computational overhead include convex relaxations such as $L1$-norm minimization [3, 7, 22], greedy approaches like orthogonal matching pursuit (OMP) [21, 25], and many flavors of iterative hard-thresholding (IHT) [5, 6] Variants of these algorithms find practical relevance in numerous disparate domains, including feature selection [9, 11], outlier removal [8, 14], compressive sensing [2], and source localization [1, 19] ## 3 Conclusion We presented a hierarchical version of the deterministic mixture of PCA and presented results on CIFAR10 images. We also provided algorithms allowing GPU computation for large scale dataset and speed. The main novelty comes from the deterministic formulate of the probabilistic mixture of PCA which allows easier use as it is known in general that MPPCA is unstable for large scale problems. 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1902.01953	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 55849, "num_imgs": 14, "llama3_tokens_count": 16362 }	[ "content_image/1902.01953/x1.png", "content_image/1902.01953/x2.png", "content_image/1902.01953/x3.png", "content_image/1902.01953/x4.png", "content_image/1902.01953/x5.png", "content_image/1902.01953/x6.png", "content_image/1902.01953/x7.png", "content_image/1902.01953/x8.png", "content_image/1902.01953/x9.png", "content_image/1902.01953/x10.png", "content_image/1902.01953/x11.png", "content_image/1902.01953/x12.png", "content_image/1902.01953/x13.png", "content_image/1902.01953/x14.png" ]	# Superconductivity under pressure in the Dirac semimetal PdTe${}_{2}$ H. Leng h.leng@uva.nl Van der Waals - Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands A. Ohmura Pacific Rim Solar Fuel System Research Center, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan Faculty of Science, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan L. N. Anh International Training Institute for Materials Science, Hanoi University of Science and Technology, 1 Dai Co Viet Road, Ha Noi, Vietnam F. Ishikawa Faculty of Science, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan T. Naka National Institute for Materials Science, Sengen 1-2-1, Tsukuba, Ibaraki 305-0047, Japan Y. K. Huang Van der Waals - Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands A. de Visser a.devisser@uva.nl Van der Waals - Zeeman Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands February 27, 2024 ###### Abstract The Dirac semimetal PdTe${}_{2}$ was recently reported to be a type-I superconductor ($T_{c}=$1.64 K, $\mu_{0}H_{c}(0)=13.6$ mT) with unusual superconductivity of the surface sheath. We here report a high-pressure study, $p\leq 2.5$ GPa, of the superconducting phase diagram extracted from ac-susceptibility and transport measurements on single crystalline samples. $T_{c}(p)$ shows a pronounced non-monotonous variation with a maximum $T_{c}=$1.91 K around 0.91 GPa, followed by a gradual decrease to 1.27 K at 2.5 GPa. The critical field of bulk superconductivity in the limit $T\to 0$, $H_{c}(0,p)$, follows a similar trend and consequently the $H_{c}(T,p)$-curves under pressure collapse on a single curve: $H_{c}(T,p)=H_{c}(0,p)[1-(T/T_{c}(p))^{2}]$. Surface superconductivity is robust under pressure as demonstrated by the large superconducting screening signal that persists for applied dc-fields $H_{a}>H_{c}$. Surprisingly, for $p\geq 1.41$ GPa the superconducting transition temperature at the surface $T_{c}^{S}$ is larger than $T_{c}$ of the bulk. Therefore surface superconductivity may possibly have a non-trivial nature and is connected to the topological surface states detected by ARPES. We compare the measured pressure variation of $T_{c}$ with recent results from band structure calculations and discuss the importance of a Van Hove singularity. ## I Introduction The family of layered transition metal dichalcogenides attracts much attention, because of the wide diversity of fascinating electronic properties. One of the present-day research interests is the possibility to realize novel quantum states as a result of the topological non-trivial nature of the electronic band structure Soluyanov _et al._ (2015); Huang _et al._ (2016); Yan _et al._ (2017); Bahramy _et al._ (2018). Especially, it has been proposed that these materials host a generic coexistence of type-I and type-II three dimensional Dirac fermion states Bahramy _et al._ (2018). An interesting example in this respect is PdTe${}_{2}$ that has been classified as a type-II Dirac semimetal following a concerted examination of _ab-initio_ electronic structure calculations and angle-resolved photoemission spectroscopy (ARPES) experiments Yan _et al._ (5); Fei _et al._ (2017); Noh _et al._ (2017); Bahramy _et al._ (2018); Clark _et al._ (2018). In a type-II Dirac semimetal the Hamiltonian breaks Lorentz invariance because the energy dispersion relations, _i.e._ the Dirac cone, are tilted Soluyanov _et al._ (2015). The Dirac point is then the touching point between the electron and hole pockets and a nearly flat band may form near the Fermi level. Moreover, PdTe${}_{2}$ is a superconductor below 1.6 K Guggenheim _et al._ (1961); Leng _et al._ (2017), which solicits the intriguing question whether superconductivity is promoted by the nearly flat band and consequently has a topological nature Fei _et al._ (2017). Topological non-trivial superconductors attract much interest since it is predicted these may host protected Majorana zero modes at the surface (for recent reviews see Refs. Sato and Fujimoto, 2016; Sato and Ando, 2017). This in turn offers a unique design route to make devices for topological quantum computation. Superconductivity in PdTe${}_{2}$ was discovered in 1961 (Ref. Guggenheim _et al._, 1961), but was not investigated in detail until 2017, when Leng _et al._ Leng _et al._ (2017) carried out comprehensive magnetic and transport experiments on single-crystals. Unexpectedly, dc-magnetization measurements, $M(H)$, revealed that PdTe${}_{2}$ is a bulk type-I superconductor, which was further embodied by the observation of the differential paramagnetic effect in the ac-susceptibility measured in applied magnetic dc-fields. The critical field $H_{c}(T)$ follows the standard quadratic temperature variation with $\mu_{0}H_{c}(0)=13.6$ mT. The possibility of type-I superconductivity in Dirac materials was recently investigated by Shapiro _et al_. Shapiro _et al._ (2018) employing a microscopic pairing theory for an arbitrary tilt parameter of the Dirac cone. For PdTe${}_{2}$ these authors concluded type-I superconductivity is feasible for a tilt parameter $k=2$. Another interesting aspect of PdTe${}_{2}$ is the observation of surface superconductivity, as evidenced by large screening currents in the ac-susceptibility for applied dc-fields $H_{a}>H_{c}$ Leng _et al._ (2017). The critical field for surface superconductivity $H_{c}^{S}$ does not follow the standard Saint-James - de Gennes expression $H_{c3}=2.39\times\kappa H_{c}$ Saint-James and de Gennes (1963), where $\kappa$ is the Ginzburg-Landau parameter. This led to the proposal Leng _et al._ (2017) that superconductivity of the surface sheath might have a topological nature and originates from topological surface states detected by ARPES Yan _et al._ (5); Noh _et al._ (2017). More recently, specific heat Amit and Singh (2018) and magnetic penetration depth Salis _et al._ (2018); Teknowijoyo _et al._ (2018), measurements have been conducted. These confirm conventional weak-coupling Bardeen-Cooper-Schrieffer (BCS) superconductivity, with a full gap in the bulk. At the same time zero-field scanning tunneling microscopy (STM) and spectroscopy (STS) experiments Das _et al._ (2018); Clark _et al._ (2018) lend further support for the absence of in-gap states, which seems to rule out topological superconductivity at the surface. Dominant $s$-wave superconductivity was also concluded from tunneling spectroscopy experiments on side junctions Voerman _et al._ (2019). Nonetheless, the uncommon type-I behavior for a binary compound, and the unexplained superconductivity of the surface sheath, justify a further in-depth examination of the superconducting properties of PdTe${}_{2}$. We here report the results of a high-pressure investigation of the superconducting phase diagram of PdTe${}_{2}$ single crystals ($p\leq 2.5$ GPa). Combined resistivity and ac-susceptibility measurements show $T_{c}$ increases at low pressures, then passes through a maximum of 1.91 K around 0.91 GPa, and subsequently decreases at higher pressure. The critical field for $T\to 0$, $H_{c}(0,p)$, follows a similar behavior and consequently the $H_{c}(T)$-curves at different pressures collapse on a single curve. Under pressure superconductivity maintains its type-I character. Surface superconductivity is robust under pressure as demonstrated by the large superconducting screening signal that persists for applied dc-fields $H_{a}>H_{c}$. Surprisingly, for $p\geq 1.41$ GPa the superconducting transition temperature of the surface, $T_{c}^{S}$, is larger than $T_{c}$ of the bulk. Therefore surface superconductivity may possibly have a non-trivial nature and is related to the topological surface states detected by ARPESYan _et al._ (5); Noh _et al._ (2017); Clark _et al._ (2018). The initial increase of $T_{c}$ with pressure is at variance with the smooth depression predicted by recent electronic structure calculations Xiao _et al._ (2017). ## II Experiment The crystals used for our high pressure study were taken from the single-crystalline boule prepared by the modified Bridgman technique Lyons _et al._ (1976) and characterized in Ref. Leng _et al._, 2017. Powder X-ray diffraction confirmed the trigonal CdI${}_{2}$ structure (spacegroup $P\bar{3}m1$ Thomassen (1929). Scanning electron microscopy (SEM) with energy dispersive x-ray (EDX) spectroscopy showed the proper 1:2 stoichiometry within the experimental resolution of 0.5%. Laue backscattering was used to orient the crystals. Standard four-point resistance measurements were performed in a Physical Property Measurement System (PPMS, Quantum Design) at temperatures down to 2 K. The resistivity, $\rho(T)$, of our crystals shows metallic behavior. A typical trace in the temperature range 2-300 K is shown in Fig. 1. The residual resistance ratio $R$(300K)/$R$(2K) = 30. Electrical resistance, $R(T,H)$, and ac-susceptibility, $\chi_{ac}(T,H)$, measurements under high-pressure were performed utilizing a clamp-type piston-cylinder cell, which has a double-layer made of Cu-Be and NiCrAl alloys. The single crystal sizes for $R(T,H)$ and $\chi_{ac}(T,H)$ were $\sim 2.3\times 1.0\times 0.18$ mm${}^{3}$ and $\sim 2.9\times 1.0\times 0.67$ mm${}^{3}$, respectively. Both samples were mounted on a plug and loaded into a Teflon capsule together with coils and a pressure-transmitting medium, Daphne oil 7373, for hydrostatic compression. A schematic drawing of the plug with samples and coil is shown in Fig. 1. The generated pressure in the capsule relating to each load was estimated from the calibration data for this cell, which was obtained from the pressure variations of superconducting transition temperatures of lead and tin in previous experiments Slooten _et al._ (2009); Bay _et al._ (24). We carried out the compression experiments on the crystals twice, first up to a pressure of 1.24 GPa and in a second run up to 2.49 GPa. Typical experimental conditions are as follows. The high-pressure cell was compressed at room temperature and then cooled down to about 0.3 K using a ${}^{3}$He refrigerator (Oxford Instruments Heliox VL). Electrical resistivity was measured by a resistance bridge (Linear Research LR-700) using a low-frequency ac method with an excitation current $I=300\mu$A. In order to investigate the field-suppression of $T_{c}$, a magnetic field was applied along the current, parallel to the $a$-axis. For ac-susceptibility measurements, a small cylinder, composed of an excitation coil and a pick-up coil in which the sample is situated, was prepared. The in-phase and out-of-phase signals were detected in the driving field $\mu_{0}H_{ac}=0.0047$ mT with a frequency of $f_{ac}=313$ Hz using a lock-in amplifier (EG&G Instruments Model 7260). Measurements were made in zero field and in applied dc-fields using a superconducting magnet. Special care was taken to reduce the remnant field of the superconducting magnet to close to zero, since our PdTe${}_{2}$ crystals show type-I superconductivity. <figure><img src="content_image/1902.01953/x1.png"><figcaption>Figure 1: Left: Resistivity of PdTe2 measured with the current in the basalplane. Right: Pressure plug with samples and χac coils mounted (schematic).</figcaption></figure> Overall the resistivity, $\rho(T)$, measured in the temperature range 2-300 K showed little variation with pressure and remained metallic. However, the absolute $\rho$-value at 300 K decreases smoothly with respect to pressure to about 80% of the ambient pressure value at the highest pressure 2.49 GPa. ## III Results ### Pressure-temperature phase diagram <figure><img src="content_image/1902.01953/x2.png"><figcaption>Figure 2: Resistance and ac-susceptibility (normalized to −1 in thesuperconducting state) of single-crystalline PdTe2 as a function oftemperature around Tc at pressures up to 2.49 GPa as indicated. The data weretaken in two pressure runs (see text): dashed-dotted lines for the first run(p-values listed above the curves) and solid lines for the second run(p-values listed adjacent to the curves). The yellow curve in both panels wastaken at 1.08 GPa. The curves at 0 GPa were measured after releasing thepressure in the second run. TRc is determined from the onset ofsuperconductivity in R(T) as indicated for p=2.49 GPa by the thin solid lines.For p≥1.41 GPa the onset of diamagnetic screening in χac(T) is attributed tosurface superconductivity at TSc, and the further drop signals bulksuperconductivity at Tχc, as indicated for p=2.49 GPa. See text.</figcaption></figure> <figure><img src="content_image/1902.01953/x3.png"><figcaption>Figure 3: Pressure variation of the superconducting transition temperature ofPdTe2 as determined from resistance, TRc (blue symbols), and from ac-susceptibility, Tχc (red symbols). TSc denotes surface superconductivity (openand closed green symbols). Open symbols are determined by extrapolation. Notethat for p≥1.41 GPa TSc>Tbulkc=Tχc (see text).</figcaption></figure> The overall results of the two pressure runs are reported in Fig. 2. In the first run data were taken at pressures of 0.25, 0.58, 0.91 and 1.24 GPa. Here the normal state resistance $R_{N}\simeq 70~{}\mu\Omega$. For the second run new voltage contacts were made on the crystal resulting in $R_{N}\simeq 60~{}\mu\Omega$. The applied pressures are 0.75, 1.08, 1.41, 1.74, 2.07 and 2.49 GPa. We remark the zero-pressure data were measured after releasing the pressure. Also, the value of the ac-susceptibility differed somewhat between different cool downs and between the two pressure runs. For clarity all the $\chi_{ac}$ data in the lower panel of Fig. 2 are normalized to $-1$ in the superconducting state. The resistance curves around $T_{c}$ at ambient pressure and $p=0.25$ GPa show a double structure which becomes more pronounced with increasing pressure. However, for $p\geq 1.08$ GPa the superconducting transition is sharp. We attribute the double structure in $R(T)$ at low pressures to parts of the crystal responding differently to pressure, because of an inhomogeneity, rather than to a pressure gradient. We remark that previous resistance experiments on crystals taken from the same single-crystalline boule revealed a single sharp superconducting transition at ambient pressure Leng _et al._ (2017). A similar behavior is observed in $\chi_{ac}(T)$ with relatively sharp, single transitions at pressures of 1.08 and 1.24 GPa. However, for $p\geq 1.41$ GPa the transition becomes structured again with an onset temperature of superconductivity larger than $T_{c}$ deduced from the resistivity curves (top panel). As we will demonstrate in the next Section, at these pressures the initial screening step is attributed to surface superconductivity Leng _et al._ (2017), while the ensuing second step with a full diamagnetic screening is attributed to bulk superconductivity. The first important result is that superconductivity is enhanced under pressure with a maximum value $T_{c}=$ 1.91 K around 0.91 GPa and a gradual depression of $T_{c}$ at higher pressures. This is illustrated in Fig. 3, where we trace $T_{c}^{R}(p)$ extracted from the resistance data. Here we use the onset transition temperatures determined by extrapolation of the $R(T)$-curves just below $T_{c}$ to the normal state plateau values, as shown for the 2.49 GPa curve in Fig. 2. The same analysis for the $\chi_{ac}(T)$-data shows $T_{c}^{\chi}(p)$ tracks $T_{c}^{R}(p)$ closely up to 1.24 GPa, see Fig. 3. However, for $p\geq 1.41$ GPa it is the second, lower in temperature, transition in $\chi_{ac}(T)$ that is attributed to bulk superconductivity and tracks $T_{c}^{R}(p)$. The agreement between $T_{c}^{\chi}(p)$ and $T_{c}^{R}(p)$ obtained with different techniques on two different crystals is good. Lastly, the temperature of surface superconductivity, $T_{c}^{S}(p)$, is traced in Fig. 3. For $p\leq 1.41$ GPa $T_{c}^{S}(p)$ is obtained from the field-temperature phase diagrams by extrapolating $T_{c}^{S}(H)$ to zero field, as reported in Ref. Leng _et al._, 2017 and presented in the following Section. For $p\geq 1.41$ GPa we take $T_{c}^{S}(p)$ from the onset of the upper transition in $\chi_{ac}(T)$. This tells us the transition temperatures for the bulk and surface have a distinct pressure variation, and for $p\geq 1.41$ GPa $T_{c}^{S}>T_{c}^{bulk}$. This underpins surface superconductivity in PdTe${}_{2}$ is a unique, robust feature. <figure><img src="content_image/1902.01953/x4.png"><figcaption>Figure 4: Upper panel: Resistance of PdTe2 as a function of temperature at apressure p=0.25 GPa measured in applied magnetic fields μ0Ha∥I∥a. Curves fromright to left are taken in fields of 0, 1, 2, 3, 5, 7, 9, 13, 16, 24, 35, 50,65, 80, 95, 110, 125, 140, 155 and 180 mT. Lower panel: Ac-susceptibility atp=0.25 GPa measured in applied magnetic fields. Curves from right to left in 0mT to 14 mT with 1 mT steps and in 16.5, 19, 21.5, 23, 27 and 30 mT.</figcaption></figure> ### Field-temperature phase diagram In order to investigate the pressure dependence of the superconducting phase diagram in the $H$-$T$ plane we have measured at each pressure the resistance and ac-susceptibility in applied dc-fields, $H_{a}$. A typical data set taken at $p=0.25$ GPa is shown in Fig. 4. In the lower panel with $\chi_{ac}$-data the zero-field curve shows $T_{c}=1.63$ K. In small applied fields a peak appears just below $T_{c}$ due to the differential paramagnetic effect (DPE). This peak signals the field induced intermediate state Leng _et al._ (2017). It shifts to lower temperatures with increasing field and for higher fields is progressively depressed because of an additional screening signal that precedes the DPE peak. The additional screening is attributed to superconductivity of the surface sheath Leng _et al._ (2017). Partial screening is still visible at 27 mT, but has nearly vanished at $\mu_{0}H_{a}=30$ mT down to 0.3 K. Consequently, in the limit $T\to 0$$H_{c}^{S}(0)>H_{c}(0)$. In the upper panel, with $R(T)$ data, the transition is first rapidly depressed with field up to $\mu_{0}H_{a}\approx 13$ mT, but then the depression rate decreases, the transition broadens and signals of superconductivity persist up to $\mu_{0}H_{a}\approx 180$ mT. We remark this field is much larger than $H_{c}(0)$ or $H_{c}^{S}(0)$. The robustness of superconductivity in resistance measurements was also observed at ambient pressure, with a critical field, $H_{c}^{R}(0)$, equal to $\sim 0.3$ T Leng _et al._ (2017). <figure><img src="content_image/1902.01953/x5.png"><figcaption>Figure 5: Left panel: Critical field Hc(T) for type-I superconductivity inPdTe2 at pressures between 0 and 2.49 GPa as indicated. The solid linesrepresent Hc(T)=Hc(0)[1−(T/Tc)2] at different pressure, where Tc=Tχc is thebulk superconductivity transition temperature extracted from the χac-data inapplied fields. Right panel: Reduced plot h∗=(Hc(T)/Tc)/(−dHc/dT)\|Tc versusT/Tc. The solid line represents h∗=0.5×[1−t2]. See text.</figcaption></figure> In the following paragraphs we present the $H$-$T$ phase diagrams determined from the $R(T)$- and $\chi_{ac}(T)$-data in applied fields, measured up to 2.49 GPa. The phase diagram at 0.25 GPa is extracted from Fig. 4. Additional data sets are presented in the Supplemental Material (SM) [25]. In Fig. 5 we present the critical field for bulk superconductivity $H_{c}(T)$. The data are obtained by tracing the $T_{c}^{\chi}$-values as a function of the applied field. The solid lines in Fig. 5 represent $H_{c}(T)=H_{c}(0)[1-(T/T_{c})^{2}]$ at different pressures, where $T_{c}=T_{c}^{\chi}$. The quadratic temperature variation is consistent with type-I superconductivity. In fact all the data under pressure collapse on one single curve, $h^{}(t)$, as shown in the right panel of Fig. 5. Here the standard expression for plotting $H_{c}(T)$ in a reduced form is applied, with $h^{}=(H_{c}(T)/T_{c})/(-dH_{c}/dT)\|_{T_{c}}$ where $t=T/T_{c}$ Bay _et al._ (26). For a type-I superconductor $h^{}(0)=0.5$. The collapsed curve $h^{}(t)$ shows type-I superconductivity persists over the whole pressure range. <figure><img src="content_image/1902.01953/x6.png"><figcaption>Figure 6: Superconducting phase diagram of PdTe2 deduced from ac-susceptibility at a pressure of 0.25 GPa (left), 1.08 GPa (middle) and 2.07GPa (right), for Ha in the basal plane. Bulk type-I superconductivity is foundbelow the critical field Hc(T). The data points (red solid symbols) follow thestandard quadratic temperature variation Hc(T)=Hc(0)[1−(T/Tχc)2] (red lines).Surface superconductivity is found below HSc(T) (green solid symbols). Thetransition temperature, TSc, is determined by extrapolating HSc(T) to Ha=0(green lines). The values of the bulk Tχc and surface TSc are indicated byarrows. Note that at the highest pressure TSc>Tχc.</figcaption></figure> Next we show how superconductivity of the surface sheath develops with pressure. Hereto we have traced $T_{c}^{S}(H)$ obtained from the $\chi_{ac}$-curves in applied fields in Fig. 6. Phase diagrams at 0.25, 1.08 and 2.07 GPa are presented. At 0.25 GPa we start to observe the (partial) diamagnetic screening due to the surface at a finite value $H_{a}\approx 5$ mT (Fig. 4, lower panel). The corresponding $T_{c}^{S}(H)$ points are traced in the left panel of Fig. 6. By extrapolating $T_{c}^{S}(H)$ to zero field we obtain $T_{c}^{S}(0)$. In the same panel we have plotted $H_{c}(T)$ for bulk superconductivity as well. We find $T_{c}^{S}(0)<T_{c}^{\chi}(0)$, just like reported previously at ambient pressure Leng _et al._ (2017). However, upon further increasing the pressure the phase lines $H_{c}(T)$ and $H_{c}^{S}(T)$ move apart and do no longer intersect for $p\geq 1.41$ GPa, in which case $T_{c}^{S}(0)>T_{c}^{\chi}(0)$. This is illustrated for $p=2.07$ GPa in the right panel of Fig. 6. The distinct pressure variation of $T_{c}^{S}$ and $T_{c}^{\chi}$ demonstrates once more that surface superconductivity is not of the standard Saint-James - de Gennes type Saint-James and de Gennes (1963). We discuss the robustness and nature of this phenomenon in the next Section. <figure><img src="content_image/1902.01953/x7.png"><figcaption>Figure 7: Superconducting phase diagram of PdTe2 constructed from resistancemeasurements in the H-T plane at different pressures, as indicated. For1.3-1.9 K the data points HRc(T) denote bulk superconductivity. Below 1.3 K(partial) superconductivity persists resulting in a critical field HRc(0) of≃0.2 T. The blue solid line compares the data to the Werthamer-Helfand-Hohenberg model (see text).</figcaption></figure> Finally we show in Fig. 7 the $H$-$T$ phase diagrams determined from the transport data at pressures up to 2.49 GPa. At each pressure we investigated the depression of superconductivity by measuring $R(T)$ in fixed applied fields. The $R(T)$-data for 0.25 GPa are shown in the upper panel in Fig. 4. Additional data sets are reported in the SM [25]. In all cases superconductivity is first depressed rapidly in small fields, and $H_{c}^{R}(T)$ tracks $H_{c}(T)$ for bulk superconductivity as deduced from $\chi_{ac}$ (see Fig. 5). The $H_{c}^{R}(T)$-data in Fig. 7 show this behavior is restricted to the temperature range 1.3-1.9 K. Below 1.3 K the transition in $R(T)$ broadens and traces of superconductivity are visible up to $\sim 0.2$ T. By tracing in Fig. 7 the onset temperature for superconductivity from $R(T)$ in fixed magnetic fields below 1.3 K, we observe a steady increase of $H_{c}^{R}(T)$. A comparison with the Werthamer-Helfand-Hohenberg (WHH) model Werthamer _et al._ (1966) indicates the data extrapolate to $H_{c}^{R}(0)\simeq 0.2$ T for $T\to 0$. We remark that for the crystal studied in Ref. Leng _et al._, 2017 this value is larger, $\simeq 0.3$ T. Interestingly, $H_{c}^{R}(T)$ below 1.3 K is almost pressure independent, which shows the superconducting transition in resistance for $H_{a}>H_{c}$ is not closely connected to surface superconductivity as was proposed in Ref. Leng _et al._, 2017. ## IV Analysis and Discussion The mechanical and electronic properties of PdTe${}_{2}$ under pressure have been investigated theoretically by several groups Soulard _et al._ (2005); Xiao _et al._ (2017); Lei _et al._ (2017). The only experimental high-pressure study carried out so far is by Soulard _et al_. Soulard _et al._ (2005) who conducted high-pressure X-ray diffraction experiments at room temperature and 300 ${}^{\circ}$C to investigate the possiblity of a structural phase transition. They found that an abrupt change in the interatomic distances occurs above $p=15.7$ GPa at room temperature, but the volume _versus_ pressure curve exhibits no discontinuity. Under pressure the unit cell volume decreases by 17.6% at the maximum applied pressure of 27 GPa, and the $c/a$ ratio decreases from 1.27 to 1.24 at 27 GPa. A bulk modulus, $B_{0}$, of 102 GPa was derived from the experimental data. This value is to be compared with 71.2 GPa (74.2 GPa) derived from first principle calculations by Lei _et al_. Lei _et al._ (2017) at 300 K (0 K). Xiao _et al_. Xiao _et al._ (2017) computed the optimized lattice parameters as a function of pressure, which are slightly overestimated compared to the experimental data Soulard _et al._ (2005). Overall, these studies indicate there is no structural transition in the modest pressure range up to 2.5 GPa in our experiments. For a layered material the change in the $c/a$-ratio is normally an important control parameter for the electronic properties. However, for PdTe${}_{2}$ this change is very tiny and 0.2% at most up to 2.5 GPa Soulard _et al._ (2005). In the following we focus on the superconducting properties. ### Bulk superconductivity A major result is the non-monotonous variation of $T_{c}$ with pressure reported in Fig. 3. $T_{c}$ first increases to 1.91 K at 0.91 GPa and then is gradually depressed. We first compare the experimental results with theoretical calculations. The evolution of superconductivity with pressure was investigated theoretically by Xiao _et al_. Xiao _et al._ (2017). The authors used the Allen-Dynes-modified McMillan equation to calculate $T_{c}$, with the characteristic phonon frequency $\omega_{log}$, the electron-phonon coupling constant $\lambda$ and the Coulomb pseudopotential $\mu^{}\simeq 0.1$ as input parameters. Combined electronic structure and phonon-density of states calculations show a gradual decrease of $\lambda$ and an increase of $\omega_{log}$ (blue shift), but overall the calculated $T_{c}$ decreases from 2.0 K at ambient pressure to 0.6 K at 10 GPa. Note the calculated $T_{c}$ at $p=0$ is larger than our experimental value of 1.6 K. While a decrease to 0.6 K at 10 GPa is within bounds of the extrapolation of $T_{c}(p)$ in Fig. 3, the calculations by Xiao _et al_. Xiao _et al._ (2017) clearly do not capture the initial increase of $T_{c}$ and its maximum value at 0.91 GPa. The superconducting properties of PdTe${}_{2}$ were also investigated by Kim _et al_. Kim _et al._ (2018) employing the same McMillan formalism. Their phonon band structure calculations show the electron-phonon interaction is dominated by the optical $O_{1,2}$ and $O_{3}$ phonon modes. Furthermore, they emphasize the importance of a saddle-point van Hove singularity (vHs) close to the Fermi energy. The computed $T_{c}$ is 1.79 K at ambient pressure. The importance of a vHs is further illustrated by the case of PtTe${}_{2}$, which is isoelectronic with PdTe${}_{2}$ but does not show superconductivity. Here the vHs-band has a broad dispersion along $k_{z}$ leading to a lower density of states at the Fermi level and absence of superconductivity Kim _et al._ (2018). Calculations for PdTe${}_{2}$ with a 15% volume contraction, which corresponds to a pressure of $\sim$20 GPa, indicate the vHs band moves close to the Fermi level Kim _et al._ (2018), which would produce a higher $T_{c}$. However, this is at variance with the experimental data presented in Fig. 3. <figure><img src="content_image/1902.01953/x8.png"><figcaption>Figure 8: Relative change of the superconducting transition temperature,(Tc−Tc(0))/Tc(0)), as a function of the relative volume change (V−V0)/V0. Redsymbols: PdTe2 under pressure, this work; blue symbols: AuxPd1−xTe2, Ref. Kudo_et al._, 2016; magenta symbol: CuxPdTe2, Ref. Hooda, M. K. and Yadav, C. S.,2018; green symbols: calculated, Ref. Xiao _et al._, 2017.</figcaption></figure> Another way to tune $T_{c}$ besides pressure is via doping or substitution. Recently, it was demonstrated that Cu intercalation enhances $T_{c}$ to a maximum value of 2.6 K in Cu${}_{x}$PdTe${}_{2}$ Yan _et al._ (33); Ryu (2015); Hooda, M. K. and Yadav, C. S. (2018) for $x=0.06$. Upon intercalation the volume contracts, but changes are minute: $\Delta V/V=-$0.07% for $x=0.04$ Hooda, M. K. and Yadav, C. S. (2018), which corresponds to an applied pressure of 0.07 GPa. This shows Cu intercalation cannot be equated to chemical pressure in tuning superconductivity. The same holds for the substitution series (Au${}_{x}$Pd${}_{1-x}$)Te${}_{2}$ Kudo _et al._ (2016). Upon alloying with Au, $T_{c}$ increases up to 4.65 K for $x=0.40$. Simultaneously, the volume _increases_ by 2.5%, which corresponds to a _negative_ pressure of $\sim$2.5 GPa. The experimental and calculated variation of $T_{c}$ with pressure and doping are summarized in Fig. 8. Here we trace the relative change of $T_{c}$ as a function of the relative volume change, $(V-V_{0})/V_{0}$, where a bulk modulus of 102 GPa is used Soulard _et al._ (2005). Although $T_{c}$ generally decreases with a smaller volume, the experimentally observed positive $dT_{c}/dp$ for PdTe${}_{2}$ up to 0.91 GPa is at odds with this trend. In an attempt to shed further light on the pressure variation of $T_{c}$, we have conducted Hall effect measurements on two PdTe${}_{2}$ crystals under pressure up to 2.07 GPa [25]. At the lowest pressure of 0.25 GPa the carrier concentration, $n$, amounts to 1.5-1.7$\times 10^{22}$ cm${}^{-3}$ at 2 K. It varies quasi-linearly with pressure resulting in an increase of $\sim$20% at 2.07 GPa. No anomalous behavior is observed around 0.9 GPa. In the most simple model the increase of $n$ is expected to result in an increase of the density of states at the Fermi level and a monotonous enhancement of $T_{c}$. The non-monotonous variation of $T_{c}$ indicates the density of states and the electron phonon-coupling constant are affected in an intricate manner by doping and/or pressure. Possibly this is a result from band structure subtleties that have not been probed in the coarse-grained calculations carried out so far Soulard _et al._ (2005); Xiao _et al._ (2017); Lei _et al._ (2017). In order to access the electronic band structure under pressure, a quantum oscillations study is highly desirable. The feasibility to observe the Shubnikov - de Haas effect and the de Haas - van Alphen effect at ambient pressure has been demonstrated in Refs. Dunsworth, 1975; Fei _et al._, 2017; Zheng _et al._, 2018. In the same context, small structural modifications that might influence $T_{c}$, such as changes in the $z$-coordinate of Te atoms in the unit cell that would affect the $O_{1,2}$ and $O_{3}$ phonon modes, cannot be excluded based on the X-ray diffraction experiment with a first pressure point at 2.2 GPa Soulard _et al._ (2005). This calls for high-precision low-pressure ($p\leq 2.5$ GPa) single-crystal X-ray diffraction measurements. ### Surface superconductivity The distinct pressure variation of the superconducting transition temperature of the surface sheath, $T_{c}^{S}$, and of the bulk, $T_{c}^{\chi}$, reported in Figs. 3 and 6, is an extraordinary result. We recall this feature is derived from the ac-susceptibility curves measured in fixed magnetic fields at eleven different pressures. Selected data sets at 0.25 GPa are shown in Fig. 4 and at 1.08 and 2.07 GPa in the SM [25]. The data show how type-I superconductivity in the bulk, probed by the DPE-peaks in small applied dc-fields, is progressively depressed with field, while surface superconductivity is observed for $H_{a}>H_{c}$ (see also Ref. Leng _et al._, 2017). Upon increasing the pressure, the DPE peak is more rapidly depressed compared to surface screening. At 2.07 GPa the DPE effect is - already in the lowest applied fields - almost completely screened by the surface [25]. Hence for $p\geq 1.41$ GPa $T_{c}^{S}>T_{c}^{\chi}$. This is further underpinned by the observation that $H_{c}(T)$, defined by $T_{c}^{\chi}(H)$, follows the quadratic temperature variation at all pressures, characteristic for bulk type-I superconductivity (Fig. 5). Note that $T_{c}^{S}$ is defined as the onset temperature for the diamagnetic signal due to surface superconductivity, while the transition itself may become very broad. $H_{c}^{S}(p)$ has a maximum near 0.9 GPa, similar to $H_{c}(p)$, as reported in the SM [25]. When the $H_{c}^{S}(T,p)$ data is traced in the reduced form $h^{}(t)$ the data do not collapse on a single curve as, see SM [25]. Instead the trend is that the values $h^{}(t)$ increase with respect to pressure, which indicates the superconducting pairing interaction changes in a non-trivial way. The distinct $H_{c}(T)$- and $H_{c}^{S}$-curves and their dissimilar pressure dependence strongly suggest surface and bulk superconductivity are independent phenomena and not tightly connected, in contrast to the familiar Saint James - de Gennes surface superconductivity Saint-James and de Gennes (1963). It remains tempting to relate surface superconductivity in PdTe${}_{2}$ to topological surface states detected by ARPESYan _et al._ (5); Noh _et al._ (2017); Clark _et al._ (2018). These surface states could possibly be investigated by STM experiments in small applied fields ($H_{a}>H_{c}$). The STM experiments performed so far were predominantly directed to probe bulk superconductivity Das _et al._ (2018); Clark _et al._ (2018). Moreover, for the spectra taken in a magnetic field the intermediate state that occurs below $H_{c}$ for a finite demagnetization factor was not taken into account. In the resistance measurements (partial) superconductivity is observed up to about 0.2 T for $T\to 0$ (Fig. 7), a value that largely exceeds $H_{c}(0)$ and $H_{c}^{S}(0)$. The enhanced $H_{c}^{R}(T)$-curves below 1.3 K are quasi pressure independent. By extrapolating the data in this field range to $H_{a}\to 0$ with the WHH function a pressure independent $T_{c}=1.2$ K is found. Since $T_{c}^{S}$ has a pronounced pressure variation the resistive superconducting transitions measured in this field range are not connected to surface superconductivity. Note that for the crystal studied in Ref. Leng _et al._, 2017 it was concluded that the transport experiment does probe surface superconductivity, but these experiments were performed at ambient pressure only. The persistence of superconductivity in resistance measurements in field is puzzling. Normally such an effect is attributed to filamentary superconductivity. Its pressure independence indicates it might not be intrinsic to PdTe${}_{2}$. ## V Summary and conclusions We have carried out a high-pressure transport and ac-susceptibility study of superconductivity in the type-I superconductor PdTe${}_{2}$ ($T_{c}=1.64$ K). $T_{c}$ shows a pronounced variation with pressure: it increases at low pressure, then passes through a maximum of 1.91 K around 0.91 GPa, and subsequently decreases smoothly up to the highest pressure measured, $p_{max}=2.5$ GPa. The critical field, $H_{c}$, follows a similar behavior, leading the $H_{c}(T)$-curves at different pressures to collapse on a single universal curve with the characteristic quadratic in temperature depression of $H_{c}$ for type-I superconductivity. Type-I superconductivity is robust under pressure. In view of the absence of structural modifications in our pressure range and the minute change of the $c/a$-ratio Soulard _et al._ (2005), the non-monotonous variation of $T_{c}$ indicates an intricate role of the dominant phonon frequency, the electron-phonon-coupling parameter and Coulomb pseudopotential used to compute $T_{c}$ with help of the McMillan formula. This effect has not been captured by band structure calculations so far Xiao _et al._ (2017); Kim _et al._ (2018), notably the electron band structure calculations predict a smooth decrease of $T_{c}$ under pressure Xiao _et al._ (2017). This calls for more elaborate and detailed calculations for pressures up to $p_{max}=2.5$ GPa. The unusual surface superconductivity, first reported at ambient pressure Leng _et al._ (2017), persists under pressure. Surprisingly, for $p\geq 1.41$ GPa the superconducting transition temperature for the surface $T_{c}^{S}$ exceeds $T_{c}$ of the bulk. This tells us surface and bulk superconductivity are distinct phenomena. This is further confirmed by the observation that the phase lines $H_{c}(T)$ and $H_{c}^{S}(T)$ move apart under pressure and no longer intersect for $p\geq 1.41$ GPa. We propose surface superconductivity possibly has a non-trivial nature and originates from topological surface states detected by ARPESYan _et al._ (5); Noh _et al._ (2017); Clark _et al._ (2018). This calls for quantum-oscillation experiments under pressure, possibly enabling one to follow the pressure evolution of the bulk electronic structure and topological surface states. In the same spirit it will be highly interesting to extend the experiments to higher pressures, especially because a pronounced change in the electronic properties of PdTe${}_{2}$ is predicted to occur in the range 4.7-6.1 GPa: the type-II Dirac points disappear at 6.1 GPa, and a new pair of type-I Dirac points emerges at 4.7 GPa Xiao _et al._ (2017). 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Supplemental Material for “Superconductivity under pressure in the Dirac semimetal PdTe${}_{2}$” 1. Resistance and ac-susceptibility measurements in field In order to investigate the response of the superconducing phase of PdTe${}_{2}$ to an applied magnetic field and to construct the field-temperature phase diagram electrical resistivity and ac-susceptibility measurements were carried out. The resistance as a function of temperature, $R(T)$, was measured using a sensitive resistance bridge (model Linear Research LR700) in a four-point geometry by a low-frequency ac-method with an excitation current $I=300~{}\mu$A. The ac-susceptibility was measured by placing the crystal in a small coil-set with an excitation and pick-up coil, mounted inside the pressure cell. The excitation field was $\mu_{0}H_{ac}=0.0047$ mT. The in-phase and out-of-phase signals were recorded at a frequency of $f_{ac}=313$ Hz using a lock-in amplifier (EG&G Instruments Model 7260). The applied dc-field, directed in the basal plane of the crystal, was produced by a superconducting magnet. Special care was taken to reduce the remnant field of the superconducting magnet to close to zero, since the PdTe${}_{2}$ crystals show type-I superconductivity. Measurements were carried out at eleven different pressures. The data at 0.25 GPa are reported in the main text. In Fig. S1 and Fig. S2 selected data sets at 1.08 and 2.07 GPa, respectively, are presented. <figure><img src="content_image/1902.01953/x9.png"><figcaption>Figure S1: Upper panel: Resistance of PdTe2 as a function of temperature at apressure p=1.08 GPa measured in applied magnetic fields μ0Ha∥I∥a. Curves fromright to left are taken in fields of 0, 1, 2, 3, 4, 5, 7, 9, 11, 14.5, 26, 40,60, 80, 100, 120, 140 and 160 mT. Lower panel: Ac-susceptibility at p=1.08 GPameasured in applied magnetic fields. Curves from right to left in 0 mT to 8 mTwith 1 mT steps and in 10, 12, 14.5, 18, 22, 26, 30, 34, 38 and 40 mT.</figcaption></figure> <figure><img src="content_image/1902.01953/x10.png"><figcaption>Figure S2: Upper panel: Resistance of PdTe2 as a function of temperature at apressure p=2.07 GPa measured in applied magnetic fields μ0Ha∥I∥a. Curves fromright to left are taken in fields of 0, 1, 2, 4, 9, 19, 28, 40, 60, 80, 100,120, 140 and 160 mT. Lower panel: Ac-susceptibility at p=2.07 GPa measured inapplied magnetic fields. Curves from right to left in 0 mT to 7 mT with 1 mTsteps and in 10, 13, 16, 19, 22, 25 and 28 mT.</figcaption></figure> We first discuss the data at 1.08 GPa. The transition to $R=0$ in small fields is sharp and rapidly depressed up to $\sim 11$ mT. This signals bulk type-I superconductivity, and by extrapolating the data in the temperature range 1.3-1.9 K using a quadratic temperature variation we estimate a critical field $H_{c}^{R}(0)=22$ mT (see Fig. 7 in the main text and Fig. S4). For larger fields superconductivity is depressed at a much lower rate, the transition broadens and becomes incomplete for $H_{a}\geq 120$ mT. Signs of superconductivity in $R(T)$ persist up to 200 mT for $T\to 0$. $H_{c}^{R}(T)$ extracted from the upper panel in Fig. S1 is reported in Fig. 7 in the main text. The $\chi_{ac}$-data, plotted in the lower panel of Fig. S1, shows the peak due to the differential paramagnetic effect (DPE) is rapidly depressed with field. The DPE peak is due to the intermediate state in the bulk of the type-I superconductor. From the shift of the DPE peak we determine the $H_{c}$ phase boundary. However, for $H_{a}>H_{c}$ large screening signals persist. This we attribute to superconductivity of the surface sheath Leng _et al._ (2017) with a critical field $H_{c}^{S}$. A sizeable screening is still observed at 40 mT. The phase boundaries $H_{c}$ and $H_{c}^{S}$ derived from $\chi_{ac}$ are reported in Fig. 6 (middle panel) in the main text. The resistance data at 2.07 GPa, shown in the upper panel of Fig. S2, compare well to the data at 1.08 GPa, except superconductivity in zero field is further depressed from $T_{c}^{R}=1.85$ K at 1.08 GPa to 1.41 K at 2.07 GPa. Consequently, the extrapolated critical field for bulk superconductivity is reduced to $H_{c}^{R}(0)=19$ mT. The $\chi_{ac}$-data plotted in the lower panel, however, show a very different behavior compared to the data at 1.08 GPa. The DPE peak is reduced at 2.07 GPa and appears in the data in applied dc-fields well below the initial diamagnetic step. This we attribute to the notion that surface screening precedes screening due to bulk superconductivity. Thus $T_{c}^{S}>T_{c}^{\chi}$, where $T_{c}^{\chi}$ is the bulk superconducting transition temperature. The data points extracted from Fig. S2 in this manner define the phase boundaries $H_{c}(T)$ and $H_{c}^{S}(T)$ reported in Fig. 6 (right panel) in the main text. Screening at the surface is not complete and amounts to 60% only. Note the DPE peak is no longer observed for $H_{a}>10~{}$mT, and $H_{c}(T)$ (defined by $T_{c}^{\chi}(H)$) follows the quadratic temperature variation for the bulk type-I superconducting phase. 2. Critical field of surface superconductivity At each pressure we have constructed the $H_{c}^{S}(T)$ phase boundary. In an attempt to collapse all the $H_{c}^{S}(T,p)$-data on a single curve a plot of $h^{}(t)$ is presented in Fig. S3, where $h^{}=(H_{c}^{S}(T)/T_{c}^{S})/(-dH_{c}^{S}/dT)\|_{T_{c}^{S}}$ and $t=T/T_{c}^{S}$. Note that for pressures up to 1.24 GPa $T_{c}^{S}$ and the initial slope $-dH_{c}^{S}/dT\|_{T_{c}^{S}}$ are determined by extrapolation, as shown in Fig. 6 (main text) for 0.25 and 1.08 GPa. This introduces some uncertainty in the data, but the overall trend is that $h^{}(0)$ increases with respect to pressure. This indicates the superconducting pairing interaction changes in a non-trivial way. Leng _et al_. Leng _et al._ (2017) reported that the $H_{c}^{S}(T)$-curve at ambient pressure follows a quadratic temperature variation. Such a behavior is absent for the present crystal. Instead $H_{c}^{S}(T)$ rather shows a downward or upward curvature near $t=0.7-0.8$. <figure><img src="content_image/1902.01953/x11.png"><figcaption>Figure S3: Critical field HSc(T) for superconductivity of the surface sheathin PdTe2 at pressures between 0 and 2.49 GPa as indicated. The data areplotted in the reduced form h∗=(HSc(T)/TSc)/(−dHSc/dT)\|TSc versus t=T/TSc.</figcaption></figure> 3. Pressure variation of the critical field* In Fig. S4 the pressure variation of the critical fields $H_{c}$ and $H_{c}^{R}$ in the limit $T\to 0$ and at $T=0.3$ K for $H_{c}^{S}$ is presented. The field $H_{c}(0)$ is representative of bulk superconductivity. It is determined from ac-susceptibility with help of the expression $H_{c}(T)=H_{c}(0)[1-(T/T_{c})^{2}]$, where $T_{c}=T_{c}^{\chi}$. The $H_{c}(T)$-curves measured at eleven different pressures are reported in Fig. 5 of the main text. $H_{c}^{R}(0)$ is determined from the data in the temperature range 1.3-1.9 K in Fig. 7 (main text) by extrapolating $T\to 0$, using the quadratic temperature variation with $T_{c}=T_{c}^{R}$. Note the temperature range in which $H_{c}^{R}(T)$ represents type-I superconductivity and follows a quadratic temperature variation is small, since below $T=1.3$ K $H_{c}^{R}(T)$ shows a pronounced upturn (see Fig. 7 in the main text). Consequently, the fit brings about an uncertainty in $H_{c}^{R}(0)$, which explains the overestimated values compared to $H_{c}(0)$. $H_{c}^{S}(0)$ represents the critical field at $T=0.3$ K for superconductivity of the surface sheath determined by ac-susceptibility. For all three data sets in Fig. S4 a maximum in the critical field as a function of pressure is observed near 0.9-1.2 GPa. <figure><img src="content_image/1902.01953/x12.png"><figcaption>Figure S4: Pressure variation of the critical field of PdTe2. Hc (red symbols)represents bulk type-I superconductivity determined by χac measurements in thelimit T→0. HRc (blue symbols) is determined from resistance measurements byextrapolating the initial low field HRc(T)-data to 0 K using a quadratictemperature variation. HSc (red symbols) represents surface superconductivityat the lowest temperature, T=0.3 K, as extracted from χac .</figcaption></figure> 4. Hall-effect measurements <figure><img src="content_image/1902.01953/x13.png"><figcaption>Figure S5: Symmetrized Hall resistance multiplied by the crystal thickness tas a function of magnetic field for two PdTe2 crystals at pressures rangingfrom 0.25 to 2.07 GPa, as indicated. The temperature is 2 K.</figcaption></figure> <figure><img src="content_image/1902.01953/x14.png"><figcaption>Figure S6: Carrier concentration as a function of pressure for two PdTe2crystals, at temperatures of 2, 10, 50, 150 and 300 K, as indicated. The dataat ambient pressure are taken from a third crystal.</figcaption></figure> The Hall effect was measured on two PdTe${}_{2}$ crystals in a piston-cylinder clamp cell developed for the Physical Property Measurement System (PPMS, Quantum Design) at nine different pressures up to 2.07 GPa. The sample space is 4.4 mm in diameter and $\sim$15 mm in height. Two crystals were placed in two stages along a compression axis perpendicular to the sample plane. The sample size (length $\times$ width $\times$ thickness) amounts to $2.8\times 1.4\times 0.08$ mm${}^{3}$ and $2.9\times 1.0\times 0.19$ mm${}^{3}$ for crystal 1 and 2, respectively. The current was applied in the basal-plane of the crystals, whereas the magnetic field was applied along the trigonal axis, perpendicular to the sample plane. Data were collected at temperatures of 2, 10, 50, 150 and 300 K in magnetic fields up to 8 T. Measuremenst were carried out for two field polarities, $B^{+}$ and $B^{-}$, and the Hall resistance, $R_{H}$, was obtained by symmetrizing: $R_{H}=(R_{B^{+}}-R_{B^{-}})/2$. In Fig. S5 we show $t\times R_{H}$ as a function of the applied field at 2 K at different pressures. Here $t$ is the sample thickness. $R_{H}(B)$ is a non-linear function indicating the presence of several charge carrier bands, expected from Fermi surface measurements Dunsworth (1975); Zheng _et al._ (2018). For crystal 1 the Hall resistance goes through a deep minimum and changes sign in the field range 6-8 T. For crystal 2 the minimum is less pronounced. We estimate the carrier concentration, $n$, from the initial linear slope of $R_{H}(B)$. The results are traced in Fig. S6. Upon lowering the temperature from 300 K to 2 K, $n$ drops typically by 20% and 50% for crystal 1 and 2, respectively. At 2 K, $n$ amounts to 1.5-1.7$\times 10^{22}$ cm${}^{-3}$ at 0.25 GPa. It varies quasi-linearly with pressure and has increased by $\sim$20% at the highest pressure. No anomalous behavior is observed around 0.9 GPa, where $T_{c}(p)$ has a maximum. We also measured the Hall resistance at ambient pressure on a third crystal. The resulting carrier concentration is 0.8$\times 10^{22}$ cm${}^{-3}$ at 2 K. which is about a factor two smaller compared to the values for crystal 1 and 2.
1504.03132	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 54885, "num_imgs": 6, "llama3_tokens_count": 18845 }	[ "content_image/1504.03132/x1.png", "content_image/1504.03132/x2.png", "content_image/1504.03132/x3.png", "content_image/1504.03132/x4.png", "content_image/1504.03132/x5.png", "content_image/1504.03132/x6.png" ]	# Learning in Neural Networks Based on a Generalized Fluctuation Theorem Takashi Hayakawa takashi.hayakawa@riken.jp RIKEN Brain Science Institute, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Toshio Aoyagi Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan February 25, 2024 ###### Abstract Information maximization has been investigated as a possible mechanism of learning governing the self-organization that occurs within the neural systems of animals. Within the general context of models of neural systems bidirectionally interacting with environments, however, the role of information maximization remains to be elucidated. For bidirectionally interacting physical systems, universal laws describing the fluctuation they exhibit and the information they possess have recently been discovered. These laws are termed _fluctuation theorems_. In the present study, we formulate a theory of learning in neural networks bidirectionally interacting with environments based on the principle of information maximization. Our formulation begins with the introduction of a generalized fluctuation theorem, employing an interpretation appropriate for the present application, which differs from the original thermodynamic interpretation. We analytically and numerically demonstrate that the learning mechanism presented in our theory allows neural networks to efficiently explore their environments and optimally encode information about them. pacs: 05.40.-a, 84.35.+i, 87.19.lo, 89.70.-a _Introduction_ : The neural systems of animals are prominent as highly efficient systems for processing information concerning the external environment. Many authors have argued that the learning capability of neural systems is information-theoretically optimal by showing that several features of neural activity can be accounted for by positing information maximization (Infomax) in the learning of sensory signals and intrinsic dynamics in neural circuits Linsker (1988); Bell and Sejnowski (1995, 1997); Tanaka _et al._ (2009); Hayakawa _et al._ (2014). However, Infomax has not yet been investigated in a general context. In particular, although a real neural system interacts bidirectionally with its environment, not only receiving sensory signals and organizing intrinsic dynamics accordingly, but also generating motor outputs that influence the environment, Infomax learning has not been clearly formulated in this context (see Fig.1(a)). For example, the formulation of Infomax must be generalized in order to facilitate its application to the following type of model. One of the standard models of the interaction between neural systems and environments employs the Markov decision process. In this model, we consider discrete-time ($t\in\mathbf{Z}$) stationary Markovian dynamics of a stochastic neural network with $N$ binary-valued neurons $x^{t}\in\{0,1\}^{N}$ interacting with an environment $y^{t}$ that takes values in a discrete state space $\mathcal{Y}$Sallans and Hinton (2004); Ackley _et al._ (1985); Smolensky (1986). The neural elements $\{x_{i}^{t}\}$ ($1\leq i\leq N$) receive inputs from the environment in such a manner that $x^{t}$ realizes values stochastically according to a conditional probability $\pi(x^{t}\|y^{t})$. This conditional probability depends on the model parameters, such as the synaptic strength, and it changes slowly during the learning process through the adjustment of the model parameters. Then, the state of the environment at the next timestep, $y^{t+1}$, is obtained stochastically with a transition probability $\mu(y^{t+1}\|y^{t},x^{t})$ that is determined by the current states of the neural network and environment. <figure><img src="content_image/1504.03132/x1.png"><figcaption>Figure 1: (Color). (a) Interactions between a neural system and itsenvironment, (b) Representation of the dynamics as a causal network.</figcaption></figure> For interacting physical systems, there are recently discovered universal laws called _fluctuation theorems_Jarzynski (1997); Crooks (1999); Evans _et al._ (1993); Sagawa and Ueda (2010); Seifert (2012); Ito and Sagawa (2013) that relate nonequilibrium physical quantities to informational quantities such as mutual information. In particular, the dynamics of the neural network and the environment mentioned above are represented in the form of a causal network with regard to which a generalized version of the fluctuation theorem has been investigated Ito and Sagawa (2013) (Fig.1(b)). Because fluctuation theorems describe informational quantities for interacting systems, it is natural to hypothesize that they may provide a description of a key aspect of the learning behavior exhibited by neural systems. Although information thermodynamic considerations have been investigated in the context of learning systems in a few pioneering studies Still _et al._ (2012); Still (2014), it has not been determined whether informational quantities are actually maximized in such systems in some systematic way. In this Letter, we study this question. We derive a novel type of Infomax learning, starting from the version of the integral fluctuation theorem presented in Ito and Sagawa (2013), which provides the following inequality relating the average entropy production $\mathrm{E}[\sigma]$ of the neural network and the transfer entropy $\mathrm{I}_{x\to y}$ from the neural network to its environment: \[\mathrm{I}_{x\to y}\geq-\mathrm{E}[\sigma],\ \ \ \sigma \equiv\log\frac{\pi(x^{t+1}\|y^{t+1})}{\pi(x^{t}\|y^{t+1})}.\] (1) Throughout this article, the expectation value $\mathrm{E}$ is taken with respect to the stationary distribution $p_{s}$ of the dynamics unless otherwise noted. The transfer entropy is defined as the conditional mutual information Schreiber (2000): \[\mathrm{I}_{x\to y}\equiv\mathrm{I}[y^{t+1};x^{t}\|y^{t}].\] (2) As in information theory Cover and Thomas (2012), the (conditional) mutual information between two variables is defined as the change in the (conditional) entropy of one of the two variables owing to the inclusion of the other variable as a conditioning variable. Explicitly, we have $\mathrm{I}[y^{t+1};x^{t}\|y^{t}]=\mathrm{H}[y^{t+1}\|y^{t}]-\mathrm{H}[y^{t+1}\|x ^{t},y^{t}]$. The (conditional) entropy is defined in terms of the stationary distribution as $\mathrm{H}[y^{t+1}\|y^{t}]=-\mathrm{E}[\log p_{s}(y^{t+1}\|y^{t})]$. The quantity $\mathrm{I}_{x\to y}$ represents the amount of information that the neural system possesses about the future state of the environment. Thus, from the point of view of Infomax, it is a reasonable hypothesis that maximizing this quantity is an effective learning mechanism. However, it is necessary for the calculation of $\mathrm{I}_{x\to y}$ to directly estimate the transition probability of the environment, $\mu$, and this estimation apparently cannot be carried out by the neural network itself. Its lower bound, $-\mathrm{E}[\sigma]$, on the other hand, can be computed within the neural network, because $\sigma$ is determined by the transition probability of the neural network, $\pi$, alone. With these in mind, it is natural to conjecture that neural systems attempt to optimize their acquisition of information about the future by adjusting $\pi$ in such a manner to maximize the quantity $-\mathrm{E}[\sigma]$. However, note that the equality in the relation, $\mathrm{I}_{x\to y}\geq-\mathrm{E}[\sigma]$, is not generally realized, and thus the maximization of $-\mathrm{E}[\sigma]$ does not necessarily imply the maximization of $\mathrm{I}_{x\to y}$. In the next, we show how consideration of a generalized entropy production, allows us to overcome this problem. _Generalized Fluctuation Theorem_ : We prove the following inequality below: \[\mathrm{I}_{x\to y}\geq-\mathrm{E}[\Theta_{\nu}^{t}].\] (3) Here, we define the following generalized forms of the entropy production in terms of a conditional distribution $\nu$: \[\Theta_{\nu}^{t}\equiv\log\frac{\pi(x^{t+1}\|y^{t+1})}{\nu(x^{t}\|x ^{t+1},y^{t})},\ \hskip-1.422638pt\widehat{\Theta}_{\nu}^{t}\equiv\Theta_{\nu} ^{t}\hskip-1.13811pt+\log\frac{p_{s}(x^{t}\|y^{t})}{p_{s}(x^{t+1}\|y^{t+1})}.\] (4) We can regard $\nu$ as representing physical quantities computed in the neural system on the basis of $x^{t},y^{t},y^{t+1}$ and adjusted through learning (see supplemental materials). First, we have the apparent identity \[\exp\left[\log p_{s}(x^{t}\|y^{t})+\log\pi(x^{t+1}\|y^{t+1})- \widehat{\Theta}_{\nu}^{t}\right]\] (5) \[=\exp\left[\log p_{s}(x^{t+1}\|y^{t+1})+\log\nu(x^{t}\|x^{t+1},y^{t })\right].\] Multiplying both sides by $p_{s}(y^{t+1},y^{t})$ and summing them over relevant random variables, we obtain a generalized form of the fluctuation theorem: \[\mathrm{E}\left[e^{-I_{tr}^{t+1}-\widehat{\Theta}_{\nu}^{t}} \right]=1,\ \ I_{tr}^{t+1}\equiv\log\frac{p_{s}(y^{t+1}\|x^{t},y^{t})}{p_{s}(y^ {t+1}\|y^{t})}.\] (6) Applying Jensen’s inequality ($\exp\mathrm{E}[F(Z)]<\mathrm{E}[\exp F(Z)]$, which applies to any random variable $Z$ and any function $F$) to Eq.(6) gives \[\mathrm{E}\left[-I_{tr}^{t+1}-\widehat{\Theta}_{\nu}^{t}\right] \leq 0.\] (7) Noting that $\mathrm{E}[\widehat{\Theta}_{\nu}^{t}]=\mathrm{E}[\Theta_{\nu}^{t}]$ and $\mathrm{I}_{x\to y}=\mathrm{E}[I_{tr}^{t+1}]$, we have the inequality in Eq.(3). It is found that, for fixed $\pi$, the right-hand side of Eq.(3) is maximal if and only if \[\nu(x^{t}\|x^{t+1},y^{t})=p_{s}(x^{t}\|x^{t+1},y^{t}).\] (8) Furthermore, we can prove that the equality in Eq.(3) holds if and only if, in addition to Eq.(8), the mutual information takes the maximal value for the fixed $\pi$ and hence satisfies $\mathrm{I}[x^{t};y^{t}]=\mathrm{H}[y^{t}]$, under suitable conditions (see supplemental materials). If the neural network has sufficient capacity, it is expected that there is some optimal $\pi$ that maximizes both $\mathrm{I}_{x\to y}$ and $\mathrm{I}[x^{t};y^{t}]$, simultaneously. In this case, the above analysis implies that the optimal $\pi$ is obtained by maximizing $-\mathrm{E}[\Theta_{\nu}^{t}]$ with respect to $\pi$ and $\nu$. In conclusion, we find that, for a neural network with a large capacity, the maximization of $-\mathrm{E}[\Theta_{\nu}^{t}]$ leads to the maximization of $\mathrm{I}_{x\to y}$ and $\mathrm{I}[x^{t};y^{t}]$. Because the maximization of $\mathrm{I}[x^{t};y^{t}]$ served as the definition of Infomax in previous studies Linsker (1988); Bell and Sejnowski (1995, 1997), the maximization of $-\mathrm{E}[\Theta_{\nu}^{t}]$ provides a generalized Infomax. _Structures of Neural Networks_ : To maximize $-\mathrm{E}[\Theta_{\nu}^{t}]$, the neural network must be able to adjust $\pi$ and $\nu$ to optimal conditional distributions through learning. For this purpose, in the remainder of this Letter, we parameterize $\pi(x^{t}\|y^{t})$ as \[\pi(x^{t}\|y^{t}) =\] (9) \[\pi_{i}(x_{i}^{t} = 1\|y^{t},\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}})=g(e_{i}^{t}),\] \[e_{i}^{t} = \sum_{1\leq j\leq M_{\ell}}\rho_{ij}\{g(\xi_{(\ell),j}^{t})-\frac {1}{2}\}-h_{0},\] \[\xi_{(\ell),j}^{t} = \sum_{1\leq k\leq d}v_{jk}^{(\ell)}y_{k}^{t}+\sum_{1\leq k\leq N_ {\ell-1}}w_{jk}^{(\ell)}x_{k}^{t}-h_{j}^{(\ell)}.\] (10) Here, $g(e_{i}^{t})$ is the logistic function $(1+\tanh(e_{i}^{t}))/2$. Equation(10) describes the situation in which each neuron computes its own transition probability, $\pi_{i}$, through the intermediate units $\xi_{(\ell),j}^{t}$, with the adjustable parameters $\rho_{ij},v_{jk}^{(\ell)},w_{jk}^{(\ell)}$ and $h_{j}^{(\ell)}$ and the constant parameter $h_{0}$. These parameters represent the synaptic strengths and intrinsic properties of the neurons and intermediate units. Note that we assume a layered structure of the system, as illustrated in Fig.2, in which neuron $x_{i}^{t}$ in layer $\ell$ receives an input $g(e_{i}^{t})$ from the neurons in layers $1$ through $\ell-1$ and the environment through the $\ell$-th intermediate layer. It is known that an arbitrary continuous mapping of $\{y_{k}^{t}\}_{k=1}^{d}$ and $\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}}$ to $e_{i}^{t}$ can be approximated by the last two lines in Eq.(10) to arbitrary precision if the number of the intermediate units, $M_{\ell}$, is sufficiently large Funahashi (1989). Thus, any conditional probability of the form given in Eq.(9) can be represented in terms of $g(e_{i}^{t})$, as in Eq.(10) . Increasing the number of layers of the neural network increases its capability to represent various conditional probabilities. We believe that the capability to represent a wide variety of conditional probabilities will allow for realization of the optimal $\pi$, and therefore such capability is necessary for our purposes. We model $\nu$ in the same way as $\pi$. Note that $\nu$ is not used for the realization of neural states. We consider the situation in which only the values of $\log\nu(x^{t}\|x^{t+1},y^{t})$ are calculated through some biological mechanisms based on the realized states, $x^{t}$, $x^{t+1}$ and $y^{t}$ (see supplemental materials for details regarding $\nu$). <figure><img src="content_image/1504.03132/x2.png"><figcaption>Figure 2: (Color online). An illustration of the layered neural networks forthe modeling of π (ν is modeled in a similar manner).</figcaption></figure> _A Simple Model of Animals Learning to Explore Environments_ : We have shown that neural systems can maximize the transfer entropy and mutual information through a learning mechanism based on a generalized fluctuation theorem. In order to characterize the present learning mechanism, we must clarify the role of the maximization of the transfer entropy in biological contexts, while that of the mutual information has been investigated in previous studies Linsker (1988); Bell and Sejnowski (1995, 1997). In the following sections, we show that the maximization of the transfer entropy can be understood as a mechanism for the active exploration by an animal of its environment. In order to clearly demonstrate this effect in biological contexts, we introduce a learning problem in which an animal seeks to obtain rewards (e.g., food, water, etc.) through active exploration. Concretely, an animal with a neural system represented by the state $x^{t}$ moves around in a two-dimensional grid. At each position in the grid, a value of a reward associated with that position is defined (Fig.3(a)). Specifically, in each timestep, the animal takes either one step or zero steps, with the number and direction determined by the values of the specialized neurons, as shown in Fig.3(b). The state of the environment, $y^{t}$, is specified by the position of the animal and the status of reward configuration in the grid. At each timestep, the animal “receives” the reward $r^{t}=r(y^{t})$ at its present position. As shown in Fig.3(a), at most positions in the grid, the reward takes a negative value fixed throughout the simulation. Such a negative reward is interpreted as a punishment. The size of the punishment is minimal in the center of the grid and increases in each direction moving away from the center. At eight (fixed) positions in the outer region of the grid, there are positive rewards. The value of each is initially $R$. If such a positive reward is visited by the animal, the reward at the position is 0 for the subsequent 100 timesteps and then reset to $R$. The animal receives inputs from the environment as twelve real variables $\{y_{k}^{t}\}\ (1\leq k\leq 12)$. The inputs $\{y_{k}^{t}\}$ consist of the coordinate values of the animal’s position in the grid $(k=1,2)$, the presence or absence of a reward $R$ at the animal’s current position ($y_{3}^{t}=1\ \mathrm{or}\ 0$) and the values of the rewards at all positions within one step of the current position $(4\leq k\leq 12)$, as shown in Fig.3(c). This set of values allows the animal to predict the immediate consequence of its movement. Initially, the model parameters that determine $\pi$ are set in such a way that the animal primarily attempts to avoid negative rewards, mimicking the innate behavior of real animals (see supplemental materials). With this model, it is very natural to consider maximization of the average reward, $\mathrm{E}[r^{t}]$, by adjusting the animal’s behavior represented by $\pi$, because animals must do so for survival. This maximization problem is called a _reinforcement learning problem_. However, in general, it is known that algorithms that simply maximize $\mathrm{E}[r^{t}]$ do not reach an optimal outcome in most realistic situations because there is a lack of new experiences, unless some mechanisms for active exploration are included Sutton and Barto (1998); Wiering and van Otterlo (2012). In the present case, in order to obtain the rewards $R$ to realize a larger $\mathrm{E}[r^{t}]$, the animal must possess a mechanism that allows it to explore the outer region and tolerate the punishment incurred there. In the following, we show that maximization of the transfer entropy in addition to the average reward provides this mechanism. <figure><img src="content_image/1504.03132/x3.png"><figcaption>Figure 3: (Color). A simple model of an animal exploring in a two-dimensionalgrid: (a) the configuration of reward in the grid, (b) animal’s movementaccording to the values of the specialized neurons, (c) surrounding rewardvalues as input variables.</figcaption></figure> We consider the following learning problem: \[\max\ \left(\mathrm{I}_{x\to y}+\beta\mathrm{E}[r^{t}] \right),\] (11) where $\beta\geq 0$. First, we theoretically analyze the optimal $\pi$ for the above problem. Since the neural control over the environment is deterministic in the above model; that is, for given $x^{t}$ and $y^{t}$, we have $\mu(y^{t+1}\|x^{t},y^{t})=1\ \mathrm{or}\ 0$, the optimization of $\pi(x^{t}\|y^{t})$ reduces to that of $\alpha(y^{t+1}\|y^{t})\equiv\sum_{x^{t}}\mu(y^{t+1}\|x^{t},y^{t})\pi(x^{t}\|y^{t})$. As we know from the basic theory of reinforcement learning, it is helpful for analysis of the maximization problem treated here to consider the following functions of $y\in\mathcal{Y}$: \[V_{r,\alpha}^{(\gamma)}(y) =\] \[V_{I,\alpha}^{(\gamma)}(y) =\] (12) where $\gamma$ is a parameter satisfying $0\leq\gamma<1$. The above quantities with $\gamma\to 1$ represent the average amounts of “excess reward” and “excess information”, obtained from the initial state $y$ until the system has relaxed into the steady state. This is analogous to the definition of the “excess heat” in steady-state thermodynamics Oono and Paniconi (1998); Sasa and Tasaki (2006). With these limits, we can prove that the learning problem, Eq.(11), has a unique optimal distribution $\alpha^{}$ of the following form (see supplemental materials): (13) Inspecting Eq.(13), we understand that the animal shows the following three types of behaviors determined by the value of $\beta$. In the case with finite $\beta(>0)$, the animal moves with high probability in a direction for which large future reward is expected, and with small (but non-zero) probability in a direction for which small future reward is expected. It is known that such exploratory behavior, with (infrequent) excursions in directions with low expected payoff, is necessary for neural systems to find larger rewards Sutton and Barto (1998); Wiering and van Otterlo (2012). Contrastingly, in the case with $\beta\rightarrow\infty$, the optimal behavior is deterministic, and exploration is stifled. In the case with $\beta=0$, the animal is completely insensitive to the values of reward. Hence, we see behavior that is a compromise between the drive to explore and the drive to acquire large rewards, represented by $\mathrm{I}_{x\to y}$ and $\mathrm{E}[r^{t}]$, respectively. _Numerical Simulations_ : In order to confirm the theoretical results obtained in the above, we carried out simulations in which we maximized $\mathrm{E}[\beta r^{t}-\Theta_{\nu}^{t}]$ by applying a stochastic gradient algorithm to the model depicted in Fig.3 (see supplemental materials for the algorithms and discussion of its biological counterparts). It is expected that this maximization will result in the maximization in Eq.(11). We first examine the case with $\beta=\infty$, i.e., that in which the animal attempts to maximize $\mathrm{E}[r^{t}]$ ($R=600$). In this case, we observe that the environmental entropy, $\mathrm{H}[y^{t}]$, decreases monotonically and that $\mathrm{E}[r^{t}]$ becomes fixed at zero (Fig.4(b),(c)). This indicates that the animal has learned only to avoid the outer areas and remains for all times at the origin. Hence, the learning has essentially failed. By contrast, setting $\beta=0.1$ and $R=0$, we observe that $-\mathrm{E}[\Theta_{\nu}^{t}]$ increases monotonically in Fig.4(a). We also observe that $\mathrm{H}[y^{t}]$ increases in a similar manner to $-\mathrm{E}[\Theta_{\nu}^{t}]$ and that $\mathrm{I}[x^{t};y^{t}]$ almost realizes the maximal value, and satisfies $\mathrm{I}[x^{t};y^{t}]=\mathrm{H}[y^{t}]$ (Fig.4(b)). Hence, we have confirmed that the maximization of $-\mathrm{E}[\Theta_{\nu}^{t}]$ leads to exploration and maximization of $\mathrm{I}[x^{t};y^{t}]$, as theoretically predicted above. Finally, with $\beta=0.1$ and $R=600$, we find that the animal is able to increase $\mathrm{E}[r^{t}]$ through the exploration (Fig.4(c)). <figure><img src="content_image/1504.03132/x4.png"><figcaption>Figure 4: (Color). The values of (a) −E[Θtν] (the red line is out of range),(b) H[yt] and I[xt;yt], and (c) E[rt], during the course of learning. Neuralnetworks with L=4, N1=30, N2=60, N3=62, N4=64, M1=M2=120, and M3=M4=60 weresimulated with β=∞,0.1, and R=0,600.</figcaption></figure> _Conclusion_ : We have shown on the basis of theoretical and numerical analysis that assuming that the learning process exhibited by neural systems is based on a principle described by a generalized fluctuation theorem, this system will learn an effective form of exploring behavior (by maximizing the transfer entropy, $\mathrm{I}_{x\to y}$) and acquiring information about its environment (by maximizing the mutual information, $\mathrm{I}[x^{t};y^{t}]$). Although informational quantities other than the transfer entropy have been considered as mechanisms for the exploration Azar _et al._ (2012); Peters _et al._ (2010); Still and Precup (2012); Ay _et al._ (2012); Bialek _et al._ (2001), it has not been elucidated how those quantities are maximized in neural systems. We believe that use of the transfer entropy as a mechanism for exploration is more plausible, because the present learning mechanism can be utilized for it. Although the demonstration is limited to the case of Markovian environmental dynamics and neural networks without memory, this work will be generalized to more complex systems in the near future using the foundation laid by the present work. This work was supported by JST CREST from MEXT. ## Supplemental Materials _Proof of The Maximization of The Mutual Information, $\mathrm{I}[x^{t};y^{t}]$_ : In this section, we prove that the maximization of the mutual information, $\mathrm{I}[x^{t};y^{t}]$, and Eq.(8) are equivalent to equality in Eq.(3), under suitable conditions. First, we note that we can replace $\nu(x^{t}\|x^{t+1},y^{t})$ by $\nu(x^{t}\|x^{t+1},y^{t},y^{t+1})$ in Eqs.(3), (4), (5), (6) and (7). In this case, equality in Eq.(3) follows from equality in the Jensen inequality ($\exp F(Z)=\mathrm{E}[\exp F(Z)]$ with probability 1): \[e^{-I_{tr}^{t+1}-\widehat{\Theta}_{\nu}^{t}}=\mathrm{E}[e^{-I_{ tr}^{t+1}-\widehat{\Theta}_{\nu}^{t}}]=1.\] (14) This implies $-I_{tr}^{t+1}-\widehat{\Theta}_{\nu}^{t}=0$ with probability 1. By rearranging terms, we have \[\nu(x^{t}\|x^{t+1},y^{t},y^{t+1}) = p_{s}(x^{t}\|y^{t+1},y^{t},x^{t+1}).\] (15) Hence, by including $y^{t+1}$ as a conditioning variable of $\nu$, we can easily obtain the equality. However, by reducing the number of conditioning variables, we can also obtain the maximization of the mutual information, $\mathrm{I}[x^{t};y^{t}]$, as we noted in the main text. We prove this in the following. First, we obtain an explicit expression of the optimal $\nu(x^{t}\|x^{t+1},y^{t})$ in Eq.(8) from the following inequality: \[\mathrm{E}\left[\log\frac{\nu(x^{t}\|y^{t},x^{t+1})}{p_{s}(x^{t}\|y ^{t},x^{t+1})}\right]=\sum_{y^{t},x^{t+1}}p_{s}(y^{t},x^{t+1})\sum_{x^{t}}p_{s }(x^{t}\|y^{t},x^{t+1})\log\frac{\nu(x^{t}\|y^{t},x^{t+1})}{p_{s}(x^{t}\|y^{t},x^ {t+1})}\leq 0.\] (16) The above inequality is derived from the inequality $F-1\geq\log F$ for positive real $F$, and thus, the optimality condition in Eq.(8) is obtained from the equality, $F-1=\log F\leftrightarrow F=1$ with probability 1: \[\nu(x^{t}\|x^{t+1},y^{t}) = p_{s}(x^{t}\|y^{t},x^{t+1}).\] (17) Then, in order to analyze the equality condition of Eq.(3) for $\nu(x^{t}\|x^{t+1},y^{t})$, we calculate the difference $\Delta$ between the values of $-\mathrm{E}[\Theta_{\nu}^{t}]$ as calculated with Eqs. (17) and (15), writing \[\Delta=\mathrm{E}\left[\log\frac{p_{s}(x^{t}\|y^{t},y^{t+1},x^{t+1 })}{p_{s}(x^{t}\|y^{t},x^{t+1})}\right]=\mathrm{I}[y^{t+1};x^{t}\|x^{t+1},y^{t}].\] (18) Because $-\mathrm{E}[\Theta_{\nu}^{t}]=\mathrm{I}_{x\to y}$ in the case with Eq.(15), we obtain \[\mathrm{I}_{x\to y}+\mathrm{E}[\Theta_{\nu}^{t}]=\mathrm{ I}[y^{t+1};x^{t}\|x^{t+1},y^{t}],\] (19) in the case with Eq.(17). Next, we show that the condition \[\mathrm{I}[y^{t+1};x^{t}\|x^{t+1},y^{t}]=0,\] (20) is equivalent to the maximization of the mutual information, \[\mathrm{I}[x^{t};y^{t}]=\mathrm{H}[y^{t}],\] (21) on the assumption that the environmental state space $\mathcal{Y}$ is not too finely partitioned in comparison with the precision of neural control over the environment. Precisely, we assume that there is no coarse-grained partition $\mathcal{Y^{\prime}}$ of $\mathcal{Y}$ such that the neural control has the same precision on the two partitions $\mathcal{Y}$ and $\mathcal{Y}^{\prime}$ of the environmental state space. We also assume that $p_{s}(y^{t+1}\|y^{t})\neq 1$ for all $y^{t}$ and $y^{t+1}$, which holds for most sets of values of the model parameters in a general model. A coarse-grained partition $\mathcal{Y}^{\prime}$ is a set of subcollections of $\mathcal{Y}$ such that $y^{\prime}\cap y^{\prime\prime}=\emptyset$ for any $y^{\prime}\neq y^{\prime\prime}\in\mathcal{Y}^{\prime}$ and $\cup_{y^{\prime}\in\mathcal{Y}^{\prime}}y^{\prime}=\mathcal{Y}$. For any such coarse-grained partition $\mathcal{Y}^{\prime}$, we require \[\mathrm{I}[y^{t+1,\prime};x^{t}\|y^{t}]\neq\mathrm{I}[y^{t+1};x^{t }\|y^{t}],\ y^{t+1}\in y^{t+1,\prime}\in\mathcal{Y}^{\prime},\] (22) where we have defined the random variable $y^{t+1,\prime}$, which takes values in $\mathcal{Y}^{\prime}$ with $y^{t+1}\in y^{t+1,\prime}$. Under this assumption, we first show that the conditional mutual information, \[\mathrm{I}[y^{t+1};x^{t+1}\|y^{t}]=\sum_{x^{t+1},y^{t+1},y^{t}}p_{ s}(y^{t},y^{t+1},x^{t+1})\log\frac{p_{s}(y^{t+1}\|x^{t+1},y^{t})}{p_{s}(y^{t+1} \|y^{t})},\] (23) must be maximal. Note that the conditional mutual information takes its maximal value and hence satisfies \[\mathrm{I}[x^{t+1};y^{t+1}\|y^{t}]=\mathrm{H}[y^{t+1}\|y^{t}]\] (24) if and only if $p_{s}(y^{t+1}\|x^{t+1},y^{t})=1$ with probability 1. Thus, to obtain the desired result, we show that $p_{s}(y^{t+1}\|x^{t+1},y^{t})>0$ for multiple $y^{t+1}$ with some $y^{t}=y_{0}$ and $x^{t+1}=x_{0}$ contradicts Eq.(20). First, we define the set \[\overline{y}=\{y^{t+1}\in\mathcal{Y}\|p_{s}(y^{t+1}\|x^{t+1}=x_{0}, y^{t}=y_{0})>0\},\] (25) and a coarse-grained partition of $\mathcal{Y}$ as \[\mathcal{Y}^{\prime}=\{\{y\}\}_{y\in{\mathcal{Y}\setminus \overline{y}}}\cup\{\overline{y}\}.\] (26) Here, $\mathcal{Y}\setminus\overline{y}$ is the relative complement of $\overline{y}$ in $\mathcal{Y}$, which consists of all the elements of $\mathcal{Y}$ that are not contained in $\overline{y}$. The assumption in Eq.(22) requires \[\mathrm{I}[y^{t+1};x^{t}\|y^{t}]-\mathrm{I}[y^{t+1,\prime};x^{t}\|y ^{t}] = \mathrm{I}[y^{t+1},y^{t+1,\prime};x^{t}\|y^{t}]-\mathrm{I}[y^{t+1, \prime};x^{t}\|y^{t}]\] (27) \[= \mathrm{I}[y^{t+1};x^{t}\|y^{t+1,\prime},y^{t}]\] \[> 0,\] where the first equality holds because $y^{t+1}$ uniquely determines $y^{t+1,\prime}$ and thus the additional inclusion of $y^{t+1,\prime}$ in the first term does not affect the value of the conditional mutual information. Also, Eq.(20) implies \[\mathrm{I}[y^{t+1};x^{t}\|y^{t},x^{t+1}]=\mathrm{I}[y^{t+1};x^{t}\| y^{t+1,\prime},y^{t},x^{t+1}]=0.\] (28) Now, recall that the inclusion of additional conditioning variables (in this case, $y^{t+1,\prime}$) always reduces the value of the conditional mutual information. The right-hand side of the above equation can be written as \[\mathrm{E}\left[\log\frac{p_{s}(y^{t+1}\|x^{t},y^{t+1,\prime},y^{t },x^{t+1})}{p_{s}(y^{t+1}\|y^{t+1,\prime},y^{t},x^{t+1})}\right]\] (29) \[= \mathrm{E}\left[\log\left\{\frac{p_{s}(x^{t+1}\|y^{t+1})p_{s}(y^{t +1}\|y^{t+1,\prime},y^{t},x^{t})}{\sum_{\widetilde{y}^{t+1}\in\mathcal{Y}}p_{s} (x^{t+1}\|\widetilde{y}^{t+1})p_{s}(\widetilde{y}^{t+1}\|y^{t+1,\prime},y^{t},x^ {t})}\frac{\sum_{\widehat{y}^{t+1}\in\mathcal{Y}}p_{s}(x^{t+1}\|\widehat{y}^{t+ 1})p_{s}(\widehat{y}^{t+1}\|y^{t+1,\prime},y^{t})}{p_{s}(x^{t+1}\|y^{t+1})p_{s}( y^{t+1}\|y^{t+1,\prime},y^{t})}\right\}\right]\] \[= 0,\] with the dummy variables $\widetilde{y}^{t+1}$ and $\widehat{y}^{t+1}$ having the same (conditional) distributions as $y^{t+1}$. The above equality requires that the argument of the logarithm be 1 with probability 1, since $F-1\geq\log F$, $F-1=\log F\leftrightarrow F=1$ and \[-\mathrm{E}\left[\log\frac{p_{s}(y^{t+1}\|x^{t},y^{t+1,\prime},y^{ t},x^{t+1})}{p_{s}(y^{t+1}\|y^{t+1,\prime},y^{t},x^{t+1})}\right]\] (30) \[= \sum_{x^{t},y^{t+1,\prime},y^{t},x^{t+1}}p_{s}(x^{t},y^{t+1, \prime},y^{t},x^{t+1})\sum_{y^{t+1}}p_{s}(y^{t+1}\|x^{t},y^{t+1,\prime},y^{t},x ^{t+1})\log\frac{p_{s}(y^{t+1}\|y^{t+1,\prime},y^{t},x^{t+1})}{p_{s}(y^{t+1}\|x^ {t},y^{t+1,\prime},y^{t},x^{t+1})}\] \[\leq \sum_{x^{t},y^{t+1,\prime},y^{t},x^{t+1}}p_{s}(x^{t},y^{t+1, \prime},y^{t},x^{t+1})\sum_{y^{t+1}}p_{s}(y^{t+1}\|x^{t},y^{t+1,\prime},y^{t},x ^{t+1})\left(\frac{p_{s}(y^{t+1}\|y^{t+1,\prime},y^{t},x^{t+1})}{p_{s}(y^{t+1}\| x^{t},y^{t+1,\prime},y^{t},x^{t+1})}-1\right)\] \[= 0.\] Hence, noting that $p_{s}(x^{t+1}=x_{0}\|y^{t+1})>0$ for $y^{t+1}\in\overline{y}$, we have \[\frac{p_{s}(y^{t+1}\|y^{t+1,\prime}=\overline{y},y^{t},x^{t})}{p_{ s}(y^{t+1}\|y^{t+1,\prime}=\overline{y},y^{t})}=\frac{\sum_{\widehat{y}^{t+1} \in\mathcal{Y}}p_{s}(x^{t+1}=x_{0}\|\widehat{y}^{t+1})p_{s}(\widehat{y}^{t+1}\|y ^{t+1,\prime}=\overline{y},y^{t})}{\sum_{\widetilde{y}^{t+1}\in\mathcal{Y}}p_{ s}(x^{t+1}=x_{0}\|\widetilde{y}^{t+1})p_{s}(\widetilde{y}^{t+1}\|y^{t+1,\prime}= \overline{y},y^{t},x^{t})},\ \ \ \forall y^{t+1}\in\overline{y}.\] (31) Furthermore, Eq.(31) with Eq.(27) implies \[\frac{p_{s}(y^{t+1}\|y^{t+1,\prime}=\overline{y},y^{t},x^{t})}{p_{ s}(y^{t+1}\|y^{t+1,\prime}=\overline{y},y^{t})}=c\neq 1,\] (32) for all $y^{t+1}\in\overline{y}$ and some $y^{t}$ and $x^{t}$, noting \[\frac{p_{s}(y^{t+1}\|y^{t+1,\prime},y^{t},x^{t})}{p_{s}(y^{t+1}\|y^ {t+1,\prime},y^{t})}=1,\ \ \ \forall y^{t+1,\prime}\in\mathcal{Y}^{\prime} \setminus\{\overline{y}\}.\] (33) Here, note that $c=1$ in Eq.(32) with Eq.(33) implies $\mathrm{I}[y^{t+1};x^{t}\|y^{t+1,\prime},y^{t}]=0$, violating the assumption in Eq.(27). However, this implies \[1=\sum_{y^{t+1}\in\overline{y}}p_{s}(y^{t+1}\|y^{t+1,\prime}= \overline{y},y^{t},x^{t})=c\sum_{y^{t+1}\in\overline{y}}p_{s}(y^{t+1}\|y^{t+1, \prime}=\overline{y},y^{t})=c\neq 1.\] (34) This contradiction completes the proof of the maximality of the conditional mutual information, Eq.(24). Next, we show the equivalence of the maximality of the conditional mutual information and the maximality of the mutual information. As we have discussed, Eq.(24) implies \[p_{s}(y^{t+1}\|x^{t+1},y^{t})=\frac{p_{s}(y^{t}\|y^{t+1})p_{s}(y^{ t+1}\|x^{t+1})}{\sum_{\widetilde{y}^{t+1}\in\mathcal{Y}}p_{s}(y^{t}\|\widetilde{ y}^{t+1})p_{s}(\widetilde{y}^{t+1}\|x^{t+1})}=1,\] (35) with probability 1. Then, the assumption $p_{s}(y^{t+1}\|y^{t})\neq 1$ implies that $p_{s}(y^{t}\|y^{t+1})$ is positive for multiple $y^{t+1}$. Thus, the condition Eq.(35) implies $p_{s}(y^{t+1}\|x^{t+1})=1$ with probability 1, or equivalently, that the mutual information, $\mathrm{I}[x^{t};y^{t}]$, is maximal and hence satisfies \[\mathrm{I}[x^{t};y^{t}]=\mathrm{H}[y^{t}].\] (36) Conversely, the equality in Eq.(36) implies \[\mathrm{I}[y^{t+1};x^{t}\|x^{t+1},y^{t}] = \mathrm{H}[y^{t+1}\|x^{t+1},y^{t}]-\mathrm{H}[y^{t+1}\|x^{t},x^{t+1 },y^{t}]\] (37) \[\leq \mathrm{H}[y^{t+1}\|x^{t+1}]\] \[= \mathrm{H}[y^{t}]-\mathrm{I}[x^{t};y^{t}]\] \[= 0.\] Thus we have recovered the condition Eq.(20). This completes the proof that equality in Eq.(3) is equivalent to Eq.(8) and Eq.(21). In the above proof, we note that different definitions of $\nu$ also lead to the maximization of the mutual information $\mathrm{I}[x^{t};y^{t}]$ in different manners, although we do not note this point in the main text for the simplicity of presentation. Concretely, we consider the model described by the causal network in Fig.5 by splitting $x^{t}$ into $x_{(1)}^{t}$ and $x_{(2)}^{t}$, and define a generalized entropy production as \[\Theta_{\nu}^{t}=\log\frac{\pi(x_{(1)}^{t+1}\|y^{t+1})}{\nu(x_{(1) }^{t}\|x_{(2)}^{t},y^{t+1})}.\] (38) Then, in the same manner as above, we can prove that the equality, $-\mathrm{E}[\Theta_{\nu}^{t}]=\mathrm{I}_{x_{(1)}\to y}$, implies the maximization of the mutual information, $\mathrm{I}[x_{(2)}^{t};y^{t,\prime}]=\mathrm{H}[y^{t,\prime}]$, in some coarse-grained partition $\mathcal{Y}^{\prime}$ that satisfies \[\mathrm{I}[x_{(1)}^{t};y^{t,\prime}\|y^{t+1}]=\mathrm{I}[x_{(1)}^{ t};y^{t}\|y^{t+1}].\] (39) Further results in this direction will be investigated in the future reports. <figure><img src="content_image/1504.03132/x5.png"><figcaption>Figure 5: Representation of the dynamics with an additional neural network ztas a causal network.</figcaption></figure> _Modeling of $\nu$ with a Neural Network_ : We compute $\nu$ in the same way as $\pi$, explicitly writing \[\nu(x^{t}\|x^{t+1},y^{t}) = \prod_{\ell=1}^{L}\prod_{i=N_{\ell-1}+1}^{N_{\ell}}\nu_{i}(x_{i}^ {t}\|\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}},x^{t+1},y^{t}),\] \[\nu_{i}(x_{i}^{t} = 1\|\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}},x^{t+1},y^{t})=g(f_{i}^{t+1}),\] \[f_{i}^{t+1} = \sum_{1\leq j\leq M_{\ell}}\kappa_{ij}\{g(\eta_{(\ell),j}^{t+1})- \frac{1}{2}\}-m_{0},\] \[\eta_{(\ell),j}^{t+1}=\hskip-4.267913pt\sum_{1\leq k\leq N}z_{jk} ^{(\ell)} x_{k}^{t+1} +\hskip-7.113189pt\sum_{1\leq k\leq N_{\ell-1}}\hskip-7.113189ptu _{jk}^{(\ell)}x_{k}^{t}+\hskip-4.267913pt\sum_{1\leq k\leq d}q_{jk}^{(\ell)}y_ {k}^{t}-m_{j}^{(\ell)}.\] (40) Here, the neurons in the $\ell$-th layer receive inputs from $\{x_{k}^{t}\}_{1\leq k\leq N_{\ell-1}}$, $x^{t+1}$ and $y^{t}$ through the intermediate units, $\eta_{(\ell),j}^{t+1}$, with the adjustable parameters $\kappa_{ij}$, $z_{jk}^{(\ell)}$, $u_{jk}^{(\ell)}$, $q_{jk}^{(\ell)}$ and $m_{j}^{(\ell)}$ and the constant parameter $m_{0}$. This computation may seem strange, because here the neurons receive inputs from the future states. However, this is not problematic, because the goal of the computation is not to realize the states of the neural network but to calculate the value of $\log\nu(x^{t}\|x^{t+1},y^{t})$. Consider the following situation for this computation, for example. The intermediate units in the $\ell$-th layer receive inputs at time $t+1$ from $x^{t+1}$ and also from $\{x_{k}^{t}\}_{1\leq k\leq N_{\ell-1}}$ and $y^{t}$ through some time-delay mechanisms. These intermediate units send outputs $\{g(\eta_{(\ell),j}^{t+1})\}_{1\leq j\leq M_{\ell}}$ to the neurons in the $\ell$-th layer. At this time, the $i$-th neuron in this layer possesses memory of its own state at time $t$, $x_{i}^{t}$, through some mechanism. Then, the $i$-th neuron can compute the value of $\nu_{i}(x_{i}^{t}\|\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}},x^{t+1},y^{t})$ as a function of $x_{i}^{t}$ and $\{g(\eta_{(\ell),j}^{t+1})\}_{1\leq j\leq M_{\ell}}$. The value of $\log\nu$ is the sum of these values of $\nu_{i}$ over the neurons in the neural network. _Proofs of the Relations Used in the Theoretical Analysis of the Reinforcement Learning Problem_ : In this section, our goal is to prove Eq.(13). First, we define the following functions called “value functions” in the field of reinforcement learning: \[\widehat{V}_{r,\alpha}^{(\gamma)}(y) = \mathrm{E}\left[\sum_{s=1}^{\infty}\gamma^{s}r^{t+s}\Big{\|}y^{t}= y\right],\] \[\widehat{V}_{I,\alpha}^{(\gamma)}(y) = \mathrm{E}\left[\sum_{s=1}^{\infty}\gamma^{s}I_{tr}^{t+s}\Big{\|}y ^{t}=y\right],\] \[\widehat{V}_{\alpha}^{(\gamma)}(y) = \widehat{V}_{I,\alpha}^{(\gamma)}(y)+\beta\widehat{V}_{r,\alpha}^ {(\gamma)}(y).\] (41) Then, we can write the learning problem Eq.(11) in terms of the value functions as \[\mathrm{I}_{x\to y}+\beta\mathrm{E}[r^{t}]=\lim_{\gamma \to 1}(1-\gamma)\widehat{V}_{\alpha}^{(\gamma)}(y),\ \ \ \ \forall y \in\mathcal{Y}.\] (42) By definition, the value function satisfies the following recursive relation called the “Bellman equation”: \[\widehat{V}_{\alpha}^{(\gamma)}(y^{t}) = \sum_{y^{t+1}\in\mathcal{Y}}\gamma\alpha(y^{t+1}\|y^{t})\] \[\times \{ \beta r(y^{t+1})-\log\alpha(y^{t+1}\|y^{t})+\widehat{V}_{\alpha}^{ (\gamma)}(y^{t+1})\}.\] (43) Next, we show that for fixed $\gamma$, it is known that an optimal control $\alpha^{}(y^{t+1}\|y^{t})$ maximizes the value function at all $y\in\mathcal{Y}$, in comparison with suboptimal controls. Explicitly, for any control $\alpha$, the following inequality holds: \[\widehat{V}^{(\gamma)}_{\alpha^{}}(y)\geq\widehat{V}^{(\gamma)}_ {\alpha}(y),\ \ \ \ \forall y\in\mathcal{Y}.\] (44) In order to prove Eq.(44), we consider the following operator called a backup operator, operating on functions of the environmental state $y^{t}$: \[B\phi(y^{t})=\max_{\alpha(y^{t+1}\|y^{t})}\sum_{y^{t+1}\in \mathcal{Y}}\alpha(y^{t+1}\|y^{t})\{\gamma r(y^{t+1})-\gamma\log\alpha(y^{t+1}\| y^{t})+\gamma\phi(y^{t+1})\}.\] (45) We first show that this operation results in contraction in the space of functions of environmental states $y^{t}$ with respect to max norm: \[\parallel\phi\parallel_{\infty}=\max_{y^{t}\in\mathcal{Y}}\|\phi(y ^{t})\|.\] (46) For two functions $\phi_{1}$ and $\phi_{2}$, a fixed $\alpha(y^{t+1}\|y^{t})$, and an operator $B_{\alpha}$ defined as \[B_{\alpha}\phi(y^{t})=\sum_{y^{t+1}\in\mathcal{Y}}\alpha(y^{t+1} \|y^{t})\{\gamma r(y^{t+1})-\gamma\log\alpha(y^{t+1}\|y^{t})+\gamma\phi(y^{t+1})\},\] (47) we have \[\parallel B_{\alpha}\phi_{1}-B_{\alpha}\phi_{2}\parallel_{\infty} = \parallel\sum_{y^{t+1}}\alpha(y^{t+1}\|y^{t})\gamma\{\phi_{1}(y^{t +1})-\phi_{2}(y^{t+1})\}\parallel_{\infty}\] (48) \[\leq \max_{y^{t}\in\mathcal{Y}}\left\{\sum_{y^{t+1}}\|\alpha(y^{t+1}\|y^ {t})\|\right\}\parallel\gamma(\phi_{1}-\phi_{2})\parallel_{\infty}\] \[= \gamma\parallel\phi_{1}-\phi_{2}\parallel_{\infty}.\] Then, with the distribution $\alpha^{(i),}(y^{t+1}\|y^{t})$ maximizing $B_{\alpha}\phi_{i}(y^{t})$$(i=1,2)$, we have \[\parallel B\phi_{1}-B\phi_{2}\parallel_{\infty} = \parallel B_{\alpha^{(1),}}\phi_{1}-B_{\alpha^{(2),}}\phi_{2} \parallel_{\infty}\] (49) \[\leq \max_{i}\parallel B_{\alpha^{(i),}}\phi_{1}-B_{\alpha^{(i),}} \phi_{2}\parallel_{\infty}\] \[\leq \gamma\parallel\phi_{1}-\phi_{2}\parallel_{\infty}.\] This proves that the backup operation yields a contraction of the space of functions on the environmental state space $\mathcal{Y}$, and that there is a unique fixed point of this operation in this space of functions. Because the backup operation always increases the values of any value function at any point in $\mathcal{Y}$, we have Eq.(44). Hence, when we consider the maximality condition of $\widehat{V}_{\alpha}^{(\gamma)}(y)$ with respect to $\alpha(y^{t+1}\|y^{t})$, it is sufficient to consider the stationarity condition of $B\widehat{V}_{\alpha}^{(\gamma)}(y^{t})$ by differentiating it with respect to $\alpha(y^{t+1}\|y^{t})$ and simply putting the derivative of $\widehat{V}_{\alpha}^{(\gamma)}(y^{t+1})$ to be zero. Solving the stationarity condition with the Lagrange multiplier corresponding to $\sum_{y^{t+1}}\alpha(y^{t+1}\|y^{t})=1$, we obtain \[\alpha^{}(y^{t+1}\|y^{t}) \propto \exp[\beta\{r(y^{t+1})+\widehat{V}_{r,\alpha^{}}^{(\gamma)}(y^{t +1})\}+\widehat{V}_{I,\alpha^{}}^{(\gamma)}(y^{t+1})].\] (50) The optimal condition for the learning problem, the maximization of Eq.(42), is obtained by taking the limit $\gamma\to 1$ in Eq.(50). In order to avoid divergence, we need to replace the value functions in Eq.(50) with the functions defined in Eq.(12) that represent the “excess reward” and “excess information”. _Derivation and Biological Plausibility of the Learning Rule_ : In the gradient ascent method used for the simulation, we update each parameter $\theta\in\{\rho_{ij},v_{jk}^{(\ell)},w_{jk}^{(\ell)},h_{j}^{(\ell)},\kappa_{ij },z_{jk}^{(\ell)},u_{jk}^{(\ell)},q_{jk}^{(\ell)},m_{j}^{(\ell)}\}$ as follows: \[\theta^{t+1} = \theta^{t}+\epsilon\left\{\tau(\beta r^{t+1}-\Theta_{\nu}^{t}) \psi_{\theta}^{t}-\frac{\partial}{\partial\theta}\Theta_{\nu}^{t}\right\}\] \[\psi_{\theta}^{t} = \frac{1}{\tau}\frac{\partial}{\partial\theta}\log\pi(x^{t+1}\|y^{t +1})+\left(1-\frac{1}{\tau}\right)\psi_{\theta}^{t-1}.\] (51) Here, the constant $\tau$ is a positive real number that is large compared with the mixing time of the dynamics. We set $\epsilon$ to such a small value that the change in the model parameters in each update does not affect the stationarity on a time scale of $\tau$. Then, in the above learning rule, the expectation value of the change in the parameter $\theta$ in each update is equal to the gradient of $\mathrm{E}[\beta r^{t}-\Theta_{\nu}^{t}]$ with respect to $\theta$ as we show below. Thus, we can regard the learning rule as a stochastic gradient ascent algorithm to maximize $\mathrm{E}[\beta r^{t}-\Theta_{\nu}^{t}]$. In the gradient ascent method, we must calculate the gradient of the following quantity with respect to $\theta$: \[\mathrm{E}[\beta r^{t}-\Theta_{\nu}^{t}]=\sum_{x^{t},x^{t+1},y^{t },y^{t+1}}p_{s}(y^{t})\pi(x^{t}\|y^{t})\mu(y^{t+1}\|y^{t},x^{t})\pi(x^{t+1}\|y^{t +1})\{\beta r(y^{t+1})-\Theta_{\nu}^{t}\}.\] In this calculation, we find that differentiation of the stationary distribution $p_{s}(y^{t})$ is apparently intractable, while differentiation of the other components is easily carried out. We note, however, that we do not need to differentiate the stationary distribution explicitly, assuming that the stationary distribution is a smooth function of any model parameter $\theta$. In this case, small changes in $p_{s}(y^{t-\tau})$ for $\tau\gg 1$ vanish at $t$ and $t-1$, and thus terms including the derivatives of $p_{s}(y^{t-\tau})$ are negligible (see also Baxter and Bartlett (1999)). Thus, we can compute the gradient as follows: \[\frac{\partial}{\partial\theta}\mathrm{E}[\beta r(y^{t+1})-\Theta _{\nu}^{t}]\] (52) \[= \lim_{\tau\rightarrow\infty}\sum_{y^{t-\tau},x^{t-\tau},\cdots,y^ {t+1},x^{t+1}}p_{s}(y^{t-\tau})\frac{\partial}{\partial\theta}\left[\pi(x^{t- \tau}\|y^{t-\tau})\prod_{s=0}^{\tau}\mu(y^{t-s+1}\|x^{t-s},y^{t-s})\pi(x^{t-s+1} \|y^{t-s+1})\{\beta r(y^{t+1})-\Theta_{\nu}^{t}\}\right]\] \[= \lim_{\tau\rightarrow\infty}\sum_{y^{t-\tau},x^{t-\tau},\cdots,y^ {t+1},x^{t+1}}p_{s}(y^{t-\tau},x^{t-\tau},\cdots,y^{t+1},x^{t+1})\sum_{s=0}^{ \tau+1}\frac{\frac{\partial}{\partial\theta}\pi(x^{t-s+1}\|y^{t-s+1})}{\pi(x^{t -s+1}\|y^{t-s+1})}\{\beta r(y^{t+1})-\Theta_{\nu}^{t}\}\] \[+\lim_{\tau\rightarrow\infty}\sum_{y^{\tau},x^{t-\tau},\cdots,y^{ t+1},x^{t+1}}p_{s}(y^{t-\tau},x^{t-\tau},\cdots,y^{t+1},x^{t+1})\frac{\partial }{\partial\theta}\{\beta r(y^{t+1})-\Theta_{\nu}^{t}\}\] \[= \lim_{\tau\rightarrow\infty}\mathrm{E}\left[\{\beta r(y^{t+1})- \Theta_{\nu}^{t}\}\sum_{s=0}^{\tau+1}\frac{\partial}{\partial\theta}\log\pi(x^ {t-s+1}\|y^{t-s+1})\right]-\mathrm{E}\left[\frac{\partial}{\partial\theta} \Theta_{\nu}^{t}\right].\] Note that the third equality follows from $\frac{\partial}{\partial\theta}r(y^{t+1})=0$. In order to decompose the expectation values into time-stepwise quantities, we introduce the auxiliary variable $\psi_{\theta}^{t}$, defined through \[\psi_{\theta}^{t}=\frac{1}{\tau}\frac{\partial}{\partial\theta} \log\pi(x^{t+1}\|y^{t+1})+(1-\frac{1}{\tau})\psi_{\theta}^{t-1},\ \ \ \mathrm{ and}\ \ \ \psi_{\theta}^{t}=0\ \ (t\leq 0).\] (53) Then, we have \[\psi_{\theta}^{t}=\frac{1}{\tau}\sum_{s=0}^{\infty}(1-\frac{1}{ \tau})^{s}\frac{\partial}{\partial\theta}\log\pi(x^{t-s+1}\|y^{t-s+1}).\] If the process under consideration is stationary, $\psi_{\theta}^{t}$ approaches the long-time average of $\frac{\partial}{\partial\theta}\log\pi(x^{t+1}\|y^{t+1})$ as $\tau\rightarrow\infty$ and $t/\tau\rightarrow\infty$. Similarly, assuming that the correlation of $\beta r(y^{t+1})-\Theta_{\nu}^{t}$ with $\frac{\partial}{\partial\theta}\log\pi(x^{t-\tau+1}\|y^{t-\tau+1})$ is small for $\tau\gg 1$ and that $T\gg\tau$, we have \[\mathrm{E}\left[\{\beta r(y^{t+1})-\Theta_{\nu}^{t}\}\sum_{s=0}^{ \tau+1}\frac{\partial}{\partial\theta}\log\pi(x^{t-s+1}\|y^{t-s+1})\right] \approx \frac{\tau}{T}\sum_{t=1}^{T}(\beta r(y^{t+1})-\Theta_{\nu}^{t}) \psi_{\theta}^{t}.\] (54) Then, applying a well-known argument in stochastic approximation theory (Robbins and Monro, 1951), we obtain the learning rule given in Eq.(51) as a stepwise approximation of the gradient in Eq.(52). Finally, we derive the exact form of the learning rule with respect to several $\theta$ and present its interpretation. Note that Eq.(51) is composed of $\log\pi(x^{t+1}\|y^{t+1})$, $\log\nu(x^{t}\|x^{t+1},y^{t})$ and their derivatives with respect to $\theta$. First, we show that these components are easily calculated in a neuron-wise manner. Note that $\Theta_{\nu}^{t}$, $\log\pi(x^{t+1}\|y^{t+1})$ and $\log\nu(x^{t}\|x^{t+1},y^{t})$ are decomposed as \[\Theta_{\nu}^{t} = \log\pi(x^{t+1}\|y^{t+1})-\log\nu(x^{t}\|x^{t+1},y^{t})\] (55) \[= \sum_{\ell=1}^{L}\sum_{i=N_{\ell-1}+1}^{N_{\ell}}\log\pi_{i}(x_{i }^{t+1}\|y^{t+1},\{x_{k}^{t+1}\}_{k=1}^{N_{\ell-1}})-\sum_{\ell=1}^{L}\sum_{i=N _{\ell-1}+1}^{N_{\ell}}\log\nu_{i}(x_{i}^{t}\|\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}}, x^{t+1},y^{t})\] \[= \sum_{i}\log\chi(e_{i}^{t+1},x_{i}^{t+1})-\sum_{i}\log\chi(f_{i}^ {t},x_{i}^{t}),\] where \[\chi(a,b) = \left\{\begin{array}[]{ccc}g(a)&,&\mathrm{if\ \ }b=1,\\ -g(a)&,&\mathrm{if\ \ }b=0.\end{array}\right.\] Then, the derivatives of $\Theta_{\nu}^{t}$, $\log\pi(x^{t+1}\|y^{t+1})$ and $\log\nu(x^{t}\|x^{t+1},y^{t})$ (with respect to $v_{jk}^{(\ell)},\rho_{ij},z_{jk}^{(\ell)},\kappa_{ij}$, for example) are calculated as follows. First, denoting the derivative of $\chi(a,b)$ with respect to $a$ as $\chi_{a}(a,b)$, \[\frac{\partial}{\partial v_{jk}^{(\ell)}}\log\pi(x^{t+1}\|y^{t+1}) = \sum_{i=N_{\ell-1}+1}^{N_{\ell}}\frac{\partial}{\partial v_{jk}^{ (\ell)}}\log\pi_{i}(x_{i}^{t+1}\|y^{t+1},\{x_{k}^{t+1}\}_{k=1}^{N_{\ell-1}})\] (57) \[= \sum_{i=N_{\ell-1}+1}^{N_{\ell}}\frac{\chi_{a}(e_{i}^{t+1},x_{i}^ {t+1})}{\chi(e_{i}^{t+1},x_{i}^{t+1})}\rho_{ij}g^{\prime}(\xi_{(\ell),j}^{t+1} )y_{k}^{t+1}.\] \[\frac{\partial}{\partial\rho_{ij}}\log\pi(x^{t+1}\|y^{t+1}) = \frac{\partial}{\partial\rho_{ij}}\log\pi_{i}(x_{i}^{t+1}\|y^{t+1} ,\{x_{k}^{t+1}\}_{k=1}^{N_{\ell-1}})\] (58) \[= \frac{\chi_{a}(e_{i}^{t+1},x_{i}^{t+1})}{\chi(e_{i}^{t+1},x_{i}^{ t+1})}g(\xi_{(\ell,j)}^{t+1}).\] \[\frac{\partial}{\partial z_{jk}^{(\ell)}}\log\nu(x^{t}\|x^{t+1},y^ {t}) = \sum_{i=N_{\ell-1}+1}^{N_{\ell}}\frac{\partial}{\partial z_{jk}^{ (\ell)}}\log\nu_{i}(x_{i}^{t}\|\{x_{k}^{t}\}_{k=1}^{N_{\ell-1}},x^{t+1},y^{t})\] (59) \[= \sum_{i=N_{\ell-1}+1}^{N_{\ell}}\frac{\chi_{a}(f_{i}^{t+1},x_{i}^ {t})}{\chi(f_{i}^{t+1},x_{i}^{t})}\kappa_{ij}^{(\ell)}g^{\prime}(\eta_{(\ell), j}^{t+1})x_{k}^{t+1}.\] \[\frac{\partial}{\partial\kappa_{ij}}\log\nu(x^{t}\|x^{t+1},y^{t}) = \frac{\partial}{\partial\kappa_{ij}}\log\nu_{i}(x_{i}^{t+1}\|\{x_{ k}^{t}\}_{k=1}^{N_{\ell-1}},x^{t+1},y^{t})\] (60) \[= \frac{\chi_{a}(f_{i}^{t+1},x_{i}^{t})}{\chi(f_{i}^{t+1},x_{i}^{t} )}g(\eta_{(\ell,j)}^{t+1}).\] It should be noted that calculations of the derivatives involve quantities only for related neurons and intermediate units. For example, the derivative with respect to $v_{jk}^{(\ell)}$ used only information regarding $y_{k}^{t+1}$, $\xi_{(\ell),k}^{t+1}$ and $\{\mu_{i}^{t+1},x_{i}^{t+1},\rho_{ij}\}$ of the $i$-th neuron to which the $j$-th intermediate unit is connected. Thus, we can regard the change in the synaptic strength $v_{jk}^{(\ell)}$ as being determined by the local interactions at the synapse on the $j$-th intermediate unit. Continuing with this line of argument, we can obtain even more realistic forms of learning rules for actual neural systems. However, we do not go into detail here, because the argument becomes quite complicated and is beyond the scope of the current study. _Initial Values of Model Parameters and Values of Learning Parameters Used in the Simulation_ : In the numerical simulation of our model of learning, we used initial values of the model parameters that results in behavior in which the animal primarily attempts to avoid negative reward, mimicking innate behavior of real animals. We set the values of the model parameters involved in the inputs to the movement-related neurons as shown in Fig.6. A neuron controlling motion in one of four directions receives connections with relatively strong positive weights, $\rho_{0}$, from a specialized intermediate unit (for example, from $\xi_{(4),1}^{t}$ to $x_{N}^{t}$). The intermediate unit receives connections from the environmental variables that take the values of the rewards within one step of the animal’s position, $y_{k}^{t}$ ($4\leq k\leq 12$), with the weight-values $v_{0}$, $-v_{0}$ and $0$, as illustrated in Fig.6. These initial values of the weight parameters make the neurons controlling motion take a value of 1 when relative amounts of the reward in the corresponding direction are large. We chose the other weight parameters with small random values in accordance with the following: $\rho_{ij}\sim[-0.05:0.05]\ (i\leq N_{2})$; $v_{ij}^{(\ell)}\sim[-0.05:0.05]\ (\ell=1,2)$; $\rho_{ij}=0$ if $N_{2}<i\leq N_{4}=N$ and $(i,j)\neq(N,1),(N-1,2),(N-2,1),(N-3,2)$; $v_{ij}^{(\ell)}=0$ ($\ell=3,4$ except the red and blue synaptic weights in Fig.6); $w_{ij}^{(\ell)}\sim[-0.05:0.05]\ (\ell=1,2)$; $w_{ij}^{(\ell)}=0\ (\ell=3,4)$; $h_{0}=\log 20$; $h_{j}^{(\ell)}=0$; $\kappa_{ij}\sim[-0.05:0.05]$; $z_{jk}^{(\ell)}\sim[-0.05:0.05]$; $u_{jk}^{(\ell)}\sim[-0.05:0.05]$; $q_{jk}^{(\ell)}\sim[-0.05:0.05]$; $m_{j}^{(\ell)}=0$; $m_{0}=0$. In the updates of the model parameters according to Eq.(51), we used the following (fixed) values of learning parameters: $\epsilon=3.0\times 10^{-5}$; $\tau=50$. <figure><img src="content_image/1504.03132/x6.png"><figcaption>Figure 6: (Color online). 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1708.05742	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 19695, "num_imgs": 4, "llama3_tokens_count": 5467 }	[ "content_image/1708.05742/Jacobi.png", "content_image/1708.05742/x1.png", "content_image/1708.05742/barparticles.png", "content_image/1708.05742/Vlos.png" ]	# New insights on the origin of the High Velocity Peaks in the Galactic Bulge J. G. Fernández-Trincado Departamento de Astronomía, Casilla 160-C, Universidad de Concepción, Concepción, Chile email: jfernandezt@astro-udec.cl and/or jfernandezt87@gmail.com A. C. Robin Institut Utinam, CNRS UMR 6213, Université Bourgogne-Franche-Comté, OSU THETA Franche-Comté, Observatoire de Besançon, BP 1615, 25010 Besançon Cedex, France E. Moreno Instituto de Astronomía, Universidad Nacional Autónoma de México, Apdo. Postal 70264, México D.F., 04510, México A. Pérez-Villegas Max-Planck-Institut für Extraterrestrische Physik, Gießenbachstraße, 85748 Garching, Germany B. Pichardo${}^{3}$ ###### Abstract We provide new insight on the origin of the cold high-V${}_{\rm los}$ peaks ($\sim$200 kms${}^{-1}$) in the Milky Way bulge discovered in the APOGEE commissioning data (Nidever et al. 2012). Here we show that such kinematic behaviour present in the field regions towards the Galactic bulge is not likely associated with orbits that build the boxy/peanut (B/P) bulge. To this purpose, a new set of test particle simulations of a kinematically cold stellar disk evolved in a 3D steady-state barred Milky Way galactic potential, has been analysed in detail. Especially bar particles trapped into the bar are identified through the orbital Jacobi energy $E_{J}$, which allows us to identify the building blocks of the B/P feature and investigate their kinematic properties. Finally, we present preliminary results showing that the high-V${}_{\rm los}$ features observed towards the Milky Way bulge are a natural consequence of a large-scale _midplane_ particle structure, which is unlikely associated with the Galactic bar. T proceeding of the SF2A 2017. 7 pages, 2 Fig he Galaxy, bulge, disk, kinematics and dynamics, galaxies structure, numerical methods ## 1 Introduction The discovery by Nidever et al. (2012) of cold high-velocity peaks ($\sim$ 200 km s${}^{-1}$) in the Apache Point Observatory Galactic Evolution Experiment (APOGEE) commissioning data across the Galactic bulge $l=$ {4, 14} and $b=$ {-2, 2} and confirmed by the High-Order Kinematic Moments by Zasowski et al. (2016), suggests that there may be a significant non-axisymmetric structure that dominates the bulge regions (e.g., Robin et al. 2012; Wegg & Gerhard 2013, for instance), which has turned the study and characterization of a B/P bulge (e.g., Portail et al. 2015; Simion et al. 2017, among others) into a very active research field. Most of models that attempt to explain the high-velocity peaks observed toward the Milky Way bulge suggest that these features are most likely bulge stars on bar orbits, i.e., orbits in 2:1 and/or higher order resonant family (see e.g., Aumer & Schönrich 2015; Molloy et al. 2015). Recently, alternative scenarios have been proposed that do not invoke any family of bar resonant orbits linked with the building blocks of the B/P feature. Additionally, it has been suggested that the high-velocity peaks may be the product of a kiloparsec-scale nuclear stellar disk in the Galactic bulge (Debattista et al. 2015). Also, the recent study by Li et al. (2014) suggests that these kinematics features might be an artifact due to small number statistics. With these issues in mind, we expect this preliminary contribution will help improve the current understanding on the origin of the cold high-velocity peaks. In this work, we qualitatively analyzed a set of numerical simulations of a synthetic Milky Way Galaxy made up of the superposition of many composite stellar populations already described and analyzed in Fernández-Trincado (7). Using numerical simulations from Fernández-Trincado (7), we began a pilot project aimed to provide an alternative scenario for the origin of high-V${}_{\rm los}$ feature in the bulge to look for possible orbital energy imprints of the cold high-V${}_{\rm los}$. <figure><img src="content_image/1708.05742/Jacobi.png"><figcaption>Figure 1: Face-on view of the simulated cold stellar thin disk in the inertialreference frame where the bar is at an angle of 20 degrees from the Sun-GCline of sight. Colors indicate ⟨EJ⟩ in units of 100 km2 s−2. The white dashedcircle indicates the corotation radius (6.5 kpc), which is in good agreementwith results from the literature to explain the Hercules group Pérez-Villegaset al. (2017); and the white circle marks the solar radius (8 kpc) and thepresent-day solar position (black star symbol). The black contours refer tothe surface density distribution for the entire sample of 1×106 particles int=15 Gyr.</figcaption></figure> ## 2 The Galactic Model We use the galactic dynamic software _GravPot16_ in order to carry out a comprehensive orbital study of particles in the inner region of the Milky Way. For a more detailed discussion about _GravPot16_, we refer the readers to a forthcoming paper (Fernández-Trincado et al. in preparation). Here we summarize the backbone of _GravPot16_. Using the new version of the Besançon Galaxy Model, in good agreement with many observations, we computed a semi-analytical steady-state 3D gravitational potential of the Milky Way, observationally and dynamically constrained. The model is primarily made up of the superposition of several composite stellar components, where the density profiles in cylindrical coordinates, $\rho_{i}$(R, Z), are the same as those proposed in Robin et al. (2003, 2012, 2014), i.e., a B/P bulge, a Hernquist stellar halo, seven stellar Einasto thin disks with spherical symmetry in the inner regions, two stellar sech${}^{2}$ thick disks, a gaseous exponential disk, and a spherical structure associated with the dark matter halo. A new formulation for the global potential, $\Phi$(R, Z), of this Milky Way density model, $\Sigma\rho_{i}$(R, Z), will be described in detail in a forthcoming paper (Fernández-Trincado, et al. in preparation). $\Phi$(R, Z) has been rescaled to the Sun’s galactocentric distance. The Sun is located at $R_{\odot}=$ 8.0 kpc, and the local rotation velocity is assumed to be $\Theta_{0}({\rm R_{\odot}})=244.5$ km s${}^{-1}$, given by Sofue (2015). Here, we briefly describe the bar’s structural parameters, such as recommended by Fernández-Trincado (7) from dynamical constraints using the BRAVA data set (Kunder et al. 2012): We assume a total mass for the bar of 1.1$\times$10${}^{10}$ M${}_{\odot}$, an angle of 20 degrees for the present-day orientation of the major axis of the bar and an angular velocity, $\Omega_{bar}=35$ km s${}^{-1}$ kpc${}^{-1}$, consistent with the recent estimate of Portail et al. (2015), and a cut-off radius $R_{c}=3.28$ kpc (e.g., Robin et al. 2012). Additionally, it should be noted that the non-axisymmetric configuration of our dynamic model has been extensively employed to predict stellar orbits (see Fernández-Trincado et al. 2016, 2017), and/or orbital parameters for a large set of APOGEE-TGAS sources (see Abolfathi et al. 2017; Tang et al. 2017). For a more detailed discussion, we refer the readers to a forthcoming paper (Anders et al. 3, 2). [FOOTNOTE:][ENDFOOTNOTE] <figure><img src="content_image/1708.05742/x1.png"><figcaption>Figure 2: Orbits viewed face-on (top), side-on (middle) and meridional(bottom) in the non-inertial reference frame where the bar is at rest. Column1 and column 2 show the typical orbital configuration found for bar-trappedparticles (EJ<EboundaryJ), while the column 3 and 4 show the orbitalconfiguration for not-bar-trapped particles (EJ>EboundaryJ). The top bluelabel indicates the orbital eccentricity and its respective Jacobi energy(EJ).</figcaption></figure> ## 3 Test particle simulations First, we ran controled particle simulations to mimic one of the cold stellar thin disk described in the Besan¸con population synthesis model (disk in the age range 7 to 10 Gyr; see e.g., Robin et al. 2003). To this purpose we use the _GravPot16_ code in its axisymmetric and non-axisymmetric configuration. We adopt a similar strategy as described in Romero-Gómez et al. (2015) and Martinez-Medina et al. (2016). The test particles are initially involved in a steady axisymmetric potential model over long integration time (in this work we adopt a integration time of 10 Gyr) to ensure that the initial disk particle distribution reaches a state of relaxation within the background potential. Then, the boxy bar structure grows adiabatically into the simulations during a period of time of 2 Gyr. Once the bar potential is introduced into the system, we increase the integration time during a period of time long enough ($>3$ Gyr) to avoid transient effects. The initial conditions for the particle velocities are assigned using the Besançon population synthesis model disc kinematics fitted to RAVE and TGAS data (see e.g., Robin et al. 2017). It is important to note that our initial conditions are based on locally self-consistent recipes, but it is not guaranteed to be fully self-consistent globally, and will thus be slightly relaxed before turning on the non-axisymmetric potential (e.g., Fernández-Trincado 7). Secondly, after 15 Gyr integration time in the above potential we record the Jacobi energy per unit mass, $E_{J}$†, in the bar frame for all particles in a box of $\pm$3.5 kpc $\times$ 2.5 kpc $\times$ 2 kpc, which is thought to have high chance to contain orbits trapped into the bar structure as illustrated in Fig. 1 and Fig. 2 (first and second column). With the Jacobi energy distribution in the box above mentioned we determined the boundary between bar-trapped particles ($E_{J}<E_{J}^{boundary}$) and not-bar-trapped particles ($E_{J}>E_{J}^{boundary}$) by identifying the trough in the Jacobi energy distribution ($E_{J}^{boundary}\sim$ -2.7$\times$10${}^{5}$ km${}^{2}$ s${}^{-2}$). The results are briefly described in §4. [FOOTNOTE:†][ENDFOOTNOTE] <figure><img src="content_image/1708.05742/barparticles.png"><figcaption>Figure 3: Kernel Density Estimate (KDE) smoothed distributions of bar-trappedparticles (left column) and not-bar-trapped particles (right column) of thesimulated cold stellar thin disk in the non-inertial reference frame where thebar is at rest. Density distribution for the entire sample viewed face-on(first row) and side-on (second row).</figcaption></figure> ## 4 Results and Concluding Remarks Figure 3 plots the Kernel Density Estimate (KDE) smoothed distributions for the bar-trapped particles (first column). In particular we note that the B/P feature is carried largely by particles having a Jacobi energy $E_{J}<E_{J}^{boundary}$. In our numerical simulations, all the building blocks of the B/P bulge structure are composite of different orbits existing at energies smaller than the boundary energy, $E_{J}<E_{J}^{boundary}$, in particular diverse resonant orbits (i.e., family of tube orbits; $x_{1}v_{1}$: banana orbits, etc) which generate a strong peanut shape at shorter radii on the side-on projection (column 1, row 2 in the same figure). This Galactic B/P structure accounts for $\sim$34% of the particles of the bulge (within $\sim 5$ kpc) (e.g., Fernández-Trincado 7). Figure 3 also plots the KDE smoothed distributions for the not-bar-trapped particles (second column), which do not show the B/P shape. The not-bar-trapped particles, those orbits existing at energies larger than the boundary energy, ($E_{J}>E_{J}^{boundary}$) in this model consist mostly of low eccentricity orbits, which dominate the mid-plane (see column 2 and 3 in Figure 2) and accounts for $\sim$66% of the particles of the bulge. ### An alternative explanation for the kinematics feature at high $V_{los}$ It is important to note that here we provide the kinematic predictions for orbits existing at energies smaller or greater than the boundary energy. Detailed azimuthal projections were already analyzed in Fernández-Trincado (7) confirming the presence of the cold high-V${}_{\rm los}$ peaks extending to Galactic longitude $l$$\sim$ 10 degree, which are absent a few kiloparsecs off the mid-plane, indicating that orbits with Jacobi energy (a mid-plane hosting more particles at $E_{J}>E_{J}^{boundary}$) responsible for the feature do not extend this far off-plane, as also shown in second column in Figure 3. Figure 4 plots the predicted line-of-sight velocity distributions (LOSVDs –here called V${}_{\rm los}$) in the Galacto-centric restframe. At $E_{J}<E_{J}^{boundary}$ the V${}_{\rm los}$ distribution has a single peak dominated by bar-trapped-particles, hosting more particles at $V_{los}<$ 150 km s${}^{-1}$. There are also few particles that have high V${}_{\rm los}$ ($\sim$ 200 km s${}^{-1}$) and are likely associated with the high-velocity tail of the resonant bar-supporting 2:1 orbits (see Molloy et al. 2015). At $E_{J}>E_{J}^{boundary}$ the V${}_{\rm los}$ have developed two peaks, with particles moving at significantly larger velocities ($~{}200$ km s${}^{-1}$) in the mid-plane, dominated by not-bar-trapped particles, but remain well below the circular velocity of the galaxy. These two high velocity peaks are more prominent than the low-V${}_{\rm los}$ peak developed by bar-trapped particles. ### Conclusion We have made an attempt to explain the presence of the cold high velocity peaks in the bulge. It is important to note, that we account for the composite nature of the bulge in our simulations. The dependence of V${}_{\rm los}$ with $l$ and $b$ has not been shown in the present work, but extensively studied by Fernández-Trincado (7). The right panel of Figure 4 shows color-coded maps of the average $V_{los}$, $\langle V_{los}\rangle$, for the entire sample of our simulated cold stellar thin disk in Galactic coordinates. In a similar manner as in Debattista et al. (2015) our numerical approach is capable of producing the peak velocities at orbit tangent points with the characteristic _winged_ pattern of the velocity fields. Lastly, we conclude that the most natural interpretation of the high velocity features towards the Galactic bulge is that they are likely not dominated by orbits at $E_{J}<E_{J}^{boundary}$ that build the B/P bulge, but may be a consequence of families of orbits at $E_{J}>E_{J}^{boundary}$ and low orbital eccentricities in the mid-plane that do not support the bar structure. Our Milky Way potential model fine-tuned to observations is able to explain the velocity distributions in most APOGEE fields in the bulge, without invoking the presence of any nuclear disk in the inner $\sim$ 1 kpc as pointed out in Debattista et al. (2015). The advantage of our numerical approach is that the test particles have evolved in a realistic Milky Way potential inheriting the information on both density and kinematics, and the particles in statistical equilibrium with the potential imposed (e.g., Romero-Gómez et al. 2015; Martinez-Medina et al. 2016; Fernández-Trincado 7). It should be noticed that we find very similar $V_{los}$ distributions to those in APOGEE, without any adjustment parameters, but without applying the observation selection function. Hence we shall verify this point in the near future. The high precision of the _Gaia_ mission will provide the 6D phase space needed to confirm our orbital interpretations and to compute the orbital Jacobi energy beyond $\sim$ 5 kpc from the Sun. <figure><img src="content_image/1708.05742/Vlos.png"><figcaption>Figure 4: Left: Vlos histograms normalised to unit peak showing a dual-peakstructure for the not-bar-trapped particles (black dashed line) and anunimodal distribution for the bar-trapped particles (blue line), using 20 kms−1 binning. Right: Kinematics map of the simulated cold stellar thin disk inGalactic coordinates for the entire sample of 1×106 particles. The whitecircles and black contours levels are identical to those in Figure 1. Colorsindicate ⟨Vlos⟩.</figcaption></figure> ###### Acknowledgements. We thank Merce Romero Gómez, Fran¸coise Combes and Famaey Benoit for comments on an earlier version of this paper. J.G.F-T gratefully acknowledges partial financial support from the SF2A in order to attend the SF2A-2017 meeting held in Paris in July 2017, and the Chilean BASAL Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA) grant PFB- 06/2007. We also acknowledge the support of the UTINAM Institute of the Université de Franche-Comté, Région de Franche-Comté and Institut des Sciences de l’Univers (INSU) for providing HPC resources on the Cluster Supercomputer Mésocentre de calcul de Franche-Comté. E.M, A.P.V and B.P acknowledge support from UNAM/PAPIIT grant IN105916 and IN114114. Funding for the _GravPot16_ software has been provided by the Centre national d’études spatiale (CNES) through grant 0101973 and UTINAM Institute of the Université de Franche-Comte, supported by the Region de Franche-Comte and Institut des Sciences de l’Univers (INSU). The _GravPot16_ Web site is at https://fernandez-trincado.github.io/GravPot16/. ## References * Abolfathi et al. (2017) Abolfathi, B., Aguado, D. S., Aguilar, G., et al. 2017, ArXiv e-prints * Anders et al. (2017a) Anders, F., Queiroz, A. B., Chiappini, C., et al. 2017a, in preparation * Anders et al. (2017b) Anders, F., Queiroz, A. B., Chiappini, C., et al. 2017b, in IAU Symposium, Vol. 334, Rediscovering our Galaxy, ed. C. Chiappini, I. Minchev, E. Starkenburg, & M. Valentini * Aumer & Schönrich (2015) Aumer, M. & Schönrich, R. 2015, , 454, 3166 * Debattista et al. (2015) Debattista, V. P., Ness, M., Earp, S. W. F., & Cole, D. R. 2015, , 812, L16 * Fernández-Trincado et al. (2017) Fernández-Trincado, J. G., Zamora, O., Garcia-Hernandez, D. A., et al. 2017b, ArXiv e-prints * Fernández-Trincado (2017a) Fernández-Trincado, J. G. 2017a, Université Bourgogne Franche-Comté Phd Thesis, 1, 174 * Fernández-Trincado et al. (2016) Fernández-Trincado, J. G., Robin, A. C., Moreno, E., et al. 2016, , 833, 132 * Kunder et al. (2012) Kunder, A., Koch, A., Rich, R. M., et al. 2012, , 143, 57 * Li et al. (2014) Li, Z.-Y., Shen, J., Rich, R. M., Kunder, A., & Mao, S. 2014, , 785, L17 * Martinez-Medina et al. (2016) Martinez-Medina, L. A., Pichardo, B., Moreno, E., Peimbert, A., & Velazquez, H. 2016, , 817, L3 * Molloy et al. (2015) Molloy, M., Smith, M. C., Evans, N. W., & Shen, J. 2015, , 812, 146 * Nidever et al. (2012) Nidever, D. L., Zasowski, G., Majewski, S. R., et al. 2012, , 755, L25 * Pérez-Villegas et al. (2017) Pérez-Villegas, A., Portail, M., Wegg, C., & Gerhard, O. 2017, , 840, L2 * Pichardo et al. (2004) Pichardo, B., Martos, M., & Moreno, E. 2004, , 609, 144 * Portail et al. (2015) Portail, M., Wegg, C., Gerhard, O., & Martinez-Valpuesta, I. 2015, , 448, 713 * Robin et al. (2017) Robin, A. C., Bienaymé, O., Fernández-Trincado, J. G., & Reylé, C. 2017, ArXiv e-prints * Robin et al. (2012) Robin, A. C., Marshall, D. J., Schultheis, M., & Reylé, C. 2012, , 538, A106 * Robin et al. (2003) Robin, A. C., Reylé, C., Derrière, S., & Picaud, S. 2003, , 409, 523 * Robin et al. (2014) Robin, A. C., Reylé, C., Fliri, J., et al. 2014, , 569, A13 * Romero-Gómez et al. (2015) Romero-Gómez, M., Figueras, F., Antoja, T., Abedi, H., & Aguilar, L. 2015, , 447, 218 * Simion et al. (2017) Simion, I. T., Belokurov, V., Irwin, M., et al. 2017, ArXiv e-prints * Sofue (2015) Sofue, Y. 2015, , 67, 75 * Tang et al. (2017) Tang, B., Fernández-Trincado, J. G., Geisler, D., et al. 2017, submitted, 1, 1 * Wegg & Gerhard (2013) Wegg, C. & Gerhard, O. 2013, , 435, 1874 * Zasowski et al. (2016) Zasowski, G., Ness, M. K., García Pérez, A. E., et al. 2016, , 832, 132
1903.09222	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 76259, "num_imgs": 11, "llama3_tokens_count": 25297 }	[ "content_image/1903.09222/x1.png", "content_image/1903.09222/x4.png", "content_image/1903.09222/x7.png", "content_image/1903.09222/x8.png", "content_image/1903.09222/x9.png", "content_image/1903.09222/x10.png", "content_image/1903.09222/x11.png", "content_image/1903.09222/x12.png", "content_image/1903.09222/x13.png", "content_image/1903.09222/x14.png", "content_image/1903.09222/x15.png" ]	# Gravitational form factors of light mesons Adam Freese afreese@anl.gov Argonne National Laboratory, Lemont, Illinois 60439, USA Ian C. Cloët icloet@anl.gov Argonne National Laboratory, Lemont, Illinois 60439, USA ###### Abstract We calculate the gravitational form factors of the pion, sigma meson, and rho meson in the Nambu–Jona-Lasinio (NJL) model of quantum chromodynamics. The canonical energy-momentum tensor (EMT) is used in their derivation, allowing the possibility of an antisymmetric contribution when the hadron has intrinsic spin. We show that the asymmetric graviton vertex arising from the canonical EMT satisfies a simpler Ward-Takahashi identity (WTI) than the symmetric graviton vertex of the Belinfante EMT. The necessity of fully dressing the graviton vertex through the relevant Bethe-Salpeter equation is demonstrated for observing both the WTI and a low-energy pion theorem. Lastly, we calculate static moments of the meson EMT decompositions, obtaining predictions for the meson mass radii. We find light cone mass radii of 0.27 fm for the pion, 0.32 fm for the sigma, and 0.39 fm for the rho. For the pion and rho, these are smaller than the light cone charge radii, respectively 0.51 fm and 0.45 fm, while we have a sigma charge radius of zero. Our light cone pion mass radius agrees with a phenomenological extraction from KEKB data. ## I Introduction In recent years, the energy-momentum tensor (EMT) of hadrons has become an increasingly popular object of study in our quest to better understand hadronic structure—and, in turn, quantum chromodynamics (QCD) itself. (See Refs. Leader and Lorcé (2014); Polyakov and Schweitzer (2018) and references therein.) Understanding the EMT can help address such fundamental questions as where does the mass of the proton come from, and where is all of its spin? It also opens new avenues for exploration, including not just spatial distributions of energy, momentum, and angular momentum Polyakov (2003); Leader and Lorcé (2014); Lorcé (4); Lorcé _et al._ (2018), but also the distribution of pressure and shear forces Polyakov and Shuvaev (2002); Polyakov (2003); Polyakov and Schweitzer (2018); Lorcé _et al._ (2019). It has been remarked that the EMT introduces a new intrinsic global quantity in addition to mass and spin, called the “D-term” Polyakov and Shuvaev (2002); Polyakov and Schweitzer (2018). The D-term, which quantifies the strength of the forces binding the hadron together, is not constrained by conservation laws nor by representation theory of the Lorentz group. However, in certain cases it may be constrained by other considerations—as it is with Nambu-Goldstone bosons Novikov and Shifman (1981); Voloshin and Zakharov (1980); Polyakov and Weiss (1999). Since the pion plays a vital role in QCD as the Nambu-Goldstone boson of dynamical chiral symmetry breaking (DCSB) Nambu and Jona-Lasinio (11, 12), understanding how its D-term comes to be constrained may play as important a role in fully grasping QCD as understanding the origin of proton mass and spin. DCSB has come to be understood as one of the central features of QCD, and is intimately involved in the observed mass of hadrons Bashir _et al._ (2012). Accordingly, understanding the pion is vital to understanding QCD, and pion structure has long been a major topic of study in hadron physics. Its EMT in particular has recently been computed on the lattice Brommel (2007); Shanahan and Detmold (2019). Additionally, it has become possible through dispersive analysis of KEKB data for the reaction $\gamma^{}\gamma\rightarrow\pi^{0}\pi^{0}$ to extract an empirical parametrization of the pion EMT in the timelike region Kumano _et al._ (2018). The prospect of comparison to empirical values makes model calculations of the pion EMT especially important now. In this work, we investigate the EMT of the pion and of other mesons in the Nambu–Jona-Lasinio (NJL) model Vogl and Weise (1991); Klevansky (1992); Hatsuda and Kunihiro (1994) of QCD. The NJL model is a Poincaré covariant quantum field theory that successfully reproduces low-energy properties of QCD such as DCSB. Additionally, since the model does not contain gluons, we can defer issues regarding gauge invariance of the EMT. These properties make the NJL model an ideal framework in which to investigate the EMT of the pion, especially aspects of the EMT that arise from DCSB. The sigma and rho mesons are also investigated as a point of contrast, with the latter serving to illustrate the ways in which spin manifests in the EMT. This work is organized into the following sections. We first give an overview of the NJL model in Sec. II. We then develop the formalism needed to calculate the meson EMTs in Sec. III, where the formalism is cast as a study of quark-graviton and meson-graviton interactions. In Sec. IV, we develop the formalism needed to calculate the axial form factors of the rho meson, which are needed to numerically demonstrate a correspondence between certain gravitational and axial form factors. Results for the meson EMTs are given in Sec. V, where we also explore implications of our results. Finally, we conclude in Sec. VI and give a brief outlook, including the prospects of performing similar calculations in QCD. ## II The NJL model The Nambu–Jona-Lasinio (NJL) model, originally proposed as a theory of elementary nucleons Nambu and Jona-Lasinio (11, 12), is used as a low-energy effective field theory that models the dynamical chiral symmetry breaking of quantum chromodynamics through a four-fermi contact interaction Vogl and Weise (1991); Klevansky (1992); Hatsuda and Kunihiro (1994). It has successfully been applied to modeling the physical properties of both mesons Vogl and Weise (1991); Klevansky (1992); Cloët _et al._ (2014); Ninomiya _et al._ (2017) and baryons Ishii _et al._ (22, 23, 1995); Cloët _et al._ (2014). Accordingly, we employ the NJL model in this work to calculate the energy-momentum tensor of light mesons. The two-flavor NJL model Lagrangian we use is Cloët _et al._ (2014): \[\mathcal{L} =\overline{\psi}(i\overleftrightarrow{\not{\partial}}-\hat{m}) \psi+\frac{1}{2}G_{\pi}[(\overline{\psi}\psi)^{2}-(\overline{\psi}\gamma_{5} \bm{\tau}\psi)^{2}]-\frac{1}{2}G_{\omega}(\overline{\psi}\gamma_{\mu}\psi)^{2} -\frac{1}{2}G_{\rho}[(\overline{\psi}\gamma_{\mu}\bm{\tau}\psi)^{2}+(\overline {\psi}\gamma_{\mu}\gamma_{5}\bm{\tau}\psi)^{2}]\] \[+\frac{1}{2}G_{\eta}[(\overline{\psi}\bm{\tau}\psi)^{2}-( \overline{\psi}\gamma_{5}\psi)^{2}]-\frac{1}{2}G_{f}(\overline{\psi}\gamma_{ \mu}\gamma_{5}\psi)^{2}-\frac{1}{2}G_{T}[(\overline{\psi}i\sigma^{\mu\nu}\psi) ^{2}-(\overline{\psi}i\sigma^{\mu\nu}\bm{\tau}\psi)^{2}]\,,\] (1) where $\hat{m}=\mathrm{diag}[m_{u},m_{d}]$ is the current quark mass matrix (where we take $m_{u}=m_{d}\equiv m$ in this work), $\tau_{i}$ are the isospin matrices, and the $G_{i}$ are four-fermi coupling constants. The expression in Eq. (1) is invariant under $\mathrm{SU}(2)_{V}$ and $\mathrm{SU}(2)_{A}$ transformations, but $\mathrm{U}(1)_{A}$ symmetry requires the additional constraints $G_{\eta}=G_{\pi}$ and $G_{T}=0$, which we will assume in this work. We take the Lagrangian in Eq. (1) to symbolically represent a Lagrangian that has already been Fierz symmetrized, so that only direct terms need to be calculated in the interaction kernel (See Ref. Ishii _et al._ (1995) for a detailed description of the Fierz symmetrization procedure.) The NJL model dynamically generates a dressed quark mass $M$ when $G_{\pi}>G_{\mathrm{critical}}$. This dynamical mass generation is described by the gap equation: \[M=m+2iG_{\pi}(2N_{c})\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}} \mathrm{Tr}_{D}\left[S(k)\right]=m+8iG_{\pi}(2N_{c})\int\frac{\mathrm{d}^{4}k} {(2\pi)^{4}}\frac{M}{k^{2}-M^{2}+i0}\,,\] (2) where the trace is over the Clifford matrix structure (with $2N_{c}$ having come from the color and isospin traces already). <figure><img src="content_image/1903.09222/x1.png"><figcaption>Figure 1: Diagrammatic depiction of the inhomogeneous Bethe-Salpeter equationfor the two-body T-matrix in the NJL model.</figcaption></figure> Mesons appear in the NJL model as bound state poles in the quark-antiquark T-matrix, which itself can be found from solving an inhomogeneous Bethe-Salpeter equation, depicted diagrammatically in Fig. 1. Poles corresponding to mesons with the quantum numbers of various mesons are present in the NJL T-matrix. Our interest in this work lies primarily with the pion, sigma, and rho, but we also consider the $f_{1}$ meson as a means of determining the coupling constant $G_{f}$. The T-matrices for these four mesons can be written¹: [FOOTNOTE:1][ENDFOOTNOTE] \[i{\mathcal{T}_{\pi}(q)}_{\alpha\beta,\gamma\delta} =\frac{-2iG_{\pi}}{1+2G_{\pi}\Pi_{PP}(q^{2})}(\gamma_{5}\tau_{i}) _{\alpha\beta}(\gamma_{5}\tau_{i})_{\gamma\delta}\] (3) \[i{\mathcal{T}_{\sigma}(q)}_{\alpha\beta,\gamma\delta} =\frac{2iG_{\pi}}{1-2G_{\pi}\Pi_{SS}(q^{2})}(1)_{\alpha\beta}(1)_ {\gamma\delta}\] (4) \[i{\mathcal{T}_{\rho}(q)}_{\alpha\beta,\gamma\delta} =\frac{-2iG_{\rho}}{1+2G_{\rho}\Pi_{VV}(q^{2})}\left[g^{\mu\nu}+2 G_{\rho}\Pi_{VV}(q^{2})\frac{q^{\mu}q^{\nu}}{q^{2}}\right](\gamma_{\mu}\tau_{i })_{\alpha\beta}(\gamma_{\nu}\tau_{i})_{\gamma\delta}\] (5) \[i{\mathcal{T}_{f_{1}}(q)}_{\alpha\beta,\gamma\delta} =\frac{-2iG_{f}}{1+2G_{f}\Pi_{AA}^{(T)}(q^{2})}\left[g^{\mu\nu}+2 G_{f}\Pi_{AA}^{(T)}(q^{2})\frac{q^{\mu}q^{\nu}}{q^{2}}\right](\gamma_{\mu} \gamma_{5})_{\alpha\beta}(\gamma_{\nu}\gamma_{5})_{\gamma\delta}\,,\] (6) where the bubble diagrams are defined in App. A. Since the mesons appear as poles in these T-matrices, their masses are given by the conditions: \[1+2G_{\pi}\Pi_{PP}(m_{\pi}^{2}) =0\] (7) \[1-2G_{\pi}\Pi_{SS}(m_{\pi}^{2}) =0\] (8) \[1+2G_{\rho}\Pi_{VV}(m_{\rho}^{2}) =0\] (9) \[1+2G_{f}\Pi_{AA}^{(T)}(m_{f_{1}}^{2}) =0\,.\] (10) The residues at the poles can be interpreted as effective quark-meson coupling constants, which allow us to find the properly normalized homogeneous Bethe-Salpeter vertex functions: \[\Gamma_{\pi}^{i} =\sqrt{Z_{\pi}}\gamma_{5}\tau_{i}\] (11) \[\Gamma_{\sigma}^{i} =\sqrt{Z_{\sigma}}\] (12) \[\Gamma_{\rho}^{i} =\sqrt{Z_{\rho}}\gamma^{\mu}\tau_{i}\,,\] (13) where \[Z_{\pi}^{-1}=-\frac{\partial}{\partial q^{2}}\Pi_{PP}(q^{2}) \bigg{\|}_{q^{2}=m_{\pi}^{2}}\] (14) \[Z_{\sigma}^{-1}=+\frac{\partial}{\partial q^{2}}\Pi_{SS}(q^{2}) \bigg{\|}_{q^{2}=m_{\sigma}^{2}}\] (15) \[Z_{\rho}^{-1}=-\frac{\partial}{\partial q^{2}}\Pi_{VV}(q^{2}) \bigg{\|}_{q^{2}=m_{\rho}^{2}}\,.\] (16) The NJL model is non-renormalizable, owing to the four-fermi contact interaction. To fully define the model, it is necessary to introduce a regularization scheme. We follow Refs. Ebert _et al._ (1996); Hellstern _et al._ (1997); Cloët _et al._ (2014) in using proper time regularization, with both an infrared and an ultraviolet regulator. The regularization proceeds formally through the substitution: \[\frac{1}{X^{n}}=\frac{1}{(n-1)!}\int_{0}^{\infty}\mathrm{d}\tau\, \tau^{n-1}e^{-\tau X}\rightarrow\frac{1}{(n-1)!}\int_{1/\Lambda_{\mathrm{UV}}^ {2}}^{1/\Lambda_{\mathrm{IR}}^{2}}\mathrm{d}\tau\,\tau^{n-1}e^{-\tau X}\,.\] (17) Only the UV regulator is necessary to make the model finite, but the presence of an IR regulator ensures the two-body T-matrix is always real, and prevents the decay of mesons into two quarks. As is customary with non-remormalizable theories, these regulators are kept finite as additional model parameters. We adopt the NJL model parameters used in previous work Cloët _et al._ (2014), along with a value for $G_{f}$ which produces an $f_{1}$ pole in the T-matrix with mass 1.28 GeV. The model parameters we use are given in Tab. 1. ΛIR \| ΛUV \| M \| Gπ=Gη \| Gρ \| Gω \| Gf \| GT ---\|---\|---\|---\|---\|---\|---\|--- 0.240 \| 0.645 \| 0.4 \| 19.0 \| 11.0 \| 10.4 \| 0.82 \| 0 Table 1: NJL model parameters used in this work. All but Gf are adopted from Ref. Cloët _et al._ (2014). Gf is determined by requiring an f1 pole at mf1=1.28 GeV in the NJL T-matrix. Couplings are in units GeV−2, while the dressed quark mass M and regulators are in units GeV. ## III Gravitation and the EMT The energy-momentum tensor (EMT) has long had strong ties to gravitation. The equivalence between inertial and gravitational mass has long suggested that energy is the charge upon which gravitation acts, and this equivalence principle is currently canonized as one of the central premises of general relativity (and its extensions, such as Einstein-Cartan theory Cartan (1923, 1924)). Accordingly, it is helpful in theoretical investigations of the EMT—even if gravitation is not the intended topic of study—to consider the physics of graviton coupling. If the EMT is the current upon which gravitons act, then one can constrain the EMT of a system of interest using traditional field theoretic methods that have been applied to other currents such as the electromagnetic current. Just as with coupling to a photon, one can obtain a Ward-Takahashi identity for graviton coupling and solve Dyson-Schwinger equations for the interaction between a graviton and a fully dressed field in a strongly-interacting, non-perturbative regime. ### Gravitational Ward-Takahashi identity The gravitational Ward-Takahashi identity (GWTI) is given by Brout and Englert (1966): \[\Delta_{\mu}\Gamma^{\mu\nu}_{G}(p^{\prime},p)=p^{\nu}S^{-1}(p^{ \prime})-p^{\prime\nu}S^{-1}(p)\,.\] (18) This relation has applicability to fields of general spin if the canonical EMT is used as the source of the gravitational field. A proof of the general validity of Eq. (18) can be found in App. B. The canonical EMT for which Eq. (18) has general applicability naturally arises as the Noether current associated with spacetime translation symmetry. This current is not symmetric in its indices for particles with intrinsic spin Leader and Lorcé (2014); Lorcé (30); Florkowski and Ryblewski (2018), as a consequence of the generalized angular momentum \[M^{\mu\alpha\beta}=x^{\alpha}T^{\mu\beta}-x^{\beta}T^{\mu\alpha} +S^{\mu\alpha\beta}\] (19) being a conserved quantity—in particular, the Noether current associated with Lorentz transformations. Here $S^{\mu\alpha\beta}$ is the intrinsic generalized angular momentum, and as a consequence of the conservation laws $\partial_{\mu}M^{\mu\alpha\beta}=0$ and $\partial_{\mu}T^{\mu\nu}=0$, one has $T^{\alpha\beta}-T^{\beta\alpha}=-\partial_{\mu}S^{\mu\alpha\beta}$. For particles with spin, this is not expected to be zero, so the EMT is not expected to be symmetric. It is possible to obtain a symmetric, “Belinfante-improved” EMT by adding the divergence of a superpotential to the EMT derived through Noether’s theorem Leader and Lorcé (2014). The resulting tensor is also a conserved current, but there is no guarantee that it encodes exactly the same physical properties as the Noether current associated with spacetime translations. It has been observed that adding a total derivative to the EMT can in fact change the values of measurable quantities such as the D-term Hudson and Schweitzer (2017). We thus choose not to add any total derivatives to the EMT in this work. One reason for preferring the Belinfante-improved EMT over the canonical EMT is that general relativity assumes the EMT to be symmetric, and spacetime to accordingly be torsion-free. However, we do not a priori know whether spacetime is really torsion-free, and whether gravitons can couple to the antisymmetric component of the EMT². Since in this work we consider graviton coupling only as a theoretical means of developing the formalism for calculating the EMT—which is empirically accessed through other means, such as DVCS Ji (1997) and hadron pair production in diphoton collisions Kumano _et al._ (2018)—we find it most prudent to consider the graviton as capable of containing torsion and proceed to use the canonical, asymmetric EMT. [FOOTNOTE:2][ENDFOOTNOTE] As remarked above, one major consequence of using the canonical rather than the Belinfante EMT is that Eq. (18) holds in general. One can readily confirm this for simple examples of particles with spin, such as an elementary free fermion, for which the gravitational vertex is: \[\vbox{\hbox{\includegraphics[scale=0.5]{graviton_qq.pdf}}}=\gamma _{G}^{\mu\nu}\left(k+\frac{\Delta}{2},k-\frac{\Delta}{2}\right)=\gamma^{\mu}k^ {\nu}-g^{\mu\nu}(\not{k}-m)\,.\] (20) On the other hand, the symmetrization of the vertex $\gamma_{G}^{\{\mu\nu\}}$—which arises from using the Belinfante EMT as the gravitating current in the spin-half case—does not obey Eq. (18). In fact, several authors DeWitt (1967); Bessler _et al._ (1969) have found that the symmetrized graviton vertex obeys a different Ward-Takahashi identity with a more complicated structure: \[\Delta_{\mu}\Gamma^{\{\mu\nu\}}_{G}(p^{\prime},p)=p^{\nu}S^{-1}(p ^{\prime})-p^{\prime\nu}S^{-1}(p)+\frac{1}{2i}\Delta_{\mu}\left[S^{-1}(p^{ \prime})\Sigma^{\mu\nu}-\Sigma^{\mu\nu}S^{-1}(p)\right]\,,\] (21) where $\Sigma^{\mu\nu}$ is the generator of Lorentz transforms. We consider the fact that it obeys a simpler WTI one of the virtues of the canonical EMT. ### Graviton-quark vertices in NJL model Application of Noether’s theorem to the full NJL model Lagrangian of Eq. (1) gives the following EMT: \[T^{\mu\nu}(x)=\bar{\psi}(x)(i\gamma^{\mu}\overleftrightarrow{ \partial}^{\nu})\psi(x)-g^{\mu\nu}\bar{\psi}(x)(i\overleftrightarrow{\not{ \partial}}-m)\psi(x)-\frac{1}{2}g^{\mu\nu}\sum_{\Omega}G_{\Omega}\big{(}\bar{ \psi}(x)\Omega\psi(x)\big{)}\big{(}\bar{\psi}(x)\Omega\psi(x)\big{)}\,,\] (22) which contains the non-linear contact interaction terms. The presence of a four-fermi interaction term in the EMT entails the existence of a five-point vertex function (4 quarks and 1 graviton line) in addition to the usual three-point vertex (2 quarks and 1 graviton). At the level of truncation we are considering here, it is not necessary to dress the five-point vertex. We do not dress the four-fermi contact interaction, after all, and it would be inconsistent to dress the graviton’s interaction with the contact interaction while not dressing the interaction itself. The effective Feynman rule for the five-point vertex can be read off from the EMT in Eq. (22) directly: \[\vbox{\hbox{\includegraphics[scale=0.5]{graviton_qqqq.pdf}}}= \gamma_{Gqqqq}^{\mu\nu}=-g^{\mu\nu}\sum_{\Omega}2G_{\Omega}\Omega\otimes\Omega\,.\] (23) We shall see presently that the existence of this five-point vertex has non-trivial consequences, including both a vacuum condensate contribution to the Bethe-Salpeter equation for the three point vertex (see second-from-the-right diagram of Fig. 2), and the existence of a “bicycle diagram” in the calculation of bound state EMTs (see rightmost diagram of Fig. 3). Crucially, these diagrams are both necessary for energy-momentum conservation to be observed. <figure><img src="content_image/1903.09222/x4.png"><figcaption>Figure 2: Inhomogeneous Bethe-Salpeter equation for the dressed quark-graviton vertex.</figcaption></figure> For the fully-dressed three-point vertex, we must solve an inhomogeneous Bethe-Salpeter equation, depicted in Fig. 2. In contrast to the photon vertex BSE, there are two driving terms. The second diagram from the right arises because it is possible for the five-point vertex to contribute to the dressing of the three-point vertex. Most interestingly, this new driving term is directly proportional to the vacuum condensate, which—as a consequence of the gap equation, Eq. (2)—is proportional to the mass dressing $(M-m)$. In particular, we have: \[\vbox{\hbox{\includegraphics[scale=0.5]{graviton_qq_condensate. pdf}}}=2iG_{\pi}(2N_{c})g^{\mu\nu}\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}} \mathrm{Tr}_{D}\left[S(k)\right]=g^{\mu\nu}(M-m)\,,\] (24) meaning that the two driving terms together contribute $\gamma^{\mu}k^{\nu}-g^{\mu\nu}(\not{k}-M)$ to the dressed graviton vertex. When we take into consideration that the NJL model propagator takes the same functional form as the bare propagator, but with the current quark mass replaced by the dressed mass, we observe the remarkable property that the driving terms in the graviton vertex BSE already satisfy the gravitational WTI on their own. This means the rightmost diagram in Fig. 2 must contribute a quantity transverse to the momentum transfer $\Delta$. Additionally, since the NJL interaction kernel contains no explicit momentum dependence and the $k$ dependence in the dressed kernel is integrated over in the loop, the rightmost diagram can only depend on $\Delta$. These considerations, and the fact that the EMT is $\mathsf{P}$ and $\mathsf{T}$ even, gives us the following most general form for the dressed three-point vertex: \[\vbox{\hbox{\includegraphics[scale=0.5]{graviton_qq_dressed.pdf}} }=\Gamma_{Gqq}^{\mu\nu}\left(k+\frac{\Delta}{2},k-\frac{\Delta}{2}\right)= \gamma^{\mu}k^{\nu}-g^{\mu\nu}(\not{k}-M)+\frac{\Delta^{\mu}\Delta^{\nu}- \Delta^{2}g^{\mu\nu}}{4M}C_{Q}(t)+\frac{i\epsilon^{\mu\nu\Delta\sigma}\gamma_{ \sigma}\gamma_{5}}{4}D^{\prime}_{Q}(t)\,.\] (25) The functions $C_{Q}(t)$ and $D^{\prime}_{Q}(t)$ can be determined by algebraically solving the BSE. We find: \[C_{Q}(t) =\frac{-8G_{\pi}\Pi_{SG}(t)}{1-2G_{\pi}\Pi_{SS}(t)}\] (26) \[D^{\prime}_{Q}(t) =\frac{2G_{f}\Pi_{AA}^{(T)}(t)}{1+2G_{f}\Pi_{AA}^{(T)}(t)}\,,\] (27) with the bubbles given in App. A. Remarkably, $C_{Q}(t)$ has a pole in the timeline region when $t=m_{\sigma}^{2}$, and $D_{Q}^{\prime}(t)$ has a timelike pole when $t=m_{f_{1}}^{2}$, by comparison to the pole conditions in Eqs. (8,10). ### EMT decomposition and gravitational form factors Now that we have constructed the necessary formalism, we proceed to consider matrix elements of the NJL model EMT between momentum and spin eigenstates of a particular hadron, viz., $\langle p^{\prime},\lambda^{\prime}\|T_{\mu\nu}(0)\|p,\lambda\rangle$. For conciseness, we refer to this matrix element as the EMT of the hadron. <figure><img src="content_image/1903.09222/x7.png"><figcaption>Figure 3: Diagrams contributing to meson-graviton coupling.</figcaption></figure> To begin, we must consider three diagrams when calculating the EMT of a meson in the NJL model. These diagrams are given in Fig. 3. The first two (triangle) diagrams are completely analogous to the diagrams contributing to the electromagnetic or axial vector current. The third (bicycle) diagram is new to graviton interactions, and is a consequence of the equivalence principle. The four-fermi contact interaction is a sort of potential energy in the NJL model, and gravitation couples to all energy in exactly the same way. The EMT of any hadron can be decomposed into a finite number of independent algebraic structures, each multiplied by a Lorentz-invariant function only of the invariant momentum transfer $t=(p^{\prime}-p)^{2}$. These functions are called gravitational form factors (GFFs), and are analogous to the electromagnetic form factors appearing in decompositions of the electromagnetic current $\langle p^{\prime},\lambda^{\prime}\|j_{\mu}(0)\|p,\lambda\rangle$. Like with electromagnetic form factors, the number of GFFs depends on the spin of the hadron, and this number increases with increasing spin. The most general form the spin-zero EMT can take for a particular parton flavor $a$ is Polyakov and Schweitzer (2018): \[\langle p^{\prime}\mid T^{a}_{\mu\nu}(0)\mid p\rangle =2P_{\mu}P_{\nu}A_{a}(t)+\frac{1}{2}(\Delta_{\mu}\Delta_{\nu}- \Delta^{2}g_{\mu\nu})C_{a}(t)+2m_{\pi}^{2}\bar{c}_{a}(t)g_{\mu\nu}\,,\] (28) where $a=q,g$, $P=(p+p^{\prime})/2$ is the average momentum between initial and final states, and $\Delta=p^{\prime}-p$ is the momentum transfer. The following sum rules follow from conservation of energy and momentum: \[\sum_{a=q,g}\bar{c}_{a}(t)=0\] (29) \[\sum_{a=q,g}A_{a}(0)=1\,.\] (30) Since the NJL model does not contain gluons, the former sum rule translates for us to $\bar{c}_{q}(t)=0$. Another rule, following from a low-energy pion theorem Novikov and Shifman (1981); Voloshin and Zakharov (1980); Polyakov and Weiss (1999), is: \[\lim_{m_{\pi}\to 0}\sum_{a=q,g}C_{a}(0)=-1\,.\] (31) This is a rule that holds exactly only in the chiral limit, and is a consequence of the pion being the Nambu-Goldstone boson of chiral symmetry breaking. Since the NJL model is a model of dynamical chiral symmetry breaking, we should observe Eq. (31) when taking the pion mass to zero. Moreover, one expects a quantity close to (although not exactly) $-1$ even at the physical pion mass. The most general form the spin-one EMT can take is Cosyn _et al._ (2019); Polyakov and Sun (2019): \[\langle p^{\prime},\lambda^{\prime}\mid T^{a}_{\mu\nu}(0)\mid p,\lambda\rangle =-2P_{\mu}P_{\nu}\left[(\epsilon^{\prime}\epsilon)\mathcal{G}^{a }_{1}(t)-\frac{(\Delta\epsilon^{\prime})(\Delta\epsilon)}{2m_{\rho}^{2}} \mathcal{G}^{a}_{2}(t)\right]\] \[-\frac{1}{2}(\Delta_{\mu}\Delta_{\nu}-\Delta^{2}g_{\mu\nu})\left[ (\epsilon^{\prime}\epsilon)\mathcal{G}^{a}_{3}(t)-\frac{(\Delta\epsilon^{ \prime})(\Delta\epsilon)}{2m_{\rho}^{2}}\mathcal{G}^{a}_{4}(t)\right]+P_{\{ \mu}\left(\epsilon^{\prime}_{\nu\}}(\Delta\epsilon)-\epsilon_{\nu\}}(\Delta \epsilon^{\prime})\right)\mathcal{G}^{a}_{5}(t)\] \[+\frac{1}{2}\left[\Delta_{\{\mu}\left(\epsilon^{\prime}_{\nu\}}( \Delta\epsilon)+\epsilon_{\nu\}}(\Delta\epsilon^{\prime})\right)-\epsilon_{\{ \mu}^{\prime}\epsilon_{\nu\}}\Delta^{2}-g_{\mu\nu}(\Delta\epsilon^{\prime})( \Delta\epsilon)\right]\mathcal{G}^{a}_{6}(t)\] \[+\epsilon_{\{\mu}^{\prime}\epsilon_{\nu\}}m_{\rho}^{2}\mathcal{G }^{a}_{7}(t)+g_{\mu\nu}m_{\rho}^{2}(\epsilon^{\prime}\epsilon)\mathcal{G}^{a} _{8}(t)+\frac{1}{2}g_{\mu\nu}(\Delta\epsilon^{\prime})(\Delta\epsilon) \mathcal{G}^{a}_{9}(t)\] \[+P_{[\mu}\left(\epsilon^{\prime}_{\nu]}(\Delta\epsilon)-\epsilon _{\nu]}(\Delta\epsilon^{\prime})\right)\mathcal{G}^{a}_{10}(t)+\Delta_{[\mu} \left(\epsilon^{\prime}_{\nu]}(\Delta\epsilon)+\epsilon_{\nu]}(\Delta\epsilon ^{\prime})\right)\mathcal{G}^{a}_{11}(t)\,,\] (32) where $A^{\{\mu\nu\}}=\frac{1}{2}(A^{\mu\nu}+A^{\nu\mu})$ and $A^{[\mu\nu]}=\frac{1}{2}(A^{\mu\nu}-A^{\nu\mu})$. The following sum rules follow from energy-momentum conservation, and the assumption that the symmetric and antisymmetric components of the EMT are separately conserved³: [FOOTNOTE:3][ENDFOOTNOTE] \[\sum_{a=q,g}\mathcal{G}_{7}^{a}(t)=\sum_{a=q,g}\mathcal{G}_{8}^{a }(t)=\sum_{a=q,g}\mathcal{G}_{9}^{a}(t)=\sum_{a=q,g}\mathcal{G}_{11}^{a}(t)=0\,,\] (33) but since there are no gluons in the NJL model, each of these form factors should be identically zero for the quarks. We have two additional sum rules, the first also following from energy-momentum conservation, and the latter from angular momentum conservation: \[\sum_{a=q,g}\mathcal{G}_{1}^{a}(0)=1\] (34) \[\sum_{a=q,g}\mathcal{G}_{5}^{a}(0)=2\,.\] (35) Lastly, there is a correspondence between the GFF $\mathcal{G}_{10}(t)$ for a spin-one hadron is related to its isoscalar axial form factors: \[\mathcal{G}_{10}^{q}(t) =-\widetilde{G}_{1}^{q}(t)+\frac{t}{m_{\rho}^{2}}\widetilde{G}_{2 }^{q}(t)\,.\] (36) This correspondence is a consequence of the QCD equation of motion Leader and Lorcé (2014): \[i\overline{\psi}\gamma^{[\mu}\overleftrightarrow{\partial}^{\nu] }\psi=-\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}\partial_{\rho}\Big{(}\overline{ \psi}\gamma_{\sigma}\gamma_{5}\psi\Big{)}\,.\] (37) However, Eq. (37) also holds in the NJL model (see App. C for a proof), and therefore Eq. (36) is also expected to hold in the NJL model. ### Energy-momentum tensor of free quarks Although quarks are not asymptotic states in the NJL model owing to the introduction of an infrared regulator, we may ask what the EMT and gravitational form factors of a free dressed quark would look like if we were to remove this regulator. The most general from that the EMT of a spin-half particle can take is Lorcé _et al._ (2019): (38) Using the quark-graviton vertex in Eq. (25) and placing the quark on-shell gives, once the quark flavors have been summed over, $A_{Q}(t)=1$, $B_{Q}(t)=0$, $C_{Q}(t)$ as given in Eq. (26), $\bar{c}_{Q}(t)=0$, and $D_{Q}(t)=-1+D_{Q}^{\prime}(t)$, with $D_{Q}^{\prime}(t)$ given in Eq. (27). This allows the functions $C_{Q}(t)$ and $-1+D_{Q}^{\prime}(t)$ we found by solving the BSE to be interpreted as gravitational form factors of dressed quarks. <figure><img src="content_image/1903.09222/x8.png"><figcaption>Figure 4: The gravitational form factors CQ(t) and DQ(t) of a dressed quarkin the NJL model. AQ(t)=1 and BQ(t)=0 are not shown here, since they aretrivial (undressed) in this model.</figcaption></figure> We have plotted in Fig. 4 the two non-trivial GFFs of a dressed quark, leaving out the trivial $A_{Q}(t)=1$ and $B_{Q}(t)=0$, using the NJL model parameters given in Tab. 1. In Fig. 4, both $C_{Q}(t)$ and $D_{Q}(t)$ can be seen to approach their elementary values of $0$ and $-1$, respectively. It should be noted that $C(t)=0$ for an elementary free Dirac particle Hudson and Schweitzer (2018), and that a finite outcome for this form factor can only be produced by interaction dynamics. ## IV Axial form factors In order to check Eq. (36) within the NJL model, we must also calculate the axial form factors of the rho meson. Since gravitation is an isoscalar interaction, we look specifically at the isoscalar axial vector current. The matrix element of the axial vector current takes the following most general form for a spin-one particle Frederico _et al._ (1992); Berger _et al._ (2001): \[\langle p^{\prime},\lambda^{\prime}\|J_{5}^{\mu}(0)\|p,\lambda\rangle =-2i\epsilon^{\mu\epsilon^{\prime}\epsilon P}\widetilde{G}_{1}(t )-2i\epsilon^{\mu\Delta P\sigma}\frac{\epsilon_{\sigma}(\epsilon^{\prime} \Delta)-\epsilon^{\prime}_{\sigma}(\epsilon\Delta)}{m_{\rho}^{2}}\widetilde{G }_{2}(t)\,.\] (39) In order to calculate the matrix element in this equation, we must evaluate the triangle diagrams in Fig. 5 with the axial vector vertex for a dressed quark, and this vertex itself must be found through an inhomogeneous Bethe-Salpeter equation. In this section, we will find the axial form factors of the rho by first finding the requisite axial vector vertex, and in the process explore its properties and demonstrate that it satisfies the axial Ward-Takahashi identity. <figure><img src="content_image/1903.09222/x9.png"><figcaption>Figure 5: Diagrams contributing to the axial vector current of a meson.</figcaption></figure> ### Axial vector and pseudoscalar vertices <figure><img src="content_image/1903.09222/x10.png"><figcaption>Figure 6: Diagrammatic representation of the inhomogeneous Bethe-Salpeterequation for either the pseudoscalar or axial vector vertex.</figcaption></figure> We begin by finding the dressed axial vector and pseudoscalar vertices in the NJL model. The relevant currents are isoscalar, and the driving terms for the axial vector and pseudoscalar vertices are respectively $\gamma^{\mu}\gamma_{5}$ and $\gamma_{5}$. We denote the dressed vertices $\Gamma_{5}^{\mu}(p^{\prime},p)$ for the axial vector and $\Gamma_{5}(p^{\prime},p)$ for the pseudoscalar current. The most general forms that the dressed vertices can take are: \[\Gamma_{5}^{\mu}(p^{\prime},p) =a_{1}(t)\gamma^{\mu}\gamma_{5}+\frac{\Delta^{\mu}}{M}\gamma_{5}a _{2}(t)+\frac{\Delta^{\mu}\not{\Delta}}{t}\gamma_{5}a_{3}(t)\] (40) \[\Gamma_{5}(p^{\prime},p) =g_{1}(t)\gamma_{5}+\frac{\not{\Delta}}{M}g_{2}(t)\,,\] (41) where $a_{i}(t)$ and $g_{i}(t)$ are off-shell form factors to be solved for algebraically. The resulting solutions to the two BSEs are: \[g_{1}(t)=\frac{1+2G_{f}\Pi_{AA}^{(L)}(t)}{\mathcal{D}_{A}(t)}\,, \qquad g_{2}(t)=\frac{2G_{f}\Pi_{PA}(t)}{\mathcal{D}_{A}(t)}\] (42) \[a_{1}(t)=\frac{1}{1+2G_{f}\Pi_{AA}^{(T)}(t)}\,,\qquad a_{2}(t)= \frac{-2G_{\eta}\Pi_{PA}(t)}{\mathcal{D}_{A}(t)}\,,\] (43) \[a_{3}(t)=\frac{-2G_{f}}{1+2G_{f}\Pi_{AA}^{(L)}(t)}\left[\Big{(} \Pi_{AA}^{(L)}(t)-\Pi_{AA}^{(T)}(t)\Big{)}a_{1}(t)+\frac{t}{M^{2}}\Pi_{AP}(t)a _{2}(t)\right]\,,\] (44) where the bubbles are given in App. A, and \[\mathcal{D}_{A}(t) =\big{(}1+2G_{\eta}\Pi_{PP}(t)\big{)}\big{(}1+2G_{f}\Pi_{AA}^{(L) }(t)\big{)}-4G_{\eta}G_{f}\frac{t}{M^{2}}\Pi_{AP}(t)\Pi_{PA}(t)\,.\] (45) ### Axial vector Ward-Takahashi identity The axial vector and pseudoscalar vertices we have found should satisfy an axial vector Ward-Takahashi identity, which can be stated as Adler (1969): \[\Delta_{\mu}\Gamma_{5}^{\mu}(p^{\prime},p)=S^{-1}(p^{\prime}) \gamma_{5}+\gamma_{5}S^{-1}(p)+2m\Gamma_{5}(p^{\prime},p)\,.\] (46) Here, $m$ is the current quark mass rather than the dressed quark mass. One can immediately observe that: \[a_{1}(t)+a_{3}(t) =\frac{1+2G_{\eta}\Pi_{PP}(t)}{\mathcal{D}_{A}(t)}\] (47) \[\Delta_{\mu}\Gamma_{5}^{\mu}(p^{\prime},p) =\not{\Delta}\gamma_{5}\Big{(}a_{1}(t)+a_{3}(t)\Big{)}+\frac{t}{M }\gamma_{5}a_{2}(t)=\left[\frac{1+2G_{\eta}\Pi_{PP}(t)}{\mathcal{D}_{A}(t)} \not{\Delta}-\frac{t}{M}\frac{2G_{\eta}\Pi_{PA}(t)}{\mathcal{D}_{A}(t)}\right] \gamma_{5}\] (48) \[S^{-1}(p^{\prime})\gamma_{5}+\gamma_{5}S^{-1}(p) =\not{\Delta}\gamma_{5}-2M\gamma_{5}\] (49) \[2m\Gamma_{5}(p^{\prime},p) =2m\left[\frac{1+2G_{f}\Pi_{AA}^{(L)}(t)}{\mathcal{D}_{A}(t)}+ \frac{2G_{f}\Pi_{PA}(t)}{\mathcal{D}_{A}(t)}\frac{\not{\Delta}}{M}\right] \gamma_{5}\,,\] (50) meaning the axial vector WTI requires that: \[\frac{m}{M}=\frac{1+2G_{\eta}\Pi_{PP}(t)-\mathcal{D}_{A}(t)}{2G_{ f}\Pi_{PA}(t)}=\frac{2\mathcal{D}_{A}(t)-2G\eta\frac{t}{M^{2}}\Pi_{PA}(t)}{1+2 G_{f}\Pi_{AA}^{(L)}(t)}\,.\] (51) This can be proved, but requires a little work to show. The equality between the second and third expressions in Eq. (51) can be shown using a little algebra using $\Pi_{PA}(t)=-\Pi_{AP}(t)$ and $\Pi_{AA}^{(L)}(t)=-2\Pi_{PA}(t)$. The equality between the first and second expressions requires: \[\frac{m}{M}\] \[=1+2G_{\eta}\Pi_{PP}(t)+G_{\eta}\frac{t}{M^{2}}\Pi_{AP}(t)\] (52) The remaining bubbles can be found to evaluate to: \[\Pi_{PP}(t) =4i(2N_{c})\int_{0}^{1}\mathrm{d}x\,\int\frac{\mathrm{d}^{4}k}{(2 \pi)^{4}}\left[-\frac{1}{k^{2}-M^{2}+x(1-x)t+i0}+\frac{2x(1-x)t}{[k^{2}-M^{2}+ x(1-x)t+i0]^{2}}\right]\] (53) \[\Pi_{AP}(t) =-4i(2N_{c})\int_{0}^{1}\mathrm{d}x\,\int\frac{\mathrm{d}^{4}k}{( 2\pi)^{4}}\frac{M^{2}}{[k^{2}-M^{2}+x(1-x)t+i0]^{2}}\,,\] (54) where we leave the regularization scheme unspecified and implicit, except to assume that several basic operations are allowed, namely: (1) differentiation and integration with respect to variables other than $k$ commutes with the $k$ integral, and (2) one can cancel factors of $k^{2}-B(M,x,t)$ between the numerator and denominator and still obtain the same result. These restrictions, it should be noted, are met by the proper time regularization scheme we employ. With these rules, we can find that: \[\frac{t}{M^{2}}\Pi_{AP}(t)+2\Pi_{PP}(t) =-4i(2N_{c})\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\int_{0}^{1} \mathrm{d}x\,\left[\frac{2}{k^{2}-M^{2}+x(1-x)t+i0}+\frac{1-4x(1-x)t}{[k^{2}-M ^{2}+x(1-x)t+i0]^{2}}\right]\] \[=-8i(2N_{c})\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\int_{0}^{1} \mathrm{d}x\,\left[\frac{1}{k^{2}-M^{2}+x(1-x)t+i0}-\frac{x(1-2x)t}{[k^{2}-M^{ 2}+x(1-x)t+i0]^{2}}\right]\] \[=-8i(2N_{c})\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\int_{0}^{1} \mathrm{d}x\,\frac{\partial}{\partial x}\left[\frac{x}{k^{2}-M^{2}+x(1-x)t+i0}\right]\] \[=-8i(2N_{c})\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\frac{1}{k^{2}- M^{2}+i0}\,.\] (55) Comparison with the gap equation, Eq. (2), gives us: \[\frac{t}{M^{2}}\Pi_{AP}(t)+2\Pi_{PP}(t)=-\left(\frac{M-m}{G_{\pi} M}\right)\,,\] (56) and therefore the axial vector WTI requires that: \[\frac{m}{M}=1-\frac{G_{\eta}}{G_{\pi}}\frac{M-m}{M}\,,\] (57) which holds true if and only if $G_{\eta}=G_{\pi}$. This requirement is not surprising, since $\mathrm{U}(1)_{A}$ invariance is the symmetry responsible for the partial conservation of isoscalar axial symmetry, and therefore for the isoscalar axial vector WTI. It was remarked in Cloët _et al._ (2014) that $G_{\eta}=G_{\pi}$ is a necessary condition for the NJL model Lagrangian (1) to satisfy $\mathrm{U}(1)_{A}$ symmetry, and the necessity of this equality in the final step of our proof is simply a manifestation of that fact. ## V Results We now present results for the pion and rho meson EMTs as calculated in the two-flavor NJL model. Since the EMT is an isoscalar quantity, and since we take $m_{u}=m_{d}$ in this work, the up and down quark contributions to the EMT are equal, and we present the meson EMT obtained after summing over both quark flavors. ### Pion and sigma <figure><img src="content_image/1903.09222/x11.png"><figcaption>Figure 7: Gravitational form factors of the pion and sigma meson.</figcaption></figure> The GFFs of the pion and sigma meson have been plotted in Fig. 7. One can see immediately that the expected constraints appear to hold. In particular, we have $A_{\pi,\sigma}(0)=1$ and $C_{\pi}(0)\approx-1$. Not included in the plot are the results $\bar{c}_{\pi,\sigma}(t)=0$, which are found to hold exactly. In addition, we find that in the chiral limit, $C_{\pi}(0)=-1$ exactly, satisfying the low-energy pion theorem of Eq. (31). It’s worth stressing the importance of the “quark D term” [$C_{Q}(t)$ in Eq. (25), not $D_{Q}(t)$] in satisfying the low-energy pion theorem. This has implications for the calculation of generalized parton distributions (GPDs). The leading-twist GPDs $H_{\pi}^{q,g}(x,\xi,t)$ are related to GFFs through the second Mellin moment⁴, viz., [FOOTNOTE:4][ENDFOOTNOTE] \[\int_{-1}^{1}\mathrm{d}x\,H_{\pi}^{a}(x,\xi,t) =A_{\pi}^{a}(t)+\xi^{2}C_{\pi}^{a}(t)\,,\] (58) which is a specific case of the more general property that Mellin moments of GPDs produce matrix elements of local operators. If a bare non-local operator is used to calculate the GPDs, then its moments can produce only matrix elements of bare local operators. For the second moment in particular, this will give the “$++$” component of the bare three-point graviton vertex. As a consequence, if one does not dress the non-local operator defining the GPDs, an incorrect value will be inferred for $C_{a}(t)$. To illustrate this, we define “partially dressed” gravitational form factors as those that incorporate the vacuum condensate diagram, but for which $C_{Q}(t)$ is set to zero. We note that we need to include the vacuum condensate diagram for consistency, as otherwise the gravitational WTI is violated. Additionally, this diagram has no bearing on the Mellin moments of leading-twist GPDs, since its contribution contains a factor $g^{\mu\nu}$, and $g^{++}=0$. On the other hand, $C_{Q}(t)$ should appear in the Mellin moment with a factor $\xi^{2}$, so we expect $C_{\pi}(t)$ to be incorrect with this term missing. <figure><img src="content_image/1903.09222/x12.png"><figcaption>Figure 8: The “D-term” form factor Cπ(t) for the pion. (Solid curve) Fullcalculation. (Dashed curve) With the quark D-term CQ(t) neglected.</figcaption></figure> In Fig. 8, we show the difference between $C_{\pi}(t)$ as calculated in the NJL model with and without the quark D-term $C_{Q}(t)$ folded in. The contrast is actually stark, with about two-thirds of the pion D-term coming from the quark dressing. We repeat and reemphasize that the “partially dressed” curve in this plot is the $C_{\pi}(t)$ that one would find from taking the second Mellin moment of the leading-twist pion GPD if it were calculated using a bare non-local operator, and that this speaks to the necessity of dressing operators in GPD calculations. We illustrate this point further by finding analytic expressions for $C_{Q}(0)$ and $C_{\pi}(0)$ within proper time regularization. We find: \[C_{\pi}(t=0) =\frac{1}{3}\frac{\int_{0}^{1}\mathrm{d}x\,\int\mathrm{d}\tau \left\{-[1-6C_{Q}(0)]\frac{1}{\tau}+[1+6C_{Q}(0)]\frac{x(1-x)m_{\pi}^{2}}{M^{2 }-x(1-x)m_{\pi}^{2}}\right\}e^{-(M^{2}-x(1-x)m_{\pi}^{2})\tau}}{\int_{0}^{1} \mathrm{d}x\,\int\mathrm{d}\tau\left\{\frac{1}{\tau}+\frac{x(1-x)m_{\pi}^{2}}{ M^{2}-x(1-x)m_{\pi}^{2}}\right\}e^{-(M^{2}-x(1-x)m_{\pi}^{2})\tau}} \xrightarrow[m_{\pi}\to 0]{}-\frac{1}{3}+2C_{Q}(0)\] (59) \[C_{Q}(t=0) =-\frac{1}{3}\frac{G_{\pi}\frac{M^{2}}{\pi^{2}}\int_{0}^{1} \mathrm{d}x\,\int\mathrm{d}\tau\frac{1}{\tau}}{\frac{m}{M}+G_{\pi}\frac{M^{2}} {\pi^{2}}\int_{0}^{1}\mathrm{d}x\,\int\mathrm{d}\tau\frac{1}{\tau}} \xrightarrow[m\to 0]{}-\frac{1}{3}\,,\] (60) where the $\tau$ integration is from $\Lambda_{\mathrm{UV}}^{-2}$ to $\Lambda_{\mathrm{IR}}^{-2}$. One can indeed see that $C_{\pi}(0)=-1$ exactly in the chiral limit. Additionally, we instead get exactly $-\frac{1}{3}$ if the contribution from $C_{Q}(0)$ is neglected—meaning a full two-thirds of the expected value come from the quark dressing. The necessity for fulling dressing the relevant operator to observe the low-energy pion theorem—either the three-point graviton vertex or the bilocal light cone operator—draws an analogy to other results of dynamical chiral symmetry breaking, such as the dressing of quark mass. These non-perturbative phenomena both require a self-consistent solution of the relevant Dyson-Schwinger equations to manifest, and both result in significant changes to physical, observable quantities that are not constrained by conservation laws. This can be contrasted with electric charge, which remains unaltered between the bare electromagnetic vertex and the dressed vertex found by solving the inhomogeneous BSE. In contrast to the pion, the sigma meson is not a Nambu-Goldstone boson and its “D-term” is not constrained in any a priori fashion. One can observe that $C_{\sigma}(0)=-2.27\not\approx-1$. On another point, it is worth mentioning that the vanishing of $\bar{c}_{\pi,\sigma}(t)$ is possible only when all three diagrams have been included. Each of the diagrams on its own makes a non-zero contribution to the spin-zero meson EMT. The triangle diagrams contribute $\frac{1}{2}Z_{\pi}\Pi_{PP}(m_{\pi}^{2})$ or $\frac{1}{2}Z_{\sigma}\Pi_{SS}(m_{\sigma}^{2})$ each to $\bar{c}_{\pi}(t)$ or $\bar{c}_{\sigma}(t)$, respectively (with direct application of the WTI of Eq. (18) being the most straightforward way to find this result), while the bicycle diagram contributes $2G_{\pi}Z_{\pi}\Big{(}\Pi_{PP}(m_{\pi}^{2})\Big{)}^{2}$ or $-2G_{\sigma}Z_{\sigma}\Big{(}\Pi_{SS}(m_{\sigma}^{2})\Big{)}^{2}$ to the same form factor. The three diagrams sum to \[2m_{\pi}^{2}\bar{c}_{\pi}(t)=Z_{\pi}\Pi_{PP}(m_{\pi}^{2})\left[1 +2G_{\pi}\Pi_{PP}(m_{\pi}^{2})\right]=0\] (61) \[2m_{\sigma}^{2}\bar{c}_{\sigma}(t)=Z_{\sigma}\Pi_{SS}(m_{\pi}^{2 })\left[1-2G_{\sigma}\Pi_{SS}(m_{\sigma}^{2})\right]=0\,,\] (62) which both vanish because of the pion and sigma mass-shell conditions in Eqs. (7,8). #### v.1.1 Spin-zero meson mass radius With the EMT of the spin-zero mesons in hand, it is also possible to study their static mechanical properties. It is conventional in much of the literature Polyakov (2003); Lorcé _et al._ (2018) to define a Breit frame EMT through: (63) where the requirement that $\mathbf{P}=\mathbf{p}^{\prime}-\mathbf{p}=0$ gives us $P^{0}=\sqrt{m_{\pi}^{2}+\bm{\Delta}^{2}/4}$ and $\Delta^{0}=0$. One can then evaluate moments of $\langle T_{\mu\nu}\rangle_{\mathrm{Breit}}(\mathbf{r})$ to obtain static properties such as the mass radius. Interpretation of Eq. (63) as an actual spatial distribution has been called into question Miller (2019). Despite this, one could still formally define a mass radius in terms of the Breit frame “density,” as is often done with the proton charge radius. Nonetheless, we run into problems for the pion. The limit $\bm{\Delta}\to 0$ must be taken when evaluating multipole moments, which requires the existence of a rest frame for the particle. Thus, Eq. (63) cannot be used in the chiral limit. One may alternatively define a two-dimensional spatial distribution using light cone quantization Dirac (1949), where the spatial dimensions are the transverse light cone coordinates. The transverse spatial distribution is then: \[\langle T^{a}_{\mu\nu}\rangle_{\mathrm{LC}}(\mathbf{r}_{\perp}) \equiv\int\frac{\mathrm{d}^{2}\bm{\Delta}_{\perp}}{2P^{+}(2\pi)^{2}}e^{-i(\bm{ \Delta}_{\perp}\mathbf{r}_{\perp})}\langle p^{\prime}\,\|\,T^{a}_{\mu\nu}(0)\,\| \,p\rangle\bigg{\|}_{\Delta^{+}=0}\,,\] (64) where the requirement $\Delta^{+}=0$ implies $P^{+}=p^{+}=p^{\prime+}$. Since in light cone quantization $P^{+}$ is kinematic and $P^{-}$ is dynamical Dirac (1949); Brodsky _et al._ (1998)⁵, $P^{+}$ has no dependence on $\bm{\Delta}$, and thus no dependence on $t$, by contrast to $P^{0}$ in the Breit frame. We shall see the significance of this presently. [FOOTNOTE:5][ENDFOOTNOTE] The mass radius can be found as the mean value of either $\mathbf{r}^{2}$ weighted by $\frac{1}{2P^{0}}\langle T^{00}\rangle_{\mathrm{Breit}}(\mathbf{r})$, or $\mathbf{r}_{\perp}^{2}$ weighted by $\frac{1}{2P^{+}}\langle T^{++}\rangle_{\mathrm{LC}}(\mathbf{r}_{\perp})$. We find: \[\langle r^{2}\rangle_{\mathrm{Breit}} =\sum_{a=q,g}\lim_{\bm{\Delta}\to 0}-\frac{1}{P^{0}} \nabla_{\Delta}^{2}\left[\frac{1}{2P^{0}}\langle p^{\prime}\,\|\,T_{a}^{00}(0) \,\|\,p\rangle\bigg{\|}_{\Delta^{0}=0}\right]=6\frac{\mathrm{d}A_{\pi,\sigma}(t) }{\mathrm{d}t}\bigg{\|}_{t=0}-\frac{3}{4m_{\pi,\sigma}^{2}}\left[A_{\pi,\sigma} (0)+2C_{\pi,\sigma}(0)\right]\] (65) \[\langle r_{\perp}^{2}\rangle_{\mathrm{LC}} =\sum_{a=q,g}\lim_{\bm{\Delta}\to 0}-\frac{1}{P^{+}} \nabla_{\Delta_{\perp}}^{2}\left[\frac{1}{2P^{+}}\langle p^{\prime}\,\|\,T_{a}^ {++}(0)\,\|\,p\rangle\bigg{\|}_{\Delta^{+}=0}\right]=4\frac{\mathrm{d}A_{\pi, \sigma}(t)}{\mathrm{d}t}\bigg{\|}_{t=0}\,.\] (66) These radii differing (beyond a factor of the number of spatial dimensions) ultimately amounts to the mass form factor being different in the Breit frame and on the light cone, respectively $P^{0}A(t)$ and $P^{+}A(t)$. In the relevant frames, $P^{0}$ has $t$ dependence while $P^{+}$ does not. This is analogous to the electric charge distribution of the nucleon Miller (2007) or deuteron Carlson and Vanderhaeghen (2009) differing between the Breit frame and the light cone. In the case of the nucleon, the Sachs form factor $G_{E}(t)$ describes the electric charge distribution in the Breit frame, while the Dirac form factor $F_{1}(t)$ does instead on the light cone. A remarkable property of the Breit frame radius (65) is that it remains finite even for point particles, for which $\frac{\mathrm{d}A(t)}{\mathrm{d}t}=0$. By contrast, the light cone radius (66) is zero for point particles. The former can be understood as owing to spatial distributions not being invariant under Lorentz boosts, while the latter is due to the fact that transverse boosts in light cone coordinates are Galilean Brodsky _et al._ (1998). In more detail, spatial distributions in the Breit frame are found by integrating over a collection of reference frames where the system of interest is in motion by different amounts, but without applying any corrections to counteract Lorentz contractions. These relativistic corrections are intrinsically accounted for by using light cone coordinates, however. \| Breit frame \| Light cone \| Empirical light cone Kumano _et al._ (2018) [see text] ---\|---\|---\|--- Pion \| 1.28 \| 0.27 \| 0.26∼0.32 Sigma \| 0.56 \| 0.32 \| Table 2: Mean squared mass radius of spin-zero mesons in the NJL model, both using the Breit frame and light cone prescriptions. For the pion, an empirical value for the light cone mass radius extracted from KEKB data is included for comparison. All values are in fm. Because of the $\mathcal{O}(m_{\pi}^{-2})$ term in Eq. (65), the Breit frame radius of the pion blows up in the chiral limit. On the other hand, Eq. (66) remains finite at zero pion mass. Both equations can be used at physical pion mass, but produce staggeringly different values for the mass radius. Numerical values can be found in Tab. 2. For the empirical value of the pion mass radius, we look to the extraction in Kumano _et al._ (2018), where a dispersive analysis of KEKB data for $\gamma^{}\gamma\rightarrow\pi^{0}\pi^{0}$ was done to extract gravitational form factors. The formula used in Kumano _et al._ (2018) was the same as our Eq. (66) for the squared light cone pion mass radius, but with a factor 6 instead of 4. We have thus scaled down the range of $0.32\sim 0.39$ fm reported in Kumano _et al._ (2018) by a factor $\sqrt{2/3}$. The NJL model result for this radius agrees with the empirical range, but falls on the low end. In Ref. Cloët _et al._ (2014), the Breit frame charge radius of the pion was found to be 0.62 fm, which is significantly smaller than the Breit frame mass radius we have found. However, as we have discussed, the Breit frame mass radius made artificially large by $\mathcal{O}(m_{\pi}^{-1})$ terms that are not present in the light cone mass radius. The pion charge radius of Ref. Cloët _et al._ (2014), when scaled by $\sqrt{2/3}$, gives a light cone charge radius of 0.51 fm, which is instead larger than the light cone mass radius. The Breit frame and light cone prescriptions for radii thus suggest strongly divergent pictures of the relative distribution of mass and charge in the pion. Since doubt has been cast on the interpretation of the Breit frame radius as the moment of an actual density Miller (2019), the picture painted by light cone coordinates seems more plausible. Because the form factor $A_{\sigma}(t)$ falls off faster than $A_{\pi}(t)$ (see Fig. 7), the light cone mass radius of the sigma is larger than that of the pion. On the other hand, because of its greater mass, the sigma meson has a smaller Breit frame radius than the pion. There is a stark difference between the results in these two frames, again demonstrating the magnitude and importance of correctly accounting for relativistic effects—by, for instance, using light cone coordinates in defining spatial distributions. Finally, we remark again that getting the correct value of $C_{\pi}(0)$ by fully dressing the quark-graviton vertex is vital here. If one neglects the “quark D-term,” or equivalently obtains the GFFs through Mellin moments of bare GPDs, so that $C_{\pi}(0)\approx-\frac{1}{3}$, then one finds $\langle r^{2}\rangle_{\mathrm{Breit}}<0$ at the physical pion mass—an obvious absurdity that violates the weak energy condition Hawking and Ellis (2011). ### Rho meson The rho meson, as a spin-one hadron, has many more GFFs than the pion. There are 11 GFFs total, with 4 of these being non-conserved Cosyn _et al._ (2019). As required from the lack of gluons in the NJL model, the four non-conserved GFFs vanish: $\mathcal{G}_{7}=\mathcal{G}_{8}=\mathcal{G}_{9}=\mathcal{G}_{11}=0$. There are thus seven non-zero GFFs to consider. Three of the rho GFFs have direct analogues to the pion GFFs, in particular, $\mathcal{G}_{1}(t)\sim A_{\pi}(t)$, $\mathcal{G}_{3}(t)\sim C_{\pi}(t)$, and $\mathcal{G}_{8}(t)\sim-2\bar{c}_{\pi}(t)$. The behavior of these form factors is remarkably similar to those of the pion. Firstly, we find $\mathcal{G}_{1}(0)=1$, as is required by momentum conservation. Additionally, we curiously find $\mathcal{G}_{3}(0)\approx-1$, and even more curiously find that this becomes exactly $-1$ in the chiral limit. Since the rho meson is not a Nambu-Goldstone boson, we do not know of any theorems requiring that this be the case, as we did for $C_{\pi}(0)$. Lastly, as with $\bar{c}_{\pi}(t)$ for the pion, we find that including the contribution of the bicycle diagram (rightmost diagram in Fig. 3) is necessary for $\mathcal{G}_{8}(t)$ to fully vanish. Among the new conserved form factors, $\mathcal{G}_{5}(t)$ describes the spatial distribution of total angular momentum. An angular momentum sum rule Abidin and Carlson (2008) requires that $\mathcal{G}_{5}(0)=2$, and we satisfy this sum rule in the NJL model. The non-conserved form factor $\mathcal{G}_{7}(t)$ also contributes to this spatial distribution, but vanishes in the NJL model. The remaining form factors contribute only to higher multipole moments of the energy-momentum tensor. <figure><img src="content_image/1903.09222/x13.png"><figcaption>Figure 9: The six non-zero gravitational form factors of the rho mesonappearing in the symmetric component of its energy-momentum tensor.</figcaption></figure> Of the seven non-zero GFFs, the six $\mathcal{G}_{1-6}(t)$ appear in the symmetric component of the energy-momentum tensor. These six form factors are of special phenomenological interest, since they can be found from second Mellin moments of leading-twist generalized parton distributions Cosyn _et al._ (2018, 2019), which can be measured in hard exclusive reactions such as DVCS and DVMP. These six GFFs have been plotted in Fig. 9. \| √⟨r2⟩mass \| √⟨r2⟩elec. \| Qmass \| Qelec. ---\|---\|---\|---\|--- Breit frame \| 0.45 \| 0.67 \| -0.0224 \| -0.0200 Light cone \| 0.39 \| 0.45 \| \| Table 3: Static properties of the rho meson in the NJL model. Electromagnetic properties taken from the BSE calculation (without pion cloud effects) for ρ+ from Cloët _et al._ (2014). All radii are in fm, the mass quadrupole moment is in units of mρ-fm2, and the electric quadrupole moment is in e-fm2. In Cosyn _et al._ (2019), the Breit frame multipole moments of the spin-one EMT were found. The mean squared mass radius and gravitational quadrupole moment are: \[\langle r^{2}\rangle_{\mathrm{mass}} =6\frac{\mathrm{d}\mathcal{G}_{1}(t)}{\mathrm{d}t}\bigg{\|}_{t=0}+ \frac{1}{m_{\rho}^{2}}\left[-\frac{7}{4}\mathcal{G}_{1}(0)-\mathcal{G}_{2}(0)- \frac{3}{2}\mathcal{G}_{3}(0)+\mathcal{G}_{5}(0)+\frac{1}{2}\mathcal{G}_{6}(0)\right]\] (67) \[\mathcal{Q}_{\mathrm{mass}} =\frac{1}{m_{\rho}}\left[-\mathcal{G}_{1}(0)-\mathcal{G}_{2}(0)+ \mathcal{G}_{5}(0)+\frac{1}{2}\mathcal{G}_{6}(0)\right]\,,\] (68) where we have neglected the non-conserved form factors, since they are zero in the NJL model. Since the Breit frame density cannot be literally interpreted as an actual spatial density Miller (2019), we also determine the light cone transverse mass radius as a point of contrast, which we find to be: \[\langle r_{\perp}^{2}\rangle_{\mathrm{LC}} =4\frac{\mathrm{d}\mathcal{G}_{1}(t)}{\mathrm{d}t}\bigg{\|}_{t=0}+ \frac{1}{m_{\rho}^{2}}\left[\frac{2}{3}\mathcal{G}_{1}(0)-\frac{2}{3}\mathcal{ G}_{2}(0)-\frac{1}{3}\mathcal{G}_{5}(0)-\frac{1}{3}\mathcal{G}_{6}(0)\right]\,.\] (69) Since the light cone density is a two-dimensional quantity, we will defer exploration of the quadrupole moment—both quantitative and conceptual—to a future work on the light cone interpretation of the EMT. The numerical values have been computed and tabulated in Tab. 3, along with the equivalent electric (Coulomb) quantities Cloët _et al._ (2014). The quadrupole moments are remarkably close when comparable units are used, but the electric charge radius is larger than the mass radius, for both the Breit frame and light cone radii. This suggests a highly inhomogeneous distribution of electric charge in the rho meson, as could occur (for instance) in a configuration with a small negatively-charged core surrounded by a shell of positive charge. Curiously, this occurs even for a positive rho meson—consisting of positively charged up and anti-down quarks—but we remark that the dressed quarks themselves have spatially extended electric charge distributions. <figure><img src="content_image/1903.09222/x14.png"><figcaption>Figure 10: The gravitational form factor G10(t) of the rho meson, comparedwith the rho axial form factors. The form factor ~G2(t) has been weighted byt/m2ρ to make it comparable to the other form factors on the same plot.</figcaption></figure> The conserved, non-zero GFF $\mathcal{G}_{10}(t)$ appears in the antisymmetric component of the EMT. In both QCD and the NJL model, this GFF has a special relationship with the axial form factors, given in Eq. (36). To demonstrate this correspondence, we plot the relevant form factors in Fig. 10. Because of this relation, $\mathcal{G}_{10}(t)$ encodes information about the distribution of quark spin, with the quantity $-\frac{1}{2}\mathcal{G}_{10}(0)=0.526$ giving the proportion of the rho meson’s total angular momentum carried by quark spin. If we take $G_{f}=0$, we instead get $0.556$, a number consistent with previous work on the rho meson in the NJL model Ninomiya _et al._ (2017) that implicitly assumed $G_{f}=0$. ## VI Conclusions In this work, we have found the gravitational form factors appearing in the decompositions of the canonical energy-momentum tensor for the pion, sigma, and tho mesons in the NJL model. In the process of obtaining these, we proved a gravitational Ward-Takahashi identity for the canonical EMT of fields with arbitrary spin, which takes a simpler form than the equivalent WTI for the Belinfante EMT. This simpler WTI makes consistency cross-checks between the dressed graviton vertex and dressed propagator within model calculations easier, and will also extend to models with spin-one constituents. We found that the non-linear four-fermi interaction of the NJL model entails a five-point gravitational vertex (with four quark lines and one graviton line) in addition to the usual three-point vertex (with two quark lines and one graviton). This finding is a result of the equivalence principle, and is necessary for conservation of energy and momentum to be observed. In solving the Bethe-Salpeter equation for the dressed three-point vertex, we found it was necessary to include contributions from the five-point vertex for the gravitational WTI to be satisfied. Additionally, the five-point vertex contributed to meson EMT calculations directly in the form of a bicycle diagram, which was needed in order for the non-conserved GFFs $\bar{c}(t)$ and $\mathcal{G}_{8}(t)$ to vanish. The necessity of the five-point vertex suggests difficulties in the prospect of calculating gravitational form factors in QCD. The QCD Lagrangian contains quark-gluon, three-gluon, four-gluon, and (in a covariant gauge) ghost-gluon vertices in its Lagrangian, and the equivalence principle—encoded by the presence of $-g^{\mu\nu}\mathcal{L}$ in the EMT—requires that the graviton be able to couple directly to every one of these vertices. Inclusion of all these graviton interactions is likely necessary for energy-momentum conservation to be observed, just as inclusion of the five-point vertex was in the NJL model. Moreover, consistent solution of Dyson-Schwinger equations for the dressings of these vertices is likely necessary for the gravitational WTI to be satisfied. The meson GFF results demonstrate several fascinating properties of not just the mesons themselves, but the field theoretical framework which gives rise to them. A low-energy pion theorem requires $C_{\pi}(0)\approx-1$, with the approximation becoming exact in the chiral limit. The NJL model correctly reproduces this (including the exactness in the chiral limit), and we found fully dressing the three-point graviton vertex by solving its inhomogeneous Bethe-Salpeter equation to be necessary to reproduce this behavior—an observation that has implications for model calculations of generalized parton distributions of the pion. We also observed the importance of correctly accounting for relativistic effects when describing spatial properties of light mesons, especially the pion—and more especially when considering the chiral limit. The Breit frame mass radius of the pion is significantly larger than its transverse light cone radius, only the latter of which is even finite in the chiral limit. We concur with previous literature that light cone coordinates are necessary to meaningfully define spatial distributions. We find that the light cone mass for the pion predicted by the NJL model agrees with a phenomenological extraction from KEKB data. In future work, we plan to extend the methods developed here to baryons. The results found here will be directly applicable within a quark-diquark model, where scalar and axial vector diquarks mimic closely the structure of pions and rho mesons. Additionally, the gravitational WTI we derived can be used as a consistency cross-check for the off-shell diquark-graviton vertex. ## Acknowledgements We would like to thank Rafael Badui, Wim Cosyn, Sabrina Cotogno, and Cédric Lorcé for illuminating discussions that helped contribute to our investigation. This work was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, contract no. DE-AC02-06CH11357. AF was supported by an LDRD initiative at Argonne National Laboratory under Project No. 2017-058-N0. ## Appendix A Bubbles in the NJL model The bubbles are defined using the following convention: \[\Pi_{XY}(s)=i(2N_{c})\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}} \mathrm{Tr}_{D}\left[\Omega_{X}S(k)\Omega_{Y}S(k-p)\right]\] (70) where $\Omega_{X}$ and $\Omega_{Y}$ represent particular vertices that appear in the bubble diagrams. A diagrammatic depiction of the bubble is given in Fig. 11, where the factor $i^{2}$ from the propagators cancels the factor $(-1)$ appearing because the diagram has a closed fermion loop. An overall factor $i$ is included in the definition (70) since this factor typically appears in diagrammatic equations involving bubbles as sub-diagrams, and this factor additionally makes the bubbles purely real below the two-particle production threshold. Specific bubbles that appear in this work are defined as follows: \[\Pi_{PP}(q^{2}) =(2N_{c})i\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{Tr}_{D} \left[\gamma_{5}S(k)\gamma_{5}S(k-q)\right]\] (71) \[\Pi_{SS}(p^{2}) =(2N_{c})i\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{Tr}\left[ S(k)S(k-p)\right]\] (72) \[\left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}\right)\Pi_{VV}(q^{2}) =(2N_{c})i\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{Tr}_{D} \left[\gamma^{\mu}S(k)\gamma^{\nu}S(k-q)\right]\] (73) \[\left(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}}\right)\Pi_{AA}^{(T) }(q^{2})+\frac{q^{\mu}q^{\nu}}{q^{2}}\Pi_{AA}^{(L)}(q^{2}) =(2N_{c})i\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{Tr}_{D} \left[\gamma^{\mu}\gamma_{5}S(k)\gamma^{\nu}\gamma_{5}S(k-q)\right]\] (74) \[\left(\frac{p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}}{M}\right)\Pi_{SG}(p^{ 2}) =(2N_{c})i\int\frac{\mathrm{d}^{4}k}{(2\pi)^{4}}\mathrm{Tr}\left[ S(k)\gamma_{Gqq}^{\mu\nu}(k,k-p)S(k-p)\right]\,.\] (75) <figure><img src="content_image/1903.09222/x15.png"><figcaption>Figure 11: Bubble diagram.</figcaption></figure> ## Appendix B Proof of the gravitational Ward-Takahashi identity We here prove the gravitational Ward-Takahashi identity given in Eq. (18), first proved for spin-zero fields in Brout and Englert (1966), is true for fields with arbitrary spin, provided the graviton couples to the canonical energy-momentum tensor. We must start with a definition of the fully dressed 3-point gravitational vertex function $\Gamma^{\mu\nu}(p^{\prime},p)$. The definition we use is: \[\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,e^{-i[(py) -(p^{\prime}z)+(qx)]}\langle 0\|\mathtt{T}\{T^{\mu\nu}(x)\phi_{r}(y)\phi_{s}(z) \}\|0\rangle\] (76) where $p$ and $p^{\prime}$ are respectively the initial and final momentum of a quantum of a field $\phi$, $S$ is the fully-dressed propagator of said field, $T^{\mu\nu}$ is the canonical energy-momentum tensor (EMT), and we explicitly notate the internal degrees of freedom of $\phi$ with indices $r,s$. We begin by contracting the left-hand side of Eq. (76) with $q_{\mu}$. Let us use $L^{\mu\nu}_{rs}(p^{\prime},p,q)$ as shorthand for the entire LHS, for better use of space. We have: \[q_{\mu}L^{\mu\nu}_{rs}(p^{\prime},p,q) =\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,q_{\mu}e^ {-i[(py)-(p^{\prime}z)+(qx)]}\langle 0\|\mathtt{T}\{T^{\mu\nu}(x)\phi_{r}(y) \phi_{s}(z)\}\|0\rangle\] \[=i\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,\Big{(} \partial_{\mu}^{(x)}e^{-i[(py)-(p^{\prime}z)+(qx)]}\Big{)}\langle 0\|\mathtt{T} \{T^{\mu\nu}(x)\phi_{r}(y)\phi_{s}(z)\}\|0\rangle\] \[=-i\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,e^{-i[( py)-(p^{\prime}z)+(qx)]}\Big{(}\partial_{\mu}^{(x)}\langle 0\|\mathtt{T}\{T^{ \mu\nu}(x)\phi_{r}(y)\phi_{s}(z)\}\|0\rangle\Big{)}\,.\] (77) Now, in differentiating this time-ordered product, we note that $\partial_{\mu}T^{\mu\nu}(0)=0$ as a consequence of Noether’s theorem. (We note that differentiation with respect to the first index is crucial here, as conservation with respect to the second index is not guaranteed.) Thus, any $x$ dependence in the time-ordered product can only come from the time-ordering itself. We have, explicitly: \[\partial_{\mu}^{(x)}\langle 0\|\mathtt{T}\{T^{\mu\nu}(x)\phi_{r}(y )\phi_{s}(z)\}\|0\rangle =\langle 0\|\mathtt{T}\left\{[T^{0\nu}(x),\phi_{r}(y)]\delta(x_{0} -y_{0})\phi_{s}(z)+[T^{0\nu}(x),\phi_{s}(z)]\delta(x_{0}-z_{0})\phi_{r}(y) \right\}\|0\rangle\] \[=\langle 0\|\mathtt{T}\left\{i\partial^{\nu}_{(y)}\phi_{r}(y)\phi_ {s}(z)\delta^{4}(x-y)+i\partial^{\nu}_{(z)}\phi_{s}(z)\phi_{r}(y)\delta^{4}(x- z)\right\}\|0\rangle\,,\] (78) where we have used the canonical commutation relation \[[T^{0\mu}(x),\phi_{r}(y)]\delta(x_{0}-y_{0})=i\partial^{\mu}\phi_ {r}(y)\delta^{4}(x-y)\,,\] (79) from which $P^{\mu}=\int\mathrm{d}^{3}x\,T^{0\mu}(x)$ follows. (Note that such a commutation relation does not hold for $T^{\mu 0}(x)$, giving us a second point at which privileging the first index is important.) From the definition of the propagator, we obtain: \[\partial_{\mu}^{(x)}\langle 0\|\mathtt{T}\{T^{\mu\nu}(x)\phi_{r}(y )\phi_{s}(z)\}\|0\rangle=i\partial^{\nu}_{(y)}S_{rs}(y-z)\delta^{4}(x-y)+i \partial^{\nu}_{(z)}S_{rs}(y-z)\delta^{4}(x-z)\,.\] (80) This enables us to find: \[q_{\mu}L^{\mu\nu}_{rs}(p^{\prime},p,q) =\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,e^{-i[(py )-(p^{\prime}z)+(qx)]}\Big{(}\partial^{\nu}_{(y)}S_{rs}(y-z)\delta^{4}(x-y)+ \partial^{\nu}_{(z)}S_{rs}(y-z)\delta^{4}(x-z)\Big{)}\] \[=-\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,e^{-i[( py)-(p^{\prime}z)+(qx)]}\Big{(}\partial^{\nu}_{(z)}S_{rs}(y-z)\delta^{4}(x-y)+ \partial^{\nu}_{(y)}S_{rs}(y-z)\delta^{4}(x-z)\Big{)}\] \[=-\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,e^{-i[( py)-(p^{\prime}z)+(qx)]}\Big{(}-ip^{\prime\nu}S_{rs}(y-z)\delta^{4}(x-y)+ip^{ \nu}S_{rs}(y-z)\delta^{4}(x-z)\Big{)}\,.\] (81) At this point, we compare the RHS of Eq. (76) and integrate over $q$. We have: \[\int\frac{\mathrm{d}^{4}q}{(2\pi)^{4}}q_{\mu}L^{\mu\nu}_{rs}(p^{ \prime},p,q) =-i\big{(}iS_{ss^{\prime}}(p^{\prime})\big{)}\Gamma^{\mu\nu}_{s^{ \prime}r}(p^{\prime},p)\big{(}iS_{r^{\prime}r}(p)\big{)}\] \[=i\int\mathrm{d}^{4}x\,\mathrm{d}^{4}y\,\mathrm{d}^{4}z\,e^{-i[( py)-(p^{\prime}z)]}\Big{(}p^{\prime\nu}S_{rs}(y-z)\delta^{4}(x-y)\delta^{4}(x) -p^{\nu}S_{rs}(y-z)\delta^{4}(x-z)\delta^{4}(x)\Big{)}\] \[=i\int\mathrm{d}^{4}y\,\Big{(}p^{\prime\nu}S_{rs}(-y)e^{i(p^{ \prime}y)}-p^{\nu}S_{rs}(y)e^{-i(py)}\Big{)}\] \[=i\Big{(}p^{\prime\nu}S_{rs}(p^{\prime})-p^{\nu}S_{rs}(p)\Big{)}\,.\] (82) Now, by cancelling out the factors $i$, multiplying by the inverse propagators, and dropping the $r$ and $s$ indices, we get Eq. (18), as required. ## Appendix C Proof of axial-gravitational correspondence in NJL model To prove Eq. (37) holds in the NJL model, we follow the derivation in Sec. 3.2 of Leader and Lorcé (2014) for QCD. Using the NJL model Lagrangian (1), we obtain the following equations of motion: \[i\overrightarrow{\not{\partial}}\psi =\left[m+2i\sum_{\Omega}G_{\Omega}(\overline{\psi}\Omega\psi) \Omega\right]\psi\] (83) \[i\overline{\psi}\overleftarrow{\not{\partial}}\equiv\hat{M}\psi =-\overline{\psi}\left[m+2i\sum_{\Omega}G_{\Omega}(\overline{\psi }\Omega\psi)\Omega\right]=\overline{\psi}\hat{M}\,.\] (84) From here, we use Eqs. (129) of Leader and Lorcé (2014) with $A_{\mu}=0$, which can be stated: \[\sigma^{\mu\nu}\overrightarrow{\not{\partial}} =2\gamma^{[\nu}\overrightarrow{\partial}^{\mu]}+i\epsilon^{\mu\nu \rho\sigma}\gamma_{\sigma}\gamma_{5}\overrightarrow{\partial}_{\rho}\] \[\overleftarrow{\not{\partial}}\sigma^{\mu\nu} =2\overleftarrow{\partial}^{[\nu}\gamma^{\mu]}+i\epsilon^{\mu\nu \rho\sigma}\gamma_{\sigma}\gamma_{5}\overleftarrow{\partial}_{\rho}\,.\] (85) With this, we find that: \[2\overline{\psi}\gamma^{[\nu}i\overleftrightarrow{\partial}^{\mu ]}\psi=\overline{\psi}[\hat{M},\sigma^{\mu\nu}]\psi+\epsilon^{\mu\nu\rho\sigma }\partial_{\rho}\Big{(}\overline{\psi}\gamma_{\sigma}\gamma_{5}\psi\Big{)}\,.\] (86) What remains to be shown is that $\overline{\psi}[\hat{M},\sigma^{\mu\nu}]\psi=0$. Using the identities: \[[1,\sigma^{\mu\nu}] =0\] (87) \[=0\] (88) \[=2i\Big{(}g^{\mu\pi}\gamma^{\nu}-g^{\nu\pi}\gamma^{\mu}\Big{)}\] (89) \[=2i\Big{(}g^{\mu\pi}\gamma^{\nu}-g^{\nu\pi}\gamma^{\mu}\Big{)} \gamma_{5}\] (90) \[=2i\Big{(}\sigma^{\pi\mu}g^{\rho\nu}+\sigma^{\rho\nu}g^{\pi\mu}- \sigma^{\pi\nu}g^{\rho\mu}-\sigma^{\rho\mu}g^{\pi\nu}\Big{)}\,,\] (91) we find: \[[\hat{M},\sigma^{\mu\nu}] =-2iG_{\omega}\overline{\psi}\gamma^{[\mu}\psi\gamma^{\nu]}-2iG_{ f}\overline{\psi}\gamma^{[\mu}\gamma_{5}\psi\gamma^{\nu]}\gamma_{5}-2iG_{\rho} \left(\overline{\psi}\gamma^{[\mu}\tau_{i}\psi\gamma^{\nu]}\tau_{i}+\overline{ \psi}\gamma^{[\mu}\gamma_{5}\tau_{i}\psi\gamma^{\nu]}\gamma_{5}\tau_{i}\right)\] \[-4iG_{T}\Big{(}(\overline{\psi}i\sigma^{\pi[\mu}\psi)-(\overline{ \psi}i\sigma^{\pi[\mu}\tau_{i}\psi)\tau_{i}\Big{)}\sigma^{\nu]\pi}\] (92) Finally, sandwiching this between $\psi$ and $\overline{\psi}$, we get \[\overline{\psi}[\hat{M},\sigma^{\mu\nu}]\psi =-2iG_{\omega}\overline{\psi}\gamma^{[\mu}\psi\overline{\psi} \gamma^{\nu]}\psi-2iG_{f}\overline{\psi}\gamma^{[\mu}\gamma_{5}\psi\overline{ \psi}\gamma^{\nu]}\gamma_{5}\psi-2iG_{\rho}\left(\overline{\psi}\gamma^{[\mu} \tau_{i}\psi\overline{\psi}\gamma^{\nu]}\tau_{i}\psi+\overline{\psi}\gamma^{[ \mu}\gamma_{5}\tau_{i}\psi\overline{\psi}\gamma^{\nu]}\gamma_{5}\tau_{i}\psi\right)\] \[-4iG_{T}\Big{(}(\overline{\psi}i\sigma^{\pi[\mu}\psi)(\overline{ \psi}i\sigma^{\nu]\pi}\psi)-(\overline{\psi}i\sigma^{\pi[\mu}\tau_{i}\psi)( \overline{\psi}i\sigma^{\nu]\pi}\tau_{i}\psi)\Big{)}\,,\] (93) an expression which is clearly both symmetric and antisymmetric under the swap $(\mu\leftrightarrow\nu)$, and which is therefore zero. 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1512.02016	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 46777, "num_imgs": 4, "llama3_tokens_count": 13696 }	[ "content_image/1512.02016/model.png", "content_image/1512.02016/apfig.png", "content_image/1512.02016/burnin.png", "content_image/1512.02016/apburnin.png" ]	# Discriminative Nonparametric Latent Feature Relational Models with Data Augmentation Bei Chen${}^{\dagger}$, Ning Chen${}^{\ddagger}$, Jun Zhu${}^{\dagger}$, Jiaming Song${}^{\dagger}$, Bo Zhang${}^{\dagger}$ ${}^{\dagger}$Dept. of Comp. Sci. & Tech., State Key Lab of Intell. Tech. & Sys., Center for Bio-Inspired Computing Research, ${}^{\ddagger}$MOE Key lab of Bioinformatics, Bioinformatics Division and Center for Synthetic & Systems Biology, TNList, Tsinghua University, Beijing, 100084, China {chenbei12@mails., ningchen@, dcszj@, sjm12@mails., dcszb@}tsinghua.edu.cn Corresponding authors. ###### Abstract We present a discriminative nonparametric latent feature relational model (LFRM) for link prediction to automatically infer the dimensionality of latent features. Under the generic RegBayes (regularized Bayesian inference) framework, we handily incorporate the prediction loss with probabilistic inference of a Bayesian model; set distinct regularization parameters for different types of links to handle the imbalance issue in real networks; and unify the analysis of both the smooth logistic log-loss and the piecewise linear hinge loss. For the nonconjugate posterior inference, we present a simple Gibbs sampler via data augmentation, without making restricting assumptions as done in variational methods. We further develop an approximate sampler using stochastic gradient Langevin dynamics to handle large networks with hundreds of thousands of entities and millions of links, orders of magnitude larger than what existing LFRM models can process. Extensive studies on various real networks show promising performance. ## Introduction Link prediction is a fundamental task in statistical network analysis. For static networks, it is defined as predicting the missing links from a partially observed network topology (and some attributes if exist). Existing approaches include: 1) Unsupervised methods that design good proximity/similarity measures between nodes based on network topology features [18], e.g., common neighbors, Jaccard’s coefficient [25], Adamic/Adar [1], etc; 2) Supervised methods that learn classifiers on labeled data with a set of manually designed features [19, 11, 27]; 3) others [4] that use random walks to combine the network structure information with node and edge attributes. One possible limitation for such methods is that they rely on well-designed features or measures, which can be time demanding to get and/or application specific. Latent variable models [12, 13, 5] have been widely applied to discover latent structures from complex network data, based on which prediction models are developed for link prediction. Although these models work well, one remaining problem is how to determine the unknown number of latent classes or features. A typical way using model selection, e.g., cross-validation or likelihood ratio test [20], can be computationally prohibitive by comparing many candidate models. Bayesian nonparametrics has shown promise in bypassing model selection by imposing an appropriate stochastic process prior on a rich class of models [3, 10]. For link prediction, the infinite relational model (IRM) [15] is class-based and uses Bayesian nonparametrics to discover systems of related concepts. One extension is the mixed membership stochastic blockmodel (MMSB) [2], which allows entities to have mixed membership. [22] and [32] developed nonparametric latent feature relational models (LFRM) by incorporating Indian Buffet Process (IBP) prior to resolve the unknown dimension of a latent feature space. Though LFRM has achieved promising results, exact inference is intractable due to the non-conjugacy of the prior and link likelihood. One has to use Metropolis-Hastings [22], which may have low accept rates if the proposal distribution is not well designed, or variational inference [32] with truncated mean-field assumptions, which may be too strict in practice. In this paper, we develop discriminative nonparametric latent feature relational models (DLFRM) by exploiting the ideas of data augmentation with simpler Gibbs sampling [23, 24] under the regularized Bayesian inference (RegBayes) framework [31]. Our major contributions are: 1) We use the RegBayes framework for DLFRM to deal with the imbalance issue in real networks and naturally analyze both the logistic log-loss and the max-margin hinge loss under a unified setting; 2) We explore data augmentation techniques to develop a simple Gibbs sampling algorithm, which is free from unnecessary truncation and assumptions that typically exist in variational approximation methods; 3) We develop an approximate Gibbs sampler using stochastic gradient Langevin dynamics, which can handle large networks with hundreds of thousands of entities and millions of links (See Table 1), orders of magnitude larger than what the existing LFRM models [22, 32] can process; and 4) Finally, we conduct experimental studies on a wide range of real networks and the results demonstrate promising results of our methods. ## Nonparametric LFRM Models We consider static networks with $N$ entities. Let $Y$ be the $N\times N$ binary link indicator matrix, where $y_{ij}=1$ denotes the existence of a link from entity $i$ to $j$, and $y_{ij}=-1$ denotes no link from $i$ to $j$. $Y$ is not fully observed. <figure><img src="content_image/1512.02016/model.png"><figcaption>Figure 1: The graphical structure of LFRM.</figcaption></figure> Our goal is to learn a model from the partially observed links and predict the values of the unobserved entries of $Y$. Fig. 1 illustrates a latent feature relational model (LFRM), where each entity is represented by $K$ latent features. Let $Z$ be the $N\times K$ feature matrix, each row is associated with an entity and each column corresponds to a feature. We consider the binary features¹: If entity $i$ has feature $k$, then $z_{ik}=1$, otherwise $z_{ik}=0$. Let $Z_{i}$ be the feature vector of entity $i$, $U$ be a $K\times K$ real-valued weight matrix, $\eta=\textrm{vec}(U)$ and $Z_{ij}=\textrm{vec}(Z_{i}^{\top}Z_{j})$, where $\textrm{vec}(A)$ is a vector concatenating the row vectors of matrix $A$. Note that $\eta$ and $Z_{ij}$ are column vectors, while $Z_{i}$ is a row vector. Then the probability of the link from entity $i$ to $j$ is [FOOTNOTE:1][ENDFOOTNOTE] \[p(y_{ij}=1\|Z_{i},Z_{j},U)=\sigma\left(Z_{i}UZ_{j}^{\top}\right)=\sigma\left( \eta^{\top}Z_{ij}\right),\] (1) where$\sigma(x)\!\!=\!\!\frac{1}{1+\exp(-x)}$ is the sigmoid function. We assume that links are conditionally independent given $Z$ and $U$, then the link likelihood is$p(Y\|Z,U)=\prod_{(i,j)\in\mathcal{I}}p(y_{ij}\|Z_{i},Z_{j},U)$, where $\mathcal{I}$ is the set of training links (observed links). In the above formulation, we assume that the dimensionality of the latent features $K$ is known a priori. However, this assumption is often unrealistic especially when dealing with large-scale applications. The conventional approaches that usually need a model selection procedure (e.g., cross validation) to choose an appropriate value by trying on a large set of candidates can be expensive and often require extensive human efforts on guiding the search. Recent progress on Bayesian optimization [28] provides more effective solution to searching for good parameters, but still needs to learn many models under different configurations of the hyper-parameter $K$. In this paper, we focus on the nonparametric Bayesian methods [10] for link prediction. The recently developed nonparametric latent feature relational models (LFRM) [22] leverage the advancement of Bayesian nonparametric methods to automatically resolve the unknown dimensionality of the feature space by applying a flexible nonparametric prior. It assumes that each entity $i$ has an infinite number of binary features, that is $Z_{i}\in\{0,1\}^{\infty}$, and the Indian Buffet Process (IBP) [10] is used as a prior of $Z$ to produce a sparse latent feature vector for each entity. We treat the weight matrix $U$ as random and put a prior on it for fully Bayesian inference. Then with Bayes’ theorem, the posterior distribution is \[q(Z,U\|Y)\propto p_{0}(Z)p_{0}(U)p(Y\|Z,U),\] (2) where the prior $p_{0}(Z)$ is an IBP and $p_{0}(U)$ is often assumed to be an isotropic Gaussian prior. ### Discriminative LFRM Models The conventional Bayesian inference as above relies on Bayes’ rule to infer the posterior distribution. In fact, this procedure can be equivalently formulated as solving an optimization problem. For example, the Bayes posterior in Eq. (2) is equivalent to the solution of the following problem: \[\min_{q(Z,U)\in\mathcal{P}}{\!\!\rm{KL}}(q(Z,U)\|\|p_{0}(Z,U))\!-\! \mathbb{E}_{q}[\log p(Y\|Z,U)],\] (3) where $\mathcal{P}$ is the space of well-defined distributions and ${\rm{KL}}(q\|\|p)$ is the Kullback-Leibler (KL) divergence from $q$ to $p$. Such an optimization view has inspired the development of regularized Bayesian inference (RegBayes) which solves: \[\min_{q(Z,U)\in\mathcal{P}}{\rm{KL}}(q(Z,U)\|\|p_{0}(Z,U))+c\cdot \mathcal{R}(q(Z,U)),\] (4) where $\mathcal{R}(q)$ is a posterior regularization defined on the target posterior distribution and $c$ is a non-negative regularization parameter that balances the prior part and the posterior regularization part. We refer the readers to [31] for more details on a generic representation theorem of the solution and its application [30, 21] to learn latent feature models for classification. Below, we explore the ideas to develop effective latent feature relational models for link prediction. Although we could define an averaging classifier and make predictions using the sign rule $\hat{y}_{ij}\!\!=\!\!\textrm{sign}(\mathbb{E}_{q}[Z_{i}UZ_{j}^{\top}\!])$, the resulting problem needs to be approximately solved by truncated variational methods, which can be inaccurate in practice. Here, we propose to define a Gibbs classifier, which admits simple and efficient sampling algorithms that are guaranteed to be accurate. Our Gibbs sampler randomly draws the latent variables $(Z,U)$ from the unknown but pre-assumed to be given posterior distribution $q(Z,U)$. Once $Z$ and $U$ are given, we can make predictions using the sign rule $\hat{y}_{ij}\!\!=\!\!\textrm{sign}(Z_{i}UZ_{j}^{\top})$ and measure the training error $r(Z,U)=\sum_{(i,j)\in\mathcal{I}}{\mathbb{I}}(y_{ij}\neq\hat{y}_{ij})$, where ${\mathbb{I}}(\cdot)$ is an indicator function. Since the training error is non-smooth and non-convex, it is often relaxed by a well-behaved loss function. Let $\omega_{ij}=Z_{i}UZ_{j}^{\top}$, two well-studied examples are the logistic log-loss $r_{1}$ and the hinge loss $r_{2}$: \[r_{1}(Z,U) = -\sum_{(i,j)\in\mathcal{I}}\log p(\tilde{y}_{ij}\|Z_{i},Z_{j},U),\] \[r_{2}(Z,U) = \sum_{(i,j)\in\mathcal{I}}(\ell-{y}_{ij}\omega_{ij})_{+},\] where $p(\tilde{y}_{ij}\|Z_{i},Z_{j},U)=\frac{e^{\omega_{ij}\tilde{y}_{ij}}}{1+e^{ \omega_{ij}}}$, $(x)_{+}:=\max(0,x)$, $\ell$ is the pre-defined cost to penalize a wrong prediction, and $\tilde{y}_{ij}=(y_{ij}+1)/2$ so that $0$ refers to a negative link instead of $-1$. To account for the uncertainty of the latent variables, we define the posterior regularization as the expected loss: With these posterior regularization functions, we can do the RegBayes as in problem (4), where the parameter $c$ balances the influence between the prior distribution (i.e., $\mathrm{KL}$ divergence) and the observed link structure (i.e., the loss term). We define the un-normalized pseudo link likelihood: \[\varphi_{1}(\tilde{y}_{ij}\|Z_{i},Z_{j},U) = \frac{(e^{\omega_{ij}})^{c\tilde{y}_{ij}}}{(1+e^{\omega_{ij}})^{c }},\] (5) \[\varphi_{2}({y}_{ij}\|Z_{i},Z_{j},U) = \exp(-2c(\ell-{y}_{ij}\omega_{ij})_{+}).\] (6) Then problem (4) can be written in the equivalent form: \[\min_{q(Z,U)\in\mathcal{P}}{\!\!\!\rm{KL}}(q(Z,U)\|\|p_{0}(Z,U))-\mathbb{E}_{q}[ \log\varphi(Y\|Z,U)],\] (7) where $\varphi(Y\|Z,U)=\prod_{i,j\in\mathcal{I}}\varphi(y_{ij}\|Z_{i},Z_{j},U)$ and $\varphi$ can be $\varphi_{1}$ or $\varphi_{2}$. Then the optimal solution of (4) or (7) is the following posterior distribution with link likelihood: \[q(Z,U\|Y)\propto p_{0}(Z)p_{0}(U)\varphi(Y\|Z,U).\] (8) Notice that if adopting the logistic log-loss, we actually obtain a generalized pseudo-likelihood which is a powered form of likelihood in Eq. (1). For real networks, positive links are often highly sparse as shown in Table 1. Such sparsity could lead to serious imbalance issues in supervised learning, where the negative examples are much more than positive examples. In order to deal with the imbalance issue in network data and make the model more flexible, we perform RegBayes by controlling the regularization parameter. For example, we can choose a larger $c$ value for the fewer positive links and a relatively smaller $c$ for the larger negative links. This strategy has shown effective in dealing with imbalanced data in [6, 32]. We will provide experiments to demonstrate the benefits of RegBayes on dealing with imbalanced networks when learning nonparametric LFRMs. ## Gibbs Sampling with Data Augmentation As we do not have a conjugate prior on $U$, exact posterior inference is intractable. Previous inference methods for nonparametric LFRM use either Metropolis-Hastings [22] or variational techniques [32] which can be either inefficient or too strict in practice. We explore the ideas of data augmentation to give the pseudo-likelihood a proper design, so that we can directly obtain posterior distributions and develop efficient Gibbs sampling algorithms. Specifically, our algorithm relies on the following unified representation lemma. Lemma 1.: _Both $\varphi_{1}$ and $\varphi_{2}$ can be represented as_ \[\varphi(y_{ij}\|Z_{i},Z_{j},U)\propto\int_{0}^{\infty}\!\!\exp{ \Big{(}\!\kappa_{ij}\omega_{ij}\!-\!\frac{\rho_{ij}\omega_{ij}^{2}}{2}\Big{)}} \!\phi(\lambda_{ij}){\rm{d}}\lambda_{ij},\] _where for $\varphi_{1}$ we have_ \[\kappa_{ij}=c(\tilde{y}_{ij}-\frac{1}{2}),~{}\rho_{ij}=\lambda_{ij},~{}\phi( \lambda_{ij})=\mathcal{PG}(\lambda_{ij};c,0);\] _while for $\varphi_{2}$, let $\gamma_{ij}=\lambda_{ij}^{-1}$, we have_ \[\kappa_{ij}=c{y}_{ij}(1+c\ell\gamma_{ij}),~{}\rho_{ij}=c^{2}\gamma_{ij},~{} \phi(\lambda_{ij})=\mathcal{GIG}(\frac{1}{2},1,c^{2}\ell^{2}).\] We have used $\mathcal{PG}$ to denote a Polya-Gamma distribution [24] and $\mathcal{GIG}$ to denote a generalized inverse Gaussian distribution. We defer the proof to Appendix A², which basically follows [24, 23] with some algebraic manipulation on re-organizing the terms. [FOOTNOTE:2][ENDFOOTNOTE] ``` Init: draw $Z$ from IBP, $U$ from $\mathcal{N}(0,\nu^{-2})$; set $\lambda=1$. for $iter=1,2,\dots,L$ do for $n=1,2,\dots,N$ do draw $\{z_{nk}\}_{k=1}^{K}$ from Eq. (11); draw $k_{n}$ using Eq. (12). if $k_{n}>0$ then update $K\gets K+k_{n}$, update new weights; end if end for draw $U$ using Eq. (13) and draw $\lambda$ using Eq. (14). end for ``` Algorithm 1* Gibbs sampler for DLFRM ### Sampling Algorithm Lemma 1 suggests that the pseudo-likelihood $\varphi$ can be considered as the marginal of a higher dimensional distribution that includes the augmented variables $\lambda$: \[\psi(\lambda,Y\|Z,U)\!\propto\!\!\prod_{(i,j)\in\mathcal{I}}\exp{ \left(\kappa_{ij}\omega_{ij}-\frac{\rho_{ij}\omega_{ij}^{2}}{2}\right)}\phi( \lambda_{ij}),\] (9) which is a mixture of Gaussian components of $U$ once $Z$ is given, suggesting that we can effectively perform Gibbs sampling if a conjugate Gaussian prior is imposed on $U$. We also construct the complete posterior distribution: \[q(Z,U,\lambda\|Y)\propto p_{0}(Z)p_{0}(U)\psi(\lambda,Y\|Z,U),\] (10) such that our target posterior $q(Z,U\|Y)$ is a marginal distribution of the complete posterior. Therefore, if we can draw a set of samples $\{(Z_{t},U_{t},\lambda_{t})\}_{t=1}^{L}$ from the complete posterior, by dropping the augmented variables, the rest samples $\{(Z_{t},U_{t})\}_{t=1}^{L}$ are drawn from the target posterior $q(Z,U\|Y)$. This technique allows us to sample the complete posterior via a Gibbs sampling algorithm, as outlined in Alg. 1 and detailed below. For $Z$: We assume the Indian Buffet Process (IBP) prior on the latent feature $Z$. Although the total number of latent features is infinite, every time we only need to store $K$ active features that are not all zero in the columns of $Z$. When sampling the $n$-th row, we need to consider two cases, due to the nonparametric nature of IBP. First, for the active features, we sample $z_{nk}(k=1,...,K)$ in succession from the following conditional distribution \[q(z_{nk}\|Z_{-nk},\eta,\lambda)\propto p(z_{nk})\psi(\lambda,Y\|Z_{-nk},\eta,z_{ nk}),\] (11) where $p(z_{nk}=1)\propto m_{-n,k}$ and $m_{-n,k}$ is the number of entities containing feature $k$ except entity ${n}$. Second, for the infinite number of remaining all-zero features, we sample $k_{n}$ number of new features and add them to the $n$th row. Then we get the new $N\times(K+k_{n})$ matrix $Z^{}$ which becomes old when sampling the $(n+1)$-th row. Every time when the number of features changes, we also update $U$ and extend it to a $(K+k_{n})\times(K+k_{n})$ matrix $U^{}$. Let $Z^{\prime}$ and $U^{\prime}$ be the parts of $Z^{}$ and $U^{}$ that correspond to the $k_{n}$ new features. Also, we define $\eta^{\prime}=\textup{vec}(U^{\prime})$. During implementation, we can delete the all-zero columns after every resampling of $Z$, but here we ignore it. Let $\eta‘$ follow the isotropic Normal prior $\mathcal{N}(0,\nu^{-2})$. Now the conditional distribution for $k_{n}=0$ is $p(k_{n}=0\|Z,\eta,\lambda)=p_{0}(k_{n})$, and the probability of $k_{n}\not=0$ is \[p(k_{n}\not=0\|Z,\eta,\lambda)\!=\!p_{0}(k_{n})\|\Sigma\|^{\frac{1} {2}}\nu^{D}\!\exp{\Big{(}\frac{1}{2}\mu^{\top}\Sigma^{-1}\mu\Big{)}},\] (12) where $p_{0}(k_{n})=\textup{Poisson}\left(k_{n};\frac{\alpha}{N}\right)$ is from the IBP prior, $D=2k_{n}K+k_{n}^{2}$ is the dimension of $\eta^{\prime}$ and the mean $\mu=\Sigma(\sum_{(i,j)\in\mathcal{I}}(\kappa_{ij}-\rho_{ij}\omega_{ij})Z_{ij}^ {\prime})$ , covariance $\Sigma=(\sum_{(i,j)\in\mathcal{I}}\rho_{ij}Z_{ij}^{\prime}{Z_{ij}^{\prime}}^{ \top}+{\nu^{2}}I)^{-1}$. We compute the probabilities for $k_{n}\!=\!0,1,...,K_{max}$, do normalization and sample from the resulting multinomial. Here, $K_{max}$ is the maximum number of features to add. Once we have added $k_{n}\!(\not=\!\!\!0)$ new features, we should also sample their weights $\eta^{\prime}$, which follow a $D$ dimensional multivariate Gaussian, in order to resample the next row of $Z$. For $U$: After the update of $Z$, we resample $U$ given the new $Z$. Let $\tilde{D}=K\times K$ and $\eta$ follow the isotropic Normal prior $p_{0}(\eta)=\prod_{d=1}^{\tilde{D}}\mathcal{N}(\eta_{d};0,\nu^{-2})$. Then the posterior is also a Gaussian distribution \[q(\eta\|\lambda,Z)\propto p_{0}(\eta)\psi(\lambda,Y\|Z,\eta)= \mathcal{N}(\eta;\tilde{\mu},\tilde{\Sigma}),\] (13) with the mean $\tilde{\mu}=\tilde{\Sigma}(\sum_{(i,j)\in\mathcal{I}}\kappa_{ij}Z_{ij})$ and the convariance $\tilde{\Sigma}=(\sum_{(i,j)\in\mathcal{I}}\rho_{ij}Z_{ij}Z_{ij}^{\top}+{\nu^{2 }}I)^{-1}$. For $\boldsymbol{\lambda}$: Since the auxiliary variables are independent given the new $Z$ and $U$, we can draw each $\lambda_{ij}$ separately. From the unified representation, we have \[q(\lambda_{ij}\|Z,\eta)\propto\exp{\Big{(}\kappa_{ij}\omega_{ij}- \frac{\rho_{ij}\omega_{ij}^{2}}{2}\Big{)}}\phi(\lambda_{ij}).\] (14) By doing some algebra, we can get the following equations. For $\varphi_{1}$, $\lambda_{ij}$ still follows a Polya-Gamma distribution $q(\lambda_{ij}\|Z,\eta)=\mathcal{PG}(\lambda_{ij};c,\omega_{ij})$, from which a sample can be efficiently drawn. For $\varphi_{2}$, $\lambda_{ij}$ follows a generalized inverse Gaussian distribution $q(\lambda_{ij}\|Z,U,Y)=\mathcal{GIG}(\frac{1}{2},1,c^{2}\zeta_{ij}^{2})$, where $\zeta_{ij}=\ell-y_{ij}\omega_{ij}$. Then $\gamma_{ij}:=\lambda_{ij}^{-1}$ follows an inverse Gaussian distribution $q(\gamma_{ij}\|Z,U,Y)=\mathcal{IG}(\frac{1}{c\|\zeta_{ij}\|},1)$, from which a sample can be easily drawn in a constant time. ### Stochastic Gradient Langevin Dynamics Alg. 1 needs to sample from a $K^{2}$-dim Gaussian distribution to get $U$, where $K$ is the latent feature dimension. This procedure is prohibitively expensive for large networks when $K$ is large (e.g., $K>40$). To address this problem, we employ stochastic gradient Langevin dynamics (SGLD) [29], an efficient gradient-based MCMC method that uses unbiased estimates of gradients with random mini-batches. Let $\theta$ denote the model parameters and $p(\theta)$ is a prior distribution. Given a set of i.i.d data points $\mathcal{D}=\{x_{i}\}_{i=1}^{M}$, the likelihood is $p(\mathcal{D}\|\theta)=\prod_{i=1}^{M}p(x_{i}\|\theta)$. At each iteration $t$, the update equation for $\theta$ is: \[\Delta\theta_{t}\!=\!\frac{\epsilon_{t}}{2}\Big{(}\nabla\log p(\theta_{t})+ \frac{M}{m}\!\!\sum_{x_{i}\in{\mathcal{D}_{t}}}\!\!\nabla\log p(x_{i}\|\theta_{ t})\Big{)}+\delta_{t},\] (15) where $\epsilon_{t}$ is the step size, ${\mathcal{D}_{t}}$ is a subset of ${\mathcal{D}}$ with size $m$ and $\delta_{t}\sim\mathcal{N}(0,\epsilon_{t})$ is the Gaussian noise. When the stepsize is annealed properly, the Markov chain will converge to the true posterior distribution. Let ${\mathcal{I}_{t}}$ be a subset of ${\mathcal{I}}$ with size $m$. We can apply SGLD to sample $\eta$ (i.e., $U$). Specifically, according to the true posterior of $\eta$ as in Eq. (13), the update rule is: \[\Delta\eta_{t}\!=\!\frac{\epsilon_{t}}{2}\Big{(}-{\nu^{2}}\eta_{t}+\frac{\| \mathcal{I}\|}{m}\!\!\sum_{(i,j)\in{\mathcal{I}_{t}}}\!\!(\kappa_{ij}-\rho_{ij} \omega_{ij})Z_{ij}\Big{)}+\delta_{t},\] (16) where $\delta_{t}$ is a $K^{2}$-dimensional vector and each entry is a Gaussian noise. After a few iterations, we will get the approximate sampler of $\eta$ (i.e., $U$) very efficiently. Dataset \| NIPS \| Kinship \| WebKB \| AstroPh \| Gowalla ---\|---\|---\|---\|---\|--- Entities \| 234 \| 104 \| 877 \| 17,903 \| 196,591 Positive Links \| 1,196 \| 415 \| 1,608 \| 391,462 \| 1,900,654 Sparsity Rate \| 2.2% \| 4.1% \| 0.21% \| 0.12% \| 0.0049% Table 1: Statistics of datasets. ## Experiments We present experimental results to demonstrate the effectiveness of DLFRM on five real datasets as summarized in Table 1, where NIPS contains $234$ authors who have the most coauthor-relationships with others from NIPS $1$-$17$; Kinship includes $26$ relationships of $104$ people in the Alyawarra tribe in central Australia; WebKB contains $877$ webpages from the CS departments of different universities, where the dictionary has $1,703$ unique words; AstroPh contains collaborations between $17,903$ authors of papers submitted to Arxiv Astro Physics in the period from Jan. 1993 to Apr. 2003 [17]; and Gowalla contains $196,591$ people and their friendships on Gowalla social website [7]. All these real networks have very sparse links. We evaluate three variants of our model: (1) DLFRM: to overcome the imbalance issue, we set $c^{+}=10c^{-}=c$ as in [32], where $c^{+}$ is the regularization parameter for positive links and $c^{-}$ for negative links. We use a full asymmetric weight matrix $U$; (2) stoDLFRM: the DLFRM model that uses SGLD to sample weight matrix $U$, where the stepsizes are set by $\epsilon_{t}=a(b+t)^{-\gamma}$ for log-loss and AdaGrad [9] for hinge loss; (3) diagDLFRM: the DLFRM that uses a diagonal weight matrix $U$. Each variant can be implemented with the logistic log-loss or hinge loss, denoted by the superscript $l$ or $h$. We randomly select a development set from training set with almost the same number of links as testing set and choose the proper hyper-parameters, which are insensitive in a wide range. All the results are averaged over $5$ runs with random initializations and the same group of parameters. ### Results on Small Networks We first report the prediction performance (AUC scores) on three relatively small networks. For fair comparison, we follow the previous settings to randomly choose $80\%$ of the links for training and use the remaining $20\%$ for testing. AUC score is the area under the Receiver Operating Characteristic (ROC) curve; higher is better. #### NIPS Coauthorship Prediction Table 2 shows the AUC scores on NIPS dataset, where the results of baselines (i.e., LFRM, IRM, MMSB, MedLFRM and BayesMedLFRM) are cited from [22, 32]. We can see that both DLFRM${}^{l}$ and DLFRM${}^{h}$ outperform all other models, which suggests that our exact Gibbs sampling with data augmentation can lead to more accurate models than MedLFRM / BayesMedLFRM that uses the variational approximation methods with truncated mean-field assumptions. The stoDLFRMs obtain comparable results to DLFRMs, which suggests that approximate sampler for $\eta$ using SGLD is very effective. With SGLD, we can improve efficiency without sacrificing performance which we will discuss later with Table 4. Furthermore, diagDLFRM${}^{l}$ and diagDLFRM${}^{h}$ also perform well, as they beat all other methods except (sto)DLFRMs. By using a lower dimensional $\eta$ derived from the diagonal weight matrix $U$, diagDLFRM has the advantage of being computationally efficient, as shown in Fig. 3(d). The good performance of stoDLFRMs and diagDLFRMs suggests that we can use SGLD with a full weight matrix or simply use a diagonal weight matrix on large-scale networks. Models \| NIPS \| Kinship ---\|---\|--- MMSB \| 0.8705 ± − \| 0.9005 ± 0.0022 IRM \| 0.8906 ± − \| 0.9310 ± 0.0023 LFRM rand \| 0.9466 ± − \| 0.9443 ± 0.0018 LFRM w / IRM \| 0.9509 ± − \| 0.9346 ± 0.0013 MedLFRM \| 0.9642 ± 0.0026 \| 0.9552 ± 0.0065 BayesMedLFRM \| 0.9636 ± 0.0036 \| 0.9547 ± 0.0028 DLFRMl \| 0.9812 ± 0.0013 \| 0.9650 ± 0.0032 stoDLFRMl \| 0.9804 ± 0.0007 \| 0.9673 ± 0.0044 diagDLFRMl \| 0.9717 ± 0.0031 \| 0.9426 ± 0.0028 DLFRMh \| 0.9806 ± 0.0027 \| 0.9640 ± 0.0023 stoDLFRMh \| 0.9787 ± 0.0012 \| 0.9657 ± 0.0031 diagDLFRMh \| 0.9722 ± 0.0021 \| 0.9440 ± 0.0038 Table 2: AUC on the NIPS coauthorship and Kinship dataset. #### Kinship Multi-relation Prediction For multi-relational Kinship dataset, we consider the “single” setting [22], where we infer an independent set of latent features for each relation. The overall AUC is obtained by averaging the results of all relations. As shown in Table 2, both (sto)DLFRM${}^{l}$ and (sto)DLFRM${}^{h}$ outperform all other methods, which again proves the effectiveness of our methods. Furthermore, the diagonal variants also obtain fairly good results, close to the best baselines. Finally, the better results by the discriminative methods in general demonstrate the effect of RegBayes on using various regularization parameters to deal with the imbalance issue; Fig. 3(b) provides a detailed sensitivity analysis. #### WebKB Hyperlink Prediction We also examine how DLFRMs perform on WebKB network, which has rich text attributes [8]. Our baselines include: 1) Katz: a proximity measure between two entities—it directly sums over all collection of paths, exponentially damped by the path length to count short paths more heavily [18]; 2) Linear SVM: a supervised learning method using linear SVM, where the feature for each link is a vector concatenating the bag-of-words features of two entities; 3) RBF-SVM: SVM with the RBF kernel on the same features as the linear SVM. We use SVM-Light [14] to train these classifiers; and 4) MedLFRM: state-of-the-art methods on learning latent features for link prediction [32]. Note that we don’t compare with the relational topic models [5, 6], whose settings are quite different from ours. Table 3 shows the AUC scores of various methods. We can see that: 1) both MedLFRM and DLFRMs perform better than SVM classifiers on raw bag-of-words features, showing the promise of learning latent features for link prediction on document networks; 2) DLFRMs are much better than MedLFRM³, suggesting the advantages of using data augmentation techniques for accurate inference over variational methods with truncated mean-field assumptions; and 3) both stoDLFRMs and diagDLFRMs achieve competitive results with faster speed. [FOOTNOTE:3][ENDFOOTNOTE] Models \| WebKB ---\|--- Katz \| 0.5625 ± − Linear SVM \| 0.6889 ± − RBF SVM \| 0.7132 ± − MedLFRM \| 0.7326 ± 0.0010 DLFRMl \| 0.8039 ± 0.0057 stoDLFRMl \| 0.8044 ± 0.0058 diagDLFRMl \| 0.7954 ± 0.0085 DLFRMh \| 0.8002 ± 0.0073 stoDLFRMh \| 0.7966 ± 0.0013 diagDLFRMh \| 0.7900 ± 0.0056 Table 3: AUC scores on the WebKB dataset. ### Results on Large Networks We now present results on two much larger networks. As the networks are much sparser, we randomly select $90\%$ of the positive links for training and the number of negative training links is $10$ times the number of positive training links. The testing set contains the remaining $10\%$ of the positive links and the same number of negative links, which we uniformly sample from the negative links outside the training set. This test setting is the same as that in [16]. <figure><img src="content_image/1512.02016/apfig.png"><figcaption>Figure 2: AUC scores on the AstroPh dataset.</figcaption></figure> {tabu} X[0.6l]—X[1.2c]—X[1.2c] Models & DLFRMl &stoDLFRMl Sample Z &16312.0 (25.58%)& 32095.9 (95.18%) Sample U & 47389.9 (74.32%) &1516.4 (4.50%) Sample λ & 65.7 (0.10%) & 109.0 (0.32%) Table 4: Split of training time (sec) on AstroPh dataset. #### AstroPh Collaboration Prediction Fig. 2 presents the test AUC scores, where the results of the state-of-the-art nonparametric models aMMSB (assortative MMSB) and aHDPR (assortative HDP relational model, a nonparametric generalization of aMMSB) are cited from [16]. We can see that DLFRMs achieve significantly better AUCs than aMMSB and aHDPR, which again demonstrates that our models can not only automatically infer the latent dimension, but also learn the effective latent features for entities. Furthermore, stoDLRMs and diagDLFRMs show larger benefits on the larger networks due to the efficiency. As shown in Table 4, the time for sampling $U$ is greatly reduced with SGLD. It only accounts for $4.50\%$ of the whole time for stoDLFRM${}^{l}$, while the number is $74.32\%$ for DLFRM${}^{l}$. {tabu} X[0.7l]—X[0.9c]—X[1c] Models & AUC & Time (sec) CN & 0.8823 ± − &12.3 ± 0.3 Jaccard & 0.8636 ± − &11.7 ± 0.5 Katz &0.9145 ± − &8336.9 ± 306.9 stoDLFRMl &0.9722 ± 0.0013&220191.4 ± 4420.2 diagDLFRMl &0.9680 ± 0.0009&7344.5 ± 943.7 Table 5: AUC scores on Gowalla dataset. #### Gowalla Friendship Prediction Finally, we test on the largest Gowalla network, which is out of reach for many state-of-art methods, including LFRM, MedLFRM and our DLFRMs without SGLD. Some previous works combine the geographical information of Gowalla social network to analyze user movements or friendships [7, 26], but we are not aware of any fairly comparable results for our setting of link prediction. Here, we present the results of some proximitiy-measure based methods, including common neighbors (CN), Jaccard coefficient, and Katz. As the network is too large to search for all the paths, we only concern the paths that shorter than $4$ for Katz. As shown in previous results and Fig. 3(d), DLFRMs with logistic log-loss are more efficient and have comparable results of DLFRMs with hinge loss, so we only show the results of stoDLFRM${}^{l}$ and diagDLFRM${}^{l}$. The AUC scores and training time are shown in Table 5. We can see that stoDLFRM${}^{l}$ outperforms all the other methods and diagDLFRM${}^{l}$ obtain competitive results. Our diagDLFRM${}^{l}$ gets much better performance than the best baseline with less time. It shows that our models can also deal with the large-scale networks. <figure><img src="content_image/1512.02016/burnin.png"><figcaption></figcaption></figure> ### Closer Analysis We use NIPS network as an example to provide closer analysis. Similar observations can be obtained in larger networks (e.g., AstroPh in Appendix B), but taking longer time to run. #### Sensitivity to Burn-In Fig. 3(a) shows the test AUC scores w.r.t. the number of burn-in steps. We can see that all our variant models converge quickly to stable results. The diagDLFRM${}^{l}$ is a bit slower, but still within $150$ steps. These results demonstrate the stability of our Gibbs sampler. #### Sensitivity to Parameter $c$ To study how the regularization parameter $c$ handles the imbalance in real networks, we change the value of $c^{+}\!/c^{-}$ for DLFRM${}^{l}$ from $1$ to $15$ (with all other parameters selected by the development set); and report AUC scores in Fig. 3(b). The first point (i.e., $c^{+}=c^{-}=1$) corresponds to LFRM with our Gibbs sampler, whose lower AUC demonstrates the effectiveness of a larger $c^{+}\!/c^{-}$ to deal with the imbalance issue. We can see that the AUC score increases when $c^{+}\!/c^{-}$ becomes larger and the prediction performance is stable in a wide range (e.g., $6<c^{+}\!/c^{-}<12$). How large $c^{+}\!/c^{-}$ a network needs depends on its sparsity. A rule of thumb is that the sparser a network is, the larger $c^{+}\!/c^{-}$ it may prefer. The results also show that our setting ($c^{+}=10c^{-}$) is reasonable. #### Latent Dimensions Fig. 3(c) shows the number of latent features automatically learnt by variant models. We can see that diagDLFRMs generally need more features than DLFRMs because the simplified weight matrix $U$ doesn’t consider pairwise interactions between features. Moreover, DLFRM${}^{h}$ needs more features than DLFRM${}^{l}$, possibly because of the non-smoothness nature of hinge loss. The small variance of each method suggests that the latent dimensions are stable in independent runs with random initializations. #### Running Time Fig. 3(d) compares the training time. It demonstrates all our variant models are more efficient than MedLFRM and BayesMedLFRM [32] that use truncated mean-field approximation. Compared to DLFRM${}^{l}$, DLFRM${}^{h}$ takes more time to get the good AUC. The reason is that DLFRM${}^{h}$ often converges slower (see Fig. 3(a)) with a larger latent dimension $K$ (see Fig. 3(c)). stoDLFRMs are more effective as we have discussed before. diagDLFRMs are much more efficient due to the linear increase of training time per iteration with respect to $K$. The testing time for all the methods are very little, omitted due to space limit. Overall, DLFRMs improve prediction performance and are more efficient in training, compared with other state-of-the-art nonparametric LFRMs. ## Conclusions and Future Work We present discriminative nonparametric LFRMs for link prediction, which can automatically resolve the unknown dimensionality of the latent feature space with a simple Gibbs sampler using data augmentation; unify the analysis for both logistic log-loss and hinge loss; and deal with the imbalance issue in real networks. Experimental results on a wide range of real networks demonstrate superior performance and scalability. 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Let $X$ follow a Polya-Gamma distribution, denoted by $X\sim\mathcal{PG}(a,b)$, that is \[X=\frac{1}{2\pi^{2}}\sum_{d=1}^{\infty}\frac{g_{d}}{(d-1/2)^{2}+ b^{2}/(4\pi^{2})},\] (17) where $a>0$ and $b\in\mathcal{R}$ are parameters and each $g_{d}\sim\mathcal{G}(a,1)$ is an independent Gamma random variable. The main result of [24] provides an alternative expression for the form of $\varphi_{1}$ in Eq. (5) by incorporating an augmented variable $\lambda$: \[\!\!\!\!\!\!\!\varphi_{1}(\tilde{y}_{ij}\|Z_{i},Z_{j},U)\!=\!\frac {1}{2^{c}}\!\int_{0}^{\infty}\!\!\!\!\exp\!\!{\left(\!\kappa_{ij}\omega_{ij}\! -\!\frac{\lambda_{ij}\omega_{ij}^{2}}{2}\right)}\!\phi(\lambda_{ij}){\rm{d}} \lambda_{ij},\] (18) where $\kappa_{ij}=c(\tilde{y}_{ij}-\frac{1}{2})$ and $\phi(\lambda_{ij})=\mathcal{PG}(\lambda_{ij};c,0)$. For the case with hinge loss, we take the advantage of data augmentation for support vector machines [23] and $\varphi_{2}$ in Eq. (6) can be represented as a scale mixture of Gaussian distributions: \[\!\!\!\!\!\varphi_{2}({y}_{ij}\|Z_{i},Z_{j},U)\!=\!\!\!\int_{0}^{ \infty}\!\!\!\!\!\frac{1}{\sqrt{2\pi\lambda_{ij}}}\exp\!{\left(\!-\frac{{( \lambda_{ij}+c\zeta_{ij})}^{2}}{2\lambda_{ij}}\!\right)}{\rm{d}}\lambda_{ij},\] (19) where $\zeta_{ij}=\ell-{y}_{ij}\omega_{ij}$ and $\lambda_{ij}$ is the augmented variable. By reformulating similar terms in Eq. (19), we have: \[\!\!\!\varphi_{2}({y}_{ij}\|Z_{i},Z_{j},U) \!\propto\int_{0}^{\infty}\frac{1}{\sqrt{2\pi\lambda_{ij}}}\exp\! \Big{(}\!-\frac{1}{2}\Big{(}\frac{c^{2}\ell^{2}}{\lambda_{ij}}+\lambda_{ij} \Big{)}\Big{)}\] (20) \[\exp\Big{(}{c{y}_{ij}\Big{(}1+\frac{c\ell}{\lambda_{ij}}\Big{)} \omega_{ij}-\frac{c^{2}\omega_{ij}^{2}}{2\lambda_{ij}}\Big{)}}{\rm{d}}\lambda_ {ij}\] \[\!\!\propto\!\!\int_{0}^{\infty}\!\!\exp\Big{(}\kappa_{ij}\omega_ {ij}-\frac{\rho_{ij}\omega_{ij}^{2}}{2}\Big{)}\phi(\lambda_{ij}){\rm{d}} \lambda_{ij}.\] where $\kappa_{ij}=c{y}_{ij}(1+c\ell\lambda_{ij}^{-1}),~{}\rho_{ij}=c^{2}\lambda_{ij} ^{-1}$ and $\phi(\lambda_{ij})=\mathcal{GIG}(\frac{1}{2},1,c^{2}\ell^{2})$. Given the results of Eq. (18) and Eq. (20), Lemma 1 holds true. ∎ ### Appendix B: Closer Analysis on AstroPh dataset <figure><img src="content_image/1512.02016/apburnin.png"><figcaption></figcaption></figure> Here, we provide more closer analysis on AstroPh dataset which is much larger than the NIPS dataset. #### Sensitivity to Burn-In Fig. 4(a) shows the AUC scores on testing data with respect to the number of burn-in steps on AstroPh dataset. We can observe that all our variant models converge quickly to stable results, similar as on NIPS dataset. Our DLFRMs with full weight matrix (e.g., DLFRM${}^{l}$, DLFRM${}^{h}$, stoDLFRM${}^{l}$ and stoDLFRM${}^{h}$) converge quickly within $10$ steps. The diagDLFRMs need more steps to converge, but still within $40$ steps to converge to stable results. These results demonstrate the stability of our Gibbs sampling algorithm. #### Sensitivity to Parameter $c$ We analyze how the regularization parameter $c$ handles the imbalance in real networks using diagDLFRM${}^{l}$, which is very efficient (see Fig. 4(d)). Following the settings on NIPS dataset, we change the ratio of $c^{+}\!/c^{-}$ for diagDLFRM${}^{l}$ from $1$ to $15$ with all the parameters selected by the development set. As shown in Fig. 4(b), the AUC score increases when $c^{+}\!/c^{-}$ becomes larger and the prediction performance is stable in a wide range (e.g., $6<c^{+}\!/c^{-}<12$). These observations again demonstrate that using a larger $c^{+}$ than $c^{-}$ can effectively deal with the imbalance issue and our setting ($c^{+}=10c^{-}$) is reasonable. #### Latent Dimensions Our variant models take the advantage of nonparametric technique to automatically learn the dimension of the latent features as shown in Fig. 4(c). We can see that diagDLFRMs generally need more features than DLFRMs because the simplified weight matrix $U$ does not consider pairwise interactions between features. Moreover, DLFRM${}^{h}$ needs more features than DLFRM${}^{l}$, possibly because of the non-smoothness nature of hinge loss. The small variance of each method suggests that the latent dimensions are stable in independent runs with random initializations. #### Running Time The training time of our variant models on AstroPh dataset is shown in Fig. 4(d). We can see that for this relatively large network (with tens of thousands of entities and millions of links), the least time we need to obtain the good AUC score is only about $7\times 10^{3}$ seconds. As on NIPS dataset, DLFRM${}^{h}$ takes more time for training than DLFRM${}^{l}$ and this phenomenon is more obvious here due to the scalability of the network. The reason is that DLFRM${}^{h}$ often converges slower (see Fig 4(a)) with a larger latent dimension $K$ (see Fig. 4(c)). As discussed before, stoDLFRMs are more effective. When a full weight matrix $U$ is used, training time per iteration increases exponentially with respect to $K$. Therefore, diagDLFRMs are much more efficient due to the linear increase of training time per iteration with respect to $K$. Overall, DLFRMs are stable and improve prediction performance efficiently .
1103.4128	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 25842, "num_imgs": 2, "llama3_tokens_count": 8229 }	[ "content_image/1103.4128/x1.png", "content_image/1103.4128/x2.png" ]	# Quantum Fluctuations in Dipolar Bose Gases Aristeu R. P. Lima Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany Axel Pelster Fachbereich Physik, Universität Duisburg-Essen, Lotharstrasse 1, 47048 Duisburg, Germany February 28, 2024 ###### Abstract We investigate the influence of quantum fluctuations upon dipolar Bose gases by means of the Bogoliubov-de Gennes theory. Thereby, we make use of the local density approximation to evaluate the dipolar exchange interaction between the condensate and the excited particles. This allows to obtain the Bogoliubov spectrum analytically in the limit of large particle numbers. After discussing the condensate depletion and the ground-state energy correction, we derive quantum corrected equations of motion for harmonically trapped dipolar Bose gases by using superfluid hydrodynamics. These equations are subsequently applied to analyze the equilibrium configuration, the low-lying oscillation frequencies, and the time-of-flight dynamics. We find that both atomic magnetic and molecular electric dipolar systems offer promising scenarios for detecting beyond mean-field effects. pacs: 03.75.Hh,03.75.Kk _Introduction_ – Bose-Einstein condensates (BEC) with the anisotropic and long-range dipole-dipole interaction (DDI) have received much attention, specially after the condensation of ${}^{52}$Cr in 2005 [1]. This investigation pioneered a series of experiments which led to a robust understanding of the DDI on a mean-field level. Experimental successes include the direct observation of the DDI in the time-of-flight (TOF) dynamics [2; 3], the stabilization of a purely dipolar gas [4], and the observation of a d-wave Bose-nova explosion [5]. In the meantime, the DDI has also been observed even in ${}^{87}$Rb [6] and evidences of it have been found in ${}^{7}$Li [7]. Recently, an important experiment has been realized, in which the influence of the DDI upon the oscillation frequencies of ${}^{52}$Cr has been studied [8]. Parallel to these experiments, much theoretical work has been pursued [9]. For instance, building on the previous construction of the dipolar pseudo-potential [10], a solution of the mean-field Gross-Pitaevskii (GP) equation was obtained [11; 12], which accounts quantitatively for the static as well as the dynamic properties of trapped dilute ${}^{52}$Cr gases [2; 3; 8]. Nowadays, dipolar interactions in magnetic systems are, therefore, considered to be relatively well understood in terms of the GP mean-field theory. Nonetheless, highly magnetic atoms, such as dysprosium [13], or strongly polar heteronuclear molecules, exemplified by ${}^{40}$K${}^{87}$Rb [14; 15; 16], are expected to push the understanding of dipolar systems beyond the edge of mean-field theory’s domain of validity. Already in the case of pure contact interaction, ultracold quantum gases have been the scenario for investigating many-body theories. Theoretically, much work has been carried out based on the Lee-Huang-Yang (LHY) correction for the equation of state [17], when implemented in terms of superfluid hydrodynamics [18; 19], or on quantum Monte Carlo simulations [20]. Experimentally, quantum fluctuations have been studied in setups, which favor their enhancement, such as in optical lattices [21], or in the presence of Feshbach resonances in both bosonic [22] and fermionic [23] systems. In particular, the importance of the latter experiment should be stressed, as it achieved a precision level which is capable of distinguishing the predictions of Bardeen-Cooper-Schrieffer (BCS) mean-field theory from QMC results for the oscillation frequencies of low-lying modes at the BEC side of the BEC-BCS crossover. This demonstrates unequivocally that collective modes provide an adequate testing ground for many-body theories of interacting quantum gases. In this letter, we theoretically study quantum fluctuations in dipolar Bose gases in the large particle number regime. Within the realm of the Bogoliubov-de Gennes (BdG) theory, we first calculate the condensate depletion and the ground-state energy. Then, by applying superfluid hydrodynamics, we discuss the quantum corrections to both static and dynamic properties of the system focusing on the observability of these effects. _BdG Theory_ – Consider a Bose gas of $N$ dipoles with mass $M$ in a cylinder symmetric trapping potential $U_{\rm tr}({\bf x})=M\omega_{x}^{2}(x^{2}+y^{2}+{\lambda}^{2}z^{2})/2$ with the trap aspect ratio $\lambda$. The corresponding Hamilton operator reads \[\hat{H}\!=\!\int\!\!{\rm d}{{}^{3}x}\hat{\Psi}^{\dagger}({\bf x})\!\left[h_{0} \!+\!\int\!\frac{{\rm d}{{}^{3}y}}{2}\hat{\Psi}^{\dagger}({\bf y})V_{\rm int} \!\left({\bf x}\!-\!{\bf y}\right)\hat{\Psi}({\bf y})\right]\!\hat{\Psi}({\bf x})\] (1) with $h_{0}=-{\hbar^{2}\nabla^{2}}/{2M}+U_{\rm tr}({\bf x})$. Moreover, $\hat{\Psi}^{\dagger}({\bf x})$ and $\hat{\Psi}({\bf x})$ denote the usual creation and annihilation field operators, respectively. Considering the dipoles to be aligned along the $z$ axis, the interaction potential reads \[V_{\rm int}({\bf x})=\frac{4\pi\hbar^{2}a_{s}}{M}\!\left[\delta({\bf x})\!+\! \frac{3a_{\rm dd}}{4\pi a_{s}\|{\mathbf{x}}\|^{3}}\left(1\!-\!3\frac{z^{2}}{\|{ \mathbf{x}}\|^{2}}\right)\right],\] (2) where $a_{s}=Mg/4\pi\hbar^{2}$ is the s-wave scattering length and $a_{\rm dd}$ denotes a length scale which characterizes the dipolar interaction, irrespective whether it is of magnetic or electric nature. At this point, it is useful to introduce the dimensionless relative interaction strength $\epsilon_{\rm dd}=a_{\rm dd}/a_{s}$. In order to study dipolar Bose gases beyond the mean-field approximation, we follow the Bogoliubov prescription and decompose the field operator according to $\hat{\Psi}({\bf x})=\Psi({\bf x})+\delta\hat{\psi}({\bf x})$, where $\Psi({\bf x})$ is the condensate wave function and $\delta\hat{\psi}({\bf x})$ accounts for the quantum fluctuations. By considering only the leading term in this expansion, one obtains the GP theory of dipolar BECs, which was exactly solved in the Thomas-Fermi (TF) approximation [11; 12]. In order to go beyond the GP theory, we insert the Bogoliubov prescription into the grand-canonical Hamiltonian $\hat{K}=\hat{H}-\mu\hat{N}$, with the chemical potential $\mu$ and the number operator $\hat{N}$, and retain terms up to second order in the fluctuations. Then we decompose the fluctuating field operator according to $\delta\hat{\psi}({\bf x})={\sum_{\nu}}^{\prime}\left[{\mathcal{U}}_{\nu}({\bf x })\hat{\alpha}_{\nu}+{\mathcal{V}}_{\nu}^{}({\bf x})\hat{\alpha}_{\nu}^{ \dagger}\right]$. Here, ${\mathcal{U}}_{\nu}({\bf x})$ and ${\mathcal{V}}_{\nu}({\bf x})$ are the Bogoliubov amplitudes, whereas $\hat{\alpha}_{\nu}^{\dagger}$ and $\hat{\alpha}_{\nu}$ denote bosonic creation and annihilation operators. The index $\nu$ is associated with excited energy levels and, therefore, the sum excludes the $\nu=0$ mode, which represents the condensate. By requiring the Hamiltonian to be diagonal, we derive the BdG equations \[\!\!\!\!\int\!\!{\rm d}{{}^{3}y}\!\!\left(\begin{array}[]{cc}\!\! H_{{\mathcal{U}},{\mathcal{U}}}\!\left({{\bf x},{\bf y}}\right)&\!\!H_{{ \mathcal{U}},{\mathcal{V}}}\!\left({{\bf x},{\bf y}}\right)\\ \!\!H^{}_{{\mathcal{U}},{\mathcal{V}}}\!\left({{\bf x},{\bf y}}\right)&\!\!H^ {}_{{\mathcal{U}},{\mathcal{U}}}\!\left({{\bf x},{\bf y}}\right)\end{array}\! \!\right)\!\!\left(\begin{array}[]{c}\!\!{\mathcal{U}}_{\nu}({\bf y})\\ \!\!{\mathcal{V}}_{\nu}({\bf y})\end{array}\!\!\right)\!=\!\varepsilon_{\nu}\! \!\left(\begin{array}[]{c}\!\!\!{\mathcal{U}}_{\nu}({\bf x})\\ \!\!\!-\!{\mathcal{V}}_{\nu}({\bf x})\end{array}\!\!\!\right)\!\!,\] (3) where the diagonal matrix elements are given according to $H_{{\mathcal{U}},{\mathcal{U}}}\!\left({{\bf x},{\bf y}}\right)\!=\!\delta\! \left({\bf x}\!-\!{\bf y}\right)\!H_{\rm Fl}({\bf y})\!+\Psi^{}({\bf y})\!V_{ \rm int}\!\left({\bf x}\!-\!{\bf y}\right)\Psi({\bf x})$ and the off-diagonal ones according to $H_{{\mathcal{U}},{\mathcal{V}}}\left({{\bf x},{\bf y}}\right)=\Psi({\bf y})V_{ \rm int}\left({\bf x}-{\bf y}\right)\Psi({\bf x})$. In these expressions, an important role is played by the functionally diagonal part, which is given by the fluctuation Hamiltonian \[H_{\rm Fl}({\bf x})=h_{0}-\mu+\int{\rm d}{{}^{3}x^{\prime}}\Psi^{}({\bf x}^{ \prime})V_{\rm int}\left({\bf x}-{\bf x}^{\prime}\right)\Psi({\bf x}^{\prime}).\] (4) Moreover, in the framework of the BdG theory, the operator $\hat{\alpha}_{\nu}$ annihilates the ground state $\|0\rangle$, so that excitations are interpreted as quasi-particles with energy $\varepsilon_{\nu}$. In general, solving the BdG equations exactly is very difficult and numerically arduous in the case of long-range interactions [24]. Nonetheless, they provide an adequate starting point for investigating quantum fluctuations in most cases of interest. Typically, the number of particles is large enough, so that we can neglect the condensate kinetic energy and its wave function becomes $\Psi({\bf x})=\sqrt{n_{0}({\bf x})}$, where $n_{0}({\bf x})$ denotes the TF condensate density. Moreover, in this regime, a semiclassical approximation for the excitations is appropriate. Let $q_{\nu}\left({\bf x}\right)$ be either ${\mathcal{U}}_{\nu}\left({\bf x}\right)$ or ${\mathcal{V}}_{\nu}\left({\bf x}\right)$. The approximation is implemented through $\varepsilon_{\nu}\rightarrow\varepsilon\left({\bf x},{\bf k}\right)$, $\sum_{\nu}\rightarrow\int{\rm d}^{3}k/(2\pi)^{3}$, and ${q}_{\nu}\rightarrow{q}\left({\bf x},{\bf k}\right)e^{i{\bf k}\cdot{\bf x}}$, where ${q}\left({\bf x},{\bf k}\right)$ is a slowly varying function of ${\bf x}$[25]. With this, the free Hamiltonian becomes $h_{0}\left({\bf x},{\bf k}\right)=\hbar^{2}{\bf k}^{2}/2M+U_{\rm tr}({\bf x})$. Despite the simplification, we still cannot solve Eqs. (3) because of their non-local character. The semiclassical approximation also offers a procedure to deal with this problem. Consider the exchange term in Eq. (3), which, in general, reads $I_{\rm Ex}\equiv\int{{\rm d}^{3}y}H_{{\mathcal{U}},{\mathcal{V}}}\left({{\bf x },{\bf y}}\right){q}_{\nu}\left({\bf y}\right)$. Within the local density approximation (LDA), it becomes $I_{\rm Ex}\approx q({\bf x},{\bf k})n_{0}({\bf x}){V}_{\rm int}({\bf k})$, with ${V}_{\rm int}({\bf k})=g[1+\epsilon_{\rm dd}(3\cos^{2}\theta-1)]$ and $\cos\theta=\hat{\bf k}\cdot\hat{\bf e}_{z}$. This is the leading order in a systematic gradient expansion [26]. To estimate the error of ignoring the next order, we replace the gradients as ${\nabla}_{\bf x}\to 1/R_{\rm TF},{\nabla}_{\bf k}\to 1/K_{c}$, with $R_{\rm TF}$ the mean Thomas-Fermi radius and $K_{c}$ the momentum scale defined by the speed of sound. In terms of the condensate density at the trap center, one finds the LDA to be valid for $3\times[N^{2}a_{s}^{3}n_{0}({\bf 0})]^{\frac{1}{6}}\times[1+\epsilon_{\rm dd} \left(3\cos^{2}\theta-1\right)]^{\frac{1}{2}}\gg 1$. For $\epsilon_{\rm dd}<1$, this is commonly met in BEC experiments. One of the important physical features brought about by quantum fluctuations is the condensate depletion, i.e., particles are expelled from the condensate to excited states by interactions. The depletion density is, in general, given by $n({\bf x})-n_{0}({\bf x})=\sum_{\nu}^{\prime}{\mathcal{V}}_{\nu}^{}({\bf x}){ \mathcal{V}}_{\nu}({\bf x})$ and reduces in the present semiclassical theory to \[\Delta n({\bf x})=\frac{8n({\bf x})}{3}{\mathcal{Q}}_{3}(\epsilon_{\rm dd}) \sqrt{\frac{n({\bf x})a_{s}^{3}}{\pi}}.\] (5) Here, $n({\bf x})$ denotes the total density of the gas and the auxiliary function ${\mathcal{Q}}_{l}(x)=\int_{0}^{1}{{\rm d}u}\left(1-x+3\,x\,u^{2}\right)^{l/2}$ describes the contribution of the DDI. In particular, the function ${\mathcal{Q}}_{3}$ is an increasing monotonic function of the relative interaction strength $\epsilon_{\rm dd}$, which is bounded between ${\mathcal{Q}}_{3}(0)=1$ and ${\mathcal{Q}}_{3}(1)=3\sqrt{3}/4\approx 1.30$. For ${}^{52}$Cr, the most extensively studied dipolar system, the magnetic moment of $6\,\mu_{\rm B}$ with the Bohr magneton $\mu_{\rm B}$ and the s-wave scattering length $a_{s}\approx 100~{}a_{0}$ with the Bohr radius $a_{0}$ yield the relative interaction strength $\epsilon_{\rm dd}\approx 0.16$. Then, the gas parameter $n({\bf x})a_{s}^{3}$ evaluated at the trap center is typically of the order $n({\bf 0})a_{s}^{3}\approx 10^{-4}$[3]. For a homogeneous system with this value of the gas parameter, the fractional depletion amounts to $\Delta N/N\approx 1.5\,\%$ and is not significantly affected by the DDI. The presence of quantum fluctuations also gives rise to a shift of the ground-state energy. The present BdG theory yields the following correction to the energy density \[\frac{\Delta E({\bf x})}{n({\bf x})}=\frac{64}{15}gn({\bf x}){\mathcal{Q}}_{5} (\epsilon_{\rm dd})\sqrt{\frac{n({\bf x})a_{s}^{3}}{\pi}},\] (6) where ${\mathcal{Q}}_{5}$ is also a monotonic function and grows from ${\mathcal{Q}}_{5}(0)=1$ up to ${\mathcal{Q}}_{5}(1)=3\sqrt{3}/2\approx 2.60$. Thus, physical quantities directly related to the ground-state energy offer much better perspectives for observing dipolar quantum fluctuations. Indeed, with the help of Eq. (6) a corresponding LHY equation of state for a homogeneous system can be derived, which leads to the Beliaev sound velocity [27] both in the absence [18] and in the presence of the DDI. Note that both functions ${\mathcal{Q}}_{3}$ and ${\mathcal{Q}}_{5}$ become imaginary for $\epsilon_{\rm dd}>1$. This arises from the fact that a homogeneous dipolar gas is only stable for $\epsilon_{\rm dd}\leq 1$, otherwise the DDI dominates over the contact interaction and its attractive part leads to collapse. _Superfluid Hydrodynamics_ – At $T=0$, the BdG theory states irrespective of the two-particle interaction that the whole sample is superfluid though not all particles are condensed. Thus, the system can be studied by means of the superfluid hydrodynamics. Indeed, by writing the superfluid velocity as ${\bf v}=\nabla\chi$, the superfluid hydrodynamic equations can be obtained by extremizing the action [28] \[{\mathcal{A}}[n,\chi]=-\int{\rm d}{t}{{\rm d}^{3}x}{n}\left\{M\left[\dot{\chi} +\frac{1}{2}\nabla\chi^{2}\right]+e\!\left[n\right]\right\}\] (7) with respect to the density $n({\bf x},t)$ and the velocity potential $\chi({\bf x},t)$. The energy density of the system as a functional of the particle density $e\!\left[n\right]$ contains a mean-field component, which reads in TF approximation \[e_{\rm MF}=U_{\rm tr}({\bf x})\!+\!\frac{g}{2}n({\bf x},t)\!+\!\int\!\!{{\rm d }^{3}x^{\prime}}\frac{V_{\rm dd}({\bf x}\!-\!{\bf x^{\prime}})}{2}n({\bf x^{ \prime}}\!,t),\] (8) and a quantum correction, which is given in the case of a dipolar Bose gas by the rhs of Eq. (6). Let us, now, extremize the resulting action (7) within a variational approach. To this end, we adopt a harmonic ansatz for the velocity potential $\chi({\bf x},t)=[\alpha_{x}(t)x^{2}+\alpha_{y}(t)y^{2}+\alpha_{z}(t)z^{2}]/2$ and a parabolic ansatz for the particle density \[n({{\bf x}},t)=\frac{15N}{8\pi\overline{R}^{3}(t)}\left[1-\sum\limits_{i=x,y,z }\frac{x^{2}_{i}}{R_{i}^{2}(t)}\right]\] (9) for $n({{\bf x}},t)\geq 0$ and $n({{\bf x}},t)=0$ elsewhere. The bar denotes geometric average. Note that the particular density profile in (9) corresponds to the exact mean-field TF solution [11; 12]. This is consistent with the BdG theory, which assumes that interactions are not too strong. In order to derive quantum corrected equations of motion for the TF radii ${R}_{i}$, we eliminate, at first, the variational parameters for the velocity potential through their equations of motion $\dot{\alpha}_{i}=\dot{R}_{i}/R_{i}$. Then, in the case of a cylinder symmetric trap, we obtain \[\!\!\!\!\!\!\!\!\!\ddot{R}_{x} \!\!= \!-\omega_{x}^{2}R_{x}\!+\!\frac{15gN/4\pi}{MR_{x}\overline{R}^{3 }}\!\!\left[1-\epsilon_{\rm dd}A\!\left(\frac{R_{x}}{R_{z}}\right)\!+\!\frac{ \beta}{\overline{R}^{\frac{3}{2}}}\right]\!,\] \[\!\!\!\!\!\!\!\!\!\ddot{R}_{z} \!\!= \!-\omega_{z}^{2}R_{z}\!+\!\frac{15gN/4\pi}{MR_{z}\overline{R}^{3 }}\!\!\left[1+2\epsilon_{\rm dd}B\!\left(\frac{R_{x}}{R_{z}}\right)\!+\!\frac{ \beta}{\overline{R}^{\frac{3}{2}}}\right]\!,\] (10) together with the auxiliary functions \[\!\!\!\!\!\!\!\!\!A\left(x\right) = 1+\frac{3}{2}\frac{{x}^{2}f_{s}\left(x\right)}{{x}^{2}-1},\quad B \left(x\right)=1+\frac{3}{2}\frac{f_{s}\left(x\right)}{{x}^{2}-1}.\] (11) Here, $f_{s}(x)$ denotes the anisotropy function \[f_{s}(x) \equiv \frac{1+2x^{2}}{1-x^{2}}-\frac{3x^{2}\tanh^{-1}\sqrt{1-x^{2}}}{(1 -x^{2})^{3/2}},\] (12) which arises for both BECs [11; 29] and nonsuperfluid fermionic dipolar gases [30]. In Eqs. (10), the quantum correction is governed by the parameter $\beta=\gamma{\mathcal{Q}}_{5}(\epsilon_{\rm dd})\sqrt{a_{s}^{3}N}$, with the constant $\gamma={{3^{\frac{3}{2}}\cdot 5^{\frac{3}{2}}\cdot 7}/{2^{\frac{13}{2}}}} \approx 4.49$. Consequently, setting $\beta=0$ in Eqs. (10) leads to the exact mean-field TF results [11; 12]. The beyond mean-field equations of motion for dipolar Bose gases (10) are the main result of the present letter. They allow for investigating both the static and the dynamic properties of these systems. To this end, we solve them by considering $\beta$ as a small perturbation, as this is consistent with the assumption of small fluctuations. <figure><img src="content_image/1103.4128/x1.png"><figcaption>Figure 1: (color online.) Quantum correction to the oscillation frequencies.</figcaption></figure> _Results_ – As is well known, the DDI affects the system in an anisotropic manner. Thus, in contrast to the case of a Bose gas with contact interaction alone, quantum fluctuations lead to a correction in the gas aspect ratio in equilibrium. To investigate this effect quantitatively, we set the rhs of Eqs. (10) to zero and solve them up to first order in $\beta$. Thereby, one finds that the TF radii are given according to $R_{i}=R_{i}^{0}+\delta R_{i}$, where $R_{i}^{0}$ denotes the mean-field value and $\delta R_{i}$ represents the quantum correction. Thus, the beyond mean-field aspect ratio is $\kappa\equiv{(R_{x}^{0}+\delta R_{x})}/({R_{z}^{0}+\delta R_{z}})\approx\kappa ^{0}\left(1+\delta\kappa\right)$. Usually, the correction to the aspect ration in the present experiments is quite small. For ${}^{52}$Cr [3], for example, one has at most ${\delta\kappa}\lesssim 10^{-2}$, which is too small to be detected. For stronger interactions, this turns out to be an important effect, since it implies a characteristic dependence on the trap geometry due to the DDI. Consider, e.g., the experimental values for particle number $N=3\times 10^{4}$ and trap frequencies $\overline{\omega}\approx 2\pi\times 660$ Hz as in Ref. [3], but instead of chromium values, use $a_{s}=190~{}a_{0}$ and $\epsilon_{\rm dd}=0.7$. This could represent a dysprosium sample, which is heavier than chromium and possesses a larger magnetic moment of $10~{}\mu_{\rm B}$ yielding the dipolar length $a_{\rm dd}\approx 133~{}a_{0}$. Then, the correction for the aspect ratio would be about ${\delta\kappa}\approx 27~{}\%$ for moderate cigar-like traps. Let us, now, analyze the low-lying excitations. To this end, we apply for the TF radii the ansatz $R_{i}(t)=R_{i}(0)+\eta_{i}\sin\left({\Omega t}+\varphi\right)$ and obtain by linearizing Eqs. (10) a matrix equation which leads to the frequencies of the monopole and of the quadrupole modes up to first order in $\beta$. In Fig. 1, we plot the resulting corrections to the oscillation frequencies as functions of the trap aspect ratio $\lambda$ in units of $\tilde{\delta\Omega}=({63\sqrt{\pi}}/{128})\sqrt{a_{s}^{3}~{}n({\bf 0})}$ for $\epsilon_{\rm dd}=0.16$. The dashed lines represent the Pitaevskii-Stringari result for contact interaction alone [18], which we immediately recover by setting $\epsilon_{\rm dd}=0$. Fig. 1 shows a dependence on the trap aspect ratio, which is substantially different from the contact case and stems from the quantum correction for the aspect ratio. Thus, tuning the trap aspect ratio could be used to identify the effects of the DDI at the many-body level in ultracold Bose gases. This, however, requires stronger interactions. Indeed, the oscillation frequency of the intermediate mode of ${}^{52}$Cr has recently been investigated in a triaxial trap [8]. For that experiment we estimate the quantum correction of the oscillation frequencies to be of the order $\tilde{\delta\Omega}\approx 3\times 10^{-3}$, due to the small particle number and weak trap. For more favorable values of these parameters, such as those in Ref. [3], one already obtains $\tilde{\delta\Omega}\approx 1~{}\%$. For the previous hypothetical dysprosium setup, one could even have $\tilde{\delta\Omega}\approx 4~{}\%$, which should render these effects observable. <figure><img src="content_image/1103.4128/x2.png"><figcaption>Figure 2: (color online.) Aspect ratio as a function of time.</figcaption></figure> For the TOF dynamics, the quantum correction of the equilibrium TF radii also plays an important role. Indeed, by inserting the expansion $R_{i}(t)=R_{i}^{0}(t)+\delta R_{i}(t)$, one derives a set of coupled differential equations for the mean-field TF radii $R_{i}^{0}(t)$ and their corrections $\delta R_{i}(t)$. For a system consisting of ${}^{52}$Cr atoms, no signal of quantum fluctuations can be seen in the present TOF analysis and the mean-field theory provides again a good agreement with the experimental results [2; 3]. For stronger interactions, however, the situation changes. In Fig. 2, we plot the ratio $\kappa(t)={[R_{x}^{0}(t)+\delta R_{x}(t)]}/{[R_{z}^{0}(t)+\delta R_{z}(t)]}$ as a function of time for the previous dysprosium setup. Then, clear differences with respect to the mean-field TOF dynamics show up, specially at large times. To investigate the dependence of the asymptotic aspect ratio on the system parameters we have settled the dipolar length at $a_{\rm dd}\approx 133~{}a_{0}$ and have varied the s-wave scattering length $a_{s}$, so as to vary the relative strength from $\epsilon_{\rm dd}=0.2$ up to $\epsilon_{\rm dd}=1.0$. Our results are shown in the inset of Fig. 2, where the asymptotic value $[\kappa(\infty)-\kappa^{0}(\infty)]/\kappa^{0}(\infty)$ is plotted against $\epsilon_{\rm dd}$ for different trap aspect ratios. The results reasonably deviate from the mean-field values, so that quantum fluctuations should be observable in the TOF dynamics of Bose gases of highly magnetic atoms. _Conclusion_ – Applying the BdG theory in the limit of large particle numbers, we obtained analytic expressions for the condensate depletion and the ground-state energy of dipolar Bose gases. This allowed us to study key properties of the system such as low-lying excitations and TOF dynamics beyond the mean-field approximation. We find that both polar molecules and highly magnetic atoms are realistic scenarios for observing unprecedented beyond mean-field dipolar physics. We thank J. Dietel and L. Santos for useful discussions and the DFG for financial support (KL256/53-1). ## References * [1] A. 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Glaum _et al._, _Phys. Rev. Lett._98, 080407 (2007); K. Glaum and A. Pelster, _Phys. Rev. A_76, 023604 (2007). * [30] A. R. P. Lima and A. Pelster, _Phys. Rev. A_81, 021606(R) (2010); ibid. 81, 063629 (2010).
1809.00726	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 18049, "num_imgs": 0, "llama3_tokens_count": 5172 }	[]	# The Swampland, Quintessence and the Vacuum Energy M.C. David Marsh Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, United Kingdom _Email:_ m.c.d.marsh@damtp.cam.ac.uk ###### Abstract It has recently been conjectured that string theory does not admit de Sitter vacua, and that quintessence explains the current epoch of accelerated cosmic expansion. A proposed, key prediction of this scenario is time-varying couplings in the dark sector, induced by the evolving quintessence field. We note that cosmological models with varying couplings suffer from severe problems with quantum corrections, beyond those shared by all quintessence models. The vacuum energy depends on all masses and couplings of the theory, and even small variations of parameters can lead to overwhelmingly large corrections to the effective potential. We find that quintessence models with varying parameters can be realised in consistent quantum theories by either: 1) enforcing exceptional levels of fine-tuning; 2) realising some unknown mechanism that cancels all undesirable contributions to the effective potential with unprecedented accuracy; or 3) ensuring that the quintessence field couples exclusively to very light states, and does not backreact on heavy fields. ## I Introduction An important question in fundamental physics is what distinguishes general effective field theories from those that can be consistently realised in quantum gravity. Inspired by examples of compactifications from string theory, the authors of [1] conjectured that quantum gravity severely restricts the effective scalar potential, $V$, of the low-energy theory: \[\|\nabla V\|\geq c\,V\,,\] (1) for a positive constant $c\sim{\cal O}(1)$ and in units where $M_{\mathrm{Pl}}=1/\sqrt{8\pi G}=1$. If true, equation (1) has far-reaching implications [1, 2, 3, 4, 5, 6, 7]. Most notably, equation (1) forbids local de Sitter critical points (see also [8]) and forces the current period of accelerated expansion to be realised through particular models of quintessence ¹. Reference [2] argued that such models can be naturally realised in string theory where slowly rolling moduli fields can support the accelerated expansion. [FOOTNOTE:1][ENDFOOTNOTE] Some well-known restrictions on quintessence were discussed in [2, 4, 10, 6, 7]. Very light scalar fields coupled to the Standard Model can mediate long-range forces, which are severely constrained by precision tests of the equivalence principle. Moreover, scalar fields that modify the masses and couplings of the Standard Model are constrained by astronomical observations. Finally, models of quintessence require not only that the value of the scalar potential is very small, but so must its gradient. In reference [2], the absence of observed variations in the Standard Model parameters were interpreted as evidence for comparatively stronger couplings between the quintessence scalar and some fields in the dark sector. This is not a direct consequence of equation (1), but is arguably natural as such a scenario can be realised in string theory through branes, e.g. of type IIB or F-theory. For example, the quintessence field may control the volume of the cycle where dark matter originates, so that its evolution leads to variations in dark matter couplings. In the cosmology literature, models realising dark energy/dark matter interactions are usually referred to as ‘interacting dark energy’ [11]. The purpose of this note is to recall that a cosmic scalar field, $\phi$, that cause variations in couplings and masses suffer from _severe problems_ when considered in quantum field theory [12, 13, 14, 15]. The basic argument (reviewed in detail below) is that small variations in couplings cause large variations in the vacuum energy. For example, a variation in a fine-structure constant $\alpha(\phi)=\bar{\alpha}+\delta\alpha$ to which matter with large mass $M$ is coupled leads to a variation of the vacuum energy that is schematically of the form, \[\delta\rho_{\rm vac}\sim\delta\alpha(\phi)\,M^{4}\,.\] (2) This is a contribution to the low-energy effective potential of $\phi$ that can overwhelm any naive quintessence potential. This makes it very challenging to promote cosmological models of varying ‘constants’ into consistent quantum theories. In this note, we apply these arguments to the recently proposed quintessence models of [2], and find that they can only be realised under certain restrictive conditions. ## II The vacuum energy and varying parameters The one-loop Coleman-Weinberg potential for a general field theory in four-dimensional flat space is given by [16, 17, 18], \[\delta V =\frac{1}{(8\pi)^{2}}\left[\Lambda^{4}\,{\rm STr}(M^{0})\ln\left( \frac{\Lambda^{2}}{\mu^{2}}\right)+2\Lambda^{2}\,{\rm STr}(M^{2})\right.\] (3) \[\left.+{\rm STr}\left(M^{4}\right)\ln\left(\frac{M^{2}}{\Lambda^{ 2}}\right)+\ldots\right]\,,\] where $\mu$ is scale parameter and $\Lambda$ the cut-off scale. The supertrace is given by ${\rm STr}(M^{n})=\sum_{i}(-1)^{2j_{i}}(2j_{i}+1)m_{i}^{n}$ where $j_{i}$ is the spin of the different particles with mass eigenvalues $m_{i}$. The first term is always field-independent, vanishes for spontaneously broken supersymmetric theories, and is only relevant for the original cosmological constant problem. In spontaneously broken supergravities, the supertrace is generically non-vanishing for $n>0$, but in some special ‘no-scale’ supergravities, the $n=2$ term can vanish even after supersymmetry breaking [19]. In this note, we will conservatively consider only the third term, which is only logarithmically sensitive to the model-dependent cutoff. Including also higher loop-order corrections, we may write these contributions as, \[\delta V=\frac{1}{(8\pi)^{2}}\sum_{i}c_{i}\,m_{i}^{4}+\ldots\,,\] (4) where the coefficients $c_{i}(\alpha)$ depend on the coupling constants of the theory, and absorb any logarithmic factors. Accounting for loop-factors, $\partial_{\alpha}^{p}c_{i}\sim{\cal O}((4\pi)^{-p})$. The effective quintessence potential below the scale $\Lambda$ is then given by the sum of the bare contribution $V_{0}(\phi)$ and the loop corrections: $V=V_{0}+\delta V$. It is now easy to understand the particular problems associated with quintessence models with varying parameters. A change in the Standard Model fine-structure constant of the order of the current observational limit, $\delta\alpha/\alpha\sim 10^{-6}$, leads to a change in the vacuum energy of the order of (cf. [12]), \[\delta V\sim(170\,{\rm MeV})^{4}=3\times 10^{43}\,\rho_{0}\,,\] (5) where we have taken ${\rm Max}(m_{i})=m_{\rm top}=173$ GeV and the vacuum energy is $\rho_{0}=(2.3\times 10^{-3}\,{\rm eV})^{4}$. Since $\alpha$ is field dependent, this is a highly disruptive contribution to the effective potential of $\phi$². [FOOTNOTE:2][ENDFOOTNOTE] ## III The flatness of the quintessence potential Even if the quintessence is completely decoupled from the Standard Model, small changes in the parameters of the dark sector can lead to overwhelmingly large contributions to the quintessence effective potential. An important question is then: _given the vacuum energy contribution of equation (4), how can the quintessence potential be sufficiently flat?_ Excessive fine-tuning is one option. In order for the potential to be sufficiently flat, not only the value of the potential and its gradient need to be tuned, but also higher-orders in the Taylor expansion around the present-day value. Suppose that an evolution of the quintessence field by $\delta\phi$ causes a variation, $\delta\alpha_{\rm tot}$, in a dark sector coupling constant under which matter of mass $m_{i}$ is charged. Imposing that the value of the potential over this range does not exceed $\rho_{0}$ then requires cancellations of the $k$:th order in a Taylor expansion to at least one part in [13], \[\left(\frac{m_{i}^{4}}{(8\pi)^{2}\rho_{0}}\right)\left(\frac{\delta\alpha_{\rm tot }}{4\pi}\right)^{k}\,,\] (6) for $k\leq k_{\rm max}={\rm floor}\left[\ln(\frac{m_{i}^{4}}{(8\pi)^{2}\rho_{0}}))/ \ln\left(\frac{4\pi}{\delta\alpha_{\rm tot}}\right)\right]$. For example, if the dark matter has mass $m=100\,{\rm GeV}$ and is charged under the gauge group with a coupling constant that changes by $\delta\alpha_{\rm total}=10^{-2}$, the quintessence potential need to be tuned up to order $k_{\rm max}=16$. The total fine-tuning (on top of that required by the original cosmological constant problem) is the product of the required fine-tuning of the individual operators and is given by [13]: \[{\mathfrak{f}}=\left(\frac{m^{4}}{(8\pi)^{2}\rho_{0}}\right)^{\frac{1}{2}(k_{ \rm max}-1)}=1\times 10^{388}\,.\] (7) In the absence of this tuning, $\phi$ could get stuck in a de Sitter minimum or rapidly evolve towards a crunch. A second option is that a new symmetry or mechanism cancels all undesirable contributions to the potential with exceptional accuracy. Unbroken global supersymmetry sets ${\rm STr}(M^{n})=0$[21], but this is no longer true in supergravity [22, 23]. A new cancellation mechanism may be intrinsically string theoretic and appear unexpected from a supergravity viewpoint. While this is an intriguing possibility, no such mechanism has yet been identified in the low-energy theories arising from string compactifications. The discovery of such a mechanism through careful string theory calculations would strengthen the case for the cosmology proposed in [2], and possibly for the correctness of equation (1). A third option is that $\phi$ couples exclusively to very light states, so that the equation (4) gives a negligible contribution to the effective potential of $\phi$ (cf. [15] for an example). This may be realised rather naturally if $m_{i}^{4}(\phi)\ll V_{0}(\phi)$ as $\phi$ descends the quintessence potential. In the present era, such fields should be no more massive than $m_{i}\lesssim 4\times 10^{-2}\,{\rm eV}$ to allow for ${\cal O}(1)$ changes in parameters without spoiling the quintessence potential. This solution is comparatively appealing, but has two important caveats. First, to convincingly realise such a mechanism one must demonstrate that the contribution from the second term of the Coleman-Weinberg potential (3) is negligible. If the cutoff $\Lambda$ is close to the Planck scale, this may require the stricter limit $m_{i}\lesssim{\cal O}(H_{0})$, in which case $\phi$ can only couple to other quintessence fields. Second, the parametrically large hierarchy between particle physics mass scales and the vacuum energy requires that $\phi$ interacts extremely weakly with other moduli. For example, if the evolution of $\phi$ changes the total volume of the compactification or the string coupling constant, the spectrum of massive states will change, and the vacuum energy problem is re-introduced. To illustrate how sharp this decoupling must be, suppose that a Standard Model gauge coupling, $g$, is controlled by a volume modulus, ${\cal V}_{\rm SM}=1/g^{2}$. For concreteness, we take $\alpha=1/25$, as is appropriate for Grand Unified Theories. As $\phi$ evolves, ${\cal V}_{\rm SM}$ must stay fixed to a high accuracy, or the Standard Model vacuum energy corrections dominate over the quintessence potential. This requires, \[\frac{\delta{\cal V}_{\rm SM}}{{\cal V}_{\rm SM}}<6\times 10^{-51}\,,\] (8) where we have again set $m=m_{\rm top}$ and require $\delta V_{\rm SM}<\rho_{0}$. Such a rigidity of the Standard Model cycle can be challenging to realise when all fields are (at least gravitationally) coupled, and $\phi$ evolves substantially. In closing, we recall that the conjecture (1) is violated if the potential for $\phi$ is additively combined with the Standard Model Higgs potential, $V_{H}=\lambda_{H}\left(\|H\|^{2}-v^{2}\right)^{2}$, and evaluated at $H=0$[4]. After first identifying this issue, reference [4] considered a simple modification of the coupling between the Higgs field and $\phi$ that avoids this problem: \[V_{0}=e^{-\lambda\phi}\left(V_{H}(H)+\Lambda\right)\,.\] (9) We note that equation (9) leads to substantial variations in the Higgs sector parameters, and consequently to large quantum corrections to the potential. These models must then realise either of the first two options identified in this paper to explain the present-day accelerated expansion through quintessence. ## IV Conclusion The drastically simple condition (1) has been proposed to delineate the ‘swampland’ of theories that cannot be embedded into any consistent theory of quantum gravity. The current status of this conjecture is highly uncertain and controversial [24, 3, 25, 4, 26, 5, 27, 28, 6, 7], in particular as detailed calculations demonstrating the failures of apparent counter examples are still lacking. Equation (1) excludes de Sitter vacua, but is compatible with certain models of quintessence. A key prediction of reference [2] is that such models cause cosmological variations in the couplings of dark matter and other dark sector fields. In this note, we have considered the theoretical implications of this proposed cosmology, and we have shown that they suffer from severe quantum instability problems. Variations in the couplings of massive states lead to large contributions to the vacuum energy that must be cancelled to an incredible accuracy. This instability problem is distinct from the cosmological constant problem as well as the regular fine-tuning problem of quintessence models. We have shown that if the quintessence models of [2] are realised in nature, one out of three conditions must hold: 1) the theory is incredibly fine-tuned; 2) there is a new, fantastic mechanism that surpasses even supersymmetry in taming dangerous quantum corrections; or 3) the quintessence field couples only to light states. 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1906.06620	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31060, "num_imgs": 5, "llama3_tokens_count": 7438 }	[ "content_image/1906.06620/x1.png", "content_image/1906.06620/x2.png", "content_image/1906.06620/x3.png", "content_image/1906.06620/x4.png", "content_image/1906.06620/x5.png" ]	# Joint Visual-Textual Embedding for Multimodal Style Search Gil Sadeh Amazon Lab126 gilsadeh@amazon.com &Lior Fritz Amazon Lab126 liorf@amazon.com &Gabi Shalev Amazon Lab126 shalevg@amazon.com &Eduard Oks Amazon Lab126 oksed@amazon.com ###### Abstract We introduce a multimodal visual-textual search refinement method for fashion garments. Existing search engines do not enable intuitive, interactive, refinement of retrieved results based on the properties of a particular product. We propose a method to retrieve similar items, based on a query item image and textual refinement properties. We believe this method can be leveraged to solve many real-life customer scenarios, in which a similar item in a different color, pattern, length or style is desired. We employ a joint embedding training scheme in which product images and their catalog textual metadata are mapped closely in a shared space. This joint visual-textual embedding space enables manipulating catalog images semantically, based on textual refinement requirements. We propose a new training objective function, Mini-Batch Match Retrieval, and demonstrate its superiority over the commonly used triplet loss. Additionally, we demonstrate the feasibility of adding an attribute extraction module, trained on the same catalog data, and demonstrate how to integrate it within the multimodal search to boost its performance. We introduce an evaluation protocol with an associated benchmark, and compare several approaches. ## 1 Introduction Recently, the ability to embed representations of images and text in a joint space was studied thoroughly for many tasks. Among which are image annotation and search [14, 2], zero-shot recognition [19, 16, 4, 18], robust image classification [4], image description generation [10], visual question-answering [17] and more. Vector arithmetic properties have been demonstrated lately as a surprising artifact of learning semantic embedding spaces. Mikolov et al. [15] showed that a learned _word2vec_ embedding space can capture semantic vector arithmetics, such as: “Paris” - “France” +“Italy” = “Rome”. Kiros et al. [13] demonstrated a similar phenomenon in multimodal visual-semantic embedding spaces, in which, with linear encoders, the learned embedding space captures multimodal regularities. For instance, given $f_{I}$, a representing vector of an image of a blue car, $f_{I}$ - “blue” + “red” yields a representing vector of a red car image. This paper refers to the specific, fine-grained, task of visual-textual multimodal search in the fashion domain. Example queries and their retrieved results can be seen in Figure 1. We believe this type of application can greatly impact the customer shopping experience, by enabling intuitive and interactive search refinements. With this technology, browsing large fashion catalogs and finding specific products can become easier and less frustrating. We consider training a visual-textual joint embedding model in an end-to-end manner, based on images and textual metadata of catalog products. We propose a training objective function which we refer to as Mini-Batch Match Retrieval (MBMR). Each mini-batch consists of matching and non-matching image-text pairs. We compute the cosine similarity of each pair, and maximize matching samples similarities with cross-entropy loss, as done in [21]. However, unlike [21], which assigns an embedding vector per each category, in our retrieval task the notion of category does not exist. Instead, we learn an embedding for each item (image and text) and try to classify the correct pair from the mini-batch reference set. We demonstrate the superiority of this approach over the commonly used triplet loss. In addition, we explore the task of visual fashion attribute extraction, utilizing the noisy catalog data alone, without additional annotation effort. A pool of possible fashion attributes is extracted from frequent words in the catalog metadata, and a multi-label classifier is trained to extract the correct ones given a product image. We demonstrate that, although the catalog-based labels are noisy, attribute extraction produces satisfying results. We propose and evaluate several approaches for multimodal search. The first approach leverages the query-arithmetic phenomenon of visual-textual joint embeddings. A second approach utilizes our learned attribute extraction module, for soft textual filtering, alongside visual search, based on our visual-semantic embedding space. Finally, we propose a combined approach, which leverages both the joint embedding based query-arithmetic property and soft attribute filtering. This approach yields a considerable performance improvement over the other methods. <figure><img src="content_image/1906.06620/x1.png"><figcaption>Figure 1: Examples of typical multimodal search queries and their topretrieved results.</figcaption></figure> ## 2 Related Work Image recognition classifiers treat labels as disconnected and unrelated, resulting in visual recognition systems that cannot transfer semantic information about learned labels to unseen words or phrases. Early visual-textual joint embedding works addressed this problem by mapping image-word pairs, where the words corresponded to image labels or attributes. Weston et al. [20] trained a joint embedding model of both images and their labels, by employing an online learning-to-rank algorithm. Frome et al. [4] leveraged textual data to learn semantic relationships between labels by explicitly mapping images into a common visual-semantic embedding space. They showed this approach leads to more semantically reasonable errors and significantly improved zero-shot predictions. More recent works attempted to map images and their textual descriptions into a common embedding space. Klein et al. [14] employed Canonical Correlation Analysis (CCA) [8] to learn projections of precomputed image and caption features, onto a joint embedding space, for cross-modal retrieval tasks. Kiros et al. [13] employed a triplet-based ranking loss in order to learn a similar embedding space for images and text, for caption generation and ranking tasks. Karpathy et al.[11] worked on a finer level, embedding fragments of images and sentences jointly with a max-margin based objective. In the fashion domain, Han et al. [6] learned a similar visual-semantic embedding for product images and their corresponding textual descriptions. They combined this joint embedding in their outfit recommendation engine, so that it is agnostic to the input type (image, text or a combination of both). Several works considered the task of manipulating attributes for fashion search. Zhao et al. [22] trained a network to jointly optimize attribute classification loss and triplet ranking loss, over image triplets, for facilitating precise attribute manipulation and image retrieving. The network learned, in a supervised manner, to modify the intermediate image representation based on the desired manipulation. Kenan et al. [1] proposed learning attribute specific representations by leveraging weakly-supervised localization, in order to manipulate attributes during fashion search. M. Günel et al. [5] proposed a GAN-based solution for language guided image manipulation, where the generator performs feature-wise linear modulation between visual features and desired natural language descriptions. Zhao et al. [23] proposed a Multi-Task Learning (MTL) system to jointly train an image captioning and attribute extraction model. They demonstrated how the auxiliary attribute extraction task resulted in better image representation and improved performance in the original captioning task. ## 3 Data The data used for training the joint embedding model consists of 0.5M fashion products from a retail website. Each product item has associated image and textual metadata. The catalog has a very diverse range of products, and includes rich and relatively accurate metadata. The actual search, and its evaluation, are performed on an larger set of 1.5M catalog items (only tops, bottoms and dresses) from a different retail website. Although the textual metadata of our training catalog is relatively clean and accurate compared to other catalogs, there still exists noise and variability in the textual metadata. Similar items can have very different textual descriptions, while non-similar items may have relatively similar descriptions. Moreover, textual metadata may be lacking in details frequently. <figure><img src="content_image/1906.06620/x2.png"><figcaption>Figure 2: Joint Embedding: A ResNet-18 CNN extracts visual features from theimage with an additional fully connected (FC) layer which projects thesefeatures to the joint space. The textual encoder sums the word embeddings ofall relevant words in the textual metadata. Attribute Extraction: Anadditional network branch extracts attribute probabilities from the imagerepresentation. It utilizes the catalog textual metadata as ground-truthattribute labels.</figcaption></figure> ## 4 Training Figure 2 illustrates the architecture of our model. The basic joint embedding model consists of two main branches, an image encoder and a text encoder. Image encoding is based on a ResNet-18 [7] deep convolutional neural network (CNN), followed by an additional fully connected layer which projects the visual feature vector to the same space as the textual encoding. Text encoding is done by summing the word embeddings of all input words. The text is treated as a bag-of-words, rather than an ordered sequence of words, since it is accumulated from several metadata fields, and may contain a mixture of sentences and individual keywords. For attribute extraction, we add a third branch to this joint embedding architecture. The branch consists of a fully connected layer, followed by a sigmoid activation function for multi-label classification. The input to this branch is the image feature vector, $f_{I}$, and the output is a vector of attribute probabilities, $p_{\bm{w}}(I)$. The size of the attribute probability vector is determined by the vocabulary size, $\|V\|$. The model is trained end-to-end. That is, both encoder branches are trained jointly. The ResNet weights are initialized by a pre-trained ImageNet [3] model. The word embeddings are based on _word2vec_, and are trained on product titles. Word embeddings that do not appear in this set are initialized randomly. The fully connected layer parameters are initialized with PCA over the extracted ImageNet features. We also fix the ResNet weights at the begining of training, and unfreeze only the two top Residual blocks after two epochs. We use the Adam [12] optimizer, with an exponentially decaying learning rate schedule. We have found that all of these settings are helpful in order to improve convergence and reduce overfitting. Our training objective is composed of two loss terms. A Mini-Batch Match Retrieval (MBMR) loss, $\mathcal{L}_{MBMR}$, for the task of learning a joint embedding space, and a multi-label cross-entropy loss, $\mathcal{L}_{a}$, for attribute extraction. The final objective is a weighted sum of both loss terms. ### Textual Metadata Preprocessing In order to clean and normalize the textual metadata we use several preprocessing steps when building our vocabulary. (1) Tokenization – divide the raw description text into a set of tokens. (2) Stemming – normalize words to their base form, in order to avoid multiple word variations with the same visual meaning. (3) Part-Of-Speech (POS) based filtering – identify noun and adjective tokens, which are more likely to have visual significance, and ignore the rest. (4) Word frequency thresholding – words that appear less times in the dataset than some hard cut-off threshold are removed, thus reducing noise and avoiding an unnecessarily large vocabulary. We set our threshold to $500$. These preprocessing steps determine the vocabulary, $V$, of our model. Its size, $\|V\|$, also affects the number of parameters in the word embeddings and attribute extraction fully connected layer. ### Mini-Batch Match Retrieval Objective The objective of the joint-embedding training procedure should encourage matching (non-matching) image-text pairs to be as close (distant) as possible to (from) each other, in the common embedding space. To achieve this, we propose the following Mini-Batch Match Retrieval (MBMR) objective. In our training setting, each mini-batch consists of $N$ product items, $\left\{I_{i},T_{i}\right\}_{i=1}^{N}$, where $I_{i}$ is an image, and $T_{i}$ is its corresponding textual metadata. For each image embedding in the batch, $f_{I}$, and text embedding in the batch, $f_{T}$, we compute their cosine similarity, \[S_{I,T}=\dfrac{f_{I}\cdot f_{T}}{\\|f_{I}\\|\\|f_{T}\\|}.\] (1) We then define the probability of image $I_{i}$ to match description $T_{j}$ as, \[P(T_{j}\:\|\:I_{i})=\dfrac{\exp\left\{S_{I_{i},T_{j}}/\tau\right\}}{\sum_{k}{ \exp\left\{S_{I_{i},T_{k}}/\tau\right\}}},\] (2) where $\tau$ is a temperature parameter. The probability of $T_{i}$ to match image $I_{j}$, is calculated similarly, \[P(I_{j}\:\|\:T_{i})=\dfrac{\exp\left\{S_{I_{j},T_{i}}/\tau\right\}}{\sum_{k}{ \exp\left\{S_{I_{k},T_{i}}/\tau\right\}}}.\] (3) The final objective is obtained by applying cross-entropy for every query image and text in the batch, \[\mathcal{L}_{MBMR}=-\sum_{i}{\log P(T_{i}\:\|\:I_{i})}-\sum_{i}{\log P(I_{i}\:\| \:T_{i})}.\] (4) ### Attributes Extraction Since our model learns to bridge the gap between images and text, it is natural to expect it to be able to provide out-of-the-box attribute extraction just by computing cosine-similarities between images and words. In practice, however, this leads to noisy results, due to the following reasons. First, not all words are equally visually grounded. Some words are very visually dominant, while others may have very little (if any) visual significance, and may exist only due to imperfect textual preprocessing. Second, word frequencies vary significantly. Some attributes appear in almost every item description in the catalog, like garment types, while others appear very rarely. This data behavior can be considered as noisy labels for our attribute extractor. In order to create a more robust attribute extraction model, we add another branch to the model which consists of a fully connected layer that projects image embeddings to the vocabulary size, $\|V\|$, followed by a sigmoid function. The outputs of this branch, $\left\{\hat{p}_{w}(I)\right\}$, are approximations of the probabilities for each word $w$ in the vocabulary to belong to image $I$. The ground-truth labels are determined by the existence of words in the product textual metadata. An additional loss term is added for this multi-label classification task. During inference, we take the following additional steps in order to obtain reliable attribute extraction. We compute a per-word threshold, by optimizing the F-score on the validation set. This threshold, $thr_{w}$, is used to define a classification score, \[\tilde{p}_{w}(I)={\rm sigmoid}\left(\dfrac{\hat{p}_{w}(I)-{\rm thr}_{w}}{{\rm thr }_{w}}\right).\] (5) Additionally, we compute a cosine-similarity score between word and image features, \[S_{w,I}=\dfrac{f_{w}\cdot f_{I}}{\\|f_{w}\\|\\|f_{I}\\|},\] (6) where $f_{w}$ and $f_{I}$ are the word and image embeddings, in the joint space, respectively. Finally, we average the classification score, $\tilde{p}_{w}(I)$, and the clipped cosine-similarity score, $S_{w,I}$, in order to obtain the final probability that word $w$ is a characteristic of image $I$, \[p_{w}(I)=\dfrac{\tilde{p}_{w}(I)+\max{(S_{w,I},0)}}{2}.\] (7) In order to approximate the probability of a desired and undesired attribute set $\bm{w}=\left\{\bm{w}^{+},\bm{w}^{-}\right\}$, in the multimodal search scenario, we follow Bayes rule, under the independence assumption, \[p_{\bm{w}}(I)=\prod_{w^{+}\in\bm{w}^{+}}{p_{w^{+}}(I)}\prod_{w^{-}\in\bm{w}^{- }}{(1-p_{w^{-}}(I))}.\] (8) <figure><img src="content_image/1906.06620/x3.png"><figcaption>Figure 3: Attributes extraction examples. We list the top 6 extractedattributes, for each catalog image, according to their probabilities pw(I).</figcaption></figure> ## 5 Multimodal Refinement Search ### Query Arithmetic Approach During inference, the text and image encoders can yield image and textual query feature vectors which lay in a common embedding space. These feature vectors can be used to search for products, with similar visual or textual properties, in a dedicated catalog. The similarity metric used for matching the query and catalog items is, as in the training phase, cosine similarity. The catalog image and textual features can be precomputed offline once. Ideally speaking, the fact that visual and textual modalities share the same embedding space, combined with the linear nature of the text encoder, enables performing arithmetic operations (as in _word2vec_) in order to manipulate the desired search query. This enables searching for visually similar products with some different properties, defined textually, by simply adding (subtracting) desired (undesired) textual features to (from) the product visual feature vector. That is, for a given query image, $I$, and a desired and undesired attribute set, $\bm{w}=\left\{\bm{w}^{+},\bm{w}^{-}\right\}$, the new mutlimodal query $q$ can be defined by, \[q=f_{I}+f_{T},\] (9) \[f_{T}=\sum_{w^{+}\in\bm{w}^{+}}{f_{w^{+}}}-\sum_{w^{-}\in\bm{w}^{-}}{f_{w^{-}}},\] (10) where $f_{I}$ is the image embedding, and $f_{T}$ is the linear combination of desired and undesired word embeddings. The similarity score, $S$, between the query and reference catalog items, is defined as the cosine similarity between $q$ and the reference visual features $f_{I_{r}}$. ### Attribute Filtering Approach An alternative approach for multimodal search is filtering out all catalog items which are not consistent with the textual query. Then, the search score can be calculated based on visual similarity alone. This approach can be formulated as follows. \[S=\dfrac{q\cdot f_{I_{r}}}{\\|q\\|\\|f_{I_{r}}\\|}\cdot\mathbbm{1}{(w\in T_{r}\;\; \;\forall w\in\bm{w}^{+})}\cdot\mathbbm{1}{(w\notin T_{r}\;\;\;\forall w\in\bm {w}^{-})},\] (11) where $q=f_{I}$, $T_{r}$ is the set of words in the reference textual metadata, and $\bm{w}^{+}$ ($\bm{w}^{-}$ ) is the set of desired (undesired) properties. This approach should work well given an ideal catalog, with complete and error-free textual metadata. However, this is not the case in most catalogs. Hence, we derive a soft filtering method based on attribute extraction probabilities, \[S=\dfrac{q\cdot f_{I_{r}}}{\\|q\\|\\|f_{I_{r}}\\|}\cdot p_{\bm{w}}(I_{r}),\] (12) where $q=f_{I}$ and $p_{\bm{w}}(I_{r})$ is the probability of the textual desired and undesired properties in the reference image $I_{r}$. ### Combined Approach We attempt to combine both previously described methods into a single robust one. We do so by using the soft attribute filtering along with the query arithmetic based search. The motivation of incorporating attribute filtering is to better meet the textual manipulation criteria. Since attribute filtering is soft and noisy, it is not enough to use it with visual search alone (as in 5.2), as it will encourage retrieval of visually similar items without considering the textual manipulation. The exact formulation is as follows. \[S=\dfrac{q\cdot f_{I_{r}}}{\\|q\\|\\|f_{I_{r}}\\|}\cdot p_{\bm{w}}(I_{r}),\] (13) where $q=f_{I}+f_{T}$, as in 5.1. ## 6 Evaluation <figure><img src="content_image/1906.06620/x4.png"><figcaption>Figure 4: Comparison of top-K validation accuracy convergence, duringtraining, between triplet loss, MBMR loss and a multi-task objective composedof MBMR loss and multi-label cross-entropy loss for attribute extraction.</figcaption></figure> For evaluation purposes we automatically constructed a benchmark of multimodal queries. Query product images (of tops, bottoms and dresses) were randomly sampled, and assigned with desired and undesired textual requirements out of a pool of common fashion attributes. The pool consisted of $110$ fashion attributes from $5$ major categories: color, pattern, neckline, style and garment type. Textual requirements can specify either adding, removing or replacing specific properties to or from the query image. The final benchmark consists of 1500 queries (300 for each attribute category), after manual verification and query filtering. A commonly used metric in information retrieval tasks is the normalized Discounted Cumulative Gain (nDCG) [9]. The DCG metric measures ranking quality, which cumulates the relevance of the top-$K$ retrieved items per query, while penalizing them differently based on their rank. \[{\rm DCG_{K}}=\sum_{i=1}^{K}\dfrac{rel_{i}}{log_{2}{(i+1)}},\] (14) where $rel_{i}$ is the relevance of the reference item ranked in place $i$ by the model. The relevance is given by some oracle. The nDCG normalizes the DCG metric by the Ideal-DCG value (IDCG), which is calculated similarly to the DCG, over an ideally sorted reference list. For IDCG, we used an upper-bound approximation which assumes our reference corpus contains $K$ items with $rel=1$. In order to evaluate multimodel search performance, two aspects need to be accounted for, visual and textual. A perfect result would meet the textual criteria while still being as visually similar as possible to the query image. We develop two nDCG metrics, with relevance scores based on a visual oracle and a textual oracle. Our final, multimodal, metric is a simple geometric mean of both nDCG scores. * Visual nDCG (V-nDCG): Based on visual relevance, which is extracted from a baseline visual search model. This purely visual model was trained with triplet loss on catalog images, where for each query image a different image of the same item was considered as a positive sample and images of different items were considered as negative samples. The relevance is the cosine similarity between reference and query visual features, extracted from this baseline model. * Textual nDCG (T-nDCG): Based on presence (absence) of desired (undesired) query words in the reference textual metadata. The relevance is defined by the rate of criteria that are met. A desired word criterion is considered as met if the reference metadata includes the word. An undesired word criterion is met if the reference metadata does not include the word. * Multimodal (MM): MM$\triangleq\sqrt{\text{V-nDCG}\cdot\text{T-nDCG}}$. These metrics are somewhat noisy, and may be inaccurate in specific cases, such as incomplete and inaccurate metadata or inaccuracies caused by the baseline visual search model. However, on the corpus level we observe that they are stable and reliable enough to serve as evaluation metrics, and help us compare different methods. ## 7 Experimental Results We compare our described Mini-Batch Match Retrieval (MBMR) objective with a triplet loss, as utilized in [13]. Figure 4 shows the convergence of top-5 and top-20 validation accuracy during the joint embedding training procedure. The top-$K$ accuracy metric measures the rate of images and text descriptions for which the actual matching pair was ranked, based on cosine similarity, within the top $K$ references out of the entire validation set, which consists of 23.5K items. In our experiments, the mini-batch size was set to $160$, the MBMR temperature $\tau$ to 0.025 and the triplet loss margin to 0.2. We believe that top-$K$ accuracy is a good metric for this task, as in retrieval tasks we usually mostly care about the top retrieved results. It can be seen that the MBMR objective leads to faster and superior convergence over triplet loss. Additionally, it can be seen that multi-task training, with the additional attribute extraction branch and corresponding loss, slightly increases performance. We follow our evaluation protocol for multimodal search, as described in Section 6, and compare the following methods: Soft Attribute Filtering (SAF), Query Arithmetic (QA) and their combination (QA+SAF). It can be seen in Table 1 that although there is a clear trade-off between the visual and textual metrics (V-nDCG and T-nDCG), on the overall multimodal (MM) metric, the combined approach (QA+SAF) outperforms all others significantly. These conclusions are further reinforced by our qualitative visualization and analysis of the results, as can be seen in Figure 5. \| SAF \| QA \| QA+SAF ---\|---\|---\|--- \| V-nDCG \| T-nDCG \| MM \| V-nDCG \| T-nDCG \| MM \| V-nDCG \| T-nDCG \| MM Color \| 0.726 \| 0.407 \| 0.543 \| 0.8 \| 0.413 \| 0.574 \| 0.621 \| 0.591 \| 0.605 Pattern \| 0.769 \| 0.407 \| 0.559 \| 0.818 \| 0.426 \| 0.59 \| 0.672 \| 0.543 \| 0.604 Neckline \| 0.77 \| 0.572 \| 0.663 \| 0.806 \| 0.527 \| 0.651 \| 0.68 \| 0.628 \| 0.653 Style \| 0.761 \| 0.464 \| 0.594 \| 0.815 \| 0.401 \| 0.572 \| 0.676 \| 0.563 \| 0.617 Garment \| 0.785 \| 0.27 \| 0.46 \| 0.828 \| 0.221 \| 0.427 \| 0.696 \| 0.486 \| 0.581 Overall \| 0.76 \| 0.419 \| 0.564 \| 0.813 \| 0.397 \| 0.568 \| 0.669 \| 0.561 \| 0.612 Table 1: Evaluation results: we report V-nDCG, T-nDCG and MM metrics, and compare the Soft Attribute Filtering (SAF), the naive linear Query-Arithmetic (QA) and the combined (QA+SAF) methods. Queries are split by the type of textual criteria. ## 8 Conclusions In this paper, we explored the task of multimodal fashion search. We proposed utilizing a visual-textual joint embedding model for this task, suggested an alternative training objective and demonstrated its effectiveness. We explored and evaluated several approaches to leverage this joint-embedding model for the multimodal search task. Unlike previous works, our method does not require direct supervised data of images before and after the textual manipulation. Moreover, our training and evaluation methods are all performed over noisy, not well structured, catalog data. <figure><img src="content_image/1906.06620/x5.png"><figcaption>Figure 5: Qualitative results of top 3 retrieved items for example querieswith Soft Attribute Filtering (SAF), Query Arithmetics (QA) and combinedapproach (QA+SAF).</figcaption></figure> ## References * [1] Kenan E Ak, Ashraf A Kassim, Joo Hwee Lim, and Jo Yew Tham. 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Large scale image annotation: learning to rank with joint word-image embeddings. _Machine learning_, 81(1):21–35, 2010. * [21]Nicolai Wojke and Alex Bewley. Deep cosine metric learning for person re-identification. In _2018 IEEE winter conference on applications of computer vision (WACV)_, pages 748–756. IEEE, 2018. * [22] Bo Zhao, Jiashi Feng, Xiao Wu, and Shuicheng Yan. Memory-augmented attribute manipulation networks for interactive fashion search. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_, pages 1520–1528, 2017. * [23] Wei Zhao, Benyou Wang, Jianbo Ye, Min Yang, Zhou Zhao, Ruotian Luo, and Yu Qiao. A multi-task learning approach for image captioning. In _IJCAI_, pages 1205–1211, 2018.
1303.1774	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 39947, "num_imgs": 13, "llama3_tokens_count": 11948 }	[ "content_image/1303.1774/Fig1.png", "content_image/1303.1774/Fig2.png", "content_image/1303.1774/Fig3.png", "content_image/1303.1774/Fig4.png", "content_image/1303.1774/Fig5.png", "content_image/1303.1774/Fig6.png", "content_image/1303.1774/Fig7.png", "content_image/1303.1774/Fig8.png", "content_image/1303.1774/Fig9.png", "content_image/1303.1774/Fig10.png", "content_image/1303.1774/Fig11.png", "content_image/1303.1774/Fig12.png", "content_image/1303.1774/Fig13.png" ]	# Angle-Resolved Photoemission from Cuprates with Static Stripes Tonica Valla Condensed Matter Physics and Materials Science Department, Brookhaven National Lab, Upton, NY 11973, USA ###### Abstract 25 years after discovery of high-temperature superconductivity (HTSC) in La${}_{2-x}$Ba${}_{x}$CuO${}_{4}$ (LBCO), the HTSC continues to pose some of the biggest challenges in materials science. Cuprates are fundamentally different from conventional superconductors in that the metallic conductivity and superconductivity are induced by doping carriers into an antiferromagnetically ordered correlated insulator. In such systems, the normal state is expected to be quite different from a Landau-Fermi liquid - the basis for the conventional BCS theory of superconductivity. The situation is additionally complicated by the fact that cuprates are susceptible to charge/spin ordering tendencies, especially in the low-doping regime. The role of such tendencies on the phenomenon of superconductivity is still not completely clear. Here, we present studies of the electronic structure in cuprates where the superconductivity is strongly suppressed as static spin and charge orders or “stripes”” develop near the doping level of $x=1/8$ and “outside” of the superconducting dome, for $x<0.055$. We discuss the relationship between the “stripes”, superconductivity, pseudogap and the observed electronic excitations in these materials. keywords: superconductivity, stripes, photoemission † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction The underdoped side of cuprate phase diagram is full of amazing features that do not exist in conventional superconductors. One example is a normal-state gap (pseudogap), which exists above the temperature of the superconducting transition $T_{c}$. It is generally believed and observed that the magnitude of the pseudogap monotonically decreases with increasing doping, whereas $T_{c}$ moves in the opposite direction in the underdoped regime as shown in Fig. 1[1; 2]. The origin of the pseudogap and its relationship to superconductivity is one of the most important open issues in physics of HTSC and represents the focal point of current theoretical interest [3; 4; 5; 6]. In one view, the pseudogap is a pairing (superconducting) gap, reflecting a state of Cooper pairs without global phase coherence. The superconducting transition then occurs at some lower temperature when phase coherence is established [7; 8]. In an alternative view, the pseudogap represents another state of matter that competes with superconductivity. However, the order associated with such a competing state has not been unambiguously identified. Candidates were found in neutron scattering studies, where an incommensurate spin order was detected inside vortices [9] and in <figure><img src="content_image/1303.1774/Fig1.png"><figcaption>Figure 1: A generic phase diagram of cuprate superconductors. In theunderdoped region, the excitation gap Δ⋆ increases while Tc decreases withreduced doping</figcaption></figure> scanning tunnelling microscopy (STM) experiments where a charge ordered state, energetically very similar to the superconducting state, has been found in the vortex cores, in the ‘pseudogap’ regime above $T_{c}$ and in patches of underdoped material in which the coherent conductance peaks were absent [10; 11; 12]. These charge/spin superstructures - “stripes” are particularly prominent in 214 family of cuprates (LBCO and La${}_{2-x}$Sr${}_{x}$CuO${}_{4}$ (LSCO)), where a _static_ spin (and charge) superstructure forms at low temperatures in the “spin glass” phase ($0.02<x<0.055$), and around the “1/8 anomaly” ($x\sim 0.125$). Superconductivity is notably absent in both regimes, indicating that it competes with such orders. However, the ordering tendencies exist in most cuprates and extend well into the overdoped region. For example, neutron scattering studies show that incommensurate spin structure forms as soon as the carriers are doped into the system, exists within the whole superconducting regime and disappears together with superconductivity at $x\sim 0.3$[13]. Incommensuration from the antiferromagnetic wave vector ($\pi/a,\pi/a$) grows roughly in proportion with doping till $x\sim 1/8$ and tends to saturate after that. When these super-structures are static, superconductivity is either completely absent ($x<0.055$) or strongly suppressed (near $x=1/8$). In the former case the spin superstructure is diagonal, while in the latter it is parallel to the Cu-O bond [14] as shown in Fig. 2. The charge order (with half-wavelength of the spin order) has been found so far only in the latter case [15]. <figure><img src="content_image/1303.1774/Fig2.png"><figcaption>Figure 2: Doping dependence of incommensurability (a), and angle (relative tothe Cu-Cu direction) (b) of the spin superstructure measured in LSCOFujita2002 . Arrows indicate regions where the superstructure is static</figcaption></figure> ## 2 Parallel Stripes <figure><img src="content_image/1303.1774/Fig3.png"><figcaption>Figure 3: (a) Spin Tranquada2004 and (b) charge Abbamonte2005 superstructurein LBCO at x=1/8.</figcaption></figure> LBCO exhibits a sharp drop in superconducting transition temperature, T${}_{c}\to 0$, when doped to $\sim 1/8$ holes per Cu site ($x=1/8$), while having almost equally ‘strong’ superconducting phases, with T${}_{Cmax}\sim 30$ K at both higher and lower dopings [17]. Therefore, the $x=1/8$ case represents an ideal system to study the ground state of the pseudogap as the ‘normal’ state extends essentially to $T=0$. In scattering experiments (Fig. 3) on single crystals, static stripes with spin period of 8 unit cells [16; 18] and a charge period of 4 unit cells have been detected at low temperatures [15] . While superconductivity is strongly reduced at $x=1/8$, metallic behaviour seems to be preserved [19; 20; 21]. Optical studies have detected a loss of spectral weight at low frequencies with simultaneous narrowing of a Drude component, suggesting the development of an anisotropic gap [22]. <figure><img src="content_image/1303.1774/Fig4.png"><figcaption>Figure 4: Photoemission from LBCO at x=1/8 Valla2006 . (a) Intensity at theFermi level (Fermi surface). Arrows correspond to the momentum linesrepresented in (b) and (c). (b) Photoemission intensity as a function ofbinding energy along the momentum lines indicated in (a). (c) Energydistribution curves (EDCs) of spectral intensity integrated over a smallinterval around kF along the two lines in k-space shown in (b). The arrowrepresents the shift of the leading edge. The spectra were taken at T=16 K.</figcaption></figure> Figure 4 shows the photoemission spectra from LBCO at $x=1/8$ in the ordered state ($T=16$ K) [23]. The momentum distribution of the photoemission intensity from the energy window of ±10 meV around the Fermi level is shown within the Brillouin zone (Fig. 4a). From these contours, the Fermi surface is extracted. The area enclosed by the Fermi line corresponds to $x=0.115\pm 0.015$, in good agreement with the nominal doping levels, signaling that the bulk property has been probed. As shown in Fig. 4b and c, we have detected an excitation gap in photoemission spectra from this material with a magnitude that depends on the $k$ position on the Fermi surface, vanishing at the node and with maximum amplitude near the antinode. <figure><img src="content_image/1303.1774/Fig5.png"><figcaption>Figure 5: Magnitude of single-particle gap in LBCO at x=1/8 and x=0.095 as afunction of an angle around the Fermi surface at T=16 K (a) and as a functionof temperature, for two characteristic Fermi points, for the x=1/8 sample (b)Valla2006 .</figcaption></figure> In the detailed $k$-dependence for several samples (Fig. 5a), two unexpected properties are uncovered. First, gaps in all samples have magnitudes consistent with $d$-wave symmetry, even though superconductivity is essentially nonexistent in LBCO at $x=1/8$. Second, the gap in LBCO is larger at $x=1/8$ than at $x=0.095$. This finding contradicts a common belief that the excitation gap in cuprates monotonically increases as the antiferromagnetic (AF) phase is approached. The momentum-resolved picture from ARPES is consistent with the STM data obtained from the same parent crystal used for ARPES. In Fig. 6a, a typical STM topographic image of a cleaved LBCO surface is shown. In addition, the differential conductance ($dI/dV$) spectra were taken at many points in a wide range of energies and averaged over the whole field of view. The resulting conductance spectrum (Fig. 6b) shows a symmetric V-like shape at low energies, with zero-DOS falling exactly at the Fermi level, which is consistent with a pairing $d$-wave gap. The magnitude of this gap, $\Delta_{0}\approx 20$ meV, as determined from the breaks in $dI/dV$ curve agrees with the maximal gap $\Delta_{0}$ measured in photoemission. <figure><img src="content_image/1303.1774/Fig6.png"><figcaption>Figure 6: (a) STM topographic image of the cleaved LBCO sample at 4.2 K. (b) Atunneling conductance spectrum averaged over the area shown in (a). A V-likeprofile of DOS for energies \|ω\|<20 meV is consistent with a d-wave gapobserved in ARPES Valla2006 .</figcaption></figure> <figure><img src="content_image/1303.1774/Fig7.png"><figcaption>Figure 7: Doping dependence of Δ0 in LBCO (triangles) Valla2006 and LSCO(circles and squares) Zhou2004 ; Damascelli2003 .</figcaption></figure> Our study provides the evidence for a $d$-wave gap in the normal ground state of a cuprate material. Previous studies on underdoped Bi${}_{2}$Sr${}_{2}$CaCu${}_{2}$O${}_{8+\delta}$ (BSCCO) were always affected by the superconductivity: The disconnected “Fermi arcs” were seen, shrinking in length as $T$ was lowered below pseudogap temperature $T^{\star}$ and collapsing onto (nodal) points below $T_{c}$[25; 26; 27]. As a result of this abrupt intervention of superconductivity, it was not clear whether the pseudogap ground state would have a Fermi arc of finite length or a nodal point or whether it would be entirely gapped. In LBCO, the absence of superconductivity at $x=1/8$ has enabled us to show that the normal state gap has isolated nodal points in the ground state. This result points to the pairing origin of the pseudogap. With increasing $T$, a finite-length Fermi arc forms, as suggested in Fig. 5b, in accord with results on BSCCO [26; 27]. <figure><img src="content_image/1303.1774/Fig8.png"><figcaption>Figure 8: STM Hanaguri2004 and ARPES Shen2005 on lightly dopedCa2−xNaxCuO2Cl2. (a) differential conductance and (b) its Fourier transform inSTM reveal a checkerboard superstructure with q=1/4(2π/a0) and q=3/4(2π/a0).(c) ARPES on the same material shows that the Fermi surface can be nested bythe same vectors.</figcaption></figure> <figure><img src="content_image/1303.1774/Fig9.png"><figcaption>Figure 9: Doping dependence of the Fermi surface in 214 cuprates. (a-c) LBCO,at x=0.095, x=1/8 and x=0.165 Valla2007 . (d) LSCO, for several dopings in therange 0.03<x<0.3 Yoshida2006</figcaption></figure> <figure><img src="content_image/1303.1774/Fig10.png"><figcaption>Figure 10: A compilation of relevant wave vectors from neutron and x-rayscattering and ARPES on 214 materials. a) Doping dependence of the antinodalkF, as indicated in (b) by the arrow, in LBCO (triangles) Valla2006 and LSCO(circles) Yoshida2006 . The wavevector for charge order (diamond)Abbamonte2005 and the incommensurability δ from (π,π) point from neutron-scattering experiments Fujita2002 ; Tranquada2004 (squares) are also shown.The gray bar represents the boundary between the “diagonal” and “parallel”spin superstructures and the onset of superconductivity. (b) A sketch ofrelevant vectors in the k-space.</figcaption></figure> What might be the origin of the observed d-wave gap in LBCO if superconductivity is absent? Neutron and $x$-ray scattering studies on the same crystal have identified a static spin order and a charge order [15; 16]. Therefore, it would be tempting to assume that at least a portion of the measured gap is due to the charge/spin order, in analogy with conventional 2-dimensional (2D) charge-density-wave (CDW) and spin-density wave (SDW) systems. It has been suggested that in cuprates the charge/spin ordered state forms in a way where carriers doped into the AF insulator segregate into one-dimensional (1D) charge rich structures (‘stripes’) separated by the charge poor regions of a parent antiferromagnet [5; 18; 32; 33]. Some of the predicted spectral signatures of such 1D-like structures would be the states that are relatively narrow in momentum, but broad in energy [33], in a good agreement with the experiments (see Fig. 4 and ref. [34; 35], for example). However, questions have often been raised on how to reconcile these unidirectional structures with an apparent 2D Fermi surface and a gap with $d$-wave symmetry. In the more conventional view, doped carriers are delocalized in the planes, forming a 2D Fermi surface that grows in proportion with carrier concentration. The charge/spin ordered state may then be formed in the particle-hole channel by nesting of Fermi surface segments, producing a divergent electronic susceptibility and a Peierls-like instability and pushing the system into a lower energy state with a single particle gap at nested portions of the Fermi surface. Therefore, if this is a relevant scenario for the origin of charge/spin order in cuprates, there should be favourable nesting conditions on (portions of) the FS and these conditions would have to change with doping in a way that is consistent with changes observed in (neutron and $x$-ray) scattering experiments. An example of a cuprate where such a ‘nesting’ scenario is proposed to be at play is Ca${}_{2-x}$Na${}_{x}$CuO${}_{2}$Cl${}_{2}$ (CNCOC) [29]. STM studies have detected checkerboard-like modulations in local DOS on the surface of this material, with $4a\times 4a$ periodicity, independent of doping [28]. Subsequent ARPES studies on the same system have shown a Fermi surface with a nodal arc and truncated anti-nodal segments [29]. The anti-nodal segments can be efficiently nested by $q_{CDW}=2k_{F}=\pi/(2a)$ (and $3\pi/(2a)$) - the same wave vectors observed in STM for charge superstructure, making the nesting scenario viable. However, as the area enclosed by the Fermi surface in Fig. 8c is much smaller than the nominal doping would require (corresponding to the electron doping), it is suggestive that the nesting might be at play at the surface of CNCOC, but the physics in bulk of this material could be very different. If we apply the same nesting scenario to LBCO at $x=1/8$, we obtain $q_{CDW}\sim 4k_{F}$$(=\pi/2a)$, for charge order, instead of $2k_{F}$ nesting, suggested to be at play in CNCOC. Moreover, from the doping dependence of the Fermi surface in both LBCO and LSCO (Fig. 9), the nesting of anti-nodal segments would produce a wavevector that shortens with doping, opposite of that observed in neutron scattering studies in terms of magnetic incommensurability. This is illustrated in Fig. 10, where we compile the doping dependences of the anti-nodal $2k_{F}$ for LBCO [23] and LSCO [31] samples and wave-vectors measured in scattering experiments [14; 16; 15]. There is another, more fundamental problem with the ‘nesting’ scenario: any order originating from nesting (particle-hole channel) would open a gap only on nested segments of the Fermi surface, preserving the non-nested regions. The fact that only four gapless points (nodes) remain in the ground state essentially rules out nesting as an origin of pseudogap. In addition, a gap caused by conventional charge/spin order would be pinned to the Fermi level only in special cases [36]. The observation that it is always pinned to the Fermi level (independent of $k$-point, as measured in ARPES and of doping level, as seen in STM on different materials) and that it has $d$-wave symmetry undoubtedly points to its pairing origin - interaction in the particle-particle singlet channel. Note that, in contrast to the low-energy pairing gap, STM at higher energies shows a DOS suppressed in a highly asymmetric manner, indicating that some of the ‘nesting’-related phenomena might be at play at these higher energies (Fig. 6b). The surprising anti-correlation of the low-energy pairing gap and $T_{c}$ over some region of the phase diagram suggests that in the state with strongly bound Cooper pairs, the phase coherence is strongly suppressed by quantum phase fluctuations. Cooper pairs are then susceptible to spatial ordering and may form various unidirectional [18; 5] or 2D [37; 28] superstructures. Quantum phase fluctuations are particularly prominent in cases where such superstructures are anomalously stable. Some of these effects have been observed in transport properties [20; 21], but the connection to the specific models has not been firmly established. For some of the theoretically proposed structures, the quantum phase fluctuations are strongest at the doping of $1/8$, in general agreement with our results: $1/8$ represents the most prominent ‘magic fraction’ for a checkerboard-like ‘CDW of Cooper pairs’ [37], and it locks the ‘stripes’ to the lattice in a unidirectional alternative. The presence of nodes in the ground state of the pseudogap represents a new decisive test for validity of models proposed to describe such structures. The more recent ARPES studies [38; 39] have suggested that the apparent deviations from the perfect (cos$k_{x}-$cos$k_{y}$)/2 dependence of the single-particle gap in some underdoped cuprates point to the co-existence of two gaps: the pairing one, that resides near the node and scales with $T_{c}$ and the second one, that originates from the competing charge/spin orders, resides near the anti-nodes and scales with $T^{\star}$. We note that in the case of LBCO, this picture does not seem to make much sense, since $T_{c}\to 0$ would imply that the near-nodal, ‘pairing’ gap also goes to zero, opposite of the experimental observation [23; 39]. ## 3 Diagonal Stripes <figure><img src="content_image/1303.1774/Fig11.png"><figcaption>Figure 11: (a) LBCO Fermi surface at x=0.04. The nodal kF is measured from(π/2,π/2) point as indicated in (b) and its doping dependence in the “spinglass” regime is shown in (c) (red diamonds). Data for LSCO from ref. Zhou2004; Yoshida2006 are represented by blue diamonds. Also shown is theincommensurability δ measured in neutron scattering close to the onset ofsuperconductivity (gray area) where the spin structure rotates from diagonal(x<0.06, open triangles) to parallel (x>0.06, open circles) direction relativeto the Cu-O bond.</figcaption></figure> There is another important evidence for a close relationship between superconductivity and static charge/spin superstructures at $x=0.055$ where superconductivity just appears. There, a peculiar transition has been also observed in neutron scattering experiments on LSCO samples [14; 40; 41]: a static incommensurate scattering near AF wave vector with incommensurability $\delta\propto x$ rotates by $45^{\circ}$, from being diagonal to Cu-O bond ($x\leq 0.055$) to being parallel to it ($x\geq 0.055$). More recently, similar features were also observed in LBCO, with the similar doping dependence, indicating the common nature of this transition in the 214 family of cuprates [42]. The transition from diagonal (static) to parallel (dynamic) ‘stripes’ coincides with the transition seen in transport [43] where ”insulator” turns into superconductor for $x>0.055$. Although the in-plane transport becomes ”metallic” at high temperatures even at 1% doping, at low temperature there is an upturn in $\rho_{ab}$, indicating some localization. This upturn is only present for $x<0.055$, where the diagonal spin incommensurability exists. The moment diagonal points in the neutron scattering experiments disappear (rotate), the upturn in resistivity vanishes, and superconductivity appears. The early ARPES experiments on LSCO in this low doping regime [24; 44] have shown that the first states appearing at the Fermi level are those near the nodal line, whereas the rest of the Fermi surface is affected by a large gap of similar $d$-wave symmetry as the superconducting gap. A closer look at ARPES results from this region of the phase diagram uncovers that the diagonal incommensurability $\delta$ seen in neutron scattering is closely related to the increase in $k_{F}$ of nodal states [24; 44]. <figure><img src="content_image/1303.1774/Fig12.png"><figcaption>Figure 12: a) ARPES spectrum from the nodal line of the x=0.04 LBCO sample. Asmall gap (Δ≈3 meV) indicates that the “diagonal” stripes act as theconventional spin-density-waves (SDW). b) SDW gap</figcaption></figure> In Fig. 11, we show the Fermi surface of an LBCO sample at $x=0.04$. The nodal span of the Fermi surface, $2k_{F}$, that can be precisely extracted from these measurements, shows the same doping dependence as the diagonal incommensurability from neutron scattering. Our results from $x=0.02$ and $x=0.04$ samples show that $\delta=k_{F}-\pi/2$ in this low-doping regime, as can be seen from Fig. 11c. At higher doping levels, the Fermi surface grows more in the anti-nodal region, while the nodal $k_{F}$ saturates near $x\approx 0.07$ (Fig. 9(a-d)), similar to the evolution seen in LSCO [24]. The relationship $\delta=k_{F}-\pi/2$ in the ‘spin-glass’ phase suggests that the diagonal incommensurate scattering may be understood in terms of conventional spin-density wave (SDW) picture, originating from 2$k_{F}$ nesting of nodal states. Such SDW would open the gap and localize the nodal states at low temperatures, preventing the superconductivity from occurring, in agreement with transport [43; 45] and optical [22; 46] studies. Indeed, we have detected a small gap ($\Delta\approx 3$ meV) at the nodal point of the $x=0.04$ sample, as can be seen in Fig. 12, leaving the whole Fermi surface gapped (Fig. 13). Similar gap has been also observed in LSCO samples at similar doping levels [47] This reaffirms our proposal that the nodal states are affected by the “diagonal” stripes in a similar way as they would be in the case of the conventional, Fermi-surface nesting induced SDW. The observed nodal SDW (or ‘diagonal stripe’) gap has dramatic consequences on the transport properties in this doping regime of the phase diagram. While it is clear that the near-nodal states are responsible for metallic normal state transport at these low doping levels, their role in superconductivity is not adequately appreciated. We think that they actually play a crucial role in superconductivity itself: when the diagonal SDW vanishes, the nodal states are released and superconductivity follows immediately. The role of near nodal states in superconductivity might be only a secondary one: the one of phase coherence propagators, where due to the small superconducting gap $\Delta(k)$, these states have large coherence length $\xi(k)=v_{F}(k)/\|\Delta(k)\|$, and are able to stabilize the global superconducting phase [48] in the presence of inhomogeneities observed in STM in underdoped Bi${}_{2}$Sr${}_{2}$CaCu${}_{2}$O${}_{8+\delta}$[12; 49; 50]. However, it might as well be that the superconductivity is a relatively weak order parameter residing only in the near-nodal region, disrupted by something much larger, with similar symmetry that dominates the rest of the Fermi surface in the underdoped regime [51; 52; 53]. <figure><img src="content_image/1303.1774/Fig13.png"><figcaption>Figure 13: Magnitude of single-particle gap in LBCO at x=0.04 as a function ofan angle around the Fermi surface at T=12 K. The whole Fermi surface is gappedat this doping level.</figcaption></figure> ## 4 Conclusions In conclusion, I have pointed out that experimental results on cuprate superconductors indicate a very strong connection between spin susceptibility measured in neutron scattering and single-particle properties measured in ARPES and superconductivity itself. The interplay of spin fluctuations and superconductivity is particularly evident at the onset ($x\approx 0.05$) of superconductivity, where the static “”diagonal” incommensurate spin density wave gives away to superconductivity and at $x=1/8$, where the freezing of spin fluctuations and the appearance of static “parallel” stripes suppresses superconducting transition temperature almost to zero. We have found that in the ground state of LBCO at $x=1/8$, a system with static spin and charge orders [15; 16] and no superconductivity, the $k$-dependence of the single-particle gap looks the same as the superconducting gap in superconducting cuprates: it has magnitude consistent with $d$-wave symmetry and vanishes at four nodal points on the Fermi surface. Furthermore, the gap, measured at low temperature, has a doping dependence with a maximum at $x\approx 1/8$, precisely where the charge/spin order is established between two adjacent superconducting domes. These findings reveal the pairing origin of the “pseudogap” and imply that the most strongly bound Cooper pairs at $x\approx 1/8$ are most susceptible to phase disorder and spatial ordering [7; 18; 37]. In the low-doping regime ($x<0.05$), we have uncovered the tight relationship between the ‘diagonal stripes’ and the single-particle spectral features in the near-nodal region of the Fermi surface in 214 cuprates. 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1408.2940	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 61627, "num_imgs": 0, "llama3_tokens_count": 21695 }	[]	# Optimal preconditioners for Nitsche-XFEM discretizations of interface problems Christoph Lehrenfeld Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D-52056 Aachen, Germany (lehrenfeld@igpm.rwth-aachen.de). Arnold Reusken Institut für Geometrie und Praktische Mathematik, RWTH-Aachen University, D-52056 Aachen, Germany (reusken@igpm.rwth-aachen.de). ###### Abstract In the past decade, a combination of _unfitted_ finite elements (or XFEM) with the Nitsche method has become a popular discretization method for elliptic interface problems. This development started with the introduction and analysis of this Nitsche-XFEM technique in the paper [A. Hansbo, P. Hansbo, Comput. Methods Appl. Mech. Engrg. 191 (2002)]. In general, the resulting linear systems have very large condition numbers, which depend not only on the mesh size $h$, but also on how the interface intersects the mesh. This paper is concerned with the design and analysis of optimal preconditioners for such linear systems. We propose an additive subspace preconditioner which is optimal in the sense that the resulting condition number is independent of the mesh size $h$ and the interface position. We further show that already the simple diagonal scaling of the stifness matrix results in a condition number that is bounded by $ch^{-2}$, with a constant $c$ that does not depend on the location of the interface. Both results are proven for the two-dimensional case. Results of numerical experiments in two and three dimensions are presented, which illustrate the quality of the preconditioner. AMS subject classifications. 65N12, 65N30 Key words. ellitic interface problem, extended finite element space, XFEM, unfitted finite element method, Nitsche method, preconditioning, space decomposition ## 1 Introduction Let $\Omega\in\mathbb{R^{d}}$, $d=2,3$, be a polygonal domain that is subdivided in two connected subdomains $\Omega_{i}$, $i=1,2$. For simplicity we assume that $\Omega_{1}$ is strictly contained in $\Omega$, i.e., $\partial\Omega_{1}\cap\partial\Omega=\emptyset$. The interface between the two subdomains is denoted by $\Gamma=\partial\Omega_{1}\cap\partial\Omega_{2}$. We are interested in interface problems of the following type: \[-\mathop{\rm div}(\alpha\nabla u)= \,f \text{in}~{}~{}\Omega_{i}, i=1,2,\] (1.1a) \[[\![\alpha\nabla u\cdot\mathbf{n}]\!]_{\Gamma}= \,0 \text{on}~{}~{}\Gamma,\] (1.1b) \[[\![\beta u]\!]_{\Gamma}= \,0 \text{on}~{}~{}\Gamma,\] (1.1c) \[u= \,0 \text{on}~{}~{}\partial\Omega.\] (1.1d) Here $\mathbf{n}$ is the outward pointing unit normal on $\Gamma=\partial\Omega_{1}$, $[\![\cdot]\!]$ the usual jump operator and $\alpha=\alpha_{i}>0$, $\beta=\beta_{i}>0$ in $\Omega_{i}$ are piecewise constant coefficients. In general one has $\alpha_{1}\neq\alpha_{2}$. If $\beta_{1}=\beta_{2}=1$, this is a standard interface problem that is often considered in the literature [7, 5, 4, 20]. For $\beta_{1}\neq\beta_{2}$ this model is very similar to models used for mass transport in two-phase flow problems [2, 1, 16, 17, 11]. Without loss of generality we assume $\beta_{i}\geq 1$. The interface condition in (1.1c) is then usually called the Henry interface condition. Note that if $\beta_{1}\neq\beta_{2}$, the solution $u$ is discontinuous across the interface. If $\beta_{1}=\beta_{2}$ and $\alpha_{1}\neq\alpha_{2}$ the first (normal) derivative of the solution is discontinuous across $\Gamma$. In the setting of two-phase flo ws one is typically interested in moving interfaces and instead of (1.1) one uses a time-dependent mass transport model. In this paper, however, we restrict to the simpler stationary case. In the past decade, a combination of _unfitted_ finite elements (or XFEM) with the Nitsche method has become a popular discretization method for this type of interface problems. This development started with the introduction and analysis of this Nitsche-XFEM technique in the paper [7]. Since then this method has been extended in several directions, e.g., as a fictitious domain approach, for the discretization of interface problems in computational mechanics, for the discretization of Stokes interface problems and for the discretization of mass transport problems with moving interfaces, cf. [3, 8, 9, 13, 14, 15, 10]. Almost all papers on this subject treat applications of the method or present discretization error analyses. Efficient iterative solvers for the discrete problem is a topic that has hardly been addressed so far. In general, solving the resulting discrete problem efficiently is a challenging task due to the well-known fact that the conditioning of the stiffness matrix is sensitive to the position of the interface relative to the mesh. If the interface cuts elements in such a way that the ratio of the areas (volumes) on both sides of the interface is very large, the stiffness matrix becomes (very) ill-conditioned. Recently, for stabilized versions of the Nitsche-XFEM method condition number bounds of the form $ch^{-2}$, with a constant $c$ that is independent of how the interface $\Gamma$ intersects the triangulation, have been derived [3, 10, 20]. In [10] an inconsistent stabilization is used to guarantee LBB-stability for the pair of finite element spaces used for the Stokes interface problem. This stabilization also improves the conditioning of the stiffness matrix, leading to a $ch^{-2}$ condition number bound. In [20] a stabilized variant of the Nitsche-XFEM for the problem (1.1) is considered. For this method an $ch^{-2}$ condition number bound is derived. In this paper we consider the original Nitsche-XFEM method from [7] for the discretization of (1.1), without any stabilization. In [7] for this method optimal discretization error bounds are derived. We prove that after a simple diagonal scaling the condition number is bounded by $ch^{-2}$, with a constant $c$ that is independent of how the interface $\Gamma$ intersects the triangulation. We prove that an _optimal preconditioner_, i.e. the condition number of the preconditioned matrix is independent of $h$ and of how the interface $\Gamma$ intersects the triangulation, can be constructed from approximate subspace corrections. If in the subspace spanned by the continuous piecewise linears one applies a standard multigrid preconditioner and in the subspace spanned by the discontinuous finite element functions that are added close to the interface (the xfem basis functions) one applies a simple Jacobi diagonal scaling, the resulting _additive subspace preconditioner is optimal_. The latter is the main result of this paper. The analysis uses the very general theory of subspace correction methods [18, 19]. Our analysis applies to the two-dimensional case ($d=2$), but we expect that a very similar optimality result holds for $d=3$. This claim is supported by results of numerical experiments that are presented. The results derived in this paper also hold (with minor modifications) if in (1.1b), (1.1c) one has a nonhomogeneous right-hand side. In such a case one has to modify the right-hand side functional in the variational formulation, but the discrete linear operators that describe the discretization remain the same. The outline of this paper is as follows. In section 2 the Nitsche-XFEM method from [7] for the discretization of (1.1) is described. In section 3 we study the direct sum splitting of the XFEM space into three subspaces, namely a subspace of continuous piecewise linears, and two subspaces of xfem functions on both sides of the interface. In Theorem 3.3, which is the main result of this paper, we prove that this is a uniformly stable splitting. Following standard terminology (as in [18, 19]) we introduce an additive subspace preconditioner in section 4. Based on the stable splitting property the quality of the preconditioner (i.e., the condition number of the preconditioned matrix) can easily be analyzed. In section 5 we present results of some numerical experiments, both for $d=2$ and $d=3$. ## 2 The Nitsche-XFEM discretization In this section we describe the Nitsche-XFEM discretization, which can be found at several places in the literature [7, 4]. Let $\{\mathcal{T}_{h}\}_{h>0}$ be a family of shape regular simplicial triangulations of $\Omega$. A triangulation $\mathcal{T}_{h}$ consists of simplices $T$, with $h_{T}:={\rm diam}(T)$ and $h:=\max\{\,h_{T}~{}\|~{}T\in\mathcal{T}_{h}\}$. The triangulation is _unfitted_. We introduce some notation for _cut elements_, i.e. elements $T\in\mathcal{T}_{h}$ with $\Gamma\cap T\neq\emptyset$. The subset of these cut elements is denoted by $\mathcal{T}_{h}^{\Gamma}:=\{T\in\mathcal{T}_{h}~{}\|~{}T\cap\Gamma\neq\emptyset\}$. To simplify the presentation and avoid technical details we assume that for all $T\in\mathcal{T}_{h}^{\Gamma}$ the intersection $\Gamma_{T}:=T\cap\Gamma$ does not coincide with a subsimplex of $T$ (a face, edge or vertex of $T$). Hence, we assume that $\Gamma_{T}$ subdivides $T$ into two subdomains $T_{i}:=T\cap\Omega_{i}$ with ${\rm meas}_{d}(T_{i})>0$. We further assume that there is at least one vertex of $T$ that is inside domain $\Omega_{i},~{}i=1,2$. In the analysis we assume that $\mathcal{T}_{h}^{\Gamma}$ is _quasi-uniform_. Let $V_{h}\subset H_{0}^{1}(\Omega)$ be the standard finite element space of continuous piecewise linears corresponding to the triangulation $\mathcal{T}_{h}$ with zero boundary values at $\partial\Omega$. Let $\{\mathbf{x}_{j}~{}\|~{}j=1,\ldots n\}$, with $n=\dim V_{h}$, be the set of internal vertices in the triangulation. The index set is denoted by $\mathcal{J}=\{1,\ldots,n\}$. Let $(\phi_{j})_{j\in\mathcal{J}}$ be the nodal basis functions in $V_{h}$, where $\phi_{j}$ corresponds to the vertex with index $j$. Let $\mathcal{J}_{\Gamma}:=\{\,j\in\mathcal{J}~{}\|~{}~{}\|\Gamma\cap{\rm supp}(\phi_ {j})\|>0\,\}$ be the index set of those basis functions the support of which is intersected by $\Gamma$. The Heaviside function $H_{\Gamma}$ has the values $H_{\Gamma}(x)=0$ for $x\in\Omega_{1}$, $H_{\Gamma}(x)=1$ for $x\in\Omega_{2}$. Using this, for $j\in\mathcal{J}_{\Gamma}$ we define an enrichment function $\Phi_{j}(x):=\|H_{\Gamma}(x)-H_{\Gamma}(\mathbf{x}_{j})\|$. We introduce additional, so-called _xfem basis functions_$\phi_{j}^{\Gamma}:=\phi_{j}\Phi_{j}$, $j\in\mathcal{J}_{\Gamma}$. Note that $\phi_{j}^{\Gamma}(\mathbf{x}_{k})=0$ for all $j\in\mathcal{J}_{\Gamma},\,k\in\mathcal{J}$. Furthermore, for $j\in\mathcal{J}_{\Gamma}$ and $\mathbf{x}_{j}\in\Omega_{1}$, we have ${\rm supp}(\phi_{j}^{\Gamma})\subset\bar{\Omega}_{2}$ and for $\mathbf{x}_{j}\in\Omega_{2}$, we have ${\rm supp}(\phi_{j}^{\Gamma})\subset\bar{\Omega}_{1}$. Related to this, the index set $\mathcal{J}_{\Gamma}$ is partitioned in $\mathcal{J}_{\Gamma,2}:=\{j\in\mathcal{J}_{\Gamma}~{}\|~{}\mathbf{x}_{j}\in \Omega_{1}\}$ and $\mathcal{J}_{\Gamma,1}:=\mathcal{J}_{\Gamma}\setminus\mathcal{J}_{\Gamma,2}=\{ j\in\mathcal{J}_{\Gamma}~{}\|~{}\mathbf{x}_{j}\in\Omega_{2}\}$. Hence, for $j\in\mathcal{J}_{\Gamma,i}$ the xfem basis function $\phi_{j}^{\Gamma}$ has its support in $\bar{\Omega}_{i}$, $i=1,2$. The XFEM space is defined by \[V_{h}^{\Gamma}:=V_{h}\oplus V_{h,1}^{x}\oplus V_{h,2}^{x}=V_{h}\oplus V_{h}^{x }\quad\mbox{with }V_{h,i}^{x}:={\rm span}\{\,\phi_{j}^{\Gamma}~{}\|~{}j\in \mathcal{J}_{\Gamma,i}\,\},\] (2.1) and $V_{h}^{x}:=V_{h,1}^{x}\oplus V_{h,2}^{x}$. Remark 2.1.: The XFEM space $V_{h}^{\Gamma}$ can also be characterized as follows: $v_{h}\in V_{h}^{\Gamma}$ if and only if there exist finite element functions $v_{1},v_{2}\in V_{h}$ such that $(v_{h})_{\|\Omega_{i}}=(v_{i})_{\|\Omega_{i}}$, $i=1,2$. From this characterization one easily derives optimal approximation properties of the XFEM space for functions that are piecewise smooth, cf. [7, 12]. In the literature, e.g., [7, 4], discretization with the space $V_{h}^{\Gamma}$ is also called an _unfitted finite element method_. An $L^{2}$-stability property of the basis $(\phi_{j})_{j\in\mathcal{J}}\cup(\phi_{j}^{\Gamma})_{j\in\mathcal{J}_{\Gamma}}$ of $V_{h}^{\Gamma}$ is given in [12]. For the discretization of the equation (1.1) in the XFEM space we first introduce some notation for scalar products. The $L^{2}$ scalar product is denoted by $(u,v)_{0}:=\int_{\Omega}uv\,dx$. Furthermore we define \[(u,v)_{1,\Omega_{1\!,2}}:=(\nabla u,\nabla v)_{L^{2}(\Omega_{1})}+(\nabla u, \nabla v)_{L^{2}(\Omega_{2})},\quad u,v\in H^{1}(\Omega_{1,2}):=H^{1}({\Omega_ {1}\cup\Omega_{2}}),\] with the semi-norm denoted by $\|\cdot\|_{1,\Omega_{1\!,2}}=(\cdot,\cdot)_{1,\Omega_{1\!,2}}^{\frac{1}{2}}$ and norm $\\|\cdot\\|_{1,\Omega_{1\!,2}}:=(\\|\cdot\\|_{0}^{2}+\|\cdot\|_{1,\Omega_{1\!,2}}^{2 })^{\frac{1}{2}}$. On the interface we introduce the scalar product \[(f,g)_{\Gamma}:=\int_{\Gamma}fg\,ds\] (2.2) and the mesh-dependent weighted $L^{2}$ scalar product \[(f,g)_{\frac{1}{2},h,\Gamma}:=h^{-1}\int_{\Gamma}fg\,ds.\] (2.3) The Nitsche-XFEM discretization of the interface problem (1.1) reads as follows: Find $u_{h}\in V_{h}^{\Gamma}$ such that \[\begin{split}&(\alpha\beta u_{h},v_{h})_{1,\Omega_{1\!,2}}-(\{\! \!\{\!\alpha\nabla u_{h}\cdot\mathbf{n}\!\}\!\!\},[\![\beta v_{h}]\!])_{\Gamma }-(\{\!\!\{\!\alpha\nabla v_{h}\cdot\mathbf{n}\!\}\!\!\},[\![\beta u_{h}]\!])_ {\Gamma}\\ &+(\lambda[\![\beta u_{h}]\!],[\![\beta v_{h}]\!])_{\frac{1}{2},h ,\Gamma}=(\beta f,v_{h})_{0}\quad\text{for all}~{}~{}\,v_{h}\in V_{h}^{\Gamma} .\end{split}\] (2.4) Here we used the average $\{\!\!\{\!w\!\}\!\!\}:=\kappa_{1}w_{1}+\kappa_{2}w_{2}$ with an element-wise constant $\kappa_{i}=\frac{\|T_{i}\|}{\|T\|}$. This weighting in the averaging is taken from the original paper [7]. The stabilization parameter $\lambda\geq 0$ should be taken sufficiently large, $\lambda>c_{\lambda}\max\{\alpha_{i}\}_{i=1,2}$, with a suitable constant $c_{\lambda}$ only depending on the shape regularity of $T\in\mathcal{T}_{h}$. Discretization error analysis for this method is available in the literature. In [7] optimal order discretization error bounds are derived for the case $\beta_{1}=\beta_{2}=1$. The case $\beta_{1}\neq\beta_{2}$ is treated in [15]. For the development and analysis of preconditioners for the discrete problem, without loss of generality we can restrict to the case $\beta_{1}=\beta_{2}=1$. This is due to the following observation. We note that (also if $\beta_{1}\neq\beta_{2}$) we have $\beta v_{h}\in V_{h}^{\Gamma}$ iff $v_{h}\in V_{h}^{\Gamma}$. Thus, by rescaling the test functions $v_{h}$ and with $\tilde{\alpha}:=\alpha\beta^{-1}$ the problem (2.4) can be reformulated as follows: Find $\tilde{u}_{h}=\beta u_{h}\in V_{h}^{\Gamma}$ such that \[\begin{split}&(\tilde{\alpha}\tilde{u}_{h},v_{h})_{1,\Omega_{1\!, 2}}-(\{\!\!\{\!\tilde{\alpha}\nabla\tilde{u}_{h}\cdot\mathbf{n}\!\}\!\!\},[\![ v_{h}]\!])_{\Gamma}-(\{\!\!\{\!\tilde{\alpha}\nabla v_{h}\cdot\mathbf{n}\!\}\! \!\},[\![\tilde{u}_{h}]\!])_{\Gamma}\\ &+(\lambda[\![\tilde{u}_{h}]\!],[\![v_{h}]\!])_{\frac{1}{2},h, \Gamma}=(f,v_{h})_{0}\quad\text{for all}~{}~{}\,v_{h}\in V_{h}^{\Gamma}.\end{split}\] (2.5) The stiffness matrices corresponding to (2.4) and (2.5) are related by a simple basis transformation. In the remainder of the paper we only consider the preconditioning of the stiffness matrix corresponding to (2.5). Via the simple basis transformation the solution to (2.5) directly gives a solution to (2.4). Remark 2.2.: In certain situations it may be (e.g., due to implementational aspects) less convenient to transform the discrete problem (2.4) into (2.5). If one wants to keep the original formulation, it is easy to provide an (optimal) preconditioner for it, given a preconditioner for the transformed problem (2.5). We briefly explain this. Let $(\psi_{j})_{1\leq j\leq m}$ denote the basis for $V_{h}^{\Gamma}$, and $\mathbf{A}$, $\tilde{\mathbf{A}}$ the stiffness matrices w.r.t. this basis of the problems (2.4) and (2.5), respectively. Let $\mathbf{T}$ be the matrix representation of the mapping $v_{h}\to\beta^{-1}v_{h}$, for $v_{h}\in V_{h}^{\Gamma}$, i.e., the $i$-th row of $\mathbf{T}$ contains the coefficients $t_{i,k}$ such that $\beta^{-1}\psi_{i}=\sum_{k=1}^{m}t_{i,k}\psi_{k}$. Then the relation $\tilde{\mathbf{A}}=\mathbf{T}\mathbf{A}\mathbf{T}^{T}$ holds. Given a preconditioner $\tilde{\mathbf{C}}$ for $\tilde{\mathbf{A}}$, we define $\mathbf{C}:=\mathbf{T}^{T}\tilde{\mathbf{C}}\mathbf{T}$ as precondition er for $\mathbf{A}$. Due to the equality of spectra, $\sigma(\mathbf{C}\mathbf{A})=\sigma(\tilde{\mathbf{C}}\tilde{\mathbf{A}})$, the quality of $\mathbf{C}$ as a preconditioner for $\mathbf{A}$ is the same as the quality of $\tilde{\mathbf{C}}$ as a preconditioner for $\tilde{\mathbf{A}}$. We introduce a compact notation for the symmetric bilinear form used in (2.5). For convenience we write $\alpha$ instead of $\tilde{\alpha}$, and we assume a global constant value for $\lambda$: (2.6) This bilinear form is well-defined on $V_{h}^{\Gamma}\times V_{h}^{\Gamma}$. For the analysis we introduce the bilinear form and corresponding norm defined by \[\|\!\|\!\|u\|\!\|\!\|_{h}^{2}=\|u\|_{1,\Omega_{1\!,2}}^{2}+\lambda\\|[\![u]\!]\\|_{\frac {1}{2},h,\Gamma}^{2},\quad u\in V_{h}^{\Gamma}.\] (2.7) In [7] it is shown that, for $\lambda$ sufficiently large, the norm corresponding to the Nitsche bilinear form is uniformly equivalent to $\|\!\|\!\|\cdot\|\!\|\!\|_{h}$: \[a_{h}(u,u)\sim\|\!\|\!\|u\|\!\|\!\|_{h}^{2}\quad\text{for all}~{}~{}u\in V_{h}^{ \Gamma}.\] (2.8) Here and in the remainder we use the symbol $\sim$ to denote two-sided inequalities with constants that are _independent of h and of how the triangulation is intersected by the interface $\Gamma$_. The constants in these inequalities may depend on $\alpha$ and $\lambda$. We also use $\lesssim$ to denote one-sided estimates that have the same uniformity property. In the remainder we assume that $\lambda>0$ is chosen such that (2.8) holds. ## 3 Stable subspace splitting We will derive an optimal preconditioner for the bilinear form in (2.6) using the theory of _subspace correction methods_. Two excellent overview papers on this topic are [18, 19]. The theory of subspace correction methods as described in these overview papers is a very general one, with applications to multigrid and to domain decomposition methods. We apply it for a relatively very simple case with three disjoint spaces. We use the notation and some main results from [19]. It is convenient to adapt our notation to the one of the abstract setting in [19]. The three subspaces in (2.1) are denoted by $\mathcal{W}_{0}=V_{h}$, $\mathcal{W}_{i}=V_{h,i}^{x}$, $i=1,2$. Thus we have the direct sum decomposition (3.1) Below $u=u_{0}+u_{1}+u_{2}\in\mathcal{S}$ always denotes a decompositon with $u_{l}\in\mathcal{W}_{l}$, $l=0,1,2$. For the norm induced by the bilinear form $a_{h}(\cdot,\cdot)$ we use the notation \[\\|u\\|_{h}:=a_{h}(u,u)^{\frac{1}{2}},\quad u\in\mathcal{S}.\] Recall that this norm is uniformly equivalent to $\|\!\|\!\|\cdot\|\!\|\!\|_{h}$, cf. (2.8). In theorem 3.3 below we show that the splitting in (3.1) is stable w.r.t. the norm $\\|\cdot\\|_{h}$. The result in the next theorem is the key point in our analysis. We show that the splitting of $\mathcal{S}$ into $\mathcal{W}_{0}$ and the subspace spanned by the xfem basis functions $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ is stable. For this we restrict to the two-dimensional case $d=2$. We use a transformation of certain patches to a reference patch on $[0,1]^{2}$. We first describe this transformation. We construct a subdivision of $\mathcal{T}_{h}^{\Gamma}$ into patches $\{\omega_{k}\}$ as follows, cf. Figure 3.1. We first define a subset $\mathcal{E}$ of all edges that are intersected by $\Gamma$. Consider an edge $E_{1}$ which is intersected by $\Gamma$ such that one vertex $V_{1}$ is in $\Omega_{1}$ and the other, $V_{1}^{\ast}$, is in $\Omega_{2}$. We define this edge as the first element in $\mathcal{E}$. Now fix one direction along the interface and going in this direction along $\Gamma$ we get an ordered list of all edges intersected by $\Gamma$. As last edge in this list we include the starting edge $E_{1}$. As the next edge $E_{2}\in\mathcal{E}$ we take the first one after $E_{1}$ (in the list) that has no common vertex with $E_{1}$. As $E_{3}\in\mathcal{E}$ we take the first one after $E_{2}$ that has no common vertex with $E_{2}$, etc.. To avoid technical details we assume that the final edge $E_{N_{\mathcal{E}}}$ included in $\mathcal{E}$ coincides with $E_{1}$. By construction we get a numbering of certain vertices as in the left part of Figure 3.1: edge $E_{j}$ has vertices $V_{j}\in\Omega_{1}$, $V_{j}^{\ast}\in\Omega_{2}$. [FIGURE:S3.F1][ENDFIGURE] The elements between two edges $E_{k},E_{k+1}\in\mathcal{E}$ form the patch $\omega_{k}$. The patches $\{\omega_{k}\}_{1\leq k\leq N_{\omega}}$, with $N_{\omega}=N_{\mathcal{E}}-1$, form a disjoint partitioning of $\mathcal{T}_{h}^{\Gamma}$. We define the extended patch $\omega_{k}^{e}$ by adding the neighboring elements which are not in $\mathcal{T}_{h}^{\Gamma}$, i.e., $\omega_{k}^{e}:=\omega_{k}\cup\{T\in\mathcal{T}_{h}\setminus\mathcal{T}_{h}^{ \Gamma}~{}\|~{}\text{$T$ has a common edge with a}~{}T^{\prime}\in\omega_{k}\}$. The part of the interface $\Gamma$ contained in $\omega_{k}^{e}$ is denoted by $\Gamma_{k}$. The triangulation (and corresponding domain) formed by the union of the extended patches $\omega_{k}^{e}$ is denoted by $\mathcal{T}_{h}^{\Gamma,e}$. Note that every element $T\in\mathcal{T}_{h}^{\Gamma,e}$ can appear in at most two patches $\omega_{k}^{e}$. Further note that the number of elements within each extended patch $\omega_{k}^{e}$ is uniformly bounded due to shape regularity of $\mathcal{T}_{h}$. For each extended patch $\omega_{k}^{e}$ there exists a piecewise affine transformation $\Phi_{k}:\omega_{k}^{e}\rightarrow\mathbb{R}^{2}$ such that $\Phi_{k}(\omega_{k})=[0,1]^{2}$. Accordingly we denote a transformed patch by $\hat{\omega}$ and $\hat{\omega}^{e}$. Theorem 3.1.: _Take $d=2$. The following holds:_ \[\\|u_{0}\\|_{h}^{2}+\\|w\\|_{h}^{2}\lesssim\\|u_{0}+w\\|_{h}^{2}\quad\text{for all}~ {}u_{0}\in\mathcal{W}_{0},~{}w\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}.\] (3.2) Proof.: Due to norm equivalence the result in (3.2) is equivalent to: \[\|\!\|\!\|u_{0}\|\!\|\!\|_{h}^{2}+\|\!\|\!\|w\|\!\|\!\|_{h}^{2}\lesssim\|\!\|\!\|u_{0}+w\|\!\| \!\|_{h}^{2}\quad\text{for all}~{}u_{0}\in\mathcal{W}_{0},~{}w\in\mathcal{W}_{1 }\oplus\mathcal{W}_{2}.\] For $w\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}$ we have $w=0$ on $\Omega\setminus\mathcal{T}_{h}^{\Gamma,e}$, and $\mathcal{T}_{h}^{\Gamma,e}$ is partitioned into patches $\omega^{e}_{k}$. Hence, it suffices to prove \[\|\!\|\!\|u_{0}\|\!\|\!\|_{h,\omega^{e}_{k}}^{2}+\|\!\|\!\|w\|\!\|\!\|_{h,\omega^{e}_{k}}^ {2}\lesssim\|\!\|\!\|u_{0}+w\|\!\|\!\|_{h,\omega^{e}_{k}}^{2}\quad\text{for all}~{}u _{0}\in\mathcal{W}_{0},~{}w\in\mathcal{W}_{1}\oplus\mathcal{W}_{2}.\] (3.3) We use the transformation to the reference patch $\hat{\omega}^{e}$ described above. On the reference patch we have transformed spaces $\hat{\mathcal{W}}_{0}$ (continuous, piecewise linears) and $\hat{\mathcal{W}}_{1}\oplus\hat{\mathcal{W}}_{2}$. The functions in $\hat{\mathcal{W}}_{1}$ ($\hat{\mathcal{W}}_{2}$) are piecewise linear on the part of the patch below (above) the interface $\hat{\Gamma}$, zero on the line segment $y=0$ ($y=1$) and zero on the part of the patch above (below) the interface $\hat{\Gamma}$. The norm $\|\!\|\!\|u\|\!\|\!\|_{h,\omega^{e}_{k}}$ and the induced norm $\|\!\|\!\|\hat{u}\|\!\|\!\|_{\hat{\omega}^{e}_{k}}=\big{(}(\nabla\hat{u},\nabla\hat{ u})_{L^{2}(\hat{\omega}_{k}^{e})}+\lambda([\![\hat{u}]\!],[\![\hat{u}]\!])_{L^ {2}(\hat{\Gamma}_{k})}\big{)}^{\frac{1}{2}}$, with $\hat{u}=u\circ\Phi_{k}^{-1}$ on $\hat{\omega}_{k}^{e}$, are uniformly equivalent, because the constants in this norm equivalence are determined only by the condition number of the piecewise affine transformation between $\omega^{e}_{k}$ and $\hat{\omega}_{k}^{e}$. Note that neither the spaces $\hat{\mathcal{W}}_{l}$ nor the norm $\|\!\|\!\|\cdot\|\!\|\!\|_{\hat{\omega}^{e}_{k}}$ depend on $h$ (the $h$-dependence is implicit in the piecewise affine transformation). The reference patches $\hat{\omega}_{k}^{e}$ all have the same geometric structure, cf. Figure 3.1. These patches have (due to shape regularity of $\mathcal{T}_{h}$) a uniformly bounded number of vertices on the line segment that connects the vertices $V_{i}$, $V_{i+1}$ (or $V_{i}^{\ast}$, $V_{i+1}^{\ast}$). In the rest of the proof a generic reference patch and its extension are denoted by $\hat{\omega}$ and $\hat{\omega}^{e}$, respectively. The interface segment that is intersected by $\hat{\omega}$ is denoted by $\hat{\Gamma}$. We conclude that for (3.3) to hold it is sufficient to prove \[\|\!\|\!\|u_{0}\|\!\|\!\|_{\hat{\omega}^{e}}^{2}+\|\!\|\!\|w\|\!\|\!\|_{\hat{\omega}^{e}}^ {2}\leq K\|\!\|\!\|u_{0}+w\|\!\|\!\|_{\hat{\omega}^{e}}^{2}\quad\text{for all}~{}u_{ 0}\in\hat{W}_{0},~{}w\in\hat{W}_{1}\oplus\hat{W}_{2},\] (3.4) with a constant $K$ that is independent of how the patch $\hat{\omega}$ is intersected by the interface $\hat{\Gamma}$. Note that $(\nabla u_{0},\nabla w)_{L^{2}({\hat{\omega}}^{e}\setminus\hat{\omega})}=([\![ u_{0}]\!],[\![w]\!])_{L^{2}(\hat{\Gamma})}=0$ for $u_{0}\in\hat{W}_{0}$ and $w\in\hat{W}_{1}\oplus\hat{W}_{2}$. Hence, \[\|\!\|\!\|u_{0}+w\|\!\|\!\|_{\hat{\omega}^{e}}^{2}=\|\!\|\!\|u_{0}\|\!\|\!\|_{\hat{\omega} ^{e}}^{2}+\|\!\|\!\|w\|\!\|\!\|_{\hat{\omega}^{e}}^{2}+2(\nabla u_{0},\nabla w)_{L^{ 2}(\hat{\omega})},\quad u_{0}\in\hat{W}_{0},~{}w\in\hat{W}_{1}\oplus\hat{W}_{2}\] holds. Thus it suffices to prove the strengthened Cauchy-Schwarz inequality \[(\nabla u_{0},\nabla w)_{L^{2}(\hat{\omega})}\leq C^{\ast}\|\!\|\!\|u_{0}\|\!\|\!\|_ {\hat{\omega}^{e}}\|\!\|\!\|w\|\!\|\!\|_{\hat{\omega}^{e}}\quad\text{for all}~{}u_{0 }\in\hat{W}_{0},~{}w\in\hat{W}_{1}\oplus\hat{W}_{2},\] (3.5) with a uniform constant $C^{\ast}<1$. The proof of (3.5) is divided into three steps, namely a strengthened Cauchy-Schwarz inequality related to the $x$-derivative, a suitable Cauchy-Schwarz inequality related to the $y$-derivative and then combining these estimates. Step 1. The following holds: \[\|(u_{x},w_{x})_{L^{2}(\hat{\omega})}\|\leq c_{0}\\|u_{x}\\|_{L^{2}(\hat{\omega}^{ e})}\\|w_{x}\\|_{L^{2}(\hat{\omega})}\quad\text{for all}~{}u\in W_{0},~{}w\in \hat{W}_{1}\oplus\hat{W}_{2},\] (3.6) with a uniform constant $c_{0}<1$. From the Cauchy-Schwarz inequality we get $\|(u_{x},w_{x})_{L^{2}(\hat{\omega})}\|\leq\\|u_{x}\\|_{L^{2}(\hat{\omega})}\\|w_{x }\\|_{L^{2}(\hat{\omega})}$. Within the patch $\hat{\omega}=\{T_{i}\}$ the $x$-derivative $u_{x}$ is piecewise constant and $u_{x}\|_{T_{i}}=u_{x}\|_{T_{i,N}}$ for the neighboring triangle $T_{i,N}\in\hat{\omega}^{e}\setminus\hat{\omega}$. This implies $\\|u_{x}\\|_{L^{2}(T_{i})}\leq\hat{c}\\|u_{x}\\|_{L^{2}(T_{i}\cup T_{i,N})}$, with $\hat{c}<1$ depending only on shape regularity. Thus we obtain $\\|u_{x}\\|_{L^{2}(\hat{\omega})}\leq c_{0}\\|u_{x}\\|_{L^{2}(\hat{\omega}^{e})}$, with a uniform constant $c_{0}<1$, which yields (3.6). Step 2. The following holds: \[\begin{split}\|(u_{y},w_{y})_{L^{2}(\hat{\omega})}\|& \leq\min\{c_{1}\\|u_{x}\\|_{L^{2}(\hat{\omega})},\\|u_{y}\\|_{L^{2}(\hat{\omega})} \}\\|w_{y}\\|_{L^{2}(\hat{\omega})}\\ &+c_{2}\\|u_{y}\\|_{L^{2}(\hat{\omega})}\\|[\![w]\!]\\|_{L^{2}(\hat{ \Gamma})}\quad\text{for all}~{}u\in W_{0},~{}w\in\hat{W}_{1}\oplus\hat{W}_{2}, \end{split}\] (3.7) with suitable uniform constants $c_{1},c_{2}$. Let $\{T_{i}\}$ be the set of triangles that form $\hat{\omega}$ and let these be ordered such that ${\rm meas}_{1}(T_{i}\cap T_{i+1})>0$. We denote the interior edges by $e_{i}=T_{i}\cap T_{i+1}$. To show (3.7) we start with partial integration \[\begin{split}\Big{\|}\int_{\hat{\omega}}u_{y}w_{y}\,dx\Big{\|}& =\Big{\|}\sum_{T_{i}}\int_{\partial T_{i}}n_{T_{i},y}\,u_{y}w\,ds+ \int_{\hat{\Gamma}_{T_{i}}}n_{\Gamma,y}\,u_{y}[\![w]\!]\,ds\Big{\|}\\ &\leq\sum_{e_{i}}\Big{\|}[\![u_{y}]\!]_{e_{i}}\Big{\|}\Big{\|}\int_{ e_{i}}w\,ds\Big{\|}+\\|u_{y}\\|_{L^{2}(\hat{\Gamma})}\\|[\![w]\!]\\|_{L^{2}(\hat{ \Gamma})}\end{split}\] (3.8) where for the edges of $\partial T_{i}$ that lie on $\partial\hat{\omega}=\partial[0,1]^{2}$ we used $w=0$ for $y\in\{0,1\}$ and $n_{T_{i},y}=0$ for $x\in\{0,1\}$. To proceed we need technical estimates to bound $[\![u_{y}]\!]_{e_{i}}$ and $\int_{e_{i}}w\,ds$. For those estimates we exploit propertries of the geometry of $\hat{\omega}$. First consider $u\in\hat{W}_{0}$ along an interior edge $e_{i}\not\in\partial\hat{\omega}$ and denote the unit tangential vector to $e_{i}$ by $\mathbf{\tau}=(\tau_{x},\tau_{y})$. For $\tau$ we have $\|\tau_{y}\|\geq 1/\sqrt{2}\geq\|\tau_{x}\|$. Due to continuity of $u$ along $e_{i}$ there holds $[\![\nabla u]\!]_{e_{i}}\cdot\mathbf{\tau}=0$, which implies Thus we obtain \[\left\|[\![u_{y}]\!]_{e_{i}}\right\|\leq c\,\min\{\\|u_{x}\\|_{L^{2}(T_{i}\cup T_{ i+1})},\\|u_{y}\\|_{L^{2}(T_{i}\cup T_{i+1})}\,\}.\] (3.9) Next, we consider $w=w_{1}+w_{2}\in\hat{W}_{1}\oplus\hat{W}_{2}$ along the interior edge $e_{i}$. Let $T_{i}$ be a triangle adjacent to $e_{i}$. Without loss of generality we assume that two vertices of $T_{i}$ are in $\hat{\Omega}_{1}$ and we thus have $(w_{1})_{x}=0$ on $T_{i}$. We denote the vertices of $e_{i}$ by $\mathbf{x}_{i}=e_{i}\cap\partial\hat{\omega}\cap\hat{\Omega}_{i},~{}i=1,2$ and the intersection point by $\mathbf{x}_{\Gamma}=e_{i}\cap\hat{\Gamma}$ and define the distances $d_{i}=\\|\mathbf{x}_{i}-\mathbf{x}_{\Gamma}\\|_{2},~{}i=1,2$. As $w$ is piecewise linear along $e_{i}$, zero at $\mathbf{x}_{1}$, and $(w_{1})_{x}=0$ on $T_{i}$, we have $w_{1}(\mathbf{x}_{\Gamma})=\pm d_{1}\tau_{y}(w_{1})_{y}$. Furthermore: \[\int_{e_{i}}w\,ds=\frac{1}{2}d_{1}w_{1}(\mathbf{x}_{\Gamma})+\frac{1}{2}d_{2}w _{2}(\mathbf{x}_{\Gamma})=\frac{1}{2}(d_{1}+d_{2})w_{1}(\mathbf{x}_{\Gamma})- \frac{1}{2}d_{2}[\![w]\!](\mathbf{x}_{\Gamma}).\] We also have the geometrical information $d_{1}\leq d_{1}+d_{2}\leq\sqrt{2}$, $d_{1}\leq c\|T_{i}\|^{\frac{1}{2}}$, $\|\hat{\Gamma}_{T_{i}}\|\leq\sqrt{2}$ and $d_{2}\leq c\|\hat{\Gamma}_{T_{i}}\|^{\frac{1}{2}}$. Because $[\![w]\!]$ is linear along $\hat{\Gamma}_{T_{i}}$ there also holds . Using these results we get \[\Big{\|}\int_{e_{i}}w\,ds\Big{\|}\leq c\\|w_{y}\\|_{L^{2}(T_{i})}+c\\|[\![w]\!]\\|_{ L^{2}(\hat{\Gamma}_{T_{i}})}.\] (3.10) From (3.9) and (3.10) we obtain \[\sum_{e_{i}}\Big{\|}[\![u_{y}]\!]_{e_{i}}\Big{\|}\Big{\|}\int_{e_{i}}w\,ds\Big{\|} \leq c\\|u_{y}\\|_{L^{2}(\hat{\omega})}\\|[\![w]\!]\\|_{L^{2}(\hat{\Gamma})}+c\\|u_ {x}\\|_{L^{2}(\hat{\omega})}\\|w_{y}\\|_{L^{2}(\hat{\omega})}.\] (3.11) Combining (3.8), (3.11) and the Cauchy-Schwarz inequality $\Big{\|}\int_{\hat{\omega}}u_{y}w_{y}\,dx\Big{\|}\leq\\|u_{y}\\|_{L^{2}(\hat{ \omega})}\\|w_{y}\\|_{L^{2}(\hat{\omega})}$ results in (3.7). Step 3. The following holds: (3.12) for all $u\in W_{0},~{}w\in\hat{W}_{1}\oplus\hat{W}_{2}$, with a uniform constant $C^{\ast}<1$. The proof combines the preceding results. We define $\alpha_{x}=\\|u_{x}\\|_{L^{2}(\hat{\omega}^{e})}$, $\beta_{x}=\\|w_{x}\\|_{L^{2}(\hat{\omega})}$, $\alpha_{y}=\\|u_{y}\\|_{L^{2}(\hat{\omega})}$, $\beta_{y}=\\|w_{y}\\|_{L^{2}(\hat{\omega})}$, $\gamma=\\|[\![w]\!]\\|_{L^{2}(\hat{\Gamma})}$. Then we have with (3.6), (3.7) and $\theta=\frac{\alpha_{x}^{2}}{\alpha_{x}^{2}+\alpha_{y}^{2}}$, $\alpha=(\alpha_{x}^{2}+\alpha_{y}^{2})^{\frac{1}{2}}$ and $\beta=(\beta_{x}^{2}+\beta_{y}^{2}+\lambda\gamma^{2})^{\frac{1}{2}}$ \[\|(\nabla u,\nabla w)_{L^{2}(\hat{\omega})}\| \leq c_{0}\alpha_{x}\beta_{x}+\min\{c_{1}\alpha_{x},\alpha_{y}\} \beta_{y}+c_{2}\alpha_{y}\gamma\] \[\leq(c_{0}^{2}\alpha_{x}^{2}+\min\{c_{1}^{2}\alpha_{x}^{2},\alpha _{y}^{2}\}+c_{2}^{2}\alpha_{y}^{2}\lambda^{-1})^{\frac{1}{2}}(\beta_{x}^{2}+ \beta_{y}^{2}+\lambda\gamma^{2})^{\frac{1}{2}}\] \[\leq(c_{0}^{2}\theta+\min\{c_{1}^{2}\theta,1-\theta\}+c_{2}^{2}(1 -\theta)\lambda^{-1})^{\frac{1}{2}}\alpha\beta.\] One easily sees that $c_{0}^{2}\theta+\min\{c_{1}^{2}\theta,1-\theta\}\leq\frac{c_{0}^{2}+c_{1}^{2}} {1+c_{1}^{2}}<1$. For sufficiently large $\lambda$ ($\lambda>\frac{1+c_{1}^{2}}{c_{2}^{2}(1-c_{0}^{2})}$) (3.12) follows for a suitable uniform constant $C^{\ast}<1$. The result (3.12) directly implies (3.5) and thus the estimate (3.2) holds for $\lambda$ sufficiently large. For different values $\lambda\geq\lambda^{\ast}$, with $\lambda^{\ast}$ the critical value for which the norm equivalence (2.8) holds, the norms $\\|\cdot\\|_{h}$ (depending on $\lambda$) are equivalent, with equivalence constants depending only on $\lambda$. This implies that (3.2) holds for any $\lambda\geq\lambda^{\ast}$. ∎ In the next lemma we derive the stable splitting property of $\mathcal{W}_{1}\oplus\mathcal{W}_{2}$. Lemma 3.2.: _The following holds:_ \[\\|u_{l}\\|_{h} \sim\|u_{l}\|_{1,\Omega_{l}}\quad\text{for all}~{}u_{l}\in\mathcal{ W}_{l}\quad\text{and}~{}l=1,2,\] (3.13) \[\\|u_{1}\\|_{h}^{2}+\\|u_{2}\\|_{h}^{2} \lesssim\\|u_{1}+u_{2}\\|_{h}^{2}\quad\text{for all}~{}u_{1}+u_{2} \in\mathcal{W}_{1}\oplus\mathcal{W}_{2}.\] (3.14) Proof.: Take $l=1$. We have \[\\|u_{1}\\|_{h}^{2}\sim\|\!\|\!\|u_{1}\|\!\|\!\|_{h}^{2}=\|u_{1}\|_{1,\Omega_{1}}^{2}+ \lambda\\|[\![u_{1}]\!]\\|_{\frac{1}{2},h,\Gamma}^{2}\sim\|u_{1}\|_{1,\Omega_{1}}^ {2}+h^{-1}\\|u_{1}\\|_{L^{2}(\Gamma)}^{2}.\] (3.15) This implies $\|u_{1}\|_{1,\Omega_{1}}\lesssim\\|u_{1}\\|_{h}$. Next we show \[h^{-1}\\|u_{1}\\|_{L^{2}(\Gamma)}^{2}\lesssim\|u_{1}\|_{1,\Omega_{1}}^{2}.\] (3.16) For this, we represent $\Gamma$ locally as the graph of a function $\psi$, with a local coordinate system $(\xi,\eta)$ as in Figure 3.2. [FIGURE:S3.F2][ENDFIGURE] Then we can write \[u_{1}(\xi,\psi(\xi)) =\underbrace{u_{1}(\xi,\psi(0))}_{=0}+\int_{0}^{\psi(\xi)}\frac{ \partial u_{1}}{\partial\eta}(\xi,\eta)\,d\eta,\] and thus \[u_{1}(\xi,\psi(\xi))^{2} =\Big{\|}\int_{0}^{\psi(\xi)}\frac{\partial u_{1}}{\partial\eta}( \xi,\eta)\,d\eta\Big{\|}^{2}\leq\underbrace{\|\psi(\xi)\|}_{\leq ch}\int_{0}^{ \psi(\xi)}(\frac{\partial u_{1}}{\partial\eta}(\xi,\eta))^{2}\,d\eta.\] Integration over $\xi$ yields (3.16). In combination with (3.15) this yields $\\|u_{1}\\|_{h}^{2}\lesssim\|u_{1}\|_{1,\Omega_{1}}$, which completes the proof of (3.13). We now consider the result in (3.14). Due to $\\|\cdot\\|_{h}\sim\|\!\|\!\|\cdot\|\!\|\!\|_{h}$ is suffices to prove \[\|\!\|\!\|u_{1}\|\!\|\!\|_{h}^{2}+\|\!\|\!\|u_{2}\|\!\|\!\|_{h}^{2}\lesssim\|\!\|\!\|u_{1}+u_ {2}\|\!\|\!\|_{h}^{2}\quad\text{for all}~{}u_{1}+u_{2}\in\mathcal{W}_{1}\oplus \mathcal{W}_{2}.\] (3.17) The scalar product corresponding to $\|\!\|\!\|\cdot\|\!\|\!\|_{h}$ is denoted by $(\cdot,\cdot)_{\ast}$, i.e. $(u,v)_{\ast}=(u,v)_{1,\Omega_{1\!,2}}+\lambda([\![u]\!],[\![v]\!])_{\frac{1}{2 },h,\Gamma}$. From $(u_{1},u_{2})_{1,\Omega_{1\!,2}}=0$ it follows that \[\|(u_{1},u_{2})_{\ast}\|=\|\lambda([\![u]\!],[\![v]\!])_{\frac{1}{2},h,\Gamma}\| \leq\lambda h^{-1}\\|u_{1}\\|_{L^{2}(\Gamma)}\\|u_{2}\\|_{L^{2}(\Gamma)}.\] Using the results in (3.16), (3.13) we get, with a suitable constant $c$ and for arbitrary $\delta\in(0,1)$: \[\|(u_{1},u_{2})_{\ast}\| \leq(1-\delta)\lambda h^{-1}\\|u_{1}\\|_{L^{2}(\Gamma)}\\|u_{2}\\|_{L ^{2}(\Gamma)}+\delta c\lambda\|u_{1}\|_{1,\Omega_{1}}\|u_{2}\|_{1,\Omega_{2}}\] \[\leq\max\{1-\delta,\delta c\lambda\}\|\!\|\!\|u_{1}\|\!\|\!\|_{h}\|\!\|\! \|u_{2}\|\!\|\!\|_{h}.\] By choosing a suitable $\delta$, we obtain the strengthened Cauchy-Schwarz inequality \[\|(u_{1},u_{2})_{\ast}\|\leq C^{\ast}\|\!\|\!\|u_{1}\|\!\|\!\|_{h}\|\!\|\!\|u_{2}\|\!\|\!\|_ {h}\quad\text{for all}~{}u_{1}\in\mathcal{W}_{1},\,u_{2}\in\mathcal{W}_{2},\] with a constant $C^{\ast}<1$, independent of $h$ and of how the triangulation is intersected by $\Gamma$. This result is equivalent to the one in (3.17). ∎ As a direct consequence of the stable splitting properties derived above we obtain the following main result. Theorem 3.3.: _Take $d=2$. There exists a constant $K$, independent of $h$ and of how the triangulation is intersected by $\Gamma$, such that_ \[\\|u_{0}\\|_{h}^{2}+\\|u_{1}\\|_{h}^{2}+\\|u_{2}\\|_{h}^{2}\leq K\\|u_{0}+u_{1}+u_{2} \\|_{h}^{2}\quad\text{for all}~{}~{}u=u_{0}+u_{1}+u_{2}\in\mathcal{S}.\] Proof.: Combine the result in (3.2) with the one in (3.14). ∎ ## 4 An optimal preconditioner based on approximate subspace corrections We describe and analyze an additive subspace decomposition preconditioner using the framework given in [19]. For this we first introduce some additional notation. Let $Q_{l}:\mathcal{S}\to\mathcal{W}_{l}$, $l=0,1,2$, be the $L^{2}$-projection, i.e., for $u\in\mathcal{S}$: \[(Q_{l}u,w_{l})_{0}=(u,w_{l})_{0}\quad\text{for all}~{}w_{l}\in\mathcal{W}_{l}.\] The bilinear form $a_{h}(\cdot,\cdot)$ on $\mathcal{S}$ that defines the discretization can be represented by the operator $A:\,\mathcal{S}\to\mathcal{S}$: \[(Au,v)_{0}=a_{h}(u,v)\quad\text{for all}~{}u,v\in\mathcal{S}.\] (4.1) The discrete problem (2.5) has the compact representation $Au=f_{Q}$, where $f_{Q}$ is the $L^{2}$-projection of the given data $f\in L^{2}(\Omega)$ onto the finite element space $\mathcal{S}$. The Ritz approximations $A_{l}:\mathcal{W}_{l}\to\mathcal{W}_{l}$, $l=0,1,2$, of $A$ are given by \[(A_{l}u,v)_{0}=(Au,v)=a_{h}(u,v)\quad\text{for all}~{}u,v\in\mathcal{W}_{l}.\] Note that these are symmetric positive definite operators. In the preconditioner we need symmetric positive definite approximations $B_{l}:\mathcal{W}_{l}\to\mathcal{W}_{l}$ of the Ritz operators $A_{l}$. The spectral equivalence of $B_{l}$ and $A_{l}$ is described by the following: \[\gamma_{l}(B_{l}u,u)_{0}\leq(A_{l}u,u)_{0}\leq\rho_{l}(B_{l}u,u)_{0}\quad\text {for all}~{}u\in\mathcal{W}_{l},\] (4.2) with strictly positive constants $\gamma_{l}$, $\rho_{l}$, $l=0,1,2$. The _additive subspace preconditioner_ is defined by \[C=\sum_{l=0}^{2}B_{l}^{-1}Q_{l}.\] (4.3) For the implementation of this preconditioner one has to solve (in parallel) three linear systems. The operator $Q_{l}$ is not (explicitly) needed in the implementation, since if for a given $z\in\mathcal{S}$ one has to determine $d_{l}=B_{l}^{-1}Q_{l}z$, the solution can be obtained as follows: determine $d_{l}\in\mathcal{W}_{l}$ such that \[(B_{l}d_{l},v)_{0}=(z,v)_{0}\quad\text{for all}~{}v\in\mathcal{W}_{l}.\] The theory presented in [19] can be used to quantify the quality of the preconditioner $C$. Theorem 4.1.: _Define $\gamma_{\min}=\min_{l}\gamma_{l}$, $\rho_{\max}=\max_{l}\rho_{l}$. Let $K$ be the constant of the stable splitting in Theorem 3.3. The spectrum $\sigma(CA)$ is real and_ \[\sigma(CA)\subset\big{[}\frac{\gamma_{\min}}{K},3\rho_{\max}\big{]}\] _holds._ Proof.: We recall a main result from [19] (Theorem 8.1). If there are strictly positive constants $K_{1},K_{2}$ such that \[K_{1}^{-1}\sum_{l=0}^{2}(B_{l}u_{l},u_{l})\leq\\|u_{0}+u_{1}+u_{2}\\|_{h}^{2} \leq K_{2}\sum_{l=0}^{2}(B_{l}u_{l},u_{l})\quad\text{for all}~{}u_{l}\in \mathcal{W}_{l}\] is satisfied, then $\sigma(CA)\subset[K_{1}^{-1},K_{2}]$ holds. For the lower inequality we use Theorem 3.3 and (4.2), which then results in \[\\|u_{0}+u_{1}+u_{2}\\|_{h}^{2}\geq K^{-1}\sum_{l=0}^{2}\\|u_{l}\\|_{h}^{2}=K^{-1} \sum_{l=0}^{2}(A_{l}u_{l},u_{l})_{0}\geq\frac{\gamma_{\min}}{K}\sum_{l=0}^{2}( B_{l}u_{l},u_{l})_{0}.\] For the upper bound we note \[\\|u_{0}+u_{1}+u_{2}\\|_{h}^{2}\leq 3\sum_{l=0}^{2}\\|u_{l}\\|_{h}^{2}=3\sum_{l=0} ^{2}(A_{l}u_{l},u_{l})_{0}\leq 3\rho_{\max}\sum_{l=0}^{2}(B_{l}u_{l},u_{l})_{0}.\] Now we apply the above-mentioned result with $K_{1}=K/\gamma_{\min}$ and $K_{2}=3\rho_{\max}$. ∎ The result in Theorem 3.3 yields that the constant $K$ is independent of $h$ and of how the triangulation intersects the interface $\Gamma$. It remains to choose appropriate operators $B_{l}$ such that $\gamma_{\rm min}$ and $\rho_{\max}$ are uniform constants, too. We first consider the approximation $B_{0}$ of the Ritz-projection $A_{0}$. Note that the finite element functions in $\mathcal{W}_{0}=V_{h}$ are continuous across $\Gamma$. This implies that \[(A_{0}u,v)=a_{h}(u,v)=(\alpha u,v)_{1,\Omega_{1\!,2}}=(\alpha\nabla u,\nabla v )_{0}\quad\text{for all}~{}~{}u,v\in\mathcal{W}_{0}.\] Hence, $A_{0}$ is a standard finite element discretization of a Poisson equation (with a discontinuous diffusion coefficient $\alpha$). As a preconditioner $B_{0}$ for $A_{0}$ we can use a standard symmetric multigrid method (which is a multiplicative subspace correction method). From the literature [6, 18, 19] we know that for this choice of $B_{0}$ we have spectral inequalities as in (4.2), with $\rho_{0}=1$ and a constant $\gamma_{0}>0$ that is independent of $h$ and of how $\Gamma$ intersects the triangulation. It remains to find an appropriate preconditioner $B_{l}$ of $A_{l}$, $l=1,2$. For this we propose the simple Jacobi method, i.e., diagonal scaling as a preconditioner for $A_{l}$, $l=1,2$. We first introduce the operator $B_{l}$ that represents the Jacobi preconditioner. Recall that $\mathcal{W}_{l}={\rm span}\{\phi_{j}^{\Gamma}~{}\|~{}j\in\mathcal{J}_{\Gamma,l}\}$. Elements $u,v\in\mathcal{W}_{l}$ have unique representations $u=\sum_{j\in\mathcal{J}_{\Gamma,l}}\alpha_{j}\phi_{j}^{\Gamma}$, $v=\sum_{j\in\mathcal{J}_{\Gamma,l}}\beta_{j}\phi_{j}^{\Gamma}$. In terms of these representations the Jacobi preconditioner is defined by \[(B_{l}u,v)_{0}=\sum_{j\in\mathcal{J}_{\Gamma,l}}\alpha_{j}\beta_{j}a_{h}(\phi_ {j}^{\Gamma},\phi_{j}^{\Gamma}),\quad u,v\in\mathcal{W}_{l},~{}l=1,2.\] (4.4) Note that $a_{h}(\phi_{j}^{\Gamma},\phi_{j}^{\Gamma})$ are diagonal entries of the stiffness matrix corresponding to $a_{h}(\cdot,\cdot)$. The result in the next lemma shows that this diagonal scaling yields a robust preconditioner for the Ritz operator $A_{l}$. Lemma 4.2.: _For the Jacobi preconditioner $B_{l}$ there are strictly positive constants $\gamma_{l}$, $\rho_{l}$, independent of $h$ and of how the triangulation is intersected by $\Gamma$ such that_ \[\gamma_{l}(B_{l}u,u)_{0}\leq(A_{l}u,u)_{0}\leq\rho_{l}(B_{l}u,u)_{0}\quad\text {for all}~{}u\in\mathcal{W}_{l},~{}l=1,2,\] (4.5) _holds._ Proof.: Take $u=\sum_{j\in\mathcal{J}_{\Gamma,l}}\alpha_{j}\phi_{j}^{\Gamma}\in\mathcal{W}_{l}$. For each $T\in\mathcal{T}_{h}^{\Gamma}$ we define $T_{l}=T\cap\Omega_{l}$, and for each $T_{l}$ we denote by $V(T_{l})$ the set vertices of $T$ that are _not_ in $\Omega_{l}$. Note that $V(T_{l})\neq\emptyset$ and $V(T_{l})$ does not contain all vertices of $T$. Using (3.13) and the construction of the xfem basis functions we get \[\begin{split}(B_{l}u,u)_{0}&=\sum_{j\in\mathcal{J}_{ \Gamma,l}}\alpha_{j}^{2}a_{h}(\phi_{j}^{\Gamma},\phi_{j}^{\Gamma})\sim\sum_{j \in\mathcal{J}_{\Gamma,l}}\alpha_{j}^{2}\|\phi_{j}^{\Gamma}\|_{1,\Omega_{l}}^{2} \\ &=\sum_{T\in\mathcal{T}_{h}^{\Gamma}}\sum_{j\in V(T_{l})}\alpha_{ j}^{2}\|\phi_{j}^{\Gamma}\|_{1,T_{l}}^{2}\sim\sum_{T\in\mathcal{T}_{h}^{\Gamma}} \sum_{j\in V(T_{l})}\alpha_{j}^{2}\\|\nabla(\phi_{j})_{\|T}\\|_{2}^{2}\|T_{l}\|. \end{split}\] (4.6) Using (3.13) and the fact that $\nabla u$ is a constant vector on each $T_{l}$ we get, with $\\|\cdot\\|_{2}$ the Euclidean vector norm, \[(A_{l}u,u)_{0}=\\|u\\|_{h}^{2}\sim\|u\|_{1,\Omega_{l}}^{2}=\sum_{T\in\mathcal{T}_{ h}^{\Gamma}}\\|\nabla u\\|_{L^{2}(T_{l})}^{2}=\sum_{T\in\mathcal{T}_{h}^{\Gamma} }\|T_{l}\|\\|(\nabla u)_{\|T_{l}}\\|_{2}^{2}.\] (4.7) Now note that $(\nabla u)_{\|T_{l}}=\sum_{j\in V(T_{l})}\alpha_{j}(\nabla\phi_{j}^{\Gamma})_{\| T_{l}}=\sum_{j\in V(T_{l})}\alpha_{j}\nabla(\phi_{j})_{\|T}$. Because $V(T_{l})$ does not contain all vertices of $T$, the vectors in the set $\{(\nabla\phi_{j})_{\|T}~{}\|~{}j\in V(T_{l})\}$ are independent and the angles between the vectors depend only on the geometry of the triangulation $\mathcal{T}_{h}$. This implies that \[\\|(\nabla u)_{\|T_{l}}\\|_{2}^{2}\,\sim\sum_{j\in V(T_{l})}\alpha_{j}^{2}\\| \nabla(\phi_{j})_{\|T}\\|_{2}^{2}.\] Combining this with the results in (4.6) and (4.7) completes the proof. ∎ Remark 4.1.: Instead of an optimal multigrid preconditioner in the subspace $\mathcal{W}_{0}=V_{h}$, one can also use a simpler (suboptimal) Jacobi preconditioner, i.e. $B_{0}$ analogous to (4.4). For this choice the spectral constants in (4.2) are $\gamma_{0}\sim h^{2}$ and $\rho_{0}\sim 1$. The three subspaces are disjoint and thus if one applies a Jacobi preconditioner in the three subspaces, the additive subspace preconditioner $C$ in (4.3) coincides with a Jacobi preconditioner for the operator $A$. From Theorem 4.1 we can conclude that $\kappa(CA)\leq ch^{-2}$ holds, with a constant $c$ independent on $h$ and the cut position. Similar uniform $\mathcal{O}(h^{-2})$ condition number bounds have recently been derived in the literature, cf. [20] and [4]. In these papers, however, for obtaining such a bound an additional stabilization term is added to the bilinear form $a_{h}(\cdot,\cdot)$. Our analysis shows that although the condition number of the stiffness matrix corresponding to $a_{h}(\cdot,\cdot)$ does not have a uniform (w.r.t. the interface cut) bound $ch^{-2}$, a simple diagonal scaling results in a matrix with a spectral condition number that is bounded by $ch^{-2}$, with a constant $c$ that is independent of how $\Gamma$ is intersected by the triangulation. We note that adding a stabilization as treated [4] may have a positive effect not only on the condition number, but also on robustness of the discretization w.r.t. large jumps in the diffusion coefficient. Remark 4.2.: The assumption $d=2$ is essential only in the proof of Theorem 3.1. Concerning a generalization to $d=3$ we note the following. Firstly, it is not obvious how the subdivision into patches $\omega_{k}$ can be generalized to three space dimensions. Secondly, if $d=2$ then for every element within the reference patch $\hat{\omega}$ we know that the local finite element space on $T\cap\Omega_{i}$ is one-dimensional which is exploited to characterize the one-sided limit at the interface. In three dimensions the local finite element space can be two-dimensional on both parts $T\cap\Omega_{i},~{}i=1,2$ such that it is not obvious how to generalize the proof of Theorem 3.1. Nevertheless, we expect that the result of Theorem 3.3, hence also the results on the additive subspace preconditioner, hold in three space dimensions. This claim is supported by the results of a numerical example with $d=3$, presented in section 5.2. Remark 4.3.: For ease of presentation, all dependencies on $\alpha$, especially on the jumps in $\alpha$, have been absorbed in the constants that appear in the estimates. The results in neither Lemma 3.2, Theorem 3.1 nor Lemma 4.2 are robust with respect to jumps in $\alpha$. We illustrate the dependence of the quality of the subspace preconditioner on the jumps in $\alpha$ in a numerical example in section 5.1. Remark 4.4.: Instead of the _additive_ preconditioner $C$ in (4.3), one can also use a _multiplicative_ version, cf. [19]. The optimality of this multiplicative variant, which can be used as a solver or a preconditioner, can easily be derived using the framework given in [19] and the results presented above. ## 5 Numerical experiments In this section results for different subspace correction preconditioners are presented. We consider a discrete interface problem of the form: determine $u_{h}\in V_{h}^{\Gamma}$ such that \[a_{h}(u_{h},v_{h})=(f,v_{h})_{0}\quad\text{for all}~{}~{}v_{h}\in V_{h}^{ \Gamma},\] with $a_{h}(\cdot,\cdot)$ as in(2.6). We take test problems with $d=2$ and $d=3$. The resulting stiffness matrix, which is the matrix representation of the operator $A$ in (4.1), is denoted by $\mathbf{A}$. The matrices corresponding to the Ritz approximations $A_{0}$ (projection on $V_{h}$) and $A_{x}$ (projection on $V_{h}^{x}$) are denoted by $\mathbf{A}_{0}$ and $\mathbf{A}_{x}$, respectively. The diagonal matrices ${\rm diag}(\mathbf{A})$, ${\rm diag}(\mathbf{A}_{0})$${\rm diag}(\mathbf{A}_{x})$ are denoted by $\mathbf{D}_{\mathbf{A}}$, $\mathbf{D}_{0}$ and $\mathbf{D}_{x}$, respectively. Furthermore, $\mathbf{C}_{0}$ denotes a preconditioner for $\mathbf{A}_{0}$, for instance a multigrid preconditioner or $\mathbf{C}_{0}=\mathbf{D}_{0}$. We define the block preconditioners \[\mathbf{B}_{\mathbf{A}}:=\left(\begin{array}[]{cc}\mathbf{A}_{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{A}_{x}\end{array}\right),\quad\mathbf{B}_{\mathbf{D}}:= \left(\begin{array}[]{cc}\mathbf{A}_{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{D}_{x}\end{array}\right),\quad\mathbf{B}_{\mathbf{C}}:= \left(\begin{array}[]{cc}\mathbf{C}_{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{D}_{x}\end{array}\right).\] (5.1) The matrix $\mathbf{B}_{\mathbf{A}}$ corresponds to an additive subspace preconditioner with exact subspace corrections, $\mathbf{B}_{\mathbf{D}}$ has an exact correction in $V_{h}$ and an approximate diagonal subspace correction in $V_{h}^{x}$, and $\mathbf{B}_{\mathbf{C}}$ has approximate subspace corrections in all subspaces. In the following we study the performance of these preconditioners, in particular their robustness w.r.t. both the variation in the mesh size $h$ and the location of the interface. We also ilustrate the dependence of the condition numbers on $\lambda$ and the diffusivity ratio $\alpha_{1}/\alpha_{2}$. In section 5.1 we consider a two-dimensional example with a challenging configuration in the sense that many elements in the mesh have small cuts. This setting allows for a detailed study of the dependencies on $h$, $\alpha_{1}/\alpha_{2}$ and $\lambda$. In the second example in section 5.2 we consider a three-dimensional analog and apply a multigrid preconditioner $\mathbf{C}_{0}$ for $\mathbf{A}_{0}$. ### Two-dimensional test case The domain is the unit square $\Omega=[0,1]^{2}$ with an interface $\Gamma$ which is a square with corners that are rounded off. A sketch is displayed in Figure 5.1 (left). The rounded square is centered around $\mathbf{x}_{0}$, it is denoted as $\Omega_{1}$. We set the dimensions to $l=0.2$ and $r=0.05$. In the implementation a piecewise linear approximation of $\Gamma$ is used. To investigate conditioning of the system, we consider a situation with many small cuts. [FIGURE:S5.F1][ENDFIGURE] To this end we use a uniform triangulation of $\Omega$ and set $\mathbf{x}_{0}=(0.5,0.5)+\varepsilon(1,1)$ with a “shift parameter” $\varepsilon=2^{-20}$. In this configuration almost all cut elements $T\in\mathcal{T}_{h}^{\Gamma}$ have very small cuts (cf. right sketch in Figure 5.1). A similar test case has been considered in [3] as “sliver cut case”. We use four levels of uniform refinement denoted by L1,..,L4. The diffusion parameters are fixed to $(\alpha_{1},\alpha_{2})=(1.5,2)$. Note that we consider $\beta_{1}=\beta_{2}=1$, but the problem is equivalent to every combination of Henry and diffusion parameters which fulfill $(\alpha_{1}/\beta_{1},\alpha_{2}/\beta_{2})=(1.5,2)$. The Nitsche stabilization parameter is set to $\lambda=4\bar{\alpha}$ with $\bar{\alpha}=\frac{1}{2}(\alpha_{1}+\alpha_{2})=1.75$. As a right-hand side source term we choose $f=1$ in $\Omega_{1}$ and $f=0$ in $\Omega_{2}$. In the tables below we present results for the spectral condition number of the preconditioned matrix. We also include the iteration number of the CG method, applied to the preconditioned system, needed to reduce the starting residual by a factor of $10^{6}$. \| \| L1 \| L2 \| L3 \| L4 ---\|---\|---\|---\|---\|--- κ(B−1AA) \| (its.) \| 4.98×100 \| (13) \| 4.95×100 \| (13) \| 4.82×100 \| (12) \| 4.82×100 \| (11) κ(B−1DA) \| (its.) \| 5.12×100 \| (13) \| 5.06×100 \| (13) \| 4.94×100 \| (12) \| 4.94×100 \| (11) κ(D−1AA) \| (its.) \| 2.78×101 \| (22) \| 1.11×102 \| (40) \| 4.42×102 \| (73) \| 1.77×103 \| (127) Table 5.1: Condition number and iteration counts of CG method (λ=4¯α, α1/α2=0.75). In Table 5.1 the condition numbers corresponding to the block preconditioners $\mathbf{B}_{\mathbf{A}}$, $\mathbf{B}_{\mathbf{D}}$ and $\mathbf{D}_{\mathbf{A}}$ are displayed for four different levels of refinement. The condition number of $\mathbf{A}$ is above $10^{7}$ and the number of CG iterations without preconditioning is above $2000$ on all four levels. We observe that the condition numbers of $\mathbf{B}_{\mathbf{A}}$ and $\mathbf{B}_{\mathbf{D}}$ are essentially independent on the mesh size $h$. From further experiments we observe that the condition number of $\mathbf{A}$ severely depends on the shift parameter, the results for the block preconditioners however remain essentially the same. This is in agreement with the results derived in section 4. Also the Jacobi preconditioner $\mathbf{D}_{\mathbf{A}}$ behaves as expected. With decreasing mesh size $h$, for the condition number we observe $\kappa(\mathbf{D}_{\mathbf{A}}^{-1}\mathbf{A})\sim h^{-2}$. λ/¯α \| 4×100 \| 4×101 \| 4×102 \| 4×103 ---\|---\|---\|---\|--- κ(B−1AA) \| (its.) \| 4.95×100 \| (13) \| 2.50×100 \| (9) \| 2.29×100 \| (7) \| 2.27×100 \| (6) κ(B−1DA) \| (its.) \| 5.06×100 \| (13) \| 2.14×101 \| (13) \| 2.07×102 \| (14) \| 2.07×103 \| (15) κ(D−1AA) \| (its.) \| 1.11×102 \| (40) \| 9.49×101 \| (36) \| 2.07×102 \| (38) \| 2.07×103 \| (44) Table 5.2: Condition number and iteration counts of CG method (level L2, α1/α2=0.75). For these preconditioners, with a fixed mesh (level L2) the dependence on $\lambda$ is shown in Table 5.2. The results suggest that the estimate in Theorem 3.1 is essentially independent on $\lambda$. The condition number $\kappa(\mathbf{B}_{\mathbf{A}}^{-1}\mathbf{A})$ even slightly decreases for increasing $\lambda$. The diagonal preconditioning of the xfem block $\mathbf{A}_{x}$, however, results in a linear dependence on $\lambda$. Hence, diagonal preconditioning of $\mathbf{A}_{x}$ is not robust w.r.t. $\lambda$. Despite the increasing condition number, the CG iteration counts seem to stay almost constant. A similar behavior can be observed for the Jacobi preconditioner $\mathbf{D}_{\mathbf{A}}$. α1/α2 \| 7.5×10−1 \| 7.5×100 \| 7.5×101 \| 7.5×102 ---\|---\|---\|---\|--- κ(B−1AA) \| (its.) \| 4.95×100 \| (13) \| 1.13×101 \| (20) \| 5.54×101 \| (26) \| 5.21×102 \| (28) κ(B−1DA) \| (its.) \| 5.06×100 \| (13) \| 1.29×101 \| (20) \| 9.87×101 \| (28) \| 9.61×102 \| (26) κ(D−1AA) \| (its.) \| 1.11×102 \| (40) \| 6.33×102 \| (45) \| 5.90×103 \| (50) \| 5.86×104 \| (72) Table 5.3: Condition number and iteration counts of CG method (level L2, λ=4¯α). In Table 5.3 we illustrate the behavior of the preconditioners for increasing diffusivity ratios. We observe that for all three preconditioners the corresponding condition number has a roughly linear dependence on $\alpha_{1}/\alpha_{2}$. We conclude that the stability estimate in Theorem 3.1 is not robust with respect to variation in $\alpha_{1}/\alpha_{2}$. The increase of the CG iteration counts, however, is only very mild. ### Three-dimensional test case We consider a setup in three dimensions very similar to the one used in section 5.1. The domain is the unit cube $\Omega=[0,1]^{3}$ with a cube that is rounded off as the dividing interface. The cube, denoted as $\Omega_{1}$, is centered around $\mathbf{x}_{0}=(0.5,0.5,0.5)+\varepsilon(1,1,1)$ with a small “shift parameter” $\varepsilon=2^{-20}$. The dimensions of the cube are chosen as in section 5.1 ($l=0.2$, $r=0.05$) and a uniform triangulation of $\Omega$ is used. We use seven levels of uniform refinement denoted by L0,..,L6 where the coarsest level (L0) is a $2\!\times\!2\!\times\!2$-grid. The diffusion parameters are fixed to $(\alpha_{1},\alpha_{2})=(1,3)$. Note that we consider $\beta_{1}=\beta_{2}=1$. The Nitsche stabilization parameter is set to $\lambda=5\bar{\alpha}$ with $\bar{\alpha}=\frac{1}{2}(\alpha_{1}+\alpha_{2})=2$. As a right-hand side source term we choose $f=1$ in $\Omega_{1}$ and $f=0$ in $\Omega_{2}$. We investigate the performance of the CG method preconditioned with $\mathbf{B}_{\mathbf{C}}$, cf. (5.1). For the preconditioner $\mathbf{C}_{0}$ of $\mathbf{A}_{0}$ we use a standard multigrid method. In this multigrid preconditioner we apply one V-cycle with a damped Jacobi (damping factor $0.8$) iteration as pre- and post-smoother. In Table 5.4 the iteration counts that were needed to reduce the initial residual by a factor of $10^{6}$ for the levels L2 to L6 are shown. On level L6 we have approximately two million unknowns. \| L2 \| L3 \| L4 \| L5 \| L6 ---\|---\|---\|---\|---\|--- CG iterations \| 22 \| 25 \| 27 \| 29 \| 32 Table 5.4: Iteration counts of multigrid-preconditioned CG method (λ=5¯α, α2/α1=3). We observe that the iteration counts stay essentially bounded such that the effort for solving the linear systems is $\mathcal{O}(N)$ with $N$ the number of degrees of freedom, i.e. $\mathbf{B}_{\mathbf{C}}$ is an optimal preconditioner. The mild increase in iteration numbers further decreases if the Jacobi preconditioner $\mathbf{D}_{x}$ used in the subspace $V_{h}^{x}$ is replaced by a symmetric Gauss-Seidel preconditioner. For this choice we obtain the numbers 21,23,23,25,27 for the levels L2 to L6. ### Acknowledgement The authors gratefully acknowledge funding by the German Science Foundation (DFG) within the Priority Program (SPP) 1506 “Transport Processes at Fluidic Interfaces”. ## References * [1]D. Bothe, M. Koebe, K. Wielage, J. 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0904.0483	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 22616, "num_imgs": 3, "llama3_tokens_count": 6071 }	[ "content_image/0904.0483/x1.png", "content_image/0904.0483/x2.png", "content_image/0904.0483/x4.png" ]	# Impact of a viscous liquid drop Robert D. Schroll Physics Department and the James Franck Institute, The University of Chicago, 929 E. 57th St., Chicago, Illinois 60637 Christophe Josserand Stéphane Zaleski UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France Wendy W. Zhang Physics Department and the James Franck Institute, The University of Chicago, 929 E. 57th St., Chicago, Illinois 60637 February 23, 2024 ###### Abstract We simulate the impact of a viscous liquid drop onto a smooth dry solid surface. As in experiments, when ambient air effects are negligible, impact flattens the falling drop without producing a splash. The no-slip boundary condition at the wall produces a boundary layer inside the liquid. Later, the flattening surface of the drop traces out the boundary layer. As a result, the eventual shape of the drop is a “pancake” of uniform thickness except at the rim, where surface tension effects are significant. The thickness of the pancake is simply the height where the drop surface first collides with the boundary layer. pacs: 47.55.D-,47.55.dr,47.15.Cb The impact of a liquid drop onto a dry solid surface lies at the heart of many important technological processes Rein (1993); Yarin (2006), from the application of a thermal spray Herman et al. (2000); Jiang et al. (2001); Fauchais et al. (2004); Li and Li (2004); McDonald et al. (2007); Molta and Moreira (2007); Shinoda et al. (2007) to atomization of fuel in a combustion chamber Williams (1958); Chigier (1976); Lefebvre (1989); Molta and Moreira (2007). Recent experiments revealed the splash formed when a low-viscosity liquid, such as water or ethanol, first collides with a dry smooth wall at several m/s owes its existence entirely to the presence of air Xu et al. (2005); Quéré (2005); Xu et al. (2007); Xu (2007). These results are motivating new studies on the large-scale deformations created by impact when air effects are absent as well as how a splash forms Jepsen et al. (2006); Duez et al. (2007); Mukherjee and Abraham (2007); Subramani et al. (2007); Deegan et al. (2008); Kannan and Sivakumar (2008); Pepper et al. (2008). Here we focus on the impact of a viscous liquid drop when air effects are absent. Recent experiments show that reducing the ambient gas pressure also suppresses the splash of a silicone oil drop. However, the form of the splash is very different. While the splash from a low-viscosity liquid develops within a few 10 $\mu$s of impact, the splash from a silicone oil develops slowly, becoming evident only after most of the liquid drop has fallen and flattened into a thin pancake Stevens et al. (2007); Driscoll et al. (2008). We use an axisymmetric Volume-of-Fluid (VOF) code to simulate the impact at reduced ambient pressure Lafaurie et al. (1994); Josserand and Zaleski (2003); Josserand et al. (2005). Our results show that a boundary layer, corresponding to a thin region where the radial flow created by impact adjusts to the no-slip condition at the wall, is created by the impact. The boundary layer has uniform thickness. As impact nears its end, the drop surface flattens onto the boundary layer, evolving into a pancake of uniform thickness. The Volume-of-Fluid simulation solves the Navier-Stokes equations, together with constraint of incompressibility, for both the liquid interior and the gas exterior at reduced pressure. Physically appropriate boundary conditions, in particular the Laplace pressure jump across the surface due to surface tension, are enforced Lafaurie et al. (1994). In a typical run, an initially spherical liquid drop with radius $a$ collides with a dry, smooth solid surface at an impact speed $U_{0}$ of several m/s. The ambient air density $\rho_{g}$ is kept so small that $2$-fold changes in in the value of $\rho_{g}$ have little effect on the liquid dynamics. The bottom surface of the liquid drop is not broken upon impact, which corresponds to maintaining an apparent contact angle of $180^{\circ}$ where the liquid rim meets the wall (Fig. 2a) [29]. We require that the velocity field inside the drop satisfies both the no-flux and no-slip boundary conditions at the solid wall. Figure 1 plots successive drop surface shapes (outlined in white) against snapshots from an experiment by Driscoll & Nagel under the same impact conditions Driscoll et al. (2008). We have rescaled time by the impact time-scale $\tau\equiv a/U_{0}$. The simulation agrees very well with the experiment. After the drop hits the wall, the liquid liquid inside the drop is diverted outwards, forming a thin sheet which expands radially along the wall (Fig. 1b). Later, the falling drop flattens, evolving towards a shape resembling a pancake with a thickened outer rim (Fig. 1c). By $t=7.4\tau$, a considerable time after the drop has stopped falling, the expanding drop attains its maximum extent (Fig. 1d). After that point in time, surface tension causes the drop in the simulation to retract inwards and reform into a spherical shape, a dynamics we do not analyze. <figure><img src="content_image/0904.0483/x1.png"><figcaption>Figure 1: Impact of a viscous silicone oil drop at 4 m/s onto a smooth, drysubstrate at reduced ambient pressure (34 kPa). Surface profiles (white) fromsimulated impact are overlaid against snap shots from an experiment. From leftto right, the successive times are t=0, τ, 2τ and 7.4τ where τ≡a/U0 is theimpact time-scale. Here Re≈1280 and We≈2280. Photos courtesy of Driscoll &Nagel.</figcaption></figure> For this impact, the liquid drop has radius $a=0.16$ cm and the impact speed $U_{0}$ is $4$ m/s. The liquid is a low-molecular-weight silicone oil with dynamic viscosity $\mu_{L}=9.4$ cP, density $\rho_{L}=0.94$ g/cm${}^{3}$ and surface tension $\sigma=21$ dynes/cm. The exterior fluid corresponds to air at $34$ kPa, with density $\rho_{g}=4.4\times 10^{-4}$ g/cm${}^{3}$ and dynamic viscosity $\mu_{g}=1.8\times 10^{-2}$ cP. The simulated impact occurs in a cylindrical tube with radius $R=6a$ and height $H=6a$. The experiment uses a larger tube. Changing $R$ does not change the impact results. In describing the results, we use a cylindrical coordinate system, where the tube is centered at $r=0$ and the wall lies at $z=0$. To correlate different impacts, we non-dimensionalize all the length-scales by $a$, the velocities by $U_{0}$ and the time-scales by $\tau$. Since neither the tube dimensions nor the air properties affect the liquid dynamics reported here, the outcomes depend on only two dimensionless parameters: the Reynolds number $Re\equiv{2\rho_{L}U_{0}a}/{\mu_{L}}$ and the Weber number $We\equiv{2\rho_{L}U_{0}^{2}a}/{\sigma}$. The impact in Fig. 1 correspond to $Re\approx 1280$ and $We\approx 2280$. How the thin liquid sheet ejected by impact evolves over time is more difficult to characterize in the experiments. Previous studies Chandra and Avedisian (1991); Clanet et al. (2004) proposed a simple estimate for the eventual thickness $h$ of the pancake. The idea is that if the only mechanism that can arrest the outward radial expansion is viscous dissipation. We then assume that the impact energy is dissipated by a radial flow of strength $U_{0}$ in a liquid pancake of thickness $h$ and maximal extent $R_{\rm max}$. Balancing the dissipation against the initial kinetic energy yields \[\frac{\mu_{L}U_{0}}{h^{2}}(\pi R^{2}_{\rm max})R_{\rm max}\approx\frac{\rho_{L }U^{2}_{0}}{2}\left(\frac{4\pi a^{3}}{3}\right)\ .\] (1) This energy balance, together with volume conservation $4\pi a^{3}/3\approx\pi R^{2}_{\rm max}h$, predicts that the dimensionless pancake thickness $h/a$ should scale as $Re^{-2/5}$. This feature has been confirmed by previous experiments Clanet et al. (2004). Here we find that the eventual thickness of the pancake formed by impact from our simulation for different values of $U_{0}$, $\sigma$ and $\mu_{L}$ are entirely consistent with this estimate (see supplementary materials). While successful, the scaling estimate for $h$ does not tell us when the characteristic pancake thickness $h$ first emerges. Nor does it explain why the top surface of the drop evolves into a pancake of uniform thickness. To address these questions, we examine results from the simulation. This time we switch to a reference frame $x=r-R_{h}(t)$. The radial location $R_{h}$ corresponds to a point where the rounded rim joins onto the rest of the drop. Within the $O(\tau)$ time window, the simulation shows that $R_{h}(t)\approx\sqrt{4aU_{0}t}$, so that the reference frame $x$ decelerates over time. Figure 2a displays the surface profiles at the leading edge at different moments. Initially ($t=0.3\tau$) a thin collar is ejected from a nearly spherical drop. As time goes on, a local minimum develops in the interface profile, separating the drop profile into a rounded rim region where surface tension effects are important and a downward sloping profile which becomes more and more gently sloped over time. As time goes on the profile flattens to the left of the minimum. By $t=4.9\tau$, a pancake of uniform thickness is apparent. At the rim, surface tension acts to slow the expanding liquid sheet, causing liquid to accumulate, consistent with results from previous studies Keller et al. (1995); deGennes et al. (2003); Fedorchenko et al. (2005). The flattening dynamics, however, has not been examined previously. To quantify its progress, we define an onset time $t_{\rm onset}$. This is the time when the height of the interface at the local minimum first equals the eventual pancake thickness $h$. In Fig. 2b we plot $t_{\mathrm{onset}}$ as a function of the impact parameters. Non-dimensionalizing $t_{\mathrm{onset}}$ by the impact time-scale $\tau=a/U_{0}$ produces essentially flat curves with respect to both $Re$ and $We$. In other words, $t_{\mathrm{onset}}$ is simply controlled by the kinematics of impact, with no apparent dependence on either the liquid viscosity or surface tension. As a comparison we plot $t_{\rm max}$, the time when the drop attains its maximum extent and then retracts due to surface tension. This quantity has a strong dependence on surface tension, or $We$. Since $t_{\mathrm{onset}}$ is considerably shorter than $t_{\rm max}$, there is still appreciable liquid motion inside the drop when the first flattening begins. This suggests that the emergence of $h$ is not related to whether the kinetic energy has been sufficiently dissipated, but instead depends on the kinematics. <figure><img src="content_image/0904.0483/x2.png"><figcaption>Figure 2: Time evolution of ejected liquid sheet. (a) Leading-edge evolutionin the co-moving frame x=r−Rh(t) . Here we reduced the impact speed to U0=2m/s and increased the liquid viscosity μL=0.2 poise to generate a thickerpancake. Other parameters unchanged. (b) Onset time tonset (open symbols) andtmax, the time of maximum extent (closed symbols) as a function of the Webernumber (We≡ρLU20a2/σ). The different symbols correspond to U0=2–8 m/s (∘),σ=5.25–84 dynes/cm (▽), and μ=10–50 cP (△). Inset plots onset time tonset vs.Re≡2aU0/νL.</figcaption></figure> To get some insight into the kinematic origin of $h$, we examine the vorticity field. Prior to impact, the drop is falling with a spatially-uniform downward velocity, so the vorticity is $0$ everywhere inside the drop. After impact, the no-flux condition at the wall causes the liquid previously falling downwards to be diverted into a radially expanding flow. This expansion flow speeds up as it moves away from the centerline, reaches a peak at $R_{h}$, and then slows down as it enters the rim. This radial expansion also adjusts, via viscous effects, to the no-slip boundary condition at the solid wall. As a result of this adjustment, vorticity is generated in the liquid layer nearest to the solid wall. At any moment, the amount of vorticity generated to ensure zero slip at the wall is dictated by the strength of the radial expansion flow. Because the simulated impact is axisymmetric, only the azimuthal component of the vorticity is nonzero, i.e. $\mbox{\boldmath$\omega$}=\omega(r,z,t){\bf e}_{\theta}$ where $\omega\equiv\partial u_{z}/\partial r-\partial u_{r}/\partial z$. For high Reynolds number flows this adjustment takes place inside a narrow boundary layer. In Fig. 3a we plot the wall value of the vorticity $\omega_{0}(r,t)$ [35]. Since the radial outflow is largest near the outer edge, the vorticity also has a maximum near $R_{h}$. As the impact proceeds, the downward fall of the liquid drop slows, slowing the expansion and thus reducing the magnitude of $\omega_{0}$. <figure><img src="content_image/0904.0483/x4.png"><figcaption>Figure 3: Vorticity evolution after impact. (a) Value of the vorticity at thewall as a function of radial distance r vor . From top to bottom, the profilesare taken at t=0.3τ, 0.9τ, 1.6τ and 3.0τ. (b) The 20%, 50%, and 90% contoursof the vorticity distribution.</figcaption></figure> We next outline the spatial extent of the boundary layer. This task is complicated by the fact that the absolute size of the vorticity is strongly correlated with the strength of the radial expansion flow. Since the radial expansion flow, generated due to the no-flux condition, varies with $r$, vorticity is generated at different rates at different spatial locations. Moreover, at a given location, the vorticity production rate slows over time because the radial expansion slows. Thus contours of absolute vorticity do not provide clear indications for the spatial extent of the boundary layer. We side-step this complication by normalizing $\omega(r,z,t)$, the vorticity distribution in the bulk of the liquid, by the “wall” value $\omega_{0}(r,t)$. This essentially strips away variations in the vorticity distribution due to the varying speed of the radial expansion flow. For simple high Reynolds number flows, such as a uniform flow past a solid wall, as well as the boundary layer created by a straining flow towards a solid wall Acheson (1990), this procedure correctly reproduces the boundary layer structure that emerges from asymptotic analysis. In Fig. 4b we plot contours of the normalized vorticity distribution. In each snapshot, the solid lines within the liquid drop correspond to contours where $\omega(r,z,t)/\omega_{0}(r,t)=90\%$ (lowest curve), $50\%$ and $20\%$ (highest curve). At early times ($t=0.3\tau$), the boundary layer delineated by the contours is a pancake-shaped region inside the liquid drop. Except at the outermost edge, the top surface of the liquid drop is widely separated from the boundary layer. As impact proceeds, the boundary layer extends radially and thickens slightly, but retains its pancake shape. At $t=t_{\mathrm{onset}}$, the top surface of the drop collides with the boundary layer. After the collision, the surface at the collision location ceases to decrease in height. At the same time, the rest of the drop surface continues to fall downwards, bringing more and more portions of the surface into collision with the boundary layer. The result is a “front” that flattens inwards radially, tracing out the pancake shaped boundary layer. The idea that the eventual pancake thickness $h$ first emerges when the top surface of the drop collides with the boundary layer suggests a slightly different scaling relation for the eventual thickness $h$. If the boundary layer simply thickens diffusively prior to its collision with the top surface, then $h$ should scale as $\sqrt{\nu_{L}t_{\mathrm{onset}}}$, where $\nu_{L}$ is the kinematic viscosity of the liquid. Since $t_{\mathrm{onset}}$ is simply $a/U_{0}$ (see Fig. 2), we see that $h/a$ should obey the Blasius scaling $Re^{-1/2}$. This is also consistent with our simulation results, basically because our range of $Re$ is too limited to resolve a scaling exponent of $1/2$ from the previously proposed $2/5$ scaling (See supplementary materials). Before concluding, we comment on how these results on impact without air effects may relate to splash formation when air is present. In the simulated impacts, air effects are negligible and the boundary layer always remains attached to the wall. As $t$ approaches $2\tau$, the radial expansion slows and the pressure gradient within the liquid is essentially zero. In this time window, the boundary layer is not securely attached to the wall. Any external perturbation that adds an adverse pressure gradient, _e.g._ increased resistance from the air at larger ambient pressures, may be enough to cause the outer edge of the boundary layer to separate from the wall. Since the surface profile is coupled to the boundary layer, the separating boundary layer may peel the thin liquid layer away from the wall, forming the beginning of a corona. Simulations to check this idea are underway [37]. In conclusion, we have simulated the impact of a viscous oil drop when the ambient air pressure is reduced to a very low value, so that impact at several m/s does not produce a splash. Results on the large-scale shape deformation agree quantitatively with measurements from available experiments. The simulation reveals that the thin, spatially uniform pancake shape that a falling drop gets flattened into owes its existence to the boundary layer in the liquid drop created by impact. The authors are grateful to Cacey Stevens, Michelle Driscoll, and Sidney Nagel for helpful discussions and for sharing unpublished data from their experiments. We thank Leo Kadanoff, Margo Levine, Alex Obakov, David Quéré, and Tom Witten for encouragement and helpful comments. 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1004.3594	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 42780, "num_imgs": 7, "llama3_tokens_count": 13627 }	[ "content_image/1004.3594/x1.png", "content_image/1004.3594/x3.png", "content_image/1004.3594/x4.png", "content_image/1004.3594/x5.png", "content_image/1004.3594/x6.png", "content_image/1004.3594/x7.png", "content_image/1004.3594/x10.png" ]	# Capability of LHC to discover supersymmetry with $\sqrt{s}=$7 TeV and 1 fb${}^{-1}$ Howard Baer${}^{a}$, Vernon Barger${}^{b}$, Andre Lessa${}^{a}$ and Xerxes Tata${}^{b,c}$ ${}^{a}$Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA ${}^{b}$Dep’t of Physics, University of Wisconsin, Madison, WI 53706, USA ${}^{c}$Dept. of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822, US E-mail: , , , baer@nhn.ou.edu barger@pheno.wisc.edu lessa@nhn.ou.edu tata@phys.hawaii.edu ###### Abstract: We examine the capability of the CERN Large Hadron Collider to discovery supersymmetry (SUSY) with energy $\sqrt{s}=7$ TeV and integrated luminosity of about 1 fb${}^{-1}$. Our results are presented within the paradigm minimal supergravity model (mSUGRA or CMSSM). Using a 6-dimensional grid of cuts for optimization of signal to background– including missing $E_{T}$– we find for $m_{\tilde{g}}\sim m_{\tilde{q}}$ an LHC reach of $m_{\tilde{g}}\sim 800,\ 950,\ 1100$ and 1200 GeV for 0.1, 0.3, 1 and 2 fb${}^{-1}$, respectively. For $m_{\tilde{g}}\ll m_{\tilde{q}}$, the reach is instead near $m_{\tilde{g}}\sim 480,\ 540,\ 620$ and 700 GeV, for the same integrated luminosities. We also examine the LHC reach in the case of very low integrated luminosity where missing $E_{T}$ may not be viable. We focus on the multi-muon, multi-lepton (including electrons) and dijet signals. Although the LHC reach without $E_{T}^{\rm miss}$ is considerably lower in these cases, it is still substantial: for 0.3 fb${}^{-1}$, the dijet reach in terms of gluino mass is up to 600 GeV for very low $m_{0}$, while the dilepton reach is to gluino masses of $\sim 500$ GeV over a range of $m_{0}$ values. Supersymmetry Phenomenology, Supersymmetric Standard Model, Large Hadron Collider † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction The CERN Large Hadron Collider (LHC) has recently begun to generate data from proton-proton collisions at $\sqrt{s}=7$ TeV. The plan is to run for much of the next two years, with a goal of accumulating $\sim 1$ fb${}^{-1}$ of usable data. This initial run will be followed by a shut down for a year or so for various upgrades, followed by a turn-on at or near design energy of $\sqrt{s}=14$ TeV. The discovery capability of LHC with $\sqrt{s}=14$ TeV (LHC14), has been investigated for several new physics scenarios, where supersymmetry (SUSY) [1] is frequently used as a canonical example [2]. In the paradigm minimal supergravity (mSUGRA or CMSSM) model [3] based on local supersymmetry [4], the LHC14 reach with 100 fb${}^{-1}$ was found to extend to $m_{\tilde{g}}\sim 3.1$ TeV for $m_{\tilde{q}}\sim m_{\tilde{g}}$, and to $m_{\tilde{g}}\sim 1.8$ TeV, for $m_{\tilde{q}}\gg m_{\tilde{g}}$. As LHC turn-on drew near, the question turned to how well LHC could do in its initial stages, at very low integrated luminosity, and perhaps before the LHC detectors are fully calibrated, so that the canonical SUSY signature – the presence of mult-jet plus large missing $E_{T}$ ($E_{T}^{\rm miss}$) – is not fully viable. In Ref. [5, 6, 7], it was emphasized that SUSY could be discovered at LHC even without using $E_{T}^{\rm miss}$, by focusing instead on events with large multiplicity of isolated leptons. In Ref. [6], it was shown that LHC could discover SUSY in the dijet channel, using new kinematic variables, even without viable $E_{T}^{\rm miss}$. In a previous study [7], we investigated the supersymmetry discovery potential of LHC with $\sqrt{s}=10$ TeV (LHC10), the energy at which the machine was then expected to operate, both with and without the use of $E_{T}^{\rm miss}$, and compared it to the reach of LHC14. After Ref. [7] appeared, the decision was made to operate LHC at half its design energy of $\sqrt{s}=7$ TeV (LHC7). Furthermore, the additional year of LHC down-time allowed the various detectors to amass millions of cosmic muon events. This array of cosmic data allowed the experiments to make progress on important issues of detector alignment, tracking and calibration. At the end of 2009, the first proton-proton collisions were recorded in the CMS and ATLAS (and ALICE and LHC-b) detectors at center-of-mass energies of 900 GeV and 2.36 TeV. Initial analyses of these events show remarkably good agreement between Monte Carlo expectations and the actual data, including the (very low energy) $E_{T}^{\rm miss}$ spectrum [8, 9]. By March 30, 2010, the first $pp$ collisions were recorded at $\sqrt{s}=7$ TeV. At this time, millions of $pp$ collision events at 7 TeV have been recorded, including various multi-jet events, and even candidate leptonically decaying $W$ events. In light of the CERN decision to perform a major collider run at $\sqrt{s}=7$ TeV with $\sim 1$ fb${}^{-1}$ of integrated luminosity, it is reasonable to re-calculate the SUSY reach using the revised run parameters. In this paper, we evaluate the discovery capability of LHC7 for SUSY particles and display it as a reach plot in the $m_{0}-m_{1/2}$ plane of the mSUGRA model. The parameter space of the model is given by \[m_{0},\ m_{1/2},\ A_{0},\ \tan\beta,\ sign(\mu),\] (1) where $m_{0}$ is a common GUT scale soft SUSY breaking (SSB) scalar mass, $m_{1/2}$ is a common GUT scale SSB gaugino mass, $A_{0}$ is a common GUT scale trilinear SSB term, $\tan\beta$ is the ratio of Higgs field vevs, and $\mu$ is the superpotential Higgs mass term, whose magnitude, but not sign, is constrained by the electroweak symmetry breaking minimization conditions. At each model parameter space point, many simulated collider events are generated and compared against SM backgrounds with the same experimental signature [10]. A 6-dimensional grid of cuts are then employed to enhance the SUSY signal over SM backgrounds, and the signal is deemed observable if it satisfies pre-selected criteria for observability. Based on previous studies [2], we include in our analysis the following channels: * $jets+E_{T}^{\rm miss}$ (no isolated leptons), * $1\ell+jets+E_{T}^{\rm miss}$, * two opposite-sign isolated leptons (OS)$+jets+E_{T}^{\rm miss}$, * two same-sign isolated leptons (SS)$+jets+E_{T}^{\rm miss}$, * $3\ell+jets+E_{T}^{\rm miss}$. We evaluate the reach for various values of integrated luminosities ranging from 0.1 fb${}^{-1}$ to 2 fb${}^{-1}$, that may be relevant at LHC7. While the initial reports of detector performance at $\sqrt{s}\sim 0.95-2.36$ TeV are encouraging, we should keep in mind that the initial agreement between data and event simulation has been obtained at low luminosity and CM energies, and only for relatively simple event topologies with $E_{T}^{\rm miss}\stackrel{{<}}{{\sim}}40$ GeV and with limited total scalar $E_{T}$ in the events. Since fake $E_{T}^{\rm miss}$ grows with the total scalar energy in hadron collider events, it is still unclear how accurate the $E_{T}^{\rm miss}$ measurements will be at very high values of $E_{T}^{\rm miss}\sim 100-500$ GeV. With this in mind, we include a separate conservative low luminosity reach analysis where we do not make use of any $E_{T}^{\rm miss}$ information. We also present results with only reliable isolated muon identification,¹ since misidentification of jets as electrons could be problematic at very early stages in the analysis. We also present our no-$E_{T}^{\rm miss}$ results in the case where both $e$s and $\mu$s are reliably identified. In these cases, with limited detector performance the LHC reach, though more limited, still extends considerably beyond present limits. [FOOTNOTE:1][ENDFOOTNOTE] The remainder of this paper is organized as follows. In Sec. 2, we present details of our SUSY signal and SM background calculations. In Sec. 3, we show LHC7 reach plots using the complete anticipated detector performance, including reliable $E_{T}^{\rm miss}$ resolution and electron ID, for integrated luminosities from 0.1-2 fb${}^{-1}$. Our full analysis plots include scans over a vast grid of possible cut values, so signal/background is optimized in various regions of model parameter space. In Sec. 4, we present SUSY discovery reach plots in the more conservative scenario where reliable $E_{T}^{\rm miss}$ measurement may not be attainable, including the case that reliable $e$ ID may also not yet be possible. We also show reach results for acollinear dijet production via the Randall-Tucker-Smith analysis [6]. We conclude with a summary of our results in Sec. 5. ## 2 Standard model background and signal calculations Because our analysis covers several search channels, we include in our background calculations all relevant $2\to n$ processes for the multi-lepton and multi-jet searches. However, since we restrict our results to the first LHC physics run ($\lesssim$ 2 fb${}^{-1}$ and $\sqrt{s}=7$ TeV) we can ignore processes such as $pp\to VVV$ ($V=W^{\pm},Z$), for which the cross section is too small to be relevant. In order to obtain a proper statistical representation of our background and signal events, we generate (at least) the equivalent of 1 fb${}^{-1}$ of events for each process (except for our QCD samples). For the simulation of the background events, we use AlpGen and MadGraph to compute the hard scattering events and Pythia [11] for the subsequent showering and hadronization. For the final states containing multiple jets (namely $Z(\to ll,\nu\nu)+jets$, $W(\to l\nu)+jets$, $b\bar{b}+jets$, $t\bar{t}+jets$, $Z+b\bar{b}+jets$, $Z+t\bar{t}+jets$, $W+b\bar{b}+jets$, $W+t\bar{t}+jets$ and QCD), we use the MLM matching algorithm [12] to avoid double counting. All the processes included in our analysis are shown in Table 1 as well as their total cross-sections, number of events generated and event generator used. The signal events were generated using Isajet 7.79 [13] which, given an mSUGRA parameter set, generates all $2\to 2$ SUSY processes in the right proportion, and decays the sparticles to lighter sparticles using the appropriate branching ratios and decay matrix elements, until the sparticle decay cascade terminates in the stable LSP, assumed here to be the lightest neutralino. Using Prospino [14], we plot in Fig. 1 the NLO gluino and squark production cross-sections for the LHC at 7 TeV for the case of _a_) $m_{\tilde{q}}=m_{\tilde{g}}$ and _b_) $m_{\tilde{q}}=2m_{\tilde{g}}$. In frame _a_), we see that for low $m_{\tilde{g}}\stackrel{{<}}{{\sim}}500$ GeV, the total strongly interacting sparticle pair production cross section exceeds $10^{4}$ fb, so that with 1 fb${}^{-1}$ of integrated luminosity, there could be cases where over $10^{4}$ sparticle pair production events are created at LHC during the first run! For $m_{\tilde{q}}\sim m_{\tilde{g}}$, $\tilde{q}\tilde{g}+\tilde{q}\tilde{q}$ production are the dominant sparticle production mechanisms, whereas for $m_{\tilde{q}}\sim 2m_{\tilde{g}}$ the total SUSY cross section is dominated by $\tilde{g}\tilde{g}$ production and is somewhat smaller. <figure><img src="content_image/1004.3594/x1.png"><figcaption>Figure 1: Squark and gluino production cross-sections at NLO for LHC7 as afunction of m~g. In frame a) we show the cross-sections for m~q=m~g, whileframe b) has m~q=2m~g.</figcaption></figure> For event generation, we use a toy detector simulation with calorimeter cell size $\Delta\eta\times\Delta\phi=0.05\times 0.05$ and $-5<\eta<5$ . The HCAL (hadronic calorimetry) energy resolution is taken to be $80\%/\sqrt{E}+3\%$ for $\|\eta\|<2.6$ and FCAL (forward calorimetry) is $100\%/\sqrt{E}+5\%$ for $\|\eta\|>2.6$, where the two terms are combined in quadrature. The ECAL (electromagnetic calorimetry) energy resolution is assumed to be $3\%/\sqrt{E}+0.5\%$. We use the cone-type Isajet [13] jet-finding algorithm to group the hadronic final states into jets. Jets and isolated lepton are defined as follows: * Jets are hadronic clusters with $\|\eta\|<3.0$, $R\equiv\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}\leq 0.4$ and $E_{T}(jet)>50$ GeV. * Electrons and muons are considered isolated if they have $\|\eta\|<2.0$, $p_{T}(l)>10$ GeV with visible activity within a cone of $\Delta R<0.2$ about the lepton direction, $\Sigma E_{T}^{cells}<5$ GeV. * We identify hadronic clusters as $b$-jets if they contain a B hadron with $E_{T}(B)>$ 15 GeV, $\eta(B)<$ 3 and $\Delta R(B,jet)<$ 0.5. We assume a tagging efficiency of 60$\%$ and light quark and gluon jets can be mis-tagged as a $b$-jet with a probability 1/150 for $E_{T}\leq$ 100 GeV, 1/50 for $E_{T}\geq$ 250 GeV, with a linear interpolation for 100 GeV $\leq E_{T}\leq$ 250 GeV We point out the following technical improvements to our previous analyses [7]: * QCD events are now generated in $E_{T}$ bins for the hardest jet; this gives a better statistical representation for the high $E_{T}(j)$ events. * Our current analysis uses Isajet 7.79 for event generation. The version 7.79 SUSY spectrum calculation includes threshold corrections at each distinct decoupling squark and slepton mass value, whereas previous Isajet versions implemented all squark threshold corrections at a common scale $m_{\tilde{u}_{L}}$ and all sleptons at a common scale $m_{\tilde{e}_{L}}$[15]. Furthermore, previous Isajet versions included two-loop RGE running for the MSSM only from the $M_{SUSY}$ scale up to $M_{GUT}$; Isajet 7.79 also includes two-loop RGE running from $M_{Z}$ up to $M_{SUSY}$ (for more details see Ref. [16]). * We consider $b$-jet tagging to improve the optimized reach of the LHC. \| \| Cross \| number of ---\|---\|---\|--- SM process \| Generator \| section \| events QCD: 2, 3 and 4 jets (40 GeV<ET(j1)<100 GeV) \| AlpGen \| 2.6×109 fb \| 26M QCD: 2, 3 and 4 jets (100 GeV<ET(j1)<200 GeV) \| AlpGen \| 3.9×108 fb \| 44M QCD: 2, 3 and 4 jets (200 GeV<ET(j1)<500 GeV) \| AlpGen \| 1.6×107 fb \| 16M QCD: 2, 3 and 4 jets (500 GeV<ET(j1)<3000 GeV) \| AlpGen \| 9.4×104 fb \| 0.3M t¯t: t¯t \+ 0, 1 and 2 jets \| AlpGen \| 1.6×105 fb \| 5M b¯b: b¯b \+ 0, 1 and 2 jets \| AlpGen \| 8.8×107 fb \| 91M Z \+ jets: Z/γ(→l¯l,ν¯ν) \+ 0, 1, 2 and 3 jets \| AlpGen \| 8.6×106 fb \| 13M W \+ jets: W±(→lν) \+ 0, 1, 2 and 3 jets \| AlpGen \| 1.8×107 fb \| 19M Z \+ t¯t: Z/γ(→l¯l,ν¯ν) \+ t¯t \+ 0, 1 and 2 jets \| AlpGen \| 53 fb \| 0.6M Z \+ b¯b: Z/γ(→l¯l,ν¯ν) \+ b¯b \+ 0, 1 and 2 jets \| AlpGen \| 2.6×103 fb \| 0.3M W \+ b¯b: W±(→all) \+ b¯b \+ 0, 1 and 2 jets \| AlpGen \| 6.4×103 fb \| 9M W \+ t¯t: W±(→all) \+ t¯t \+ 0, 1 and 2 jets \| AlpGen \| 1.8×102 fb \| 9M W \+ tb: W±(→all) \+ ¯tb(t¯b) \| AlpGen \| 6.8×102 fb \| 0.025M t¯tt¯t \| MadGraph \| 0.6 fb \| 1M t¯tb¯b \| MadGraph \| 1.0×102 fb \| 0.2M b¯bb¯b \| MadGraph \| 1.1×104 fb \| 0.07M WW: W±(→lν)+W±(→lν) \| AlpGen \| 3.0×103 fb \| 0.005M WZ: W±(→lν)+Z(→all) \| AlpGen \| 3.4×103 fb \| 0.009M ZZ: Z(→all)+Z(→all) \| AlpGen \| 4.0×103 fb \| 0.02M Table 1: Background processes included in this LHC7 study, along with their total cross sections and number of generated events. All light (and b) partons in the final state are required to have ET>40 GeV. For QCD, we generate the hardest final parton jet in distinct bins to get a better statistical representation of hard events. For Wtb production, additional multi-jet production is only via the parton shower because the AlpGen calculation including all parton emission matrix elements is not yet available. For this process, we apply the cut \|m(Wb)−mt\|≥5 GeV to avoid double counting events from real t¯t production. ## 3 Optimized reach of the LHC utilizing $E_{T}^{\rm miss}$ As noted in Sec. 1, preliminary results from minimum bias events in $pp$ collisions at $\sqrt{s}=0.9$ and 2.36 TeV already show good reconstruction of the $E_{T}^{\rm miss}$ spectrum for low missing $E_{T}$ out to $E_{T}^{\rm miss}\sim 35$ GeV. As the experiments accumulate data, the reconstruction algorithms will be fully tested and refined, and soon $E_{T}^{\rm miss}$ should become a reliable variable for detecting SUSY events. With this in mind, we examine the SUSY reach of LHC7 including $E_{T}^{\rm miss}$ and also isolated electrons in the analysis, even for small integrated luminosities. Certainly by the time the integrated luminosity exceeds $\sim 0.5-1$ fb${}^{-1}$, we expect the detector to be very well understood, leading us to optimize the reach by looking simultaneously at various multi-jets and multi-lepton channels. As in Ref. [7], we define the signal to be observable if $S\geq max\left[5\sqrt{B},\ 5,\ 0.2B\right]$ where $S$ and $B$ are the expected number of signal and background events, respectively, for an assumed value of integrated luminosity. The requirement $S\geq 0.2B$ is imposed to avoid the possibility that a _small_ signal on top of a _large_ background could otherwise be regarded as statistically significant, but whose viability would require the background level to be known with exquisite precision in order to establish a discovery. Our optimization procedure selects the channel which maximizes $S/\sqrt{S+B}$, used as the figure of merit for the statistical significance of the signal. The grid of cuts used in our optimized analysis is: * $E_{T}^{\rm miss}>$ 100 - 1000 GeV (in steps of 100 GeV), * $n(jets)\geq$ 2, 3, 4, 5 or 6, * $n(b-jets)\geq$ 0, 1, 2 or 3, * $E_{T}(j_{1})>$ 50 - 300 GeV (in steps of 50 GeV) and 400-1000 GeV (in steps of 100 GeV) (jets are ordered $j_{1}-j_{n}$, from highest to lowest $E_{T}$), * $E_{T}(j_{2})>$ 50 - 200 GeV (in steps of 30 GeV) and 300, 400, 500 GeV, * $n(\ell)=$ 0, 1, 2, 3, OS, SS and inclusive channel: $n(\ell)\geq$ 0. (Here, $\ell=e,\ \mu$). * 10 GeV$\leq m(\ell^{+}\ell^{-})\leq 75$ GeV or $m(\ell^{+}\ell^{-})\geq 105$ GeV (for the OS, same flavor (SF) dileptons only), * transverse sphericity $S_{T}>0.2$. <figure><img src="content_image/1004.3594/x3.png"><figcaption>Figure 2: The optimized SUSY reach of LHC7 for different integratedluminosities combining the different channels described in the text. The fixedmSUGRA parameters are A0=0, tanβ=45 and μ>0. Gluino mass contours (dashed,dark grey) are shown by the dashed, dark grey curves. Higgs mass contours(dash-dotted purple) are also shown for mh=111 and 114 GeV. The shaded greyarea is excluded due to stau LSPs (left side of figure) or no electroweaksymmetry breaking (right side of figure), while the shaded grey area marked“LEP excluded” is excluded by non-observation of a sparticle signal from LEP2searches.</figcaption></figure> We show in Fig. 2 the optimized discovery reach of LHC7. We also show gluino isomass curves and the SM Higgs mass bound contours as obtained using the Isasugra routines in Isajet, together with contours of $m_{h}=111$ and 114 GeV. While limits from Higgs searches at LEP2 imply $m_{h}>114.4$ GeV for a SM-like Higgs boson, we also show the $m_{h}\sim 111$ GeV contour as a conservative indicator of the Higgs limit in the mSUGRA model to incorporate an approximate $\pm 3$ GeV uncertainty in the theoretical calculation of $m_{h}$. We see in Fig. 2 that with only 0.1 fb${}^{-1}$ of integrated luminosity, experiments at the LHC will be able to explore well beyond current Tevatron bounds, reaching $m_{\tilde{g}}\sim 800$ GeV for $m_{\tilde{q}}\simeq m_{\tilde{g}}$ in the low $m_{0}$ part of the figure. The precise reach will be determined by background levels in different channels (many of which will be able to be obtained directly from the data as discussed in Ref. [7]). The gluino mass reach for $m_{\tilde{g}}\sim m_{\tilde{q}}$ extends up to 950 (1100) ((1200)) GeV for 0.3 (1) ((2)) fb${}^{-1}$ of integrated luminosity, respectively! For heavy squarks (large $m_{0}$ region), the reach is still at the level of $m_{\tilde{g}}\simeq 540$ (650) ((700)) GeV for 0.3 (1) ((2)) fb${}^{-1}$. We emphasize here that the reach in Fig. 2 has been obtained at LO using the rates as given by Isajet. If instead, we scale the $\tilde{q}\tilde{q}+\tilde{q}\tilde{g}+\tilde{g}\tilde{g}$ cross section to its NLO value as given by Prospino [14] (the scaling factor varies between 1.3-2.5 depending on where we are in the plane), and scale the SM background cross sections where available to their NLO values using MCFM [17], the reach in $m_{1/2}$ is _increased_ by about 5% for low $m_{0}$ values, and by as much as 15-20% for high values of $m_{0}$. We have checked that if we also include fluctuations of the background using the procedure used by ATLAS [18], and include a 50% systematic uncertainty [19] that we add in quadrature to the statistical uncertainty of the background, the reach in $m_{1/2}$ is _reduced_ from its value in Fig. 2, the reduction being just a few percent for an integrated luminosity of 1 fb${}^{-1}$, and almost 25% for 100 pb${}^{-1}$ at low values of $m_{0}$. In Fig. 3, we show the optimized reach restricted to the $n(\ell)=0$, $n(b)\geq 0$ channel. We see that the $0\ell$ multi-jet + $E_{T}^{\rm miss}$ channel– which has the largest cross section of all the signal channel – essentially saturates the reach, except for tiny regions at large $m_{0}$ and integrated luminosities $\geq 1$ fb${}^{-1}$. While the greatest LHC reach occurs in the multijet$+E_{T}^{\rm miss}$ channel, it is important to note that even for very low integrated luminosities there should be a signal in several different channels if the new physics is supersymmetry as manifested by the mSUGRA model framework. With this in mind, in Fig. 4 we compare the 1 fb${}^{-1}$ optimized reaches in the $n(\ell)=1,\ OS,\ SS,\ 3\ell$ channels (all with $n(b)\geq 0$) against the $n(b)\geq 2$ channel (with $n(\ell)=0$). The presence of the multilepton channels not only will lend confidence that one is indeed seeing SUSY cascade decays, but also sparticle mass information may be extracted, _e.g_, the $m(\ell^{+}\ell^{-})$ mass edge [20, 7] conveys information on the $m_{\widetilde{Z}_{2}}-m_{\widetilde{Z}_{1}}$ mass difference, or on sleptons masses. <figure><img src="content_image/1004.3594/x4.png"><figcaption>Figure 3: The optimized SUSY reach of LHC7 with different integratedluminosities for the n(ℓ)=0, n(b)≥0 channel. The fixed mSUGRA parameters areA0=0, tanβ=45 and μ>0. Gluino mass contours (dashed, dark grey) are shown bythe dashed, dark grey curves. Higgs mass contours (dash-dotted purple) arealso shown for mh=111 and 114 GeV. The shaded grey area is excluded due tostau LSPs or no electroweak symmetry breaking, while the shaded area marked“LEP excluded” is excluded by direct LEP bounds on sparticle masses.</figcaption></figure> <figure><img src="content_image/1004.3594/x5.png"><figcaption>Figure 4: The optimized reach for 1 fb−1 restricted to mutileptons (n(ℓ)=1,OS, SS, 3ℓ, with n(b)≥0) or multi b-jets (n(b)≥2, with n(ℓ)=0) channels. Thefixed mSUGRA parameters are A0=0, tanβ=45 and μ>0. Gluino mass contours(dashed, dark grey) are shown by the dashed, dark grey curves. The shaded greyarea is excluded due to stau LSPs or no electroweak symmetry breaking, whilethe shaded area marked “LEP excluded” is excluded by direct LEP bounds onsparticle masses.</figcaption></figure> ### Identifying the light Higgs boson in SUSY cascade events at LHC7 We note that while discovery of SUSY particles may be possible during the first run of the LHC, detection of a SM-like Higgs boson using conventional production and decay modes will require much higher integrated luminosity, primarily because an observable signal occurs only via its sub-dominant decay modes. However, it is also possible to detect the lightest SUSY Higgs boson via its dominant $h\to b\bar{b}$ decay when it is produced via cascade decays of gluinos and squarks [21]. The idea is to produce $\tilde{g}$ and $\tilde{q}$ at a large rate, and look for $\tilde{q}\to q\widetilde{Z}_{2}$ or $\tilde{g}\to q\bar{q}\widetilde{Z}_{2}$ production followed by $\widetilde{Z}_{2}\rightarrow\widetilde{Z}_{1}h$ decay, in a $E_{T}^{\rm miss}$ event sample designed to pick our SUSY events over SM backgrounds. If $m_{\widetilde{Z}_{2}}>m_{\widetilde{Z}_{1}}+m_{h}$, then the latter decay mode becomes kinematically allowed and usually dominates the $\widetilde{Z}_{2}$ decay branching fractions. Then, one might search for a $b\bar{b}$ mass bump within the SUSY signal sample. As an example, we generate gluino and squark pair production events at the mSUGRA point $m_{0},\ m_{1/2},\ A_{0},\ \tan\beta,\ sign(\mu)=330\ {\rm GeV},\ 330\ {\rm GeV },\ 0,\ 10,\ (+)$, and apply the cuts: * $n(j)\geq 4$, $n(b)\geq 2$, $n(l)=0$, $pT(j_{1})>100$ GeV, $S_{T}>0.2$ and $E_{T}^{\rm miss}>250$ GeV For this set of cuts $t\bar{t}+jets$ is the dominant background, which is partially reduced by the isolated lepton veto. We construct the di-$b$-jet invariant mass of the two hardest $b$-jets, and plot the distribution in Fig. 5. The signal plus background is shown by the red histogram, while background is shown in blue. For these hard cuts the signal stands out above background, but for only 1 fb${}^{-1}$ of integrated luminosity, there would be only about 3 signal events in the peak region. However, as more events are gathered, gradually a signal should begin clustering in the vicinity of the Higgs mass. If the LHC7 run goes exceptionally well and 2-3 fb${}^{-1}$ of integrated luminosity is accrued, or if the data from ATLAS and CMS detectors can be effectively combined, then evidence for the Higgs in SUSY signal events might be found. For higher values of $m_{1/2}$ and $m_{0}$, the signal should decrease, and more integrated luminosity will be required. If $m_{1/2}$ is lowered, then the $\widetilde{Z}_{2}\rightarrow\widetilde{Z}_{1}h$ mode will close. There will then be no Higgs boson signal as $\widetilde{Z}_{2}$ instead decays via $\widetilde{Z}_{2}\rightarrow\widetilde{Z}_{1}Z$ or possibly $\widetilde{Z}_{2}\rightarrow\tilde{f}f$ ($f$ is a SM fermion) or via 3-body decay modes, leading to other signatures that may be searched for. <figure><img src="content_image/1004.3594/x6.png"><figcaption>Figure 5: Invariant mass of di-b-jet pair from SUSY plus BG events (redhistogram) and SM background, after cuts listed in the text, for the mSUGRApoint m0,m1/2,A0,tanβ,sign(μ)=330 GeV, 330 GeV, 0, 10, (+).</figcaption></figure> ## 4 Early SUSY discovery at $\sqrt{s}=7$ TeV without utilizing $E_{T}^{\rm miss}$ In previous analyses [5, 6, 7], it has been shown that even without utilizing $E_{T}^{\rm miss}$ and with an integrated luminosity of just $\sim 0.1$fb${}^{-1}$, experiments at LHC10 or LHC14 could detect SUSY signals in both the multimuon as well as in the acollinear dijet channels, for parameter regions beyond the reach of the Fermilab Tevatron. Our objective in this section is to check that this is still possible for the case of LHC7, and if so, delineate the portion of mSUGRA parameter space can be explored. ### Multilepton channels For early SUSY discovery using multiple isolated leptons in lieu of $E_{T}^{\rm miss}$, we use the following set of cuts: * $C_{\rm lep}$: * Jet cuts: $n(jets)\geq 4$ with $E_{T}(j_{1})\geq 100$ GeV, $E_{T}(j)\geq 50$ GeV, * $S_{T}\geq 0.2$, * $Z$-veto cuts: 10 GeV$\leq m(\ell^{+}\ell^{-})\leq 75$ GeV or $m(\ell^{+}\ell-)\geq 105$ GeV (for OS/SF dileptons only) We show results for the conservative case of $\ell=\mu$ only, as well as for the more optimistic case $\ell=e$ or $\mu$, to cover the likely possibility that electrons will also be identifiable in the early stage of LHC7. The _multi-lepton channel_ is further divided in opposite sign dileptons, same sign dileptons and trileptons. In Fig. 6, the LHC discovery reach for the _a_) OS dimuon, _b_) SS dimuon and _c_) trimuon signals with no $E_{T}^{\rm miss}$ cuts are shown by the colored shaded regions for 0.1, 0.33, 1 and 2 fb${}^{-1}$ of integrated luminosity. We have checked that the trimuon signal in frame _c_) is below the 5 event level for all but one scanned point located in the tiny orange triangle in the $m_{0}-m_{1/2}$ plane in the last frame of the figure, even for an integrated luminosity as high as 1 fb${}^{-1}$. Thus, unlike the situation at LHC10 [7] where the highest multimuon reach was obtained in the trimuon channel, _there is no reach in this channel at LHC7._ If reliable electron ID in jetty events is possible early in the LHC run and we can include isolated $e$s as well as $\mu$s, the signal in the trilepton channel is roughly eight times larger than with muons alone (assuming the same acceptance and detection efficiency for electrons and muons). In this case, the reach via trileptons again exceeds the reach for OS and SS dileptons for integrated luminosity values of $\sim$ 1 fb${}^{-1}$. The following other features from the figure are worth noting. 1. Due to the reduced cross-sections, there is no reach for 0.1 fb${}^{-1}$ in the multi-muon channels. As in the case of LHC10, the larger signal cross section for OS dimuons implies that the earliest reach is obtained in the OS dimuon channel, but the SS dimuon channel with its larger $S:B$ ratio, yields the greater reach (in $m_{\tilde{g}}$), which, at its maximum extends up to $m_{\tilde{g}}\sim 550$ GeV for $m_{\tilde{g}}\lesssim m_{\tilde{q}}$, with 1 fb${}^{-1}$ of integrated luminosity. After the $C_{\rm lep}$ cut, $t\bar{t}$ and $Z^{}/\gamma^{}(\to l\bar{l})$ are the main SM backgrounds for OS dileptons, while the SS dilepton background is dominated by $t\bar{t}$ only. 2. When electrons are included in the multilepton channels, the reach increases considerably, with a tiny region of parameter space being accessible even for 0.1 fb${}^{-1}$ of integrated luminosity. The large increase in the trilepton channel (due to the inclusion of electrons) and its tiny background (dominated by $t\bar{t}$ and $t\bar{t}Z$) makes this the best channel for larger integrated luminosities. At the 1 fb${}^{-1}$ level, the reach extends up to $m_{\tilde{g}}\sim 680$ GeV for $m_{\tilde{g}}\lesssim m_{\tilde{q}}$. Also, larger values of $m_{0}$ become accessible. 3. While the reach in the OS and SS channels (both for dimuons and dileptons) are background limited, the trilepton reach is limited by its signal cross-section, with a total background $\lesssim 0.5$ fb. <figure><img src="content_image/1004.3594/x7.png"><figcaption>Figure 6: SUSY reach of the LHC at √s=7 TeV for different luminosities via a)OS-dimuon (dilepton) events, b) SS-dimuon (dilepton) and a) trimuon(trilepton) events using the cuts Clep for l=μ (l=μ,e) introduced in the text.The fixed mSUGRA parameters are A0=0, tanβ=45 and μ>0. Gluino mass contours(dashed, dark grey) are shown by the dashed, dark grey curves. Higgs masscontours (dash-dotted purple) are also shown for mh=111 and 114 GeV. Theshaded grey area is excluded due to stau LSPs or no electroweak symmetrybreaking, while the shaded area marked “LEP excluded” is excluded by directLEP bounds on sparticle masses.</figcaption></figure> ### Acollinear dijet channel The discovery potential of the acollinear dijet channel, suggested as a discovery mode in Ref. [6], is shown in Fig. 7. We adopt the set of cuts: * $C_{\rm dijet}$: * $n(jets)=2$, * $E_{T}(j)\geq 50$ GeV, * $E_{T}(j_{1})+E_{T}(j_{2})\geq 650$ GeV, * $\alpha\equiv E_{T}(j_{2})/m(j_{1}j_{2})>0.1$, * $\Delta\phi(j_{1},j_{2})<2.4$, * number of isolated leptons $n(\ell)=0$.² [FOOTNOTE:2][ENDFOOTNOTE] As expected, this channel is most effective at low $m_{0}$ where $\tilde{q}_{R}$ decays mainly via $\tilde{q}_{R}\to q\widetilde{Z}_{1}$. The signal rapidly degrades as $m_{0}$ increases, where squarks and gluinos then decay to multiple jets and/or leptons via SUSY cascades decays [22]. The reach extends up to $m_{\tilde{g}}\sim 900$ GeV for 1 fb${}^{-1}$ and low values of $m_{0}$. As at LHC10 [7], this channel complements the multi-lepton channel in that for small $m_{0}$, the dijet reach extends to larger values of $m_{1/2}$ whereas the multilepton channel probes larger values of $m_{0}$. <figure><img src="content_image/1004.3594/x10.png"><figcaption>Figure 7: SUSY reach of the LHC at √s=7 TeV for different luminosities viathe dijet channel using the cuts Cdijet. The fixed mSUGRA parameters are A0=0,tanβ=45 and μ>0. Gluino mass contours (dashed, dark grey) are shown by thedashed, dark grey curves. Higgs mass contours (dash-dotted purple) are alsoshown for mh=111 and 114 GeV. The shaded grey area is excluded due to stauLSPs or no electroweak symmetry breaking, while the shaded area marked “LEPexcluded” is excluded by direct LEP bounds on sparticle masses.</figcaption></figure> ## 5 Summary and conclusions With the first $pp$ collisions at $\sqrt{s}=7$ TeV, the era of LHC exploration of the TeV energy scale has begun. In this paper, we have calculated the LHC7 reach for supersymmetric particles assuming an integrated luminosity in the vicinity of $\sim 1$ fb${}^{-1}$. The good agreement that the CMS and Atlas collaborations find [8, 9] between Monte Carlo simulations and the very early LHC data at $\sqrt{s}=0.9$ and 2.36 TeV indicates that analyses including reliable $E_{T}^{\rm miss}$ resolution as well as electron ID may be viable very early. Our main result is shown in Fig. 2: we find that with just $\sim 1$ fb${}^{-1}$ of data – as anticipated in the first run of the LHC – gluinos up to 1.1 TeV (650 GeV) should be accessible if $m_{\tilde{q}}\sim m_{\tilde{g}}$ ($m_{\tilde{q}}\gg m_{\tilde{g}}$). Such a large reach for SUSY, even with half the design energy and very low integrated luminosity, illustrates the sheer discovery power of a three-and-a-half fold increase of the CM energy of the LHC over the Tevatron. Our results are succintly summarized in Table 2 where we show the optimized reach of the LHC at $\sqrt{s}=7$ TeV and also at its design energy of 14 TeV, taking $m_{\tilde{q}}\sim m_{\tilde{g}}$. While the current plan is to ramp the energy to 14 TeV after the machine upgrade following the first run, it is entirely possible that the LHC may have to be run at a lower energy of 10-13 TeV if the required training of the magnets cannot be completed during the shutdown. To facilitate the interpolation of the LHC SUSY reach at these slightly reduced energies, we have also included the reach of LHC10 from Ref. [7] in Table 2. \| 0.1 fb−1 \| 0.33 fb−1 \| 1 fb−1 \| 2 fb−1 ---\|---\|---\|---\|--- √s= 7 TeV \| 0.8 TeV \| 0.9 TeV \| 1.1 TeV \| 1.2 TeV √s= 10 TeV \| 1.0 TeV \| 1.1 TeV \| 1.4 TeV \| 1.5 TeV √s= 14 TeV \| 1.3 TeV \| 1.6 TeV \| 1.8 TeV \| 2.0 TeV Table 2: The optimized SUSY reach of the LHC within the mSUGRA model expressed in terms of the gluino mass for integrated luminosity values of 0.1, 0.33, 1 and 2 fb−1 at √s= 7 TeV, 10 TeV and 14 TeV, assuming m~q∼m~g. The results for 10 and 14 TeV are obtained from Ref. [7] Ultimately, the proper utilization of $E_{T}^{\rm miss}$ in SUSY searches will require an understanding of the high energy tail of its distribution at values well beyond where reconstruction algorithms have been tested (even allowing for the scaling with the increased CM energy to 7 TeV). Taking a conservative view that it may well take time (and data) before detectors are understood well enough for $E_{T}^{\rm miss}$ analyses to be reliably performed, we have also shown the LHC7 reach using mutimuons, multileptons and dijets channels, with no $E_{T}^{\rm miss}$ cuts. In this case, the LHC7 reach is of course more limited, but still substantial: it extends up to $m_{\tilde{g}}\sim 550$ GeV (680 GeV) in the dimuon (dilepton) channel for 1 fb${}^{-1}$ of integrated luminosity and, even if squarks are very heavy, up to 500-600 GeV in the trilepton channel. In the case where $m_{\tilde{q}}\sim m_{\tilde{g}}$, the LHC7 reach in the acollinear dijet channel, extends to $m_{\tilde{g}}\sim 900$ GeV for 1 fb${}^{-1}$. To conclude, the long-awaited search for physics beyond the SM has begun in earnest at the LHC. Although the machine is operating at just half its design energy, at least within the context of discovery of squarks and gluinos of supersymmetry, LHC experiments in their first run should be able to probe far beyond current limits whether or not reliable $E_{T}^{\rm miss}$ determination or electron ID is available. If, as it appears, $E_{T}^{\rm miss}$ can be reliably used early on in LHC analyses, experiments should be able to access SUSY gluinos and squarks as heavy as $\sim 1$ TeV with just 1 fb${}^{-1}$ of data, for the case of comparable sparticle masses. ###### Acknowledgments. We thank Graham Ross for urging us to perform this study. We thank Michael Schmitt and Sridhar Dasu for helpful discussions. We thank M. Mangano for helpful comments on the MLM matching algorithm. We also thank JoAnne Hewett for urging us to consider how the systematic error on the data-driven backgroud estimates would affect the reach. 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1411.4052	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 99135, "num_imgs": 12, "llama3_tokens_count": 30331 }	[ "content_image/1411.4052/x1.png", "content_image/1411.4052/x3.png", "content_image/1411.4052/x4.png", "content_image/1411.4052/x7.png", "content_image/1411.4052/x8.png", "content_image/1411.4052/x10.png", "content_image/1411.4052/x11.png", "content_image/1411.4052/x13.png", "content_image/1411.4052/x16.png", "content_image/1411.4052/x17.png", "content_image/1411.4052/x19.png", "content_image/1411.4052/x20.png" ]	# On the signature of the baryon-dark matter relative velocity in the two and three-point galaxy correlation functions Zachary Slepian and Daniel J. Eisenstein Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 E-mail: zslepian@cfa.harvard.eduE-mail: deisenstein@cfa.harvard.edu ###### Abstract We develop a configuration-space picture of the relative velocity between baryons and dark matter that clearly explains how it can shift the BAO scale in the galaxy-galaxy correlation function. The shift occurs because the relative velocity is non-zero only within the sound horizon and thus adds to the correlation function asymmetrically about the BAO peak. We further show that in configuration space the relative velocity has a localized, distinctive signature in the three-point galaxy correlation function (3PCF). In particular, we find that a multipole decomposition is a favorable way to isolate the relative velocity in the 3PCF, and that there is a strong signature in the $l=1$ multipole for triangles with 2 sides around the BAO scale. Finally, we investigate a further compression of the 3PCF to a function of only one triangle side that preserves the localized nature of the relative velocity signature while also nicely separating linear from non-linear bias. We expect that this scheme will substantially lessen the computational burden of finding the relative velocity in the 3PCF. The relative velocity’s 3PCF signature can be used to correct the shift induced in the galaxy-galaxy correlation function so that no systematic error due to this effect is introduced into the BAO as used for precision cosmology. ## 1 Introduction The baryon acoustic oscillation (BAO) method uses the imprint of sound waves in the early Universe on the clustering of galaxies today as a sensitive probe of the Universe’s expansion history (see Weinberg et al. 2013 for a recent review). This in turn constrains the dark energy equation of state, which offers insight into dark energy’s fundamental nature (Albrecht et al. 2006 for a review; Copeland et al. 2006; Li et al. 2011 for model compendia; for recent work on specific models, see e.g. Dutta & Scherrer 2008; Chiba 2009; Chiba et al. 2009; Gott & Slepian 2011; De Boni et al. 2011; Chiba et al. 2013; Slepian et al. 2014). The BAO method’s accuracy depends on precisely modeling how the sound waves frozen in at high redshift imprint on galaxy clustering today, and hence how baryons and DM combine to form these galaxies. A potentially important effect on early generations of galaxies is the supersonic relative velocity between baryons and DM at decoupling ($z\sim 1020)$, recently presented by Tseliakhovich & Hirata (2010). The relative velocity is sourced by the difference in the behavior of baryons and dark matter before decoupling. Prior to decoupling, the baryons and photons form a tightly coupled fluid, locked together by Thomson scattering (linking electrons to photons) and the Coulomb force (linking protons to electrons). This fluid undergoes acoustic oscillations, or sound waves, that propagate to roughly 150 Mpc comoving before halting as electron-photon scattering drops precipitously and decoupling occurs (Peebles & Yu 1970; Sunyaev & Zel’dovich 1970; Bond & Efstathiou 1984, 1987; Holtzmann 1989; Hu & Sugiyama 1996; Eisenstein & Hu 1998). The scale at which these waves halt is termed the sound horizon. Given an isolated overdense region, baryons nearer to it than the sound horizon are kept in rough hydrostatic equilibrium by the radiation pressure and so do not infall. In contrast, baryons more distant than the sound horizon fall towards the overdensity. Meanwhile, DM on _all_ scales infalls gravitationally. Consequently, the baryons and DM differ in behavior below the sound horizon, resulting in a relative velocity at decoupling on these scales. It is believed that the relative velocity can modulate the formation of the first galaxies in the Universe on scales similar to the sound horizon (we describe this more below). Since these galaxies are the progenitors of those we observe today, galaxy clustering today may retain a memory of this effect. Yoo et al. (2011) analyze how such a memory might cause a shift in the galaxy-galaxy correlation function by which the sound horizon scale today is measured, an idea also hinted at in Tseliakhovich & Hirata 2010 and Dalal et al. 2010. Given the high precision of current and impending BAO surveys such as BOSS, even a modest, order $1\%$ systematic source of error could significantly bias the inferred cosmological parameters. Therefore it is essential to understand how the relative velocity can induce this shift and how, if the shift is indeed present, it can be corrected. While Yoo et al. (2011) state that the correlation function can shift, their analysis presents results in Fourier space, showing the power spectrum and, importantly, finding that the bispectrum can be used to remove the relative velocity effect from the power spectrum. Here, we focus on configuration space, for several reasons. First, complicated behavior in the power spectrum and bispectrum often has a simple interpretation in configuration space (Bashinsky & Bertschinger 2001, 2002). Our work shows that the relative velocity is indeed simple in configuration space: it is non-zero only within the sound horizon. Our work therefore makes it clear that any effect on the correlation function is primarily on sub-horizon scales. It is adding or subtracting from the correlation function only inward of the BAO peak that shifts the peak in or out in scale. Our configuration space approach also offers the new result that, for extracting the relative velocity from the three-point galaxy correlation function (3PCF), Legendre polynomials are an excellent angular basis (Szapudi 2004 and Pan & Szapudi 2005 first suggested such a basis for general measurements of the bispectrum). Because the relative velocity has compact support in configuration space, we can additionally integrate over one side-length of the triangles entering the 3PCF for lengths where the relative velocity has support. This produces a novel compression scheme that improves the chances for detecting the relative velocity in the 3PCF while easing the computational demands of such an effort. Indeed, this compression scheme is not the only practical advantage of a configuration space approach. The bispectrum is challenging to measure accurately on a cut sky, because survey boundaries break the translational symmetry implicit in a Fourier decomposition. They also impose some miminum wavenumber below which the Fourier representation is truncated, leading to Gibbs phenomenon ringing in the bispectrum. In contrast, in configuration space, the 3PCF can be measured straightforwardly and cut-sky effects corrected by use of an estimator (see e.g. Kayo et al. 2004; Szapudi 2004; Szapudi and Szalay 1998), though at some computational cost (McBride et al. 2011; Marin et al. 2013). Indeed, Pan and Szapudi (2005) have already measured the monopole moment of the 3PCF in Two-degree-Field Galaxy Redshift Survey (2dFGRS), showing the feasibility of this approach. In the remainder of the Introduction, we give greater detail on the physical mechanisms by which the relative velocity may affect galaxy clustering today: how does the relative velocity effect the first galaxies to form, and how might galaxies today retain a memory of these distant progenitors? The relative velocity affects the formation of early, low-mass haloes, but precisely how remains an open question. In their initial paper presenting the relative velocity, Tseliakhovich & Hirata (2010) predict suppression of halos with $M\lesssim 10^{6}\;M_{\odot}$ due to the relative velocity. Naoz et al. (2012) find this in simulations as well, though simulations by Richardson et al. (2013) find only a small effect in halo number density by $z\sim 20$. In those halos that do form, the gas content is lowered (Dalal et al. 2010; Tseliakhovich et al. 2011; Fialkov et al. 2012; Naoz et al. 2013). In simulations, Maio et al. (2011) find that star formation in low mass halos is suppressed, though Stacy et al. (2011) argue that the later-time star formation is not strongly affected. The minimum cooling mass for star formation via molecular hydrogen lines also may be raised in simulations (Greif et al. 2011, though Stacy et al.’s earlier work argues it is not). O’Leary and McQuinn (2012) simulate structure formation to show that the relative velocity has a substantial effect on the first mini-halos’ accretion history. Barkana (2013) points out that there may be additional dynamical effects, such as asymmetric disruption of accreting gas filaments and formation of supersonic wakes by halos moving in regions of high relative velocity. Bovy & Dvorkin (2013) suggest that for reasons such as these, star formation in small DM halos may be suppressed enough to resolve the over-prediction of small halos in $\Lambda{\rm CDM}$ simulations. It is believed that the modulation of early, low-mass halos by the relative velocity as discussed above will affect the subsequent formation of the higher-mass halos we observe today, perhaps through feedback channels such as altering the metal abundance or supernovae rate (Yoo et al. 2011). Since these links are not known in detail, it is simply assumed that the relative velocity biases the galaxy overdensity with some amplitude $b_{v}$, to be fit from the data. This will be our approach here as well. Finally, numerous studies have also developed rich small-scale consequences of the relative velocity, though that will not be our focus here. To give only a few examples, Naoz & Narayan (2013) show that the relative velocity modifies the Biermann battery picture of magnetogenesis, while Tanaka et al. (2013), Tanaka & Li (2014), and Latif et al. (2014) consider the impact on primordial supermassive black hole formation. Much work has also investigated the consequences of the relative velocity for the 21 cm radiation field, e.g. Visbal et al. (2012); McQuinn & O’Leary (2012). A detailed recent review of work on the relative velocity is Fialkov (2014). This paper is structured as follows. In §2, we lay out our approach and assumptions. In §3, we present the structure of the relative velocity due to a point perturbation (the Green’s function), and in §4 we compute the shift the relative velocity induces in the correlation function. §5 discusses this shift and shows how it can be traced back to the compact support of the Green’s function. §6 presents the 3PCF at one vertex of a triangle of galaxies, and §7 connects this with the sum over all vertices that we observe and shows how the 3PCF may be compressed to maximize the signal. §8 concludes. An Appendix presents mathematical results that we use in the paper to accelerate the numerical calculations of §4. ## 2 Approach and assumptions We begin by presenting our bias model and then outline how the spatial structure of the relative velocity (which this model requires as an input) can be found using a Green’s function approach. Throughout this paper, we use linear perturbation theory in configuration space and neglect redshift-space distortions. We model the low-redshift galaxy overdensity, denoted by $\delta_{{\rm g}}$, as being biased by the square of the relative velocity normalized by its mean value, following Yoo et al. (2011). We then use perturbation theory to compute the correlation function and 3PCF (§4 & §6 respectively). Writing the relative velocity as $\vec{v}_{{\rm bc}}=\vec{v}_{{\rm b}}-\vec{v}_{{\rm c}}$ (baryon velocity minus dark matter velocity) and $\sigma_{{\rm bc}}=\left<\|\vec{v}_{{\rm bc}}\|^{2}\right>^{1/2}$ (the root mean square value), we define the dimensionless $v_{{\rm s}}^{2}=\|\vec{v}_{{\rm bc}}\|^{2}/\sigma_{{\rm bc}}^{2}$ and expand the galaxy overdensity in $v_{{\rm s}}^{2}-1$ to ensure $\left<\delta_{{\rm g}}\right>=0$. Mathematically, this is \[\delta_{{\rm g}}\left(\vec{r}\right)=\delta_{{\rm g},b_{v}=0}\left(\vec{r} \right)+b_{v}\left[v_{{\rm s}}^{2}\left(\vec{r}\right)-1\right],\] (1) where \[\delta_{{\rm g},b_{v}=0}\left(\vec{r}\right)=b_{1}\delta_{{\rm m}}\left(\vec{r }\right)+b_{2}\left[\delta_{{\rm m}}^{2}\left(\vec{r}\right)-\left<\delta_{{ \rm m}}^{2}\right>\right]\] (2) captures the standard perturbation theory linear and non-linear bias in the galaxy overdensity.¹$b_{v}$ in equation (1) is an unknown bias coefficient that, as discussed in $\lx@sectionsign 1$, encodes how strongly the relative velocity affects galaxy formation. $\delta_{{\rm m}}$ is the matter overdensity. [FOOTNOTE:1][ENDFOOTNOTE] We next outline how we compute the spatial structure of the relative velocity, a required input in our bias model (1). Since the true primordial density field at a given location is not known $a$$priori$, we need to be able to compute the $v_{s}^{2}$ generated by an arbitrary density field.² We therefore find the relative velocity due to a point perturbation and then integrate it against the true density field. Though this latter is not known $a$$priori$, its statistical properties are. As we will only be considering expectation values over $v_{s}^{2}$, and thus over the density field, this is sufficient. [FOOTNOTE:2][ENDFOOTNOTE] Since the response to an impulse is called the Green’s function, we denote the relative velocity due to a point perturbation by $\vec{v}_{{\rm G}}=v_{{\rm G}}\hat{r}$. By symmetry, it must always point radially outward from the density point sourcing it. It points outward because DM infalls under gravity and baryons are static or pushed outwards by radiation pressure. We now define the Green’s function implicitly: \[\vec{v}_{{\rm bc}}\left(\vec{r},z\right)=\int v_{{\rm G}}\left(r_{1},z\right) \delta_{{\rm pri}}\left(\vec{r}+\vec{r}_{1}\right)\hat{r}_{1}d^{3}\vec{r}_{1}.\] (3) To linear order, the relative velocity at redshift $z$ due to a primordial density field $\delta_{{\rm pri}}\left(\vec{r}\right)$ is found by integrating $\delta_{{\rm pri}}$ against the Green’s function. Isotropy demands that $\left<\vec{v}_{{\rm bc}}\right>=0$. Notice that $\vec{v}_{{\rm bc}}$ represents the dipole moment of the density field weighted by $v_{{\rm G}}$, suggesting that multipole expansions will be natural moving forwards. Figure 1 portrays schematically the use of the Green’s function to compute the relative velocity (and its square) due to an arbitary density field. The Green’s function formalism above makes it evident that in our bias model, the square of the relative velocity contributes to the correlation function only beginning at fourth order in the perturbed quantities. One overdensity is required to source a relative velocity field, as shown in the lefthand panel of Figure 1, and to produce $v_{{\rm bc}}^{2}$ (equivalently, $v_{{\rm s}}^{2}),$ two overdensities are needed, as shown in the righthand panel of Figure 1. For a Gaussian random field, all odd moments vanish, meaning the velocity contributes to the correlation function beginning only at fourth order. To obtain all of the fourth order contributions to the correlation function, we must expand $\delta_{{\rm m}}$ to second order in equation (2): \[\delta_{{\rm m}}\left(\vec{r}\right)=\delta\left(\vec{r}\right)+\delta^{\left( 2\right)}\left(\vec{r}\right).\] (4) Here and throughout, $\delta$ is the linear density field while $\delta^{\left(2\right)}$ is the second-order density field, which is $\mathcal{O}\left(\delta^{2}\right)$. <figure><img src="content_image/1411.4052/x1.png"><figcaption>Figure 1: Illustrations of the computation of the relative velocity field andits square norm. The X is a “dummy” density point to be integrated over. Theleft panel shows how →vbc is evaluated at the dot by integrating the densityfield over all space weighted by the Green’s function vG. The darker shadingindicates that r2vG peaks at rs, a result discussed further in Figure 2. →v2bc(right panel) is evaluated analogously but using two copies of the densityfield each weighted by the Green’s function. In practice, because of theGreen’s function’s structure (see Figure 2), only density points within ∼rs ofthe dot significantly contribute to the relative velocity there.</figcaption></figure> Finally, we close this section with four further points about our perturbation theory framework. First, we note a subtlety of our bias model. $\left<v_{s}^{2}\right>=1$ and so if we carried through the expansion of the galaxy overdensity to higher orders in $v_{s}^{2},$ we would expect these terms to contribute to the correlation function at order unity times some combinatoric factor. For instance, in the limit that $\xi\to\delta_{{\rm D}}^{\left[3\right]}$ (a 3-D Dirac delta function) one can compute explicitly that a term $\left<v_{s}^{4}\left(0\right)\delta^{2}\left(\vec{r}\right)\right>\approx 4$ appears in $\xi_{{\rm gg}}\left(\vec{r}\right)$ if equation (1) is taken out to $v_{s}^{4}$. Since there are potentially an arbitrary number of these terms, one might ask if our expansion converges. However, physically, it is likely that the dimensionless parameter of importance for galaxy formation is $\|\vec{v}_{{\rm bc}}\|^{2}/\sigma_{{\rm g}}^{2},$ where $\sigma_{{\rm g}}$ is some unknown, redshift-dependent circular velocity or velocity dispersion for a typical galaxy. We expect that $\left<\|\vec{v}_{{\rm bc}}\|^{2}/\sigma_{{\rm g}}^{2}\right>\ll 1,$ so that an expansion in powers of this quantity would converge. Our expansion, now with the coefficient of $v_{s}^{2n}$ labeled by $b_{vn}$, may be rewritten in terms of $\|\vec{v}_{{\rm bc}}\|^{2}/\sigma_{{\rm g}}^{2}$ by taking (5) The $b_{vn\sigma_{\rm g}}$ are coefficients of an expansion in terms of powers of $\|\vec{v}_{{\rm bc}}\|^{2}/\sigma_{{\rm g}}^{2}$ and are assumed to be all intrinsically on the same order of magnitude. Solving for $b_{vn}$ shows that it must fall rapidly with $n$ and our expansion converges. Second, we justify the use of linear perturbation theory. Though perturbation theory does not provide highly accurate fits to simulation results on small scales ($\lesssim 20\;{\rm Mpc}$), the large scales ($\sim 150\;{\rm Mpc}$) relevant for the BAO have remained roughly linear down to the present day (for discussion of non-linear effects, see Smith et al. 2003; Seo et al. 2008; Sherwin & Zaldarriaga 2012, though see also Roukema et al. 2014). The primary effect of what non-linear evolution has occurred is to broaden the BAO peak in the galaxy-galaxy correlation function, not to shift its center. As Eisenstein et al. (2007a) show, the peak position is robust in configuration space. Further, modern BAO surveys (e.g. Anderson et al. 2014) use reconstruction to compute the peculiar velocity field implied by a given density field and reverse it, thus allowing analysis to be performed on a density field that is linear to even better approximation (Eisenstein et al. 2007b; Seo et al. 2008). These considerations justify our use of linear perturbation theory to compute how the relative velocity effect shifts the BAO peak. It is unambiguous to calculate the lowest order change in the correlation function and 3PCF the velocity produces. As we discuss above, higher order corrections should quickly become negligible. One can debate the precise details of the no-velocity correlation function, but the addition from the velocity to any model chosen can be accurately computed in perturbation theory. Third, when computing the expansion of the density field to second order, we only consider effects generated by gravity. For example, we neglect effects due to couplings of radiation and matter. Naoz & Barkana (2005) point out that on small scales, the sound speed varies spatially after recombination due to density-dependent Compton heating (see also Naoz et al. (2011)). This will not affect our conclusions because the BAO scale is dominated by the relativistic sound speed prior to decoupling. Another effect generated by coupling of radiation and matter is the impact of inhomogeneities in the intergalactic medium on the Lyman-$\alpha$ emission observed from galaxies (Wyithe & Djikstra 2011). This can create an additional clustering signal on large scales that would need to be properly accounted for if one wished to use the techniques in this work on a Lyman-$\alpha$ selected galaxy sample such as might be found using the Hobby-Eberly Telescope Dark Energy Experiment (HETDEX). Finally, we consider the effects of redshift-space distortions (see Hamilton 1998 for a review). Peculiar velocities systematically alter galaxy clustering even on large scales (Kaiser 1987; Bernardeau et al. 2002), introducing a strong directional dependence. These distortions do not substantially alter the Green’s function picture, because galaxy positions in redshift space are shifted by much less than the acoustic scale. However, the distortions can alter the resulting correlations because the true large-scale correlations are also small. These effects can be accurately treated in cosmological simulations, and we expect that studies of observational data would want to compare to full simulations. However, we note that our analysis will average over triangles irrespective of their orientation to the line of sight. While not optimal as regards information content, such averages do tend to reduce the effects of redshift distortions on large scales. For example, the redshift-space spherically averaged two-point correlation function on large scales is primarily a rescaling of the real-space result, with a mild extra broadening of the acoustic peak. We similarly expect that our orientation-averaged 3PCF results will be only mildly changed by redshift-space distortions. Furthermore, the reconstruction of the linear density field discussed above can also correct redshift-space distortions on the scales relevant for this work by introducing a factor $1+f$, where $f=d(\ln D)/d(\ln a)$, $D$ is the linear growth function, and $a$ is the scale factor. This factor represents the additional squashing along the line-of-sight (Eisenstein et al. 2007b). This technique should allow removal of the redshift-space distortions on the large scales most significant for the signature presented here. ## 3 Deriving the Green’s function We now seek to obtain an explicit expression for the Green’s function. We begin with the linear theory continuity equation in configuration space. $a$ is the scale factor and we use comoving positions and velocities. An overdot denotes a derivative with respect to time. We have \[a^{-1}\nabla\cdot\vec{v}\left(\vec{r}\right)=-\dot{\delta}\left(\vec{r}\right),\] (6) which Fourier transforms to \[-ia^{-1}\vec{k}\cdot\tilde{\vec{v}}(\vec{k})=-\dot{\tilde{\delta}}(\vec{k}),\] (7) where a tilde denotes a 3-D Fourier transform given by \[\tilde{f}(\vec{k})=\int d^{3}\vec{r}f\left(\vec{r}\right)e^{i\vec{k}\cdot\vec{ r}}\] (8) with inverse transform \[f\left(\vec{r}\right)=\int\frac{d^{3}\vec{k}}{\left(2\pi\right)^{3}}\tilde{f}( \vec{k})e^{-i\vec{k}\cdot\vec{r}}.\] (9) For growing modes, $\tilde{\vec{v}}$ is parallel to $\vec{k}$ so \[\tilde{\vec{v}}(\vec{k})=-\frac{i\vec{k}}{k^{2}}H\left(z\right)\frac{\partial \tilde{\delta}(\vec{k})}{\partial z}.\] (10) This means \[\tilde{\vec{v}}_{{\rm bc}}(\vec{k},z)=-\frac{i\vec{k}}{k^{2}}H\left(z\right) \frac{\partial(\tilde{\delta}_{{\rm b}}-\tilde{\delta}_{{\rm c}})}{\partial z} =-iT_{{\rm vbc}}\left(k,z\right)\tilde{\delta}_{{\rm pri}}(\vec{k})\hat{k},\] (11) where subscript c is for CDM, b for baryons, vbc for relative velocity, and we define the relative velocity transfer function \[T_{{\rm vbc}}\left(k,z\right)=\frac{H\left(z\right)}{k}\frac{\partial}{ \partial z}\left[T_{{\rm b}}\left(k,z\right)-T_{{\rm c}}\left(k,z\right)\right].\] (12) $T_{{\rm b}}$ and $T_{{\rm c}}$ are the baryon and CDM transfer functions, which give the evolution of each mode with redshift via \[\tilde{\delta}_{{\rm b}}(\vec{k},z)=T_{{\rm b}}\left(k,z\right)\tilde{\delta}_ {{\rm pri}}(\vec{k}),\;\tilde{\delta}_{{\rm c}}(\vec{k},z)=T_{{\rm c}}\left(k, z\right)\tilde{\delta}_{{\rm pri}}(\vec{k}).\] (13) $\tilde{\delta}_{{\rm pri}}$ is the primordial density perturbation related to the primordial power spectrum $P_{\rm pri}$ by $\left<\tilde{\delta}_{{\rm pri}}\left(\vec{k}\right)\tilde{\delta}_{{\rm pri}} ^{}\left(\vec{k}^{\prime}\right)\right>=\left(2\pi\right)^{3}\delta_{{\rm D}} ^{\left[3\right]}\left(\vec{k}-\vec{k^{\prime}}\right)P_{{\rm pri}}\left(k\right)$, with $P_{{\rm pri}}=Ak^{n_{s}}$.³ We emphasize that the relative velocity transfer function maps the primordial density field to a velocity field at some redshift $z,$ so $T_{{\rm vbc}}$ always acts on $\tilde{\delta}_{{\rm pri}}$. [FOOTNOTE:3][ENDFOOTNOTE] We now obtain the configuration space Green’s function, defined implicitly by equation (3). Using the Fourier representation of $\vec{v}_{{\rm bc}}$ (11) we have \[\vec{v}_{{\rm bc}}\left(\vec{r},z\right) =\int\frac{d^{3}\vec{k}}{\left(2\pi\right)^{3}}e^{-i\vec{k}\cdot \vec{r}}\left[-iT_{{\rm vbc}}\left(k,z\right)\tilde{\delta}_{{\rm pri}}\left( \vec{k}\right)\hat{k}\right]\] (14) \[=\int d^{3}\vec{r}_{1}\hat{r}_{1}v_{{\rm G}}\left(r_{1},z\right) \delta_{{\rm pri}}\left(\vec{r}+\vec{r}_{1}\right).\] We then rewrite $\tilde{\delta}_{{\rm pri}}\left(\vec{k}\right)=\int d^{3}\vec{q}e^{i\vec{k} \cdot\vec{q}}\delta_{{\rm pri}}\left(\vec{q}\right)$ to find \[\int d^{3}\vec{q}\delta_{{\rm pri}}\left(\vec{q}\right)\int\frac{ d^{3}\vec{k}}{\left(2\pi\right)^{3}}e^{i\vec{k}\cdot\vec{q}-i\vec{k}\cdot\vec{ r}}\left[-iT_{{\rm vbc}}\left(k,z\right)\hat{k}\right]\] (15) \[=\int d^{3}\vec{r}_{1}\hat{r}_{1}v_{{\rm G}}\left(r_{1},z\right) \delta_{{\rm pri}}\left(\vec{r}+\vec{r}_{1}\right).\] Changing variables on the left-hand side via $\vec{q}=\vec{r}-\vec{r}_{1}$ and then equating the resulting integrands over $d^{3}\vec{r}_{1}$, we have \[v_{{\rm G}}\left(r_{1},z\right)\hat{r}_{1}=\int\frac{d^{3}\vec{k}}{\left(2\pi \right)^{3}}e^{-i\vec{k}\cdot\vec{r}_{1}}\left[iT_{{\rm vbc}}\left(k,z\right) \hat{k}\right],\] (16) which, projecting onto $\hat{r}_{1}$, results in \[v_{{\rm G}}\left(r_{1},z\right) =\int\frac{k^{2}dk}{2\pi^{2}}\int_{-1}^{1}\frac{d\mu}{2}\mu e^{- ikr_{1}\mu}\left[iT_{{\rm vbc}}\left(k,z\right)\right]\] (17) \[=\int\frac{k^{2}dk}{2\pi^{2}}j_{1}\left(kr_{1}\right)T_{{\rm vbc} }\left(k,z\right).\] Above, $\mu=\hat{k}\cdot\hat{r}_{1}$ and $j_{1}$ is the spherical Bessel function of order one. We now have the desired velocity Green’s function. Noting that its Fourier transform is closely related to the velocity transfer function, for notational consistency we define \[\tilde{v}_{{\rm G}}\left(k,z\right)=\int 4\pi r^{2}drj_{1}\left(kr\right)v_{{ \rm G}}\left(r,z\right)=T_{{\rm vbc}}\left(k,z\right)\] (18) and use $\tilde{v}_{\rm G}$ going forward. In practice, we compute $v_{\rm G}$ by transforming $\tilde{v}_{\rm G}$ using equation (17). Thus we must first compute $\tilde{v}_{{\rm G}}$. Using a flat $\Lambda CDM$ cosmology with $\Omega_{{\rm b}}h^{2}=0.0226$, $\Omega_{{\rm c}}h^{2}=0.112$, $n_{s}=0.96$, and $H_{0}=70\;{\rm km/s/Mpc}$, we output transfer functions $T_{{\rm b}}$ and $T_{{\rm c}}$ from CAMB (Lewis 2000) on a grid equally spaced in $\log k$ with 5,000 divisions per decade from $k=6.95\times 10^{-5}$ to $10.50$. To approximate $\partial T_{{\rm b}}/\partial z$ and $\partial T_{{\rm c}}/\partial z$ (cf. equation (12)), we discretize the derivative at each redshift $z$ with $\Delta z=0.10z$. To avoid ringing due to the finiteness of our grid in Fourier space, we use a smoothing $\exp\left[-k^{2}\right]$ to evaluate the integral (17) (and all analogous integrals over $dk$ in what follows). Figure 2 shows the Green’s functions for $v_{{\rm b}},\;v_{{\rm c}},$ and $v_{{\rm bc}}$ at $z\sim 1020$. We have multiplied each by $r^{2}$ for two reasons. First, for a random distribution of densities, a spherical shell of radius $r$ will contribute as $r^{2}$ when integrated over volume. Second, this weighting renders the fine structure more apparent. The most striking feature of Figure 2 is the compact support of the $v_{{\rm bc}}$ Green’s function. This occurs because for $r>r_{{\rm s}}$, $v_{{\rm bc}}\to 0$, as radiation pressure cannot support the baryons against gravitational infall. Also salient is that $v_{{\rm bc}}\approx v_{{\rm c}}$ for $r\lesssim 0.9r_{{\rm s}}$: baryons are in hydrostatic equilibrium with $v_{{\rm b}}\approx 0$. There is a bump in the baryon velocity at $r_{s}$ due to the outgoing baryon-photon overdensity from the BAO. Meanwhile, the DM infalls. Inside $r_{{\rm s}},$ the DM infalls as roughly $\sim 1/r$ rather than $1/r^{2}$ because the baryon-photon fluid’s contribution to the potential, which, during radiation domination, overshadows that of the DM overdensity at the origin, is diluted. For a test particle at $r<r_{\rm s}$, some of the baryon-photon overdensity is outside a Gaussian sphere of radius $r$, and hence does not contribute to the gravitational force felt by the particle. This is equivalent to the fact that modes inside the horizon grow less quickly than those outside the horizon during radiation domination, and is why the DM transfer function turns over for $k\sim k_{{\rm eq}},$$k_{{\rm eq}}$ the wavenumber entering the horizon at matter-radiation equality. Using the continuity equation in Fourier space (equation (7)), slower growth of a given mode implies a lower velocity field for that mode, so modes inside the horizon indeed have a lower velocity field than those outside the horizon. <figure><img src="content_image/1411.4052/x3.png"><figcaption>Figure 2: Relative velocity Green’s function vG, the relative velocityvbc=vb−vc due to a δ(3)D density perturbation at the origin. We show theGreen’s function at z∼1020, as at this epoch the relative velocity freezes in(cf. §1). Here and throughout the paper, green will denote the Green’sfunction, plot symbol X will pertain to functions involving the relativevelocity, red will be baryons, and blue CDM. Note that for r>rs≃150Mpc,baryons and DM infall at the same speed. In contrast, inside rs the baryonsare mostly locked in hydrostatic equilibrium while the DM infalls roughly as1/r due to the dilution of the radiation density perturbation as this latterexpands with time (see §3 for further discussion). The slight bump in thebaryon velocity represents baryons in motion as the fluid compresses andexpands at the sound horizon. The neutrinos, which travel at roughly c and soare outside rs, smooth the transition to zero vbc outside the sound horizon,as does Silk damping (Silk 1968).</figcaption></figure> ## 4 Analysis of the two-point correlation function ### The shift in $\xi_{\rm gg}$ We now wish to compute $\xi_{v}$, the relative velocity contribution to the correlation function. We define $\xi_{{\rm gg}}$ as the full galaxy-galaxy correlation function including $\xi_{v}$ and denote the late-time linear matter correlation function $\xi$. To compute $\xi_{v}$, we use the galaxy overdensity bias model of $\lx@sectionsign 2$. Numerical subscripts denote spatial positions: $\delta_{1}\equiv\delta(\vec{r}_{1})$. For more compact notation, we also define $\delta_{v}=v_{{\rm s}}^{2}-1$; note this is second-order in $\delta$. We have the velocity contributions to the product $\delta_{{\rm g}}\left(\vec{r}_{1}\right)\delta_{{\rm g}}\left(\vec{r}_{2}\right)$: \[\left[\delta_{{\rm g}}\left(\vec{r}_{1}\right)\delta_{{\rm g}} \left(\vec{r}_{2}\right)\right]_{v}=b_{1}b_{v}\left(\delta_{{\rm m}1}\delta_{v 2}+\delta_{{\rm m}2}\delta_{v1}\right)+b_{2}b_{v}\] (19) \[\times\left(\delta_{{\rm m}1}^{2}\delta_{v2}+\delta_{{\rm m}2}^{2 }\delta_{v1}-\left<\delta_{{\rm m}}^{2}\right>\left[\delta_{v1}+\delta_{v2} \right]\right)+b_{v}^{2}\delta_{v1}\delta_{v2}.\] Defining $r=\big{\|}\vec{r}_{2}-\vec{r}_{1}\big{\|}$ and noting that terms in $\delta_{1}\delta_{v2}$ vanish because we assume a Gaussian random field, we have \[\xi_{v}\left(r\right)\equiv\left<\left[\delta_{{\rm g}}\left(\vec {r}_{1}\right)\delta_{{\rm g}}\left(\vec{r}_{2}\right)\right]_{v}\right>=\xi_{ v1}+\xi_{v2}+\xi_{vv}\] (20) The factors of 2 come from invoking homogeneity so that $\left<\delta_{1}^{(2)}\delta_{v2}\right>=\left<\delta_{2}^{(2)}\delta_{v1}\right>$ and $\left<\delta_{1}^{2}\delta_{v2}\right>=\left<\delta_{2}^{2}\delta_{v1}\right>.$ Moving forward we will often denote the term proportional to $b_{1}$ by $\xi_{v1}$ and analogously for the terms in $b_{2}$ and $b_{v}$, as indicated above. We have also dropped terms above fourth order. Using the definition of $\delta_{v}$, we can simplify the three terms in $\xi_{v}$ to: \[\xi_{v1}\left(r\right)\equiv 2b_{1}b_{v}\left[\left<\delta_{1}^{(2)}v_{s2}^{2} \right>-\left<\delta^{(2)}\right>\right],\] (21) \[\xi_{v2}\left(r\right)\equiv 2b_{2}b_{v}\left[\left<\delta_{1}^{2}v_{s2}^{2} \right>-\xi\left(0\right)\right],\] (22) \[\xi_{vv}\left(r\right)\equiv b_{v}^{2}\left[\left<v_{s1}^{2}v_{s2}^{2}\right>- 1\right].\] (23) For equation (22), we have replaced $\left<\delta^{2}\right>$ by $\xi\left(0\right)$. These three terms (ignoring the constant offsets) are shown schematically in Figure 3. Spheres indicate a velocity squared, while the solid square shows the density field squared and the solid triangle represents the second-order density field $\delta^{(2)}$. The dotted lines show the correlations that remain after the constants above are subtracted off. Note that these expectation values involve products of four values of the linear density field at different locations. Since the linear density field is a Gaussian Random Field, we can use Wick’s theorem to simplify. The constants subtracted above are just generated by the presence of $\left<\delta_{{\rm m}}^{2}\right>$ in our model for $\delta_{{\rm g}}$, and ultimately cancel when Wick’s theorem is applied. Finally, we need to compute $v_{s}^{2}$ since it enters equations (21)-(23). We have \[v_{s}^{2}\left(\vec{r}\right) =\frac{1}{\sigma_{{\rm bc}}^{2}\left(z\right)}\int d^{3}\vec{r}_{ 1}d^{3}\vec{r}_{2}v_{{\rm G}}\left(r_{1},z\right)v_{{\rm G}}\left(r_{2},z\right)\] (24) \[\times\delta_{{\rm pri}}\left(\vec{r}_{1}+\vec{r}\right)\delta_{{ \rm pri}}\left(\vec{r}_{2}+\vec{r}\right)\hat{r}_{1}\cdot\hat{r}_{2},\] with $v_{{\rm G}}$ given by equation (17). See the right panel of Figure 1 for a visual reminder of how $v_{s}^{2}$ is calculated from the Green’s function. For $v_{s}^{2}$, we in turn require $\sigma_{{\rm bc}}^{2}\equiv\left<\vec{v}_{{\rm bc}}\cdot\vec{v}_{{\rm bc}}\right>$, which can be found by writing $\vec{v}_{{\rm bc}}\left(\vec{r}\right)$ in the Fourier basis using equations (14) and (15), squaring, and taking expectation values. One then uses the relation between $\tilde{\delta}_{\rm pri}$ and $P_{\rm pri}$ given in §3 to simplify the integrals and finds \[\sigma_{{\rm bc}}^{2}\left(z\right)=\int\frac{k^{2}dk}{2\pi^{2}}\tilde{v}_{{ \rm G}}^{2}\left(k,z\right)P_{{\rm pri}}\left(k\right),\] (25) in agreement with Dalal et al. (2010). Note that after decoupling, the relative velocity is no longer sourced and will therefore decay as $a^{-1}$, as will ${\rm\sigma_{{\rm bc}}}.$ Hence after decoupling $v_{s}^{2}\left(\vec{r}\right)$ is redshift-independent. <figure><img src="content_image/1411.4052/x4.png"><figcaption>Figure 3: Illustrations of the three contributions to ξv. The top panel isξv1(r) (equation (21)), the middle panel ξv2(r) (equation (22)), and thebottom panel ξvv(r) (equation (23)). The symbols are as explained in Figure 1;the dotted lines show the correlations that remain after simplifying usingWick’s theorem. The triangle (top panel) is where δ(2)(→r) is evaluated, byintegrating over the entire surrounding density field, while the square(middle panel) is where δ2(→r) is evaluated. These diagrams make theconvolutional structure of this calculation evident; it is to be furtherdiscussed in §4.2.</figcaption></figure> ### $\xi_{v}$ as convolutions We now show that $\xi_{v}$ can be expressed as convolutions of various functions of the linear density field with a 6-dimensional velocity kernel we define below. This latter preserves the radial structure of the Green’s function, and this insight will support our analysis in §5 of why the BAO peak shifts. We begin with $\xi_{v2}$ (equation (22)) because it is the simplest. The disconnected part of $\left<\delta_{1}^{2}v_{s2}^{2}\right>$ is $\left<\delta_{1}^{2}\right>\left<v_{s2}^{2}\right>=\xi\left(0\right)$ and so cancels off. Focusing on the remaining terms from Wick’s theorem, represented by the dotted lines in Figure 3, we find \[\xi_{v2}\left(r\right)=4b_{2}b_{v}\int_{34}\xi_{\times}\left(\big{\|}\vec{r}- \vec{r}_{3}\big{\|}\right)\xi_{\times}\left(\big{\|}\vec{r}-\vec{r}_{4}\big{\|} \right)V_{3,4},\] (26) where the factor of $2$ relative to equation (22) comes from Wick’s theorem. To connect with Figure 3, note that we have put the origin at the black dot (i.e. used homogeneity to set $\vec{r}_{2}=0$ in equation (22)). We have defined the 6-dimensional velocity kernel \[V_{a,b}\equiv v_{G}\left(r_{a}\right)v_{G}\left(r_{b}\right)\left(\hat{r}_{a} \cdot\hat{r}_{b}\right)/\sigma_{{\rm bc}}^{2}.\] (27) This kernel generates $v_{s}^{2}\left(0\right)$ from integrals over $\delta_{{\rm pri,}a}$ and $\delta_{{\rm pri,}b}$ for all $\vec{r}_{a}$ and $\vec{r}_{b}$ within roughly $r_{{\rm s}}$ of the origin. Note that it is redshift-independent. We have also defined $\xi_{\times}$: a heterosynchronous correlation of the low-redshift linear theory matter density field with the primordial density field: \[\xi_{\times}\left(\vec{r}\right)=\int\frac{d^{3}\vec{k}}{\left(2\pi\right)^{3} }e^{-i\vec{k}\cdot\vec{r}}P_{\times}\] (28) where \[P_{\times}\equiv Ak^{n_{s}}T_{{\rm m}}\left(k,z=0\right).\] (29) The matter transfer function is $T_{{\rm m}}=(\Omega_{{\rm c}}T_{{\rm c}}+\Omega_{{\rm b}}T_{{\rm b}})/(\Omega_ {{\rm c}}+\Omega_{{\rm b}})$. Note that $P_{\times}$ is not the standard power spectrum, which would use the square of the transfer function. Rather, is is a cross power spectrum between the low-redshift linear theory matter field (we use $z=0$) and the primordial density field. We define the 6-D convolution as \[\left[f_{1,2}\star g_{1,2}\right]\left(\vec{R}\right)=\int d^{3}\vec{r}_{1}d^{ 3}\vec{r}_{2}f\left(\vec{r}_{1},\vec{r_{2}}\right)g\left(\vec{r}_{a}-\vec{r}_{ 1},\vec{r}_{b}-\vec{r}_{2}\right),\] (30) where $\vec{R}=\left(\vec{r}_{a},\vec{r}_{b}\right)$ is a 6-D vector and $\vec{r}_{a}$ and $\vec{r}_{b}$ are 3-D vectors. With this definition, it is immediate to write \[\xi_{v2}\left(r\right)=4b_{2}b_{v}\left[V_{3,4}\star\xi_{\times 3}\xi_{\times 4 }\right]\left(\vec{r},\vec{r}\right).\] (31) We now show that $\xi_{vv}$ (equation (23)) can also be expressed as a convolution. Canceling the disconnected part and then evaluating $\xi_{vv}$ (see Figure 3), we find \[\xi_{vv}\left(r\right) =2b_{v}^{2}\int_{3456}\xi_{{\rm pri}}\left(\big{\|}\vec{r}+\vec{r} _{3}-\vec{r}_{5}\big{\|}\right)\xi_{{\rm pri}}\left(\big{\|}\vec{r}+\vec{r}_{4}- \vec{r}_{6}\big{\|}\right)\] \[\times V_{3,4}V_{5,6},\] (32) where again the factor of $2$ relative to equation (23) is from Wick’s theorem. Note that for $\xi_{vv}=b_{v}^{2}\left<\delta_{v1}\delta_{v2}\right>$ we are correlating a velocity field with a velocity field, and so we must use only $\delta_{{\rm pri}}$, leading to $\xi_{{\rm pri}}=\left(2\pi\right)^{-3}\int d^{3}\vec{k}e^{-i\vec{k}\cdot\vec{r }}P_{{\rm pri}}$ in the expression above. Let us first consider the integrals over $d^{3}\vec{r}_{3}d^{3}\vec{r}_{4}$ above. Flipping the signs of $\vec{r}_{3}$ and $\vec{r}_{4,}$ which leaves the Jacobian unchanged, and applying the definition (30), we obtain \[\int_{34}\xi_{{\rm pri}}\left(\big{\|}\vec{r}+\vec{r}_{3}\big{\|}\right)\xi_{{ \rm pri}}\left(\big{\|}\vec{r}+\vec{r}_{4}\big{\|}\right)V_{3,4}=\left[V_{34} \star\xi_{{\rm pri}3}\xi_{{\rm pri}4}\right](\vec{r},\vec{r}).\] (33) Inserting this in equation (32), we have \[\xi_{vv}\left(r\right)=2b_{v}^{2}\int_{56}V_{5,6}\left[V_{34}\star\xi_{{\rm pri }3}\xi_{{\rm pri}4}\right](\vec{r}-\vec{r}_{5},\vec{r}-\vec{r}_{6}).\] (34) Applying again the definition (30), we find \[\xi_{vv}\left(r\right)=2b_{v}^{2}\left[V_{5,6}\star\left[V_{3,4}\star\xi_{{\rm pri }3}\xi_{{\rm pri}4}\right]\left(\vec{r}_{5},\vec{r}_{6}\right)\right]\left( \vec{r},\vec{r}\right).\] (35) Finally, we turn to $\xi_{v1}$ (equation (21)), the most difficult term to evaluate because it involves the second-order density field $\delta^{\left(2\right)}$. Using the same procedure as for $\xi_{v2}$ and $\xi_{vv}$, we have \[\xi_{v1}\left(r\right) =4b_{1}b_{v}\int_{3456}\xi_{\times}\left(\big{\|}\vec{r}+\vec{r}_{ 5}-\vec{r}_{3}\big{\|}\right)\xi_{\times}\left(\big{\|}\vec{r}+\vec{r}_{6}-\vec{ r}_{4}\big{\|}\right)\] \[\times F_{3,4}^{\left(2\right)}V_{5,6},\] (36) where the factor of $2$ relative to equation (21) is again from Wick’s theorem. Again, Figure 3 illustrates the relevant correlations. $F_{3,4}^{(2)}$ is the Fourier transform of the second-order kernel \[\tilde{F}^{\left(2\right)}\left(\vec{k}_{1},\vec{k}_{2}\right)= \frac{5}{7}+\frac{1}{2}\hat{k}_{1}\cdot\hat{k}_{2}\left(\frac{k_{1}}{k_{2}}+ \frac{k_{2}}{k_{1}}\right)+\frac{2}{7}\left(\hat{k}_{1}\cdot\hat{k}_{2}\right) ^{2}\] \[=\frac{17}{21}P_{0}\left(\hat{k}_{1}\cdot\hat{k}_{2}\right)+\frac {1}{2}\left(\frac{k_{1}}{k_{2}}+\frac{k_{2}}{k_{1}}\right)P_{1}\left(\hat{k}_{ 1}\cdot\hat{k}_{2}\right)\] (37) \[+\frac{4}{21}P_{2}\left(\hat{k}_{1}\cdot\hat{k}_{2}\right)\] (Goroff et al. 1986; Jain & Bertschinger 1994; Bernardeau et al. 2002 [equation (45)]). Analogously to $V_{3,4}$, an integral of two density fields $\delta_{3}$ and $\delta_{4}$ against $F_{3,4}^{(2)}$ generates a second-order density field $\delta^{(2)}(0)$. The $P_{i}$s are Legendre polynomials. Working now in Fourier space, we see that equation (36) becomes \[\xi_{v1}\left(r\right) =4b_{1}b_{v}\bigg{\{}\bigg{[}\int\frac{d^{3}\vec{k}_{1}d^{3}\vec{ k}_{2}}{\left(2\pi\right)^{6}}\tilde{F}^{(2)}\left(\vec{k}_{1},\vec{k}_{2} \right)P_{\times}\left(k_{1}\right)\] (38) \[\times P_{\times}\left(k_{2}\right)e^{-i\vec{k_{1}}\cdot\vec{r}_{ 3}}e^{-i\vec{k}_{2}\cdot\vec{r}_{4}}\bigg{]}\left(\vec{r}_{3},\vec{r}_{4} \right)\star V_{5,6}\bigg{\}}\left(\vec{r},\vec{r}\right).\] Note that the integral over $d^{3}\vec{r}_{3}d^{3}\vec{r}_{4}$ in equation (36) is the 6-D convolution $\xi_{\times}\left(\vec{r}\right)\xi_{\times}(\vec{r})\star F_{3,4}^{(2)}$, which we have rewritten as a product in Fourier space using the Convolution Theorem to obtain equation (38). In equation (38), the integral in square brackets becomes a function of $\vec{r}_{3}$ and $\vec{r}_{4}$ when evaluated; this in turn is convolved with $V_{5,6}$. For convenience, we define the integral in square brackets as \[f_{\times 3,4}\] (39) \[\times P_{\times}\left(k_{2}\right)e^{-i\vec{k_{1}}\cdot\vec{r}_{ 3}}e^{-i\vec{k}_{2}\cdot\vec{r}_{4}}\bigg{]}\left(\vec{r}_{3},\vec{r}_{4} \right).\] It is more convenient to work with this representation than to directly consider $\xi_{\times}\left(\vec{r}\right)\xi_{\times}(\vec{r})\star F_{3,4}^{(2)}$, since $F_{3,4}^{(2)}$ (the Fourier transform of $\tilde{F}^{(2)}$) is divergent without the regularization multiplication of $\tilde{F}^{(2)}$ by the smoothed cross power spectrum provides. With this notation, we now have \[\xi_{v1}(r)=4b_{1}b_{v}\left[f_{\times 3,4}\star V_{3,4}\right]\left(\vec{r}, \vec{r}\right).\] (40) Thus, from equations (31), (35), and (40), we see that all three terms in $\xi_{v}$ are just convolutions of various functions built from the correlation function with the 6-D velocity kernel (27) we have defined. This latter ultimately encodes the radial structure of the Green’s function, shown in Figure 2. We now pause to examine the limit that $\xi\to\delta_{{\rm D}}^{\left[3\right]}$, as this will offer strong physical intuition for the behavior we should see once we have numerically evaluated the equations above. In this limit, $\xi_{v2}\left(r\right)=4b_{2}b_{v}v_{G}^{2}\left(r\right)$, while $\xi_{vv}\left(r\right)=2b_{v}^{2}\left[V_{1,2}\star V_{1,2}\right]\left(\vec{r }\right)$ (unfortunately $\xi_{v1}$ is divergent in this limit). The approximate form that $v_{{\rm G}}^{2}\propto 1/r^{2}$ for $r\leq r_{{\rm s}}$ (see Figure 2) means that inside $r_{{\rm s}},$ each radial bin of volume $dV=4\pi r^{2}dr$ will give an equal contribution to $\xi_{v2}$. We thus expect that $r^{2}\xi_{v2}$ will be approximately a step function, constant for $r\leq r_{{\rm s}}$ and zero otherwise. Meanwhile, $V_{1,2}\star V_{1,2}$ is an autocorrelation, so $\xi_{vv}$ will peak at $\vec{r}=0.$ A second, lesser peak in the autocorrelation occurs at $\|\vec{r}\|=r_{{\rm s}}$, which becomes the most prominent one when we consider the volume-weighted $4\pi r^{2}\xi_{vv}\left(r\right)$. Therefore $r^{2}\xi_{vv}$ should have a well-defined peak that encodes the acoustic scale, and have support out to $\sim 2r_{{\rm s}}$. These behaviors are displayed in the lower panel of Figure 5, magenta dashed and orange X-ed curves. ### Evaluating the convolutions We now give details on how the convolutions of the previous section can be quickly evaluated. Formally, the convolutions are multidimensional integrals, and so could be computed directly via Monte Carlo methods. We avoid this by showing that in principle all of the convolutions can be analytically reduced to one-dimensional radial integrals, because the angular dependences of all functions involved are known. This is highly desirable and may be done using results we prove in the Appendix. First, we explicitly obtain $f_{\times 3,4}$; since this is also useful in computing the 3PCF, we write it down here. We have \[f_{\times 3,4}=\mathcal{H}_{0}^{-1}\left\{\frac{17}{21}P_{\times }\left(k_{1}\right)P_{{\rm}\times}\left(k_{2}\right)\right\}(r_{3},r_{4})P_{0} \left(\hat{r}_{3}\cdot\hat{r}_{4}\right)\] (41) \[-\mathcal{H}_{1}^{-1}\left\{\frac{1}{2}\left(\frac{k_{1}}{k_{2}}+ \frac{k_{2}}{k_{1}}\right)P_{\times}\left(k_{1}\right)P_{\times}\left(k_{2} \right)\right\}(r_{3},r_{4})P_{1}\left(\hat{r}_{3}\cdot\hat{r}_{4}\right)\] \[+\mathcal{H}_{2}^{-1}\left\{\frac{4}{21}P_{\times}\left(k_{1} \right)P_{\times}\left(k_{2}\right)\right\}(r_{3},r_{4})P_{2}\left(\hat{r}_{3} \cdot\hat{r}_{4}\right),\] where the $\mathcal{H}^{-1}s$, defined in the Appendix (equations (61) and (62)), are 2-D transforms composed of 1-D integrals against spherical Bessel functions (these latter integrals are closely related to Hankel transforms). Simplifying (see Appendix for formulae used to do so), we have \[f_{{\rm}\times 3,4}=\frac{17}{21}P_{0}\left(\hat{r}_{3}\cdot\hat {r}_{4}\right)\xi_{\times 3}\xi_{\times 4}-\frac{1}{2}P_{1}\left(\hat{r}_{3} \cdot\hat{r}_{4}\right)\] (42) \[\times\left[\xi_{{\rm}\times 4}^{[1-]}\xi_{\times 3}^{[1+]}+\xi_{ \times 4}^{[1+]}\xi_{{\rm}\times 3}^{[1-]}\right]+\frac{4}{21}P_{2}\left(\hat{ r}_{3}\cdot\hat{r}_{4}\right)\xi_{{\rm}\times 3}^{[2]}\xi_{{\rm}\times 4}^{[2]},\] with \[\xi_{{\rm}\times}^{[1\pm]}\left(r\right)\equiv\int_{0}^{\infty}\frac{k^{2}dk}{ 2\pi^{2}}j_{1}\left(kr\right)P_{\times}\left(k\right)k^{\pm 1}\] (43) and \[\xi_{\times}^{[2]}\left(r\right)\equiv\int_{0}^{\infty}\frac{k^{2}dk}{2\pi^{2} }j_{2}\left(kr\right)P_{\times}\left(k\right).\] (44) This reduces the terms in $\xi_{v}$ to \[\xi_{v1}\left(r\right)=4b_{1}b_{v}\sigma_{{\rm bc}}^{-2}\bigg{(} \frac{31}{35}\left[h_{1}^{-1}\left\{\tilde{v}_{{\rm G}}P_{\times}\right\}(r) \right]^{2}\] (45) \[-\frac{1}{3}\bigg{[}h_{0}^{-1}\left\{\tilde{v}_{{\rm G}}P_{{\rm} \times}k\right\}(r)h_{0}^{-1}\left\{\tilde{v}_{{\rm G}}P_{{\rm}\times}k^{-1} \right\}(r)\] \[+2h_{2}^{-1}\left\{\tilde{v}_{{\rm G}}P_{\times}k\right\}(r)h_{2} ^{-1}\left\{\tilde{v}_{{\rm G}}P_{{\rm}\times}k^{-1}\right\}(r)\bigg{]}\] \[+\frac{12}{105}\left[h_{3}^{-1}\left\{\tilde{v}_{{\rm G}}P_{{\rm} \times}\right\}(r)\right]^{2}\bigg{)},\] (46) and \[\xi_{vv}\left(r\right) =\frac{2}{3}b_{v}^{2}\sigma_{{\rm bc}}^{-4}\bigg{[}\left[h_{0}^{- 1}\left\{\tilde{v}_{{\rm G}}^{2}P_{{\rm pri}}\right\}(r)\right]^{2}\] (47) \[+2\left[h_{2}^{-1}\left\{\tilde{v}_{{\rm G}}^{2}P_{{\rm{\rm pri}} }\right\}(r)\right]^{2}\bigg{]}.\] Note that the right-hand side of each equation is a function of $r$, as $\tilde{v}_{{\rm G}}$ and $P_{\times}$ (equations (18) and (29)) are functions of $k$, and the $h_{l}^{-1}$s bring them to functions of $r$ (cf. equation (62)). The equations above allow efficient numerical evaluation of $\xi_{v}$. Finally, it should be noted that Wick’s theorem yields immediately that $\xi_{vv}\left(0\right)$$=\left(2/3\right)b_{v}^{2}$ (cf. equation (23)). Taking this limit of equation (47) explicitly cross-checks our use of the convolutional formalism: \[\xi_{vv}\left(0\right)=\frac{2}{3}b_{v}^{2}\sigma_{{\rm bc}}^{-4}\left[\int \frac{k^{2}dk}{2\pi^{2}}\tilde{v}_{{\rm G}}^{2}P_{{\rm pri}}\right]^{2}=\frac{ 2}{3}b_{v}^{2},\] (48) with the last equality using equation (25). ## 5 Explaining the BAO peak shift We begin with a descriptive sketch of why the BAO peak shifts and then move to more detailed discussion of our numerical results. First: the key aspect of $\xi_{v}$ that allows it to shift the BAO peak in the correlation function is the sharp drop to zero for $r$ greater than roughly $r_{s}$. This can be traced back to the same feature in the relative velocity Green’s function (see Figure 2). Thus $\xi_{v}$ adds $\left(b_{v}>0\right)$ or subtracts $\left(b_{v}<0\right)$ probability density from $\xi_{{\rm gg}}$ asymmetrically, doing so primarily _inwards_ of $r_{{\rm s}}$. It is this asymmetric alteration that pulls the BAO peak in or out in scale. Fundamentally, then, the shift is due to the presence of an acoustic horizon: the relative velocity is at root a pressure effect, and as such can alter correlations roughly only within the sound horizon. In greater mathematical detail, the 6-D velocity kernel $V_{1,2}$ has support only for $r\lesssim r_{s}$. Convolving $\xi_{{\rm\times}}$ and $f_{\times}$ with it act as smoothing operations that simply broaden its peak. Since the smoothing functions are so narrow, the convolutions are only non-zero essentially where the velocity kernel is non-zero. For $b_{v}>0,$ this adds probability density inwards of $r_{{\rm s}}$, while for $b_{v}<0$, this subtracts probability density inwards of $r_{{\rm s}}$. Figure 4 shows the two smoothing functions, $\xi_{{\rm\times}}$ and $f_{\times}$ (built on transforms of $\xi$), and the $v_{{\rm bc}}$ Green’s function they smooth. The behavior of the contribution from $\xi_{vv}$ is complex, as it is a double convolution, but as we learned from the $\xi_{\times}\to\delta_{\rm D}^{[3]}$ limit, it has support out to $\sim 2r_{\rm s}$. However, around the BAO peak location $\xi_{vv}$ is roughly symmetric, so it does not contribute much to shifting the peak (Figure 5). We now move to discuss each term in $\xi_{v}$ more extensively; each term is shown in Figure 5. Two of the terms are fairly simple in structure, and can be understood by recalling the limit $\xi\to\delta_{\rm D}^{[3]}$ of §4.2. $\xi_{v2}$ is roughly constant for $r<r_{\rm s}$, and drops quickly to zero outside the sound horizon. It extends slightly beyond the sound horizon; this may be traced back to the smooth drop of the velocity Green’s function (Figure 2) produced by the neutrinos and Silk damping. The small bump near the sound horizon is due to the baryons’ velocity there; this has its origin in the baryon velocity’s contribution to the Green’s function (see Figure 2). This is confirmed by the bottom panel of Figure 5, plotting the $\xi\to\delta_{\rm D}^{[3]}$ limit. Here, a bump is still present near the acoustic scale, meaning that the bump is due to structure in the velocity Green’s function and not due to any structure in $\xi_{\times}$. Since $\xi_{v2}$ is non-zero essentially solely inside $r_{\rm s}$, it adds asymmetrically to the correlation function and hence can pull the BAO bump inwards ($b_{v}>0$) or outwards ($b_{v}<0$). $\xi_{vv}$ is also fairly simple and can also be understood by recalling the limit $\xi\to\delta_{\rm D}^{[3]}$ of §4.2. From equation (35), we see that it is a double convolution, and that the first convolution produces a function identical, up to amplitude, to $\xi_{v2}$. However, this is then convolved with an additional velocity kernel, which is, unlike $\xi_{\times}$ and $f_{\times}$, rather broad. Thus there is overlap between the velocity kernel and the first convolution until nearly twice the sound horizon. However, $\xi_{vv}$ is roughly symmetric around the acoustic scale out to $\sim 20\;\rm{Mpc}$ on either side of it; this means that in the region of the BAO peak, $\xi_{vv}$ is adding symmetrically to $\xi$ and hence will contribute minimally to shifting the peak. We now discuss the last remaining term in $\xi_{v}$, $\xi_{v1}$. This term has the most complicated structure, a consequence of its generation by the second-order density field. This latter is non-local: computing $\delta^{(2)}$ requires integrating over all space (see Figure 3, top panel, and equation (36)). Hence this term requires correlating the entire linear density field with the velocity field. Nonetheless, its behavior can be understood qualitatively. The second-order density field (or the $\tilde{F}_{2}$ kernel) represents non-linear gravitational evolution, which, in our Green’s function approach, pulls mass towards the origin. Thus it is not surprising that $f_{\times}$ is rather narrow. Since $f_{\times}$ is so narrow, it will only have non-zero overlap with $V_{3,4}$ where the latter is non-zero. This explains the compact support of $\xi_{v1}$. $\xi_{v1}$ is able to become negative because $f_{\times}$ does. Note that (Figure 4) $f_{\times}$ is positive very close to the origin; this is what permits $\xi_{v1}$ to become positive at intermediate scales, $r\sim 50-150\;\rm{Mpc}$. Overall, for $b_{v}>0$, $\xi_{v1}$ mostly adds to $\xi_{v}$ inwards of the BAO bump and subtracts from it directly outside the sound horizon; this asymmetric effect can shift the BAO peak. In summary, then, we have seen that of the three terms entering $\xi_{v}$, only two make significant contributions to a peak shift. In both cases, the shift is due to the compact support of these terms, which in turn results from the narrowness of the smoothing functions combined with the compact support of the velocity Green’s function. We now consider the relative importance of these two terms when they are added together with plausible values of the linear and non-linear bias. Note that both of these terms are proportional to $b_{v}$, so different values of $b_{v}$ will not alter their relative weights. $\xi_{v1}$ is a factor of $\sim 5-10$ larger than $\xi_{v2}$, and so it dominates the total velocity contribution to $\xi_{\rm gg}$ for $b_{1}=1,\;b_{2}=0.1$. The total velocity contribution to $\xi_{\rm gg}$ in Figure 6 thus looks very similar to $\xi_{v1}$. Note however that the effects of $\xi_{vv}$ are perceptible. In particular, inside $r_{\rm s}$, the addition for $b_{v}>0$ is greater in magnitude than that for $b_{v}<0$. This is because for $b_{v}>0$$\xi_{v1}$ and $\xi_{vv}$ are both mostly positive within $r_{\rm s}$, while for $b_{v}<0$, $\xi_{v1}$ flips sign but $\xi_{vv}$ does not (compare $b_{v}=-2\%$ and $b_{v}=2\%$ curves in Figure 6). Outside $r_{\rm s}$, the case is reversed. For $b_{v}<0$, $\xi_{v1}>0$ as is $\xi_{vv}$, while for $b_{v}>0$, these two terms have opposite signs and thus interfere destructively. This point explains why the curve for $b_{v}=-2\%$ is larger in magnitude than that for $b_{v}=2\%$ outside $r_{\rm s}$. Finally, the convergence of the $b_{v}<0$ curves to those with $b_{v}>0$ at large $r$ is because the $b_{v}^{2}$ term is the only contribution for $r\gg r_{{\rm s}}$ and it is symmetric in $b_{v}$. Note also that the curves are nearly anti-symmetric under sign flip in $b_{v}$, not surprising since two of the terms manifestly have this symmetry. They are not perfectly anti-symmetric under this transformation due to the $b_{v}^{2}$ term. It is significant that the $\xi_{v1}$ term dominates the velocity addition to $\xi_{\rm gg}$ because this means one need not measure $b_{2}$ as well as one measures $b_{1}$ to obtain a good correction to the correlation function. Fortunately observationally $b_{1}$ is indeed measured better than $b_{2}$. <figure><img src="content_image/1411.4052/x7.png"><figcaption>Figure 4: Equations (31), (35), and (40) display the correction ξv to ξ as the6-D convolutions of 2 different 6-D kernels, ξ×3ξ×4 and f×3,4, with V1,2.These kernels have angular dependence, so the convolutions genuinely are fully6-D. However, we can gain heuristic intuition for their behavior by settingall angle-dependent terms to unity and considering the radial behavior of eachfunction. Each function then becomes a product of two copies of the sameradial function at r3 and r4. We plot one copy of each here, against vG, whichappears in V1,2=vG1vG2(^r1⋅^r2). The two functions convolved with vG are bothsharply peaked at r=0, so the convolution does not greatly alter vG; thereforewe expect all results involving it to have distinctive structure at theacoustic scale. As a reminder, ξ× generates ξv2 and f× generates ξv1.ξvv isgenerated by convolving V1,2 with itself; intuition may be gained for thiscase by imagining convolving two copies of the Green’s function above.</figcaption></figure> <figure><img src="content_image/1411.4052/x8.png"><figcaption>Figure 5: The top panel shows the three terms in ξv (equations (21)-(23))computed using equations (45)-(47). We have used b1=1,b2=0.1,bv=0.01. We haveadded a vertical red line at the BAO scale in the late-time linear correlationfunction, 150Mpc. The bottom panel shows the three terms in ξv in the limitξ×→δ(3)D. Here we have scaled each term to have maximum =1, as the limit thatξ×→δ[3]D requires some normalization. Formally, the limit that ξ×→δ[3]D meansP×→1, but we use a smoothing such that P→exp[−k2] to avoid introducing ringingdue to our finite grid in Fourier space. This smoothing effectivelyregularizes the divergence of f×3,4 discussed in §4.2. The three curves arediscussed in detail in §5; note that (top panel) ξv1 dominates and has supportonly roughly within rs. This is also true for ξv2. While ξvv has support outto ∼2rs, it adds to ξgg roughly symmetrically about the BAO peak location andso will not shift the peak much. Note the similarity of the two panels; thisis because ξ× is already very narrow compared to vG. The ξ×→δ[3]D limit tellsus that there is some intrinsic width to the peaks in ξv1,ξv2, and ξvv. Forinstance the trough in ξv1 around rs is ∼15Mpc wide in the bottom panel. Itwidens to ∼40Mpc in the top panel, consistent with the fact that ξ× has aboutthis width and so will smooth structure on this scale (see Figure 4).</figcaption></figure> <figure><img src="content_image/1411.4052/x10.png"><figcaption>Figure 6: ξv (equation (20)) with different values of bv. We have usedb1=1,b2=0.1. The case we trace throughout the paper is bv=1%, but even bv∼2%is allowed by current constraints (Yoo & Seljak 2013), hence its presentationhere. We have marked the BAO scale in the late-time linear correlationfunction with a red line. The inset is ξ, the linear-theory matter correlationfunction today, which we present to highlight the BAO peak that ξv can shift.The key point of this plot is that ξv alters the correlation functionasymmetrically about rs, mostly altering it inwards of rs. This creates ashift in the peak. A secondary point is the similarity with Figure 5, toppanel, ξv1 (compare black solid curves between the two figures); see §5 forfurther discussion. Finally note that the results are not exactly symmetric inthe sign of bv, especially visible at scales larger than 150Mpc where the b2vterm dominates.</figcaption></figure> ## 6 Isolating $b_{v}$ from the 3PCF: pre-cyclic computation The three-point galaxy correlation function (3PCF) describes the excess probability over random of finding three galaxies with positions $\vec{r}_{1}$, $\vec{r}_{2}$, and $\vec{r}_{3}$; that is, in a given triangle configuration. We compute it at fourth order using our bias model ((1), (2), and (4)) as \[\zeta(\vec{r}_{1},\vec{r}_{2},\vec{r}_{3})\equiv<\delta_{{\rm g}_ {1}}\delta_{{\rm g}_{2}}\delta_{{\rm g}_{3}}>=\zeta_{{\rm pc}}\left(\vec{r}_{3 }-\vec{r}_{1},\vec{r}_{2}-\vec{r}_{1}\right)\] (49) \[+\zeta_{{\rm pc}}\left(\vec{r}_{1}-\vec{r}_{2},\vec{r}_{3}-\vec{r }_{2}\right)+\zeta_{{\rm pc}}\left(\vec{r}_{2}-\vec{r}_{3},\vec{r}_{1}-\vec{r} _{3}\right).\] We now rewrite the pre-cyclic piece (subscript “pc”) as a function of two side lengths and their enclosed angle (so that numerical subscripts denote sides of the triangle rather than spatial positions): \[\zeta_{{\rm pc}}(r_{1},r_{2};\theta_{12})=b_{1}^{2}b_{v}\left[< \delta_{1}\delta_{2}\delta_{v_{3}}>-\xi_{12}\left<\delta_{v}\right>\right]+b_{ 1}^{2}b_{2}\] (50) \[\times\left[<\delta_{1}\delta_{2}\delta_{3}^{2}>-\xi_{12}\left< \delta^{2}\right>\right]+b_{1}^{3}\left[<\delta_{1}\delta_{2}\delta_{3}^{(2)}> -\xi_{12}\left<\delta^{(2)}\right>\right].\] Here, the phrase “pre-cyclic” means that we choose one vertex of the triangle of galaxies at which to define $\theta_{12}$ and the two sides enclosing it, $r_{1}$ and $r_{2}$. Note that in the product of three copies of $\delta_{g}$ needed to form the 3PCF, each of the three galaxies can contribute a $\delta_{v},$ a $\delta^{2}$ and a $\delta^{(2)}$. The pre-cyclic term is written by choosing one galaxy to contribute each of these (it may be the same one). We have chosen the third galaxy to contribute all of these more complicated terms. Since we can then take this galaxy to be at the origin, this approach simplifies the calculation. However, to connect with observations, where there is no “preferred” vertex (galaxy), we eventually must sum cyclically, giving each galaxy in the survey the chance to contribute $\delta_{v},\;\delta^{2},$ and $\delta^{(2)}$. This procedure and its results are described in §7. We now may calculate $\zeta_{{\rm pc}}$ explicitly from perturbation theory.⁴ Motivated by the fact that $\vec{v}_{{\rm bc}}$ is a weighted dipole moment of the density field, we expand the angular dependence of $\zeta_{{\rm pc}}$ using Legendre polynomials as a basis.⁵ We find [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:5][ENDFOOTNOTE] \[\zeta_{{\rm pc}}\left(r_{1},r_{2};\theta_{12}\right)=\sum_{l=0}^{2}\zeta_{{\rm pc }l}\left(r_{1},r_{2}\right)P_{l}\left(\cos\theta_{12}\right),\] (51) with the coefficients as \[\zeta_{{\rm pc}0}\left(r_{1},r_{2}\right)=\left[2b_{1}^{2}b_{2}+\frac{34}{21}b _{1}^{3}\right]\xi_{1}\xi_{2},\] (52) \[\zeta_{{\rm pc}1}\left(r_{1},r_{2}\right)\] (53) \[+2b_{1}^{2}b_{v}\left[V_{3,4}\star\xi_{\times 3}\xi_{\times 4} \right]\left(r_{1},r_{2}\right)\] and \[\zeta_{{\rm pc}2}\left(r_{1},r_{2}\right)=\frac{8}{21}b_{1}^{3}\xi_{1}^{\left[ 2\right]}\xi_{2}^{\left[2\right]}.\] (54) $\xi$ is the linear theory matter correlation function at $z=0$; $\xi^{\left[1\pm\right]}$ and $\xi^{\left[2\right]}$ are defined analogously to the earlier $\xi_{\times}^{\left[1\pm\right]}$ and $\xi_{\times}^{\left[2\right]}$ (equations (43) and (44)) as: (55) and \[\xi^{[2]}\left(r\right)=\int_{0}^{\infty}\frac{dk}{2\pi^{2}}k^{2}j_{2}\left(kr \right)P\left(k\right).\] (56) Recall that numerical subscripts on $\xi$ and $V$ indicate spatial position. Factors of 2 enter all terms above (e.g. $2b_{1}^{2}b_{2}$) due to Wick’s theorem, which reduces the four linear fields implict in each expectation value of equation (50) to a sum of products of three expectation values over two linear fields each, one of which cancels off due to the subtractions in equation (50). The two that remain are the same by homogeneity. The $b_{1}^{3}$ coefficients, $34/21$ in equation (52), $-1$ in equation (53), and $8/21$ in equation (54) can be traced back to equation (37) for the kernel generating the second-order density field, multiplied by the $2$ from Wick’s theorem. The negative sign in equation (53) relative to this kernel is because the $b_{1}^{3}$ term in equation (53) comes from a Legendre polynomial of odd order $(l=1)$ in equation (37) and so picks up a factor of $(-1)^{l}=-1$ when transformed according to equation (60). The functions appearing in the pre-cyclic term, and the products of these functions evaluated on isosceles triangles, are shown in Figure 7. In both plots the structure around $\sim r_{{\rm s}}$ should particularly be noted, as it is ultimately why the 3PCF will have signatures of both the standard BAO and the relative velocity. Finally, in the pre-cyclic 3PCF, $b_{v}$ enters only at $l=1$, the dipole term. Given that it is this term that generates $\left<v_{{\rm bc}}^{2}\right>$ in the first place, this result is not surprising. <figure><img src="content_image/1411.4052/x11.png"><figcaption>Figure 7: The top panel shows the functions appearing in the 3PCF pre-cyclicterms (equations (55) and (56)). Note the variety of structures around theacoustic scale. The bottom panel shows the products in the l=1 pre-cyclic termfor isosceles triangles (i.e. ζpc1(r1,r1)). Note that both l=1 terms havestructure at the acoustic scale; the no-velocity one inherits it from the BAO,while the velocity term inherits it from the velocity Green’s function (notethe similarity by comparing with Figure 2). We have used b1=1,bv=0.01.</figcaption></figure> Figure 8 shows the behavior of the coefficients that enter into the Legendre polynomial expansion of the pre-cyclic term at $l=1$ (equation (53)), the relevant multipole for the velocity. $\zeta_{{\rm pc1}}$ receives an increment from the velocity for roughly isosceles triangles with two sides of length $\sim r_{{\rm s}}.$ This can also be seen in the bottom panel fo Figure 7, which is a trace along the diagonals of the three panels in Figure 8. The relative velocity effect also produces a very modest decrement for triangles with one side of length $\sim r_{{\rm s}}$ and one side of length $\sim 2r_{{\rm s}}$. The part of $\zeta_{{\rm pc}1}$ due to the usual (no relative velocity) terms also has acoustic structure, with an increment for triangles with one or more side of length $\sim r_{{\rm s}}$. Adding the velocity (for $b_{v}>0$) and non-velocity contibutions to $\zeta_{{\rm pc1}}$ together produces a sharp increment for isosceles triangles with two side lengths $\sim r_{{\rm s}}$, while the decrement from the relative velocity for triangles with one side $\sim 2r_{{\rm s}}$ is so modest as to be washed out by the no-velocity contribution. <figure><img src="content_image/1411.4052/x13.png"><figcaption>Figure 8: The top panel shows the P1 coefficient with bv=0 (equation (53)).The middle panel shows the total P1 coefficient with velocity term included.The bottom panel shows the P1 coefficient due to bv alone. Note that therelative velocity subtly enhances the number of triangles with two sides ∼rsby carefully comparing the top two panels; this is made clear in the bottompanel. We have used b1=1,b2=0.1,bv=0.01 and weighted by r21r22/104Mpc4.</figcaption></figure> ## 7 Isolating $b_{v}$ from the 3PCF: cyclic summing and compression In §6, we discussed the effect of the relative velocity on the 3PCF with the simplification that we had chosen to evaluate $v_{s}^{2},\;\delta^{2},$ and $\delta^{(2)}$ at the origin. This corresponds to our knowing which galaxy is contributing these terms to the 3PCF; in practice, we do not know this. Therefore, to give each of the three galaxies in a given triplet a chance to contribute these terms, we must cyclically sum the pre-cyclic 3PCF equation (50) around the triangle specified by $r_{1,}r_{2},$ and $\theta_{12}.$ We verify our prescription for this sum by calculating the reduced 3PCF in a power-law $\xi\propto r^{-2}$ case and comparing with Bernardeau et al. (2002)’s result (their equation (159) and Figure 10). After cyclically summing, we re-project onto the basis of Legendre polynomials to find the radial coefficients for a multipole expansion of our 3PCF (the analog of equation (51)). These are \[\zeta_{l}\left(r_{1},r_{2}\right) =\frac{2l+1}{2}\int_{-1}^{1}d\mu_{12}\bigg{[}\zeta_{{\rm pc}} \left(r_{1},r_{2},\mu_{12}\right)\] (57) \[+\zeta_{{\rm pc}}\left(r_{2},r_{3},\mu_{23}\right)+\zeta_{{\rm pc }}\left(r_{3},r_{1},\mu_{31}\right)\bigg{]}P_{l}(\mu_{12}),\] with $\mu_{12}\equiv\cos\theta_{12}$. Note that $r_{3,}\;\mu_{23}$ and $\mu_{31}$ are all functions of $r_{1,}\;r_{2},$ and $\mu_{12}$, easily found using the law of cosines. The factor of $\left(2l+1\right)/2$ is necessary because ; that is, the Legendre polynomials are an orthogonal but not orthonormal basis. Where in the pre-cyclic terms, we only found terms up to $l=2$, cyclically summing introduces higher orders. Indeed, generically the cyclic sum projects onto an arbitrary number of Legendre polynomials. We present the first few modes, split by bias coefficient, in Figure 9. In Figure 9 we must apply a more complicated weighting than our usual $r^{2}$ weighting to the 3PCF. The 3PCF projections become very large in magnitude for isosceles triangles. This is because when $\mu=1$, the Legendre polynomial being projected onto becomes very large. This heavily weights squeezed triangles with zero opening angle. When these triangles are also isosceles, their third side is zero, causing rapid increase in the functions of side length entering the pre-cyclic terms as these functions go roughly as $r^{-n},\;n>0$. These triangles are precisely those we must exclude, since for one side length $\lesssim 20\;\rm{Mpc}$ we expect perturbation theory to be invalid. We have therefore suppressed the diagonal by multiplying by a Gaussian weighting $\exp[-\left(12\;{\rm Mpc}/(r_{1}-r_{2})\right)^{2}]$.⁶ We also weight by $r_{1}^{2}r_{2}^{2}/10^{4}\;{\rm Mpc}^{4}$ to make the fine structure more apparent and to capture the expected contribution of each spherical shell with volume $dV\propto r_{1}^{2}r_{2}^{2}$. [FOOTNOTE:6][ENDFOOTNOTE] As we might expect, the velocity signature is strongest in $l=1$ but has echoes in $l=0$ and additional structure in $l>1.$ In particular, the velocity bias produces in $\zeta_{1}$ an increment for triangles with two sides $\lesssim r_{{\rm s}}$ and a decrement for triangles with one side $\lesssim r_{{\rm s}}$ and one side $\gtrsim r_{s}$. This is consistent with what we might expect from the pre-cyclic term, which also has an increment and decrement for these respective configurations. Indeed, this can be roughly interpreted as a blurring of the structure present in the pre-cyclic $l=1$ velocity plot (compare Figure 8, bottom panel, with the $l=1$, $b_{v}$ panel in Figure 9). Neither of the other bias coefficients contribute such an effect to $\zeta_{1}$. The velocity bias produces in $\zeta_{2}$ a decrement for triangles with one side $\lesssim r_{{\rm s}}$ and one side $\gtrsim r_{s}$. Neither of the other bias coefficients contribute such an effect to $\zeta_{2}$. The results for $\zeta_{3}$ and $\zeta_{4}$ are very similar to those for $\zeta_{2}$. This is because the higher-order Legendre polynomials are more sharply peaked at $\mu=\pm 1$, and so the triangles with $\mu=1$ (zero opening angle) and $\mu=-1$ are weighted heavily.⁷ Thus as $l\to l+1$ for large $l$, the same small subset of triangles is dominating the projection integral (57), meaning the projections will be similar for $l$ and $l+1$. [FOOTNOTE:7][ENDFOOTNOTE] The 3PCF decomposition we have presented thus far has two independent variables: the two triangle sides $r_{1}$ and $r_{2}$. It is worthwhile to consider whether this information can be compressed into a function of one independent variable. By reducing the dimensionality of the problem, such a compression would ease handling of the covariance matrix associated with an eventual measurement, for example by accelerating the computations required in analyzing a large number of mock catalogs. Further, a clever compression might allow us to avoid entirely the squeezed limit (isosceles triangles with small opening angle so that the third side approaches zero), in which perturbation theory is not expected to be valid because two of the galaxies are very close to each other. Finally, a compression might allow us to focus on the set of triangles where the relative velocity is most pronounced: as Figure 9 and our previous discussion indicate, the relative velocity is localized to specific triangles in all $l$. In particular, on the scales expected to be better controlled observationally (i.e. those with $r_{1}$ and $r_{2}<200\;{\rm Mpc}$), the relative velocity is in a fairly condensed region of the $l=1$ multipole. With these desiderata in mind, we integrate the 3PCF’s multipole moments (displayed with weighting in Figure 9) over one triangle side, but constrain this side to be within some fraction of the side that remains a free variable. We term this approach “compression.” If we integrate over $r_{2}\in[r_{1}/3,2r_{1}/3]$ and constrain $r_{1}>50\;{\rm Mpc}$, we can avoid any triangle side’s nearing zero. The minimum of $r_{2}$ will be $16.7\;{\rm Mpc}$, and by the Triangle Inequality the minimum of the third side, $r_{3}$, will be $33.3\;{\rm Mpc}$. Thus all three triangle sides remain sufficiently large that perturbation theory should be valid. This compression scheme is shown in two different ways in Figure 10; the top panel portrays its effect in configuration space, while the bottom panel shows the region of each panel in Figure 9 integrated over. Notice that the region of integraton captures much of the area where the relative velocity is important in $l=1$. Several intervals for $r_{2}$ were considered before choosing $[r_{1}/3,2r_{1}/3]$; in an observational study the exact interval chosen might differ as optimality will somewhat depend on the signal-to-noise at different scales, an issue we do not treat in detail here. Summarizing mathematically, to obtain the compressions at each $l$ split out by bias coefficient, denoted $\bar{\zeta}_{lb_{1}},\bar{\zeta}_{lb_{2}},$ and $\bar{\zeta}_{lb_{v}}$, we compute \[\bar{\zeta}_{lx}=\frac{4\pi}{V_{{\rm shell}}}\int_{r_{1}/3}^{2r_{ 1}/3}r_{2}^{2}dr_{2}\zeta_{lx}(r_{1},r_{2}),\] (58) where $x$ ranges over $\left\{b_{1},b_{2},b_{v}\right\}$ and $V_{{\rm shell}}=(7/27)(4\pi/3)r_{1}^{3}$ is the volume of the shell being integrated over. Figure 11 shows that each $l$ we study can be used to detect the relative velocity effect, with the strongest constraint coming from $l=1,$ as might be expected given that this is the only mode where the velocity contributes in the pre-cyclic 3PCF. It is encouraging that $l=0$ through $l=2$ even show differences between the effect and no-effect models at scales $\sim 100\;{\rm Mpc},$ as this smaller side length increases the number of triangles available in a given survey volume relative to triangles with side length $r_{s},$ where the effect is most pronounced in $l=1$. Interestingly, the effect in all $l$’s studied at scales $r_{1}>r_{s}$ is substantial, though survey volume limitations may not permit strong constraints to be derived from such large scales. We have weighted the compressions by $r_{1}$ to display the finer structure as well as to simulate shot noise-limited measurements. Shot noise is inversely proportional to $\sqrt{N}$, the number of galaxies in a given spherical shell, so it scales as $1/r$. Meanwhile the signal is proportional to the number of galaxies, so $S/N\propto r$. ## 8 Discussion and conclusions Previous work by Yoo et al. (2011) has argued that the relative velocity effect can shift the scale at which the BAO peak appears in the galaxy correlation function $\xi_{{\rm gg}}$. In this work, we have shown that the relative velocity’s shift to $\xi_{{\rm gg}}$ is generated because the relative velocity itself is non-zero only within the sound horizon at decoupling (Figure 2). More precisely, we have explicitly calculated that the velocity corrections to the correlation function are all generated by convolving relatively narrow functions with a kernel that shares the radial structure of the velocity Green’s function. Hence the spatial structure, in particular the compact support, of the Green’s function is inherited by the velocity corrections to $\xi_{{\rm gg}}$. Thus, these corrections can alter the correlation function only below roughly the sound horizon.⁸ Adding or subtracting from the correlation function only below this scale can change the radius at which the BAO peak occurs; a negative velocity bias pushes it outwards while a positive bias pulls it inwards. [FOOTNOTE:8][ENDFOOTNOTE] To correct this shift and ensure that the BAO remain an accurate cosmological ruler, the relative velocity bias must be measured. Motivated by the previous work in Fourier space of Yoo et al. (2011), we have presented a configuration-space template for fitting the three-point function $\zeta$ to isolate $b_{v}.$ We have shown that the full 3PCF has robust radial signatures of the relative velocity effect that are unique and cannot come from any other bias term at the order in perturbation theory to which we work, in agreement with Yoo et al. (2011). Furthermore, we have offered a useful basis for measuring the full 3PCF and then suggested a further scheme for processing these results. This scheme should unambiguously expose the relative velocity’s signature while avoiding the regimes in which perturbation theory is expected to be inadequate. It will also likely ease handling of the covariance matrix if large numbers of mock catalogs are to be used for computing error bars. Previous attempts to constrain the relative velocity observationally have focused on Fourier space. We have already alluded to the advantages of configuration space in §1, and we revisit these points here. On the theory side, our configuration space approach has exposed the relatively simple spatial structure of the relative velocity. Fourier-space work on the bispectrum does not render transparent which triangle configurations are optimal for velocity bias constraints. Our configuration space approach immediately shows that, on scales small enough to measure with current surveys, the velocity signature is localized to a small region of triangle side lengths and a single multipole ($l=1$). This localized signature (see Figure 9) naturally suggests that an integral over the desirable region would enhance the velocity signal-to-noise, an intuition borne out by our compression scheme results (Figure 11), which, it should be noted, are weighted to reflect shot noise. On the practical side, there are also considerable advantages to working in configuration space. As we have already discussed in §1, edge-correction is much simpler in configuration space. Further, Pan & Szapudi’s (2005) measurement of the monopole moment of the 3PCF shows it is possible in practice to extract information from a multipole decomposition of the 3PCF. Looking forward, in forthcoming work we will present a fast algorithm for computing the multipole moments of the 3PCF while accounting for edge correction. This work will also address in more detail the covariance matrix of the 3PCF. Thus far, the bispectrum technique of Yoo et al. (2011) has not been used to constrain the relative velocity. Therefore three-point statistics remain an entirely unexploited means of gaining traction on the relative velocity, a situation we hope the configuration space signatures of this work will improve. Nonetheless, recent work by Yoo & Seljak (2013) has used measurements of the power spectrum in the Constant Mass (CMASS) sample of the Sloan Digital Sky Survey (SDSS) DR9 (260,000 galaxies) to compute a root-mean square shift of $0.57\%$ in the BAO peak position, showing this shift can potentially be of order the entire error budget for the BAO distance measurement. It should be noted that Yoo & Seljak’s best-fit parameter values imply no relative velocity at all (the root-mean square shift is from integrating over the probability distribution of the linear, non-linear, and velocity biases). On the other hand, a velocity bias of as large as $\sim 2\%$ (in our units; they use different values of $b_{1}$ and $b_{2}$ from our fiducial case so $b_{v}$ must be rescaled for comparison) is consistent with their measurement at one sigma.⁹ [FOOTNOTE:9][ENDFOOTNOTE] A limiting factor in their analysis is the growth of the error bars at smaller wavenumber (large scales; see their Figure 6); as our analysis shows (see our Figure 6), large scales are important for the relative velocity’s addition to the correlation function. In this context two points should be made. First, even when restricted to measurements of the power spectrum (or correlation function), controlling the error bars on large scales should have significant rewards, meaning increasing the number of galaxies used for the measurement is highly desirable. Thus use of the full sample of $\sim 1$ million galaxies in the most recent SDSS data should offer compelling new constraints. Second, the distinctiveness of the 3PCF signature, where obtaining the dipole $(l=1)$ moment already begins to isolate the velocity signature, should render the large-scale error bars less problematic even at fixed number of galaxies. Finally, a separate issue our work clarifies is how to constrain the non-linear bias. Our compressions show that the higher multipoles are extremely insensitive to the non-linear bias, while for $l=0$ and $l=1$ it contributes much more strongly than the linear bias for a given magnitude of both $(b_{1}=b_{2}=1)$ (see Figure 12). Given that the non-linear bias enters only at $l=0$ in our pre-cyclic calculation (equation (52)), we indeed expect its projection onto $l=0$ and close-by multipoles to be strongest. This suggests that measuring different multipoles of the 3PCF, compressed as we outline, should offer a robust way to separate the linear bias from the non-linear bias. As the Yoo & Seljak (2013) best-fit measurement from CMASS shows, the non-linear bias may be $\sim 0.2$ in our units (their Figure 6), in which case Figure 11 shows it would contribute $\gtrsim 20\%$ of what the linear bias does at $50\;{\rm Mpc}$ scales in $l=0,$ but negligibly in $l=2$ and up. We hope the compression scheme presented here will, independent of its utility for relative velocity measurements, provide a new method to extract the non-linear bias from 3PCF measurements. Robustly separating the non-linear bias from other effects may also prove helpful in correcting the BAO peak shift if one is found. Yoo & Seljak (2013) find that non-linear bias $b_{2}$ can also shift the BAO peak. Historically $b_{2}$ has been constrained using the 3PCF or bispectrum (Scoccimarro et al. 2001; Verde et al. 2002; Wang et al. 2004; Gaztañaga et al. 2005; Gaztañaga et al. 2009; McBride et al. 2011b; Marin 2011; Marin et al. 2013; Guo et al. 2014), and our compressions offer a particularly clear way to isolate $b_{2}$ from the linear bias $b_{1}$ and from the velocity bias $b_{v}$. This separation should aid accurate measurements of both $b_{2}$ and $b_{v}$ and ensure the peak shift can be corrected. With the percent-level constraints on the cosmic distance scale the BAO method offers through surveys such as BOSS (Anderson et al. 2014) and the concomitant limits on dark energy’s equation of state $w=p/\rho$ (Aubourg et al. 2014), understanding any sources of bias is essential. In future work, we will implement the strategy discussed here to measure $b_{v}$ using data from SDSS-III and assess whether the imprint of the relative velocity between baryons and dark matter might be present at a level relevant to modern BAO surveys. <figure><img src="content_image/1411.4052/x16.png"><figcaption>Figure 9: Cyclically summed, weighted (see equation (57) and surrounding text)3PCF split by bias coefficient and projected onto Legendre polynomials P0through P4. Each row corresponds to one multipole, and the columns are aslabeled at the top of the plot. The axes are r1 (horizontal) and r2 (vertical)in Mpc, and we have divided the linear bias (b1) plots by 10 so that allcolumns can be on the same colorbar. The weighting isr21r22/104Mpc4exp[−(12Mpc/(r1−r2))2]; we have suppressed the diagonal becauseit is dominated by squeezed triangles for which perturbation theory is notvalid. We have used b1=1,b2=0.1,bv=0.01. Note the distinctive velocitysignatures, especially in l=1: there is an increment for triangles with twosides ≲rs and a decrement for triangles with one side ≲rs and one side ≳rs.</figcaption></figure> <figure><img src="content_image/1411.4052/x17.png"><figcaption>Figure 10: Compression scheme (see also equation (58)). Each dot represents agalaxy. We integrate the cyclically summed 3PCF over all angles and over aspherical shell with r2 from (1/3)r1 to (2/3)r1. This is to capture the regionof largest signal in Figure 9 while avoiding the squeezed limit where one sideof the triangle of galaxies becomes so small as to invalidate linearperturbation theory. The region of the bv, l=1 plot being integrated over inFigure 9 is also shown above as an example of what this scheme does to eachsub-plot in Figure 9\. Note that the diagonal is suppressed in Figure 9 but wedo not apply this suppression when integrating for the compressions.</figcaption></figure> <figure><img src="content_image/1411.4052/x19.png"><figcaption>Figure 11: Compressions by integrating along a wedge in Figure 9; see Figure10 for visualization of this scheme. The no-velocity model is in dashed cyan,while the model with velocity is in purple X’s. These show that the velocityproduces distinctive signatures, especially at l=1. Worth noting also is thatl=0 and l=1 display signal even at scales r∼100Mpc<rs. These plots have beenweighted by r1 to show the finer structure and model a shot-noise limitedmeasurement, and scaled up by 105 to make the vertical axis more compact.Notice that the non-linear bias (b2) does not contribute much to any of theseplots, but especially so in the higher multipoles.</figcaption></figure> <figure><img src="content_image/1411.4052/x20.png"><figcaption>Figure 12: To show what the contribution from b2 would look like in an extremecase, we present above results with b1=1,b2=1, and bv=0.01. Even in this case,the relative velocity still has a distinct signature, especially in l=1 if wefocus on the l for which the relative velocity has a signature on the smallestscales. Taking the relative velocity aside, also note that here again we seethat the higher multipoles are much more sensitive to linear bias than non-linear bias, providing a new means to discriminate between the two and gain arobust measurement of each.</figcaption></figure> ## Acknowledgements ZS thanks Neta Bahcall, Jo Bovy, Pierre Christian, Cora Dvorkin, Anastasia Fialkov, Doug Finkbeiner, Margaret Geller, JR Gott III, Laura Kulowski, Avi Loeb, Chung-Pei Ma, Robert Marsland, Cameron McBride, Philip Mocz, Jim Moran, Stephen Portillo, Matthew Reece, Mohammadtaher Safarzadeh, David Spergel, and Harvey Tananbaum for useful discussions during this work. ZS especially thanks Harry Desmond, Meredith MacGregor, Smadar Naoz, Yuan-Sen Ting, and Anjali Tripathi for comments on the manuscript. We also thank the anonymous referee for comments that considerably improved the scientific content and presentation of this work. 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Wyithe JSB & Djikstra M, 2011, MNRAS 415:3929-3950. ## Appendix We begin by proving two theorems on the Fourier transform of a function that can be represented as a Legendre polynomial of $\hat{r}_{1}\cdot\hat{r}_{2}$ times a function of $r_{1}$ and $r_{2}$, i.e. of the form \[K^{[l]}(\vec{r}_{1},\vec{r}_{2}) =K_{{\rm r}}^{[l]}\left(r_{1},r_{2}\right)P_{l}\left(\hat{r}_{1} \cdot\hat{r}_{2}\right)\] (59) \[=K_{{\rm r1}}^{[l]}\left(r_{1}\right)K_{{\rm r2}}^{[l]}\left(r_{2 }\right)P_{l}\left(\hat{r}_{1}\cdot\hat{r}_{2}\right),\] where in the last equality we additionally assume that the radial piece $K_{{\rm r}}^{[l]}(r_{1},r_{2})$ can be split into a product of two functions with the same form. We first show that the Fourier transform of a function of this form will have the same angular dependence as the original function. We define $\tilde{K}^{[l]}(\vec{k}_{1},\vec{k}_{2})={\rm FT}\left\{K^{[l]}(\vec{r}_{1}, \vec{r}_{2})\right\}(\vec{k}_{1},\vec{k}_{2}),$ where ${\rm FT}$ denotes a 6-D Fourier transform. We prove that \[\tilde{K}^{[l]}(\vec{k}_{1},\vec{k}_{2}) =\left(-1\right)^{l}\mathcal{H}_{l}\left\{K_{{\rm r}}^{[l]}\left( r_{1},r_{2}\right)\right\}\left(k_{1},k_{2}\right)\] (60) \[\times P_{l}\left(\hat{k}_{1}\cdot\hat{k}_{2}\right).\] $\mathcal{H}_{l}$ is a 2-D transform defined as \[\mathcal{H}_{l}\left\{f\left(r_{1}\right)g\left(r_{2}\right) \right\}(k_{1},k_{2})=h_{l}\left\{f(r_{1})\right\}(k_{1})h_{l}\left\{g(r_{2}) \right\}(k_{2});\] \[h_{l}\left\{f\left(r\right)\right\}\left(k\right)=4\pi\int drr^{ 2}j_{l}\left(kr\right)f\left(r\right).\] (61) $\mathcal{H}_{l}^{-1}$ is defined analogously in terms of $h_{l}^{-1}$, given by¹⁰ [FOOTNOTE:10][ENDFOOTNOTE] \[h_{l}^{-1}\left\{\tilde{f}\left(k\right)\right\}\left(r\right)=\int\frac{k^{2} dk}{2\pi^{2}}j_{l}\left(kr\right)\tilde{f}\left(k\right).\] (62) With this notation in place, we now prove equation (60). We will need the following two identities. First, the spherical harmonic addition theorem is \[P_{l}\left(\hat{r}_{1}\cdot\hat{r}_{2}\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l} Y_{lm}\left(\hat{r}_{1}\right)Y_{lm}^{}\left(\hat{r}_{2}\right)\] (63) where the $Y_{lm}$ are spherical harmonics and star means conjugate. Second, the expansion of the plane wave in spherical harmonics is (64) The 6-D Fourier transform of $K^{[l]}(\vec{r}_{1},\vec{r}_{2})$ is \[\tilde{K}^{[l]}(\vec{k}_{1},\vec{k}_{2})=\int d^{3}\vec{r}_{1}d^{3}\vec{r}_{2} e^{i\vec{k}_{1}\cdot\vec{r}_{1}}e^{i\vec{k}_{2}\cdot\vec{r}_{2}}K_{{\rm r}}^{[ l]}(r_{1},r_{2})P_{l}(\hat{r}_{1}\cdot\hat{r}_{2}).\] (65) Using the spherical harmonic addition theorem (63) to replace $P_{l}(\hat{r}_{1}\cdot\hat{r}_{2})$ and applying the plane wave expansion (64) to replace the plane waves we have \[\tilde{K}^{[l]}(\vec{k}_{1},\vec{k}_{2})=\left(4\pi\right)^{3}\times\] (66) \[\sum_{l_{1},m_{1},l_{2},m_{2}}\sum_{m=-l}^{l}\frac{i^{l_{1}+l_{2} }}{2l+1}Y_{l_{1}m_{1}}\left(\hat{k}_{1}\right)Y_{l_{2}m_{2}}^{}\left(\hat{k}_ {2}\right)\] \[\times\int d^{3}\vec{r}_{1}d^{3}\vec{r}_{2}j_{l_{1}}\left(k_{1}r_ {1}\right)j_{l_{2}}\left(k_{2}r_{2}\right)K_{{\rm r1}}^{[l]}\left(r_{1}\right) K_{{\rm r2}}^{[l]}\left(r_{2}\right)\] \[\times Y_{lm}\left(\hat{r}_{1}\right)Y_{lm}^{}\left(\hat{r}_{2} \right)Y_{l_{1}m_{1}}^{}\left(\hat{r}_{1}\right)Y_{l_{2}m_{2}}\left(\hat{r}_{ 2}\right).\] Using the orthogonality of the spherical harmonics integrated over solid angle we have \[\tilde{K}^{[l]}(\vec{k}_{1},\vec{k}_{2})=\left(4\pi\right)^{3} \left(-1\right)^{l}\sum_{m=-l}^{l}\frac{1}{2l+1}Y_{lm}\left(\hat{k}_{1}\right) Y_{lm}^{}\left(\hat{k}_{2}\right)\] \[\times\int dr_{1}dr_{2}r_{1}^{2}r_{2}^{2}j_{l}\left(k_{1}r_{1} \right)j_{l}\left(k_{2}r_{2}\right)K_{{\rm r1}}^{[l]}\left(r_{1}\right)K_{{\rm r 2}}^{[l]}\left(r_{2}\right).\] (67) We then have the desired equation (60) using in sequence equations (63), (62), and (61). We now prove a useful result on the convolution of two functions of the form given in equation (59). The second function is \[Q^{[n]}(\vec{r}_{1},\vec{r}_{2})=Q_{{\rm r}}^{[n]}(r_{1},r_{2})P_{n}(\hat{r}_{ 1}\cdot\hat{r}_{2})\] (68) with FT \[\tilde{Q}^{[n]}(\vec{k}_{1},\vec{k}_{2})=\left(-1\right)^{n}\mathcal{H}_{n} \left\{Q_{{\rm r}}^{[n]}(r_{1},r_{2})\right\}(k_{1},k_{2})P_{n}(\hat{k}_{1} \cdot\hat{k}_{2})\] using equation (60). By the Convolution Theorem, $K^{[l]}\star Q^{[n]}$ is the inverse FT of the product of the two functions’ FTs, and using Adams’ (1878) identity for the product of 2 Legendre polynomials \[P_{k}\left(x\right)P_{l}\left(x\right)=\sum_{m=\|k-l\|}^{k+l}\left(\begin{array} []{ccc}k&l&m\\ 0&0&0\end{array}\right)^{2}\left(2m+1\right)P_{m}\left(x\right)\] (69) this product is \[\tilde{K}^{[l]}(\vec{k}_{1},\vec{k}_{2})\tilde{Q}^{[n]}(\vec{k}_{ 1},\vec{k}_{2})=\] (70) \[\left(-1\right)^{n+l}\mathcal{H}_{l}\left\{K_{{\rm r}}^{[l]}(r_{1 },r_{2})\right\}\mathcal{H}_{n}\left\{Q_{{\rm r}}^{[n]}(r_{1},r_{2})\right\}\] \[\times\sum_{m=\|l-n\|}^{l+n}\left(\begin{array}[]{ccc}l&n&m\\ 0&0&0\end{array}\right)^{2}\left(2m+1\right)P_{m}(\hat{k}_{1}\cdot\hat{k}_{2}),\] where the $2\times 3$ matrix is a Wigner 3j-symbol. Using the same approach as for equation (60) to find the inverse FT we have \[K^{[l]}(\vec{r}_{1},\vec{r}_{2})\star Q^{[n]}(\vec{r}_{1},\vec{r }_{2})=\] (71) \[\sum_{m=\|l-n\|}^{l+n}\left(-1\right)^{n+l-m}\mathcal{H}_{m}^{-1} \left\{\mathcal{H}_{l}\left\{K_{{\rm r}}^{[l]}(r_{1},r_{2})\right\}\mathcal{H} _{n}\left\{Q_{{\rm r}}^{[n]}(r_{1},r_{2})\right\}\right\}\] \[\times\left(\begin{array}[]{ccc}l&n&m\\ 0&0&0\end{array}\right)^{2}\left(2m+1\right)P_{m}\left(\hat{r}_{1}\cdot\hat{r} _{2}\right).\]
1911.06022	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 92844, "num_imgs": 2, "llama3_tokens_count": 26748 }	[ "content_image/1911.06022/figure1.png", "content_image/1911.06022/x1.png" ]	# Gauge Theories with Ultracold Atoms João C. Pinto Barros João C. Pinto Barros Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland, ¹Michele Burrello Center for Quantum Devices and Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark, ²Andrea Trombettoni CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea 265, I-34136 Trieste, Italy, ³ Michele Burrello and Andrea Trombettoni João C. Pinto Barros Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland, ¹Michele Burrello Center for Quantum Devices and Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark, ²Andrea Trombettoni CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea 265, I-34136 Trieste, Italy, ³ [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:6][ENDFOOTNOTE] ###### Abstract We discuss and review in this chapter the developing field of research of quantum simulation of gauge theories with ultracold atoms. ## 1 Introduction During the School on “Strongly Coupled Field Theories for Condensed Matter and Quantum Information Theory”, held in Natal (Brazil) in the days 2 - 14 August 2015, one of the authors gave a course on “Quantum Simulations of Gauge Fields with Ultracold Atoms”. The course was meant to be informal, at the blackboard, with time for discussions and to interact with the younger part of the audience. Subsequently, in 2017 J. C. Pinto Barros obtained the PhD in SISSA (Trieste) defending a Thesis on “Field and Gauge Theories with Ultracold Atoms”, under the supervision of Andrea Trombettoni and Marcello Dalmonte and with Michele Burrello and Enrique Rico Ortega acting as external referees. The present chapter is based on the Natal’s course and on the above mentioned PhD Thesis of J. C. Pinto Barros [1]. The latter Thesis is available at http://www.statphys.sissa.it/wordpress/?page_id=1095 The course gave an introductory discussion on lattice gauge theories, and then moved to explain how to simulate gauge potentials and gauge fields. A prior knowledge of ultracold atomic systems was assumed, even though during the lectures the corresponding concepts and notions were briefly reminded. ## 2 Gauge theories A gauge theory is a model that has a gauge symmetry. Such symmetry can be seen as a redundancy in the description of the degrees of freedom. In other words, this means that one can have two mathematically distinct solutions of the equations describing the system and nonetheless they describe the same physical situation. The paradigmatic example is classical electrodynamics. It describes the behavior of the electric field $\vec{E}\left(t,\vec{x}\right)$ and the magnetic field $\vec{B}\left(t,\vec{x}\right)$ in the presence of an electric charge density $\rho\left(t,\vec{x}\right)$ and the current density $\vec{j}\left(t,\vec{x}\right)$. The system is governed by the Maxwell equations: \[\begin{array}[]{cc}\nabla\cdot\vec{E}\left(t,\vec{x}\right)=\rho\left(t,\vec{x }\right)\,;\hfill&\nabla\times\vec{B}\left(t,\vec{x}\right)-\partial_{t}\vec{E }\left(t,\vec{x}\right)=\vec{j}\left(t,\vec{x}\right)\,;\\ \nabla\cdot\vec{B}\left(t,\vec{x}\right)=0\,;\hfill&\nabla\times\vec{E}\left(t ,\vec{x}\right)+\partial_{t}\vec{B}\left(t,\vec{x}\right)=0\,.\hfill\end{array}\] (1) In the above equations and in the rest of this Chapter, natural units shall be adopted. The homogeneous equations, which are independent of charges and currents, can be straightforwardly solved by introducing a scalar potential $\phi\left(t,\vec{x}\right)$ and a vector potential $\vec{A}\left(t,\vec{x}\right)$: \[\vec{E}\left(t,\vec{x}\right)=-\nabla\phi\left(t,\vec{x}\right)-\partial_{t} \vec{A}\left(t,\vec{x}\right)\,,\qquad\vec{B}\left(t,\vec{x}\right)=\nabla \times\vec{A}\left(t,\vec{x}\right)\,.\] (2) Using these two relations the last two equations in (1) are fulfilled and the ones from the first row can be written in terms of $\phi\left(t,\vec{x}\right)$ and $\vec{A}\left(t,\vec{x}\right)$. After a solution is found, it can be plugged in Equation 2 in order to obtain the electric and magnetic fields. However, not all different $\phi\left(t,\vec{x}\right)$ and $\vec{A}\left(t,\vec{x}\right)$ will give different electric and magnetic fields. In fact if two fields $\phi\left(t,\vec{x}\right)^{\prime}$ and $\vec{A}\left(t,\vec{x}\right)^{\prime}$ are related to another solution $\phi\left(t,\vec{x}\right)$ and $\vec{A}\left(t,\vec{x}\right)$ by: \[\phi\left(t,\vec{x}\right)^{\prime}=\phi\left(t,\vec{x}\right)+\partial_{t} \alpha\left(t,\vec{x}\right)\,,\qquad\vec{A}\left(t,\vec{x}\right)^{\prime}= \vec{A}\left(t,\vec{x}\right)-\nabla\alpha\left(t,\vec{x}\right)\,,\] (3) for some regular function $\alpha\left(t,\vec{x}\right)$, then the electric and magnetic fields, given by Equation 2, remain unchanged. This means that the solutions $\phi,\vec{A}$ and $\phi^{\prime},\vec{A}^{\prime}$ correspond to the same physical situation and therefore they are just redundant descriptions of the same physics. The transformations of Equation 3 are called gauge transformations. The existence of a gauge symmetry does not require the field to be dynamical. Consider a charged quantum particle in a background of a classical electromagnetic field. The Schrödinger equation for this system can be written as the equation in the absence of any field and “correcting” the canonical momentum $\vec{p}\rightarrow\vec{p}-e\vec{A}$. In the presence of an electromagnetic field the mechanical momentum, associated with the kinetic energy of the particle and denoted here by $\vec{\pi}$, is no longer the canonical momentum given by $\vec{p}$. The relation between them is $\vec{\pi}=\vec{p}-e\vec{A}$ which is at the core of this substitution. The same happens for the time derivative with the scalar potential $i\partial_{t}\to i\partial_{t}-e\phi$. Therefore, the Schrödinger equation reads, in the absence of any other interactions: \[\left(i\partial_{t}-e\phi\right)\psi\left(t,\vec{x}\right)=\left(-i\nabla-e \vec{A}\right)^{2}\psi\left(t,\vec{x}\right)\,.\] (4) Also this equation is invariant under the transformation 3 provided that the wave function is transformed by a position-dependent phase: \[\psi\left(t,\vec{x}\right)=e^{ie\alpha\left(t,\vec{x}\right)}\psi\left(t,\vec{ x}\right)\,.\] (5) Given the space and time dependence of this transformation, it is denoted as a local gauge symmetry. In quantum field theory an illustrative example is provided by QED. The Lagrangian is given by: \[{\cal L}=\bar{\psi}\left(\gamma^{\mu}\left(i\partial_{\mu}-eA_{\mu}\right)-m \right)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,.\] (6) Implicit sum over repeated indices is assumed. $\gamma^{\mu}$ are the gamma matrices satisfying the Clifford algebra $\left\{\gamma^{\mu},\gamma^{\nu}\right\}=2\eta^{\mu\nu}$, $\eta^{\mu\nu}$ is the Minkowski metric $\eta=\mathrm{Diag}\left(1,-1,-1,-1\right)$, $\psi$ the Dirac spinor and $\bar{\psi}=\psi^{\dagger}\gamma^{0}$. The indices $\mu$ run from $0$ to $3$ where $0$ corresponds to the time index. $A_{\mu}$ is called gauge field and the last term of the Lagrangian corresponds to its kinetic term where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Also in this case there is a local set of transformations that leave this Lagrangian invariant. Explicitly: (7) One can define the covariant derivative $D_{\mu}=\partial_{\mu}+ieA_{\mu}^{a}$ such that, under a gauge transformation, $D_{\mu}\psi\to e^{i\alpha\left(x\right)}D_{\mu}\psi$. In this way, the local gauge symmetry becomes apparent. This is an example of a $U\left(1\right)$ gauge theory: a gauge transformation is defined, at each point, by phases $\alpha\in\left[0,2\pi\right[$ which combine according to the group $U\left(1\right)$. This construction can be generalized to other gauge groups, like $\mathbb{Z}_{N}$, or even non-Abelian, like $SU\left(N\right)$, for $N$ an integer number. For example, the Kitaev toric code is a $\mathbb{Z}_{2}$ (Abelian) gauge theory [2] whereas Quantum Chromodynamics (QCD), the theory that describes strong interactions in particle physics, is a $SU\left(3\right)$ (non-Abelian) gauge theory [3; 4; 5]. In the following a brief description of non-Abelian $SU\left(N\right)$ gauge invariance in quantum field theory is provided. For more details see, for example, [6]. In order to explore these other symmetries, extra indices must be inserted (in the paradigmatic example of QCD these are the color indices). To simplify the notation, whenever $\psi$ it is used it is meant: \[\psi\equiv\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\\ \vdots\\ \psi_{n}\end{array}\right)\] (8) where each one of the $\psi_{i}$ corresponds to a (four-component in $3+1$ dimensions) Dirac spinor. Consider then a general symmetry group and a respective set of generators represented by Hermitian $n\times n$ matrices $t^{a}$. The goal is to build a Lagrangian which is invariant under the set of local transformations \[\psi\left(x\right)\to e^{i\alpha^{a}\left(x\right)t^{a}}\psi\left(x \right)\,.\] (9) This is a unitary transformation that mixes the $n$ components of the vector 8 following a $n-$dimensional representation of the gauge group element $e^{i\alpha^{a}\left(x\right)t^{a}}$. The gauge field becomes in turn a matrix which can be parametrized as $A_{\mu}^{a}t^{a}$. Under a gauge transformation the field transforms as \[A_{\mu}^{a}t^{a}\to e^{i\alpha^{a}\left(x\right)t^{a}}\left(A_{\mu}^{a }t^{a}+\frac{i}{g}\partial_{\mu}\right)e^{-i\alpha^{a}\left(x\right)t^{a}}.\] (10) Writing the covariant derivative as $D_{\mu}=\partial_{\mu}-igA_{\mu}^{a}t^{a}$ one finds $D_{\mu}\psi\left(x\right)\to e^{i\alpha^{a}\left(x\right)t^{a}}D_{\mu} \psi\left(x\right)$. In this way $\bar{\psi}\left(\gamma^{\mu}D_{\mu}-m\right)\psi$ is a gauge invariant operator which includes the fermionic kinetic term and the matter-gauge coupling. Note that $\bar{\psi}$ is to be interpreted as line vector with components $\bar{\psi}_{i}$ and $\gamma^{\mu}$ are diagonal on the color indices, i.e. act the same for every color by standard matrix multiplication $\gamma^{\mu}\psi_{i}$. In order to define the gauge field dynamics, its gauge invariant kinetic term must be inserted. A possible way to derive its form is by considering the commutator $\left[D_{\mu},D_{\nu}\right]=it^{a}F_{\mu\nu}^{a}$. Putting it differently, a general form for $F_{\mu\nu}^{a}$ can be obtained from this formula. Explicit computation yields $F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{ \mu}^{b}A_{\nu}^{c}$ where the structure constants $f^{abc}$ are given by $\left[t^{a},t^{b}\right]=it^{c}f^{abc}$ and depend only on the symmetry group. From the transformation law for the covariant derivatives, one can see that $F_{\mu\nu}^{a}F^{a\mu\nu}$ is gauge invariant. The full Lagrangian can then be written as \[{\cal L}=\bar{\psi}\left(\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{1}{4}F_{\mu\nu }^{a}F^{a\mu\nu}\] (11) The perspective of implementing these kind of models in table top experiments is very appealing. First of all, it could give answers to very fundamental questions in physics like, for example, the exploration of the phase diagram of QCD. That is certainly a long term challenge and the path envisioned towards it involves the implementation of simpler intermediate steps. While QCD has a $SU\left(3\right)$ gauge symmetry and involves $3+1$ dimensions, this does not need to be the main target. A much simpler case of a $U\left(1\right)$ gauge symmetry in $1+1$ dimensions is already of great interest. In fact, this was the target of the first experimental implementation of a lattice gauge theory [7](to be discussed in Section 4.3). Step by step one may think to be able to realize more and more complex models. It is clear that if these models are realized they become interesting on their own both theoretically and experimentally. In particular, for example, it may also be advantageous to have situations where only certain degrees of freedom live in higher dimensions keeping others in lower dimensionality [8; 9; 10] which could be used to simulate systems with long-range interactions, which have been the subject of an intense investigation in the last years [11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. ### Gauge symmetry on the lattice #### 2.1.1 Static fields Following the discussion in the previous Section, a many-body Hamiltonian in the presence of a magnetic field can be obtained by replacing the momentum components for each particle by $p_{i}\to p_{i}-eA_{i}$. On the lattice, instead, this can be approximated by the Peierls substitution where the hopping parameters become complex. This is valid in a tight-binding regime and for a slow varying magnetic field. Explicitly the kinetic term is modified according to \[K=\underset{\vec{r},j}{\sum}t_{j}\hat{a}_{\vec{r}+\hat{j}}^{\dagger}\hat{a}_{ \vec{r}}+\mathrm{h.c.}\rightarrow\underset{\vec{r},j}{\sum}t_{j}\hat{a}_{\vec{ r}+\hat{j}}^{\dagger}e^{i\theta_{j}\left(\vec{r}\right)}\hat{a}_{\vec{r}}+ \mathrm{h.c.}\] (12) In the previous equation the sum of $\vec{r}$ is taken over the lattice sites and the sum of $j$ is taken over all $d$ directions corresponding to the dimensionality of the system. The angle $\theta_{j}\left(\vec{r}\right)$ is just a phase that can depend, on general grounds, on both the direction of the hopping and the position. The key difference is that this phase here is non dynamical, so there is no kinetic term for it. This simply corresponds to allow for the hopping parameter of the particles on the lattice to be complex. Similarly to the models in continuum space, not all complex hoppings represent different physical scenarios as there is gauge invariance. In Section 3, several examples of techniques to engineer complex phases on the hopping parameters are discussed. Reviews can be found in [21; 22; 23; 24]. #### 2.1.2 Dynamical fields In order to study a dynamical quantum (lattice) gauge theory, the lattice system under analysis must include also the degrees of freedom for the gauge fields and the complex hopping parameters are therefore promoted to operators acting on these degrees of freedom. Such degrees of freedom are usually associated to the lattice edges and their kinetic term must be supplied. A constructive way to define such a system consists on taking the Lagrangian 11, write an Hamiltonian and perform a naive discretization. This offers in turn a recipe to engineer possible quantum simulations of these systems: a straightforward way of proceeding is indeed to create a system implementing the specific Hamiltonian of the lattice gauge theory, therefore it is useful to consider such theories in their Hamiltonian formulation [25]. To describe the $U\left(1\right)$ case, it is useful to introduce the following link operators, acting on the gauge degrees of freedom: \[U_{\vec{r},j}=\exp\left(ie\overset{a\left(\vec{r}+\hat{j}\right)}{\underset{a \vec{r}}{\int}}dxA_{j}\left(x\right)\right)\,,\qquad L_{\vec{r},j}=\frac{E_{ \vec{r},j}}{e}\,.\] (13) $U$ and $L$ are operators corresponding respectively to the connection and electric field of the theory (see, for example, [26] for more details). Based on these operators, we can define the Hamiltonian \[H=-\frac{i}{2a}\underset{\vec{r},j}{\sum}\left(\psi_{\vec{r}}^{\dagger}U_{\vec {r},j}\psi_{\vec{r}+\hat{j}}-\mathrm{h.c.}\right)+m\underset{n}{\sum}\psi_{ \vec{r}}^{\dagger}\psi_{\vec{r}}+\frac{ae^{2}}{2}\underset{n}{\sum}L_{\vec{r}, j}^{2}\,,\] (14) which reproduces the correct continuum theory when the _naive_ continuum limit is taken. In the expressions above $a$ is the lattice spacing, $\vec{r}$ are the lattice points and $j$ labels the links connected to it. $E_{\vec{r}j}$ is the discretized version of the electric field which is the conjugate momentum of $A_{j}$ in the continuum version. The commutation relations between the link operators are \[\left[L_{\vec{r},i},U_{\vec{r}^{\prime},j}\right]=\delta_{\vec{r}\vec{r}^{ \prime}}\delta_{ij}U_{\vec{r},i},\quad\left[L_{\vec{r},i},U_{\vec{r}^{\prime}, j}^{\dagger}\right]=-\delta_{\vec{r}\vec{r}^{\prime}}\delta_{ij}U_{\vec{r},i}^ {\dagger},\quad\left[U_{\vec{r},i},U_{\vec{r}^{\prime},j}\right]=\left[U^{ \dagger}_{\vec{r},i},U_{\vec{r}^{\prime},j}\right]=0.\] (15) We pause here to point out a couple of subtleties. The first concerns the so-called _naive_ continuum limit, obtained by simply sending $a\to 0$. While this works well for bosons, fermions suffer from the so-called "fermion doubling problem". When this limit is taken with more care each fermion flavor on the lattice gives rise to $2^{d}$ fermion flavors on the continuum, being $d$ the number of discretized dimensions. The Nielsen-Ninomiya Theorem [27; 28; 29] states that this is always the case when the fermion action is real, local and invariant under lattice translations and chiral transformations. There are alternative approaches to evade the Nielsen-Ninomiya Theorem which have their own advantages and disadvantages. A possible choice, popular among the quantum simulation community, is provided by staggered fermions [30] (also known as Kogut-Susskind fermions). The idea consists on distributing the spinor components among different lattice sites. In this way, instead of a spinor per site, one has only one fermion. Only for the Hamiltonian formulation of the $1+1D$ theory the fermion doubling problem can be completely solved in this way. In this case the Hamiltonian becomes \[H=-\frac{i}{2a}\underset{n}{\sum}\left(c_{n}^{\dagger}U_{n}c_{n+1}-\mathrm{h.c .}\right)+m\underset{n}{\sum}\left(-1\right)^{n}c_{n}^{\dagger}c_{n}+\frac{ae^ {2}}{2}\underset{n}{\sum}L_{n}^{2}.\] (16) Spinors can be reconstructed from $\psi_{n}=\left(c_{2n},c_{2n+1}\right)^{T}{/\sqrt{a_{st}}}$. In higher dimensions the most non-trivial step consists on the existence of plaquette terms or, in other words, an energy cost for magnetic fields. These are gauge-invariant terms which must be present in order to fully represent the gauge theory. The absence of these terms is related to the strong coupling limit of the theory. On a 2D square lattice, the plaquette term originating at the point $\vec{r}$ is $U_{\square}=U_{\vec{r},x}U_{\vec{r}+\hat{x},y}U_{\vec{r}+\hat{y},x}^{\dagger}U _{\vec{r},y}^{\dagger}$ consisting on the smallest loops possible to draw on the lattice. The Hamiltonian for $d$ spatial dimensions takes the form: \[\begin{split} H=&-\frac{i}{2a}\underset{\vec{r},i}{ \sum}\left(-1\right)^{r_{1}+\ldots+r_{i-1}}\left(c_{\vec{r}}^{\dagger}U_{\vec{ r},i}c_{\vec{r}+\hat{i}}-\mathrm{h.c.}\right)+m\underset{\vec{r}}{\sum}\left(- 1\right)^{r_{1}+\ldots+r_{d}}c_{\vec{r}}^{\dagger}c_{\vec{r}}\\ &+\frac{a^{2-d}e^{2}}{2}\underset{\vec{r},i}{\sum}L_{\vec{r},i}^{ 2}-\frac{a^{d-4}}{4e^{2}}\underset{\square}{\sum}\left(U_{\square}+U_{\square} ^{\dagger}\right)\end{split}\] (17) The extra alternating signs on the first term are required to obtain the correct Dirac Hamiltonian in the continuum limit with staggered fermions [30]. Another fundamental point, associated with the Hamiltonian formulation, consists on the restriction of the Hilbert space to physical states. This can be derived from the Lagrangian formulation by noting that the component $A_{0}$ is non-dynamical (there is no term $\partial_{0}A_{0}$). As a consequence it acts as a Lagrange multiplier enforcing the Gauss’ law as a constraint. Therefore, on the one spatial dimensional lattice, the physical states are defined by the relation: \[G_{\vec{r}}\left\|\Psi\right\rangle=0\] (18) for each lattice site $\vec{r}$, where \[G_{\vec{r}}=\underset{i}{\sum}\left(L_{\vec{r},j}-L_{\vec{r}-\hat{j},j}\right) -Q_{\vec{r}}\,,\] (19) where $Q_{\vec{r}}$ is the dynamical matter charge. For the $1+1D$ case, for example, $Q_{n}=c_{n}^{\dagger}c_{n}+\frac{1-\left(-1\right)^{n}}{2}$. The alternating tem, which may look odd, is related to the staggered formulation. Considering a state with no electric field. The Gauss law demands that fermions populate the odd sites while leaving the even empty. This is because the spinor degrees of freedom are distributed along the lattice. Occupied odd sites have the interpretation of a filled Dirac sea. When a fermion hops from an odd to an even site it creates a hole in the Dirac sea while creating a particle above the Dirac sea. This is interpreted as the creation of particle/anti-particle pair where the hole plays the role of an anti-particle. In the presence of gauge fields, the hopping described above must be accompanied by a change on the electric field preserving Gauss’ law, as described by the connection operator in the first term of the Hamiltonian (17). The $G_{\vec{r}}$ are also generators of the gauge transformation and can be extended for the $U\left(N\right)$ and $SU\left(N\right)$ gauge theories. To this purpose, one can consider matter fields $\psi_{\vec{r}}$ that transform under the gauge symmetries under a suitable representation of dimension $n$ of the gauge group. The generators of the gauge symmetries must therefore satisfy the relation $\left[G_{\vec{r}}^{a},\psi_{\vec{r}}\right]=t^{a}\psi_{\vec{r}}$ where $t^{a}$ are $n-$dimensional representations of the (left) group generators. In order to preserve the gauge-invariance of the Hamiltonian, the connection operators must transform like tensors under the gauge transformations and they must follow the same representation of the matter fields: $U_{\vec{r},{j}}\to e^{i\alpha_{\vec{r}}^{a}t^{a}}U_{\vec{r},{j}}e^{-i\alpha_{ \vec{r}+\hat{j}}^{a}t^{a}}$. In particular the connection is multiplied on the left side by the transformation inherited from the lattice site on its left and on the right side by the inverse of the transformation inherited from the lattice site on its right. When we deal with a non-Abelian group, it is thus useful to distinguish left and right generators for the group transformations [25], labelled by $L_{\vec{r},i}$ and $R_{\vec{r},i}$ respectively (see, for example, [26]). The local generators of the gauge transformation can therefore be defined as: \[G_{\vec{r}}^{a}=\sum_{i}\left(L_{\vec{r},i}^{a}+R_{\vec{r}-\hat{i},i}^{a} \right)+\psi_{\vec{r}}^{\dagger}t^{a}\psi_{\vec{r}}.\] (20) Finally, the lattice Hamiltonian for the non-Abelian theory will be: \[H=-\frac{i}{2a}\underset{\vec{r},i}{\sum}\left(-1\right)^{r_{1}+ \ldots+r_{i-1}}\left(\psi_{\vec{r}}^{\dagger}U_{\vec{r}i}\psi_{\vec{r}+\hat{i} }-\mathrm{h.c.}\right)+m\underset{\vec{r}}{\sum}\left(-1\right)^{r_{1}+\ldots+ r_{d}}\psi_{\vec{r}}^{\dagger}\psi_{\vec{r}}\\ +\frac{a^{2-d}g^{2}}{2}\underset{\vec{r},i,a}{\sum}\left(\left(L_ {\vec{r}i}^{a}\right)^{2}+\left(R_{\vec{r}i}^{a}\right)^{2}\right)-\frac{a^{d- 4}}{4g^{2}}\underset{\square}{\sum}\mathrm{Tr}\left(U_{\square}+U_{\square}^{ \dagger}\right).\] (21) Again, the Gauss law should be imposed on physical states $G_{\vec{r}}^{a}\left\|\Psi\right\rangle=0$. Often times, in the proceeding Sections, the matter-gauge correlated hopping will be written as $\psi_{\vec{r}}^{\dagger}U_{\vec{r}i}\psi_{\vec{r}+\hat{i}}+\mathrm{h.c.}$ rather than $i\left(\psi_{\vec{r}}^{\dagger}U_{\vec{r}i}\psi_{\vec{r}+\hat{i}}-\mathrm{h.c. }\right)$ as above. While the latter reproduces the familiar continuum Hamiltonian in the naive continuum limit, both are related by a gauge transformation. #### 2.1.3 Challenges, limitations and quantum link models Cold atom systems offer the possibility to construct Hubbard-like Hamiltonians with tunable hopping parameters and on-site interactions. However, gauge potentials and gauge fields demand more than that. When the field is static, as described in Section 2.1.1, the hopping parameters become complex. This is not readily available in simple optical lattices, but, thanks to recent experimental developments, it is nowadays possible to engineer static gauge fields, as we will discuss in Section 3. For dynamical gauge fields, as discussed in Section 2.1.2, the matter hopping and link operators must be correlated in such a way to guarantee the existence of the local gauge symmetry (at each lattice site). Such kind of hopping is not natural in a cold atomic system and a discussion on how to implement is presented in Section 4. There is, yet, a further difficulty for dynamical gauge fields. Take, for example, the commutation relations in 15 pertaining a certain link. The operator $U_{\vec{r}}$ acts as a raising operator of the electric field (or equivalently of $L_{\vec{r}}$). But, for a $U(1)$ theory, this corresponds to an infinite Hilbert space per link. Constructing such links is certainly a challenge for its implementation, even for small lattice sizes. A solution of this problem is provded by quantum link models which are characterized by a finite Hilbert space per link, without violating the required gauge symmetry. These models were introduced by Horn in 1981 [31] and were further studied in [32; 33; 34; 35; 36]. Proposed as an alternative formulation to Wilson gauge field theories on the lattice, they became an attractive realization of gauge symmetries for quantum simulation purposes. In quantum link models the link degrees of freedom are replaced by quantum spins, such that the algebra in 15 is replaced by the algebra of angular momentum. In particular this correspond to considering alterative link operators: \[L_{+\vec{r},i}=S_{\vec{r},i}^{x}+iS_{\vec{r},i}^{y}\,,\quad L_{\vec{r},i}=S_{ \vec{r},i}^{z}\] (22) where the raising and lowering operators $L_{\pm\vec{r},i}$ replace $U_{\vec{r}}$ and its conjugate. With this construction, the first two relations of 15 are still satisfied. However, the last no longer holds because $U$ and $U^{\dagger}$ do not commute any longer. In particular, $L_{\pm\vec{r}i}$ are not unitary whereas $U_{\vec{r}}$ was. Even though the algebra itself is different, the angular momentum operators can be equally used to construct a gauge theory without compromising the gauge symmetry. In particular, we can choose the dimension of the Hilbert space in each of the links to be $2S+1$ with $S$ a positive half integer (corresponding, in the spin language, to the total spin). It is expected that in the limit of large $S$ the Wilson formulation should be recovered. Explicitly one can use the following link variables $U_{\vec{r}i}\to L_{+\vec{r}i}/\sqrt{S\left(S+1\right)}$. The new non-zero commutation relation is $\left[U_{\vec{r}i},U_{\vec{r}i}^{\dagger}\right]=2L_{\vec{r}i}/S\left(S+1\right)$. In the limit of $S\rightarrow+\infty$ the right hand side goes to zero and the initial algebra is recovered. There is an analogous construction for $U\left(N\right)$ non-Abelian symmetries. One can see that the symmetry can be realized using an $SU\left(2N\right)$ algebra (note that for $N=1$ this gives, correctly, $SU\left(2\right)$). It is possible to construct the new algebra using the so-called “rishon fermions” [36]. They are written in terms of pairs of fermionic operators $l_{\vec{r},j}^{m}$ and $r_{\vec{r}+\hat{j},j}^{m}$ for each link between the sites $\vec{r}$ and $\vec{r}+\hat{j}$. These operators define additional left and right gauge modes laying on the lattice sites $\vec{r}$ and $\vec{r}+\hat{j}$, with the aim of describing the link degrees of freedom. $m$ labels their color index. We can write: \[L_{\vec{r},j}^{a}=\frac{1}{2}l_{\vec{r},j}^{m\dagger}t_{mn}^{a}l _{\vec{r},j}^{n}\,,\qquad R_{\vec{r},j}^{a}=\frac{1}{2}r_{\vec{r}+\hat{j},j}^{ m\dagger}t_{mn}^{a}r_{\vec{r}+\hat{j},j}^{n}\,,\] (23) \[E_{\vec{r},j}=\frac{1}{2}\left(r_{\vec{r}+\hat{j},j}^{m\dagger}r _{\vec{r}+\hat{j},j}^{m}-l_{\vec{r},j}^{m\dagger}l_{\vec{r},j}^{m}\right)\,,\] (24) \[U_{\vec{r},j}^{mn}=l_{\vec{r},j}^{m}r_{\vec{r}+\hat{j},j}^{n \dagger}\,.\] (25) The finiteness of the Hilbert space is a feature desirable for future quantum simulation schemes. Even though not a primary concern at this stage, it is reassuring that the effective continuum limit can be achieved even if one uses quantum link models [34]. ## 3 Simulation of gauge potentials In accordance with the previous discussion, the goal of this Section is to show specific examples on how a complex hopping parameter can be engineered. The two main strategies described will be two contrasting situations. In one external parameters are varied adibatically (Section 3.1), while in the other fast modes are integrated out (Section 3.2). ### Adiabatic change of external parameters The idea of this approach has, in its core, the tight relation between the Aharonov-Bohm phase [37] and the Berry phase which was a concept introduced by Berry in [38]. The first is the phase acquired by a particle traveling around a closed contour. At the end of the path, when it is back to the initial position, the wave function acquires a new phase which is independent of the details of how the path was done and only depending on the total magnetic flux through the contour. On the other side, the Berry phase corresponds to the phase acquired when some external parameters of the system are varied in time, “slowly”, coming back again to their initial value for a non-degenerate state. In a more precise way, the starting point is an Hamiltonian $H\left(q^{a},\lambda_{i}\right)$ where $q^{a}$ are degrees of freedom and $\lambda_{i}$ are a set of external parameters. If these parameters are varied sufficiently slowly returning, in the end, to their initial value, and if the initial state is an eigenstate non degenerate in energy, then the system will be back to its initial state. During the process, however, it will acquire a phase: \[\left\|\psi\right\rangle\underset{\mathrm{adiabatic\ change}}{\longrightarrow}e ^{i\gamma}\left\|\psi\right\rangle\,.\] (26) The phase $\gamma$ can be derived by computing the time evolution operator and subtract the “trivial” dynamical phase acquired simply due to the time evolution. Let us consider the adiabatic evolution of a system such that each energy eigenstates remain non-degenerate during the whole process. In this case, starting from one of the eigenstates of the initial Hamiltonian, the system will continuosly evolve remaining in the corresponding eigenstate, with energy $E(t)$, at each time. Therefore the dynamical phase results $e^{-i\int E\left(t\right)dt}$. The additional Berry phase reads instead: \[\gamma=\ointctrclockwise_{{\cal C}}\tilde{A}_{i}\left(\lambda\right)d\lambda_{ i}\,,\] (27) where ${\cal C}$ is the closed path in the space of the parameters $\lambda_{i}$ and $\tilde{A}_{i}$ are given by: (28) where $\left\|\phi\left(\lambda\right)\right\rangle$ are reference eigenstates taken with an arbitrary choice of their overall phases. $\tilde{A}\left(\lambda\right)$ is the Berry connection. Different choices of the reference eigenstates with different phases, for example $\|{\phi^{\prime}(\lambda)}\rangle=e^{i\alpha\left(\lambda\right)}\left\|\phi \left(\lambda\right)\right\rangle$, would just reproduce a gauge transformation on $\tilde{A}$: \[\tilde{A}_{i}\rightarrow\tilde{A}_{i}^{\prime}=\tilde{A}_{i}-\frac{\partial \alpha}{\partial\lambda_{i}}\,.\] (29) This principle can be applied in multi-level atomic systems in order to reproduce artificial gauge fields in an ultracold atomic setting. As an example, the computation can be done for a two level atom, where it is shown how this vector potential appears explicitly at the Hamiltonian level. These two levels correspond to two internal states of the atom, the ground state $\left\|g\right\rangle$ and an excited state $\left\|e\right\rangle$. The center of mass Hamiltonian, assumed diagonal on the internal states, is taken to be simply the free particle Hamiltonian. The total Hamiltonian is $H=H_{0}+U$. By a suitable shift of the energy spectrum we can assume that the ground and excited state energies are related by $E_{g}=-E_{e}$. Then $U$ can be written as \[U=\frac{\Omega}{2}\left(\begin{array}[]{cc}\cos\theta&e^{i\phi}\sin\theta\\ e^{i\phi}\sin\theta&-\cos\theta\end{array}\right)\] (30) where $\theta$ and $\phi$ may depend on the position. The frequency $\Omega$ characterizes the strength of the coupling between the two states and it is assumed to be position independent. The eigenstates of this operator, denoted to as “dressed states”, are given by: (31) with eigenvalues $\pm\hbar\Omega/2$ respectively. We assume that the initial internal state is $\left\|\chi_{1}\right\rangle$ and that the evolution is adiabatic, such that the system remains in the state $\|{\chi_{1}}\rangle$ at all times. Hence the state of the system can be described by a wave function $\left\|\psi\left(t,\vec{r}\right)\right\rangle=\varphi\left(t,\vec{r}\right) \left\|\chi_{1}\left(\vec{r}\right)\right\rangle$ where $\varphi\left(t,\vec{r}\right)$ will obey a modified Schrödinger equation due to the dependence of $\left\|\chi_{1}\left(\vec{r}\right)\right\rangle$ on the position. Plugging this into the Schrödinger equation and projecting on $\left\|\chi_{1}\left(\vec{r}\right)\right\rangle$, we find the effective Hamiltonian governing $\varphi$: (32) As expected, a vector potential $\tilde{A}_{i}\left(\vec{r}\right)$ corresponding to the Berry connection is found. Additionally a potential $\tilde{V}\left(\vec{r}\right)$ is also created and related to virtual transitions to the other state $\left\|\chi_{2}\left(\vec{r}\right)\right\rangle$. In this two level approximation, these two quantities are given by $\tilde{A}_{i}\left(\vec{r}\right)=\frac{\cos\theta-1}{2}\frac{\partial\phi}{ \partial x_{i}}$ and $\tilde{V}\left(\vec{r}\right)=\frac{\left(\nabla\theta\right)^{2}+\sin^{2} \theta\left(\nabla\phi\right)^{2}}{8m}$. Discussions about the practical implementation on optical lattices can be found in [21; 39; 40; 22]. First experimental evidence of scalar potentials in quantum optics was found in [41] and the first observation of geometric magnetic fields in cold atomic physics was done in [42]. By considering a set of degenerate or quasi-degenerate dressed states it is possible to achieve non-Abelian gauge potentials as well [21; 22]. ### Effective Hamiltonian in periodic driven system In contrast to the approach of the previous Subsection, where the creation of the magnetic field relied on a slow change in time (i.e. the particle moves slowly enough such that the position dependent internal state is followed adiabatically), the following technique relies on fast oscillations. The basic principle consists on having two very distinct timescales. A fast oscillating time dependent potential will give rise to an effective time independent Hamiltonian which will present the desired complex hopping term. A general technique was proposed in [43] and it is based on a generic time-dependent periodic Hamiltonian: \[H=H_{0}+V\left(t\right)\] (33) where all the the time dependence is relegated to $V\left(t\right)=V(t+\tau)$ where $\tau$ is the time period. $V(t)$ can be decomposed as: \[V\left(t\right)=\underset{n}{\sum}\left(V_{n+}e^{in\omega t}+V_{n-}e^{in\omega t }\right)\] (34) where $V_{n\pm}$ are operators and $\omega=2\pi/\tau$. The condition $V_{n+}=V_{n-}^{\dagger}$ guarantees the Hermiticity of the Hamiltonian. A unitary transformation $e^{iK\left(t\right)}$ generates an effective Hamiltonian given by: \[H_{\mathrm{eff}}=e^{iK\left(t\right)}He^{-iK\left(t\right)}+i\left(\frac{ \partial}{\partial t}e^{iK\left(t\right)}\right)e^{-iK\left(t\right)}\] (35) We choose a periodic operator $K\left(t\right)$ such that the effective Hamiltonian is time independent. Under this requirement, the time evolution operator can be represented as: \[U\left(t_{i}\to t_{f}\right)=e^{iK\left(t_{f}\right)}e^{-iH_{\mathrm{ eff}}\left(t_{f}-t_{i}\right)}e^{-iK\left(t_{i}\right)}\,,\] (36) and it can be shown that, at lowest order, the effective Hamiltonian can be written as [43]: \[H_{\mathrm{eff}}=H_{0}+\tau\underset{n}{\sum}\frac{1}{n}\left[V_{n+},V_{n-} \right]+{\cal O}\left(\tau^{2}\right).\] (37) This expansion relies on the small parameter $\tilde{V}\tau$ where $\tilde{V}$ is the typical energy scale of $V(t)$. This expansion turns out to be very useful in the effective description of ultracold atomic systems though care should be taken, in a case by case scenario, in order to be sure about the convergence of the series. #### 3.2.1 Lattice Shaking The lattice shaking approach consists on having an external time dependent optical potential that is changing in time in accordance to the previous description. Then a change of basis is performed for a co-moving frame that, along with a time average, will create an effective Hamiltonian with the desired complex hopping. As an example, a brief prescription is presented along the lines of the realization in a $\mathrm{Rb}$ Bose-Einstein condensate [44]. The Hamiltonian considered is the usual tight-biding Hamiltonian in 2D with the usual hopping and on-site part $H_{os}$ (by on-site it is intended one body potential and scattering terms that act in single sites). In addition, there is an extra time dependent potential: \[H=-\underset{\vec{r},j}{\sum}t_{\vec{r}j}\hat{a}_{\vec{r}+\hat{j}}^{\dagger} \hat{a}_{\vec{r}}+H_{\mathrm{os}}+\underset{\vec{r}}{\sum}v_{\vec{r}}\left(t \right)\hat{a}_{\vec{r}}^{\dagger}\hat{a}_{\vec{r}}.\] (38) The function $v_{\vec{r}}\left(t\right)$ is periodic on time with period $\tau$: $v_{\vec{r}}\left(t\right)=v_{\vec{r}}\left(t+\tau\right)$. A unitary transformation on the states is performed and plugged in on the Schrödinger equation, thus defining new states $\left\|\psi^{\prime}\right\rangle$ such that $\left\|\psi\left(t\right)\right\rangle=U\left(t\right)\left\|\psi^{\prime}\left( t\right)\right\rangle$. The Hamiltonian becomes $H^{\prime}\left(t\right)=U\left(t\right)^{\dagger}HU\left(t\right)-iU\left(t \right)^{\dagger}\dot{U}\left(t\right)$ (where the dot stands for time derivative). The transformation is given by \[U\left(t\right)=e^{-i\int_{0}^{t}dt^{\prime}\sum_{\vec{r}}v_{\vec{r}}\left(t^{ \prime}\right)\hat{a}_{\vec{r}}^{\dagger}\hat{a}_{\vec{r}}}\,.\] (39) It is straightforward to see that this transformation cancels the part of $H$ (which will be present also on $U^{\dagger}HU$) corresponding to $v_{i}\left(t\right)\hat{a}_{\vec{r}}^{\dagger}\hat{a}_{\vec{r}}$. On the other side, since this does not commute with the kinetic term, a time dependence will be inherited by the hopping terms. For a set of rapidly oscillating function $v_{i}\left(t\right)$ the Hamiltonian can be replaced by an effective one, resulting from time averaging over a period. The new hopping parameters will read: \[t_{\vec{r}j}\to t_{\vec{r}j}\left\langle e^{i\Delta v_{\vec{r}j}} \right\rangle_{\tau}\] (40) where $\left\langle\right\rangle_{\tau}$ stands for the average over a period: $\tau^{-1}\int_{0}^{\tau}dt$ and . #### 3.2.2 Laser-assisted hopping In this case the effective dynamics is induced by the coupling of the atoms on the optical lattice with a pair of Raman lasers. A fundamental ingredient consists on introducing an energy offset $\Delta$ on neighboring sites. It is enough to consider such scenario along a single direction. Considering a $2D$ lattice: \[H=-t\underset{\vec{r},j}{\sum}\left(\hat{c}_{\vec{r}+\hat{j}}^{\dagger}\hat{c} _{\vec{r}}+\mathrm{h.c.}\right)+\frac{\Delta}{2}\underset{\vec{r}}{\sum}\left( -1\right)^{x}c_{\vec{r}}^{\dagger}\hat{c}_{\vec{r}}+V\left(t\right)\] (41) where $\vec{r}=\left(x,y\right)$ runs through the lattice sites. The offset term characterized by $\Delta$ can be obtained by tilting the lattice, introducing magnetic gradients or through superlattices. The potential $V\left(t\right)$ is the result of the two external lasers that induce an electric field $E_{1}\cos\left(\vec{k}_{1}\cdot\vec{r}_{1}-\omega_{1}t\right)+E_{2}\cos\left( \vec{k}_{2}\cdot\vec{r}_{2}-\omega_{2}t\right)$. It is assumed that the frequencies are fine-tuned such that they match the offset $\omega_{1}-\omega_{2}=\Delta$. Neglecting fast oscillating terms the potential is written as: \[V\left(t\right)=2E_{1}E_{2}\underset{\vec{r}}{\sum}e^{i\left(\vec{k}_{R}\cdot \vec{r}-\Delta t\right)}c_{\vec{r}}^{\dagger}c_{\vec{r}}+\mathrm{h.c.}\] (42) with $\vec{k}_{R}=\vec{k}_{1}-\vec{k}_{2}$. Then one can get the effective Hamiltonian in two steps. First performing an unitary transformation $\exp[{-it\frac{\Delta}{2}\sum_{\vec{r}}\left(-1\right)^{x}c_{\vec{r}}^{\dagger }\hat{c}_{\vec{r}}}]$ will create oscillatory hopping terms (with $\exp\left({\pm i\Delta t}\right)$ in front). Then one may apply the previous formalism building an effective Hamiltonian using Equation 37: \[\begin{split} H=&-t\underset{x,y}{\sum}\left(\hat{c} _{x,y}^{\dagger}\hat{c}_{x,y+1}+\mathrm{h.c.}\right)\\ &-\frac{2tE_{1}E_{2}}{\Delta}\underset{x\ \mathrm{even},y}{\sum} \left[\left(e^{i\vec{k}_{R}\cdot\vec{r}}-1\right)\left(e^{i\vec{k}_{R}\cdot \vec{r}}\hat{c}_{x,y}^{\dagger}\hat{c}_{x+1,y}+e^{-i\vec{k}_{R}\cdot\vec{r}} \hat{c}_{x-1,y}^{\dagger}\hat{c}_{x,y}\right)+\mathrm{h.c.}\right]+{\cal O} \left(\Delta^{-2}\right)\end{split}\] (43) It is clear that this generates complex hopping and looking more carefully one finds that the lattice has a staggered flux. With a choice $\vec{k}_{R}=\left(\Phi,\Phi\right)$ (as also made in the experiment [45]) one can write upon a gauge transformation: \[\begin{split} H=&-t\underset{x,y}{\sum}\left(\hat{c} _{x,y}^{\dagger}\hat{c}_{x,y+1}+\mathrm{h.c.}\right)\\ &-\frac{2tE_{1}E_{2}\sin\Phi/2}{\Delta}\underset{x\ \mathrm{even} ,y}{\sum}\left[\left(e^{i\Phi y}\hat{c}_{x,y}^{\dagger}\hat{c}_{x+1,y}+e^{-i \Phi y}\hat{c}_{x-1,y}^{\dagger}\hat{c}_{x,y}\right)+\mathrm{h.c.}\right]+{ \cal O}\left(\Delta^{-2}\right)\end{split}\] (44) where it is clear that a sequence of fluxes $\pm\Phi$ alternates in the plaquettes along the $x$ direction. More refined techniques allow for the realization of systems with uniform fluxes [46]. In such systems the Chern number of the Hofstadter bands was measured in [47]. It is worth noting that other kind of one-body terms, beyond the staggered term, can be used as it was done in the first quantum simulations of this model with ultracold atoms [48; 49]. In that case a linear potential is used. These kind of approaches can be adapted to more general scenarios including different geometries and multi-component species. The latter, for example, can be achieved by introducing spin dependent potentials as done in [48]. ## 4 Simulation of gauge fields In the context of Abelian gauge theories, the goal of simulating gauge fields consists in attributing dynamics to the complex phases on the hopping parameters that were identified in the previous Section. In order to construct such dynamics one should identify degrees of freedom that will play the role of the gauge field. Several proposals have been put forward which map the gauge degrees of freedom into some other controllable variables. The platforms used include ultracold atoms, trapped ions and superconducting qubits. They may be analogue or digital quantum simulators and include Abelian or non-Abelian symmetries [50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76]. A more detailed description of two particular approaches in analogue cold atomic simulators will follow: the gauge invariance will be obtained by either penalizing with a large energy cost the non-physical states or by exploiting microscopic symmetries. The symmetries addressed will be $U\left(N\right)$ and $SU(N)$. There are other symmetries which have been explored, namely $\mathbb{Z}_{n}$[65; 54] which, in particular, can provide an alternative route towards $U\left(1\right)$ symmetry in the large $n$ limit [54] and can be addressed with similar approaches. Proposal for the realization of $\mathbb{C}P\left(N-1\right)$[77; 78] models have been put forward in [71; 72]. These models can serve as toy models for QCD and are also relevant in studying the approach to the continuum limit, in the context of D-theories, where the continuum limit is taken via dimensional reduction [35; 36]. Furthermore other formulations are possible for specific groups [79; 80; 81; 26; 82]. Gauge theories with Higgs fields have also been the target of quantum simulation proposals [83; 84; 85; 86]. Another relevant approach is the so-called quantum Zeno dynamics which takes inspiration on the quantum Zeno effect, stating that a system being continuously observed does not evolve on time. Furthermore, if the measurement commutes with a certain part of the Hamiltonian, then it can freeze a certain part of the Hilbert space but still enables the dynamics in another subspace [87]. This feature can be used in order to freeze gauge dependent quantities and let the system evolve in the gauge invariant subspace. The Hamiltonian to be implemented has the form $H_{\mathrm{noise}}=H_{0}+H_{1}+\sqrt{2\kappa}\sum_{x,a}G_{x}^{a}$ where $H_{0}$ and $H_{1}$ are time independent and are, respectively, gauge invariant and gauge variant parts of the Hamiltonian. The operators $G_{x}^{a}$ are associated to the constraint one wishes to impose $G_{x}^{a}\left\|\psi\right\rangle=0$. In the case of gauge theories $G_{x}^{a}$ are the generators of gauge transformations. An advantage of this approach, with respect to the energy punishment approach of the next Section, is that only linear terms on the generators must be imposed on the Hamiltonian (energy punishment requires quadratic terms). By other side leakage from the gauge invariant subspace of the Hilbert space happens as a function of time, which does not happen in the energy penalty approach. This approach was developed in [63]. Another approach, that was successfully implemented in the first quantum simulator of a gauge theory using trapped ions [7], is the digital quantum simulator [88]. The key idea consists on in dividing the full time evolution operator $e^{-iHt}$ into smaller pieces of sizes $\tau=t/N$ and apply time evolution of smaller parts of the Hamiltonian at a time. Consider for example an Hamiltonian which is a sum of $M$ contributions : $H=\sum_{\alpha}^{M}H_{\alpha}$. Each part $H_{\alpha}$ can represent, for example, a nearest neighbor spin interaction in which case only two spins are coupled on each $H_{\alpha}.$ For large enough $N$ one can write: \[e^{-iHt}=\left(e^{-iH\tau}\right)^{N}\simeq\left(\underset{\alpha=1}{\overset{ M}{\prod}}e^{-iH_{\alpha}\tau}\right)^{N}\] (45) Each time step can now be interpreted as an individual gate. While in the analogue simulation the great difficult lies on building the appropriate gauge invariant Hamiltonian, in digital quantum simulations that is not a problem. The difficulty lies, however, in building an efficient sequence of gates. Other then the scheme used in the first experimental realization [89], other proposals towards digital quantum simulations of lattice gauge theories have been put forward [52; 50; 61; 69; 73; 74] ### Gauge invariance from energy punishment The energy punishment approach is a quite general approach which allows for the theoretical construction of models that will exhibit a given symmetry in its low energy sector. It consists on building a Hamiltonian which does not prohibit the symmetry violation to occur but instead punishes it with a large energy. In a more concrete way, let suppose one wants to implement a set of symmetries corresponding to a set of generators $\left\{G_{x}\right\}$ commuting with each other $\left[G_{x},G_{y}\right]=0$. Furthermore consider a typical Hamiltonian $H_{0}$ which does not respect these symmetries. Then one constructs the following Hamiltonian: \[H=H_{0}+\Gamma\underset{x}{\sum}G_{x}^{2}\] (46) where $\Gamma$ is a large energy scale, meaning much larger than the energy scales involved in $H_{0}$. Since $G_{x}$ are Hermitian $G_{x}^{2}$ have non-negative eigenvalues. One can choose the lowest eigenvalue to be zero by an appropriate definition of $G_{x}$. Then, at low energy $\left(\ll\Gamma\right)$, the states will respect approximately the condition $G_{x}\left\|\psi\right\rangle\simeq 0$. If not, this would give a state automatically in an energy scale $\sim\Gamma$. It is then possible to construct an effective Hamiltonian, valid in low energy, which will respect the symmetries generated by $\left\{G_{x}\right\}$. Let $G$ be the projector operator on the subspace of the total Hilbert space obeying $G_{x}\left\|\psi\right\rangle=0$ and let $P=1-G$. Then the low energy Hamiltonian can be written as: \[H_{\mathrm{eff}}=GH_{0}G-\frac{1}{\Gamma}GH_{0}P\frac{1}{\underset{x}{\sum}G_{ x}^{2}}PH_{0}G+{\cal O}\left(\Gamma^{-2}\right)\] (47) which fulfills the symmetries. Within this framework an effective Abelian gauge theory can be constructed. In non-Abelian theories the generators of the gauge transformation do not commute and this construction fails. There are, of course, several possible drawbacks even on the theoretical level. For example the Hamiltonian 47, even though gauge invariant, may contain unwanted interactions or miss some particular terms which are present on the target system. In order to construct a quantum simulator the first task is naturally to map the degrees of freedom of the target theory into the laboratory controlled ones, in this case the atomic variables. The matter fields, which are fermionic, will naturally be described by fermionic atomic species. Regarding gauge fields, the target will be the quantum links formulation discussed in Section 2.1.3. Therefore the goal consists on building the quantum links satisfying the algebra $\left[L_{\vec{r},i},U_{\vec{r}^{\prime},j}\right]=\delta_{ij}\delta_{\vec{r} \vec{r}^{\prime}}U_{\vec{r}^{\prime},j}$ and $\left[U_{\vec{r},i},U_{\vec{r}^{\prime},j}^{\dagger}\right]=\delta_{ij}\delta_ {\vec{r}\vec{r}^{\prime}}2L_{\vec{r},i}/S\left(S+1\right)$. This can be achieved using the Schwinger representation. Given two bosonic species $b^{\left(\sigma\right)}$ with $\sigma=1,2$ which are associated to each link, one can write \[U_{\vec{r}i}=\frac{1}{\sqrt{S\left(S+1\right)}}b_{\vec{r}i}^{\left(2\right)}{} ^{\dagger}b_{\vec{r}i}^{\left(1\right)},\ L_{\vec{r}i}=\frac{1}{2}\left(b_{ \vec{r}i}^{\left(2\right)}{}^{\dagger}b_{\vec{r}i}^{\left(2\right)}-b_{\vec{r} i}^{\left(1\right)}{}^{\dagger}b_{\vec{r}i}^{\left(1\right)}\right)\] (48) Each link is loaded with a total of $2S$ bosons where $S$ is an half integer. Then one has the desired representation for the quantum links in terms of atomic variables. Now the variables are identified. One then can then build a $d$ dimensional optical lattice where fermions are allowed to hop among lattice points and in each links there are a total of $2S$ bosons. For $1D$, the target Hamiltonian is of the form: \[H=-t\underset{n}{\sum}\left(c_{n}^{\dagger}U_{n}c_{n+1}+\mathrm{h.c.}\right)+m \underset{n}{\sum}\left(-1\right)^{n}c_{n}^{\dagger}c_{n}+\frac{g^{2}}{2} \underset{n}{\sum}L_{n}^{2}\] (49) When comparing to the general structure of 21 there are two differences: the plaquette term and the position-dependent coefficient of the kinetic term. The plaquettes are naturally absent in $1D$, whereas the tunneling amplitude can be fixed by a gauge transformation $c_{n}\rightarrow\left(-i\right)^{n}c_{n}$. The Hamilton (49) has therefore the required structure and can be targeted with the Schwinger boson approach and it assumes the form: \[H=-\frac{t}{\sqrt{S\left(S+1\right)}}\underset{n}{\sum}\left(c_{n}^{\dagger}b_ {n}^{\left(\bar{\sigma}\right)}{}^{\dagger}b_{n}^{\left(\sigma\right)}c_{n}+ \mathrm{h.c.}\right)+m\underset{n}{\sum}\left(-1\right)^{n}c_{n}^{\dagger}c_{n }+\frac{g^{2}}{8}\underset{n}{\sum}\left(b_{n}^{\left(2\right)}{}^{\dagger}b_{ n}^{\left(2\right)}-b_{n}^{\left(1\right)}{}^{\dagger}b_{n}^{\left(1\right)} \right)^{2}\] (50) The two last terms can be, in principle, implemented directly using a proper tuning of the interactions between the bosons and the potential for the fermions. The first term, instead, is a correlated hopping between bosons and fermions which is obtained less easily. Furthermore the additional terms that are not gauge invariant, like $b_{n}^{\left(\bar{\sigma}\right)}{}^{\dagger}b_{n}^{\left(\sigma\right)}$ and $c_{n}^{\dagger}c_{n+\hat{i}}$, must be suppressed. This is solved by the energy punishment approach. In general the non-gauge invariant Hamiltonian with the ingredients described has the form: (51) Using the generators for the $U\left(1\right)$ gauge symmetry in Equation 19 one considers the full Hamiltonian: \[H=H_{0}+\Gamma\underset{n}{\sum}\left(L_{n}-L_{n-1}-c_{n}^{\dagger}c_{n}+\frac {1-\left(-1\right)^{n}}{2}\right)^{2}\] (52) It is crucial that one has access to the interactions that are introduced on the last term corresponding to the energy punishment. To see that this is the case it useful to be more specific about the labels $\sigma$. One can take, as in [53], the labels $\sigma=1,2$ meaning respectively left and right part of the link, which can be thought to coincide with the lattice site. In this way $b_{n}^{\left(2\right)}{}^{\dagger}b_{n}^{\left(1\right)}$ are just regular hopping terms. Furthermore it is recalled that the total number of bosons associated to each link is conserved. Therefore one can write: $L_{n}=-S+b_{n}^{\left(2\right)}{}^{\dagger}b_{n}^{\left(2\right)}=S-b_{n}^{ \left(1\right)}{}^{\dagger}b_{n}^{\left(1\right)}$. This means that terms like $L_{n}^{2}$ and and $L_{n}L_{n-1}$ can be written as a density-density interaction. Regarding the last case, recall that $b_{n}^{\left(1\right)}$ and $b_{n-1}^{\left(2\right)}$ are effectively in the same site, see Figure 1. Now Equation 47 can be applied. The number of particles in each site is a good quantum number to describe the eigenstates of $G_{x}$. The number of particles in the site $j$ are denoted by $n_{j}^{F}=c_{j}^{\dagger}c_{j}$, $n_{j}^{1}=b_{j}^{\left(1\right)}{}^{\dagger}b_{j}^{\left(1\right)}$ and $n_{j}^{2}=b_{j-1}^{\left(2\right)}{}^{\dagger}b_{j-1}^{\left(2\right)}$. The subspace of gauge invariant states is then characterized by: \[n_{j}^{F}+n_{j}^{1}+n_{j}^{2}=2S+\frac{1-\left(-1\right)^{j}}{2}\] (53) <figure><img src="content_image/1911.06022/figure1.png"><figcaption>Figure 1: Superlattice configurations for the two boson species and thefermionic one. Bosons of the species 1 at an even site 2j can only hop to 2j−1while a boson of species 2 has only access to the site 2j+1. The Figurepresents a an example of a gauge invariant state configuration (on these threesites) where Gx\|ψ⟩=0.</figcaption></figure> In the lowest order only the two last terms of 51 survive as any single hopping destroys the above relation. At the next order there are three possible virtual processes that preserve this condition. Up to some linear terms on the particle density operator, they are: 1. Boson-boson hopping: a boson hops to the neighboring site on the same link and another boson hops back. Gives rise to a boson density-density interaction. 2. Fermion-Fermion hopping: a fermion hops to a neighboring site and then hops back. Only possible if neighboring site is unoccupied and gives rise to a nearest neighbor fermion density-density interaction. 3. Boson-Fermion hopping: a fermion hops to a neighboring site and a boson belonging to the link that connects the two sites does the opposite path. Gives rise to a correlated hopping. The terms coming from i should be joined with the last term of 51 in order to form the correct kinetic term for the gauge fields. The terms in ii are somehow unwanted and correspond to a repulsion between neighbor fermions $n_{j}^{F}n_{j+1}^{F}$. Naturally, they do not spoil gauge invariance and their inclusion should not be a problem [53]. Finally the terms originating from iii give rise to the correlated hopping responsible for the matter-gauge coupling as written on the first term of 48. There is another issue which should be addressed. From the beginning it was assumed that the the number of bosons in each link is conserved. In particular this means that bosons are not allowed to pass to a neighboring link. In order to guarantee this condition in an experiment one should introduce an extra bosonic species and this is the reason that bosons in neighboring links were represented with different colors on Figure 1. Then one bosonic species is trapped on the even links and the other in the odd links. This will prevent bosonic hopping between links. A numerical study of real dynamics of the the model as well as accuracy of the effective gauge invariance obtained was also done in [53]. Finally, in a possible experimental realization, the first fundamental step is to guarantee that the system is initialized on a gauge invariant state. This can be done by loading the atoms in a deep lattice such that they are in Mott phase. Afterwards the system should evolve according to the fine tuned Hamiltonian described above (after lowering the lattice barriers). Finally measures of relevant quantities can be performed. This principle is valid in higher dimensionality where one has to face the difficulty of generating plaquette terms. This was done for the pure gauge in [90] and [59] by suitably allowing hopping between links. In the first case each link has an infinite dimensional Hilbert space that is represented by a Bose-Einstein condensate. In the second the proposal is simplified by considering a quantum link model. ### Gauge invariance from many body interaction symmetries This approach consists on building a lattice which will have the necessary local gauge invariance arising from microscopic symmetries. Specific proposals may vary significantly even though the same principle is used. For example in [76] the simulation is built upon the global symmetry conserving the total number of excitations and is achieved via a state-dependent hopping. In turn, see for example [65; 91], are built upon conservation of angular momentum. For concreteness the later approach will be described in more detail below. In the case of [62]$SU\left(N\right)$ symmetries of the ground state manifold of alkaline-earth-like atoms could be exploited in order to built non-Abelian gauge theories. Symmetries only allow for certain type of processes to occur and, by exploiting these constraints, one can build a gauge symmetry. This can be done, as said before, considering angular momentum conservation. The Schwinger model is taken as an illustrative example. Bosons, that will make up the gauge fields, are placed at the two boundaries of the links. Because the goal consists, partially, in forbidding gauge dependent terms like simple boson or fermion hopping, the lattice should be spin dependent. In this way a single hopping is forbidden as it does not conserve angular momentum. By other side one should guarantee that correlated spin between bosons and fermions is allowed. This can be achieved by a judicious choice of respective hyperfine angular momentum in each lattice site. For concreteness, consider a single link connecting two sites and a total of two bosonic ($b^{\left(1\right)}$,$b^{\left(2\right)}$) and two fermionic species ($c$,$d$). The site at the left of the link can only be populated by $c$ while the right side by $d$. Analogously the left end of the link can only be populated by $b^{\left(1\right)}$ while the right end can only be populated by $b^{\left(2\right)}$. Then the conditions described above for allowed/forbidden hopping are automatically satisfied if one chooses the hyperfine angular momentum of each atomic species to satisfy: \[m_{F}\left(d\right)-m_{F}\left(c\right)=m_{F}\left(b^{\left(1\right)}\right)-m _{F}\left(b^{\left(2\right)}\right)\] (54) It is intended that the lattice is, indeed, spin dependent so that $m_{F}\left(d\right)\neq m_{F}\left(c\right)$ and $m_{F}\left(b^{\left(1\right)}\right)\neq m_{F}\left(b^{\left(2\right)}\right)$. In other words, what this means is that the difference of angular momentum caused by a fermion hop can be exactly compensated by a bosonic hop in the opposite direction. This leads directly to the correlated hopping desired which, in fact, comes from the scattering terms between bosons and fermions. The only other allowed scattering term between fermions and bosons correspond to density-density interactions like $c^{\dagger}c\left(b^{\left(2\right)}{}^{\dagger}b^{\left(2\right)}+b^{\left(1 \right)}{}^{\dagger}b^{\left(1\right)}\right)$. These are just linear terms on the fermionic number operator due to the conservation of the total number of bosons per link. Summing over all lattice sites will give just a constant shift of the energy. The scattering terms between bosons give rise to the gauge kinetic term as before (in $1+1$ dimensions). Again, for higher dimensionality, there is a non-trivial extra step consisting on building plaquette interactions. If plaquettes are ignored and the model described above is loaded on an higher dimensional lattice the result corresponds to the strong coupling limit of the gauge theory. The plaquette terms can be achieved by the so-called loop method. It uses perturbation theory in a similar way that was used in the energy penalty approach. In order to discuss the essence of the construction of the plaquette terms, one can consider just the pure gauge theory. The target Hamiltonian is \[H_{\mathrm{target}}=\frac{g^{2}}{2}\underset{\vec{r},i}{\sum}L_{\vec{r}i}^{2}- \frac{1}{4g^{2}}\underset{\square}{\sum}\left(U_{\square}+U_{\square}^{\dagger }\right).\] (55) The description will be specialized for $2+1$ dimensions but the theoretical construction for higher dimensions is analogous. The construction of the plaquette term relies on a perturbative expansion similar to 46 but, in this case, $H_{0}$ is already a gauge invariant Hamiltonian. For reasons that will be explained below one should have two fermionic species, say $\chi$ and $\psi$, and build the trivial part of the generalization of the $1+1$ process: \[H_{0}=-t\underset{\vec{r},i}{\sum}\left(\psi_{\vec{r}}^{\dagger}U_{\vec{r}i} \psi_{\vec{r}+\hat{i}}+\chi_{\vec{r}}^{\dagger}U_{\vec{r}i}\chi_{\vec{r}+\hat{ i}}+\mathrm{h.c.}\right)+\frac{g^{2}}{2}\underset{\vec{r},i}{\sum}L_{\vec{r}i} ^{2}\,.\] (56) The fermionic species are auxiliary and in the effective model they will be integrated out. There should be no interacting term between them. Here the energy penalty must enforce the following conditions at each site $\vec{r}=\left(r_{1},r_{2}\right)$: * there is a fermion $\psi$ if both $r_{1}$ and $r_{2}$ are even * there is a fermion $\chi$ if both $r_{1}$ and $r_{2}$ are odd * no fermion otherwise The positions of these fermions is represented on Figure 2 a). This kind of constraint can be obtained, for large $\Gamma$, with a Hamiltonian of the form: \[H_{\mathrm{penalty}}=-\Gamma\underset{\vec{r}}{\sum}\left[\frac{\left(1+\left( -1\right)^{r_{1}}\right)\left(1+\left(-1\right)^{r_{2}}\right)}{4}\psi_{\vec{r }}^{\dagger}\psi_{\vec{r}}+\frac{\left(1-\left(-1\right)^{r_{1}}\right)\left(1 -\left(-1\right)^{r_{2}}\right)}{4}\chi_{\vec{r}}^{\dagger}\chi_{\vec{r}}\right]\] (57) Through perturbation theory, according to 47, one gets the plaquette terms at fourth order. This process is "cleaner" if the $U_{\vec{r}}$ in 56 are considered unitary. IN particular we may consider a unitary limit,in which the total spin of the quantum link goes to infinity: $S\rightarrow=+\infty$. Order by order: 1. Only the pure gauge part of 56 contributes, no fermionic term occurs. 2. Trivial constant contribution assuming that $U_{n}$ are unitary. The virtual process giving rise to this contribution is a single link interaction where a fermionic-bosonic correlated hopping occurs back and forth restoring the initial state. There are never fermions on the neighbor lattice site. In turn in the unitary limit there is an infinite number of bosons such that $\left[U,U^{\dagger}\right]\to 0$. In the case of finite bosonic number, extra contribution corresponding to a renormalization of the pure gauge term of 56 will appear, together with another term which can be discarded by application of the Gauss law. 3. Trivial constant contribution assuming that $U_{n}$ are unitary. Virtual contributions evolving links constitute again back and forth hopping plus a pure gauge term at any stage of the process. The extra contributions coming from considering a finite number of boson per link cannot be disregarded trivially as second order for this case. 4. Gives the desired plaquette term plus renormalization of the pure gauge term of 56 assuming that $U_{n}$ are unitary. The last case corresponds to the virtual process where a fermion goes around a plaquette and returns to the initial place. This virtual process is represented on Figure 2 b). Naturally, in the non-unitary case, more terms appear. <figure><img src="content_image/1911.06022/x1.png"><figcaption>Figure 2: Loop method for obtaining the plaquette terms. In the panel a) it isdepicted the positions of the auxiliary fermions that are used to constructthe plaquette term using gauge invariant building blocks. One of the species,say ψ, is represented in red and placed on sites with both coordinates even.In turn χ, in pink, is placed on sites with both coordinates odd. Thiscorrespond to the ground-state of 57. In the panel b) it is represented avirtual process that gives rise to a plaquette term.</figcaption></figure> Plaquette terms only appear at fourth order. However, in the unitary limit, most contributions are trivial. One can then see that it is effectively a second order contribution [92]. When one considers a finite number of bosons in the links there are extra contributions appearing which cannot be disregarded. As in the case of the energy penalty, these contributions, even though unwanted, can be tolerated as they are naturally gauge invariant. However one should guarantee that these extra contributions are not more important than the plaquette term which is the target term. That can be achieved if the coupling term is parameterized is $g^{2}$ is taken to be small in units of $t$. By taking $g^{2}\sim t^{2}/\Gamma$ one makes the unwanted terms at third order effectively of the same order as the plaquettes and unwanted terms of the fourth order effectively of higher order than the plaquettes. On top of these, an extra species of fermions can be introduced to play the role of matter fields. They will consist, in the initial Hamiltonian, to the usual correlated hopping with the bosons. Furthermore the staggered mass term (of Equation 17) should also be introduced. In the unitary case this extra piece commutes with the interacting part of 56 and no further contribution is obtained in perturbation theory. In the truncated case there is an extra (gauge invariant) correlated hopping coming at third order. Another different aspect of the introduction of dynamical fermions is that the Gauss law ($\sum_{i}L_{\vec{r}i}-L_{\vec{r}-\hat{i},i}=\mathrm{const}$) can no longer be used to trivialize terms. The divergence of the electric gets a contribution from the charge density of the dynamical fermions. Nonetheless it can still be employed and the extra charge density terms can be compensated on the initial Hamiltonian if proper fine tuning is available experimentally. In [91] it was proposed a realization of the Schwinger ($1+1$) model using a mixture of ${}^{23}\mathrm{Na}$ for the bosons and ${}^{6}\mathrm{Li}$ for the fermions as well as an extensive study on the influence of the finiteness of the number of bosons per link in that case. ### Encoding in $1+1$ fermions The case of the Schwinger model, $1+1$ Dirac fermions coupled to a gauge field, is an interesting experimental and theoretical playground. It shares some non-trivial features with QCD like confinement, chiral symmetry breaking and a topological theta vacuum [93]. However, due to its simplicity, it allows for analytical and numerical studies which may become significantly harder in more complicated theories. Furthermore it was the target of the first experimental implementation of a lattice gauge theory [7]. In the context of quantum simulations it may not only provide the entrance door towards more complicated experimental realizations but also a way of benchmarking experimental techniques. One of the reasons why this model bares an intrinsic simplicity, as mentioned previously, is the fact that the gauge fields are non-dynamical. This is reflected on the absence of plaquette terms in the Hamiltonian formulation. Furthermore the Gauss law fixes the gauge field and can be used to integrate out its degrees of freedom. This results in a long-range interacting model which will be addressed next. In the following the lattice Hamiltonian formulation is considered for $N$ lattice sites: \[H=-it\overset{N-1}{\underset{n=1}{\sum}}\left[c_{n}^{\dagger}U_{n}c_{n+1}- \mathrm{h.c}\right]+m\overset{N}{\underset{n=1}{\sum}}\left(-1\right)^{n}c_{n} ^{\dagger}c_{n}+\frac{g^{2}}{2}\overset{N-1}{\underset{n=1}{\sum}}L_{n}^{2}\] (58) Here an infinite dimensional Hilbert space per link is considered, therefore the operators $U_{n}$ are unitary and the non-trivial commutation relations on the links are given by $\left[L_{m},U_{n}\right]=U_{n}\delta_{mn}$. Equivalently the link can be written as $U_{n}=e^{i\theta_{n}}$. The Gauss law is imposed in accordance with the relations (18) and (19). This model can be formulated in terms of Pauli spin operators [94] through the Jordan-Wigner transformation: \[\left\{\begin{array}[]{c}c_{n}=\underset{l<n}{\prod}\left(i\sigma_{z}\left(l \right)\right)\sigma^{-}\left(n\right)\\ c_{n}^{\dagger}=\underset{l<n}{\prod}\left(-i\sigma_{z}\left(l\right)\right) \sigma^{+}\left(n\right)\end{array}\right.\] (59) where $\sigma_{i}\left(l\right)$ represent the Pauli matrices in the site $l$ and $\sigma^{\pm}\left(n\right)=\sigma_{x}\left(n\right)\pm i\sigma_{y}\left(n\right)$. In terms of the spins the Gauss law is determined by: \[G_{n}=L_{n}-L_{n-1}-\frac{1}{2}\left(\sigma_{z}\left(n\right)+\left(-1\right)^ {n}\right)\,.\] (60) By restricting ourselves to the physical space, through the Gauss law $G_{n}\left\|\psi\right\rangle=0$, the link variables can be almost completely eliminated. Using periodic boundary conditions ($L_{0}=L_{N}$) one finds: \[L_{n}=L_{0}+\frac{1}{2}\underset{l=1}{\overset{n}{\sum}}\left(\sigma_{z}\left( l\right)+\left(-1\right)^{n}\right)\,.\] (61) The value of $L_{0}$ is a parameter of the theory and corresponds to a background field. For simplicity it will be taken to zero at the present discussion. By using the above relations the Hamiltonian 58 can be rewritten as: \[H=t\overset{N}{\underset{n=1}{\sum}}\left[\sigma^{+}\left(n\right)e^{i\theta_{ n}}\sigma^{-}\left(n+1\right)+\mathrm{h.c}\right]+\frac{m}{2}\overset{N}{ \underset{n=1}{\sum}}\left(-1\right)^{n}\sigma_{z}\left(n\right)+\frac{g^{2}}{ 8}\overset{N}{\underset{n=1}{\sum}}\left[\underset{l=1}{\overset{n}{\sum}} \left(\sigma_{z}\left(l\right)+\left(-1\right)^{n}\right)\right]^{2}\] (62) where a trivial constant term was dropped. The remaining gauge field variable $\theta_{n}$ can be eliminated by a residual gauge transformation [95]: \[\sigma^{\pm}\left(n\right)\rightarrow\sigma^{\pm}\left(n\right)\underset{j<n}{ \prod}e^{\pm i\theta_{j}}\,.\] (63) This is a non-trivial transformation as $\theta$’s are operators. More precisely, the above relation should be seen as defining a new set of operators $\bar{\sigma}^{\pm}\left(n\right)=\sigma^{\pm}\left(n\right)\prod_{j<n}e^{\pm i \theta_{j}}$. The $\bar{\sigma}$ still respect the angular momentum algebra between each other. Therefore they are still spin operators on the sites of the lattice, despite acting non-trivially on the links. Since the links degrees of freedom are being traced out using the Gauss law, one can arrive at an effective spin model for the sites. Plugging this transformation and expanding the interaction term, the resulting model is a long-range interacting spin model: \[H=t\overset{N}{\underset{n=1}{\sum}}\left[\sigma^{+}\left(n\right)\sigma^{-} \left(n+1\right)+\mathrm{h.c}\right]+\overset{N}{\underset{n=1}{\sum}}\left( \frac{m}{2}\left(-1\right)^{n}-\frac{g^{2}}{8}\left(1-\left(-1\right)^{n} \right)\right)\sigma_{z}\left(n\right)+\frac{g^{2}}{4}\overset{N-2}{\underset{ n=1}{\sum}}\underset{l=1}{\overset{N-1}{\sum}}\left(N-l\right)\sigma_{z}\left( n\right)\sigma_{z}\left(l\right)\] (64) This is a useful formulation for quantum simulations since the total of $N$ particles and $N-1$ gauge fields are simulated by just $N$ spins (with exotic long-range interactions), thanks to the gauge invariance. The difficulty was moved towards an efficient way of implementing the long-range asymmetric interaction between spins. This Hamiltonian was implemented as a digital quantum simulator in [7] using trapped ions (${}^{40}\mathrm{Ca}^{+}$). The system was composed of four qubits. The Schwinger mechanism of pair creation of particle-antiparticle was explored, as well as real time evolution of entanglement in the system. Based on the staggering prescription in Section 2.1.2, a particle on an odd site corresponds to the vacuum and a hole as an antiparticle (the contrary holds for particles in the even sites). Following this picture the number of particles at the site $n$ is given by $\nu_{n}=\left(1-\left(-1\right)^{n}\right)/2+\left(-1\right)^{n}c_{n}^{\dagger }c_{n}$ and therefore a relevant observable is the particle density $\nu\left(t\right)=\left(2N\right)^{-1}\sum_{n}\left\langle 1+\left(-1\right)^{ n}\sigma_{z}\left(n\right)\right\rangle$. Starting from a bare vacuum ($\nu\left(0\right)=0$) it is observed a rapid increase of the particle density followed by a decrease which is due to particle/anti-particle recombination. Also the vacuum persistence $G\left(t\right)=\left\langle 0\right\|e^{-iHt}\left\|0\right\rangle$ and entanglement were evaluated. The latter is done by reconstructing the density matrix and evaluating the entanglement in one half of the system with the other half through logarithmic negativity. Entanglement is produced through particle creation that get distributed across the two halves. More detail on the simulation and experimental results can be found in [7; 89]. Future challenges include the simulation of larger systems as well higher dimensionality and non-Abelian symmetries. ###### Acknowledgements. The authors want to thank the organizers of the Natal’s school: Pasquale Sodano, Alvaro Ferraz, Kumar S. Gupta and Gordon Semenoff. 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1706.07188	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 83974, "num_imgs": 7, "llama3_tokens_count": 25562 }	[ "content_image/1706.07188/x1.png", "content_image/1706.07188/x2.png", "content_image/1706.07188/x3.png", "content_image/1706.07188/x4.png", "content_image/1706.07188/x5.png", "content_image/1706.07188/x6.png", "content_image/1706.07188/x7.png" ]	# Theory of quantum-circuit refrigeration by photon-assisted electron tunneling Matti Silveri${}^{1,2}$ matti.silveri@oulu.fi Hermann Grabert${}^{3}$ Shumpei Masuda${}^{1}$ Kuan Yen Tan${}^{1}$ Mikko Möttönen${}^{1,4}$ mikko.mottonen@aalto.fi ${}^{1}$QCD Labs, COMP Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 13500, FI-00076 Aalto, Finland ${}^{2}$Research Unit of Nano and Molecular Systems, University of Oulu, P.O. Box 3000, FI-90014 Oulu, Finland ${}^{3}$Department of Physics, University of Freiburg, D-79104 Freiburg, Germany ${}^{4}$University of Jyväskylä, Department of Mathematical Information Technology, P.O. Box 35, FI-40014 University of Jyväskylä, Finland February 27, 2024 ###### Abstract We focus on a recently experimentally realized scenario of normal-metal–insulator–superconductor tunnel junctions coupled to a superconducting resonator. We develop a first-principles theory to describe the effect of photon-assisted electron tunneling on the quantum state of the resonator. Our results are in very good quantitative agreement with the previous experiments on refrigeration and heating of the resonator using the photon-assisted tunneling, thus providing a stringent verification of the developed theory. Importantly, our results provide simple analytical estimates of the voltage-tunable coupling strength and temperature of the thermal reservoir formed by the photon-assisted tunneling. Consequently, they are used to introduce optimization principles for initialization of quantum devices using such a quantum-circuit refrigerator. Thanks to the first-principles nature of our approach, extension of the theory to the full spectrum of quantum electric devices seems plausible. ## I Introduction Superconducting quantum circuits Nakamura _et al._ (1999); Vion _et al._ (2002); Martinis _et al._ (2002); Wallraff _et al._ (1999); Manucharyan _et al._ (2009); Ristè _et al._ (2012); Devoret and Schoelkopf (2013); Barends _et al._ (2013); Ofek _et al._ (2016) are among the leading candidates of quantum technological devices for the implementation of large-scale quantum computing Chow _et al._ (2014); Kelly _et al._ (2015); Versluis _et al._ (2017) and simulations Houck _et al._ (2012), with envisioned applications of great practical value. However, fast and accurate initialization of these devices to a pure quantum state remains challenging although it is a key requirement in their efficient operation DiVincenzo (2000). A solution could be an active refrigerator Grajcar _et al._ (2008) which evacuates entropy on demand for efficient initialization. Such device may also provide a route to robust ground-state operation by reduction of errors related to thermal and non-adiabatic excitations. <figure><img src="content_image/1706.07188/x1.png"><figcaption>Figure 1: (a) Schematic diagram of a superconducting coplanar waveguide (CPW)resonator which is connected through the capacitances Cc and Cg to a normal-metal island and a transmission line with an impedance Ztr. The superconductor(S)–insulator (I)–normal-metal (N) tunnel junctions defining the island arevoltage biased through the superconducting electrodes. (b) Energy diagram ofthe photon-assisted electron tunneling. The blue arrows depict tunnelingevents leading to absorption of a photon from the coupled resonator and thered arrows correspond to emission. (c) Coupling strength γT and temperature TTof the effective thermal reservoir formed by the photon-assisted tunneling asa function of the two-junction voltage bias. The parameters correspond totypical experimental values: see Fig. 3. In the highlighted region the thermalreservoir is cooler than the electrons of the normal-metal island.</figcaption></figure> Much effort has been put into studies of incoherent tunneling of single charges in mesoscopic junctions Averin and Likharev (1986); Fulton and Dolan (1987); Ingold and Grabert (1991); Devoret _et al._ (1990); Girvin _et al._ (1990); Ingold and Nazarov (1992); Ingold _et al._ (1994); Lee and Levitov (1996); Likharev (1999); Catelani _et al._ (2011); Pekola _et al._ (2013). Whereas fully normal-metal junctions can be used as sensitive charge sensors Devoret _et al._ (1992) and primary thermometers Pekola _et al._ (1994), normal-metal–insulator–superconductor (NIS) junctions have opened an avenue for electrically refrigerating the normal-metal electron reservoirs even below the phonon bath temperature Giazotto _et al._ (2006). However, quantum devices are designed to be very well isolated from dissipative electron systems owing to the requirements of long coherence time Ofek _et al._ (2016), and hence the benefits of the NIS junction technology in quantum-circuit initialization Jones _et al._ (2013); Tuorila _et al._ (2017) are far from obvious. Although the early work on photon-assisted tunneling at NIS junctions focused on the effect of the electromagnetic circuit on the tunnel current Devoret _et al._ (1990); Girvin _et al._ (1990); Ingold and Nazarov (1992); Averin _et al._ (1990); Pekola _et al._ (2010), recent studies also demonstrate the effect of the tunneling events on the state of the circuit Zakka-Bajjani _et al._ (2010); Hofheinz _et al._ (2011); Altimiras _et al._ (2014); Stockklauser _et al._ (2015); Kubala _et al._ (2015); Bruhat _et al._ (2016); Tan _et al._ (2017); Masuda _et al._ (2016); Westig _et al._ (2017); Jebari _et al._ (2017). Importantly, a quantum-circuit refrigerator and cryogenic microwave source based on photon-assisted tunneling of electrons through NIS junctions were demonstrated in Refs. Tan _et al._, 2017; Masuda _et al._, 2016, see schematic in Fig. 1. The refrigeration occurs at junction bias voltages where the normal-metal electron needs to receive an additional energy quantum from the coupled quantum electric circuit to overcome the Bardeen–Cooper–Schrieffer energy gap in the superconductor, see Fig. 1(b)-(c). The resulting exponential tunability of the coupling strength with the bias voltage offers a promising technique to quickly initialize quantum systems on demand. Furthermore, when the junctions are biased above the superconductor gap, tunneling events which emit additional energy to the coupled quantum circuit become energetically allowed and the device can be utilized as a source of incoherent microwave radiation Masuda _et al._ (2016). In this paper, we provide a first-principles derivation of the relaxation and excitation rates induced on a superconducting resonator which is capacitively coupled to NIS junctions. This model accurately describes the physics of the quantum-circuit refrigerator and the cryogenic microwave source demonstrated in Refs. Tan _et al._, 2017; Masuda _et al._, 2016. In contrast to the previous model Tan _et al._ (2017); Masuda _et al._ (2016), we are able to capture fine details of the physical circuit, multi-photon states, and multi-photon absorption. Importantly, we put our model to an experimental test by directly comparing the recently measured radiation generated by the NIS junctions at high bias voltages Masuda _et al._ (2016) to that predicted by our model. The obtained excellent agreement verifies that our approach is valid, and encourages further extension of the theory to the full spectrum of quantum devices. This paper is organized as follows: Section II presents the physical system under study and Sec. III introduces the corresponding Hamiltonian operators. In Sec. IV, we diagonalize the system Hamiltonian and derive the tunneling-induced transition rates between the eigenstates. Section V provides a master equation for the resonator and a thermal reservoir model of the photon-assisted tunneling. Section VI is devoted to analytical approximations of the coupling strength and temperature of the thermal reservoir at different bias voltage regimes. In Sec. VII, we present optimal parameters for using the quantum-circuit refrigerator for cooling. In Sec. VIII, we study the heating regime at high bias voltages and compare our results with the measurements of Ref. Masuda _et al._, 2016 and with the previous theoretical model based on $P(E)$-theory Devoret _et al._ (1990); Girvin _et al._ (1990); Ingold and Nazarov (1992). Section IX provides our conclusions and an outlook into the future of quantum-circuit refrigeration. ## II Experimental scenario The physical system studied in this paper is illustrated in Fig. 1(a). The central element is a coplanar waveguide resonator with the fundamental resonance angular frequency $\omega_{\rm r}$. At one end, the resonator is capacitively coupled to a normal-metal island which is equipped with two identical normal-metal–insulator–superconductor junctions. At the other end, the resonator is capacitively coupled to a transmission line of characteristic impedance $Z_{\rm tr}$. A pair of NIS junctions, _i.e._, a superconductor–insulator–normal-metal–insulator–superconductor (SINIS) junction, is biased by a voltage $V_{\rm B}=2V$, where $V$ is the bias of a single NIS junction. This allows for a voltage-controlled charging and discharging of the metallic island by means of electron tunneling across the insulating barrier as illustrated in Fig. 1(b). These tunneling transitions may also involve absorption or emission of the resonator photons. Since the rate of the photon-assisted tunneling events and the relative strength between the absorptive and emissive processes is highly dependent on the bias voltage, the voltage-biased SINIS junction provides an effective means for either cooling Tan _et al._ (2017) or heating Masuda _et al._ (2016) the resonator. Thus we refer to the SINIS junction and its coupling circuitry as a quantum-circuit refrigerator. Note that inelastic tunneling processes are familiar from $P(E)$-theory accounting for the energy exchange between a tunneling electron and an electromagnetic environment in thermal equilibrium Devoret _et al._ (1990); Girvin _et al._ (1990); Ingold and Nazarov (1992). In our case, however, the resonator can be driven to a state far from equilibrium. The NIS tunnel junctions are assumed to be of high tunneling resistance $R_{\rm T}\sim 10-100$ k$\Omega$. Accordingly, the tunnel coupling can be treated as a weak perturbation and tunneling across each of the junctions can be considered independently. Thus the fundamental mode of the resonator coupled to the quantum-circuit refrigerator can be described by the effective single-junction circuit diagram shown in Fig. 2. Based on this circuit, we describe below a quantum-mechanical model for the system. ## III Model Hamiltonian <figure><img src="content_image/1706.07188/x2.png"><figcaption>Figure 2: Effective circuit diagram of the studied system. The fundamentalmode of the coplanar waveguide resonator is modeled by the lumped-elementcapacitance C and inductance L. The resonator couples to a normal-metal islandthrough an input capacitance Cc. Tunneling is depicted by the gray symbol onthe left representing the weak coupling between a normal-conducting and asuperconducting electrode. The diagram shows only one of the NIS junctionswith junction capacitance Cj and voltage bias V=VB/2. For tunneling throughthis junction, the other parallel junction acts as a capacitor, thecapacitance of which is included in the capacitance Cm of the metallic islandto the ground. An output capacitor of the capacitance Cg couples the resonatorto a transmission line with a characteristic impedance Ztr. The node fluxes atthe island and at the resonator are denoted by ΦN and Φ, respectively, and QNand Q are their conjugate charges.</figcaption></figure> The core of the system is formed by the coplanar waveguide resonator, the normal-metal island, and the capacitive coupling to the voltage-biased superconducting leads. We first establish the Hamiltonian $\hat{H}_{0}$ of the core part. Subsequently, we add the weak capacitive coupling to the transmission line and the weak tunnel coupling to the superconducting leads. ### Hamiltonian of the core circuit In our analysis below, we utilize the lumped-element circuit of Fig. 2 where we define the various physical capacitances of the system and the inductance $L$ corresponding to the fundamental resonator mode. Following the standard procedure Devoret (1997), the Lagrangian of the core circuit, $\mathcal{L}_{0}$, can be expressed in terms of the fluxes at the two nodes, _i.e._, the flux $\Phi_{\rm N}$ at the normal-metal island and the flux $\Phi$ at the resonator. We find \[\mathcal{L}_{0}= \frac{C_{\rm j}}{2}(\dot{\Phi}_{\rm N}-{V})^{2}+\frac{C_{\rm m}}{ 2}\dot{\Phi}_{\rm N}^{2}+\frac{C_{\rm c}}{2}(\dot{\Phi}_{\rm N}-\dot{\Phi})^{2}\] \[+\frac{C}{2}\dot{\Phi}^{2}-\frac{\Phi^{2}}{2L}.\] (1) Next we introduce the conjugate charges $Q_{\rm N}=\partial\mathcal{L}_{0}/\partial\dot{\Phi}_{\rm N}$ and $Q=\partial\mathcal{L}_{0}/\partial\dot{\Phi}$ and the classical Hamiltonian $H_{0}$ through the Legendre transformation $H_{0}=Q_{\rm N}\dot{\Phi}_{\rm N}+Q\dot{\Phi}-\mathcal{L}_{0}$. After quantization and neglecting irrelevant constant terms, we obtain the core Hamiltonian operator \[\hat{H}_{0}=\frac{(\hat{Q}_{\rm N}+C_{\rm j}{V})^{2}}{2C_{\rm N}}+\frac{[\hat{ Q}+\alpha(\hat{Q}_{\rm N}+C_{\rm j}{V})]^{2}}{2C_{\rm r}}+\frac{\hat{\Phi}^{2} }{2L},\] (2) where the renormalized capacitances of the island and the resonator are given by \[C_{\rm N} =C_{\rm c}+C_{\Sigma\rm{m}},\] (3a) \[C_{\rm r} =C+\alpha C_{\Sigma\rm{m}}.\] (3b) The total capacitance of the normal metal island to ground $C_{\Sigma{\rm m}}=C_{\rm m}+C_{\rm j}$ is independent of the considered junction. The capacitance ratio \[\alpha=\frac{C_{\rm c}}{C_{\rm N}}=\frac{C_{\rm c}}{C_{\rm c}+C_{\Sigma\rm{m}}},\] (4) characterizes the strength of the capacitive coupling between the normal-metal island and the resonator. Since the applied voltage causes only a constant charge shift $Q_{\rm j}=C_{\rm j}V$ in the Hamiltonian (2), we can eliminate it with a gauge transformation Koch _et al._ (2009) where states transform as $\ket{\psi^{\prime}}=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}Q_{\rm j}\hat{\Phi}_{ \rm N}}\ket{\psi}$ and the charge operator as \[\mathrm{e}^{\frac{\mathrm{i}}{\hbar}Q_{\rm j}\hat{\Phi}_{\rm N}}(\hat{Q}_{\rm N }+Q_{\rm j})\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}Q_{\rm j}\hat{\Phi}_{\rm N}}= \hat{Q}_{\rm N},\] (5) owing to the commutation rule $[\hat{\Phi}_{\rm N},\hat{Q}_{\rm N}]=\mathrm{i}\hbar$. In this gauge, the core Hamiltonian $\hat{H}_{0}$ simplifies to \[\hat{H}_{0}=\frac{\hat{Q}_{\rm N}^{2}}{2C_{\rm N}}+\frac{(\hat{Q}+\alpha\hat{Q }_{\rm N})^{2}}{2C_{\rm r}}+\frac{\hat{\Phi}^{2}}{2L},\] (6) which is independent of the applied voltage. ### Coupling to transmission line For readout and external control, the core circuit is coupled to a transmission line of characteristic impedance $Z_{\rm tr}\sim 50$$\Omega$ through an output capacitance $C_{\rm g}$. The impedance of this added element is \[Z(\omega)=Z_{\rm tr}+\frac{1}{\mathrm{i}\omega C_{\rm g}}=Z_{\rm tr}\,\frac{ \omega-\mathrm{i}\omega_{\rm RC}}{\omega},\] (7) where $\omega_{\rm RC}=1/(Z_{\rm tr}C_{\rm g})$ is the inverse of the corresponding $RC$ time. The transmission line acts as an electrodynamic environment of the core circuit. In the limit of weak coupling between the resonator and the transmission line, the influence of the environment is described by the admittance at the resonator frequency $Y(\omega_{\rm r})=Z^{-1}(\omega_{\rm r})$. The real part of $Y(\omega_{\rm r})$ determines the coupling strength, also referred to as the damping coefficient, \[\gamma_{\rm tr}=\frac{\omega_{\rm r}^{2}}{C_{\rm r}Z_{\rm tr}(\omega_{\rm r}^{ 2}+\omega_{\rm RC}^{2})}=\frac{Z_{\rm r}}{Z_{\rm tr}}\frac{\omega_{\rm r}^{3}} {\omega_{\rm r}^{2}+\omega_{\rm RC}^{2}},\] (8) where $Z_{\rm r}=\sqrt{L/C_{\rm r}}$ is the characteristic impedance of the resonator. The imaginary part of $Y(\omega_{\rm r})$ leads to a further renormalization of the resonator capacitance by $C^{\prime}_{\rm r}=C_{\rm r}+\omega_{\rm RC}/[Z_{\rm tr}(\omega^{2}_{\rm r}+ \omega_{\rm RC}^{2})]$ and the resonator frequency by \[\omega_{\rm r}^{\prime}\approx\omega_{\rm r}-\frac{Z_{\rm r}}{2Z_{\rm tr}} \frac{\omega^{2}_{\rm r}\omega_{\rm RC}}{\omega_{\rm r}^{2}+\omega_{\rm RC}^{2 }}.\] (9) Henceforth, we consider the renormalized resonator capacitance and frequency but drop the primes for notational simplicity. ### Tunneling Hamiltonian To incorporate the weak tunnel coupling between the metallic island and one of the superconducting leads, we need to introduce the quasiparticle degrees of freedom in the electrodes. The electronic Hamiltonian \[\hat{H}_{\rm el}=\hat{H}_{\rm N}+\hat{H}_{\rm S}+\hat{H}_{\rm T},\] (10) is formed by the energy of the conduction electrons in the normal metal island $\hat{H}_{\rm N}$, the energy of the quasiparticles in the superconductor $\hat{H}_{\rm S}$, and the energy related to the tunneling interaction $\hat{H}_{\rm T}$. In the gauge (5), we have Ingold and Nazarov (1992) \[\hat{H}_{\rm N}= \sum_{l\sigma}\varepsilon_{l}\hat{d}^{\dagger}_{l\sigma}\hat{d}_{ l\sigma},\] (11a) \[\hat{H}_{\rm S}= \sum_{k\sigma}(\epsilon_{k}-eV)\hat{c}^{\dagger}_{k\sigma}\hat{c} _{k\sigma}+\sum_{k}\left(\Delta_{k}\hat{c}^{\dagger}_{k\uparrow}\hat{c}^{ \dagger}_{-k\downarrow}+\text{h.c.}\right),\] (11b) \[\hat{H}_{\rm T}= \sum_{kl\sigma}\left(T_{lk}\hat{d}^{\dagger}_{l\sigma}\hat{c}_{k \sigma}\mathrm{e}^{-\mathrm{i}\frac{e}{\hbar}\hat{\Phi}_{\rm N}}+\text{h.c.} \right),\] (11c) where $\varepsilon_{l}$ denotes the energy of the normal-metal quasiparticles with wave vector $l$, spin index $\sigma\in\{\uparrow,\downarrow\}$, and annihilation operator $\hat{d}_{l\sigma}$. Similarly, for the superconductor quasiparticles, we have the energy $\epsilon_{k}$, wave vector $k$, and annihilation operator $\hat{c}_{k\sigma}$. Quasiparticles of the superconductor are coupled through the gap parameter $\Delta_{k}$. A tunneling event is associated with a change of the quantized charge $\hat{Q}_{\rm N}$ on the island. In the tunneling Hamiltonian $\hat{H}_{\rm T}$, this is implemented by the operators $\exp(\pm\mathrm{i}e\hat{\Phi}_{\rm N}/\hbar)$ where $e$ is the elementary charge Devoret _et al._ (1990). Owing to the commutation rule $[\hat{\Phi}_{\rm N},\hat{Q}_{\rm N}]=\mathrm{i}\hbar$, we have \[\mathrm{e}^{\pm\mathrm{i}\frac{e}{\hbar}\hat{\Phi}_{\rm N}}\hat{Q}_{\rm N} \mathrm{e}^{\mp\mathrm{i}\frac{e}{\hbar}\hat{\Phi}_{\rm N}}=\hat{Q}_{\rm N}\mp e.\] (12) The probability of theses charge transfer processes is proportional to the tunneling matrix elements $T_{lk}$. Next we transform the voltage bias to the operators by applying a time-dependent unitary transformation $\hat{U}_{V}(t)=\prod_{k\sigma}\exp(\mathrm{i}\frac{e}{\hbar}Vt\hat{c}^{\dagger }_{k\sigma}\hat{c}_{k\sigma})$. The Hamiltonians transform according to \[\hat{H}^{\prime}=\hat{U}_{V}^{\dagger}\hat{H}\hat{U}_{V}+\mathrm{i}\hbar( \partial_{t}\hat{U}_{V}^{\dagger})\hat{U}_{V}.\] (13) In what follows, we again drop the primed notation for simplicity. The transformation affects the terms $\hat{H}_{\rm S}$ and $\hat{H}_{\rm T}$, which become \[\hat{H}_{\rm S}= \sum_{k\sigma}\epsilon_{k}\hat{c}^{\dagger}_{k\sigma}\hat{c}_{k \sigma}+\sum_{k}\left[\widetilde{\Delta}_{k}(t)\hat{c}^{\dagger}_{k\uparrow} \hat{c}^{\dagger}_{-k\downarrow}+\hbox{h.c.}\right],\] (14a) \[\hat{H}_{\rm T}=\] (14b) The transformed gap parameter $\widetilde{\Delta}_{k}(t)=\Delta_{k}\mathrm{e}^{-\mathrm{i}\frac{2e}{\hbar}Vt}$ does not change the superconductor density of states. The charge shift operators of the transformed tunneling Hamiltonian contain a time-dependent phase arising from the applied voltage. The Hamiltonians $\hat{H}_{\rm N}$ for the normal-metal island and $\hat{H}_{0}$ for the core circuit remain unchanged. ## IV Transition rates from tunnel coupling In this section, we formulate the eigenstates of the core circuit $\hat{H}_{0}$ and investigate transitions between them induced by the tunneling Hamiltonian $\hat{H}_{\rm T}$. Transition rates corresponding to an NIS junction are derived first and then generalized for the full SINIS junction. ### Eigenstates of the core circuit In the previous sections, we introduced the different constituents of the Hamiltonian of the circuit shown in Fig. 2 in the presence of tunnel coupling \[\hat{H}=\hat{H}_{0}+\hat{H}_{\rm N}+\hat{H}_{\rm S}+\hat{H}_{\rm T}.\] (15) The coupling to the transmission line is considered separately. In the weak tunneling limit, the term $\hat{H}_{\rm T}$ can be treated as a small perturbation. In the unperturbed Hamiltonian the parts $\hat{H}_{0}$, $\hat{H}_{\rm N}$, and $\hat{H}_{\rm S}$ mutually commute, and hence we choose to diagonalize $\hat{H}_{0}$. Since the core Hamiltonian $\hat{H}_{0}$ in Eq. (6) and the island charge operator $\hat{Q}_{\rm N}$ commute, $[\hat{H}_{0},\hat{Q}_{\rm N}]=0$, they share eigenstates. First, we denote the charge eigenstates of the normal metal island by $\ket{q}$, where $q$ is an integer such that $\hat{Q}_{\rm N}\ket{q}=eq\ket{q}$. Thus the core Hamiltonian may be written as \[\hat{H}_{0}=\sum_{q=-\infty}^{\infty}\sum_{m=0}^{\infty}\ket{q\,m_{q}}\bra{q\, m_{q}}\left[E_{\rm N}q^{2}+\hbar\omega_{\rm r}\left(m+\frac{1}{2}\right)\right],\] (16) where we have introduced the charging energy of the island $E_{\rm N}=e^{2}/2C_{\rm N}$ and have diagonalized the resonator. The eigenstates of the resonator with frequency $\omega_{\rm r}=1/\sqrt{LC_{\rm r}}$ are harmonic oscillator states, \[\ket{m_{q}}=\mathrm{e}^{-\mathrm{i}\alpha q\frac{e}{\hbar}\hat{\Phi}}\ket{m},\] (17) where the subscript $q$ indicates the shift of the oscillator coordinate $Q$ by the charge on the normal metal island. For the harmonic oscillator, the charge shift affects only the eigenstates but not the energies. Furthermore, the shifted harmonic oscillator wavefunctions have nontrivial overlap which can be considered as the origin of the possibility to control the resonator state by photon-assisted tunneling. ### Tunneling at an NIS junction The eigenstates of the core circuit are linked by transitions of electrons from the superconductor to the normal-metal island and vice versa, governed by the tunneling operator \[\hat{\Theta}=\sum_{kl\sigma}T_{lk}\hat{d}^{\dagger}_{l\sigma}\hat{c}_{k\sigma}.\] (18) The tunneling Hamiltonian (14) may be expressed as \[\hat{H}_{\rm T}=\hat{\Theta}\,\mathrm{e}^{-\mathrm{i}\frac{e}{\hbar}\left(\hat {\Phi}_{\rm N}-Vt\right)}+\hbox{h.c.}.\] (19) We denote by $\ket{E}$ an eigenstate of the junction electrodes which is a product of an eigenstate of the normal-metal island $\hat{H}_{\rm N}$ and that of the superconducting lead $\hat{H}_{\rm S}$. The relevant transition matrix element is thus of the form \[\bra{E^{\prime}\!,q^{\prime}m^{\prime}_{q^{\prime}}} \hat{H}_{\rm T}\ket{E,q\,m_{q}}\] (20) \[=\] where the matrix elements of the core circuit are given by (21) The overlap between the charge-shifted resonator eigenstates can be calculated using Eq. (17). The result is expressed in terms of the generalized Laguerre polynomials $L^{\ell}_{n}(\rho)$. Abramowitz and Stegun (1972); Hollenhorst (1979); Catelani _et al._ (2011); Souquet _et al._ (2014) We find where \[M_{mm^{\prime}}^{2}=\begin{cases}\mathrm{e}^{-\rho}\rho^{\ell}\frac{m^{\prime} !}{m!}\left[L^{\ell}_{m^{\prime}}(\rho)\right]^{2},&m\geq m^{\prime},\\ &\\ \mathrm{e}^{-\rho}\rho^{-\ell}\frac{m!}{m^{\prime}!}\left[L^{-\ell}_{m}(\rho) \right]^{2},&m<m^{\prime}.\end{cases}\] (22) Above, $\ell=m-m^{\prime}$ and the interaction parameter \[\rho=\pi\alpha^{2}\frac{1}{\omega_{\rm r}C_{\rm r}R_{\rm K}}=\pi\frac{C^{2}_{ \rm c}}{C^{2}_{\rm N}}\frac{Z_{\rm r}}{R_{\rm K}},\] (23) where $R_{\rm K}=h/e^{2}$ is the von Klitzing constant defined in terms of the Planck constant $h$ and the elementary charge $e$. Note that when the interaction is strong, $\rho\gtrsim 1$, some transition overlaps $M^{2}_{mm^{\prime}}$ can vanish at the roots of the Laguerre polynomials Gramich _et al._ (2013). To evaluate the electronic matrix elements, such as $\bra{E^{\prime}}\hat{\Theta}\ket{E}$, we employ the standard approach for tunnel junctions Ingold and Nazarov (1992). Subsequently, we insert the transition matrix element in Eq. (20) into Fermi’s golden rule evaluated for the Hamiltonians (14) and trace out the electronic degrees of freedom, and hence obtain the transition rates between eigenstates of $\hat{H}_{0}$. The electronic transitions from the normal-metal island to the superconducting electrode are connected with a transition of the core circuit from a state $\ket{q\,m_{q}}$ to a state $\ket{q\!+\!1\,m^{\prime}_{q+1}}$ where the normal-metal island charge has increased by an elementary charge $e$. For these forward-tunneling transitions, we find \[\overrightarrow{\Gamma}_{q,m,m^{\prime}}(V)=\frac{M_{mm^{\prime}} ^{2}}{e^{2}R_{\rm T}}\iint\text{d}\epsilon_{k}\text{d}\varepsilon_{l}\,n_{\rm S }(\epsilon_{k})[1-f_{\rm S}(\epsilon_{k})]f_{\rm N}(\varepsilon_{l})\] \[\times\delta\left[\epsilon_{k}+E_{\rm N}(1+2q)+\hbar\omega_{\rm r }(m^{\prime}-m)-\varepsilon_{l}-eV\right].\] (24) Here, we have assumed that the tunneling matrix elements $\|T_{lk}\|$ are approximately constant over the relevant integration range around the Fermi energies and have expressed the summation $\sum_{kl\sigma}\|T_{lk}\|^{2}$ in terms of the experimentally measurable tunneling conductance $1/R_{\rm T}$. This wide-band limit is appropriate for metallic tunnel junctions. The functions $f_{\rm S}(\varepsilon)$ and $f_{\rm N}(\varepsilon)$ denote the Fermi functions in the superconducting and normal-metal electrodes, respectively. The normalized quasiparticle density of states in the superconductor is given by Dynes _et al._ (1978) \[n_{\rm S}(\varepsilon)=\left\|\textrm{Re}\left\{\frac{\varepsilon+i\gamma_{\rm D }\Delta}{\sqrt{(\varepsilon+i\gamma_{\rm D}\Delta)^{2}-\Delta^{2}}}\right\} \right\|,\] (25) where $\Delta$ is the superconductor gap parameter and the Dynes parameter $\gamma_{\rm D}$ determines the density of subgap states. Typical values for the tunnel junctions are $\gamma_{\rm D}\sim 10^{-4}$ and $\Delta\sim 200$$\mu$eV ($\Delta/h\sim 50$ GHz). It is convenient to introduce the normalized rate of forward quasiparticle tunneling for $R_{\rm T}=R_{\rm K}$ at the energy bias $E$ as \[\overrightarrow{F}(E)=\frac{1}{h}\int\text{d}\varepsilon\,n_{\rm S}( \varepsilon)[1-f_{\rm S}(\varepsilon)]f_{\rm N}(\varepsilon-E).\] (26) In terms of this rate function, the result (24) assumes an intuitive form \[\overrightarrow{\Gamma}_{q,m,m^{\prime}}(V)\!=\!M_{mm^{\prime}}^{2}\frac{R_{ \rm K}}{R_{\rm T}}\overrightarrow{F}{\textbf{(}}eV+\hbar\omega_{\rm r}\ell-E_{ q}^{+}{\textbf{)}},\] (27) where again $\ell=m-m^{\prime}$ and \[E_{q}^{\pm}=E_{\rm N}(1\pm 2q)\] (28) is the change of the charging energy of the normal metal island due to forward/backward tunneling. Likewise, we obtain the rate for the backward tunneling processes related to a transition of the core circuit from a state $\ket{q\,m_{q}}$ to a state $\ket{q\!-\!1\,m^{\prime}_{q-1}}$ as \[\overleftarrow{\Gamma}_{q,m,m^{\prime}}(V)\!=\!M_{mm^{\prime}}^{2}\frac{R_{\rm K }}{R_{\rm T}}\overleftarrow{F}{\textbf{(}}eV-\hbar\omega_{\rm r}\ell+E_{q}^{-} {\textbf{)}}\] (29) with the function \[\overleftarrow{F}(E)=\frac{1}{h}\int\text{d}\epsilon\,n_{\rm S}(\varepsilon)f_ {\rm S}(\varepsilon)\left[1-f_{\rm N}\left(\varepsilon-E\right)\right]\] (30) giving the normalized rate of the backward quasiparticle tunneling events at the energy bias $E$. ### Symmetric SINIS junction The SINIS junction consists of an SIN and an NIS junction. The above results (27)–(29) are for the NIS junction. Assuming that the tunneling resistances and junction capacitances are identical, the consideration of the SIN junction is analogous, except that the voltage $V$ is reversed, see Fig. 1. For forward tunneling events across the SIN junction, related to a transition of the core circuit from a state $\ket{q\,m_{q}}$ to a state $\ket{q\!+\!1,m^{\prime}_{q+1}}$, we obtain the forward rate \[\overrightarrow{\Gamma}^{\prime}_{q,m,m^{\prime}}(V)\!=\overrightarrow{\Gamma} _{q,m,m^{\prime}}(-V)\!\] (31) provided that the two superconducting electrodes have identical temperatures and densities of states. This symmetry holds also for the backward rate. By summing the contributions from both junctions, the total forward rate between the states $\ket{q\,m}$ and $\ket{q+1\,m^{\prime}}$ is \[\Gamma^{+}_{q,m,m^{\prime}}(V)=\sum_{\tau=\pm 1}\overrightarrow{\Gamma}_{q,m,m ^{\prime}}(\tau V).\] (32) A corresponding result holds for the total backward rate $\Gamma^{-}_{q,m,m^{\prime}}(V)$ between the states $\ket{q\,m}$ and $\ket{q-1\,m^{\prime}}$. To further evaluate the transition rates, we assume that the normal and superconducting electrodes are at the same temperature $T_{\rm N}$. Thus the normalized rate of forward tunneling transitions in Eq. (26) may be expressed as \[\overrightarrow{F}(E)=\int\text{d}\varepsilon\,n_{\rm S}(\varepsilon)\frac{f( \varepsilon-E)-f(\varepsilon)}{h\left(1-\mathrm{e}^{-E/k_{\rm B}T_{\rm N}} \right)},\] (33) from which we obtain useful identities $\overleftarrow{F}(E)=\overrightarrow{F}(-E)$ and $\overrightarrow{F}(-E)=e^{-E/k_{\rm B}T_{\rm N}}\overrightarrow{F}(E)$ relating the backward and forward rates. This allows to write the transition rates in both directions fully in terms of the normalized forward rates \[\Gamma^{\pm}_{q,m,m^{\prime}}(V)=M^{2}_{mm^{\prime}}\frac{R_{\rm K }}{R_{\textrm{T}}}\Big{[} \overrightarrow{F}\left(eV+\hbar\omega_{\rm r}\ell-E_{q}^{\pm}\right)\] (34) \[+\overrightarrow{F}\left(-eV+\hbar\omega_{\rm r}\ell-E_{q}^{\pm} \right)\Big{]},\] where $\ell=m-m^{\prime}$. The only remaining difference between the tunneling directions is in the charging energy differences $E_{q}^{\pm}=E_{\rm N}(1\pm 2q)$. In practice, the tunneling resistances and capacitances of the two junctions will differ typically by a few percent. The main effect of this asymmetry is a slight shift of the most probable charge state of the island away from the balanced value of $q=0$ Ingold _et al._ (1991); Ingold and Nazarov (1992); 52. If the charging energy associated with this shift remains small compared with the excitation energy $\hbar\omega_{\rm r}$ of the resonator, which is typically the case, the effects of a modest asymmetry of the rates (34) are very small. The theory presented here can thus be safely compared with experiments. ## V Master equation In the previous sections, we determined the transition rates between the eigenstates of the core circuit induced by a capacitively coupled SINIS junction. The weak coupling to the transmission line induces further transitions, yet, only between neighboring states of the resonator. The rates for these transitions are \[{\Gamma}^{\downarrow}_{m} =\gamma_{\rm tr}(N_{\rm tr}+1)m,\] (35a) \[{\Gamma}^{\uparrow}_{m} =\gamma_{\rm tr}\,N_{\rm tr}(m+1),\] (35b) where $N_{\rm tr}=1/[\exp(\hbar\omega_{\rm r}/k_{\rm B}T_{\rm tr})-1]$ is the mean thermal occupation factor at the reservoir temperature $T_{\rm tr}\approx 100-150$ mK and the coupling strength to the transmission line $\gamma_{\rm tr}$ is given in Eq. (8). Having determined all transition rates, we write the master equation describing the population dynamics of the core circuit. Let us denote by $p_{q,m}(t)$ the probability that the core circuit occupies the state $\ket{q\,m_{q}}$ at time $t$. This probability obeys the Pauli master equation \[\dot{p}_{q,m}={\Gamma}_{m\!-\!1}^{\uparrow}\,p_{q,m\!-\!1}+{ \Gamma}_{m\!+\!1}^{\downarrow}\,p_{q,m\!+\!1}-\left({\Gamma}_{m}^{\uparrow}+{ \Gamma}_{m}^{\downarrow}\right) p_{q,m}\] \[+\sum_{m^{\prime}=0}^{\infty}\Big{[}\Gamma^{+}_{q\!-\!1,m^{\prime },m}\,p_{q\!-\!1,m^{\prime}}+\Gamma^{-}_{q\!+\!1,m^{\prime},m} p_{q\!+\!1,m^{\prime}}\] \[-\left(\Gamma^{+}_{q,m,m^{\prime}}+\Gamma^{-}_{q,m,m^{\prime}} \right)p_{q,m^{\prime}} \Big{]}.\] (36) We are primarily interested in the steady-state solution of the resonator. Since also the dynamics of the metal island charge $\ket{q}$ is involved, solving the full master equation is a challenging problem. Fortunately, it can be simplified in specials cases as described below. ### Weak interaction and charge thermalization We begin the simplifications of the master equation by considering the magnitude of the resonator matrix elements $M_{mm^{\prime}}^{2}$ appearing in the transition rates in Eq. (34). The magnitude is governed by the interaction parameter $\rho=\pi\alpha^{2}Z_{\rm r}/R_{\rm K}$ given in Eq. (23). It characterizes how strongly the core circuit responds to a quasiparticle tunneling event. The ratio of the characteristic impedance of the resonator $Z_{\rm r}$ and the von Klitzing constant $R_{\rm K}$ is a measure for the stiffness of the harmonic oscillator potential, determining how strongly a charge shift affects the overlap of the resonator eigenstates. The magnitude of the charge shift, in turn, depends on the capacitance ratio $\alpha=C_{\rm c}/C_{\rm N}$. The capacitive coupling can be made strong Masuda _et al._ (2016) by choosing in the fabrication the input capacitance $C_{\rm c}$ to dominate over the normal metal capacitance $C_{\Sigma\rm m}$ in $C_{\rm N}=C_{\rm c}+C_{\rm\Sigma m}$. In fact, a galvanic contact Tan _et al._ (2017) realizes $\alpha=1$. Despite of strong capacitive interaction, typical geometries and material parameters of coplanar waveguide resonators render the characteristic impedance low compared to $R_{\rm K}$. Hence, with resonators, the interaction parameter has typically very low values of the order of $\rho\sim 0.001$. <figure><img src="content_image/1706.07188/x3.png"><figcaption>Figure 3: Resonator transition rates Γm,m′ of Eq. (39) as functions of thesingle-junction bias voltage. Both the one-photon processes Γ1,0 (blue solidline) and Γ0,1 (red solid line) as well as the two-photon processes Γ2,0 (bluedotted line) and Γ0,2 (red dotted line) are shown. The elastic transition rateΓ0,0 (green dashed line) exceeds the inelastic rates. The used parameterscorrespond to a typical experiment Tan _et al._ (2017); Masuda _et al._(2016): Δ=200 μeV, γD=10−4, RT=50 kΩ, TN=100 mK, Cc=1.0 pF, CΣm=10 fF,ωr/2π=7.0 GHz, and Zr=35 Ω.</figcaption></figure> Let us express the matrix elements $M_{mm^{\prime}}^{2}$ in Eq. (22) in the lowest order in $\rho$. Using $\ell=m-m^{\prime}$ and $L_{m^{\prime}}^{\ell}(0)=\binom{m}{m^{\prime}}$, we obtain \[M_{mm^{\prime}}^{2}=\begin{cases}\frac{1}{\ell!}\binom{m}{\ell}\rho^{\ell}+ \mathcal{O}\left(\rho^{\ell+1}\right),&m\geq m^{\prime},\\ \frac{1}{\|\ell\|!}\binom{m^{\prime}}{\|\ell\|}\rho^{\|\ell\|}+\mathcal{O}\left(\rho ^{\|\ell\|+1}\right),&m<m^{\prime}.\end{cases}\] (37) The small interaction parameter will imply that tunneling processes involving simultaneously several photons are suppressed as $M^{2}_{m,m\pm\ell}\propto\rho^{\ell}$ with respect to single-photon processes where $M^{2}_{m,m\pm 1}\propto\rho$. Despite of this suppression of the matrix elements, we note that photon-assisted tunneling drives multi-photon transitions, formally, without typical selection rules. In certain conditions, due to the enhancement by electron tunneling rates, the rate of absorptive two-photon transitions can exceed the emissive single-photon rate as shown in Fig. 3. In this kind of situations, the resonator state needs to be solved from the master equation (36), which takes into account all single and multi-photon processes. In this paper, however, we do not concentrate on the details of the high-photon-number processes. We observe from the master equation (36) that the charge state $\ket{q}$ is driven both by elastic and inelastic transitions. As in the case of multi-photon transitions, the single-photon inelastic transitions are suppressed with respect to elastic transitions since $M^{2}_{m,m}\propto 1-\rho$. Thus for typical experimental parameters, _i.e._, when the ratio $\hbar\omega_{\rm r}/\Delta$ is sufficiently small, the temperature low $k_{\rm B}T_{\rm N}\ll\Delta$, and the Dynes parameter $\gamma_{\rm D}$ rather large, the inelastic tunneling rates remain lower than the elastic rates in the whole relevant bias region (see Fig. 3). As shown in Appendix A, in the case of dominating elastic tunneling, the charge states of the normal-metal island rapidly approach a stationary and symmetric thermal distribution $p_{q}=\exp(-E_{\rm N}q^{2}/k_{\rm B}T_{\rm Q})/Z$, where $T_{\rm Q}$ is the effective temperature of the charge distribution and $Z=\sum_{q}\exp(-E_{\rm N}q^{2}/k_{\rm B}T_{\rm Q})$. <figure><img src="content_image/1706.07188/x4.png"><figcaption>Figure 4: (a) Spectral density of tunneling ST as a function of the angularfrequency ω and single-junction bias voltage V. The parameters equal to thoseof Fig. 3. (b) Traces from panel (a) corresponding to eV/Δ=0.1,0.5,0.9,1.3[vertical dashed lines in panel (a)]. The black dashed line is a fit to anohmic behavior ST∝ω at eV/Δ=1.3. Near the gap voltage \|eV−Δ\|≲ℏω, the spectraldensity exhibits clear non-ohmic, exponential behavior. In the region\|eV−Δ\|≳ℏω, the spectral density ST(ω) flattens to a linear, ohmic trend.</figcaption></figure> ### Master equation for the resonator The resonator states $\ket{m}$ are driven by inelastic processes or by transitions induced by the coupling to the transmission line. This latter relaxation is practically independent of the metal island charge. In the case where the elastic tunneling events dominate over the inelastic ones, the charging dynamics of the metal island is thermalized rapidly with respect to the time scales of the inelastic transitions. In an experimentally relevant regime, it suffices to consider the charge and resonator dynamics independently and it is justified to average the resonator transition rates in Eq. (34) over the stationary and symmetric thermal distribution of the charge states $p_{q}$. Consequently, we define \[{\Gamma}_{m,m^{\prime}}(V)=\sum_{q}p_{q}\left[\Gamma^{+}_{q,m,m^{\prime}}(V)+ \Gamma^{-}_{q,m,m^{\prime}}(V)\right].\] (38) With typical experimental parameters, the charging energy of the normal-metal island is much smaller than other relevant energy scales of the setup: $E_{\rm N}=e^{2}/2C_{\rm N}\ll\Delta,\hbar\omega_{\rm r},k_{\rm B}T_{\rm N}$ ($E_{\rm N}/h\sim 10$ MHz). Thus, we expand the transition rates in Eq. (38) to first order in $E_{q}$ and average over the thermal charge state distribution. Owing to the symmetry of the charge distribution, the first order effect of the charging energy differences $E_{q}^{\pm}=E_{\rm N}(1\pm 2q)$ vanishes except for a small overall bias shift by the charging energy $E_{\rm N}$. Hence, the resonator transition rate from the state $\ket{m}$ to the state $\ket{m^{\prime}}$ assumes the form \[\Gamma_{m,m^{\prime}}(V) \approx M^{2}_{mm^{\prime}}\frac{2R_{\rm K}}{R_{\rm T}}\sum_{\tau =\pm 1}\overrightarrow{F}\left(\tau eV+\hbar\omega_{\rm r}\ell-E_{\rm N}\right)\] \[=M^{2}_{mm^{\prime}}S_{\rm T}(\omega_{mm^{\prime}}).\] (39) Here, $S_{\rm T}(\omega)$ denotes the spectral density of tunneling and $\omega_{mm^{\prime}}=\omega_{\rm r}(m-m^{\prime})$ the transition frequency. The transition rates and the spectral density are shown in Figs. 3 and 4, respectively, with typical experimental parameters. Finally, we express the master equation for the population of the resonator states as \[\dot{p}_{m}={\Gamma}_{m\!-\!1}^{\uparrow}\,p_{m\!-\!1} +{\Gamma}_{m\!+\!1}^{\downarrow}\,p_{m\!+\!1}-\left({\Gamma}_{m}^ {\uparrow}+{\Gamma}_{m}^{\downarrow}\right)p_{m}\] \[+\sum_{m^{\prime}}{\Gamma}_{m^{\prime},m}\,p_{m^{\prime}}-\sum_{m ^{\prime}}{\Gamma}_{m,m^{\prime}}\,p_{m}\text{.}\] (40) Here, the terms containing $\Gamma_{m}^{\mathbin{\uparrow},\hskip 0.0pt\downarrow}$ describe the transitions caused by the transmission line and the remaining ones those due to photon-assisted tunneling. ### Single-photon thermal reservoir For the single-photon processes, we observe that $M_{m,m-1}^{2}=\rho m$, $M_{m,m+1}^{2}=\rho(m+1)$ which are exactly of the form of the matrix elements of the destruction $\hat{a}$ and creation $\hat{a}^{\dagger}$ operators of the resonator, respectively. Since the single-photon matrix elements $M_{mm\pm 1}^{2}$ are linear in the photon number $m$, the transition rates in Eq. (39) can be written in an intuitive form ¹ [FOOTNOTE:1][ENDFOOTNOTE] \[{\Gamma}_{m,m-1} =\gamma_{\rm T}(N_{\rm T}+1)m,\] (41a) \[{\Gamma}_{m,m+1} =\gamma_{\rm T}N_{\rm T}(m+1).\] (41b) These rates correspond exactly to those generated by coupling to a thermal reservoir with the coupling strength $\gamma_{\rm T}$ and the mean thermal occupation $N_{\rm T}$. The effective temperature $T_{\rm T}$ of the reservoir is defined by $N_{\rm T}=1/\left[\exp(\hbar\omega_{\rm r}/k_{\rm B}T_{\rm T})-1\right]$. The subscript T indicates that the reservoir stems from the inelastic tunneling processes either absorbing or emitting resonator photons. The coupling strength and temperature can be written as \[\gamma_{\rm T} =\bar{\gamma}_{\rm T}\frac{\pi}{\omega_{\rm r}}\sum_{\ell,\tau= \pm 1}\ell\overrightarrow{F}\left(\tau eV+\ell\hbar\omega_{\rm r}-E_{\rm N} \right),\] (42a) \[T_{\rm T}\] (42b) where, for the sake of notational brevity, we have defined \[\bar{\gamma}_{\rm T}=2\frac{C_{\rm c}^{2}}{C_{\rm N}^{2}}\frac{1}{R_{\rm T}C}= 2\frac{C_{\rm c}^{2}}{C_{\rm N}^{2}}\frac{Z_{\rm r}}{R_{\rm T}}\omega_{\rm r},\] (43) which is the coupling strength corresponding to a normal-metal–insulator–normal-metal junction. It is determined by the inverse of the $RC$ time for the two parallel junctions each with tunneling resistance $R_{\rm T}$ multiplied by the capacitive coupling fraction $C^{2}_{\rm c}/C^{2}_{\rm N}$ of the normal-metal island. Note that the coupling strength $\bar{\gamma}_{\rm T}$ has a weaker dependence on the resonator frequency than the coupling strength to the transmission line of Eq. (8) for $\omega_{\rm RC}\gg\omega_{\rm r}$. In terms of the effective temperature $T_{\rm T}$ the single-photon rates obey a detailed balance condition \[\frac{{\Gamma}_{10}}{{\Gamma}_{01}}=\frac{S_{\rm T}(\omega_{\rm r})}{S_{\rm T} (-\omega_{\rm r})}=\mathrm{e}^{\frac{\hbar\omega_{\rm r}}{k_{\rm B}T_{\rm T}}}.\] (44) In Fig. 5, we show the bias voltage dependence of the coupling strength and the effective temperature for typical experimental parameters. <figure><img src="content_image/1706.07188/x5.png"><figcaption>Figure 5: (a) The coupling strength γT and (b) the temperature TT of thethermal reservoir by photon-assisted tunneling as a function of the single-junction bias voltage. The parameters equal to those of Fig. 3 except that theresonator frequencies are ωr/2π=4.0,5.0,…,10.0 GHz corresponding to linecolors from light gray to black, respectively. The overall scaling of γT comesfrom the scaling of ¯γT introduced in Eq. (43) assuming that thecharacteristic impedance Zr is frequency independent. The coupling strength¯γT for ωr/2π=4.0 GHz is depicted by a magenta dash-dotted line. In thehighlighted region the thermal reservoir is cooler than the island electronswith temperature TN, depicted by a green dash-dotted line. The red dashedlines show the analytic results for ωr/2π=4.0 GHz in the deep subgap, thermalactivation and above the gap regions, respectively, corresponding to theanalytical results (49), (51), and (52).</figcaption></figure> In this regime where the multiphoton processes are negligible, the quantum dynamics of the circuit in Fig. 2 corresponds to a harmonic oscillator which is linearly coupled to two thermal baths: one arising from the transmission line and the other from the photon-assisted tunneling. The coupling strengths and mean thermal occupations of the baths are known. This model allows us to directly generalize Alicki (1977) the master equation (V.2) to that of the density matrix $\hat{\rho}$ of a linear resonator \[\dot{\hat{\rho}}= -\frac{\mathrm{i}}{\hbar}[\hat{H}_{\rm r},\hat{\rho}]\] (45) \[+\mathcal{D}\left(\sqrt{\gamma_{\rm tr}(N_{\rm tr}+1)}\hat{a} \right)\hat{\rho}+\mathcal{D}\left(\sqrt{\gamma_{\rm tr}N_{\rm tr}}\hat{a}^{ \dagger}\right)\hat{\rho}\] where $\mathcal{D}(\hat{c})\hat{\rho}=\hat{c}\hat{\rho}\hat{c}^{\dagger}-\frac{1}{2}( \hat{c}^{\dagger}\hat{c}\hat{\rho}+\hat{\rho}\hat{c}^{\dagger}\hat{c})$ denotes a Lindbladian dissipator and $\hat{H}_{\rm r}=\hbar\omega_{\rm r}(\hat{a}^{\dagger}\hat{a}+\frac{1}{2})$ is the Hamiltonian of the resonator. Since tunneling has no effect on the resonator energy, there is no added dephasing of the resonator in Eq. (45). The resulting steady-state occupation of the resonator is a weighted sum of the mean thermal occupation numbers \[N_{\rm r}=\frac{\gamma_{\rm tr}N_{\rm tr}+\gamma_{\rm T}N_{\rm T}}{\gamma_{\rm tr }+\gamma_{\rm T}},\] (46) and the total damping coefficient of the resonator is a direct sum the two damping coefficients \[\gamma_{\rm r}=\gamma_{\rm tr}+\gamma_{\rm T}.\] (47) Hence, the steady-state occupation of the resonator depends both on the relative temperatures and coupling strengths of the two thermal baths acting on the resonator. The coupling strength $\gamma_{\rm T}$ and the temperature $T_{\rm T}$ inherit the strong bias voltage dependence of the resonator transition rates originating from the tunneling processes, see Fig. 5. This is one of the most important results of our work. In the remainder, we will elaborate on these findings, give analytical results in parameter regions of experimental interest, and compare those with experimental data. Tuning the bias voltage near the edge of the superconductor gap, absorptive transitions become energetically favorable since emissive transitions suffer from the suppressed density of states of the superconductor, see Fig. 1(b). This implies a cold effective thermal reservoir, ideally $T_{\rm T}=T_{\rm N}/2$. Near the gap, also the coupling strength increases exponentially together with the rate of absorptive forward transitions, see Eq. (42a). In other words, near the gap one can achieve a strong coupling to a cold bath, which is the operating principle of the quantum-circuit refrigerator Tan _et al._ (2017). When tuning beyond the superconductor gap $eV\gg\Delta$, the relative difference of absorptive and emissive transitions asymptotically vanishes implying a thermal reservoir approaching an infinite temperature. This operating regime can be utilized as incoherent on-chip microwave source as recently demonstrated Masuda _et al._ (2016). The coupling strength saturates as the absolute difference between the rates approaches a constant. ## VI Regions of experimental interest Let us consider in greater detail three parameter regions of particular interest to the recent experimental studies Tan _et al._ (2017); Masuda _et al._ (2016). First, the bias region $\|eV\pm\hbar\omega_{\rm r}\|\ll\Delta$, referred to as the deep subgap, where the tunneling is dominated by the remaining superconductor subgap states characterized by the Dynes parameter $\gamma_{\rm D}$ in Eq. (25). The second region is referred to as the thermal-activation region, where the thermal excitations over the superconductor gap dominate over the subgap transitions. Typically, thermal activation becomes the most important type of transitions for $\Delta/2\lesssim eV-\hbar\omega_{\rm r}$ and $eV+\hbar\omega_{\rm r}\lesssim\Delta$, where the lower bound $\Delta/2$ depends on the electron temperature of the metals and the Dynes parameter. If the junction is biased above the superconductor gap $\Delta\lesssim eV-\hbar\omega_{\rm r}$, neither subgap states nor thermal excitations are the dominant source of excitations since electrons can tunnel through the insulating barrier just given the energy from the bias voltage. In the extreme biasing regime $eV\gg\Delta$ the shape of the superconductor density of states loses its significance. Since the matrix elements in Eq. (22) are independent of the bias voltage, we may analytically calculate the tunneling rates in Eq. (41). In the analytic considerations we ignore the charging energy $E_{\rm N}$ since with typical experimental parameters Tan _et al._ (2017); Masuda _et al._ (2016)$E_{\rm N}/\Delta\sim 10^{-3}$. ### Deep subgap In the deep-subgap region at small bias, the forward $\overrightarrow{F}(E)$ and backward tunneling $\overrightarrow{F}(-E)$ rates are of comparable magnitude. We use the expression (33) and linearize $\overrightarrow{F}(E)$ for small $E$ resulting in \[\overrightarrow{F}(E) \approx\frac{\frac{E}{k_{\rm B}T_{\rm N}}}{4h\left(1-\mathrm{e}^{ -\frac{E}{k_{\rm B}T_{\rm N}}}\right)}\int\text{d}\epsilon\ \textrm{sech}^{2} \left(\frac{\varepsilon}{2k_{\rm B}T_{\rm N}}\right)n_{\rm S}(\epsilon)\] \[\approx\frac{\gamma_{\rm D}E}{h\left(1-\mathrm{e}^{-\frac{E}{k_{ \rm B}T_{\rm N}}}\right)}.\] (48) Here the function $\textrm{sech}^{2}[\epsilon/(2k_{\rm B}T_{\rm N})]$ is peaked near the origin with width $\sim k_{\rm B}T_{\rm N}\ll\Delta$. Thus, in the deep-subgap region, the superconductor density of states of Eq. (25) can be approximated by the constant $n_{\rm s}(\epsilon)\approx\gamma_{\rm D}$. Similar arguments apply for the backward tunneling, resulting in the deep-subgap coupling strength $\gamma^{\rm dg}_{\rm T}$ and temperature $T^{\rm dg}_{\rm T}$ to be: \[\gamma^{\rm dg}_{\rm T}(V)=2\frac{C^{2}_{\rm c}}{C_{\rm N}^{2}} \frac{Z_{\rm r}}{R_{\rm T}}\omega_{\rm r}\gamma_{\rm D}=\bar{\gamma}_{\rm T} \gamma_{\rm D},\] (49a) \[T^{\rm dg}_{\rm T}(V)=\frac{\hbar\omega_{\rm r}}{k_{\rm B}}\] (49b) \[\times \left\{\log\left[\frac{\frac{eV}{\hbar\omega_{\rm r}}\sinh\left( \frac{\hbar\omega_{\rm r}}{k_{\rm B}T_{\rm N}}\right)+\cosh\left(\frac{\hbar \omega_{\rm r}}{k_{\rm B}T_{\rm N}}\right)-\mathrm{e}^{\frac{\hbar\omega_{\rm r }}{k_{\rm B}T_{\rm N}}}}{\frac{eV}{\hbar\omega_{\rm r}}\sinh\left(\frac{\hbar \omega_{\rm r}}{k_{\rm B}T_{\rm N}}\right)-\cosh\left(\frac{\hbar\omega_{\rm r }}{k_{\rm B}T_{\rm N}}\right)+\mathrm{e}^{-\frac{\hbar\omega_{\rm r}}{k_{\rm B }T_{\rm N}}}}\right]\right\}^{-1}\] See Eq. (43) for the definition of $\bar{\gamma}_{\rm T}$. Note that at zero bias, the temperature of the reservoir formed by the SINIS tunnel junction equals to the electron temperature of the metals $T_{\rm T}^{\rm dg}(0)=T_{\rm N}$. ### Thermal activation Beyond the deep-subgap region, the thermally excited electron tunneling starts to dominate. Formally, this means that the tails of the Fermi distributions reach beyond the gap of the superconductor density of states. In this region, for an analytical treatment we ignore the effects of the backward tunneling and the subgap states. Approximating the Fermi function tails by exponentials results in \[\overrightarrow{F}(E) \approx\frac{1}{h}\int_{\Delta}^{\infty}\text{d}\varepsilon\:n_{ \rm S}(\varepsilon)\exp\left(-\frac{\varepsilon-E}{k_{\rm B}T_{\rm N}}\right)\] \[=\frac{\Delta}{h}\exp\left(\frac{E}{k_{\rm B}T_{\rm N}}\right)K_{ 1}\left(\frac{\Delta}{k_{\rm B}T_{\rm N}}\right)\] \[\approx\frac{1}{h}\sqrt{\frac{\pi k_{\rm B}T_{\rm N}\Delta}{2}} \exp\left(\frac{E-\Delta}{k_{\rm B}T_{\rm N}}\right).\] (50) Here, $K_{1}[\Delta/(k_{\rm B}T_{\rm N})]$ denotes the modified Bessel function of the second kind Abramowitz and Stegun (1972), which is exponentially decaying for $\Delta\gg k_{\rm B}T_{\rm N}$. Thus the thermally activated coupling strength and the temperature of the reservoir are given by \[\gamma^{\rm th}_{\rm T}(V)=\bar{\gamma}_{\rm T}\frac{\sinh\left( \frac{\hbar\omega_{\rm r}}{k_{\rm B}T_{\rm N}}\right)}{\frac{\hbar\omega_{\rm r }}{k_{\rm B}T_{\rm N}}}\sqrt{\frac{2\Delta}{\pi k_{\rm B}T_{\rm N}}}\exp\left( {\frac{eV-\Delta}{k_{\rm B}T_{\rm N}}}\right),\] (51a) \[T^{\rm th}_{\rm T}(V)=\frac{T_{\rm N}}{2}.\] (51b) Importantly, the effective temperature of the reservoir formed by the SINIS junction is constant and equals to half of the electron temperature. This can be understood by the fact that the tunneling rates for the photon absorption and emission events are weighted by the normal-metal Fermi functions evaluated at the energy difference $2\hbar\omega_{\rm r}$. Yet each of these events causes an energy change of $\hbar\omega_{\rm r}$ in the resonator. However, the coupling strength is exponentially dependent on the distance of the bias from the superconductor gap. This originates from the fact that the thermally activated transition rates increase exponentially as the effective bias $eV\pm\hbar\omega_{\rm r}$ approaches the gap $\Delta$. This phenomenon also yields the above exponential dependence of the rates on the ratio $\hbar\omega_{\rm r}/k_{\rm B}T_{\rm N}$. The weak overall scaling of the coupling strength with the square root of the ratio of the superconductor gap to thermal energy $\sqrt{\Delta/k_{\rm B}T_{\rm N}}$ is attributed to the peak in the superconductor density of states near the gap edge. ### Above the gap Above the gap $E=eV\pm\hbar\omega_{\rm r}\gtrsim\Delta$, the dominant source of resonator transitions are the direct photon-assisted tunneling events in the forward direction from an occupied state in the normal metal to an empty state in the superconductor. In this range, backward, subgap and thermal tunneling events are negligible. Using the Sommerfeld expansion in Eq. (33) results in \[\overrightarrow{F}\left(E\right)\approx \frac{\int_{\Delta}^{E}\text{d}\varepsilon\,n_{\rm S}(\varepsilon )+\frac{(\pi k_{\rm B}T_{\rm N})^{2}}{6}\frac{\text{d}n_{\rm S}(\varepsilon)}{ \text{d}\varepsilon}\big{\|}_{\varepsilon=E}}{h\left(1-\mathrm{e}^{-E/k_{\rm B} T_{\rm N}}\right)}\] \[= \frac{\sqrt{E^{2}-\Delta^{2}}+\frac{(\pi k_{\rm B}T_{\rm N})^{2}} {6}\frac{\Delta^{2}}{(E^{2}-\Delta^{2})^{3/2}}}{h\left(1-\mathrm{e}^{-E/k_{\rm B }T_{\rm N}}\right)}\] (52) where $\Delta\gg k_{\rm B}T_{\rm N}$ and the Dynes parameter is ignored. At the asymptotic limit of large bias $eV\gg\Delta,\hbar\omega_{\rm r}$, the resulting coupling strength and the effective temperature of the reservoir become \[\gamma^{\rm ag}_{\rm T}(V)\approx \bar{\gamma}_{\rm T}\left[1+\frac{\pi^{2}}{2}\frac{\Delta^{2}k^{2 }_{\textrm{B}}T^{2}_{\rm N}}{(eV)^{4}}\right],\] (53a) \[T^{\rm ag}_{\rm T}(V)\approx \frac{eV}{2k_{\rm B}}\left[1-\frac{2\pi^{2}}{3}\frac{\Delta^{2}k^ {2}_{\textrm{B}}T^{2}_{\rm N}}{(eV)^{4}}\right].\] (53b) Asymptotically, up to the corrections from the thermal broadening, the coupling strength equals that of a fully normal-metal junction, $\bar{\gamma}_{\rm T}$, and the reservoir temperature increases linearly as $eV/(2k_{\rm B})$. ## VII Quantum-circuit refrigerator Here, we discuss how the SINIS junction can be used as a voltage-tunable refrigerator of the resonator mode. In an ideal situation, the inverse of the coupling strength $1/\gamma_{\rm T}$ is the time scale in which the refrigerated quantum circuit exponentially reaches the temperature of the refrigerator $T_{\rm T}$. Thus, to optimize the operation of the quantum-circuit refrigerator, one aims at maximizing the coupling strength $\gamma_{\rm T}$ and reaching the minimal temperature $T_{\rm T}=T_{\rm N}/2$ when the refrigerator is active. To this end, it is instructive to study the coupling strength in the thermal activation regime, _i.e._, at junction bias near and somewhat below the superconductor gap, where the thermal reservoir formed by the photon-assisted tunneling reaches its lowest effective temperature and is yet strongly coupled to the resonator mode. The value of the coupling strength at the minimal temperature is of the order of $\bar{\gamma}_{\rm T}=2\alpha^{2}Z_{\rm r}\omega_{\rm r}/R_{\rm T}$ obtained from Eq. (43) when biased near the superconductor gap, see Fig. 5. This value can be increased by reducing $R_{\rm T}$, increasing the capacitance fraction $\alpha=C_{\rm c}/C_{\rm N}$, the characteristic impedance $Z_{\rm r}$, or the frequency $\omega_{\rm r}$ of the resonator mode. Fortunately, the other optimization goal, $T_{\rm T}\approx T_{\rm N}/2$, is typically achieved in a range of bias voltages where the coupling strength is relatively close to its maximum value, see Fig. 5. Note, that a reduction of the electron temperature $T_{\rm N}$ has two positive effects for the quantum-circuit refrigerator. Naturally, it lowers the minimal temperature, but it also increases $\gamma_{\rm T}$ in the thermal activation regime. The increase occurs through the dependence on $\sinh(\hbar\omega/k_{\rm B}T_{\rm N})\sqrt{k_{\rm B}T_{\rm N}}/\hbar\omega_{ \rm r}$ in Eq. (51). When the electron temperature is decreased, the minimum reservoir temperature is achieved with smaller bias. When this shift is taken into account, the coupling strength $\gamma_{\rm T}$ at the optimal biasing point exhibits an exponential increase at lowered electron temperatures. Lower electron temperature leads also to a wider thermal activation regime being beneficial for reaching the minimal $T_{\rm T}=T_{\rm N}/2$. Furthermore, reduction of the Dynes parameter $\gamma_{\rm D}$ suppresses subgap transitions which also widens the thermal activation regime. In addition to optimization of the cooling rate, the device needs to be well decoupled when not in use. We observe from Fig. 5(a) that at $V=0$ the coupling strength reduces to the value $\gamma_{\rm T}^{\rm min}=\bar{\gamma}_{\rm T}\gamma_{\rm D}$. Typically $\gamma_{\rm T}^{\rm min}/2\pi\sim 1$ kHz which is much lower than the typical coupling strengths used to couple coplanar waveguide resonators to transmission lines. Thus the refrigerator becomes essentially decoupled from the resonator. The ratio between the maximum and minimum coupling strength scales as $\gamma^{\rm max}_{\rm T}/\gamma_{\rm T}^{\rm min}\sim\gamma_{\rm D}^{-1}$ and thus it is maximized by reducing the Dynes parameter $\gamma_{\rm D}$. ## VIII Cryogenic microwave source If the junctions are biased above the superconductor gap, the suppression of the emissive transitions due to the gap is lifted. Thus the thermal reservoir describing the photon-assisted tunneling events is hot. In this regime, the device can work as a microwave source emitting incoherent radiation through the resonator at frequency $\omega_{\rm r}$ and a spectral peak width of $\gamma_{\rm tr}+\gamma_{\rm T}$. Masuda _et al._ (2016) Let us quantitatively investigate the net power emitted from the capacitively coupled SINIS junction to the transmission line through the resonator. The resonator is coupled to a transmission line through a coupling capacitor $C_{\rm g}$. The transmission line has a characteristic impedance $Z_{\rm tr}\approx 50\ \Omega$ and mean thermal occupation $N_{\rm tr}$ of photons at the frequency $\omega_{\rm r}$ corresponding to a temperature $T_{\rm tr}$. The coupling strength $\gamma_{\rm tr}$ of the resonator to the transmission line given by Eq. (8) was derived in Sec. III.3. With these results, we consider the energy exchange at the two interfaces of the resonator. The net power flowing from the SINIS junction to the resonator is $P_{\rm T}=\hbar\omega_{\rm r}\gamma_{\rm T}(N_{\rm T}-N_{\rm r})$ and the corresponding power from the resonator to the transmission line equals $P_{\rm tr}=\hbar\omega_{\rm r}\gamma_{\rm tr}(N_{\rm r}-N_{\rm tr})$. In thermal equilibrium and in the absence of other dissipation channels for the resonator, the powers balance each other, $P_{\rm tr}=P_{\rm T}$, the fact exploited in the experiments for probing quantum-circuit refrigeration and heating Tan _et al._ (2017); Masuda _et al._ (2016). Expressing in the power $P_{\rm T}=\hbar\omega_{\rm r}\gamma_{\rm T}(N_{\rm T}-N_{\rm r})$ the resonator mean occupation $N_{\rm r}$ by means of Eq. (46), we obtain \[P_{\rm T}=\hbar\omega_{\rm r}\frac{\gamma_{\rm T}\gamma_{\rm tr}}{\gamma_{\rm T }+\gamma_{\rm tr}}(N_{\rm T}-N_{\rm tr}).\] (54) If the net output power $P_{\rm T}=P_{\rm tr}$ is positive, the resonator is heated by the voltage biased SINIS junction above the constant reference value $N_{\rm tr}$, and vice versa for negative power. Above the gap, the temperature of the thermal reservoir increases linearly with $V$ and the coupling strength saturates to $\bar{\gamma}_{\rm T}$ [Eq. (53)], implying roughly linearly increasing output power as a function of the bias voltage beyond the gap. However, increased bias voltage also heats up the electrons in the normal metal, Giazotto _et al._ (2006) which leads to an additional weak reduction of the emission power. To test the theory developed in this paper, we compute the net output power $P_{\rm T}$ of Eq. (54) based on the thermal reservoir formulation of $\gamma_{\rm T}$ and $N_{\rm T}$ in Eq. (42) using the parameters of the recent experimental work of Ref. Masuda _et al._, 2016. In Fig. 6, we achieve an excellent quantitative agreement between our theory and the experiments without fitting parameters, which provides a rigorous verification of the developed theory. ### Comparison with the $\mathbf{P(E)}$-theory <figure><img src="content_image/1706.07188/x6.png"><figcaption>Figure 6: Net output power PT as a function of the single-junction biasvoltage. Comparison of Eq. (54) providing the results of our transition ratetheory (solid line) with the measured values (filled circles) and results ofthe P(E)-theory (dashed line) from Ref. Masuda _et al._, 2016. The parametersin the models correspond to the experiments: Δ=220 μeV, γD=4×10−4, RT=12.5 kΩ,Cc=840 fF, CΣm=12 fF, ωr/2π=4.55 GHz, Zr=33.9 Ω, Cg=72.0 fF, Ztr=53.0 Ω, andTtr=180 mK. For the electron temperature, we use the bias-voltage-dependentvalues TN(V) experimentally measured in Ref. Masuda _et al._, 2016 using anadditional SINIS thermometer.</figcaption></figure> Previously, the $P(E)$-theory Ingold and Nazarov (1992) has been used to obtain the tunneling-induced transition rates in a very good agreement with the experimental data both in the cooling Tan _et al._ (2017) and heating Masuda _et al._ (2016) regimes of the biased SINIS junction. In the $P(E)$-theory one considers a junction embedded in an electromagnetic environment formed by the surrounding electric circuitry. In addition to the occupations and densities of states at either side of a junction, the probability of a tunneling event depends on the ability of the environment to absorb the excess energy or to supply the remaining energy of a tunneling quasiparticle. This ability is described by the so-called $P(E)$-function, which is typically calculated for the surrounding electric circuitry in a thermal equilibrium (see Ref. Souquet _et al._, 2014 for a nonequilibrium generalization). The transition rate theory developed here agrees with the results of the $P(E)$-theory in the regime considered in Fig. 6. In Ref. Tan _et al._, 2017, the $P(E)$-theory was applied to derive the transition rates between the two lowest levels of a resonator in the case of a galvanic contact ($\alpha=1$) between the resonator and the normal-metal island. These results also agree with our results derived in Sec. IV. However, we note that the transition rate theory is considerably more general than $P(E)$-theory. It is able to capture conveniently the fine details of the physical circuit such as non-linearities of the electromagnetic environment of the tunnel junction. Furthermore, it is not restricted to surrounding electric circuits in thermal equilibrium but is directly applicable to non-equilibrium circuits including, _e.g._, driven superconducting qubits. ## IX Outlook and conclusions We have developed a first-principles theory of quantum-circuit refrigeration by photon-assisted tunneling for a superconducting resonator. For weak interaction parameter $\rho$ characterizing the coupling strength of the tunneling transitions to the resonator, it leads to an intuitive thermal reservoir model of tunneling-induced transition rates of the resonator. This model allows us to condense the essential physics into the familiar concepts of the reservoir temperature and coupling strength to the resonator. In addition, we derived accurate analytical approximations for the temperature and the coupling strength in the experimentally relevant parameter regions. In the future, the thermal reservoir model can be straightforwardly extended to an input-output formulation Gardiner and Collett (1985); Clerk _et al._ (2010) describing, _e.g._, reflection of coherent microwaves from a refrigerated resonator. Although we focus here on the effect of the quantum-circuit refrigerator on a linear resonator, the method of deriving the resulting transition rates is independent of the quantum circuit in question. In fact, the rates are products of an integral over Fermi functions and the matrix element of the charge shift operator between the eigenstates of the quantum circuit. A different quantum circuit simply yields different matrix elements. Thus our results can likely be generalized to the full spectrum of superconducting quantum devices. For example, the superconducting transmon qubit Koch _et al._ (2007); Paik _et al._ (2011); Rigetti _et al._ (2012); Barends _et al._ (2013) seems an ideal device to be cooled by the quantum-circuit refrigerator due to its efficient transverse coupling to charge and simultaneous exponentially suppressed sensitivity of the eigenenergies to charge offsets. ###### Acknowledgements. We acknowledge useful discussions with J. Govenius, J. Goetz, J. Tuorila, and M. Partanen. This research was financially supported by European Research Council under Grant No. 681311 (QUESS), by Academy of Finland under Grant Nos. 251748, 265675, 276528, 284621, 305237, 305306, 308161, and 312300, and by Alfred Kordelin Foundation. ## Appendix A Island charge dynamics The master equation (36) shows that the charge state $\ket{q}$ of the normal-metal island is driven by both, elastic and inelastic transitions in the voltage-biased SINIS junction. Typically, the elastic transitions have a much higher rate than the inelastic ones (see Fig. 3), originating from the fact that the matrix elements for elastic events scale as $M^{2}_{mm}\propto 1-\rho$ whereas the scaling for inelastic transitions is $M^{2}_{mm^{\prime}}\propto\rho^{\|\ell\|}$ with $\ell=m-m^{\prime}\geq 1$. In this case, the master equation (36) rapidly leads to a steady-state distribution for the island charge by means of frequent elastic tunneling transitions. The resonator states equilibrate through slower inelastic processes or through transitions induced by coupling to the transmission line which is independent of the charge state. Thus we first compute the steady-state charge distribution $p_{q}$ determined by the elastic tunneling rates \[\Gamma^{\pm}_{q,m,m}(V)=M^{2}_{mm}\frac{R_{\rm K}}{R_{\rm T}}\sum_{\tau=\pm 1} \overrightarrow{F}\left(\tau eV-E^{\pm}_{q}\right),\] (55) and then compute the inelastic rates averaged over this distribution as in Eq. (38). <figure><img src="content_image/1706.07188/x7.png"><figcaption>Figure 7: Effective temperature TQ of the charge distribution of the normal-metal island as a function of the single-junction bias voltage. The parametersequal to those of Fig. 3. The solid black line shows the numerical resultbased on Eqs. (57) and (58). The filled circles show the corresponding resultbased on Eqs. (59)-(61) valid at eV≫EN,kBTN. In the deep-subgap regime, thetemperature of the charge distribution TQ is close to the electron temperatureTN=100 mK depicted by a green dash-dotted line.</figcaption></figure> Note that in Eq. (55), a change in the resonator state results in a simple scaling of the rates which affects only the speed of equilibration, not the actual steady-state distribution $p_{q}$. Thus for any resonator state indexed by $m$, the distribution can be accurately computed using the master equation \[\dot{p}_{q}(t)=\Big{[}\Gamma^{+}_{q\!-\!1,m,m}\,p_{q\!-\!1}(t)+ \Gamma^{-}_{q\!+\!1,m,m}\,p_{q\!+\!1}(t)\] \[-\left(\Gamma^{+}_{q,m,m}+\Gamma^{-}_{q,m,m}\right) p_{q}(t)\Big{]}.\] (56) This results in a steady-state distribution \[p_{q}=\frac{1}{Z}\exp\left[\sum_{q^{\prime}=0}^{q-1}\ln\left(\frac{\Gamma^{+}_ {q^{\prime},m,m}}{\Gamma^{-}_{q^{\prime}\!+\!1,m,m}}\right)\right],\] (57) where $q>0$ and $Z$ is a normalization factor. For $q<0$, we employ the symmetry $p_{-q}=p_{q}$. From Eq. (55) and the identity $E^{-}_{q+1}=-E^{+}_{q}$ we obtain \[\frac{\Gamma^{+}_{q,m,m}}{\Gamma^{-}_{q\!+\!1,m,m}}=\frac{\sum_{\tau=\pm 1} \overrightarrow{F}\left(\tau eV-E^{+}_{q}\right)}{\sum_{\tau=\pm 1} \overrightarrow{F}\left(\tau eV+E^{+}_{q}\right)}.\] (58) In Fig. 7, we show the effective temperature of the charge distribution obtained from Eqs. (57) and (58) as a function of the single-junction bias voltage $V$ using experimentally relevant parameter values. The effective temperature $T_{\rm Q}$ is defined by a least-square fit of the thermal distribution $\widetilde{p}_{\rm q}=\exp[-E_{\rm N}q^{2}/(k_{\rm B}\widetilde{T}_{\rm Q})]/Z$ to the distribution $p_{q}$ of Eqs. (57) and (58). Equation (58) shows that the steady state is determined by a ratio of tunneling processes for energies differing by $2E^{+}_{\rm q}=2E_{\rm N}(1+2q)$. In the deep subgap regime, where $eV\ll k_{\rm B}T_{\rm N}$ and $eV\lesssim E_{\rm N}$, we may use the results of Sec. VI.1 and arrive at $T_{\rm Q}^{\rm dg}=T_{\rm N}$. At higher voltages, we utilize the fact that $E_{\rm N}\ll\Delta,k_{\rm B}T_{\rm N},eV$ and expand Eq. (58) to the first order in $E^{+}_{q}$. Consequently, Eq. (57) results in a charge distribution of the thermal form \[p_{q}=\frac{1}{Z}\exp\left[-\frac{E_{\rm N}q^{2}}{k_{\rm B}T_{\rm Q}(V)}\right],\] (59) where $E_{\rm N}q^{2}$ corresponds to the energy of the charge state, $Z=\sum_{q}\exp[-E_{\rm N}q^{2}/(k_{\rm B}T_{\rm Q})]$ to the partition function, and $T_{\rm Q}(V)$ to the temperature of the distribution. The effective temperature assumes the form \[T_{\rm Q}(V)=\frac{1}{2k_{\rm B}}\left[\frac{g^{\prime}(V)}{g(V)}-\frac{1}{k_{ \rm B}T_{\rm N}\sinh\left(\frac{eV}{k_{\rm B}T_{\rm N}}\right)}\right]^{-1},\] (60) where the function $g(V)$ characterizes the difference of the Fermi functions weighted with the density of states of the superconductor as \[g(V) =\int_{-\infty}^{\infty}\text{d}\varepsilon\:n_{\rm S}( \varepsilon)\left[f(\varepsilon-eV)-f(\varepsilon)\right],\] (61a) \[g^{\prime}(V) =\frac{1}{4k_{\rm B}T_{\rm N}}\int_{-\infty}^{\infty}\text{d} \varepsilon\ \textrm{sech}^{2}\left(\frac{\varepsilon-eV}{2k_{\rm B}T_{\rm N}} \right)n_{\rm S}(\varepsilon).\] (61b) Figure 7 indicates that the first-order result of Eqs. (59) and (60) gives in general a very accurate approximation of the full numerical result of Eqs. (57) and (58). ## References * Nakamura _et al._ (1999)Y. Nakamura, Y. A. Pashkin, and J. S. 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1303.2919	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 16936, "num_imgs": 6, "llama3_tokens_count": 5247 }	[ "content_image/1303.2919/x1.png", "content_image/1303.2919/x3.png", "content_image/1303.2919/x4.png", "content_image/1303.2919/x5.png", "content_image/1303.2919/x8.png", "content_image/1303.2919/x9.png" ]	# A Monte Carlo study to measure the energy spectra of the primary cosmic-ray components at the knee using a new Tibet AS core detector array ###### Abstract A new hybrid experiment has been started by AS$\gamma$ experiment at Tibet, China, since August 2011, which consists of a low threshold burst-detector-grid (YAC-II, Yangbajing Air shower Core array), the Tibet air-shower array (Tibet-III) and a large underground water Cherenkov muon detector (MD). In this paper, the capability of the measurement of the chemical components (proton, helium and iron) with use of the (Tibet-III+YAC-II) is investigated by means of an extensive Monte Carlo simulation in which the secondary particles are propagated through the (Tibet-III+YAC-II) array and an artificial neural network (ANN) method is applied for the primary mass separation. Our simulation shows that the new installation is powerful to study the chemical compositions, in particular, to obtain the primary energy spectrum of the major component at the knee. L.M. ZHAI _et al._ energy spectra of primary proton, helium and iron M. Amenomori${}^{1}$, X. J. Bi${}^{2}$, D. Chen${}^{3}$, W. Y. Chen${}^{2}$, S. W. Cui${}^{4}$, Danzengluobu${}^{5}$, L. K. Ding${}^{2}$, X. H. Ding${}^{5}$, C. F. Feng${}^{6}$, Zhaoyang Feng${}^{2}$, Z. Y. Feng${}^{7}$, Q. B. Gou${}^{2}$, H. W. Guo${}^{5}$, Y. Q. Guo${}^{2}$, H. H. He${}^{2}$, Z. T. He${}^{4,2}$, K. Hibino${}^{8}$, N. Hotta${}^{9}$, Haibing Hu${}^{5}$, H. B. Hu${}^{2}$, J. Huang${}^{2}$, W. J. Li${}^{2,7}$, H. Y. Jia${}^{7}$, L. Jiang${}^{2}$, F. Kajino${}^{10}$, K. Kasahara${}^{11}$, Y. Katayose${}^{12}$, C. Kato${}^{13}$, K. Kawata${}^{3}$, Labaciren${}^{5}$, G. M. Le${}^{2}$, A. F. Li${}^{14,6,2}$, C. Liu${}^{2}$, J. S. Liu${}^{2}$, H. Lu${}^{2}$, X. R. Meng${}^{5}$, K. Mizutani${}^{11,15}$, K. Munakata${}^{13}$, H. Nanjo${}^{1}$, M. Nishizawa${}^{16}$, M. Ohnishi${}^{3}$, I. Ohta${}^{17}$, S. Ozawa${}^{11}$, X. L. Qian${}^{6,2}$, X. B. Qu${}^{2}$, T. Saito${}^{18}$, T. Y. Saito${}^{19}$, M. Sakata${}^{10}$, T. K. Sako${}^{12}$, J. Shao${}^{2,6}$, M. Shibata${}^{12}$, A. Shiomi${}^{20}$, T. Shirai${}^{8}$, H. Sugimoto${}^{21}$, M. Takita${}^{3}$, Y. H. Tan${}^{2}$, N. Tateyama${}^{8}$, S. Torii${}^{11}$, H. Tsuchiya${}^{22}$, S. Udo${}^{8}$, H. Wang${}^{2}$, H. R. Wu${}^{2}$, L. Xue${}^{6}$, Y. Yamamoto${}^{10}$, Z. Yang${}^{2}$, S. Yasue${}^{23}$, A. F. Yuan${}^{5}$, T. Yuda${}^{3}$, L. M. Zhai${}^{2}$, H. M. Zhang${}^{2}$, J. L. Zhang${}^{2}$, X. Y. Zhang${}^{6}$, Y. Zhang${}^{2}$, Yi Zhang${}^{2}$, Ying Zhang${}^{2}$, Zhaxisangzhu${}^{5}$, X. X. Zhou${}^{7}$ (The Tibet AS$\gamma$ Collaboration) ${}^{1}$Department of Physics, Hirosaki University, Hirosaki 036-8561, Japan ${}^{2}$Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China ${}^{3}$Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan ${}^{4}$Department of Physics, Hebei Normal University, Shijiazhuang 050016, China ${}^{5}$Department of Mathematics and Physics, Tibet University, Lhasa 850000, China ${}^{6}$Department of Physics, Shandong University, Jinan 250100, China ${}^{7}$Institute of Modern Physics, SouthWest Jiaotong University, Chengdu 610031, China ${}^{8}$Faculty of Engineering, Kanagawa University, Yokohama 221-8686, Japan ${}^{9}$Faculty of Education, Utsunomiya University, Utsunomiya 321-8505, Japan ${}^{10}$Department of Physics, Konan University, Kobe 658-8501, Japan ${}^{11}$Research Institute for Science and Engineering, Waseda University, Tokyo 169-8555, Japan ${}^{12}$Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan ${}^{13}$Department of Physics, Shinshu University, Matsumoto 390-8621, Japan ${}^{14}$School of Information Science and Engineering, Shandong Agriculture University, Taian 271018, China ${}^{15}$Saitama University, Saitama 338-8570, Japan ${}^{16}$National Institute of Informatics, Tokyo 101-8430, Japan ${}^{17}$Sakushin Gakuin University, Utsunomiya 321-3295, Japan ${}^{18}$Tokyo Metropolitan College of Industrial Technology, Tokyo 116-8523, Japan ${}^{19}$Max-Planck-Institut für Physik, München D-80805, Deutschland ${}^{20}$College of Industrial Technology, Nihon University, Narashino 275-8576, Japan ${}^{21}$Shonan Institute of Technology, Fujisawa 251-8511, Japan ${}^{22}$RIKEN, Wako 351-0198, Japan ${}^{23}$School of General Education, Shinshu University, Matsumoto 390-8621, Japan Air shower, Spectrum, Neural network, Knee region ## 1 Introduction $\;\;\;$The all-particle energy spectrum of primary cosmic rays is well discriminated by a power law dN/dE $\propto$ E${}^{-\gamma}$ over many orders of magnitude, with $\gamma$ changes sharply from 2.7 to 3.1 at about 4 PeV [1]. The break of the all-particle energy spectrum is called the ”knee”, and the corresponding energy range is called ”knee region”. Although the existence of the knee is well confirmed by experiments, there still is controversy on its origin. In order to explain the sharpness of the knee, two scenarios [2] (model A and model B) are proposed as shown in Fig.1 and Fig.2. $\;\;\;\;$In model A, an excess component is assumed to overlap the global component, and its spectrum shape suggests that it can be attributed to nearby source(s) because it is surprisingly close to the expected source spectrum of the diffuse shock acceleration. Middle composition is predicted by this model at the knee. In model B, a hard observed energy spectrum of each element from a given source is assumed. The sharp knee can be explained by a rigidity-dependent acceleration limit and the hard spectrum due to nonlinear effects. Iron-dominant composition is predicted by this model at the knee and beyond. In order to distinguish between Model A and Model B and many other models, measurements of the chemical composition around the knee, especially measurements of the primary spectra of individual component till their knee will be essentially important. Therefore, we have developed the Yangbajing Air shower Core detector (YAC) which is operated along with Tibet air-shower array (Tibet-III) and underground water Cherenkov muon detector array (MD) simultaneously, as shown in Fig.3, and the second phase of YAC is so called YAC-II. In this paper, the capability of the measurement of the chemical components with use of the (Tibet-III+YAC-II) is investigated. The simulation results using (Tibet-III+YAC-II+MD) will be reported in the near further. <figure><img src="content_image/1303.2919/x1.png"><figcaption>Figure 1: Model A: Sharp knee is attributed to extra component from nearbysource.</figcaption></figure> ## 2 A New Tibet Hybrid Experiment $\;\;\;$ The Tibet new hybrid experiment (Tibet-III+YAC-II) has been operated in Tibet, China, since August, 1st, 2011. The merit of this experiment is that the atmosphere depth of the experimental site (4300 m above sea level; 606 g/cm${}^{2}$) is close to the maximum of the air shower development with energies around the knee and the shower maximum values are almost independent of the masses of primary cosmic rays. The Tibet-III consists of 789 detectors while the YAC-II consists of 124 detectors (as shown in Fig.3). The inner 100 plastic scintillator units of YAC-II are arranged as an array (10$\times$10 grid) each with an area of 50 cm $\times$ 80 cm, with 1.875 m interval; and the outer 24 plastic scintillator units are arranged around the inner array each with an area of 50 cm $\times$ 100 cm. The outer 24 units are used to reject non core events whose shower cores are far from the YAC-II array. Each detector of YAC-II consists of lead plates with a thickness of 3.5 cm above the scintillator to convert high energy electrons and gammas to electromagnetic showers. Each unit of YAC-II is attached with two photomultipliers of high-gain (HAMAMATSU: R4125) and low-gain (HAMAMATSU: R5325) to cover the wide dynamic range from 1 MIP (Minimum Ionization Particle) to 10${}^{6}$ MIPs. The hardware of YAC-II is described in [3]. <figure><img src="content_image/1303.2919/x3.png"><figcaption>Figure 3: The Tibet-III+YAC-II+MD array. The Tibet-III consists of 789detector units, the YAC-II consists of 124 detector units. The five MDs in theblue frame are set up this year and acquiring data soon.</figcaption></figure> $\;\;\;$The area of YAC-II array is about 500 m${}^{2}$, locating near the center of the Tibet-III, and operating simultaneously with it. For an air shower event, the Tibet-III provides the arrival direction ($\theta$), the air shower age ($Age$) and the air shower size (${N_{e}}$) which is interrelated to primary energy, the YAC-II measures the high energy electromagnetic particles in the core region so as to obtain the characteristic parameters of air-shower cores. When a YAC event is triggered, its accompanying air shower is simultaneously recorded. The matching between an AS and a YAC event is made by their arrival time stamps. The air-shower direction can be estimated with an error smaller than 0.1${}^{0}$ above 100 TeV, and the primary energy resolution is estimated to be 12% at energies around 10${}^{15}$ eV by our simulation. ## 3 Simulation and Analysis $\;\;\;\;$A Monte Carlo (MC) simulation has been carried out on the development of extensive air showers in the atmosphere and the response in YAC-II and Tibet-III. The simulation code CORSIKA (version 6.024) including QGSJET2 hadronic interaction models [4] is used to generate AS events. Considering the aim of this simulation is just to check the capability of the hybrid experiment, QGSJET2 interaction model and heavy dominated (HD) [1] primary composition model are used. Primaries isotropically incident at the top of the atmosphere within the zenith angles from 0 to 60 degrees are injected into the atmosphere. The simulated data were analyzed in the same manner as in the procedure for the experimental data analysis. The electromagnetic showers in the lead layer induced by electrons or photons that hit any detector unit of the array are simulated by a subroutine which is based on the detector simulation code EPICS [5]. The detector performance, the trigger efficiency of detectors and the effective area are adequately taken into account based on the experimental conditions. Normally, the following quantities of YAC-II are used to characterize an air-shower core event: ${N_{b}}$ - the number of shower particles under the lead plate of a detector unit; ${N_{hit}}$ - the number of ”fired” detector units with ${N_{b}}$$\geq$ a given threshold value; ${N_{b}}$${}^{top}$ - the maximum burst size among fired detectors; $\sum$${N_{b}}$ - the total burst size of all fired detector units; $<R>$ - the mean lateral spread, $<R>$=$\sum$${r_{i}}$/($N_{hit}$-1); $<$${N_{b}}$$R$$>$ - the mean energy-flow spread, $<$${N_{b}}$$R$$>$=$\sum$(${N_{bi}}$$\times$${r_{i}}$)/$N_{hit}$, where ${N_{bi}}$ and ${r_{i}}$ are the burst size in the $i^{th}$ fired detector unit and the lateral distance from the air shower core to the center of the $i^{th}$ fired detector, respectively. $\;\;$ We divided the MC data into two datasets. Due to the difference of the detection efficiency, the first dataset selects events that are enriched with proton+helium origin (called tagged-I dataset), while the second dataset contains events of heavy-primary origin (called tagged-II dataset). The final select conditions for tagged-I and tagged-II are as follows: $\;\;$(1) ${N_{b}}$$\geq$ 100, $N_{hit}$$\geq$ 12, ${N_{b}}$${}^{top}$$\geq$ 3000, ${N_{e}}$$\geq$ 1$\times$10${}^{5}$; $\;\;$(2) ${N_{b}}$$\geq$ 100, $N_{hit}$$\geq$ 20, ${N_{b}}$${}^{top}$$\geq$ 1500, ${N_{e}}$$\geq$ 5$\times$10${}^{5}$. $\;\;\;$ Besides, the detector unit with ${N_{b}}$${}^{top}$ is requested to be located at inner 8$\times$8 grid for both data-sets. Then, we obtain 1.43$\times$10${}^{5}$ and 5.17$\times$10${}^{4}$ events for the tagged-I and tagged-II dataset, respectively. The detection efficiency S$\Omega$$A_{eff}$ is shown in Fig.4 . <figure><img src="content_image/1303.2919/x4.png"><figcaption>Figure 4: The detection efficiency SΩAeff of proton, proton+helium and iron.</figcaption></figure> $\;\;$ The separation of the primary mass is realized by use of a feed-forward artificial neural network (ANN[6]) method, whose applicability to our experiment was well confirmed by the MC simulation [7]. For one thing, we need to separate protons from other nuclei by training the network with a proton flag, and then separate proton+helium from other nuclei by training the network with a proton+helium flag, thus we can get the helium energy spectrum by subtracting the number of proton events from the proton+helium events, so does iron. The following 8 parameters are input to the ANN with 40 hidden nodes and 1 output unit: (1)${N_{hit}}$, (2)${N_{b}}$${}^{top}$, (3)$\sum$${N_{b}}$, (4)$<R>$, (5)$<$${N_{b}}$$R$$>$, (6)${N_{e}}$, (7)$\theta$, (8)$Age$, where the first five parameters are given by YAC-II, and the last three are obtained by Tibet-III. ## 4 Results and Discussion $\;\;\;$The ANN training results of proton, proton+helium and iron are presented in Fig.5, where average purity and selection efficiency over whole energy range are shown. The ANN output value $T$ is used to separate the primary nuclei groups. In this paper, the events with $T$$\leq$${T_{c}}$ (or $T$$\geq$${T_{c}}$ ) are regarded as proton or proton+helium group (or iron group). The value ${T_{c}}$, purity (p) and selection rate ($\varepsilon$) of events satisfying the criterion at various energy regions are summarized in Table 1. Primary Energy \| 1014−1015 eV \| 1015−1016 eV \| 1016−1017 eV ---\|---\|---\|--- \| Tc(Mod) \| p \| ε \| p \| ε \| p \| ε P \| 0.2 \| 87.6 ± 1.1 \| 45.5 ± 0.5 \| 77.3 ± 5.9 \| 19.5 ± 1.2 \| \| P+He \| 0.1 \| 96.7 ± 0.8 \| 80.8 ± 0.6 \| 87.3 ± 3.5 \| 39.1 ± 1.3 \| \| Fe \| 0.7 \| \| \| 81.6 ± 1.6 \| 55.3 ± 1.0 \| 81.6 ± 8.5 \| 69.4 ± 7.0 \| \| \| \| \| \| \| Table 1: The purity (p)(%) and the selection rate (ε)(%) of the selected primary groups. <figure><img src="content_image/1303.2919/x5.png"><figcaption>Figure 5: The ANN training results of proton, proton+helium and iron. Theaverage purity and selection rate over whole energy range of protons are 88%,46% at Tc = 0.2, while 96%, 28% for proton+helium at Tc = 0.1, 80%, 54% foriron at Tc = 0.7.</figcaption></figure> $\;\;$ The Fig.6 shows the estimated primary energy spectra of proton, helium and iron compared with the assumed ones. One can see that the assumed primary energy spectrum of proton, helium and iron are well reproduced by the estimated ones respectively, and the estimated ones could well connect with the results obtained by direct observation, as shown in Fig.7. It needs to be remarked that the iron spectra seem to be higher than the observed ones, just because of the HD model we used. The results above show that the new burst detector array is powerful to study the chemical composition, in particular, to obtain the primary energy spectrum of the major component at the knee. <figure><img src="content_image/1303.2919/x8.png"><figcaption>Figure 6: Energy spectra of primary proton, helium and iron.</figcaption></figure> <figure><img src="content_image/1303.2919/x9.png"><figcaption>Figure 7: The expected spectra of proton, helium and iron obtained by MCcompared with other experimental data.</figcaption></figure> ## 5 Acknowledgements $\;\;$This work is supported by the Chinese Academy of Sciences (H9291450S3) and the Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, CAS. The Knowledge Innovation Fund (H95451D0U2 and H8515530U1) of IHEP, China and the project Y0293900TF of NSFC also provide support to this study. ## References * [1] M. Amenomori et al., ApJ., 678:1165-1179, 2008. * [2] M. Shibata, Y. Katayose, J. Huang and D. Chen, ApJ., 716, 1076 C1083, 2010. * [3] M. Amenomori et al., ICRC 32 (HE 1.4 ID: 1241), 2011. * [4] D. Heck et al., Report FZKA 6019, 1998; J. Knapp, D. Heck et al., Report FZKA 3640, 1997; D. Heck et al., Report FZKA 5828, Forshungszentru Karldruhe, 1996. Available from http://www-ik3.fzk.deheck/corsika /physics${}_{-}$description/corsika${}_{-}$phys.html. * [5] K. Kasahara et al., http://eweb.b6.kanagawa-u.ac.jp/ Kasahara/ResearchHome/EPICSHome/Index.html. * [6] L. Lonnblad et al., Comp. Phys. Com. 81, 185 (1994). * [7] M. Amenomori et al., Physics Letters B, 632, 58-64, 2006.
1211.6813	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23963, "num_imgs": 2, "llama3_tokens_count": 7730 }	[ "content_image/1211.6813/x1.png", "content_image/1211.6813/x2.png" ]	# Opening the window to the cogenesis with Affleck-Dine mechanism in gravity mediation Ayuki Kamada Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8582, Japan Masahiro Kawasaki Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8582, Japan Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan Masaki Yamada Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan February 21, 2024 ###### Abstract The observed baryon and dark matter densities are equal up to a factor of 5. This observation indicates that the baryon asymmetry and dark matter have the same origin. The Affleck-Dine baryogenesis is one of the most promising mechanisms in this context. Q balls, which are often formed in the early Universe associated with the Affleck-Dine baryogenesis, decay both into supersymmetric particles and into quarks. Recently, it was pointed out that annihilation of squarks into quarks gives a dominant contribution to the Q-ball decay rate and the branching ratio of Q-ball decay into supersymmetric particles changes from the previous estimate. In this paper, the scenario of baryon and dark matter cogenesis from Q ball in gravity mediation is revisited in respect of the improved Q-ball decay rates. It is found that the successful cogenesis takes place when a wino with mass $0.4-1$ TeV is dark matter. pacs: 98.80.Cq, 95.35.+d, 12.60.Jv † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction The existence of the baryon asymmetry and the dark matter is a long standing challenge in cosmology and particle physics. In supersymmetric (SUSY) extensions of the Standard Model (SM), the lightest SUSY particle (LSP) is a good candidate for dark matter if the R-parity is conserved. Furthermore, the Affleck-Dine mechanism can provide the baryon asymmetry [1; 2]. In the gravity-mediated SUSY breaking model, the Affleck-Dine mechanism often predicts the formation of Q balls in the early universe [3; 4; 5; 6; 7]. The Q ball is a spherical condensate of scalar fields. It generally consists of squarks and sleptons, and eventually decays both into quarks and into SUSY particles before the Big Bang Nucleosynthesis (BBN), and the observed baryon asymmetry is released. Through the cascade decays, the SUSY particles produced by the Q-ball decay turn into LSPs, which can account for the dark matter in the Universe. In this case, the baryon asymmetry and dark matter have the same origin and the resultant ratio of baryon to dark matter can be $O(1)$ naturally [8; 4; 9; 10; 11; 12]. When we consider the case that the pair annihilation of the LSPs is ineffective and assume that the Affleck-Dine field $\phi$ takes a circular orbit in the complex $\phi$ plane, the resultant ratio of baryon to dark matter from the Q-ball decay is related only with the mass of the LSP and the branching ratio of the Q-ball decay into baryons and SUSY particles. In the previous works, the branching ratio of the Q-ball decay into SUSY particles is believed to be comparable with that into quarks [4; 9; 10; 11; 12; 13; 14]. In this case, the mass of dark matter should be $O(1)$GeV.¹ However, it was pointed out that the many body processes like the squark annihilation may be dominant and then the branching ratio may change drastically [15]. In this letter, we reexamine the branching ratio into SUSY particles in respect of the many body process. [FOOTNOTE:1][ENDFOOTNOTE] Since the effective mass of the squark inside the Q ball is smaller than that of the free squark, the Q ball cannot decay into squarks. We assume that the Q ball is kinematically allowed to decay into binos, winos (LSPs), and SM particles. When the Q-ball decay rate is saturated due to the Pauli exclusion principle [16], the branching ratio is determined only by the number of degrees of freedom in the final state. Finally, we show that the branching ratio into SUSY particles can be $O(0.01)$. By using this branching ratio, we provide a successful scenario of the baryon and dark matter cogenesis through the Q-ball decay, and show that the wino LSP with mass of $0.4-1$ TeV can naturally explains the observed baryon to dark matter ratio in the case that the pair annihilation of the LSPs is ineffective. This letter is organized as follows. In Sec. II, we briefly review the property of Q balls in gravity mediation. In Sec. III, first we compare the saturated decay and annihilations and then derive the branching ratios. In Sec. IV, we discuss the thermal history in our scenario. Sec. V is devoted to the conclusion. ## II Q ball properties in gravity mediation In SUSY extensions of the standard model, there are many flat directions in the scalar potential. The flat directions are lifted by the SUSY breaking effect, and we can take the following potential for the flat direction to see the property of the Q ball in gravity mediation: \[V=m^{2}_{\phi}\|\phi\|^{2}\left(1+K\log\frac{\|\phi\|^{2}}{M^{2}_{\rm P}}\right),\] (1) where $m_{\phi}$ is the mass of the flat direction and $M_{\rm P}$ is the reduced Planck mass($\simeq 2.4\times 10^{18}$ GeV). In gravity mediation, $m_{s}$ is the same order of the gravitino mass $m_{3/2}$. The second term in the parenthesis comes from the one-loop radiative corrections, and typically $\|K\|\sim$0.01-0.1. In many cases, the gluino loops have dominant contributions to the radiative corrections and lead to $K<0$, and then there exists a Q-ball solution [4; 17]. The energy of the Q ball $M_{Q}$, the radius $R$, the rotation speed of the field $\omega_{0}$, and the field amplitude at the center of the Q ball $\phi_{0}$ are given by \[M_{Q} \simeq m_{\phi}(\phi_{0})Q,\] (2) \[R \simeq \frac{1}{\|K\|^{1/2}m_{\phi}(\phi_{0})},\] (3) \[\omega_{0} \simeq m_{\phi}(\phi_{0}),\] (4) \[\phi_{0} \simeq (2\pi^{3/2})^{-1/2}\|K\|^{3/4}m_{\phi}(\phi_{0})Q^{1/2},\] (5) where $m_{\phi}(\phi_{0})$ is the mass defined at the energy scale $\phi_{0}$. The rotation speed $\omega_{0}$ has a further important meaning as $\omega_{0}=\text{d}M_{Q}/\text{d}Q$; i.e., the Q-ball energy per unit charge. As discussed in detail in Sec. IV, the decay temperature of Q balls should be sufficiently suppressed for the pair annihilation of LSPs to be ineffective. This indicates that the charge of Q balls should be $Q\gtrsim 10^{26}$ and thus the magnitude of the scalar field is $\phi_{0}\gtrsim 10^{13}m_{\phi}(\phi_{0})$. At this energy scale, the mass of the flat direction $m_{\phi}(\phi_{0})$ is lower than the mass of squarks at the electro-weak scale due to $K<0$, and the Q ball cannot decay into squarks. ## III Q-ball decay rates into bino-wino, and quarks The fermion production rates from the Q ball have upper bounds due to the Pauli exclusion principle [16]. The upper bound of the each massless fermion flux $\bm{j}$ from the Q-ball surface is calculated as \[\bm{n\cdot j} \lesssim 2\int\frac{\text{d}^{3}k}{(2\pi)^{3}}\theta\left(\omega_{0}/2-\| \bm{k}\|\right)\theta(\bm{k\cdot n})\bm{\hat{k}\cdot n},\] (6) \[= \frac{2}{8\pi^{2}}\int^{\omega_{0}/2}_{0}k^{2}\text{d}k=\frac{ \omega_{0}^{3}}{96\pi^{2}},\] (7) where $\bm{n}$ is the outward-pointing normal. We double the flux and take the upper limit of integration as $\omega_{0}/2$, because one of the decay products has energy less than $\omega_{0}/2$. We obtain the upper bound for the production rate from the Q ball by multiplying Eq. (7) by the area of the Q-ball surface $4\pi R^{2}$. The decay rate is saturated when $g\phi_{0}>\omega_{0}$ for the interaction $g\phi\xi\eta$ ($\xi$, $\eta$: massless fermions). The condition $g\phi_{0}>\omega_{0}$ is almost always satisfied due to the large $Q$ value (see Eq. (5)). In the case of the massive fermion $\chi$, the upper bound of the flux is lower than Eq. (7). We consider the process of squark $\to$ quark $+\chi$, and treat the quark as a massless particle. The fermion $\chi$ can obtain the energy in the range of $[m_{\chi},\omega_{0}]$, and the quark obtain the energy in the range of $[0,\omega_{0}-m_{\chi}]$. Taking this into account, we just change the integral of Eq. (7) as \[\frac{1}{8\pi^{2}}\int^{\omega_{0}-m_{\chi}}_{0}k^{2}\text{d}k,\] (8) for $\omega_{0}>m_{\chi}>\omega_{0}/2$, and as \[\frac{1}{8\pi^{2}}\left[\int^{\omega_{0}/2}_{0}k^{2}\text{d}k+\int^{\omega_{0} /2}_{m_{\chi}}k^{2}\text{d}k\right],\] (9) for $m_{\chi}<\omega_{0}/2$. Thus, the $\chi$ flux is given by \[\bm{n\cdot j}_{\chi} \simeq \frac{\omega_{0}^{3}}{96\pi^{2}}\times f(m_{\chi}/\omega_{0}),\] (10) \[f(x) \equiv \left\{\begin{array}[]{ll}4(1-x)^{3}\quad\text{ for }1/2<x<1,\\ 4[(1/2)^{3}+((1/2)^{3}-x^{3})]\quad\text{ for }x<1/2.\end{array}\right.\] (11) Q balls can also decay into quarks via heavy gluino/higgsino exchange $\phi\phi\to qq$. This reaction rate is also saturated by the Pauli exclusion principle. The detailed discussion is given in Ref. [7; 18]. The saturated flux is Eq. (7) with $\omega_{0}$ replaced by $2\omega_{0}$, which is the total energy available in this process. Thus, we obtain the each quark flux as \[\bm{n\cdot j}_{\text{quark}}\simeq\frac{(2\omega_{0})^{3}}{96\pi^ {2}}.\] (12) This is larger than Eq. (7) by a factor of 8. Notice that this flux is valid only for $M>\omega_{0}$, where $M$ is the gluino/higgsino mass, and we assume it in this letter. In Appendix, we show $N(\geq 3)$ body processes are not saturated and negligible. Now, let us compare the branching ratios of the Q-ball decay into binos, winos, and quarks. The bino or wino production rate is given by Eq. (10), while the quark production rate is given by Eq. (12). Here we should note that since the saturated production rate is determined by the Pauli exclusion principle, the total quark production rate is Eq. (12) times the number of degrees of freedom for quarks produced in the decay. We can count it once we specify the flat direction. Hereafter, we consider the flat direction $\bar{u}^{a}_{i}\bar{d}^{b}_{j}\bar{d}^{c}_{k}\epsilon_{abc}$ ($j\neq k$), where $a$, $b$, and $c$ are the color indices and $i$, $j$, and $k$ are the family indices. The Q ball can decay into all right handed quarks via gluino exchange and into all left handed quarks via higgsino exchange, because the flat direction contains all colors and, in general, all families. (Even if the flat direction does not contain all families, it can decay into all families through flavor mixings.) The $U(1)_{Y}$ charge conservation allows one up-type quarks for each two down-type quarks. The Q ball cannot directly decay into winos because the $\bar{u}\bar{d}\bar{d}$ flat direction has no tree-level interaction with winos. However, winos are produced via subsequent decays of binos if the LSP is wino. In this letter we consider winos as the LSPs. We conclude that the total decay rate of the Q ball and the branching ratios of the decay into quarks and bino are calculated as \[\sum_{\text{all}}\frac{\text{d}N}{\text{d}t} \simeq \left[8\times 36\times\frac{3}{4}+f\left(\frac{m_{\tilde{b}}}{ \omega_{0}}\right)\right]\frac{R^{2}\omega_{0}^{3}}{24\pi},\] (13) \[B_{\text{quarks}} \simeq \frac{8\times 36\times 3/4}{8\times 36\times 3/4+f(m_{\tilde{b}}/ \omega_{0})},\] (14) \[B_{\text{bino}} \simeq \frac{f(m_{\tilde{b}}/\omega_{0})}{8\times 36\times 3/4+f(m_{ \tilde{b}}/\omega_{0})}.\] (15) In Eq.(13) the factor $36$ comes from the degrees of freedom for colors (3), flavors (6) and chiralities (2), and the factor $3/4$ comes from the $U(1)_{Y}$ charge conservation. We do not include the quarks from the process of squark $\to$ quark + bino, because the quarks production rates are determined by the Pauli exclusion principle and the phase space of the quarks produced by the squark decay is a subset of that of the quarks produced by the squark annihilation. The binos eventually decay into winos (LSPs). Note that the thermal relics of winos do not overclose the Universe.² [FOOTNOTE:2][ENDFOOTNOTE] ## IV Cogenesis in gravity mediation In this section, we show that our scenario of the baryon and dark matter cogenesis works well. We consider the Affleck-Dine mechanism using the flat direction without non-renormalizable superpotential, because our scenario requires Q balls with $Q\gtrsim 10^{26}$ for the pair annihilation to be ineffective. The scalar potential for the flat direction is typically written as \[V(\phi) = (m_{\phi}^{2}-c_{H}H^{2})\|\phi\|^{2}+\frac{m_{3/2}^{2}}{M_{}^{n-2 }}(a_{m}\phi^{n}+h.c.)\] (16) \[+ \frac{H^{2}}{M_{}^{n-2}}\left(a_{H}\phi^{n}+h.c.\right)+\ldots,\] where $M_{}$ is a cut-off scale and $\ldots$ denotes higher order Planck-suppressed terms. The terms proportional to $H^{2}$ are induced via the interaction with the inflaton, and $c_{H}$, $a_{m}$, and $a_{H}$ are $O(1)$ constants. Here we assume $c_{H}>0$. Owing to the Hubble induced terms and higher order Planck-suppressed terms, the flat direction has a large expectation value during inflation $\phi\simeq M_{}$, and then begins to oscillate and rotate around $\phi=0$ when $H\simeq m_{\phi}$. Soon after the oscillation, Q balls are formed. Here we assume that the second term in Eq.(16) which kicks $\phi$ in the phase direction is large enough for $\phi$ to take a circular orbit. In this case anti-Q balls are not produced, which leads to the simple relation between baryon and dark matter densities.³ The charge of the Q ball is determined by [FOOTNOTE:3][ENDFOOTNOTE] \[Q\sim\beta\left(\frac{M_{}}{m_{\phi}}\right)^{2}\sim 3\times 10^{28}\left( \frac{M_{}}{M_{\text{P}}}\right)^{2}\left(\frac{2\text{ TeV}}{m_{\phi}(\phi_{ 0})}\right)^{2},\] (17) where $\beta=2\times 10^{-2}$ [20]. The Q-ball decay temperature is estimated as \[T_{\text{d}} = \left(\frac{90}{4\pi^{2}g_{}}\right)^{1/4}\sqrt{\Gamma_{Q}M_{ \text{P}}},\] (18) \[\simeq\] where $g_{}$ is the effective relativistic degrees of freedom at the decay time, and $\Gamma_{Q}=(1/Q)\sum_{\text{all}}\text{d}N/\text{d}t$ is the decay rate of the Q ball. In the second line of Eq. (18), we set $g_{}=10.75$ and $\sum_{\text{all}}\text{d}N/\text{d}t\sim 200\times R^{2}\omega_{0}^{3}/24\pi Q$. We find that the Q ball decays before the BBN but after the sphaleron process freezes out [21]. Winos produced from the Q-ball decay do not annihilate when the following condition is satisfied: \[Y_{\tilde{w}}^{(NT)} \ll \sqrt{\frac{45}{8\pi^{2}g_{}}}\frac{1}{\langle{\sigma v}\rangle M _{\text{P}}T_{d}},\] (19) \[\simeq 1.1\times 10^{-10}\times\left(\frac{10^{-24}{\rm cm^{3}/s}}{ \langle{\sigma v}\rangle}\right)\left(\frac{10\,{\rm MeV}}{T_{d}}\right)\ .\] (20) As mentioned above, we consider the flat direction without non-renormalizable superpotential. In this case, the Q balls dominate the Universe soon after inflation, and the baryon-to-entropy ratio is given by \[Y_{b}\simeq 10^{-10}\left(\frac{T_{\text{d}}}{10\text{ MeV}}\right)\left(\frac {2\text{ TeV}}{m_{\phi}(\phi_{0})}\right)\left(\frac{10^{4}}{\Delta}\right),\] (21) where we include the dilution factor $\Delta$. There is some mechanism to produce entropy after the reheating of the inflation, such as thermal inflation [22] and domain wall decay [23]. We do not specify the dilution mechanism and assume that the baryon asymmetry produced from the Q-ball decay is consistent with the observation. A dilution mechanism may also dilute the undesirable relics such as thermal relic of the stable bino. Thus, the successful bino LSP scenario may be realized as a simple extension of the present wino LSP scenario. However, most of the dilution mechanisms produce SUSY particles at the same time. This is a reason why we focus on the wino LSP scenario. In the case of the wino LSP, the thermal relic abundance can be ignored for $m_{\tilde{w}}\ll 1\text{ TeV}$ [24]. The baryon-to-dark matter ratio is determined only by the Q-ball decay: \[5\simeq\frac{\Omega_{\text{DM}}}{\Omega_{b}}=\frac{3m_{\tilde{w} }}{m_{N}}\frac{B_{\text{bino}}}{B_{\text{quarks}}}.\] (22) From Eqs. (11), (15), and (22), the bino and wino masses are related with each other by the following equation: \[\frac{360\text{ GeV}}{m_{\tilde{w}}}=f\left(\frac{m_{\tilde{b}}}{\omega_{0}} \right),\] (23) where $f(x)$ is defined as Eq. (11). The results are shown in Fig. 1 for the case of $m_{\tilde{w}}=m_{\tilde{b}}$. There are two solutions when $\omega_{0}\gtrsim 1.2$ TeV. In the limit of $\omega_{0}\to\infty$, two solutions are approximated to $360$ GeV and $\omega_{0}$. The resultant wino abundance is given by \[Y_{\tilde{w}}^{(NT)}\simeq 1.1\times 10^{-12}\frac{360\text{ GeV}}{m_{\tilde{w }}}.\] (24) From this and Eq. (20), we can check that the winos with mass of $0.4-1$ TeV do not annihilate when $T_{\text{d}}\lesssim 100$ MeV ($Q\gtrsim 10^{26}$). Indirect detection experiments constrain the wino mass as $m_{\tilde{w}}\gtrsim 300$ GeV [25; 26; 27]. The above predicted wino mass satisfies this constraint. <figure><img src="content_image/1211.6813/x1.png"><figcaption>Figure 1: Solutions of Eq. (23) for the case of m~w=m~b as a function ofmϕ(ϕ0). There is no solution for mϕ(ϕ0)≲1.2 TeV and are two independentsolutions for mϕ(ϕ0)≳1.2 TeV (green and blue lines). The red and magentadotted lines show the two asymptotic solutions m~w=mϕ(ϕ0) and m~w=360 GeV,respectively.</figcaption></figure> ## V conclusions We have reinvestigated the baryon and dark matter cogenesis through Q-ball decay into quarks and SUSY particles by taking into account the squark annihilation process inside the Q ball. The branching ratio of the Q-ball decay into quarks is enhanced by the number of degrees of freedom for quarks produced in the decay. We have assumed that the Q ball can decay into binos, winos, and SM particles kinematically, and considered the wino as LSP. In this case, we show that the branching into binos can be $O(0.01)$ for the $\bar{u}\bar{d}\bar{d}$ flat direction and predict that the dark matter is the wino with mass of $0.4-1$ TeV. * ## Appendix A Q-ball decay rates through the $N\geq 3$ body scattering processes Not only the decay process but also the $N$ body scattering processes can occur in the Q ball. The rate of the charge emission from the Q ball through the $N$ body scattering process can be roughly estimated as \[\left(\frac{\text{d}N}{\text{d}t}\right)_{N} \sim Q\times n_{\phi}^{N-1}\times\Gamma_{N},\] (25) \[\Gamma_{N} = \int\text{dLips}\|\mathcal{M}\|^{2}\prod_{\text{initial}}\frac{1}{2 E_{i}},\] (26) dLips \[\equiv (2\pi)^{4}\delta\left(\sum_{\text{all}}p_{j}\right)\prod_{\text{ final}}\frac{\text{d}^{3}k_{i}}{(2\pi)^{3}2E_{i}},\] (27) where $n_{\phi}\sim\omega_{0}\phi_{0}^{2}$ is the squark number density in the Q ball. Let us show that the rates of the $N$ body scattering processes are not saturated for $N\geq 3$. <figure><img src="content_image/1211.6813/x2.png"><figcaption>Figure 2: Examples of the diagrams for the N body scattering processes.</figcaption></figure> The mass of the field interacting with the Q ball is $O(\phi_{0})$, but the typical interaction energy is $O(\omega_{0})$. Thus, we can estimate the rates of the $N$ body scattering processes in the leading order of $\omega_{0}/\phi_{0}\sim Q^{-1/2}$. The number of particles in the final state should be minimized in the leading order as \[N_{\text{ext}}=\left\{\begin{array}[]{ll}N,&\text{N: even}\\ N+1,&\text{N: odd}.\end{array}\right.\] (28) Then, the number of fermion propagators can be counted as \[N_{\text{prop}}=\left\{\begin{array}[]{ll}3N/2-2,&\text{N: even}\\ 3N/2-3/2,&\text{N: odd}.\end{array}\right.\] (29) However, as shown in Fig. 2, there should be a factor of $M$ from the chirality flip, where $M$ is the Majorana gluino mass or the higgsino mass, and we assume $\omega_{0}<M\ll\phi_{0}$. The number of mass insertions is \[N_{\text{mass}}=\left\{\begin{array}[]{ll}N/2,&\text{N: even}\\ (N-1)/2,&\text{N: odd}.\end{array}\right.\] (30) The gauge boson is massless if it has no tree level interaction with the Q ball. Hereafter, we conservatively take the gauge boson as a massless field. Thus, from Eqs. (29) and (30), we can estimate $\|\mathcal{M}\|^{2}$ as \[\|\mathcal{M}\|^{2}\sim\left\{\begin{array}[]{ll}\phi_{0}^{8-6N}M^{N}\omega_{0}^ {N},&\text{N: even}\\ \phi_{0}^{6-6N}M^{N-1}\omega_{0}^{N+1},&\text{N: odd}\end{array}\right.\] (31) Here we determine the $\omega_{0}$ dependence from dimensional analysis. On the other hand, the kinematics is determined only by $\omega_{0}$. We conclude that the charge emission rates from Q ball through the $N$ body scattering process can be estimated as \[\left(\frac{dN}{dt}\right)_{N} \sim Q(\omega_{0}\phi_{0}^{2})^{N-1}\Gamma_{N},\] (32) \[\sim \left\{\begin{array}[]{ll}\omega_{0}Q^{4-2N}(M/\omega_{0})^{N},& \text{N: even}\\ \omega_{0}Q^{3-2N}(M/\omega_{0})^{N-1},&\text{N: odd}\end{array}\right.\] where we have used $Q\sim\phi_{0}^{2}/\omega_{0}^{2}$. We should compare this with the saturated emission rate from the Q ball $(\text{d}N/\text{d}t)_{\text{sat}}\sim\omega_{0}$ (see Eqs. (3) and (13)) and find that the rate is not saturated for $N\geq 3$. ###### Acknowledgements. This work is supported in part by JSPS Research Fellowship for Young Scientists (A.K.), by Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture (MEXT), Japan, No. 14102004 (M.K.), No. 21111006 (M.K.) and by World Premier International Research Center Initiative, MEXT, Japan. Numerical computation in this work was carried out in part at the Yukawa Institute Computer Facility. ## References * (1) I. Affleck and M. Dine, Nucl. Phys. B 249, 361 (1985). * (2) M. Dine, L. Randall and S. D. Thomas, Nucl. Phys. B 458, 291 (1996). * (3) A. Kusenko and M. E. Shaposhnikov, Phys. Lett. B 418, 46 (1998). * (4) K. Enqvist and J. McDonald, Phys. Lett. B 425, 309 (1998); Nucl. Phys. B 538, 321 (1999). * (5) S. Kasuya and M. Kawasaki, Phys. Rev. D 61, 041301(R) (2000). * (6) S. Kasuya and M. Kawasaki, Phys. Rev. D 62, 023512 (2000). * (7) S. Kasuya and M. 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1509.00326	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 39741, "num_imgs": 14, "llama3_tokens_count": 14226 }	[ "content_image/1509.00326/Figure1.png", "content_image/1509.00326/Figure2.png", "content_image/1509.00326/Figure3.png", "content_image/1509.00326/Figure4.png", "content_image/1509.00326/Figure5.png", "content_image/1509.00326/Figure6.png", "content_image/1509.00326/Figure7.png", "content_image/1509.00326/Figure8.png", "content_image/1509.00326/Figure9.png", "content_image/1509.00326/Figure10.png", "content_image/1509.00326/Figure11.png", "content_image/1509.00326/Figure12.png", "content_image/1509.00326/Figure13.png", "content_image/1509.00326/Figure14.png" ]	# Theoretical investigation of the pressure induced novel structure, phase transition, mechanical and electronic properties in Hf-O system. Jin Zhang Jin.Zhang.1@stonybrook.edu Department of Geosciences, Center for Materials by Design, and Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA Artem R. Oganov artem.oganov@stonybrook.edu Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, 5 Nobel St., Moscow, 143026, Russia. Department of Geosciences, Center for Materials by Design, and Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA Science and Technology on Thermostructural Composite Materials Laboratory, International Center for Materials Discovery, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia Xinfeng Li State Key Laboratory for Mechanical Behavior of Materials, School of Materials Science and Engineering, Xi’an Jiaotong University, Xi an, 710049, PR China Kan-Hao Xue School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China Zhenhai Wang Peter Grünberg Research Center, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210003, China Huafeng Dong College of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China Jin Zhang Jin.Zhang.1@stonybrook.edu Department of Geosciences, Center for Materials by Design, and Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA Artem R. Oganov artem.oganov@stonybrook.edu Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, 5 Nobel St., Moscow, 143026, Russia. Department of Geosciences, Center for Materials by Design, and Institute for Advanced Computational Science, State University of New York, Stony Brook, NY 11794-2100, USA Science and Technology on Thermostructural Composite Materials Laboratory, International Center for Materials Discovery, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia Xinfeng Li State Key Laboratory for Mechanical Behavior of Materials, School of Materials Science and Engineering, Xi’an Jiaotong University, Xi an, 710049, PR China Kan-Hao Xue School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China Zhenhai Wang Peter Grünberg Research Center, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210003, China Huafeng Dong College of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China ###### Abstract Using first-principles evolutionary simulations, we have systematically investigated phase stability in the Hf-O system at pressure up to 120 GPa. New compounds Hf${}_{5}$O${}_{2}$, Hf${}_{3}$O${}_{2}$, HfO and HfO${}_{3}$ are discovered to be thermodynamically stable at certain pressure ranges and a new stable high-pressure phase is found for Hf${}_{2}$O with space group _Pnnm_ and anti-CaCl${}_{2}$-type structure. Both _P_$\bar{6}$2_m_-HfO and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$ show semimetallic character. _Pnnm_-HfO${}_{3}$ shows interesting structure, simultaneously containing oxide O${}^{2-}$ and peroxide [O-O]${}^{2-}$ anions. Remarkably, it is _P_$\bar{6}$2_m_-HfO rather than OII-HfO${}_{2}$ that exhibits the highest mechanical characteristics among Hf-O compounds. _Pnnm_-Hf${}_{2}$O, _Imm_2-Hf${}_{5}$O${}_{2}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$ phases also show superior mechanical properties, these phases can be quenched to ambient pressure and their properties can be exploited. ## I Introduction Hafnium oxide HfO${}_{2}$ has a wide range of technological applications. In electronics industry, hafnium oxide-based material is currently used as an excellent high-_k_ gate dielectricChoi _et al._ (2011) and oxygen-deficient hafnium oxide also received additional interest for resistive-switching memoriesLin _et al._ (2011). As for other applications, even though the hardness of hafnia (HfO${}_{2}$) is not that high for it to be considered as a superhard materialAl-Khatatbeh _et al._ (2010), it still attracts attention as a potential candidate for hard oxide-based materialsChung and Sproul (2003). Unlike carbides or nitrides, oxides are more stable in the oxygen atmosphere at high temperature, which is valuable for many applications, Many oxide ceramics, especially those involving transition metals, are promising for application as hard coatings, since metal d electrons and strong bonds define their remarkable mechanical properties (high hardness, good chemical resistance, high tensile strength, and good fracture toughness)Lowther (2003). As compared to most transition metal oxide ceramics, hafnium oxide ceramics exhibit enhanced mechanical properties (higher fracture toughness) and structural stability (low thermal conductivity). Here, we want to explore all possible stable compounds of Hf-O system at pressure up to 120 GPa. Under ambient temperature, experimentsXia _et al._ (1990); Ahuja _et al._ (1993) indicated that pure Hf is stable in the $\alpha$-phase (hexagonal close-packed structure, space group: _P_6${}_{3}$/_mmc_) and transforms to $\omega$-phase (hexagonal structure, space group: _P_6/_mmm_) at 46-58 GPa and then to $\beta$-Hf (body centered cubic, space group: _Im_$\bar{3}$_m_) at 71.1 GPa-78.4 GPa. Our GGA calculated results indicate the transition pressures: $\alpha$-phase $\rightarrow$$\omega$-phase at 49 GPa and $\omega$-phase $\rightarrow$$\beta$-Hf at 70 GPa, which are in accord with above experimental results. It has been suggested that the solubility of oxygen in the octahedral interstitial sites of $\alpha$-Hf (hcp-Hf) can be as high as 20 at.%Tsuji (1997), while solubility of oxygen in $\beta$-Hf (bcc-Hf) is only 3 at.%Massalski _et al._ (1990). Several experimentalHirabayashi _et al._ (1973, 1972) and theoretical studiesRuban _et al._ (2010); Paul Burton _et al._ (2011) have investigated the interstitial oxygen in hcp-Hf. Now it is well established that three stoichiometric compositions Hf${}_{6}$O, Hf${}_{3}$O and Hf${}_{2}$O can be formed with increasing occupation of the octahedral-interstitial positions in hcp-Hf by oxygen atoms. Hf${}_{2}$O${}_{3}$ was theoretically predicted to form upon increasing the concentration of oxygen vacancies in monoclinic HfO${}_{2}$Xue _et al._ (2013). The phase sequence of HfO${}_{2}$ at ambient temperature with increasing pressure is: baddeleyite (monoclinic, space group: _P_2${}_{1}$/_c_) $\rightarrow$ orthorhombic I (orthorhombic, space group: _Pbca_, OI) $\rightarrow$ orthorhombic II (orthorhombic, space group: _Pnma_, OII)Desgreniers and Lagarec (1999); Tang _et al._ (1998); Kang _et al._ (2003). Orthorhombic OII-HfO${}_{2}$ with experimentally reported hardness between 6-13 GPaAdams _et al._ (1991) has been speculated to be much harder than the low-pressure phases (baddeleyite and OI-HfO${}_{2}$) because of its comparatively high bulk modulusLowther (2003); Ohtaka _et al._ (2001). In this study, we systematically investigate the structure and stability of Hf-O compounds up to a pressure of 120 GPa by the first-principles evolutionary algorithm USPEX. Several new stoichiometries in the Hf-O system have been predicted under high pressure. Furthermore, we verify the dynamical and mechanical stability of these new high-pressure phases at 0 GPa by calculating their phonons and elastic constants. To better understand the correlations between hardness and O content, we estimate the hardness of these phases at 0 GPa using Chen’s hardness modelChen _et al._ (2011). Quenchable high-pressure phases often possess superior mechanically properties, and we indeed find novel hafnium oxides with unusual mechanical properties. ## II Computational methodology Searching the stable high-pressure structures in Hf-O system was done using first-principles evolutionary algorithm (EA) as implemented in the USPEX codeOganov and Glass (2006); Lyakhov _et al._ (2013); Oganov _et al._ (2011) combined with ab initio structure relaxations using density functional theory (DFT) with the PBE-GGA functionalPerdew _et al._ (1996), as implemented in the VASP packageKresse and Furthmüller (1996). In our work, variable-composition structure searchesOganov _et al._ (2011) for the Hf-O system with up to 20 atoms in the unit cell were performed at 0 GPa, 10 GPa, 20 GPa, 30 GPa, 40 GPa, 50 GPa, 60 GPa, 70 GPa, 80 GPa, 90 GPa, 100 GPa, 110 GPa and 120 GPa. The initial generation of structures was produced randomly using space group symmetry, each subsequent generation was obtained by variation operators including heredity (40%), lattice mutation (20%), random (20%) and transmutation (20%). The electron-ion interaction was described by the projector-augmented wave (PAW) pseudopotentialsBlöchl (1994), with 5_p_${}^{6}$6_s_${}^{2}$5_d_${}^{4}$ and 2_s_${}^{2}$2_p_${}^{4}$ shells treated as valence for Hf and O, respectively. The generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof formPerdew _et al._ (1996) was utilized for describing exchange-correlation effects. The plane-wave energy cutoff was chosen as 600 eV and $\Gamma$-centered uniform _k_-meshes with resolution 2$\pi\times 0.06$ Å${}^{-1}$ were used to sample the Brillouin zone, resulting in excellent convergence. Phonon dispersions were calculated using the finite-displacement method with the Phonopy codeTogo _et al._ (2008). ## III Results and discussions ### Crystal structure prediction for the Hf-O system <figure><img src="content_image/1509.00326/Figure1.png"><figcaption>Figure 1: Convex hull diagrams for the Hf-O system at (a) 0 GPa, (b) 20 GPa,(c) 30 GPa, (d) 40 GPa, (e) 50 GPa, and (f) 110 GPa, respectively. Solidsquares denote stable phases while open squares represent metastable phases.</figcaption></figure> Thermodynamic convex hull, which defines stable compounds, is based on the free energies (at T = 0 K, enthalpies) of the compounds and pure elements in their stable forms. The high-pressure convex hull and pressure-composition phase diagram of the Hf-O system are depicted in Fig. 1 and Fig. 2, respectively. Besides the three well-known phases of HfO${}_{2}$ and three suboxides (_R_$\bar{3}$-Hf${}_{6}$O, _R_$\bar{3}$_c_-Hf${}_{3}$O and _P_$\bar{3}$1_m_-Hf${}_{2}$O), our structure searches found hitherto unknown compounds with new stoichiometries, including Hf${}_{5}$O${}_{2}$, Hf${}_{3}$O${}_{2}$, HfO and HfO${}_{3}$. Note that a new high-pressure phase of Hf${}_{2}$O (denoted as _Pnnm_-Hf${}_{2}$O) was also found by our searches. Recent workBurton and van de Walle (2012) indicated that Hf${}_{12}$O${}_{5}$ is a stable compound at low temperature, disproportionate above 220 K, therefore it is not expected to be observed experimentally. Our work indicates that Hf${}_{12}$O${}_{5}$ is actually stable only in the pressure range from 8 GPa to 37 GPa. <figure><img src="content_image/1509.00326/Figure2.png"><figcaption>Figure 2: Pressure-composition phase diagram of the Hf-O system.</figcaption></figure> Our calculation confirms that Hf${}_{2}$O${}_{3}$ proposed by XueXue _et al._ (2013) can exist as a metastable phase (it is dynamically and mechanically stable at 0 GPa), and shows that it should be a stable phase in the pressure range 20-23 GPa. The predicted transition from monoclinic-HfO${}_{2}$ to OI-HfO${}_{2}$ occurs at 10 GPa, which is coincides with experimental observationsDesgreniers and Lagarec (1999). The transition from OI-HfO${}_{2}$ to OII-HfO${}_{2}$ occurs at 18 GPa, which is lower than the experimental result 30-37 GPaDesgreniers and Lagarec (1999); Al-Khatatbeh _et al._ (2010) but in good agreement with other theoretical estimates of 17 GPaAl-Khatatbeh _et al._ (2010). Furthermore, our calculated result shows that OII-HfO${}_{2}$ is stable up to at least 120 GPa, which agrees with previous experimental workAl-Khatatbeh _et al._ (2010). According to our predictions, only baddeleyite-type HfO${}_{2}$ and _R_$\bar{3}$-Hf${}_{6}$O are stable at 0 GPa, in contrast with the Zr-O system (Zr${}_{6}$O, Zr${}_{3}$O, Zr${}_{2}$O, ZrO and ZrO${}_{2}$ are stable at 0 GPa)Zhang _et al._ (2015). In order to study the ordering of interstitial oxygen atoms in hcp-HfO${}_{x}$, Hirabayashi et al.Hirabayashi _et al._ (1973) used electron, neutron and X-ray diffraction to analyze single crystals containing 13.4 at % O and 15.8 at % O and found two types of interstitial superstructures: HfO${}_{\frac{1}{6}-}$ and HfO${}_{\frac{1}{6}+}$ below 600 K. The space group of HfO${}_{\frac{1}{6}-}$ reported by HirabayashiHirabayashi _et al._ (1973) is _R_$\bar{3}$, which is identical to our findings. The space group of HfO${}_{\frac{1}{6}+}$ is _P_$\bar{3}1$_c_ in Hirabayashi’ experimentHirabayashi _et al._ (1973). At 0 K and 0 GPa, our results produce three energetically competitive phases for Hf${}_{3}$O and their ordering by energy is _R_$\bar{3}$_c_-Hf${}_{3}$O (-10.075 eV/atom) $<$_P_$\bar{3}1$_c_-Hf${}_{3}$O (-10.072 eV/atom) $<$_P_6${}_{3}$22-Hf${}_{3}$O (-10.069 eV/atom). Therefore, one can note that _P_$\bar{3}1$_c_-Hf${}_{3}$O (_P_$\bar{3}1$_c_-Zr${}_{3}$O type), exhibits very close but higher energy than _R_$\bar{3}$_c_-Hf${}_{3}$O at 0 GPa and 0 K. In order to consider the effects of temperature, quasi-harmonic free-energy of _R_$\bar{3}$_c_-Hf${}_{3}$O and _P_$\bar{3}1$_c_-Hf${}_{3}$O were calculated using the Phonopy codeTogo _et al._ (2008). The results indicate that free energy of _P_$\bar{3}1$_c_-Hf${}_{3}$O decreases faster than that of _R_$\bar{3}$_c_-Hf${}_{3}$O with temperature, enabling _P_$\bar{3}1$_c_-Hf${}_{3}$O to become more stable than _R_$\bar{3}$_c_-Hf${}_{3}$O at $~{}$1000 K, thus explaining experimental result. <figure><img src="content_image/1509.00326/Figure3.png"><figcaption>Figure 3: (Color online) Equation of state of Hf2O. Our calculations were fitto a third-order Birch-Murnaghan equation of state to find B0 and B′0.</figcaption></figure> Hf${}_{2}$O undergoes a trigonal-to-orthorhombic phase transition at 58 GPa. The crystal structure of the new high-pressure phase _Pnnm_-Hf${}_{2}$O is of anti-CaCl${}_{2}$-type. The Birch-Murnaghan equation of stateBirch (1952) was used to fit the compressional behavior of the predicted Hf${}_{2}$O phases ( Fig. 3.). The third-order Birch-Murnaghan EOS is given as \[P(V)=\frac{3B_{0}}{2}\left[\left(\frac{V}{V_{0}}\right)^{-\frac{7}{3}}-\left( \frac{V}{V_{0}}\right)^{-\frac{5}{3}}\right]\left\{1+\frac{3}{4}(B_{0}^{{}^{ \prime}}-4)\left[\left(\frac{V}{V_{0}}\right)^{-\frac{2}{3}}-1\right]\right\}\] (1) Three parameters are used to describe the EOS: the volume at 0 GPa (V${}_{0}$), the bulk modulus at 0 GPa (B${}_{0}$), and the first pressure derivative of the the bulk modulus at 0 GPa (B${}_{0}^{{}^{\prime}}$). Most materials have 3 $\leq$ B${}_{0}^{{}^{\prime}}$$\leq$ 6Jeanloz (1988); Birch (1978). The B${}_{0}^{{}^{\prime}}$ of _P_$\bar{3}$1_m_-Hf${}_{2}$O and _Pnnm_-Hf${}_{2}$O is 4.0 and 3.8, respectively. ### Structure character in Hf-O compounds Table 1 lists the detailed crystallographic data of _Imm_2- Hf${}_{5}$O${}_{2}$, _Pnnm_- Hf${}_{2}$O, _C_2/_m_-Hf${}_{3}$O${}_{2}$, _P_$\bar{6}$2_m_-HfO, _C_2/_m_-Hf${}_{2}$O${}_{3}$ and _Pnnm_-HfO${}_{3}$ compounds at 0 GPa. The dynamical stabilities of all the new phases are checked by calculating phonon dispersion. As shown in Fig.4, except for HfO${}_{3}$, no imaginary phonon frequencies are found in the whole Brillouin zone at both ambient and high pressure, which means that they are dynamically stable and probably quenchable to ambient pressure. In contrast, HfO${}_{3}$ is stable only at high pressure, but at 0 GPa shows total dynamical instability and most likely decomposes. The special electronic structure of HfO${}_{3}$ will be discussed below. The weighted average lengths of Hf-Hf and Hf-O bonds in Hf-O compounds are plotted in Fig. 5. Compound \| Space group \| Enthalpy of formation \| Lattice constants \| Wyckoff positions \| x \| y \| z ---\|---\|---\|---\|---\|---\|---\|--- \| \| (eV/atom) \| (Å) \| \| \| \| Hf5O2 \| _Imm_ 2 \| -1.52 \| a=14.455 \| Hf 4c \| 0.711 \| 0.50 \| 0.566 \| \| \| c=3.141 \| Hf 2b \| 0.00 \| 0.50 \| 0.098 \| \| \| \| Hf 4c \| 0.097 \| 0.00 \| 0.594 \| \| \| c=5.082 \| O 4c \| 0.645 \| 0.00 \| 0.818 Hf2O \| _Pnnm_ \| -1.76 \| a=5.092 \| Hf 4g \| 0.263 \| 0.341 \| 0.50 \| \| \| b=5.723 \| O 2c \| 0.00 \| 0.50 \| 0.00 \| \| \| c=3.175 \| \| \| \| Hf3O2 \| _C_ 2/_m_ \| -2.04 \| a=11.967 \| Hf 4i \| 0.625 \| 0.50 \| 0.007 \| \| \| b=3.131 \| Hf 4i \| 0.465 \| 0.00 \| 0.346 \| \| \| c=11.198 \| Hf 4i \| 0.286 \| 0.00 \| 0.676 \| \| \| β=99.67∘ \| O 4i \| 0.378 \| 0.50 \| 0.607 \| \| \| \| O 4i \| 0.787 \| 0.00 \| 0.192 HfO \| _P_ ¯62 _m_ \| -2.60 \| a=5.230 \| Hf 1b \| 0.00 \| 0.00 \| 0.50 \| \| \| c=3.187 \| Hf 2c \| 0.667 \| 0.333 \| 0.00 \| \| \| \| O 3g \| 0.00 \| 0.592 \| 0.50 Hf2O3 \| _P_ ¯4 _m_ 2 \| -3.11 \| a=3.137 \| Hf 2g \| 0.00 \| 0.50 \| 0.744 \| \| \| c=5.638 \| O 2g \| 0.00 \| 0.50 \| 0.135 \| \| \| \| O 1c \| 0.50 \| 0.50 \| 0.50 HfO3 \| _Pnnm_ \| -2.22 \| a=5.554 \| Hf 4c \| 0.246 \| 0.110 \| 0.250 \| \| \| b=6.457 \| O 4c \| 0.359 \| 0.426 \| 0.250 \| \| \| c=3.307 \| O 4c \| 0.025 \| 0.339 \| 0.750 \| \| \| \| \| \| \| Table 1: Structural parameters of _Imm_ 2- Hf5O2, _Pnnm_ \- Hf2O, _C_ 2/_m_ -Hf3O2, _P_ ¯62 _m_ -HfO, _C_ 2/_m_ -Hf2O3 and _Pnnm_ -HfO3 at 0 GPa. <figure><img src="content_image/1509.00326/Figure4.png"><figcaption>Figure 4: (Color online) Calculated phonon dispersion curves for the (a)_Imm_ 2-Hf5O2 at 0 and 60 GPa (b) _Pnnm_ -Hf2O at 0 and 80 GPa (c) _C_ 2/_m_-Hf3O2 at 0 and 90 GPa (d) _P_ ¯62 _m_ -HfO at 0 and 50 GPa (e) _P_ ¯4 _m_2-Hf2O3 at 0 and 20 GPa (f) _Pnnm_ -HfO3 at 0 and 110 GPa. The solid black anddashed blue lines represent the results at zero and high pressures,respectively.</figcaption></figure> <figure><img src="content_image/1509.00326/Figure5.png"><figcaption>Figure 5: (Color online) Average bond lengths in Hf-O compounds at 0 GPa.</figcaption></figure> Structurally, hafnium oxides can be divided into four groups: suboxides with oxygen interstitials in hcp-Hf (Hf${}_{6}$O, Hf${}_{3}$O, Hf${}_{12}$O${}_{5}$ and _P_$\bar{3}$1_m_-Hf${}_{2}$O); other suboxides (Hf${}_{5}$O${}_{2}$, _Pnnm_-Hf${}_{2}$O and HfO); normal oxides (Hf${}_{2}$O${}_{3}$, HfO${}_{2}$) and oxide peroxide (HfO${}_{3}$). The octahedral sites of hcp hafnium metal are depicted in Fig.6. Oxygen atoms prefer to occupy these octahedral sites and form ordered structures _R_$\bar{3}$-Hf${}_{6}$O, _R_$\bar{3}$_c_-Hf${}_{3}$O, _R_$\bar{3}$-Hf${}_{12}$O${}_{5}$ and _P_$\bar{3}$1_m_-Hf${}_{2}$O, as shown in Fig. 7 (a) (b) (c) and (d), where Hf atom sites are omitted. The polyhedral representation of these structures is shown in Fig. 6 (e) (f) (g) and (h). Anti-CaCl${}_{2}$-type (_Pnnm_) structure of Hf${}_{2}$O can also be represented as an hcp-sublattice (distorted) of Hf atoms, where half of octahedral voids are occupied by O atoms. The structure of Hf${}_{3}$O${}_{2}$ can be considered to be defective because each layer lacks some Hf atoms to form a Hf-graphene layer. These vacancies are responsible for low values of the mechanical properties of Hf${}_{3}$O${}_{2}$. <figure><img src="content_image/1509.00326/Figure6.png"><figcaption>Figure 6: (Color online) Octahedral voids in hcp hafnium.</figcaption></figure> <figure><img src="content_image/1509.00326/Figure7.png"><figcaption>Figure 7: (Color online) Oxygen sublattice representation (arrangement ofoxygen atoms in the octahedral interstitial sites) and polyhedralrepresentation of (a)&(b) _R_ ¯3-Hf12O5 (c)&(d) _P_ ¯31 _m_ -Hf2O (e)&(f) _R_¯3-Hf6O (g)&(h) _R_ ¯3 _c_ -Hf3O. Oxygen-centered octahedra and oxygenvacancies are shown in red and pink polyhedra, respectively. Oxygen sublatticerepresentations (b, d, f, h) show only oxygen atoms (filled circles) andvacancies (open circles).</figcaption></figure> <figure><img src="content_image/1509.00326/Figure8.png"><figcaption>Figure 8: (Color online) Crystal structures of (a) _Imm_ 2-Hf5O2 (b) _P_ ¯62_m_ -HfO (c) _P_ ¯4 _m_ 2-Hf2O3 and (d) _P_ ¯4 _m_ 2-Hf2O3. O-centeredoctahedra and O-centered tetrahedra are shown in red and blue polyhedra,respectively. Large spheres-Hf atoms; small spheres-O atoms.</figcaption></figure> Similar with ZrO, the structure of HfO contains Hf-graphene layers stacked on top of each other (Zr-Zr distances within the layer are 3.01 Å, and between the layers 3.18 Å), as illustrated in Fig. 8(b), as well as additional Hf and O atoms. The structure can be represented as $\omega$-phase of Hf, intercalated with oxygen atoms. This structure, therefore, is built by a 3D-framework of short and strong Hf-O bonds, reinforced by rather strong Hf-Hf bonds. The former lead to high hardness, the latter may improve toughness due to semimetallic behavior. _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$, which was firstly proposed by XueXue _et al._ (2013), has 8-fold and 6-fold coordination of Hf atoms, as shown in Fig 8. <figure><img src="content_image/1509.00326/Figure9.png"><figcaption>Figure 9: (Color online) Crystal structure of (a) _Pnnm_ -HfO3 (b) _Pnnm_-Hf2O; (c) _Pnnm_ -HfO3 (d) _Pnnm_ -Hf2O.</figcaption></figure> _Pnnm_-HfO${}_{3}$ becomes stable at pressures above to 110 GPa. This high-pressure phase originally derives from oxygen atom dissolving in both octahedral and tetrahedral voids of a heavily distorted hcp-Hf, as shown in Fig 9(a). However, due to short distances between tetrahedral voids in the hcp structures, some O atoms form pairs and as a result HfO${}_{3}$ simultaneously contains oxide O${}^{2-}$ and peroxide [O-O]${}^{2-}$ anions, and can be described as ”oxide peroxide”. The O-O bond length in HfO${}_{3}$ is 1.44 Åat 110 GPa, which is a little smaller than the O-O bond length in peroxide [O-O]${}^{2-}$ ion with 1.47 ÅWells (1986) at ambient conditions. It seems that peroxides and oxide peroxides (e.g. Al${}_{4}$O${}_{7}$ and AlO${}_{2}$) become stabilized in many systems under pressureLiu _et al._ (2015). ### Mechanical properties of Hf-O compounds Previous studiesOhtaka _et al._ (2001); Haines _et al._ (35); Desgreniers and Lagarec (1999) suggested that dense high-pressure phase OII-HfO${}_{2}$ is quenchable to ambient conditions and has a high bulk modulus, and might be superhard (H $>$ 40 GPa). However recent studyAl-Khatatbeh _et al._ (2010) reported that the hardness of OII-HfO${}_{2}$ is well below 40 GPa and therefore this phase is not superhard. Interestingly, our systematic results not only confirm known hardness of HfO${}_{2}$ polymorphs: H(OII) $<$ H(MI) $<$ H(OI), but also suggest that HfO has the highest hardness among all hafnium oxides, see Fig. 10(f). In addition, _Pnnm_-Hf${}_{2}$O and _Imm_2-Hf${}_{5}$O${}_{2}$ also exhibit higher hardness than other Hf-O compounds, as shown in Tab 2. The hardness of Hf-O compounds does not monotonically change with O content, but a maximum at HfO. The calculated modulus _B_, shear modulus _G_, Young’s modulus _E_, Poisson’s ratio $\upsilon$, and hardness of all stable Hf$-$O compounds are depicted in Table2 and Fig.10 (for comparison, the elastic data of the high-pressure phase _Pnnm_-HfO${}_{3}$ are reported at 0 GPa although it is unstable at 0 GPa.) From Fig.10 we can conclude that the high O content in the crystal does not guarantee high hardness of Hf-O compounds and the structure plays an important role in determining mechanical properties as we discussed above. The Vickers hardness was calculated according to Chen’s modelChen _et al._ (2011): \[H_{V}=2(k^{2}G)^{0.585}-3\] (2) Compound \| \| Space group \| P \| BH \| GH \| E \| G/B \| υ \| Hv ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Hf6O \| This work \| _R_ ¯3 \| 0 \| 129.2 \| 72.8 \| 183.8 \| 0.56 \| 0.26 \| 9.55 Hf3O \| This work \| _R_ ¯3 _c_ \| 0 \| 150.3 \| 78.8 \| 201.3 \| 0.52 \| 0.28 \| 9.1 Hf5O2 \| This work \| _Imm_ 2 \| 0 \| 150.0 \| 95.3 \| 235.9 \| 0.64 \| 0.24 \| 13.9 Hf12O5 \| This work \| _R_ ¯3 \| 0 \| 163.3 \| 94.5 \| 237.7 \| 0.58 \| 0.26 \| 12.1 Hf2O \| This work \| _P_ ¯31 _m_ \| 0 \| 175.2 \| 103.1 \| 258.6 \| 0.59 \| 0.25 \| 13.2 Hf2O \| This work \| _Pnnm_ \| 0 \| 173.0 \| 110.3 \| 272.9 \| 0.64 \| 0.23 \| 15.5 Hf3O2 \| This work \| _C_ 2/_m_ \| 0 \| 154.2 \| 75.9 \| 195.6 \| 0.49 \| 0.29 \| 8.0 HfO \| This work \| _P_ ¯62 _m_ \| 0 \| 210.7 \| 128.1 \| 319.5 \| 0.61 \| 0.25 \| 16.1 Hf2O3 \| This work \| _P_ ¯4 _m_ 2 \| 0 \| 243.9 \| 127.1 \| 324.8 \| 0.52 \| 0.28 \| 12.9 HfO2 \| This work \| _P_ 21/_c_ \| 0 \| 203.6 \| 99.2 \| 256.1 \| 0.49 \| 0.29 \| 9.7 \| ExperimentOkutomi _et al._ (1984) \| \| 0 \| \| \| \| \| \| 9.9 HfO2 \| This work \| _Pbca_ \| 0 \| 225.9 \| 115.8 \| 296.6 \| 0.51 \| 0.28 \| 11.7 HfO2 \| This work \| _Pnma_ \| 0 \| 226.3 \| 93.8 \| 247.3 \| 0.41 \| 0.31 \| 7.2 \| ExperimentHaines _et al._ (1997b) \| \| 0 \| \| \| \| \| \| 6-13 HfO3 \| This work \| _Pnnm_ \| 0 \| 171.1 \| 73.6 \| 193.0 \| 0.43 \| 0.31 \| 6.2 Table 2: Calculated bulk modulus _B_ , shear modulus _G_ , Young’s modulus _E_ , Poisson’s ratio υ and hardness of Hf-O compounds, compared with literature data for HfO2 at 0 GPa. All properties are in GPa (except dimensionless G/B and υ ). <figure><img src="content_image/1509.00326/Figure10.png"><figcaption>Figure 10: (Color online) Compositional dependence of the computed mechanicalproperties of Hf-O compounds. The blue open circle represents _Pnnm_ -Hf2O;red open triangle represents OI-HfO2 ; green open square represents OII-HfO2.</figcaption></figure> We calculated the elastic anisotropy of five special phases: _P_$\bar{6}$2_m_-HfO, _Pnnm_-Hf${}_{2}$O, _Imm_2-Hf${}_{5}$O${}_{2}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$. As shown in Fig. 11, all of these five phases exhibit a moderate amount of anisotropy of Young’s modulus. The directional dependence of the Young’s modulus for hexagonal, orthorhombic, trigonal and tetragonal crystals can be calculated as: \[\frac{1}{E_{hex}}={s_{11}}{(1-l_{3}^{2})^{2}}+{s_{33}}l_{3}^{4}+(2{s_{13}}+{s_ {44}})l_{3}^{2}(1-l_{3}^{2})\] (3) \[\frac{1}{E_{ortho}}=s_{11}(l_{1})^{4}+s_{22}(l_{2})^{4}+s_{33}(l_{3})^{4}+l_{2 }^{2}l_{3}^{2}(2s_{23}+s_{44})+l_{1}^{2}l_{3}^{2}(2s_{13}+s_{55})+l_{2}^{2}l_{ 1}^{2}(2s_{12}+s_{66})\] (4) \[\frac{1}{E_{tri}}=(1-l_{3}^{2}){s_{11}}+l_{3}^{4}{s_{33}}+l_{3}^{2}(1-l_{3}^{2 })(2{s_{13}}+{s_{44}})+2{l_{2}}{l_{3}}(3l_{1}^{2}-l_{2}^{2}){s_{14}},\] (5) \[\frac{1}{E_{tetra}}=s_{11}(l_{1}^{4}+l_{2}^{4})+s_{33}l_{3}^{4}+(2s_{12}+s_{66 })l_{1}^{2}l_{2}^{2}+(2s_{13}+s_{44})l_{3}^{2}(1-l_{3}^{2})\] (6) where $s_{11}$, $s_{12}$, etc., are the elastic compliance constants and $l_{1}$, $l_{2}$, $l_{3}$ are the direction cosines of a particular crystallographic orientation to coordinate axes $x_{1}$, $x_{2}$ and $x_{3}$, respectively. <figure><img src="content_image/1509.00326/Figure11.png"><figcaption>Figure 11: (Color online) Orientational dependence of Young’s moduli (in GPa)of (a) _P_ ¯62 _m_ -HfO (b) _Pnnm_ -Hf2O (c) _Imm_ 2-Hf5O2 (d) _P_ ¯31 _m_-Hf2O and (e) _P_ ¯4 _m_ 2-Hf2O3.</figcaption></figure> ### Electronic structure of Hf-O compounds Fig. 13 shows band structures of Hf-O compounds at 0 GPa (including phases stable at both zero and high pressure). Total and partial densities of states (DOS) are presented in Fig. 14. _R_$\bar{3}$-Hf${}_{6}$O, _R_$\bar{3}$_c_-Hf${}_{3}$O, _Imm_2- Hf${}_{5}$O${}_{2}$, _R_$\bar{3}$- Hf${}_{12}$O${}_{5}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O, _Pnnm_- Hf${}_{2}$O and _C_2/_m_-Hf${}_{3}$O${}_{2}$ are predicted to be metallic with a sizable density of states at the Fermi level. The DOSs of _R_$\bar{3}$-Hf${}_{6}$O, _R_$\bar{3}$_c_-Hf${}_{3}$O, _Imm_2- Hf${}_{5}$O${}_{2}$, _R_$\bar{3}$- Hf${}_{12}$O${}_{5}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O, _Pnnm_- Hf${}_{2}$O and _C_2/_m_-Hf${}_{3}$O${}_{2}$ below E${}_{F}$ are mainly due to Hf-d and O-p orbitals, and the interactions between the Hf-d orbitals are responsible for metallicity. <figure><img src="content_image/1509.00326/Figure12.png"><figcaption>Figure 12: (Color online) ELF isosurface (ELF = 0.62) for HfO3. Blue and redatoms represent oxide O2− and peroxide [O-O]2− ions, respectively.</figcaption></figure> <figure><img src="content_image/1509.00326/Figure13.png"><figcaption>Figure 13: (Color online) Band structures of hafnium oxides at 0 GPa. TheFermi energy is set to zero.</figcaption></figure> <figure><img src="content_image/1509.00326/Figure14.png"><figcaption>Figure 14: (Color online) The normalized (per electron) total (TDOS) andpartial densities of states (PDOS) of hafnium oxides at 0 GPa. The Fermienergy is set to zero.</figcaption></figure> Unlike regular metals, semimetals possess both electronic and hole conduction, which can be seen in the band structure as overlap of the partially vacant valence band top and occupied conduction band bottom located at different points in the Brillouin ZoneXue _et al._ (2013). Moreover, the upper limit of electron and hole density for a semimetal should below 10${}^{22}$ cm${}^{-3}$. The carrier concentrations of the most common semimetals, for example Bi, Sb and As are 3$\times$10${}^{17}$ cm${}^{-3}$, 5$\times$10${}^{19}$ cm${}^{-3}$, and 2$\times$10${}^{20}$ cm${}^{-3}$Liu and Allen (1995); Ashcroft and Mermin (1976), respectively. Semimetallic behavior of _P_$\bar{6}$2_m_-HfO and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$ can be reflected in the calculated band structures, as shown in Fig. 12(h,i) and in the DOS diagrams (Fig. 14(h,i)), showing very few states at the Fermi level. For the band structure of _P_$\bar{6}$2_m_-HfO, there are small electron and hole pockets at $\Gamma$ and M, respectively, but there are no band crossings between the lowest unoccupied bands and the highest occupied bands. Both band edges are mostly derived from Hf 5d states. In the case of _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$, partially occupied valence band top and conduction band bottom correspond to different high symmetry points R and Z, respectively. The electron and hole densities can be estimated by integrating the occupation numbers in the 3D Brillouin zone. The electron and hole densities of HfO are both 1.1$\times$10${}^{20}$ cm${}^{-3}$ by integrating their occupation of the blue and green bands shown in Fig. 13(h), respectively. For _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$, the electron and hole densities are both 2.1$\times$10${}^{21}$ cm${}^{-3}$, which is very close to 1.8$\times$10${}^{21}$ cm${}^{-3}$ obtained in Xue’s workXue _et al._ (2013). The DFT band gaps of _P_2${}_{1}$/_c_-HfO${}_{2}$, _Pbca_-HfO${}_{2}$, _Pnma_-HfO${}_{2}$ and _Pnnm_-HfO${}_{3}$ are 4.01 eV, 4.18 eV, 3.36 eV and 1.92 eV, respectively, and the highest occupied states are all derived mainly from O-p orbitals, as shown in Fig. 14(j, k, l and m). Therefore, according to their electronic character, Hf-O compounds can be divided into three types: metallic, including _R_$\bar{3}$-Hf${}_{6}$O, _R_$\bar{3}$_c_-Hf${}_{3}$O, _Imm_2- Hf${}_{5}$O${}_{2}$, _R_$\bar{3}$- Hf${}_{12}$O${}_{5}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O, _Pnnm_- Hf${}_{2}$O and _C_2/_m_-Hf${}_{3}$O${}_{2}$; semimetallic, including _P_$\bar{6}$2_m_-HfO and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$; insulating or semiconducting, including _P_2${}_{1}$/_c_-HfO${}_{2}$, _Pbca_-HfO${}_{2}$, _Pnma_-HfO${}_{2}$ and _Pnnm_-HfO${}_{3}$. Electron localization function (ELF) clearly reveals special featrure of HfO${}_{3}$, the coexistence of oxide O${}^{2-}$ and peroxide [O-O]${}^{2-}$ anions (Fig. 12). The peroxide is responsible for gap states, which significantly reduce the electronic band gap of HfO${}_{2}$ (Fig. 14 (m)). To obtain further insight, we applied the Atoms in Molecules (AIM) theory developed by Bader Bader (1990). Bader charges are +2.5 for Hf, -0.68 for peroxide anion and -1.16 for oxide anion in HfO${}_{3}$ at 110 GPa, which shows a significantly ionic character of bonding. ## IV Conclusions We have systematically predicted stable compounds and crystal structures in the Hf-O system at pressures up to 120 GPa using ab initio evolutionary algorithm USPEX. Several new stable compounds, including _Imm_2-Hf${}_{5}$O${}_{2}$, _C_2/_m_-Hf${}_{3}$O${}_{2}$, _P_$\bar{6}$2_m_-HfO and _Pnnm_-HfO${}_{3}$ are found for the first time. _Pnnm_-Hf${}_{2}$O, which is the new high-pressure phase of Hf${}_{2}$O, is also discovered. HfO${}_{3}$ shows interesting structure, simultaneously containing oxide O${}^{2-}$ and peroxide [O-O]${}^{2-}$ anions. Semimetallic properties of _P_$\bar{6}$2_m_-HfO and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$ are demonstrated through their band structures, as well as low densities of conduction electrons and holes. Our results demonstrate that Hf${}_{3}$O${}_{2}$ is more ductile than other Hf-O compounds, and the hardest compound is HfO instead of OII-HfO${}_{2}$ . The superior mechanical properties of _P_$\bar{6}$2_m_-HfO, such as bulk modulus _B_, shear modulus _G_, Young’s modulus _E_ and hardness _H${}_{v}$_, can be attributed to the peculiar combination of strong Hf-O and Hf-Hf bonds. _Pnnm_-Hf${}_{2}$O, _Imm_2-Hf${}_{5}$O${}_{2}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$ also show excellent mechanical properties. Clearly, high O content is not a key factor affecting the mechanical properties of Hf-O compounds. Suboxides: Hf${}_{6}$O, Hf${}_{3}$O, Hf${}_{1}2$O${}_{5}$ and _P_$\bar{3}$1_m_-Hf${}_{2}$O based on hcp-Hf sublattice provide easy pathways for absorbing or desorbing oxygen. The recognition of the common structural features between _P_$\bar{6}$2_m_-HfO and $\omega$-Hf gives further insight into the physical properties and suggests that HfO can be made as a hard semimetallic coating on $\omega$-Hf substrate. _Pnnm_-Hf${}_{2}$O, _Imm_2-Hf${}_{5}$O${}_{2}$, _P_$\bar{3}$1_m_-Hf${}_{2}$O and _P_$\bar{4}$_m_2-Hf${}_{2}$O${}_{3}$ phases in particular can be quenched to ambient pressure and can be candidates for applications requiring mechanically strong materials. ## V acknowledgments This work was supported by the National Science Foundation (EAR-1114313), DARPA (Grants No. W31P4Q1210008), the Basic Research Foundation of NWPU (No. JCY20130114), the Natural Science Foundation of China (No. 51372203, 51332004), the Foreign Talents Introduction, the Academic Exchange Program of China (No. B08040) and the Government (No. 14.A12.31.0003) of Russian Federation. 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1612.05453	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 36829, "num_imgs": 6, "llama3_tokens_count": 12423 }	[ "content_image/1612.05453/x1.png", "content_image/1612.05453/x2.png", "content_image/1612.05453/x3.png", "content_image/1612.05453/x4.png", "content_image/1612.05453/x5.png", "content_image/1612.05453/x6.png" ]	# Probability Densities of the effective neutrino masses $m_{\beta}$ and $m_{\beta\beta}$ Andrea Di Iura diiura@fis.uniroma3.it Davide Meloni meloni@fis.uniroma3.it Dipartimento di Matematica e Fisica, Università di Roma Tre; INFN, Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome, Italy ###### Abstract We compute the probability densities of the effective neutrino masses $m_{\beta}$ and $m_{\beta\beta}$ using the Kernel Density Estimate (KDE) approach applied to a distribution of points in the $(m_{\min},m_{\beta\beta})$ and $(m_{\beta},m_{\beta\beta})$ planes, obtained using the available Probability Distribution Functions (PDFs) of the neutrino mixing angles and mass differences, with the additional constraints coming from cosmological data on the sum of the neutrino masses. We show that the reconstructed probability densities strongly depend on the assumed set of cosmological data: for $\sum_{j}m_{j}\leq 0.68$ eV at $95\%\ \mathrm{CL}$ a sensitive portion of the allowed values are already excluded by null results of experiments searching for $m_{\beta\beta}$ and $m_{\beta}$, whereas in the case $\sum_{j}m_{j}\leq 0.23$ eV at $95\%\ \mathrm{CL}$ the bulk of the probability densities are below the current bounds. keywords: † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction Although the physics of neutrino oscillation is entering a precision era, with all mixing angles and absolute values of the mass differences measured at the level of some percent, there are still questions related to the nature of neutrinos that need to be answered. Among these, we are interested in whether neutrinos are Majorana or Dirac particles and in the absolute value of their masses. As it is well known, experiments on neutrinoless double beta decays ($0\nu\beta\beta$) consider the possibility that the reaction \[(A,Z)\longrightarrow(A,Z+2)+e^{-}+e^{-}\] (1) really occurs; in the case of positive signal, we could conclude that the total lepton number is violated by two units, although the process behind the conversion of two down quarks into two up quarks would not be uniquely determined [1; 2; 3]. The $0\nu\beta\beta$-decay amplitude has the form $\mathcal{A}(0\nu\beta\beta)=m_{\beta\beta}\mathcal{M}(A,Z)$, where $\mathcal{M}(A,Z)$ is the nuclear matrix element of the decay in Eq. (1) that does not depend on the neutrino masses and mixing parameters, and $m_{\beta\beta}$ is the effective mass which, in the case of three lepton families, is given by \[m_{\beta\beta}\equiv\bigg{\|}\sum_{j}m_{j}U_{ej}^{2}\bigg{\|}= \bigg{\|}\cos^{2}\theta_{13}\Big{(}m_{1}\cos^{2}\theta_{12}+m_{2}\sin^{2}\theta _{12}e^{i\alpha}\Big{)}+m_{3}\sin^{2}\theta_{13}e^{i\beta}\bigg{\|}.\] (2) In the previous relation, $U_{ej}$ are the elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix $U_{\mathrm{PMNS}}$ that encodes the leptonic mixing angles $\theta_{ij}$, whereas the phases $\alpha$ and $\beta$ are the so-called Majorana phases (one of which eventually absorbs the $CP$ violating phase $\delta$). As it is usually done, two of the three neutrino masses $m_{j}$ in Eq. (2) can be expressed in terms of the lightest one $m_{\min}$ in a way that dependents on the supposed neutrino mass hierarchy; for Normal Ordering (NO) we have: \[m_{1}=m_{\min}\qquad m_{2}=\sqrt{m_{\min}^{2}+\Delta m^{2}_{21}}\qquad m_{3}= \sqrt{m_{\min}^{2}+\Delta m^{2}_{31}}\,,\] (3) whereas for the Inverted Ordering (IO) we set: \[m_{1}=\sqrt{m_{\min}^{2}-\Delta m^{2}_{21}-\Delta m^{2}_{32}}\qquad m_{2}= \sqrt{m_{\min}^{2}-\Delta m^{2}_{32}}\qquad m_{3}=m_{\min}\,,\] (4) so $m_{\beta\beta}$ effectively depends on the seven independent parameters $\theta_{12},\theta_{13},\Delta m^{2}_{21},\Delta m^{2}_{31},\alpha,\beta$ and $m_{\min}$. The study of the electron spectrum near the end point in the nuclear reaction ${}^{3}\mathrm{H}\to{{}^{3}\mathrm{He}}+e+\overline{\nu}_{e}$ allows, in presence of neutrino mixing, to get information on the other effective mass largely studied in the literature, $m_{\beta}$, defined by \[m_{\beta}\equiv\sqrt{\sum_{j}m_{j}^{2}\|U_{ej}\|^{2}}=\sqrt{\cos^{ 2}\theta_{13}\Big{(}m_{1}^{2}\cos^{2}\theta_{12}+m_{2}^{2}\sin^{2}\theta_{12} \Big{)}+m_{3}^{2}\sin^{2}\theta_{13}}.\] (5) Since absolute values of the PMNS matrix are taken, complex phases play no role and $m_{\beta}$ only depends on three independent observables and it is somehow correlated, although not in a simple form, with $m_{\beta\beta}$. It is customary to present such a correlation varying all mixing parameters inside their 1, 2 or 3$\sigma$ range ($[0,2\pi]$ for the Majorana phases in any case) and computing the maximum and minimum allowed value. While this procedure certainly gives insights on the possible outcomes of an experimental search, no information whatsoever can be drawn on the probability distribution of the observable itself. So, inspired by the work of [4] and [5], we computed the distributions of $m_{\beta}$ and $m_{\beta\beta}$ and the Credible Regions (CR) as obtained using the available PDFs of $\theta_{12},\theta_{13},\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$, with the additional constraints coming from cosmological data on the sum of the neutrino masses (see also Refs. [6; 7; 8] and [9]). However, unlike the procedure followed in [4] and [5], we use the KDE approach to compute PDFs of the observables in the $2D$ planes $(m_{\min},m_{\beta\beta})$ and $(m_{\beta},m_{\beta\beta})$. The use of such a procedure also allows us to save computation time which could become a critical aspect in this sort of simulations. A short summary of the numerical procedure and the data set we used to get the PDFs from the available data is done in Sect. 2 whereas a short description of the KDE method and the obtained results is done in Sect. 3. Finally in Sect. 4 we compare the PDFs derived from several choices of the Kernel function. ## 2 Numerical procedure and datasets The construction of the PDFs for $m_{\beta}$ and $m_{\beta\beta}$ passes through the extraction of the observables $p=\{\sin^{2}\theta_{12},\sin^{2}\theta_{13},\Delta m^{2}_{21},\Delta m^{2}_{3 \ell}\}$ (with $\ell=1$ for NO and $\ell=2$ for IO) from which they depend; the sampling is based on the knowledge of the likelihoods $\mathcal{L}(p)$ which in turn are functions to the single $\Delta\chi^{2}(p)$: (6) For the observables $p$ (which are only midly correlated), this information is available online at the address http://www.nu-fit.org, where the $\Delta\chi^{2}$ for the November 2016 data, based on the procedure discussed in Ref. [10], are given. Notice that a Bayesian analysis on the 2014 data set is available in Ref. [11]; the authors found that the results generally agree (at the level of one standard deviation) with those of the frequentest method, with some differences involving the atmospheric angle $\theta_{23}$ and the Dirac $CP$ violating phase. However, $\theta_{23}$ does not enter into the expressions of the effective masses and the information on the $CP$ phase is hidden by the presence of the Majorana phases. We then decided to use the most recent data set. For the sake of completeness, we report in Tab. 1 the central values and $3\sigma$ errors for all the observables relevant in neutrino oscillation for both orderings; similar values are also obtained in Ref. [12]. In addition to the oscillation data, our estimate of the PDFs also takes into account the cosmological constraints on the sum of the neutrino masses $\sum_{j}m_{j}$ coming from the Planck experiment [13]. \| Normal Ordering \| Inverted Ordering ---\|---\|--- Parameter \| Best Fit \| 3σ Range \| Best Fit \| 3σ Range sin2θ12/10−1 \| 3.06+0.12−0.12 \| 2.71 ÷ 3.45 \| 3.06+0.13−0.12 \| 2.71 ÷ 3.45 sin2θ13/10−2 \| 2.166+0.075−0.075 \| 1.934 ÷ 2.392 \| 2.179+0.076−0.076 \| 1.953 ÷ 2.408 sin2θ23/10−1 \| 4.41+0.27−0.21 \| 3.85 ÷ 6.35 \| 5.87+0.20−0.24 \| 3.93 ÷ 6.40 δ \| 4.56+0.89−1.02 \| 0 ÷ 2π \| 4.83+0.70−0.80 \| 0 ÷ 2π Δm221/10−5 [eV2] \| 7.50+0.19−0.17 \| 7.03 ÷ 8.09 \| 7.50+0.19−0.17 \| 7.03 ÷ 8.09 Δm23ℓ/10−3 [eV2] \| +2.524+0.039−0.040 \| +2.407 ÷ +2.643 \| -2.514+0.038−0.041 \| -2.635 ÷ -2.399 Table 1: Central values ± the 1σ errors and 3σ ranges for the neutrino mixing parameters as obtained in Ref. Esteban:2016qun (available at the website <http://www.nu-fit.org>). Note that in the last line ℓ=1 for NO and ℓ=2 for IO. The analysis prefers a global minimum for NO with respect to the local minimum of IO, Δχ2=χ2IO−χ2NO=0.83. The Planck Collaboration provides several likelihoods based on different assumptions among which we decide to use the following ones: * a conservative estimate (_set-1_) based on the set of data given by PLANCK TT + lowP + Lensing, which has $\sum_{j}m_{j}\leq 0.68$ eV at $95\%\ \mathrm{CL}$; * a more aggressive one (_set-2_) based on PLANCK TT + lowP + Lensing + Ext, which has $\sum_{j}m_{j}\leq 0.23$ eV at $95\%\ \mathrm{CL}$, with a maximum of the likelihood for $\sum_{j}m_{j}\sim 0.05\ \mathrm{eV}$. The acronyms used above refer to the data on the temperature power spectrum (PLANCK TT), to the Planck polarization data in the low-$\ell$ temperature (lowP), to the data on Cosmic Microwave Background lensing reconstruction (Lensing); with Ext the constraints from Baryon Acoustic Oscillations, Joint Light-curve Analysis of supernovae and the Hubble constant are indicated. For comparison purposes, we show in Tab. 2 the upper limits at 95% CL on the sum of the neutrino masses for different datasets which also include the data from the temperature-polarization cross spectrum (TE) and those from the polarization power spectrum (EE). \| Dataset \| \+ LowP \| \+ Lensing \| \+ Ext ---\|---\|---\|---\|--- ∑jmj [eV] \| TT \| 0.715 \| 0.675 \| 0.234 TT, TE, EE \| 0.492 \| 0.589 \| 0.194 Table 2: Upper bound at 95% confidence level on the sum of the neutrino masses (in eV) using the data of Ref. Ade:2015xua . With these likelihoods at our disposal, we employed the following procedure to accept or reject a given extraction of the set of observables $p$ and Majorana phases (notice that $\delta$ is not relevant because the Majorana phase $\beta$ hides any information on the Dirac $CP$ phase): * we first extract $\sin^{2}\theta_{12}$, $\sin^{2}\theta_{13}$, $\Delta m^{2}_{21}$ and $\Delta m^{2}_{3\ell}$ according to (6); the Majorana phases $\alpha$ and $\beta$ are extracted according to a flat distribution in the interval $[0,2\pi]$; * we then extract the value of $M=\sum_{j}m_{j}$ using the Planck data obtained from Fig. 30 in Ref. [13]; for NO, if $M\leq\sqrt{\Delta m^{2}_{21}}+\sqrt{\Delta m^{2}_{31}}$ (or $M\leq\sqrt{-(\Delta m^{2}_{21}+\Delta m^{2}_{32})}+\sqrt{-\Delta m^{2}_{32}}$ for IO), we reject such an $M$ and extract a new value for the sum of the neutrino masses; * once the value of $\sum_{j}m_{j}$ is accepted, we compute the lightest neutrino mass $m_{\min}$ using the relations * $m_{\min}+\sqrt{m_{\min}^{2}+\Delta m^{2}_{21}}+\sqrt{m_{\min}^{2}+\Delta m^{2} _{31}}=\sum_{j}m_{j}$ for NO * $\sqrt{m_{\min}^{2}-\Delta m^{2}_{21}-\Delta m^{2}_{32}}+\sqrt{m_{\min}^{2}- \Delta m^{2}_{32}}+m_{\min}=\sum_{j}m_{j}$ for IO . Notice that, unless the Planck distributions on $M$ are peaked around 0.06 eV assuming NO and 0.1 eV for IO (which is in fact not the case), this procedure penalizes very small values of $m_{\min}$. Thus Eqs. (2) and (5) are used to get the numerical values of $m_{\beta}$ and $m_{\beta\beta}$. We generate $\mathcal{O}(10^{6})$ realizations that satisfy the constraints discussed above. This order of magnitude is necessary to guarantee a $5\sigma$ coverage for the input parameters, as discussed in Ref. [4]. ## 3 PDF analysis The procedure outlined above produces two-dimensional histograms in the planes $(m_{\min},m_{\beta\beta})$ and $(m_{\beta},m_{\beta\beta})$. In order to compute from them the PDF and CRs, we used the Kernel Density Estimate (KDE) approach [14]. Suppose we have a $d$-dimensional vector $\mathbf{x}$ of observables of which we want to know the PDF, $f(\mathbf{x})$, and suppose also that we have $N$ different realizations of the same observables $\{\mathbf{t}_{j}\}_{j=1}^{N}$ obtained according to the procedure described above; thus $f$ is estimated from \[\hat{f}(\mathbf{x})=\frac{1}{N\prod_{k=1}^{d}h_{k}}\sum_{j=1}^{N} \left[\prod_{k=1}^{d}\mathcal{K}\left(\frac{x^{k}-t_{j}^{k}}{h_{k}}\right) \right]\,,\] (7) where $h_{k}$ is the bandwidth of the $k$-th component of the vector $\mathbf{x}$, whose estimate according to the Scott’s rule of thumb [15] is given by \[\hat{h}_{k}=\left(\frac{4}{d+4}\right)^{1/(d+4)}N^{-1/(d+4)} \sigma_{k}\,,\] (8) $\sigma_{k}$ being the standard deviation of the $k$-th observable $x^{k}$. The Scott’s rule reduces the asymptotic expected value of the integrated square errors between the actual distribution $f$ and the estimated $\hat{f}$. The positive function $\mathcal{K}$ is called _kernel_ and must satisfy the normalization condition \[\int_{\mathbb{R}^{d}}\mathrm{d}^{d}x\ \mathcal{K}(\mathbf{x})=1 \qquad\mathcal{K}(\mathbf{x})\geq 0.\] (9) A simple but equally suited kernel is the Gaussian kernel, defined as: \[\mathcal{K}(\mathbf{x})=\frac{1}{(2\pi)^{d/2}}\exp\left(-\frac{1} {2}\|\mathbf{x}\|^{2}\right)\,,\] (10) that we estimate using the same algorithm of Ref. [16]¹, based on the modified SciPy function gaussian_kde described in Ref. [17]. [FOOTNOTE:1][ENDFOOTNOTE] The results for the PDFs as a function of $\log_{10}m_{\min}\ (m_{\beta})$ and $\log_{10}m_{\beta\beta}$ at the 68%, 95% and 99% CRs obtained with the analysis performed using _set-1_ for the sum of the neutrino masses are shown in Fig. 1 in the $(m_{\min},m_{\beta\beta})$ plane and in Fig. 2 in the $(m_{\beta},m_{\beta\beta})$ plane. The analogous results for _set-2_ are shown in Figs. 3 and 4. In all planes, the excluded region for $m_{\beta\beta}$ is the area above the horizontal magenta dashed line, around $m_{\beta\beta}\geq 0.19\ \mathrm{eV}$[18] obtained using the 90% CL limit on the half-life of ${}^{76}$Ge, $T^{0\nu}_{1/2}(^{76}\mathrm{Ge})>5.2\times 10^{25}\ \mathrm{years}$, in the preliminary analysis of GERDA phase II [19]. A recent result using the ${}^{136}$Xe, $T^{0\nu}_{1/2}(^{136}\mathrm{Xe})>1.07\times 10^{26}\ \mathrm{years}$ at $90\%\ \mathrm{CL}$ obtained by the KamLAND-ZEN experiment [20], gives the lower bound indicated with green dashed lines, which excludes the region $m_{\beta\beta}\geq 0.083\ \mathrm{eV}$[18]. The bounds we quote for $m_{\beta\beta}$ are obtained according to Ref. [21], where the Authors used the results of Ref. [22] for the phase-space factor and those of Ref. [23] for the nuclear matrix elements. In our analysis we fixed the axial coupling constant of the nucleon $g_{A}=1.269$. We also outline that the large uncertainties associated to the nuclear matrix elements $\mathcal{M}(A,Z)$ can modify the prediction for decay amplitude $\mathcal{A}(0\nu\beta\beta)$; however, the impact of such effects are beyond the scope of this paper and will not be analyzed in the following. For the other observable, $m_{\beta}$, the red vertical dashed line indicates the expected sensitivity of the KATRIN experiment ($0.2\ \mathrm{eV}$ at $90\%\ \mathrm{CL}$[24], see https://www.katrin.kit.edu/128.php) and the grey vertical dashed line the expected sensitivity of the Project 8 experiment ($4\times 10^{-2}\ \mathrm{eV}$ at $90\%\ \mathrm{CL}$[25]) which has been especially designed to probe the whole IO parameter space. Notice that the most stringent upper limit on $m_{\beta}$ has been obtained by the Mainz and Troitzk experiments, $m_{\beta}\leq 2.05\ \mathrm{eV}$ at $95\%\ \mathrm{CL}$[26; 27]. To better compare the different cosmological datasets we use the same scale for the PDF densities (which are normalized to one). In the $(m_{\min},m_{\beta\beta})$ plane the maximum is fixed to be 10, while in the $(m_{\beta},m_{\beta\beta})$ plane it is 30. <figure><img src="content_image/1612.05453/x1.png"><figcaption>Figure 1: Reconstructed probability density in the (mmin,mββ) plane assumingNO (left panel) and IO (right panel) using the set-1 prior on ∑jmj. Thecredible regions are at 68% (solid lines), 95% (dashed lines) and 99% (dottedlines) for 2 dof. The horizontal pink dashed line indicates the excludedregion at 90% CL assuming the 76Ge results Agostini:2017iyd , while the greendashed line refers to the 136Xe results KamLAND-Zen:2016pfg .</figcaption></figure> <figure><img src="content_image/1612.05453/x2.png"><figcaption>Figure 2: The same as Fig. 1 but in the (mβ,mββ) plane. With vertical reddashed lines we indicate the expected sensitivity of KATRIN Osipowicz:2001sqand with vertical grey dashed lines the one of Project 8 Doe:2013jfe .</figcaption></figure> <figure><img src="content_image/1612.05453/x3.png"><figcaption>Figure 3: The same as Fig. 1, but for set-2.</figcaption></figure> <figure><img src="content_image/1612.05453/x4.png"><figcaption>Figure 4: The same as Fig. 2, but for set-2.</figcaption></figure> A close inspection at Fig. 1 shows that a large portion of the 68% CR for $m_{\beta\beta}$, the one corresponding to large $m_{\min}$, is already excluded by the KamLAND-ZEN data, for both hierarchies. In practice, this is the consequence of having a non-negligible probability that $\sum_{j}m_{j}\gtrsim 0.5\ \mathrm{eV}$, therefore the value of $m_{\min}$ can be sufficiently large to approach ${\cal O}(0.1)$. On the other hand, _set-2_ relaxes this constraint and the probability density is centered around smaller $m_{\min}$ (and consequently smaller $m_{\beta\beta}$). For both cases, the cosmological bounds on the sum of neutrino masses implies that for NO and IO the low mass region for $m_{\beta\beta}$ is strongly disfavoured. The interesting features of Figs. 2 and 4 is that, for both assumptions on the sum of the neutrino masses, the Project 8 experiment would be able to probe almost the whole allowed regions for $m_{\beta}$ at 99% level whereas KATRIN shows only a modest ability to probe the largest possible values of $m_{\beta}$, around ${\cal O}(10^{-2}-10^{-1})$ eV. Instead of discussing the effects of the cosmological bounds on the effective masses, one can also adopt an opposite point of view, asking what would be the effect on $\sum_{j}m_{j}$ of a possible measure of $m_{\beta\beta}$ at the new generation of experiments, see Refs. [28; 29; 21]. As an example, we can explore the situation that a signal for the $0\nu\beta\beta$-decay is observed at the (near future) CUORE or (next-to-near future) nEXO experiments. Following the discussion of Ref. [21] we assume an optimistic scenario where a signal is in the expected 90% experimental sensitivity region, that is $m_{\beta\beta}=0.073\pm 0.008\ \mathrm{eV}$ (assuming $g_{A}=1.269$) for CUORE [30]. Similar values can also be achieved by GERDA Phase-II [31], MAJORANA-D [32] and NEXT [33] experiments, so our discussion applies equally well to a large number of possible future experiments. In the case of nEXO experiment [34], we set $m_{\beta\beta}=0.011\pm 0.001\ \mathrm{eV}$, which is below the IO region. The results of our finding are shown in Fig. 5 where the frequency of the sum of light neutrino masses (histograms normalized to 1), after the constraints coming from $m_{\beta\beta}$, is displayed. With black dashed lines we also show the Planck PDFs for _set-1_ (upper panels) and _set-2_ (lower panels). In the first column of the plot, which refers to the case $m_{\beta\beta}=0.073\pm 0.008\ \mathrm{eV}$, we clearly see that there exists a cutoff in the distribution in the low mass region due to the fact that $m_{\beta\beta}$ cannot be arbitrarily small, with maxima around $\sum_{j}m_{j}\sim{\cal O}(0.2-0.3)$ eV for both _set-1_ and _set-2_ and for both hierarchies (NO in blue and IO in red). On the other hand, in the high mass region the distributions essentially follow the shape of the Planck priors since the assumed values of $m_{\beta\beta}$ do not impose strong constraints on $m_{\min}$. If we assume a positive signal at the nEXO experiment $m_{\beta\beta}=0.011\pm 0.001\ \mathrm{eV}$, second column of Fig. 5, we see that we cannot distinguish among different Planck datasets since the bound on the $0\nu\beta\beta$-decay effective mass constraints $\sum_{j}m_{j}$ to be of $\mathcal{O}(0.1)\ \mathrm{eV}$. <figure><img src="content_image/1612.05453/x5.png"><figcaption>Figure 5: Frequency of ∑jmj for an assumed mββ=0.073±0.008 eV (left panels)and mββ=0.011±0.001 eV (right panels), for NO (blue) and IO (red). The blackdashed lines are the Planck PDFs: set-1 in the upper panels and set-2 in thelower panels. The darkest areas under the histograms are the 68% credibleregions obtained from the cumulant distributions.</figcaption></figure> ## 4 Discussion and conclusions At first sight, the results described above seem to depend on the choice of the kernel used to estimate the PDFs. However, we have checked that adopting different functions $\mathcal{K}$ the CL regions are not altered in a significant manner. We test different kernels, provided by the scikit-learn package [35]: * _Gaussian_$\mathcal{K}(x;h)\propto\exp(-x^{2}/2h)$ ; * _tophat_$\mathcal{K}(x;h)\propto 1$ for $\|x\|\leq h$ ; * _Epanechnikov_$\mathcal{K}(x;h)\propto 1-x^{2}/h^{2}$ ; * _exponential_$\mathcal{K}(x;h)\propto\exp(-\|x\|/h)$ ; * _linear_$\mathcal{K}(x;h)\propto 1-x/h$ for $\|x\|\leq h$ ; * _cosine_$\mathcal{K}(x;h)\propto\cos(\pi x/2h)$ . The check is performed adopting the $k$_-fold cross-validation_ approach, proposed in Refs. [36; 37]; in few words, the sample of extracted points is split into $k$ smaller sets; of them, $k-1$ sets are used to estimate $f(\mathbf{x})$ according to a given _kernel_ and the resulting model is then validated on the remaining part of the dataset. In Tab. 3 we show our result for the cross-validation analysis with $m_{\rm min}$ and $m_{\beta\beta}$ as independent variables (similar results can be achieved for $m_{\beta}$ and $m_{\beta\beta}$): we analyze ten subsets with $N=\{1000,5000,10000,50000\}$ points, then we average the results. In order to investigate possible overfitting effects, each subset has been divided into two parts: a _train_ ($N_{\rm train}\approx 0.6N$) and a _test_ ($N_{\rm test}=N-N_{\rm train}$) set. We estimate the best bandwidth $\hat{h}$ using twenty $k$-folds in the train dataset. The error ${\cal E_{\rm set}}$ between the actual distribution and the kernel estimate is defined as: \[{\cal E}_{\rm set}=\sqrt{\frac{1}{N_{\rm set}}\sum_{j}^{N_{\rm set }^{1/2}}\sum_{k}^{N_{\rm set}^{1/2}}\left[f(\mathbf{x}_{j,k})-\hat{f}(\mathbf{ x}_{j,k})\right]^{2}}\,,\] (11) where _set_ can be train or test-set. The actual distribution can be obtained from the two dimensional density histogram. We assume for the histogram $N_{\rm set}^{1/2}\times N_{\rm set}^{1/2}$ bins. Notice that the normalization factor $N^{-1/2}_{\rm set}$ in the error (11) is necessary to compare datasets with different dimensions. In Fig. 6 we show our results of $\hat{f}(\mathbf{x})$ for the _set-1_ prior on the sum of neutrino masses and for all kernels introduced above (green shaded area). The PDFs are superimposed on a subset of $5\times 10^{3}$ points. As we can see, the Gaussian kernel as well as the exponential one correctly reproduce the testing dataset for both orderings (for these two cases, the green areas are concentrated below the points and they do not appear in the graphs). For the other kernels, the agreement does not appear to be as good as for the previous ones, since the PDFs extend over regions outside the subset of points. In particular, in Tab. 3 we observe that the errors ${\cal E}_{{\rm set}}$ of the Gaussian and the exponential kernels are roughly one half those of the other kernels. The cross-validation procedure is also useful to compute the best bandwidth $\hat{h}$ that minimizes the residual error between the predictions and the actual values of the sample points. Our findings are compatibles with the Scott’s rule defined in (8), see Tab. 4 for a summary of the bandwidths computed using the same data of the cross-validation analysis. Notice that in the cross-validation a single bandwidth is estimated for each kernel. For the _set-2_ our conclusions remain unaltered: the Gaussian and the exponential kernels reproduce the training dataset with a good accuracy. <figure><img src="content_image/1612.05453/x6.png"><figcaption>Figure 6: PDFs in the plane (mmin,mββ) obtained from the KDE analysis (greenshaded areas) for NO and IO datasets; the blue (red) points are a sample ofdata obtained in the numerical scan.</figcaption></figure> \| \| set-1 \| set-2 ---\|---\|---\|--- \| \| Normal Ordering \| Inverted Ordering \| Normal Ordering \| Inverted Ordering N \| Kernel \| ^h \| Etrain \| Etest \| ^h \| Etrain \| Etest \| ^h \| Etrain \| Etest \| ^h \| Etrain \| Etest 1000 \| Gaussian \| 0.060 \| 0.204 \| 0.225 \| 0.068 \| 0.378 \| 0.421 \| 0.076 \| 0.146 \| 0.181 \| 0.058 \| 0.350 \| 0.430 Tophat \| 0.537 \| 0.403 \| 0.397 \| 0.829 \| 0.606 \| 0.630 \| 0.759 \| 0.335 \| 0.346 \| 0.423 \| 0.531 \| 0.594 Epanechnikov \| 0.537 \| 0.367 \| 0.360 \| 0.829 \| 0.568 \| 0.593 \| 0.759 \| 0.305 \| 0.316 \| 0.423 \| 0.491 \| 0.556 Exponential \| 0.026 \| 0.189 \| 0.233 \| 0.026 \| 0.363 \| 0.415 \| 0.032 \| 0.142 \| 0.178 \| 0.023 \| 0.328 \| 0.423 Linear \| 0.537 \| 0.354 \| 0.347 \| 0.829 \| 0.553 \| 0.580 \| 0.759 \| 0.293 \| 0.306 \| 0.423 \| 0.477 \| 0.543 Cosine \| 0.537 \| 0.465 \| 0.459 \| 0.829 \| 0.695 \| 0.717 \| 0.759 \| 0.363 \| 0.375 \| 0.423 \| 0.625 \| 0.692 5000 \| Gaussian \| 0.043 \| 0.145 \| 0.167 \| 0.033 \| 0.301 \| 0.326 \| 0.046 \| 0.121 \| 0.144 \| 0.042 \| 0.299 \| 0.304 Tophat \| 0.778 \| 0.365 \| 0.386 \| 0.483 \| 0.582 \| 0.577 \| 0.759 \| 0.310 \| 0.343 \| 0.803 \| 0.515 \| 0.503 Epanechnikov \| 0.778 \| 0.338 \| 0.359 \| 0.483 \| 0.544 \| 0.539 \| 0.759 \| 0.289 \| 0.320 \| 0.803 \| 0.487 \| 0.476 Exponential \| 0.016 \| 0.125 \| 0.153 \| 0.014 \| 0.281 \| 0.323 \| 0.020 \| 0.101 \| 0.127 \| 0.016 \| 0.256 \| 0.277 Linear \| 0.778 \| 0.329 \| 0.349 \| 0.483 \| 0.523 \| 0.525 \| 0.759 \| 0.280 \| 0.311 \| 0.803 \| 0.476 \| 0.466 Cosine \| 0.778 \| 0.394 \| 0.417 \| 0.483 \| 0.654 \| 0.653 \| 0.759 \| 0.333 \| 0.367 \| 0.803 \| 0.588 \| 0.577 10000 \| Gaussian \| 0.031 \| 0.137 \| 0.146 \| 0.025 \| 0.295 \| 0.306 \| 0.035 \| 0.105 \| 0.127 \| 0.029 \| 0.272 \| 0.284 Tophat \| 0.583 \| 0.352 \| 0.356 \| 0.356 \| 0.567 \| 0.554 \| 0.692 \| 0.289 \| 0.319 \| 0.607 \| 0.517 \| 0.523 Epanechnikov \| 0.583 \| 0.326 \| 0.331 \| 0.356 \| 0.531 \| 0.518 \| 0.692 \| 0.270 \| 0.299 \| 0.607 \| 0.489 \| 0.497 Exponential \| 0.013 \| 0.111 \| 0.130 \| 0.012 \| 0.272 \| 0.299 \| 0.015 \| 0.084 \| 0.113 \| 0.012 \| 0.232 \| 0.255 Linear \| 0.583 \| 0.317 \| 0.322 \| 0.356 \| 0.519 \| 0.505 \| 0.692 \| 0.262 \| 0.290 \| 0.607 \| 0.479 \| 0.486 Cosine \| 0.583 \| 0.391 \| 0.399 \| 0.356 \| 0.652 \| 0.642 \| 0.692 \| 0.318 \| 0.350 \| 0.607 \| 0.578 \| 0.586 50000 \| Gaussian \| 0.019 \| 0.118 \| 0.115 \| 0.018 \| 0.252 \| 0.274 \| 0.019 \| 0.083 \| 0.089 \| 0.019 \| 0.238 \| 0.252 Tophat \| 0.544 \| 0.342 \| 0.340 \| 0.865 \| 0.584 \| 0.607 \| 0.495 \| 0.267 \| 0.275 \| 0.809 \| 0.516 \| 0.536 Epanechnikov \| 0.544 \| 0.320 \| 0.316 \| 0.865 \| 0.557 \| 0.579 \| 0.692 \| 0.270 \| 0.299 \| 0.607 \| 0.489 \| 0.497 Exponential \| 0.008 \| 0.089 \| 0.097 \| 0.007 \| 0.222 \| 0.251 \| 0.009 \| 0.068 \| 0.079 \| 0.007 \| 0.190 \| 0.212 Linear \| 0.544 \| 0.312 \| 0.308 \| 0.865 \| 0.546 \| 0.568 \| 0.495 \| 0.241 \| 0.248 \| 0.809 \| 0.484 \| 0.502 Cosine \| 0.544 \| 0.369 \| 0.367 \| 0.865 \| 0.630 \| 0.655 \| 0.495 \| 0.297 \| 0.306 \| 0.809 \| 0.562 \| 0.582 Table 3: Mean estimated bandwidth and mean errors for the train and test datasets assuming NO or IO. The results are obtained using a sample of N points performing twenty k-folds cross-validation for the train subset. The error is defined in (11). \| set-1 \| set-2 ---\|---\|--- \| Normal Ordering \| Inverted Ordering \| Normal Ordering \| Inverted Ordering N \| hmmin \| hmββ \| hmmin \| hmββ \| hmmin \| hmββ \| hmmin \| hmββ 1000 \| 0.396 \| 0.407 \| 0.393 \| 0.252 \| 0.434 \| 0.416 \| 0.466 \| 0.203 5000 \| 0.311 \| 0.316 \| 0.300 \| 0.193 \| 0.332 \| 0.319 \| 0.344 \| 0.155 10000 \| 0.279 \| 0.283 \| 0.267 \| 0.173 \| 0.292 \| 0.284 \| 0.314 \| 0.138 50000 \| 0.213 \| 0.217 \| 0.205 \| 0.132 \| 0.223 \| 0.217 \| 0.238 \| 0.105 Table 4: Mean values of the bandwidths evaluated for the Gaussian kernel using the Scott’s rule defined in (8) and the same data of Tab. 3. In conclusions, we have shown that the KDE method is an efficient tool to evaluate the PDFs of interesting physical observables. We have concentrated our efforts on two observables related to neutrino physics, namely the effective neutrino masses $m_{\beta\beta}$ and $m_{\beta}$ which will help to reveal the true nature of neutrinos and the values of their absolute masses. For them, we have computed the Credible Regions using the available PDFs on the mixing angles and mass differences, with the additional constraints coming from cosmological data on the sum of the neutrino masses. We found that the reconstructed probability densities strongly depend on the assumed set of cosmological data and, in particular, for $\sum_{j}m_{j}\leq 0.23$ eV at $95\%\ \mathrm{CL}$ the bulk of the probability densities are below the current bounds on the analyzed observables. 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1711.06092	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35, "num_imgs": 0, "llama3_tokens_count": 12 }	[]	See pages 1-last of tosem2017.pdf
1011.0428	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 72642, "num_imgs": 13, "llama3_tokens_count": 20179 }	[ "content_image/1011.0428/x1.png", "content_image/1011.0428/x2.png", "content_image/1011.0428/x3.png", "content_image/1011.0428/x4.png", "content_image/1011.0428/x5.png", "content_image/1011.0428/x6.png", "content_image/1011.0428/x7.png", "content_image/1011.0428/x8.png", "content_image/1011.0428/x9.png", "content_image/1011.0428/x10.png", "content_image/1011.0428/x11.png", "content_image/1011.0428/x12.png", "content_image/1011.0428/x13.png" ]	# A Model for Thermal Phase Variations of Circular and Eccentric Exoplanets Nicolas B. Cowan¹² , Eric Agol² , [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] ###### Abstract We present a semi-analytic model atmosphere for close-in exoplanets that captures the essential physics of phase curves: orbital and viewing geometry, advection, and re-radiation. We calibrate the model with the well-characterized transiting planet, HD 189733b, then compute light curves for seven of the most eccentric transiting planets: Gl 436b, HAT-P-2b, HAT-P-11b, HD 17156b, HD 80606b, WASP-17b, XO-3b. We present phase variations for a variety of different radiative times and wind speeds. In the limit of instant re-radiation, the light curve morphology is entirely dictated by the planet’s eccentricity and argument of pericenter: the light curve maximum leads or trails the eclipse depending on whether the planet is receding from or approaching the star at superior conjunction, respectively. For a planet with non-zero radiative timescales, the phase peak occurs early for super-rotating winds, and late for sub-rotating winds. We find that for a circular orbit, the timing of the phase variation maximum with respect to superior conjunction indicates the direction of the dominant winds, but cannot break the degeneracy between wind speed and radiative time. For circular planets the phase minimum occurs half an orbit away from the phase maximum —despite the fact that the coolest longitudes are always near the dawn terminator— and therefore does not convey any additional information. In general, increasing the advective frequency or the radiative time has the effect of reducing the peak-to-trough amplitude of phase variations, but there are interesting exceptions to these trends. Lastly, eccentric planets with orbital periods significantly longer than their radiative time exhibit “ringing” whereby the hot spot generated at periastron rotates in and out of view. The existence of ringing makes it possible to directly measure the wind speed (the frequency of the ringing) and the radiative time constant (the damping of the ringing). Subject headings: methods: data analysis — (stars:) planetary systems — ## 1. Introduction Efforts to model the atmospheres of exoplanets fall broadly into two categories: 1) radiative transfer models that make predictions about the planet’s spectral characteristics after making assumptions about the bulk energy transport in the atmosphere (numerical simulations include Iro et al., 2005; Seager et al., 2005; Barman et al., 2005; Burrows et al., 2006; Fortney et al., 2008, while Hansen 2008 developed a series of analytic radiative transfer models, and Madhusudhan & Seager 2009 introduced a Monte Carlo approach for fitting models to observed spectra); 2) hydrodynamic models that make predictions about energy recirculation given assumptions about radiative transfer. The hydrodynamic models run with a relatively simple treatment of radiative transfer may then be post-processed with more detailed radiative transfer (e.g., Burrows et al., 2010). Roughly in increasing order of sophistication, the hydrodynamic models consist of solving the: shallow water equations (Langton & Laughlin, 2007, 2008), equivalent barotropic equations (Cho et al., 2003, 18; Rauscher et al., 2008), primitive equations (Showman & Guillot, 2002; Cooper & Showman, 2005; Showman et al., 2008, 67; Menou & Rauscher, 2009; Rauscher & Menou, 2010; Thrastarson & Cho, 2010), Euler’s equations (Burkert et al., 2005; Dobbs-Dixon & Lin, 2008), or the full Navier Stokes equation (Dobbs-Dixon et al., 2010). Recently, Showman et al. (67) and Lewis et al. (2010) have combined approaches 1 and 2 by simultaneously solving the radiative transfer _and_ hydrodynamical equations. With the exceptions of the hydrodynamical models of Langton & Laughlin (2008), Lewis et al. (2010) and Beaulieu et al. (2010), as well as the radiative transfer models of Iro & Deming (2010), however, these theoretical efforts have been directed towards understanding exoplanets on circular orbits, for which the planet’s power budget is constant. Despite the difficulties in numerically modeling exoplanet atmospheres, the physical parameters that govern their large-scale behavior are thought to be: the radiative timescale (how quickly a parcel of gas radiates away its thermal energy), and the advective timescale (how long it takes for a parcel of gas to move from the planet’s day-side to its night-side). In detail, not every joule of energy is absorbed at the same depth (let alone re-radiated from that depth) and different parcels of gas will take different amounts of time to move from the day-side to the night-side. Nevertheless, lacking detailed observations of these planets, such a simple approach complements more detailed simulations by indicating —at very little computational cost— which ingredients are necessary to properly explain observed behavior. The radiative timescale is primarily determined by the depths at which incident energy from the host star is being absorbed and emitted (which are not necessarily the same). Radiative transfer models suggest that this occurs at pressures of $10^{-3}$–$10^{0}$ bars, where the radiative time scales are roughly $10^{4}$–$10^{5}$ seconds (Seager et al., 2005; Iro et al., 2005; Fortney et al., 2005; Barman et al., 2005; Burrows et al., 2006). Advection timescales are a function of planetary size and wind speed. Different hydrodynamic simulations put the wind speeds at roughly a tenth of the sound speed at the slow end, and transonic at the high end: $\sim 10^{2}$–$10^{3}$ m/s (Showman & Guillot, 2002; Cho et al., 2003; Burkert et al., 2005; Cooper & Showman, 2005, 2006; Langton & Laughlin, 2007; Thrastarson & Cho, 2010). Such wind speeds lead to advective times of $\sim 10^{5}$ s for a Jupiter-sized planet¹. Since the radiative and advective timescales are of the same magnitude, it is not immediately clear how efficient energy recirculation should be for short-period exoplanets. Indeed, hydrodynamic simulations performed by different groups (see above) do not always agree. [FOOTNOTE:1][ENDFOOTNOTE] Fortunately, there is a growing body of observations that constrain theoretical models for short-period planets. Observations of secondary eclipses in exoplanetary systems have made it possible to measure the integrated day-side flux of short-period exoplanets (e.g., Charbonneau et al., 2005; Deming et al., 2005). Observations of phase variations have made it possible to measure the brightness of planets as a function of _local time_(Harrington et al., 2006; Cowan et al., 2007; Knutson et al., 2007, 51; Snellen et al., 2009; Borucki et al., 2009; Knutson et al., 50; Crossfield et al., 2010). Ideally, there would be analytic models to bridge the gap between the detailed simulations and these cutting-edge phase observations. We presented a summary and analysis of the photometric observations of 24 transiting planets in Cowan & Agol (2010). We concluded that short-period planets exhibit a wide range of recirculation efficiencies but are all consistent with low Bond albedos ($A<0.35$). The model used in that paper, however, was too simple to predict the _shape_ of phase variations, only their amplitude. Phase function mapping (Cowan & Agol, 2008) is a largely model-independent technique, but it rests on the assumption that large-scale weather patterns on the planet do not shift with respect to the sub-stellar point. There is theoretical evidence that hot Jupiter atmospheres reach a steady-state diurnal heating pattern over a wide range of pressures that include the likely mid-IR photosphere (2.5 mbar – 19.6 bar, Cooper & Showman, 2005). But this lack of variability may be an artifact of initial conditions (Thrastarson & Cho, 2010). Nevertheless, repeat observations of HD 189733b with _Spitzer_ indicate that the 8 micron day-side variability is less than 2.7% and night-side variability is less than 17% (Agol et al., 2009; Agol et al., 2010). Thus, the steady-state hypothesis seems to be in good shape for the large-scale climate of planets on circular orbits, but it is impossible for this condition to hold true on planets with eccentric orbits². [FOOTNOTE:2][ENDFOOTNOTE] Here we develop an analytic energy transport model—complementary to hydrodynamic models—which offers important insight into the bulk energy flow in the atmospheres of hot Jupiters, and its impact on the primary observable: thermal phase variations. In § 2 we introduce our analytic model and the associated differential equation (DE). We describe general solutions and useful limits of the DE in § 3. In § 4 we explain how we compute disc-integrated light curves. We compute sample light curves in § 5, discuss our model assumptions and how they likely impact our results in § 6, and state our conclusions in § 7. ## 2. Energy Transport Model In this section, we describe a one-dimensional (longitudinal) analytic model of the heating and cooling of a planet. For now we simply state our simplifying assumptions and develop the equations governing our model. In § 6 we discuss these assumptions explicitly. We assume that all of the energy is reflected, absorbed and emitted by a single layer, which should be taken to approximate the planet’s photosphere, or emitting surface. The temperature and —for planets in edge-on orbits— the visibility of parcels of gas is greatest near the equator. Since we are interested only in disc-integrated light curves, we are principally concerned with the motion of the atmosphere near the equator. We assume that the equatorial jet stream has a constant angular velocity with latitude (i.e. solid body rotation. See also Iro et al., 2005). We define this constant effective rotation period for the planet, $\omega_{\rm rot}$, in an inertial frame. Note that from the perspective of a parcel of gas, it doesn’t matter whether this motion is a result of actual rotation of the planet, winds, or both. Winds are usually defined with respect to a rotating core. It is difficult (currently impossible) to determine the rotation rate of a gaseous exoplanet’s core, $\omega_{\rm core}$. Observationally, the only accessible quantity is the sum of the planet’s internal rotation and the winds, which we refer to as the planet’s effective rotation: $\omega_{\rm rot}=\omega_{\rm core}+\omega_{\rm wind}$. Note that the effective rotation of the planet can be trivially converted to an effective rotational period, $P_{\rm rot}=2\pi/\omega_{\rm rot}$, or an equatorial parcel velocity in an inertial frame, $v_{\rm gas}=R_{p}\omega_{\rm rot}$. For the energy budget of the parcel of gas, what matters is not the rotation frequency but rather the related _advective_ frequency, $\omega_{\rm adv}(t)=\omega_{\rm rot}-\omega_{\rm orb}(t)$, where $\omega_{\rm orb}$ is the angular velocity of the planet about its host star, which varies with time for an eccentric orbit. The advective frequency is a measure of how often the parcel of gas is subjected to stellar forcing. A planet with a tidally locked _atmosphere_³ has $\omega_{\rm adv}=0$; super-rotating winds (in the same sense as the planet’s orbital motion) have $\omega_{\rm adv}>0$; sub-rotating winds have $\omega_{\rm adv}<0$. The orbital angular velocity is a maximum at periastron: $\omega_{\rm max}=(2\pi/P)(1-e)^{-3/2}(1+e)^{1/2}$, where $P$ is the planet’s orbital period. Note that $\omega_{\rm max}$ is not the same as the pseudo-synchronous rotational frequency (see § 6 for discussion on this). In this study, we will often express $\omega_{\rm rot}$ in units of $\omega_{\rm max}$ since this immediately indicates whether the planet’s atmosphere is super-rotating or trailing when its day-side is receiving the largest incident flux. [FOOTNOTE:3][ENDFOOTNOTE] ### Energy Transport Differential Equation The flux incident on a planet changes as a function of time if the planet’s orbit is not circular: \[F(t)=\sigma T_{\rm eff}^{4}\left(\frac{R_{}}{r(t)}\right)^{2},\] (1) where $\sigma$ is the Stefan-Boltzmann constant, $T_{\rm eff}$ and $R_{}$ are the star’s effective temperature and radius, and $r(t)$ is the planet–star distance. We assume that parcels of gas move only in the E–W direction, not latitudinally, and we adopt the Lagrangian approach of tracking an individual parcel of gas and its temperature. The flux absorbed by the parcel is $(1-A)F(t)\sin\theta\max(\cos\Phi(t),0)$, where $A$ is the planet’s Bond albedo (which we assume to be constant), $\theta$ is the co-latitude of the parcel ($\theta=0$ at the north pole, $\pi/2$ at the equator, and $\pi$ at the south pole), and $\Phi$ is the angle from the sub-stellar longitude (a.k.a. local time: $\Phi=-\pi/2$ at the dawn terminator; $\Phi=0$ at the substellar meridian; $\Phi=\pi/2$ at the dusk terminator; $\Phi=\pi$ at the antistellar meridian). We further assume that all of the heat absorbed by the parcel goes into increasing its temperature, rather than doing mechanical work. The warmed gas then radiates as an ideal blackbody with temperature $T$ (This is simply Newtonian cooling, a scheme used in many of the hydrodynamic simulations listed in § 1). The differential equation governing the temperature of this parcel of gas as a function of time is: \[\frac{dT}{dt}=\frac{1}{c_{h}}\left((1-A)F(t)\sin\theta\max(\cos\Phi(t),0)- \sigma T^{4}\right),\] (2) where $c_{h}=\rho C_{P}H$ is constant everywhere on the planet, provided that the mass-density, $\rho$, specific heat capacity, $C_{P}$, and the thickness of the parcel, $H$ are planet-wide constants. Since all the incident flux is absorbed (and eventually re-radiated) in a single layer, one must take this layer to be many atmospheric scale-heights thick: including the optical and infrared photospheres. Note that even if the dynamical layer of the atmosphere is a dozen scale heights thick, it is still much smaller than the planetary radius. Setting the left side of Equation 2 to zero, we define the equilibrium temperature for a parcel of gas: \[T_{\rm eq}(\Phi,\theta,t)=\left(\frac{(1-A)F(t)\sin\theta\max(\cos\Phi(t),0)}{ \sigma}\right)^{\frac{1}{4}}.\] (3) A parcel cools if its temperature is greater than $T_{\rm eq}$; it heats up if its temperature is below $T_{\rm eq}$. The temperature extrema of the parcel (and any inflection points) thus occur when its temperature reaches $T_{\rm eq}$. Note that latitude enters the differential equation in the same way as Bond albedo. This means that locating a parcel farther from the equator is precisely equivalent to increasing its Bond albedo: $(1-A)\leftrightarrow\sin\theta$. The energy budget of an eccentric planet is dictated by what happens near periastron. Noting that the time variability of $T_{\rm eq}$ is contained in $\Phi(t)$ and $F(t)$, we define the fiducial temperature along the sub-stellar meridian at periastron (see also Hansen, 2008; Cowan & Agol, 2010): \[T_{0}=T_{\rm eff}(1-A)^{1/4}\sin^{1/4}\theta\sqrt{\frac{R_{}}{a(1-e)}}.\] (4) Following Iro et al. (2005) and Seager et al. (2005), we define the radiative time constant in the vicinity of the photosphere as \[\tau_{\rm rad}=c_{h}/\sigma T_{0}^{3}.\] (5) We define the dimensionless temperature, $\tilde{T}=T/T_{0}$, and dimensionless time⁴, $\tilde{t}=t/\tau_{\rm rad}$, and rewrite Equation 2 as [FOOTNOTE:4][ENDFOOTNOTE] \[\frac{d\tilde{T}}{d\tilde{t}}=\tilde{I}-\tilde{T}^{4},\] (6) where $\tilde{I}=\max(\cos\Phi(t),0)(a(1-e)/r(t))^{2}$ is the normalized intensity of radiation experienced by the parcel of gas. ## 3. Energy Transport Solutions Equation 6 is a first-order, non-linear ordinary differential equation (D.E.) and does not have an analytic solution on the planet’s day-side. On the planet’s night-side, $\tilde{I}\equiv 0$, and the equation admits an analytic solution: \[\tilde{T}=\tilde{T}_{\rm dusk}\left(\frac{1}{3(\tilde{t}-\tilde{t}_{\rm dusk}) \tilde{T}_{\rm dusk}^{3}+1}\right)^{1/3},\] (7) where $\tilde{T}_{\rm dusk}$ and $\tilde{t}_{\rm dusk}$ are the dimensionless temperature and time at sunset. We can —and eventually do— solve the D.E. numerically, but in the remainder of this section we consider simplifying limits that offer insight into the model behavior. Readers primarily interested in the calculated light curves may skip ahead to § 4. ### The Limits of Low and High $\tau_{\rm rad}$ The limiting case of $\tau_{\rm rad}=0$ corresponds to a planet where all the flux is re-radiated from the day-side, regardless of how rapidly the gas parcels are moving. Every parcel of gas in such a case is always right at its equilibrium temperature: $T(\Phi,\theta,t)=T_{\rm eq}(\Phi,\theta,t)$, given by Equation 3. Given our model assumptions, the $\tau_{\rm rad}\gg P_{\rm rot}$ limit corresponds to a planet that emits radiation uniformly at all longitudes. For a planet on an eccentric orbit, the limit comes in two varieties: if $P_{\rm rot}\ll\tau_{\rm rad}\ll P$, the planet instantly re-radiates all incident energy, but uniformly in longitude. This may be thought of as a planet with strong zonal winds, leading to efficient day–night heat circulation: \[T(\theta,t)=T_{\rm eff}\left(\frac{R_{}}{r(t)}\right)^{1/2}\left(\frac{1}{\pi }\right)^{1/4}\sin^{1/4}\theta(1-A)^{1/4}.\] (8) If $P_{\rm rot}\ll P\ll\tau_{\rm rad}$, the radiative time is so long that one probably cannot neglect latitudinal (meridional) transport; it is expedient to assume perfect latitudinal transport. The planet will emit uniformly everywhere on the planet and at the same rate all the time: \[T=T_{\rm eff}\left(\frac{R_{}}{2a}\right)^{1/2}(1-A)^{1/4}(1-e^{2})^{-1/8}.\] (9) ### Circular Orbit Limit In the special case of circular orbits (the majority of short-period exoplanets, for good reason), there are many useful analytic approximations that we can make to simplify the problem. Furthermore, the simpler circular case offers intuition into the behavior of our model. In the circular limit, Equation 6 can be rewritten as: \[{d\tilde{T}\over d\Phi}={1\over\epsilon}\left(\max(\cos\Phi,0)-\tilde{T}^{4} \right),\] (10) where $\epsilon=\tau_{\rm rad}\omega_{\rm adv}$ is a dimensionless constant quantifying the planet’s energy recirculation efficiency⁵, and $\Phi=\omega_{\rm adv}t$. [FOOTNOTE:5][ENDFOOTNOTE] A day in the life of a parcel of gas proceeds as shown in Figure 1. Note that we have included the $\sin^{1/4}\theta$ factor in the y-axis, so that we may plot on the same figure the heating curves for parcels at different latitudes. The two effects of a high $\epsilon$ are 1) a delay in the time of maximum temperature, and 2) higher night-time temperature. Although the observed light curve depends on both the radiative and advective timescales, the heating pattern of a parcel of gas depends only on their ratio, $\epsilon$. The night-time temperature is largely independent of latitude, $\theta$, but depends sensitively on $\epsilon$. The maximum temperature reached by a parcel, on the other hand, depends sensitively on its latitude but only weakly on $\epsilon$. Because of the equivalency of latitude and albedo, the the dashed lines in Figure 1 can be thought of as the heating curves for equatorial parcels of gas, but with $A\approx 30$%: albedo has a more important impact on the day-side heating pattern than on the night-side cooling. Finally, the delay between a parcel passing through the sub-stellar longitude and reaching its maximum temperature, $\Phi_{\rm max}$, depends more sensitively on $\epsilon$ than does the maximum temperature reached, $\tilde{T}_{\rm max}$. In the $\epsilon\to\infty$ limit, $\tilde{T}\sin^{1/4}\theta=(1/\pi)^{1/4}$, as one would expect from Equation 6. The diurnal heating patterns shown in Figure 1 are similar to those measured at the surface of Earth. In both cases parcels of gas move in and out of the sunlight: on Earth this motion is entirely due to the planet’s rotation (wind velocities are small compared to rotational velocity), while on a gaseous planet this motion is due to a combination of rotation and zonal winds. Indeed, our model may also be relevant to a rocky planet with a thin atmosphere and rapid rotation, provided that the rotational period is shorter than the lateral heat conduction timescale. <figure><img src="content_image/1011.0428/x1.png"><figcaption>Figure 1.— Temperature of a parcel of gas in the atmosphere of a short-periodplanet in a circular orbit. The plot runs from dawn (Φ=−π/2) until dusk(Φ=π/2), and back to dawn again (Φ=3π/2).</figcaption></figure> For a planet on a circular orbit, we can use the fact that the temperature extrema for a particle of gas occur when it reaches its equilibrium temperature. The temperature extrema are therefore related to their location on the planet by: \[\tilde{T}_{\rm min}=\cos^{\frac{1}{4}}\Phi_{\rm min}\] (11) and \[\tilde{T}_{\rm max}=\cos^{\frac{1}{4}}\Phi_{\rm max},\] (12) where $\Phi_{\rm min}$ and $\Phi_{\rm max}$ are the angles between the sub-stellar meridian and temperature minimum and maximum, respectively. Since $\tilde{T}_{\rm min}$ is typically much less than unity, $\Phi_{\rm min}\approx-\pi/2$ in most situations (see Figure 2), while $\tilde{T}_{\rm max}$ is close to unity so small differences in this maximum temperature correspond to significant changes in the phase offset of the maximum, as shown in Figure 3. <figure><img src="content_image/1011.0428/x2.png"><figcaption>Figure 2.— The solid line shows the minimum temperature reached by a parcel ofgas, ~Tmin, plotted against the longitude at which the minimum is reached(Φmin=−π/2 at the dawn terminator, Φmin=0 at the sub-stellar point). Thedashed line is the maximum value of ~Tmin=(1/π)1/4, reached in the ϵ→∞ limit.We show the rest of the functional form of Equation 11 as a dotted line forcompleteness.</figcaption></figure> <figure><img src="content_image/1011.0428/x3.png"><figcaption>Figure 3.— The solid line shows the maximum temperature reached by a parcel ofgas, ~Tmax, plotted against the longitude at which the maximum is reached(Φmax=0 at the sub-stellar point, Φmax=π/2 at the dusk terminator). The dashedline is the minimum value of ~Tmax=(1/π)1/4, reached in the ϵ→∞ limit. We showthe rest of the functional form of Equation 12 as a dotted line forcompleteness.</figcaption></figure> Note that the the diurnal heating pattern has a longitudinal asymmetry: parcels of gas are heated faster than they cool, so parcels East of the hot-spot tend to be warmer than those West of the hot-spot. This asymmetry manifests itself in the disc-integrated thermal phase curve of the planet: the offset in the peak of the lightcurve tends to be larger than $\Phi_{\rm max}$. In the Appendix we develop analytic approximations in the circular regime for $T_{\rm max}$, $T_{\rm dusk}$ and $T_{\rm dawn}$ (the maximum temperature reached by a parcel, its temperature at the dusk terminator, and at the dawn terminator). Those analytic approximations are compared to the numerical solutions in Figure 4. For intermediate recirculation efficiencies, $\epsilon\approx 3$, the sunset temperature has a peak. This is due to advection of heat downstream, which becomes more efficient as $\epsilon$ increases, heating the sunset terminator. However, when $\epsilon$ becomes too large, heat is distributed uniformly along a given longitude, causing the temperature to decrease again, hence the peak at intermediate $\epsilon$. <figure><img src="content_image/1011.0428/x4.png"><figcaption>Figure 4.— Dimensionless temperature, ~T=T/T0 at the hot-spot, at dusk and atdawn, as a function of recirculation efficiency, ϵ. The lines show theanalytic expressions presented in the Appendix, while the point symbols showthe result of numerically solving Equation 6. Although the analyticapproximation does a good job over a variety of ϵ, it is only applicable forplanets on circular orbits.</figcaption></figure> ## 4. Applications of Model By combining the energy transport model described above with a geometrical model of the planet’s motions (both orbital and rotational), we now produce model phase variations for an entire planet. We model the planet as a grid of gas parcels: 40 longitude and 20 latitude grid points. We run the calculation for 3 orbital periods, with 1000 time steps per orbit. We adopt a perfectly edge-on orbital geometry in all cases, but otherwise use the appropriate orbital parameters ($a$, $e$, $\omega$), planetary radius ($R_{p}$), and stellar parameters ($R_{}$, $T_{\rm eff}$, $\log g$, [Fe/H]). We can run the calculations for arbitrary wavelengths but only present here the light curves for _Spitzer_ IRAC channel 4 (8 $\mu$m). Since ours is a one-layer model, the shape of the phase variations is essentialy unchanged if one adopts a different waveband. The only free model parameters are $A$, $\tau_{\rm rad}$, and $\omega_{\rm rot}$; in practice we set $A=0$ since the albedo does not significantly affect the _shape_ of the thermal phase variations and it appears that hot Jupiters have low albedos (Rowe et al., 2008; Cowan & Agol, 2010, and references therein). All stellar and planetary data are taken from exoplanet.eu, maintained by Jean Schneider; using numbers from exoplanets.org did not perceptibly change our results. When the stellar data are not available, we have assumed typical parameters for the appropriate spectral class, and solar metallicity. Insofar as we are only concerned with the broadband mid-IR brightnesses of the stars, our results should not depend sensitively on the input stellar parameters. Using the stars’ $T_{\rm eff}$, $\log g$ and [Fe/H], we use the PHOENIX/NextGen stellar spectrum grids (Hauschildt et al., 1999) to determine their brightness temperatures at the observed frequencies. For each waveband, we determine the ratio of the stellar flux to the blackbody flux at that grid star’s $T_{\rm eff}$. We then apply this factor to the $T_{\rm eff}$ of the actual observed star. There are many computational shortcuts that one can use with this analytic model. The orbits are treated as edge-on and the planetary obliquity is assumed to be zero (this is strictly true for the core; for the atmosphere it simply means that the equatorial jet-stream flows in the East-West direction), so the D.E. need only be solved at one latitude on the day-side (e.g., the equator) and those day-side heating curves are easily adjusted for other latitudes via $T_{0}$. The D.E. can then be solved analytically on the planet’s night-side. ### Planet-Integrated Properties The observed flux ratio depends on a combination of orbital factors ($a$, $e$), planetary factors ($A$, $\tau_{\rm rad}$, $\omega_{\rm rot}$), and viewing geometry ($\omega$ and $\alpha$, the usual phase angle: $\alpha=0$ at eclipse, $\alpha=\pi$ at transit). We show schematically in Figure 5 how we combine the orbital, planetary and viewing factors to obtain disc-integrated thermal light curves. The top panel simply shows the planet’s distance from its host star; the second panel shows the planet’s orbital angular velocity; the third panel shows the equilibrium temperature at the sub-stellar point (solid line) and the highest temperature on the model planet (dotted line). The fourth panel shows the total absorbed flux (solid line) and the total emitted flux (dotted line). The fifth panel shows the planet’s illuminated fraction, $f=\frac{1}{2}(1+\cos\alpha)$. The bottom panel shows the planet/star flux ratio at 8 $\mu$m as seen from Earth. <figure><img src="content_image/1011.0428/x5.png"><figcaption>Figure 5.— Diagnostics for HD 17156b, using a model with A=0, τrad=20 hours,ωrot=2ωmax.</figcaption></figure> We start the calculations with all of the parcels at $T=T_{0}$, but the planet reaches a periodic equilibrium in a couple e-folding times (a few $\tau_{\rm rad}$). If the planet had no heat capacity, the dotted lines would perfectly track the solid lines in the third and fourth panels of Figure 5. The effect of a non-zero planetary heat capacity (non-zero $\tau_{\rm rad}$) is: the peak-to-trough swings in emitted flux are damped, and the peak in emission lags the peak in absorption. A planet on a circular orbit reaches a steady state whereby the temperature of a parcel of gas is always the same at a given latitude, $\theta$, and local time, $\Phi$. In other words, the planet’s climate is steady as seen from the host star. This condition cannot occur for an eccentric planet. Instead, we verify that our calculations satisfy a somewhat weaker steady-state condition: the parcels of gas at the same latitude, local time _and orbital phase_ should have the same temperature, orbit after orbit. Since we use radiative timescales much shorter than any of the orbital periods considered here, reaching this steady state is easily achieved by solving the heating curve D.E. for a full three orbits. The thermal contrast ratio between the planet and its host star amounts to a weighted mean of the brightness of the visible hemisphere of the planet: \[\frac{F_{p}}{F_{}}=\frac{1}{\pi B(T_{\rm b})}\left(\frac{R_{p}}{R_{}}\right) ^{2}\oint V(\phi,\theta,t)B(T)d\Omega,\] (13) where $B(T_{\rm b})$ is the star’s intensity given its brightness temperature at this wavelength, $B(T)$ is the blackbody intensity at the temperature $T(\phi,\theta,t)$, and $V(\phi,\theta,t)$ is the visibility of a region on the planet ($V=1$ at the sub-observer point, drops as the cosine of the angle from the observer, and is null on the far side of the planet. For details, see Cowan et al., 2009). ## 5. Model Light Curves Sample light curves for a planet on a circular orbit (HD 189733b) are shown in Figure 6. The minimum in the light curve occurs half an orbit from the maximum, despite the fact that the minimum temperature reached by a parcel of gas is always near the dawn terminator. <figure><img src="content_image/1011.0428/x6.png"><figcaption>Figure 6.— The 8 μm light curve for HD 189733b (Bouchy et al., 2005) for avariety of radiative and advective timescales, and assuming A=0. The verticaldashed and dotted lines mark inferior conjunction (transit) and superiorconjunction (eclipse), respectively. The transit and eclipses have beenremoved for clarity. For non-zero ϵ, the min and max of the light curve willnot necessarily occur at inferior and superior conjunction. The effectiverotation rate of the atmosphere is expressed in units of the orbitalfrequency, ωorb. The peak of the light curve occurs after superior conjunctionfor ωrot<ωorb; it occurs before superior conjunction for ωrot>ωorb.</figcaption></figure> As discussed in Agol et al. (2010), the 8 $\mu$m observations of HD 189733b (phase variations and eclipse offset) indicate an $\epsilon$ of order unity and super-rotating winds, shown by the blue line in Figure 6. Without further information, it is impossible to break the degeneracy between $\tau_{\rm rad}$ and $\omega_{\rm rot}$(this degeneracy has been pointed out in different terms in Cho et al., 18). But as mentioned in § 1, many hydrodynamical simulations of hot Jupiters indicate trans-sonic winds (Cooper & Showman, 2005; Rauscher & Menou, 2010, eg:), making for planet-crossing times of order 10${}^{5}$ seconds ($=30$ hours). This means that the radiative timescale of HD 189733b’s atmosphere is in the tens of hours. For our calculations, we adopt either the $\tau_{\rm rad}=0$ limit, or a fiducial radiative time of 20 hours, and consider three different advective schemes: $\omega_{\rm rot}=0.5\omega_{\rm orb}$ is a trailing wind; it takes parcels of gas two planetary orbits to experience one diurnal cycle. The $\omega_{\rm rot}=1.5\omega_{\rm orb}$ is a slightly super-rotating wind; it again takes parcels of gas two planetary orbits to experience one diurnal cycle. For the fast super-rotating winds ($\omega_{\rm rot}=2.0\omega_{\rm orb}$), parcels of gas experience one diurnal cycle per orbit. The sub-rotating ($\omega_{\rm rot}=0.5\omega_{\rm orb}$) and super-rotating ($\omega_{\rm rot}=1.5\omega_{\rm orb}$) cases have opposite phase offsets but identical maxima and minima. At superior conjunction they have identical contrast ratios, so one cannot distinguish between the two models with secondary eclipse depth alone. In detail, of course, the timing and shape of secondary eclipse _is_ sensitive to the day-side brightness map of the planet (Williams et al., 2006; Rauscher et al., 2007; Agol et al., 2010). Figures 7–13 show the same sample of model phase variations, but for planets on eccentric orbits. We use the same $\tau_{\rm rad}$ (0, 20 hrs) and $\omega_{\rm rot}/\omega_{\rm max}$ (0.5, 1.5, 2.0) as for HD 189733b. For consistency and ease of comparison, all planets pass through periastron at $t=0$, and 8 $\mu$m light curves are shown. In many cases _Spitzer_ light curves have been obtained at this wavelength, and since we use a single-layer model the shapes of the modelled lightcurves would be similar at other wavelengths. Furthermore, we assume $A=0$ across the board. Non-zero albedo would reduce the mean planet/star thermal contrast, but not the _shape_ of the light curves. Furthermore, light curves at multiple wavelengths are necessary to break the degeneracy between albedo and wavelength-dependence: a small DC-offset in a single waveband either indicates that the planet has reflected away a fraction of incident flux, or that the absorbed energy is being re-radiated at other wavelengths. <figure><img src="content_image/1011.0428/x7.png"><figcaption>Figure 7.— The 8 μm light curve for WASP-17b (Anderson et al., 2010) for avariety of radiative and advective timescales, and assuming A=0. The verticaldashed and dotted lines mark inferior conjunction (transit) and superiorconjunction (eclipse), respectively. The transit and eclipses have beenremoved for clarity. The effective rotation rate of the atmosphere isexpressed in units of the maximal orbital frequency, ωmax.</figcaption></figure> <figure><img src="content_image/1011.0428/x8.png"><figcaption>Figure 8.— The 8 μm light curve for Gl 436b (Butler et al., 2004) for avariety of radiative and advective timescales, and assuming A=0. The verticaldashed and dotted lines mark inferior conjunction (transit) and superiorconjunction (eclipse), respectively. The transit and eclipses have beenremoved for clarity. The effective rotation rate of the atmosphere isexpressed in units of the maximal orbital frequency, ωmax.</figcaption></figure> <figure><img src="content_image/1011.0428/x9.png"><figcaption>Figure 9.— The 8 μm light curve for HAT-P-11b (Bakos et al., 2010) for avariety of radiative and advective timescales, and assuming A=0. The verticaldashed and dotted lines mark inferior conjunction (transit) and superiorconjunction (eclipse), respectively. The transit and eclipses have beenremoved for clarity. The effective rotation rate of the atmosphere isexpressed in units of the maximal orbital frequency, ωmax.</figcaption></figure> <figure><img src="content_image/1011.0428/x10.png"><figcaption>Figure 10.— The 8 μm light curve for XO-3b (Johns-Krull et al., 2008) for avariety of radiative and advective timescales, and assuming A=0. The verticaldashed and dotted lines mark inferior conjunction (transit) and superiorconjunction (eclipse), respectively. The transit and eclipses have beenremoved for clarity. The effective rotation rate of the atmosphere isexpressed in units of the maximal orbital frequency, ωmax.</figcaption></figure> <figure><img src="content_image/1011.0428/x11.png"><figcaption>Figure 11.— The 8 μm light curve for HAT-P-2b (Bakos et al., 2007) for avariety of radiative and advective timescales, and assuming A=0. The verticaldotted and dashed lines mark superior conjunction (eclipse) and inferiorconjunction (transit), respectively. The transit and eclipses have beenremoved for clarity.</figcaption></figure> <figure><img src="content_image/1011.0428/x12.png"><figcaption>Figure 12.— The 8 μm light curve for HD 17156b (Fischer et al., 2007) for avariety of radiative and advective timescales, and assuming A=0. The verticaldotted and dashed lines mark superior conjunction (eclipse) and inferiorconjunction (transit), respectively. The transit and eclipses have beenremoved for clarity.</figcaption></figure> <figure><img src="content_image/1011.0428/x13.png"><figcaption>Figure 13.— The 8 μm light curve for HD 80606b (Naef et al., 2001) for avariety of radiative and advective timescales, and assuming A=0. The verticaldotted and dashed lines mark superior conjunction (eclipse) and inferiorconjunction (transit), respectively. The transit and eclipses have beenremoved for clarity. Models run with 3× higher time-resolution yieldsindistinguishable light curves.</figcaption></figure> ### Discussion The black lines (the $\tau_{\rm rad}=0$ case) can be understood purely geometrically: the minimum contrast ratio of zero occurs at transit (marked with a vertical dashed line), since at that phase only the planet’s night side is visible. The maximum contrast ratio —on the other hand— does not necessarily coincide with superior conjunction (the vertical dotted line) for non-zero eccentricity. If the incident flux is increasing at superior conjunction ($\|\omega\|<90^{\circ}$), the maximum in the light curve occurs somewhat late (after eclipse); if the incident flux is decreasing at superior conjunction ($90^{\circ}<\omega<270^{\circ}$), the peak occurs early (before superior conjunction). Including advection complicates this picture: super-rotating and trailing winds have the effect of shifting the peak early and late, respectively (see first Figure 6). The direction of the winds at periastron (whether $\omega_{\rm rot}$ is greater than or less than $\omega_{\rm max}$) determines the offset of the thermal phase peak from the $\tau_{\rm rad}=0$ case. We present the planets in order of increasing eccentricity, which also roughly corresponds to increasing orbital period. WASP-17b (Figure 7) has a very “circular” phase variation in the $\tau_{\rm rad}=0$ case, but the symmetry between the $\omega_{\rm rot}/\omega_{\rm max}=0.5$ and 1.5 cases is broken for non-zero radiative time. Gl 436b, HAT-P-11b and XO-3b (Figures 8, 9 and 10) show how the argument of periastron, $\omega$, is critical in determining the shape of the phase variations: the periods and eccentricities of the three planets differ, but they all have periastron shortly after superior conjunction and therefore strikingly similar light curves. For comparison, Deming et al. (2007) and Demory et al. (2007) measured 8 $\mu$m secondary eclipse depths of 5.7(8)$\times 10^{-4}$ and 5.4(7)$\times 10^{-4}$, respectively, for Gl 436b. The multi-band eclipse depths of Stevenson et al. (2010) indicate the planet has a hot day-side; this is most consistent with the $50\times$ solar metalicity model of Lewis et al. (2010), which exhibits a super-rotating equatorial jet. The 8 micron model phase curves presented by Lewis et al. (2010) are similar to our super-rotating solutions (the blue and green curves in Figure 8), and also undershoot the observed eclipse depths. HAT-P-2b, HD 17156b and HD 80606b (Figures 11, 12 and 13) have such high eccentricities that even their $\tau_{\rm rad}=0$ lightcurves look nothing like that of a circular planet. Furthermore, their orbital periods are sufficiently long that —for non-zero $\tau_{\rm rad}$— they exhibit “ringing” as the hot spot generated at periastron rotates in and out of view ($1/\omega_{\rm adv}<\tau_{\rm rad}<P$). The envelope is $\sim\exp((t-t_{\rm peri})/\tau_{\rm rad})$, and the period of the oscillations is simply the planet’s effective rotation period, $P_{\rm rot}=2\pi/\omega_{\rm rot}$. In effect, the ringing caused by flash-heating is not dissimilar to the variability seen in some hydrodynamic simulations due to precession of polar vortices (Cho et al., 2003, 18). For comparison, Langton & Laughlin (2008) predicts an 8 $\mu$m phase variation $F_{\rm max}/F_{\rm min}$ of 2.8 for HAT-P-2b, similar to our values of $\sim 2$ for super-rotating winds (see Table 1 for a summary of our results). For HD 17156b, Irwin et al. (2008) predicts $F_{\rm max}/F_{\rm min}$ of 2.9, while Iro & Deming (2010) predicts 3.3; our models predict somewhat greater phase variations, $F_{\rm max}/F_{\rm min}\sim 5$. Note that if periastron occurs near inferior conjunction, our model predicts that the maximum contrast ratio may actually be _greater_ for non-zero $\tau_{\rm rad}$ (e.g., Figure 12). Observations of HD 80606b show an 8 $\mu$m eclipse depth of 1.0(2)$\times 10^{-3}$(Laughlin et al., 2009), similar to our sub-rotating model (red line in Figure 13). For comparison, Langton & Laughlin (2008) predicts phase variations with $F_{\rm max}/F_{\rm min}=2.1$, _much_ smaller that our values of $\sim 60$. Note that they have $\tau_{\rm rad}=4.5(2)$ hours at the 8 $\mu$m photosphere, but the planet never cools to below a contrast ratio of $\sim 4\times 10^{-4}$, despite the planet’s 112 day orbit. This discrepancy is largely due to our implicit assumption of no internal heat: after spending over 100 days far from its host star, our model planet cools to $\sim 400$ K. This is still much hotter than the remnant heat of formation (50–100 K), but cooler than the 720 K night-side equilibrium temperature used for the hydrodynamic simulations of Langton & Laughlin (2008). Meanwhile, Iro & Deming (2010) predicts $F_{\rm max}/F_{\rm min}=4.2$. The peak to trough amplitudes of our models are within a factor of 2 of the Langton & Laughlin (2008) and Iro & Deming (2010) simulations. If we adopt the $\tau_{\rm rad}=4.5(2)$ hour of Langton & Laughlin (2008), we obtain peak-to-trough amplitudes of $\sim 3\times 10^{-3}$, still a factor of $\sim 2$ larger than their models. That factor of 2 is significant: it indicates that —unlike the Langton & Laughlin (2008) and Iro & Deming (2010) simulations— in our model the planet can completely radiate away it’s heat over a single planetary orbit. In general, non-zero heat capacity reduces the maximum contrast ratio and increases the minimum, making for a flatter light curve. Furthermore, the phase amplitude generally diminishes for larger advective frequencies: the peak-to-trough amplitude is smaller for $\omega_{\rm rot}=1.5\omega_{\rm max}$ than for $\omega_{\rm rot}=2\omega_{\rm max}$ across the board. For planets with $90^{\circ}<\omega<270^{\circ}$, however, a second order effect may also take place: the light curve maximum _increases_ for moderate $\tau_{\rm rad}$ as this allows the planet to hold onto absorbed power until its hot-spot has rotated into our view (see Figure 12). ## 6. Model Assumptions Agnostic about Tidal Locking: Short-period exoplanets on circular orbits should have tidally locked cores, because the tidal locking timescale is considerably shorter than the age of most planetary systems (e.g., Lubow et al., 1997). By the same token, planets on eccentric orbits should have cores in pseudo-synchronous rotation (roughly speaking, this means being tidally locked at periastron, when the tidal forces are strongest). Unlike detailed hydro simulations that use the planet’s rotation rate as an input parameter, our model is agnostic about precise prescriptions for pseudo-synchronous rotation frequency, $\omega_{\rm ps}$(e.g. Hut, 1981; Ivanov & Papaloizou, 2007). For example, at a moderate eccentricity of $e=0.3$, the Hut (1981) formulation gives $\omega_{\rm ps}\approx 0.8\omega_{\rm max}$ , while the Ivanov & Papaloizou (2007) formulation gives $\omega_{\rm ps}\approx 1.4\omega_{\rm max}$. These details about the planet’s interior are not important for our purposes, however, because the planets in our study have thick gaseous envelopes covering their cores and hydrodynamic simulations indicate that wind-speeds in the vicinity of the photosphere are comparable to the rotational velocity. As a result, the parcels of gas are probably not stationary with respect to the sub-stellar point, even for (presumably tidally locked) planets on circular orbits. As a first-order model we treat them as advecting longitudinally at a constant angular velocity. No Internal Heating: Tidal circularization is undoubtedly responsible for the generally low eccentricity of short-period planets (e.g. Ford & Rasio, 2008, and references therein). It is not clear, however, that tidal heating has an observable effect on light curves. For example, Jackson et al. (2008) calculate present-day tidal heating between $2$ and $5\times 10^{17}$ W for Gl 436b, two orders of magnitude less power than the stellar insolation, which ranges between $7$ and $13\times 10^{19}$ W. For short-period planets, the observational signature of internal heat (either remnant gravitational energy from formation or ongoing tidal heating) will likely be an inflated radius rather than additional emergent flux. If hot Jupiters exist for which the tidal heating is a sizeable fraction of the insolation, then our model will have under-estimated the zeroeth-order (DC) component of their lightcurves. By the same token, we also neglect the remnant heat of formation, which is likely less than 100 K for a Gyrs-old planet (Burrows et al., 2006). One important consequence of neglecting any internal heating and having a simple one-layer model, is that the night-side equilibrium temperature is zero. For the most part this is not important, since parcels pass through the illuminated hemisphere often enough to never get very cool. The exception is for our models of HD 80606b, where the planet spends such a long time ($>100$ days) far from its host star. It is possible that the real planet never cools to the 400 K seen in our toy model. No Heat Conduction: Our model neglects conduction of heat from one cell to its neighbors. As such, our model is only relevant when advection and radiation are the dominant energy transport mechanisms. This is probably the case for hot Jupiters, which are thought to have strong winds. If heat conduction _is_ important on hot Jupiters, then they may exhibit smaller day–night temperature contrasts and the relation between the thermal phase offset and the thermal phase amplitude will be broken (see first Equation 12). No Mechanical Work: Our model assumes that all the incident energy from the host star is either reflected or absorbed, and the energy that is absorbed only goes into increasing the temperature of the parcels of gas. In effect, this amounts to neglecting the work term in the First Law of Thermodynamics. In the steady-state, the power absorbed by the planet must be balanced by the power emitted. That is to say, the ultimate effect of work is to warm up parcels of gas (e.g., parcels speed up to super-sonic velocities, shock and deposit energy as heat; Goodman, 2009). This implicitly neglects four effects that may very well be occurring: energy may instead go into latent heat to vaporize particulates; thermal tides (Arras & Socrates, 2010); increasing the atmospheric scale height (Guillot & Showman, 2002); or speeding up winds. Any of these “work” terms would tend to dampen the temperature fluctuations experienced by a parcel of gas: they act as energy sinks near periastron and may act as energy sources near apoastron. As such our model will under-estimate gas response time, over-estimate the temperature excursions, and hence the peak-to-trough amplitude of disc-integrated lightcurves. The planet most likely to show these effects is HD 80606b, since realistically much of the energy absorbed at periastron must go into speeding up winds, inflating the atmosphere, etc. Indeed, the peak-to-trough phase amplitudes we compute for this planet are roughly a factor of 2 greater than those from hydrodynamic simulations (Langton & Laughlin, 2008). No Latitudinal Advection: Our toy model neglects advection towards the planet’s poles: effectively we have zonal but no meridional winds. At low (mbar) pressures, there almost certainly is meridional flow away from the sub-stellar point, but the low presures mean that only a small fraction of absorbed power is moved in this way. But even when meridional flows are not efficient, latitudinal heat transport may still occur in the form of eddy fluxes (e.g., Cho et al., 18; Cho, 2008), which we have also implicitly neglected. Efficient equator–pole heat transport on a planet would have the effect of reducing the DC component of the phase curve. In the limit of perfect latitudinal heat transport, the equatorial temperature is suppressed by a factor $(\pi/4)^{1/4}\approx 0.94$ and the disc-integrated temperature is only suppressed by $(3\pi^{2}/32)^{1/4}\approx 0.98$ compared to the nominal (no latitudinal transport) case. This latitudinal dilution is (necessarily) smaller than the longitudinal dilution. As pointed out in Budaj (2010), latitudinal energy transport is only important for exoplanets that are not edge-on (we get a better view of their pole), and in those cases the unknown planetary radius is a confounding factor. Single Wind Velocity: Our model uses a single angular wind velocity, which can roughly be thought of as the velocity of the equatorial jet stream. We are neglecting the jet-streams at mid-latitudes that transport energy in the opposite direction (Dobbs-Dixon et al., 2010). We have assumed that —to first order— the velocity of the equatorial jet is the only speed that affects the observed phase variations of a planet, but the speed of waves may also be important (e.g., Watkins & Cho, 2010). In any case, our $\omega_{\rm rot}$ can be thought of as the visibility-weighted mean of the competing zonal jets, which is necessarily slower than the velocity of the equatorial jet. Note that our adoption of constant _angular_ velocity amounts to solid body rotation (see also Iro et al., 2005). As far as simplifications go, solid-body rotation is attractive because it naturally produces smaller linear velocities near the planet’s poles than near the equator, a generic feature of many hydrodynamic simulations. That being said, the assumption is undoubtedly wrong in detail and one could instead parameterize the zonal flow as having a constant _linear_ rather than angular velocity: $\omega_{\rm rot}(\theta)=\omega_{\rm rot}\sin\theta$. Such a parameterization could be normalized to yield the same overall heat transport, but it would smear out thermal phase curve features, since parcels of gas flash-heated at periastron would not move in lockstep across the face of the planet. Constant Wind Velocity: We assume that the wind speed does not change with time. The existence of steady-state, E-W jet streams is supported by hydrodynamic simulations of planets on circular orbits (Showman et al., 67; Rauscher & Menou, 2010; Dobbs-Dixon & Lin, 2008, and references therein). But the simulations of Cho et al. (2003) and Thrastarson & Cho (2010) indicate that these results may be a function of initial conditions, in which case even planets on circular orbits may exhibit wandering equatorial jets. For planets on eccentric orbits, the wind speed of the equatorial jet may very well slow down as the planet’s energy budget decreases, but the timing of the primary peak should only depend on the wind speed near pericenter. If the winds damp out near apastron, then the maximum wind velocity may occur shortly after periastron, depending on the inertia of planet’s atmosphere. In effect, our model works in the limit of large inertia, so that the winds do not speed up or slow down even as the planet’s energy budget changes. There is very little energy to transport when the planet reaches apoastron, so the wind speed at that point in the orbit does not have observable consequences. Constant Albedo: Atmospheric albedo is thought to increase with decreasing temperature. This can be quite sudden if, for example, clouds form high in the atmosphere (eg, Kane & Gelino, 2010, and references therein). As mentioned in Langton & Laughlin (54), the formation of clouds on an eccentric planet as it moves away from periastron will simply exacerbate its $1/r^{2}$ energy budget, making the periastron passage even more important for thermal light curves. Note that this is the only of our model assumptions that could lead us to _under_-estimate the amplitude of phase variations. Blackbody Emitters: While we treat the parcels of gas as blackbody emitters, the _shapes_ of our lightcurves should not change much if the planet has a non-blackbody spectrum. The shape of the light curves is preserved provided that the mid-IR flux increases as a simple function of temperature. This is likely to be true on the Rayleigh-Jeans tail of the planet’s spectral energy distribution. Vertically Isothermal: Using a single temperature at each position on the planet (ie: a single-layer model) implicitly means assuming that the temperature at the emitting surface is roughly equal at all wavelengths of interest. In the case of highly irradiated planets, the day-side temperature-pressure profile is shallower than for an isolated giant planet. Depending on how much advection of heat occurs —and at what depths— this may not be true of the night-side. Observationally, the day-side brightness temperatures of hot Jupiters often differ from one infrared waveband to another. But the only hot Jupiter with thermal phase curves observed at multiple wavelengths is HD 189733b, and the _shape_ of the 8 and 24 micron phase curves are indistinguishable, given the uncertainties (Knutson et al., 51). This is notable since 1-D atmospheric models of this planet place the 8 micron photosphere at more than twice the pressure of the 24 micron photosphere. Note that this coincidence for HD 189733b only means that the _ratio_ of the radiative and advective times is the similar at the 8 and 24 micron photospheres. For eccentric planets the degeneracy between those two timescales is broken, so seeing the same morphology at two different wavebands means that _both_ the radiative and advective timescales are the same at the two photospheres. If the phase variation morphology is grossly different from one waveband to another, then one would need a model with different radiative and advective times as a function of waveband (or equivalently, a multi-layer model). ## 7. Conclusions We have presented a semi-analytic model for the heating pattern of gas in the atmosphere of a hot Jupiter. Our two physically-motivated parameters are the planet’s radiative timescale, and its wind speed. This model is meant to calculate the shape of thermal phase variations of transiting exoplanets with any eccentricity or argument of periastron. We have assumed zero albedo and blackbody-like emission throughout: these assumptions have an effect on the DC offset of our phase variations (e.g., the secondary eclipse depth), but should not significantly affect the shape of the phase variations (timing of the phase peak, peak-to-trough amplitude, etc.). Planet \| τrad \| ωrot/ωmax \| Amplitudeb \| Δtcmax ---\|---\|---\|---\|--- HD 189733b \| 0 hrs \| N/A \| 4.9×10−3 \| 0.0 hrs \| 20 hrs \| 0.5 \| 2.9 \| 5.3 hrs \| 20 hrs \| 1.5 \| 2.9 \| -5.3 hrs \| 20 hrs \| 2.0 \| 1.9 \| -7.2 hrs WASP-17b \| 0 hrs \| N/A \| 4.9×10−3 \| 0.27 hrs \| 20 hrs \| 0.5 \| 3.9 \| 5.8 hrs \| 20 hrs \| 1.5 \| 2.4 \| -2.4 hrs \| 20 hrs \| 2.0 \| 1.8 \| -4.7 hrs Gl 436b \| 0 hrs \| N/A \| 8.0×10−4 \| 1.8 hrs \| 20 hrs \| 0.5 \| 6.5 \| 5.0 hrs \| 20 hrs \| 1.5 \| 2.4 \| -24 hrs \| 20 hrs \| 2.0 \| 1.7 \| -31 hrs HAT-P-11b \| 0 hrs \| N/A \| 5.1×10−4 \| 2.6 hrs \| 20 hrs \| 0.5 \| 9.4 \| 6.7 hrs \| 20 hrs \| 1.5 \| 2.7 \| -2.2 hrs \| 20 hrs \| 2.0 \| 1.8 \| -4.7 hrs XO-3b \| 0 hrs \| N/A \| 2.6×10−3 \| 1.6 hrs \| 20 hrs \| 0.5 \| 3.9 \| 3.8 hrs \| 20 hrs \| 1.5 \| 1.5 \| -42 hrs \| 20 hrs \| 2.0 \| 1.4 \| -48 hrs HAT-P-2b \| 0 hrs \| N/A \| 2.1×10−3 \| -3.9 hrs \| 20 hrs \| 0.5 \| 4.1 \| 3.0 hrs \| 20 hrs \| 1.5 \| 2.4 \| -5.1 hrs \| 20 hrs \| 2.0 \| 2.1 \| -6.6 hrs HD 17156b \| 0 hrs \| N/A \| 5.9×10−4 \| -82 hrs \| 20 hrs \| 0.5 \| 4.9 \| -68 hrs \| 20 hrs \| 1.5 \| 5.4 \| -95 hrs \| 20 hrs \| 2.0 \| 5.3 \| -99 hrs HD 80606b \| 0 hrs \| N/A \| 3.0×10−3 \| 1.8 hrs \| 20 hrs \| 0.5 \| 63.5 \| 7.2 hrs \| 20 hrs \| 1.5 \| 56.1 \| 26 hrs \| 20 hrs \| 2.0 \| 51.0 \| 21 hrs aafootnotetext: We adopt A=0 for all these calculations. The shape of the phase variations is not affected by albedo, to first order.bbfootnotetext: For τrad=0, this refers to the maximum flux ratio, Fmax/F∗: the minimum flux ratio is always zero in that case, so the peak-to-trough amplitude is equal to the peak value. For the non-zero τrad, the amplitude refers to the _ratio_ of the maximum and minimum of the light curve, Fmax/Fmin.ccfootnotetext: Time offset between the maximum planet/star contrast and superior conjunction. Negative values mean that the maximum occurs _before_ eclipse; positive values mean that maximum occurs _after_ eclipse. Table 1Calculated Phase Variations of Eccentric Planets Our model is sufficient to fit the shape of phase variations for HD 189733b, a hot Jupiter on a circular orbit. The radiative and advective times of our “best fit” model for that planet are within the range of plausible parameters from detailed hydrodynamic simulations. We bracket these parameters and compute light curves for a selection of eccentric transiting planets. Our calculated 8 $\mu$m light curves are summarized in Table 1. Our principal findings from this numerical experiment are: 1. For planets on circular orbits, the location on the planet of the primary hot spot is intimately related to its amplitude. 2. The minimum temperature of a parcel occurs at or shortly after dawn, but the minimum in the disc-integrated phase variations occurs when the primary hotspot is on the far side of the planet rather than when the dawn terminator is facing the observer. 3. For a circular orbit, the timing of the phase variation maximum with respect to superior conjunction indicates the direction of the dominant winds, but cannot break the degeneracy between wind speed and radiative time. 4. For eccentric planets —in the limit of no advection— the light curve maximum leads or trails superior conjunction depending on whether the planet is receding from or approaching the star at superior conjunction. 5. Planets with non-zero radiative times have their thermal phase peaks offset early for super-rotating winds, and late for sub-rotating winds. 6. For planets with non-zero $\tau_{\rm rad}$, the peak-to-trough phase amplitude generally decreases with increasing $\omega_{\rm rot}$; for planets on circular orbits this can be understood in terms of the dimensionless advective efficiency, $\epsilon$. 7. For planets that are approaching their star at superior conjunction or planets on circular orbits, increasing $\tau_{\rm rad}$ has the effect of reducing the light curve maximum. 8. Planets that are approaching their star at superior conjunction also exhibit stronger phase variations if they have sub-rotating (rather than super-rotating) winds at periastron. This is because the hot spot created at periastron rotates out of view more slowly. 9. For planets receding from their star at superior conjunction, an increased $\tau_{\rm rad}$ may _increase_ the light curve maximum. 10. Eccentric planets with orbital periods significantly longer than $\tau_{\rm rad}$ exhibit “ringing” whereby the hot spot generated at periastron rotates in and out of view. 11. The existence of ringing makes it possible to explicitely measure both $\omega_{\rm rot}$ (the frequency of the ringing) and $\tau_{\rm rad}$ (the damping of the ringing). As discussed in § 6, the myriad assumptions made in our model will tend to exaggerate the amplitude of thermal phase variations. This could lead, for example, to over-estimating the radiative time if one uses our models to interpret observed phase variations. An obvious direction for the future is to couple our simple thermodynamic model to a 1-D radiative transfer code (e.g. Iro & Deming, 2010) to produce time-variable _spectra_ of eccentric planets rather than just light curves. Such a hybrid solution would require far fewer computational resources than full time-variable radiative transfer. N.B.C. acknowledges useful discussions with E. Rauscher, as well as N. Kaib for his help with orbital dynamics, and H.M. Haggard for his help with differential equations. S.L. Hawley and V.S. Meadows contributed useful comments to the manuscript. N.B.C. was supported by the Natural Sciences and Engineering Research Council of Canada. E.A. is supported by a National Science Foundation Career Grant. Support for this work was provided by NASA through an award issued by JPL/Caltech. N.B.C. acknowledges the hospitality of the Harvard-Smithsonian Center for Astrophysics, and the Kavli Institute for Theoretical Physics, where portions of this work were completed. This research was supported in part by the National Science Foundation under Grant No. NSF PHY05-51164. ## Appendix In the slow cooling regime ($\epsilon\gg 1$), the changes in temperature during each orbit are small, so one can treat Equation 10 perturbatively. The trial solution is $\tilde{T}=\tilde{T}_{0}+\delta\tilde{T}$, where $\tilde{T}_{0}=\pi^{-1/4}$ is a constant equal to the average temperature and $\delta\tilde{T}$ contains all of the variability. Substituting this into Equation 10 and noting that $\max(\cos\Phi,0)=\frac{1}{2}(\cos\Phi+\|\cos\Phi\|)$, we get: \[{d\tilde{T}\over d\Phi}={1\over\epsilon}\left(\frac{1}{2}\left[\cos{\Phi}+\| \cos{\Phi}\|\right]-\frac{1}{\pi}-4\tilde{T}_{0}^{3}\delta\tilde{T}\right),\] (14) where we have only kept terms up to first-order in $\delta\tilde{T}$. The solution to this equation is \[\tilde{T}_{\rm day} = \frac{3}{4}\tilde{T}_{0}+{\gamma\cos{\Phi}+\sin{\Phi}\over \epsilon(1+\gamma^{2})}+{e^{-\gamma\Phi}\over 2\epsilon(1+\gamma^{2})\sinh{( \pi\gamma/2)}}\] (15) \[\tilde{T}_{\rm night} = \frac{3}{4}\tilde{T}_{0}+{e^{-\gamma(\Phi-\pi)}\over 2\epsilon(1+ \gamma^{2})\sinh{(\pi\gamma/2)}},\] (16) \[\gamma = {4\tilde{T}_{0}^{3}\over\epsilon}.\] (17) Note that the day-side phase peak (middle term) has the right limits: for large $\epsilon$, the variation is as $\sin{\Phi}$ (slow-cooling regime), while for small $\epsilon$, the variations is as $\cos{\Phi}$ (instant-cooling limit). In the circular case, the longitude of the peak of the temperature, $\Phi_{\rm max}$, goes from $0$ for $\epsilon=0$ to $\pi/2$ for $\epsilon\rightarrow\infty$ (but note that in practice the parcel’s heating curve becomes flat before the the peak reaches $\pi/2$, as shown in Figure 3). We compute $\Phi_{\rm max}$ numerically, and fit $\tan(\Phi_{\rm max}(\epsilon))$ with the function \[f(\epsilon)=x_{0}(1+x_{1}\epsilon^{-x_{2}+(x_{3}/(1+x_{4}\epsilon))})^{-1},\] (18) for which we find a good fit for $x_{0}=2.9685$, $x_{1}=7.0623$, $x_{2}=1.1756$, $x_{3}=-0.2958$, and $x_{4}=0.1846$. This then gives the maximum temperature as a function of $\epsilon$ as \[\tilde{T}_{\rm max}(\epsilon)=\cos^{1/4}{\Phi_{\rm max}(\epsilon)}\approx\cos^ {1/4}{\left(\tan^{-1}{f(\epsilon)}\right)}.\] (19) The maximum temperature goes from $1$ at small $\epsilon$ (instant re-radiation) to $\pi^{-1/4}$ at large $\epsilon$ (uniform temperature). We have computed $T_{\rm max}$ numerically, shown with the asterixes in Figure 4, while Equation 19 is plotted as the solid line; as can be seen the agreement is quite good. The next critical characteristics are the temperatures at dawn and sunset — these determine the temperature of the atmosphere at the terminator, where it absorbs starlight during transit. We can estimate the sunset temperature as follows. For very small values of $\epsilon$, the flux absorbed on the day-side is emitted nearly instantly, so heating balances cooling, and $\tilde{T}_{\rm day}(\epsilon\ll 1)\approx\cos^{1/4}\Phi$. However, near sunset, $\Phi=\pi/2$, the temperature drop becomes so significant that $d\tilde{T}_{\rm day}/d\Phi$ becomes comparable to the cooling rate, $-\cos{\Phi}$. At this point the planet’s temperature is no longer determined by instant re-radiation of the absorbed heat; instead we need to take into account the cooling timescale. For small $\epsilon$ this occurs very close to $\pi/2$, so we can estimate the temperature at sunset as the time when $\epsilon d\tilde{T}_{\rm day}(\epsilon\ll 1)\approx-\cos{\Phi}$. This gives $\epsilon d[\cos^{1/4}{\Phi}]/d\Phi=-\frac{1}{4}\epsilon\cos^{-3/4}{\Phi}\sin{ \Phi}\approx-\cos{\Phi}$. Since $\Phi\approx\pi/2$ and $\tilde{T}\approx\cos^{1/4}\Phi$, this gives $\tilde{T}_{\rm dusk}\approx(\epsilon/4)^{1/7}$. This is a very weak dependence on $\epsilon$. For large values of $\epsilon$, recirculation is very efficient, so $\tilde{T}_{\rm dusk}\approx\pi^{-1/4}$. Combining these two limits, we find the approximate formula $\tilde{T}_{\rm dusk}\approx[\pi^{2}(1+y_{0}/\epsilon)^{-8}+y_{1}\epsilon^{-8/7 }]^{-1/8}$, where $y_{0}=0.69073$ and $y_{1}=7.5534$, which is plotted as the dotted line in Figure 4, as well as numerically computed data points (diamonds); again the agreement is quite good. Finally, the minimum in temperature occurs at (or very close to) the dawn terminator after the gas has cooled all night and before it starts warming again on the day side. 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1706.09463	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 27704, "num_imgs": 0, "llama3_tokens_count": 9945 }	[]	# Froggatt-Nielsen mechanism in a model with $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$ gauge group Katri Huitu¹ Niko Koivunen² Helsinki Institute of Physics and Department of Physics, P.O. Box 64 (Gustaf Hällströmin katu 2), FI-00014 University of Helsinki, Finland, [FOOTNOTE:][ENDFOOTNOTE] [FOOTNOTE:†][ENDFOOTNOTE] March 2, 2024 ###### Abstract The models with the gauge group $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$ (331-models) have been advocated to explain why there are three fermion generations in Nature. As such they can provide partial understanding of the flavour sector. The hierarchy of Yukawa-couplings in the Standard Model is another puzzle which remains without compelling explanation. We propose to use Froggatt-Nielsen -mechanism in a 331-model to explain both fundamental problems. It turns out that no additional representations in the scalar sector are needed to take care of this. The traditional 331-models predict unsuppressed scalar flavour changing neutral currents at tree-level. We show that they are strongly suppressed in our model. flavour, 331-models, Froggatt-Nielsen -mechanism HIP-2017-15/TH _Introduction._ Understanding the number of generations has been attempted using models with an extended gauge group $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$[1; 2; 3; 4; 5]. In these so-called 331-models the chiral anomalies are cancelled if the numbers of triplets and antitriplets are equal. This is possible only if the number of generations is three. Although the 331-models give a possible explanation for the number of generations, they do not shed any light on vastly different masses among the generations. One possibility is to explain the masses dynamically within the Froggatt-Nielsen -mechanism (FN) [6]. So far no attempt to combine these two approaches has been made to the best of our knowledge. We propose here a model, where a combination of scalars in the model emulate the flavon of the FN mechanism‡. We emphasize that this requires special properties from the model, as explained later. [FOOTNOTE:‡][ENDFOOTNOTE] We construct here a model, where the gauge symmetry of a 331-model is broken by three scalar triplets, which is the minimum number of triplets to break the symmetry and generate tree-level masses for all the charged fermions. An attractive feature of our model is that the $SU(3)_{L}\times U(1)_{X}$ breaking scale is at the same time the scale of flavour symmetry breaking. _Particle content in the 331-model._ The symmetry breaking of a 331-model is more involved than in the case of the Standard Model (SM), since two stage breaking is needed: $SU(3)_{L}\times U(1)_{X}\to SU(2)_{L}\times U(1)_{Y}\to U(1)_{em}$. Using the three diagonal generators $T_{3},\,T_{8}$ and $X$, the electric charge of particles can be defined as \[Q=T_{3}+\beta T_{8}+X.\] (1) Different values of $\beta$ have been considered in literature. $\beta=\pm 1/\sqrt{3}$[1; 2; 3] and $\beta=\pm\sqrt{3}$[4; 5] lead to integer charges for gauge bosons and scalars, see _e.g._[8]. The proposed mechanism will be found for $\beta=\pm\frac{1}{\sqrt{3}}$, with which there are no exotic electric charges. With $\beta=-\frac{1}{\sqrt{3}}$ only two types of scalar triplets with neutral components can be formed. In order to generate tree-level masses for all the fermions, three scalar triplets are needed [2]. In our model we use \[\eta=\left(\begin{array}[]{c}\eta^{+}\\ \eta^{0}\\ {\eta^{\prime}}^{+}\end{array}\right),\quad\rho=\left(\begin{array}[]{c}\rho^{ 0}\\ \rho^{-}\\ {\rho^{\prime}}^{0}\end{array}\right),\quad\chi=\left(\begin{array}[]{c}\chi^{ 0}\\ \chi^{-}\\ {\chi^{\prime}}^{0}\end{array}\right),\] (2) where $\eta\sim(1,3,\frac{2}{3})$ and $\rho,\chi\sim(1,3,-\frac{1}{3})$. The numbers in the parantheses label the transformation properties under the gauge group $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$. All the neutral fields can in general develop a non-zero vacuum expectation value (VEV). Degenerate minima are related to each other by $SU(3)_{L}$ rotation. Thus, the most general combination of VEVs can be written as \[\langle\eta^{0}\rangle=\frac{v^{\prime}}{\sqrt{2}},\,\langle\rho^{0}\rangle= \frac{v_{1}}{\sqrt{2}},\,\langle\rho^{\prime 0}\rangle=\frac{v_{2}}{\sqrt{2}}, \,\langle\chi^{\prime 0}\rangle=\frac{u}{\sqrt{2}}.\] (3) Here $v_{2}$ and $u$ break the $SU(3)_{L}\times U(1)_{X}\to SU(2)_{L}\times U(1)_{Y}$, and thus we expect $v_{2},u\gg v^{\prime},v_{1}$. As required by anomaly cancellation, the same number of triplets and antitriplets is needed. Here this is achieved by assigning the leptons and one of the quark families to $SU(3)_{L}$ -triplets, while two quark families belong to antitriplets. The leptons are given by \[L_{L,i}=\left(\begin{array}[]{c}\nu_{i}\\ e_{i}\\ \nu^{\prime}_{i}\end{array}\right)_{L}\sim(1,3,-\frac{1}{3}),\quad e_{R,i}\sim (1,1,-1),\] (4) where $i=1,2,3$. The fields $\nu^{\prime}_{i}$ are new neutral leptons. For $\beta=\pm\sqrt{3}$, in the minimal case the triplet includes charged lepton $e^{c}_{i}$ instead of $\nu_{i}^{\prime}$, and no charged singlets are required. However, as discussed later, the scalar structure of such model does not suit our purposes. We choose to assign first quark generation into triplet and the second and the third into antitriplet: \[Q_{L,1}=\left(\begin{array}[]{c}u_{1}\\ d_{1}\\ U\end{array}\right)_{L}\sim(3,3,\frac{1}{3}),\] (5) \[Q_{L,2}=\left(\begin{array}[]{c}d_{2}\\ -u_{2}\\ D_{1}\end{array}\right)_{L},Q_{L,3}=\left(\begin{array}[]{c}d_{3}\\ -u_{3}\\ D_{2}\end{array}\right)_{L}\sim(3,3^{\ast},0),\] \[u_{R,i},U_{R}\sim(3,1,\frac{2}{3}),\,d_{R,i},D_{R,1},D_{R,2}\sim (3,1,-\frac{1}{3}),\] where $i=1,2,3$. We have introduced new quarks $D_{1}$ and $D_{2}$ with electric charge $-1/3$ and $U$ with electric charge $2/3$. _Froggatt-Nielsen -mechanism._ Froggatt and Nielsen proposed a new symmetry, so-called flavour symmetry (_e.g._ U(1) or $Z_{N}$), to be responsible for the huge differences in fermion masses in the SM [6]. The FN charge assignment is such that the SM Yukawa-couplings are forbidden, but the following effective operator is allowed: \[c^{f}_{ij}\left(\phi/\Lambda\right)^{n^{f}_{ij}}\bar{\psi}_{L,i}^{f}Hf_{R,j}+h .c.,\] (6) where $c^{f}_{ij}$ is a dimensionless order-one number, $\Lambda$ is the scale of new physics, $\psi_{i}$, $f_{i}$ are the SM fermions, $H$ is the SM Higgs doublet and $\phi$ is a complex scalar field called flavon. The SM Yukawa couplings are generated as effective couplings when flavon gets a VEV \[y^{f}_{ij}=c^{f}_{ij}\left(\langle\phi\rangle/\Lambda\right)^{n^{f}_{ij}}.\] (7) Our purpose here is to mimick the gauge singlet flavon by a combination of the existing scalars in the model. _Froggatt-Nielsen -mechanism in the 331-model._ We introduce a global $U(1)_{FN}$-symmetry to our model. Fermions and scalars are charged under it. Two of the three triplets in Eq.(2) carry necessarily the same $U(1)_{X}$ charge. For our purposes this is crucial, since the combination $\rho^{\dagger}\chi$ is a gauge singlet. Therefore, it can act as a flavon field when developing a VEV, if the FN charge of the combination does not vanish. We take the FN charge of scalar triplets as follows: \[q_{\eta}=-1,\quad q_{\rho}=1,\quad q_{\chi}=0.\] (8) Note that the scalar representations generating tree-level masses for fermions in models with $\beta=\pm\sqrt{3}$[4; 5] cannot form an $SU(3)_{L}\times U(1)_{X}$ singlet with an FN charge. Thus in these models, an enlarged scalar sector is needed for FN mechanism. The effective flavon obtains a non-zero vacuum expectation value \[\langle\rho^{\dagger}\chi\rangle=(v_{2}u)/2.\] (9) The relevant effective operator is now \[(c_{s}^{f})_{ij}\left(\frac{\rho^{\dagger}\chi}{\Lambda^{2}}\right)^{(n^{s}_{f })_{ij}}\bar{\psi}_{L,i}^{f}Sf_{R,j}+h.c.,\] (10) where $(c_{s}^{f})_{ij}$ is a dimensionless order-one number. The fermion (anti)triplets and singlets $\bar{\psi}_{L,i}^{f}$ and $f_{R,j}$ have FN charges $q(\bar{\psi}_{L,i}^{f})$ and $q(f_{R,j})$, respectively. $S$ denotes any of the three scalar triplets $\eta$, $\rho$ or $\chi$ with which the combination is a gauge singlet. The $(n^{s}_{f})_{ij}$ is determined by the FN charge assignment, \[(n^{s}_{f})_{ij}=\left[q(\bar{\psi}_{L,i}^{f})+q(f_{R,j})+q(S)\right].\] (11) Here $v_{2}\sim u$ is responsible for both the $SU(3)_{L}\times U(1)_{X}$ breaking and of the flavour symmetry breaking, while $\Lambda$ is the scale of new physics. For simplicity we have chosen to work with charge assignments that keep $(n^{s}_{f})_{ij}$ non-negative. If $(n^{s}_{f})_{ij}$ were negative, we would include $\chi^{\dagger}\rho$ instead of $\rho^{\dagger}\chi$. As one expands (10) around the minimum one obtains \[(y^{f}_{s})_{ij}\bar{\psi}_{L,i}^{f}(S+\langle S\rangle)f_{R,j}+( n^{s}_{f})_{ij}(y^{f}_{s})_{ij}\] (12) \[\times\left[\frac{{\rho^{\prime}}^{0\ast}}{v_{2}}+\frac{{\chi^{ \prime}}^{0}}{u}+\frac{v_{1}{\chi}^{0}}{v_{2}u}\right]\sqrt{2}\bar{\psi}_{L,i} ^{f}\langle S\rangle f_{R,j}+h.c.+\cdots,\] where we have kept only the renormalizable contributions. The first term gives the fermion masses and the usual Yukawa interaction: \[(y^{f}_{s})_{ij}=(c^{f}_{s})_{ij}\left(\frac{v_{2}u}{2\Lambda^{2}}\right)^{(n^ {s}_{f})_{ij}}\equiv(c^{f}_{s})_{ij}\epsilon^{(n^{s}_{f})_{ij}}.\] (13) The other term of (12) is the artefact of the Froggatt-Nielsen -mechanism introducing a source of flavour violation into the model. The flavour violating term is suppressed by the $SU(3)_{L}$ breaking scale, $u$ and $v_{2}$, in the case of the SM fermions. Therefore, the flavour violation coming from this additional term is negligible. _Symmetry breaking and the bosonic sector._ The most general scalar potential of our model, consistent with Froggatt-Nielsen -mechanism is: \[V_{\textrm{FN}}=\mu^{2}_{1}\eta^{\dagger}\eta+\mu_{2}^{2}\rho^{ \dagger}\rho+\mu_{3}^{2}\chi^{\dagger}\chi+\lambda_{1}(\eta^{\dagger}\eta)^{2} +\lambda_{2}(\rho^{\dagger}\rho)^{2}\] \[+\lambda_{3}(\chi^{\dagger}\chi)^{2}+\lambda_{12}(\eta^{\dagger} \eta)(\rho^{\dagger}\rho)+\lambda_{13}(\eta^{\dagger}\eta)(\chi^{\dagger}\chi)\] \[+\lambda_{23}(\rho^{\dagger}\rho)(\chi^{\dagger}\chi)+\widetilde{ \lambda}_{12}(\eta^{\dagger}\rho)(\rho^{\dagger}\eta)+\widetilde{\lambda}_{13} (\eta^{\dagger}\chi)(\chi^{\dagger}\eta)\] \[+\widetilde{\lambda}_{23}(\rho^{\dagger}\chi)(\chi^{\dagger}\rho) +\sqrt{2}f(\epsilon_{ijk}\eta^{i}\rho^{j}\chi^{k}+h.c.).\] One scalar and seven pseudoscalar massless degrees of freedom are needed for the missing polarization states of the gauge bosons of the model, namely the $Z,W^{\pm}$ of the SM, new heavy charged gauge bosons, a heavy neutral gauge boson, and a non-Hermitian heavy neutral gauge boson. Thus, in the physical spectrum one has four scalars, two pseudoscalars, and two charged Higgs bosons. The global $U(1)_{FN}$ -symmetry is spontaneusly broken by the scalar field VEVs, and one of the pseudoscalars will be a massless Goldstone boson. In order to give a mass for the Goldstone boson, we add the following soft FN symmetry breaking term to the potential: \[V_{\textrm{soft}}=b(\rho^{\dagger}\chi)+h.c.\] (14) One of the scalar eigenvalues is given by light VEVs corresponding to the experimentally detected Higgs boson mass. The remaining three scalars as well as the charged Higgses are heavy with masses proportional to large VEVs and decouple. One of the physical pseudoscalars is heavy, and the mass of the pseudo-Goldstone boson, $A_{2}$, is proportional to the soft parameter $b$. _Fermion masses in the model._ The charged lepton masses are generated by the term: \[\mathcal{L}\supset y^{e}_{ij}\bar{L}^{\prime}_{L,i}\eta e^{\prime}_{R,j}+h.c.,\] (15) where prime denotes gauge eigenstate. The mass matrix is given by \[\mathcal{L}_{\ell-mass}=m^{e}_{ij}\bar{e}^{\prime}_{L,i}e^{\prime}_{R,j}+h.c. \,\,{\textrm{with}}\,\,m^{e}_{ij}=(y^{e}_{\eta})_{ij}\frac{v^{\prime}}{\sqrt{2 }}.\] The mass matrix is proportional to the Yukawa-coupling matrix. Both will be diagonalized simultaneously. As for neutrinos, with the fields in (4), one of the neutrinos gets mass only radiatively, at the same time raising degeneracy from the masses of the other two. Such an approach is demonstrated _e.g._ in [2]. This is the approach we adopt here. Tree-level masses can also be produced, if new neutral fermions $N_{R,i}$ are introduced [9]. The up-type quark masses are generated by the following terms: \[\mathcal{L}\supset \sum_{\gamma=1}^{4}(y^{u}_{\rho})_{1\gamma}\bar{Q}^{\prime}_{L,1} \rho~{}u^{\prime}_{R,\gamma}+\sum_{\gamma=1}^{4}(y^{u}_{\chi})_{1\gamma}\bar{Q }^{\prime}_{L,1}\chi~{}u^{\prime}_{R,\gamma}\] (16) \[+\sum_{\alpha=2}^{3}\sum_{\gamma=1}^{4}(y^{u}_{\eta^{\ast}})_{ \alpha\gamma}\bar{Q}^{\prime}_{L,\alpha}\eta^{\ast}~{}u^{\prime}_{R,\gamma}+h.c.\] The up-quark mass matrix elements in the gauge eigenbasis are given by \[\mathcal{L}_{up-mass}=\bar{u}^{\prime}_{L}m^{u}u^{\prime}_{R}+h.c .\quad\textrm{with}\] (17) \[m^{u}_{1\gamma}=\frac{v_{1}}{\sqrt{2}}(y^{u}_{\rho})_{1\gamma}, \quad m^{u}_{\alpha\gamma}=-\frac{v^{\prime}}{\sqrt{2}}(y^{u}_{\eta^{\ast}})_{ \alpha\gamma},\] (18) \[m^{u}_{4\gamma}=\frac{v_{2}}{\sqrt{2}}(y^{u}_{\rho})_{1\gamma}+ \frac{u}{\sqrt{2}}(y^{u}_{\chi})_{1\gamma},\] (19) where $\alpha=2,3$ and $\gamma=1,2,3,4$. The down-quark mass matrix and the terms that generate it are given in the Appendix. In 331-setting the hierachy of the matrix elements is determined by the FN charge assignment and different VEVs, in contrast to traditional FN mechanism, in which the hierarchy is set solely by the charge assignment. The hierarchy of VEVs, $u,v_{2}\gg v_{1},v^{\prime}$, greatly affects the hierarchy in the matrix. We rewrite the elements of the mass matrix to clarify the hierarchy: \[m^{u}_{1\gamma} = \frac{v^{\prime}}{\sqrt{2}}\left[\frac{v_{1}}{v^{\prime}}(c_{\rho }^{u})_{1\gamma}\right]\epsilon^{a_{1}^{u}+q(u_{R,\gamma})},\] (20) \[m^{u}_{\alpha\gamma} = \frac{v^{\prime}}{\sqrt{2}}\left[-(c_{\eta^{\ast}}^{u})_{\alpha \gamma}\right]\epsilon^{a_{\alpha}^{u}+q(u_{R,\gamma})},\quad\alpha=2,3,\] (21) \[m^{u}_{4\gamma} = \frac{v^{\prime}}{\sqrt{2}}\left[(c_{\rho}^{u})_{1\gamma}\epsilon ^{q(\rho)-q(\chi)}+(c_{\chi}^{u})_{1\gamma}\frac{u}{v_{2}}\right]\epsilon^{a_{ 4}^{u}+q(u_{R,\gamma})},\] where the quantities in square brackets are order-one numbers, and therefore the hierarchy is completely set by the powers of $\epsilon$. The $a^{u}_{\gamma}$ are : \[a^{u}_{1}=q(\bar{Q}_{L,1})+q(\rho),\quad a^{u}_{\alpha}=q(\bar{Q }_{L,\alpha})+q(\eta^{\ast}),\quad\alpha=2,3,\] \[a^{u}_{4}=(\log\epsilon)^{-1}\log\left(\frac{v_{2}}{v^{\prime}} \right)+q(\bar{Q}_{L,1})+q(\chi).\] (22) The difference between two symmetry breaking scales manifests itself as effective left-handed charges, analogous to FN charges of the left-handed fermion doublets in the original FN mechanism. This way, the textures of the diagonalization matrices can be easily obtained. _Suppression of flavour changing neutral currents._ As explicitly shown in Eq.(16), the quarks couple to multiple scalar triplets, and 331-models generally predict scalar mediated flavour changing neutral currents (FCNC) of quarks at tree-level [10], without a natural suppression mechanism. Although this is not the main motivation in our current work, we shortly discuss the suppression of FCNCs in the model, and return to this in another work [9]. The gauge sector in our model remains the same as in other 331-models, and the GIM-mechanism [11] has been studied in 331-models previously [12]. Although most of the scalars are heavy, the FCNC mediated by the lightest Higgs, $h$, or by the pseudo-Goldstone boson $A_{2}$ can be large. Here we assume that $b$ is large, which suppresses the FCNC by $A_{2}$, and concentrate on the Higgs boson. In our model the charged lepton generations are assigned to same representations. The only flavour violating leptonic couplings to the neutral scalars are due to the FN -mechanism, coming from the second term in Eq.(12). However, it is suppressed by the large $SU(3)_{L}$-breaking VEVs, and is negligible. The Yukawa interactions of quarks with the lightest Higgs boson can be written as \[\mathcal{L}_{q,Yukawa} = \frac{1}{\sqrt{2}}\bar{u}_{L}U_{L}^{u}{\Gamma^{\prime}}^{u}_{h}U_ {R}^{u\dagger}u_{R}h+\frac{1}{\sqrt{2}}\bar{d}_{L}U_{L}^{d}{\Gamma^{\prime}}^{ d}_{h}U_{R}^{d\dagger}d_{R}h,\] where the quark gauge eigenstates have been rotated to the physical states by $U_{L,R}^{u,d}$ matrices. We ignore here the heavily suppressed FN contributions to Yukawa interactions of the quarks. The physical Yukawa couplings are $\Gamma^{u}_{h}=U_{L}^{u}{\Gamma^{\prime}}^{u}_{h}U_{R}^{u\dagger}$ and $\Gamma^{d}_{h}=U_{L}^{d}{\Gamma^{\prime}}^{d}_{h}U_{R}^{d\dagger}$. By using the explicit form for the coupling matrices and approximating the $h$ eigenvector, we can write the coupling matrix in the form: \[(\Gamma^{u}_{h})_{ij} = \sqrt{2}\frac{m_{j}}{v_{SM}}\left[\delta_{ij}+\alpha_{1}(U_{L}^{u })_{i1}(U_{L}^{u\dagger})_{1j}-(U_{L}^{u})_{i4}(U_{L}^{u\dagger})_{4j}\right.\] (23) \[+\left.\alpha_{2}(U_{L}^{u})_{i1}(U_{L}^{u\dagger})_{4j}+\alpha_{ 3}(U_{L}^{u})_{i4}(U_{L}^{u\dagger})_{1j}\right],\] where $\alpha_{i}=\mathcal{O}(v_{light}/v_{heavy}),\,i=1,2,3$, with $v_{\textrm{light}}=v^{\prime},v_{1}$, and $v_{\textrm{heavy}}=u,v_{2}$. The corresponding couplings $(\Gamma^{d}_{h})_{ij}$ follow a similar pattern. We assume \[m^{q}_{i,j}\leq m^{q}_{i+1,j},\] (24) which can be ensured by a proper choice of FN charges. Rotation matrix elements then satisfy $(U^{u}_{L})_{ij}\sim\epsilon^{\lvert a^{u}_{i}-a^{u}_{j}}\rvert$, which provides additional suppression to FCNCs. For a concrete example, we set $\epsilon=0.23$, and consider the following FN charge assignments for the left-handed quark triplets: $q(Q^{c}_{L,1})=2$, $q(Q^{c}_{L,2})=1$, $q(Q^{c}_{L,3})=-1$. This charge assignment will produce the correct texture for the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The FN charges of the right-handed quarks are: $q(u_{R,1})=5$, $q(u_{R,2})=2$, $q(u_{R,3})=0$, $q(u_{R,4})=0$, $q(d_{R,1})=7$, $q(d_{R,2})=5$, $q(d_{R,3})=4$, $q(d_{R,4})=3$ and $q(d_{R,5})=2$. The up-type Yukawa-matrix texture becomes: \[\Gamma^{u}_{h}\sim\left(\begin{array}[]{llll}y_{u}&y_{c}[\alpha\epsilon^{1}]&y _{t}[\alpha\epsilon^{x}]&\epsilon^{2}\\ y_{u}[\alpha\epsilon^{1}]&y_{c}&y_{t}[\alpha\epsilon^{x}]&\epsilon^{2}\\ y_{u}[\alpha\epsilon^{x}]&y_{c}[\alpha\epsilon^{x}]&y_{t}&1\\ y_{u}[\alpha]&y_{c}[\alpha]&y_{t}[\epsilon^{x}]&\epsilon^{x}\end{array}\right),\] (25) where $x=(\log\epsilon)^{-1}\log\left(\frac{v^{\prime}}{v_{2}}\right)-2\geq 0$ due to Eq.(24), and $\alpha=\mathcal{O}(v_{light}/v_{heavy})$. The diagonal couplings of the SM quarks are the SM Yukawa-couplings and the off-diagonal elements are suppressed by the ratio of the two VEV scales and the powers of $\epsilon$. Similar texture applies for the down-type Yukawa-matrix. We find that the heavy VEVs$\sim{\cal{O}}$(5 TeV) can sufficiently suppress all the meson mixing constraints $M^{0}-\bar{M}^{0}$, $M=K,B_{d},B_{s},D$, which provide the tightest constraints in our case. The texture in Eq.(25) produces exotic quark masses that are suppressed compared to the $SU(3)_{L}$-breaking scale: $m_{U}\sim\epsilon^{2}v_{heavy}$, $m_{D_{1}}\sim\epsilon^{3}v_{heavy}$ and $m_{D_{2}}\sim\epsilon^{0}v_{heavy}$. The experimental mass limit for exotic quarks is ${\mathcal{O}}$(1 TeV) [13], and thus $v_{heavy}$ should be somewhat larger than required by the suppression of FCNCs. The texture for the SM quark masses are: $m_{u}\sim\epsilon^{8}v_{light}$, $m_{c}\sim\epsilon^{4}v_{light}$, $m_{t}\sim\epsilon^{0}v_{light}$, $m_{d}\sim\epsilon^{8}v_{light}$, $m_{s}\sim\epsilon^{5}v_{light}$ and $m_{b}\sim\epsilon^{2}v_{light}$. The CKM-matrix is not a square matrix in this model, but a $4\times 5$-matrix. \[\mathcal{L}_{gCC}=\frac{g_{3}}{\sqrt{2}}\bar{u}_{L}\gamma^{\mu}V_{CKM}^{331}d_ {L}{W}^{+}_{\mu}+h.c.\] (26) The CKM-matrix texture in our example is: \[V_{CKM}^{331}\sim\left(\begin{array}[]{ccccc}1&\epsilon^{1}&\epsilon^{3}& \epsilon^{1}\alpha&\epsilon^{3}\alpha\\ \epsilon^{1}&1&\epsilon^{2}&\alpha&\epsilon^{2}\alpha\\ \epsilon^{3}&\epsilon^{2}&1&\epsilon^{-2}\alpha&\alpha\\ \alpha&\alpha&\epsilon^{-1}\alpha&\epsilon^{-4}\alpha^{2}&\epsilon^{-2}\alpha^ {2}\\ \end{array}\right).\] (27) The $3\times 3$-block in the upper-left corner corresponds to the CKM-matrix of the Standard Model. The $W_{\mu}$-boson couples to the exotic quarks and they contribute to the neutral meson mixing. We find that the $W$-mediated meson mixing is always subleading to Higgs-mediated mixing. For a numerical example we set the $SU(3)_{L}$-breaking VEVs to be $u=48$ TeV and $v_{2}=55$ TeV, and the $SU(2)_{L}$-breaking VEVs are $v_{1}=100$ GeV and $v^{\prime}=237.05$ GeV. The exact quark mass matrices are given in the appendix and they produce the experimentally measured quark masses [13]. The exotic quark masses become: $m_{U}=5$ TeV, $m_{D_{1}}=1.295$ TeV and $m_{D_{2}}=50.9$ TeV. The physical up-type Yukawa-coupling matrix is: (28) Similar Yukawa-coupling matrix can be found for the down-type quarks. The CKM matrix is: \[\lvert V_{CKM}^{331}\rvert=\left(\begin{array}[]{ccccc}0.974&0.226&0.00332&0.0 00014&0.000048\\ 0.23&0.97&0.0434&0.00010&0.000086\\ 0.007&0.0430&0.997&0.017&0.0001\\ 0.00073&0.0016&0.057&0.00099&6.7\times 10^{-6}\\ \end{array}\right).\] (29) The SM CKM matrix elements are produced correctly at $2\sigma$ confidence level. We have checked [9] that the neutral meson mixing bounds [14] are satisfied in this example. _Conclusion._ It is interesting that Froggatt-Nielsen -mechanism, with which the hierachical structure of fermions can be realized using gauge singlet combination of triplets as an effective flavon, can be embedded in a 331-model in which also the number of generations can be understood. Furthermore, the scale of the flavour breaking is the same as the breaking scale of the symmetry of the model. In order to form an effective flavon, no new scalar triplets beyond those, which are needed to generate the tree-level masses for the particles, are necessary. We also indicate that the tree-level FCNCs are suppressed in our model. _Acknowledgements._ KH acknowledges the H2020-MSCA-RICE-2014 grant no. 645722 (NonMinimalHiggs). NK is supported by Vilho, Yrjö and Kalle Väisälä Foundation. _Appendix._ The down-type quark mass matrix is generated by the following terms in the Lagrangian: \[\mathcal{L}\supset \sum_{\gamma=1}^{5}(y^{d}_{\eta})_{1\gamma}\bar{Q}^{\prime}_{L,1} \eta~{}d^{\prime}_{R,\gamma}+\sum_{\alpha=2}^{3}\sum_{\gamma=1}^{5}(y^{d}_{ \rho^{\ast}})_{\alpha\gamma}\bar{Q}^{\prime}_{L,\alpha}\rho^{\ast}~{}d^{\prime }_{R,\gamma}\] \[+\sum_{\alpha=2}^{3}\sum_{\gamma=1}^{5}(y^{d}_{\chi^{\ast}})_{ \alpha\gamma}\bar{Q}^{\prime}_{L,\alpha}\chi^{\ast}~{}d^{\prime}_{R,\gamma}+h.c.\] The down-type quark mass matrix is: \[m^{d}_{1\gamma}=\frac{v^{\prime}}{\sqrt{2}}(c^{d}_{\eta})_{1 \gamma}\epsilon^{(h^{\eta})_{1\gamma}},m^{d}_{\alpha\gamma}=\frac{v_{1}}{\sqrt {2}}(c^{d}_{\rho^{\ast}})_{\alpha\gamma}\epsilon^{(h^{\rho^{\ast}})_{\alpha \gamma}}\] \[m^{d}_{(2+\alpha)\gamma}=\frac{v_{2}}{v_{1}}m^{d}_{\alpha\gamma} +\frac{u}{\sqrt{2}}(c^{d}_{\chi^{\ast}})_{\alpha\gamma}\epsilon^{(h^{\chi^{ \ast}})_{\alpha\gamma}},\] where $\alpha=2,3$, and \[(h^{\eta})_{1\gamma} = q(\bar{Q}_{L,1})+q(\eta)+q(d_{R,\gamma})\] \[(h^{\rho^{\ast}})_{\alpha\gamma} = q(\bar{Q}_{L,\alpha})+q(\rho^{\ast})+q(d_{R,\gamma})\] \[(h^{\chi^{\ast}})_{\alpha\gamma} = q(\bar{Q}_{L,\alpha})+q(\chi^{\ast})+q(d_{R,\gamma}).\] The order-one coefficients are restricted to the interval $\lvert c\rvert\in[0.5,5]$. The order-one coefficients for down-type quarks are: $(c^{d}_{\eta})_{11}=3.1679$, $(c^{d}_{\eta})_{12}=1.0274$, $(c^{d}_{\eta})_{13}=1.3083$, $(c^{d}_{\eta})_{14}=1.0222$, $(c^{d}_{\eta})_{15}=-1.0835$, $(c^{d}_{\rho^{\ast}})_{21}=-1.3342$, $(c^{d}_{\rho^{\ast}})_{22}=2.0875$, $(c^{d}_{\rho^{\ast}})_{23}=-0.8140$, $(c^{d}_{\rho^{\ast}})_{24}=1.3463$, $(c^{d}_{\rho^{\ast}})_{25}=1.6967$, $(c^{d}_{\rho^{\ast}})_{31}=-1.1293$, $(c^{d}_{\rho^{\ast}})_{32}=1.9257$, $(c^{d}_{\rho^{\ast}})_{33}=1.9138$, $(c^{d}_{\rho^{\ast}})_{34}=1.8382$, $(c^{d}_{\rho^{\ast}})_{35}=-0.5945$, $(c^{d}_{\chi^{\ast}})_{21}=0.9823$, $(c^{d}_{\chi^{\ast}})_{22}=-0.9842$, $(c^{d}_{\chi^{\ast}})_{23}=-1.5314$, $(c^{d}_{\chi^{\ast}})_{24}=4.4024$, $(c^{d}_{\chi^{\ast}})_{25}=-4.7233$, $(c^{d}_{\chi^{\ast}})_{31}=1.7299$, $(c^{d}_{\chi^{\ast}})_{32}=-1.0801$, $(c^{d}_{\chi^{\ast}})_{33}=-0.5451$, $(c^{d}_{\chi^{\ast}})_{34}=-4.5226$ and $(c^{d}_{\chi^{\ast}})_{35}=-3.4529$. The order-one coefficients for up-type quarks are: $(c^{u}_{\rho})_{11}=-3.3709$, $(c^{u}_{\rho})_{12}=2.4799$, $(c^{u}_{\rho})_{13}=1.1381$, $(c^{u}_{\rho})_{14}=1.5495$, $(c^{u}_{\eta^{\ast}})_{21}=-0.6744$, $(c^{u}_{\eta^{\ast}})_{22}=-1.6193$, $(c^{u}_{\eta^{\ast}})_{23}=1.0648$, $(c^{u}_{\eta^{\ast}})_{24}=1.3911$, $(c^{u}_{\eta^{\ast}})_{31}=1.3571$, $(c^{u}_{\eta^{\ast}})_{32}=1.3324$, $(c^{u}_{\eta^{\ast}})_{33}=0.9053$, $(c^{u}_{\eta^{\ast}})_{34}=1.7678$, $(c^{u}_{\chi})_{11}=1.6150$, $(c^{u}_{\chi})_{12}=0.6765$, $(c^{u}_{\chi})_{13}=2.0687$ and $(c^{u}_{\chi})_{14}=1.0458$. The scalar potential parameters used are: $\lambda_{1}=0.4$, $\lambda_{2}=0.2898$, $\lambda_{3}=0.9$, $\lambda_{12}=0.5$, $\lambda_{13}=0.5$, $\lambda_{23}=0.22$, $\widetilde{\lambda}_{12}=0.8$, $\widetilde{\lambda}_{13}=0.9$, $\widetilde{\lambda}_{23}=-0.1$ and $b=-(10~{}~{}\textrm{TeV})^{2}$. All the scalar masses are positive with these parameters. 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Koivunen, in preparation. * (10) S. L. Glashow and S. Weinberg, Phys. Rev. D 15 (1977) 1958; E. A. Paschos, Phys. Rev. D 15, 1966 (1977). * (11) S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970). * (12) H. Georgi and A. Pais, Phys. Rev. D 19, 2746 (1979); J. C. Montero, F. Pisano and V. Pleitez, Phys. Rev. D 47, 2918 (1993) [hep-ph/9212271]; M. Ozer, Phys. Rev. D 54, 4561 (1996). * (13) C. Patrignani _et al._ [Particle Data Group], Chin. Phys. C 40, no. 10, 100001 (2016). * (14) M. Bona _et al._ [UTfit Collaboration], JHEP 0803, 049 (2008) [arXiv:0707.0636 [hep-ph]].
1706.07855	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 39981, "num_imgs": 2, "llama3_tokens_count": 10319 }	[ "content_image/1706.07855/x1.png", "content_image/1706.07855/x2.png" ]	# The possibility of Scale Relativistic signatures in the Brownian motion of micro-spheres in optical traps Stephan LeBohec lebohec@physics.utah.edu [ Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112 February 19, 2024 ###### Abstract The development of a mechanics of non-differentiable paths [32] suggested by Scale Relativity [27; 28] results in a foundation of Quantum Mechanics [33; 26] including Schrödinger’s equation and all the other axioms under the assumption the path non-differentiability can be described as a Wiener process at the resolution-scale of observation. This naturally brings under question the possibility that the statistics of the dynamics of macroscopic systems fulfilling this hypothesis could fall under a _quantum-like_ description with the Planck constant replaced with some other constant, possibly system specific, and corresponding to a diffusion coefficient. The observation of such a _quantum-like_ dynamics would establish if the Scale Relativistic principle is implemented in macroscopic complex or chaotic systems. This would have major implications for the study of structure formation dynamics in various research fields. In this paper, I investigate the possibility for the detection of such an effect in the Brownian motion of a micro-sphere in an optical trap. I find that, if it exists, the observation of the transition to a _quantum-like_ regime is within reach of modern experiments. permanent address: ]201 JAMES FLETCHER BLDG. 115 S. 1400 E., Salt Lake City, UT 84112 , USA ## I Introduction and background Scale Relativity [27; 28] is centered around the proposal to extend the relativity principle to changes of resolution-scales. In other words, it revolves around the idea that, perhaps, reference frames are to be specified not only by their relative positions, orientations and motions but also by their relative resolution-scales. Then, the Scale Relativity principle is an extension of the usual relativity principle as it states that the laws of physics are the same in all reference frames independently from their relative positions, orientations, motions and resolution-scales [27; 28]. When considering usual Newtonian mechanics, this does not make much of a difference: once the trajectories are observed or considered with a resolution-scale finer than their smallest convolutions, further refining the resolution-scale does not bring anything new. Newtonian mechanics implicitly includes the assumption of differentiability for the trajectories followed by the system. Indeed, it is constructed from the consideration of infinitesimal displacements and time intervals. For example, this is the case for the notion of velocity, while, in practice, a velocity is always obtained by taking the ratio between a finite displacement and a finite time interval, without ever actually taking the limit to infinitesimal intervals. Fundamentally, the infinitesimal limit does not correspond to a physical reality as the irregularities of a non ideal environment can always be expected to start affecting the trajectory when a fine enough resolution-scale is used. Even worse than this, if the resolution-scale is fine enough, it reveals the quantum domain. Then, by virtue of Heisenberg uncertainty relations, the outcomes of the velocity measurements are increasingly affected as attempts are made to further approach the infinitesimal limit for displacements and time intervals. Indeed, R.P.Feynman and A.R.Hibbs[10], in their 1965 path integral formulation of Quantum Mechanics, identified that the _”typical paths of a quantum-mechanical particle are highly irregular on a fine scale. Thus, although a mean velocity can be defined, no mean-square velocity exists at any point. In other words, the paths are non differentiable.”_ The length of such a quantum-mechanical path can therefore be shown to diverge like the inverse of the resolution-scale with which it is being inspected. This was later described in terms of a fractal dimension [18]$D_{f}=2$ for the quantum-mechanical path [1; 14]. This is also the fractal dimension of the path followed by a particle undergoing Brownian motion [5]. So it is clear that the assumed differentiability of mechanical paths in classical mechanics breaks down at least at small resolution-scales, giving way to resolution-scale dependence, characteristic of fractal objects. It then appears that the Scale Relativity program can be implemented in mechanics by considering non differentiable or fractal paths described at a resolution-scale which becomes an explicit parameter of the equations of dynamics. This will be outlined in Section II and it is shown that, in the case of non differentiable paths resulting from a Wiener stochastic process, the equations of dynamics straightforwardly obtained are those of Quantum Mechanics. The derivation is based on the abandonment of the implicit hypothesis of differentiability but one particularly interesting aspect is that it does not depend on the statistics governing the stochastic process to be maintained all the way down to infinitesimal scales as is the case for standard Quantum Mechanics. The reasoning depends on the Wiener stochasticity only at the considered resolution-scale. The important consequence of this is that it opens up the possibility for some complex or chaotic macroscopic systems to display _quantum-like_ features in their structure or dynamics. Indeed, some complex or chaotic systems considered over some range of resolution-scales, can be effectively described in terms of a Wiener process. These are situations in which the scale relativistic approach to Quantum Mechanics is applicable with of course the Planck constant $\hbar$ replaced by a system dependent diffusion constant characterizing the stochasticity. One could talk of stochastic [22] or macroscopic quantization. In fact, soon after the publication of Schrödinger’s equation [30], it was noticed that the orbital parameters of the Solar system’s planets and moons, which constitute a chaotic system [15], are arranged in a way very similar to the orbits in Bohr’s model of the hydrogen atom [6; 19; 29]. This might be the first observation of the type of macroscopic _quantum-like_ dynamics discussed here. These analyses were recently revisited in the Scale Relativity context [24; 12] and extended to other systems [28; 25; 23]. Some of these results are impressive but it would be more convincing to observe similar dynamics in the laboratory under a controlled and reproducible environment. In this article, I am trying to identify a way to investigate this type of _quantum-like_ behavior in the laboratory. From the above comments, it is clear that the system must involve Brownian motion. In addition, an external field must be present to induce a Newtonian dynamic to confine the Brownian motion in such a way _quantum-like_ dynamics may establish itself. I propose that such a macroscopic _quantum-like_ dynamics maybe observable in the motion of micro-sphere in an optical trap[3]. The surrounding fluid is then responsible for the Brownian motion while the optical trap effectively provides the external field. In Section III, I present the modeling of such a system in order to characterize the observability of _quantum-like_ behavior neglecting dissipation whose effects are discussed in Section IV. In Section V, I review the results of recent experiments in comparison with the expectations for the scale-relativistic effect. Finally, Section VI summarizes the article and provides an outlook. Since Scale Relativity is not familiar, Section II is used to give a very brief review of its connection with quantum mechanics. ## II Quantum Mechanics from the Scale Relativistic approach There are two consequences to the abandonment of the differentiability hypothesis while considering the variation rate of a function $f(t)$. First the variation rate becomes doubled valued with possibly different variation rates before and after any considered point. Second, at least when the function is non differentiable in a dense set of points, we cannot take the limit to infinitesimal time intervals anymore. In order to retain the use of differential calculus, we may define an explicitly resolution-scale dependent function $f(t,\delta t)$. For different values of $\delta t$, this function $f(t,\delta t)$ can be thought of as different approximations resulting from the sampling at different time intervals $\delta t$ of the underlying non-differentiable function $f(t)$. We then define a double-valued and explicitly scale dependent finite differential: \[f^{\prime}_{+}(t,\delta t) = \frac{f(t+\delta t,\delta t)-f(t,\delta t)}{\delta t}{~{}~{}~{}~{ }\delta t>0};\] \[f^{\prime}_{-}(t,\delta t) = \frac{f(t+\delta t,\delta t)-f(t,\delta t)}{\delta t}{~{}~{}~{}~{ }\delta t<0}.\] We may apply this to the description of a displacement along a non-differentiable path: \[d{\bf x}_{+} = {\bf v}_{+}dt+d{\bf b}_{+}{~{}~{}~{}~{}dt>0}\] \[d{\bf x}_{-} = {\bf v}_{-}dt+d{\bf b}_{-}{~{}~{}~{}~{}dt<0}\] The first term proceeds from a _usual velocity_${\bf v}_{\pm}$,while, $d{\bf b}_{\pm}$ represents a possibly stochastic residual $\langle d{\bf b}_{\pm}\rangle=0$ accounting for the details of the path at a finer resolution-scale. This two term description formalizes the common practice of disregarding or smoothing-out the details smaller that some resolution-scale. This is fully implemented by taking the expectation values of the time-differentials _after_ and _before_ the considered instant: \[\frac{d_{+}}{dt}{\bf x}={\bf v}_{+}+\langle\frac{d{\bf b}_{+}}{dt}\rangle={\bf v }_{+}{\rm~{}~{}and~{}~{}}\frac{d_{-}}{dt}{\bf x}={\bf v}_{-}+\langle\frac{d{ \bf b}_{-}}{dt}\rangle={\bf v}_{-}\] and it is convenient to combine them linearly into a single _complex time-differential operator_ [27]: \[\frac{\hat{d}}{dt}=\frac{1}{2}\left(\frac{d_{+}}{dt}+\frac{d_{-}} {dt}\right)-\frac{i}{2}\left(\frac{d_{+}}{dt}-\frac{d_{-}}{dt}\right).\] This operator can be used to define a complex velocity: \[\mathcal{V}=\frac{\hat{d}}{dt}{\bf x}=\frac{{\bf v}_{+}+{\bf v}_{-}}{2}-i\frac {{\bf v}_{+}-{\bf v}_{-}}{2}={\bf V}-i{\bf U}\] in which the real part $\bf V$ is the _classical velocity_ while the imaginary part $\bf U$, the _kink velocity_ reveals the non differentiable nature of the path at the considered resolution-scale. We may then write the complex time-differential of a regular and indefinitely differentiable field $h({\bf x}(t,dt),t)$ along a resolution-scale dependent path ${\bf x}(t,dt)$. In doing so, we make the specific choice for the residual stochastic process $d{\bf b}_{\pm}$ to be a Wiener process with $\langle d{\bf b}_{\pm}\rangle=0$, $\langle db_{i+}\cdot db_{i-}\rangle=0$, and $\langle db_{i+}\cdot db_{j+}\rangle=\langle db_{i-}\cdot db_{j-}\rangle=2 \mathcal{D}\delta_{i,j}dt$ with $\mathcal{D}$ a diffusion coefficient. Keeping only the Taylor expansion terms of $\frac{\hat{d}h}{dt}$ that do not vanish with $dt$, the complex time-differential takes the form[27; 28; 32]: \[\frac{\hat{d}}{dt}=\frac{\partial}{\partial t}+\mathcal{V}\cdot \nabla-i\mathcal{D}\Delta\] (1) If the scale relativistic principle applies, the stochastic system must be characterized by a complex Lagrange function $\mathcal{L}({\bf x},\mathcal{V},t)$ and enforcing the stationarity of the action $\mathcal{S}=\int_{t_{1}}^{t_{2}}\mathcal{L}({\bf x},\mathcal{V},t)dt$ (while carefully keeping track of the changes in the Leibniz product rule resulting from the higher order differential term which appeared in $\frac{\hat{d}}{dt}$) results in the usual Euler-Lagrange equation with the complex time-differential operator and complex velocity replacing the usual time derivative and usual velocity: \[\nabla_{\bf x}\,\mathcal{L}-\frac{\hat{d}}{dt}\nabla_{\mathcal{V}}\,\mathcal{L }=0\] In particular, with $\mathcal{L}=\frac{1}{2}m\mathcal{V}^{2}-\Phi$ where $\Phi$ is a purely real potential energy, we recover a generalized form of Newton’s relation of dynamics \[m\frac{\hat{d}}{dt}\mathcal{V}=-\nabla\Phi.\] (2) In the stationary case, $\langle{\bf V}\rangle=0$, this generalized relation of dynamics can be turned into a Langevin equation which has been integrated numerically [20; 11; 2; 32] for various systems. In each case, the statistics of the system configuration parameters are found to follow the magnitude squared of the solutions of the time independent Schrödinger equation describing the same system with however Planck’s constant replaced by $2m\mathcal{D}$. This is particularly striking as standard Quantum Mechanics has not been invoked in the establishment of the above generalized relation of dynamics (Equation 2). The connection with standard Quantum Mechanics can be made explicit by introducing the function $\psi=\psi_{0}e^{i\mathcal{S}/\mathcal{S}_{0}}$ where $\mathcal{S}_{0}$ and $\psi_{0}$ are introduced for dimensional reasons. The complex velocity can then be expressed canonically as $\mathcal{V}=-i\frac{\mathcal{S}_{0}}{m}\nabla\ln\left(\psi/\psi_{0}\right)$ which leads to rewriting the generalized Newton’s relation of dynamics as a Schrödinger equation in which $\hbar$ is replaced with $\mathcal{S}_{0}=2m\mathcal{D}$[27; 28]: \[2im\mathcal{D}{{\partial\psi}\over{\partial t}}=-2m\mathcal{D}^{2}\Delta\psi+\Phi\psi\] I only provide an outline of this development here. Details as well as interpretive discussions of the other postulates of quantum mechanics can be found elsewhere [26; 27; 28; 32; 33]. What is important is that this does not result from anything else than the relaxation of the mechanical path differentiability hypothesis under the restriction to Wiener processes for the stochastic component. Also it does not require the stochasticity to be preserved at resolution-scales different from those considered in the observation or description of the system. The enforcement of the principle of Scale Relativity for mechanical path with a Wiener stochastic component results in this generalized Schrödinger equation. Consequently, it becomes justified to expect systems effectively evolving along this type of paths to display _quantum-like_ dynamics. This is precisely what I take under consideration in the next section with the motion of a micro-sphere in an optical trap. ## III Motion of micro-spheres in a harmonic traps Consider a sphere of radius $R$ and mass density $\rho_{I}$ in a fluid of mass density $\rho_{E}$. The inertial mass $m$ of the sphere includes an added mass term representing the inertia of the fluid the sphere displaces in the course of its motion $m={{4}\over{3}}\pi R^{3}(\rho_{I}+\rho_{E}/2)$. The micro-sphere is in a trap which I assume to be harmonic $\Phi(r)=\frac{1}{2}m\omega^{2}r^{2}$ with $\omega$ the proper frequency. In principle, the distribution of the position of the micro-sphere in the harmonic potential follows Boltzmann’s statistics $e^{-\Phi(r)/k_{B}T}$ and for the harmonic potential we can expect a Gauss distribution with the standard deviation: \[\sigma_{B}=\sqrt{{k_{B}T}\over{m\omega^{2}}}\] (3) This Boltzmann distribution does not depend on any property of the fluid other than its temperature $T$ and its mass density, which enters in the micro-sphere’s effective mass $m$. It is not achieved instantaneously. The harmonic force acting on the micro-sphere may result in a mobility limited radial drift with a speed $\langle v_{r}\rangle=-\mu m\omega^{2}r$ where the mobility is given by Stokes law as ${\mu}=\frac{1}{6\pi\eta R}$ where $\eta$ represents the fluid viscosity. It is convenient to define a relaxation time constant $\tau_{r}={{3\pi\eta R}\over{m\omega^{2}}}$. It depends on the viscosity $\eta$ but not on the temperature. After a time $t\gg\tau_{r}$, the system reaches a statistically stationary state, independently from the initial configuration. This distribution results from a Langevin dynamics equation [7] \[m\ddot{\bf r}=-m\omega^{2}{\bf r}-6\pi\eta R\dot{\bf r}+\cdots+{\bf F}_{Therm.}\] where ${\bf F}_{Therm.}$ represents the stochastic thermal force and I omit additional hydrodynamics forces which will be mentioned in Section IV. The previous section suggests that, instead, the state of the micro-sphere should be described in terms of _quantum-like_ wave functions $\psi({\bf r},t)$, solutions of a Schrödinger equation with the reduced Planck constant replaced with $2m\mathcal{D}$, \[i\mathcal{D}{{\partial\psi}\over{\partial t}}+\mathcal{D}^{2}\Delta\psi=\frac{ 1}{2}\omega^{2}r^{2}\psi+\cdots\] where I omit all hydrodynamics forces including Stoke’s drag, whose effect will be discussed in Section IV. The eigen-energies of the time independent version of this Schrödinger equation are $E_{n}=(n+\frac{1}{2})2m\mathcal{D}\omega$ in the one dimensional case. Since the micro-sphere is in a bath at a given temperature $T$, its _quantum-like_ state is a Boltzman factor weighted mixture of Hermite functions. The average _occupation number_ of each state is proportional to $e^{-(n+1/2)/\bar{n}}$, with $\bar{n}=\frac{k_{B}T}{2m\mathcal{D}\omega}$, where $k_{B}$ is Boltzmann’s constant. As the micro-sphere is subject to Brownian motion, the diffusion coefficient is given by Einstein’s kinetic theory relation [8; 31] as $\mathcal{D}=\mu k_{B}T$. So we have $\mathcal{D}={{k_{B}T}\over{6\pi\eta R}}$ and $\bar{n}=\tau_{r}\omega$. While this description is based on the application of Boltzmann’s statistics to _quantum-like_ states, if it has any validity, it should result in some departure from the Boltzmann statistics for the position and motion of the micro sphere, at least in the _quatum-like regime_ where $\bar{n}\ll 1$. Figure 1 shows the relations between the radius $R$ of a micro-sphere made of amorphous silica and the proper frequency of the trap $\omega$ for various values of $\bar{n}$ in water at a temperature of $10^{\circ}\rm C$. While $\bar{n}$ does not depend explicitly on temperature, it depends on the fluid viscosity $\eta$, which depends on temperature. For each micro-sphere radius, there is an optical trap strength above which _quantum-like_ behavior should appear. For reference, the dot represents conditions used in observations by J. Mo et al.[21] and corresponding to $\bar{n}\approx 6.7$. The same group performed other measurements under conditions corresponding to even smaller values of $\bar{n}$ which are more favorable to the observation of a scale relativistic signature as we are about to discuss. <figure><img src="content_image/1706.07855/x1.png"><figcaption>Figure 1: The silica (ρI=2.2gcm−3) micro-sphere radius R is represented as afunction of the harmonic oscillator proper frequency ω with their relationconstrained by the requirement that ¯n=0.1 (dashed line) ¯n=1 (solid blackline), ¯n=2 (grey line) and ¯n=10 (light grey line). The fluid is assumed tobe water with a density of 1gcm−3 and a 10∘C viscosity η=10−3Pas. The pointcorresponds to the system of a silica micro-sphere of diameter 3.06μm in anoptical trap with strength 188μNm−1 in water used for a test of the Maxwell-Boltzmann distributions by J.Mo et al.mo2015</figcaption></figure> If the scale relativistic _quantum-like_ description of the state of the micro-sphere undergoing Brownian motion in an optical trap is valid, the statistical distribution of the positions of the micro-sphere should correspond to the Boltzmann coefficient weighted series of the magnitude squared stationary _quantum-like_ wave functions. In the present case of a harmonic well, these wave functions are Hermite functions. In the first volume of statistical physics in the course of theoretical physics by L.D. Landau and E.M. Lifshitz [16] , this distribution is shown to be gaussian. Replacing $\hbar$ with $2m\mathcal{D}$ and making use of $\bar{n}$ in the expression of the standard deviation, we find \[\sigma_{Q}=\sigma_{B}\left[{2\bar{n}\,\tanh\left(\frac{1}{2\bar{n}}\right)} \right]^{-1/2}.\] (4) In the limit of a weak trap for which $\bar{n}\gg 1$, we see that $\sigma_{Q}\to\sigma_{B}$. In this regime, _quantum-like_ effects are not expected and the present theory reproduces the classical result. However, in the limit of a strong trap for which $\bar{n}\ll 1$, we may expect the _quantum-like_ ground state to dominate the statistics with a different dependence on the conditions: \[\lim_{\bar{n}\to 0}\sigma_{Q}=\sqrt{{{k_{B}T}\over{6\pi\eta R\omega}}}\] Starting from the classical regime and keeping all things equal, as the strength of the trap is increased, the standard deviation of the micro-sphere position decreases as $\frac{1}{\omega}$. The observation of a transition from a $\frac{1}{\omega}$ to a $\frac{1}{\sqrt{\omega}}$ dependence of the standard deviation of the position micro-sphere in the vicinity of $\omega_{0}=\frac{3\pi\eta R}{m}$ would be a signature of the actual existence of a scale relativistic or _quantum-like_ regime. This possible transition is illustrated by Figure 2 for two different silica micro-sphere radii in water. Figure 2 also shows that the observation of the transition to the _quantum-like_ regime with predominance of the ground state requires the ability to monitor the motion of the micro-sphere with a resolution that is more that three orders of magnitude smaller than the size of the microsphere. Amazingly, observations have already been performed in the domain of interest. However, before reviewing them, effects neglected in the above discussion should be discussed as they may affect the observation of the signature transition. <figure><img src="content_image/1706.07855/x2.png"><figcaption>Figure 2: The calculated standard deviation of the position of a micro-sphereof radius R=5μm and R=0.5μm is shown as a function of the harmonic trap properfrequency ω. Conditions are the same as in Figure 1. The vertical dotted linesmark the respective values of ω0 around which the transition from the standardBoltzmann statistics to quantum-like statistics may occur. The transition isrevealed by the change of slope of the curves in this Log-Log graph,indicating a change from a 1ω dependence for ω≪ω0 to a 1√ω dependence forω≫ω0. The dashed curves represent the calculated standard deviation whenviscous dissipation is taken into account as discussed in Section IV. The greydot-dashed line at the top of the figure is shown for reference. It indicatesthe radius of the micro-sphere such that ¯n=1.</figcaption></figure> ## IV Effects of dissipation and hydrodynamics The expression of $\sigma_{Q}$ (Equation 4) corresponds to an ideal harmonic oscillator while, in fact, the micro-sphere also is under the influence of a dissipative drag force. Quantizing the so-called Bateman dual system of damped and amplified oscillators, M. Blasone and P. Jizba showed [4] that, in first approximation, the quantum ground state of the under-damped harmonic oscillator to have the same form as the quantum ground state of the simple harmonic oscillator with the proper frequency of the under-damped oscillator $\tilde{\omega}=\omega\sqrt{1-(2m\omega\mu)^{-2}}$ used in the place of the proper undamped frequency $\omega$. With the Stokes’ drag as the dissipative force, we have $\tilde{\omega}=\omega\sqrt{1-\bar{n}^{2}}$. We see that the boundary of the domain within which the oscillator is under-damped corresponds to $\bar{n}=1$. This is precisely where we expected the transition to the _quantum-like_ regime. With this damped oscillator frequency, we may define the damped equivalent of $\bar{n}$ which we write $\tilde{\bar{n}}=\frac{k_{B}T}{2m\mathcal{D}\tilde{\omega}}$ and we find $\tilde{\bar{n}}=\frac{\bar{n}}{\sqrt{1-\bar{n}^{2}}}$. For $\bar{n}\ll 1$, that is far in the under-damped oscillation regime, we have $\tilde{\bar{n}}\approx\bar{n}$ and nothing is changed. When we just have $\bar{n}<1$, we see that $\tilde{\bar{n}}\geq\bar{n}$ so the dissipative effects tend to move the system away from the _quantum-like_ regime, particularly when $\bar{n}$ is close to unity. At the same time, as the dissipative term results in a smaller effective harmonic force, the resulting standard deviation $\tilde{\sigma}_{Q}$ may be larger than $\sigma_{Q}$. As an indication of this, $\sigma_{Q}(\tilde{\omega})$ is also shown on Figure 2. However, if the dissipation is taken into account, for $n>1$ the above theory is simply not applicable. Nevertheless, this does not say anything about the Scale Relativity principle being implemented in nature or not. When the drag force is important, both the motion of the micro-sphere and the motion surrounding fluid should be described using _quantum-like mechanics_. It remains that if the scale relativistic principle is implemented in nature, the above described transition between the classical domain and _quantum-like_ domain, with $\bar{n}\ll 1$, where dissipative effects are negligible, should still be observable. Other effects can complicate the comparison between the classical and _quantum-like_ regimes. When the velocity of an object immersed in a fluid changes, the boundary layer adjusts with a time delay. Correspondingly, such changes in the vorticity of the fluid affects the force acting on the object at a later time in a way depending on the object motion history, this is the Boussinesq-Basset force [9]. I find that, in the quantum regime $\bar{n}\ll 1$, it only constitute a small correction to Stokes’ drag, which we just discussed. Further more, when considering time scales shorter than the size of the object divided by the speed of sound, in principle, the finite compressibility of the fluid should also be taken into account [36]. These effects can be regarded as departures from a purely Brownian motion, corresponding to a fractal dimension $D_{F}=2$, the only one for which the Schrödinger equation is strictly valid. In principle, to be included in a Scale Relativity description, these effects would require a generalized Schrödinger equation, not restricted to strictly Markovian processes. Here I simply neglect to include these effects and we should expect that the above described transition do not necessarily follow the black curves of Figure 2 with precision but it remains that a change of behavior in the regime $\bar{n}\ll 1$ is to be expected if the Scale Relativity principle is implemented. ## V Observations Several experiments have been performed to track the motion of a micro-sphere in an optical trap [3] in a fluid. In these experiments, the light from one of the lasers forming the optical trap is collimated after scattering off the micro-sphere and directed toward a system of balanced detectors. The resulting signal depends on the position of the micro-sphere in the trap. It is digitized and recorded at a high rate for extended periods of time [17; 21]. The statistics of the data is then analyzed by comparison to the Ornstein-Uhlenbeck model [34] with the addition of the effective mass and the contribution of Basset force in liquids[13]. For measurements in a gas the data statistics is compared to the solution of the Langevin equation in an underdamped harmonic trap as given by Wang and Uhlenbeck [35]. In both cases the equipartition theorem is satisfied for instantaneous mean squared velocity and position in the harmonic well provided the effective mass $m$ of the the micro-sphere is used. In order to test for a _quantum-like_ behavior as suggested by the scale relativistic approach outlined above, we may concentrate on the distribution of the position of the micro-sphere. Table 1 summarizes the experimental parameters presented in the articles by J. Mo et al. [21] and T. Li et al. [17] and includes the calculated value of $\bar{n}$ and $\frac{\sigma_{Q}}{\sigma_{B}}-1$ for each. Both measurements with silica micro-sphere in liquids correspond to $\bar{n}>1$ and do not provide much of an opportunity to discriminate between $\sigma_{B}$ and $\sigma_{Q}$. However, the two other measurements provide $\bar{n}<1$ with appreciable differences of 6.4% and 20% respectively between $\sigma_{B}$ and $\sigma_{Q}$, which could lead to a clear detection or exclusion of the _quantum-like_ effect. Systems \| Silica inWater \| Silica inAcetone \| BaTiO3 inAcetone \| Silica inair ---\|---\|---\|---\|--- R (μm) \| 1.53 \| 1.99 \| 2.68 \| 1.50 m (10−14kg) \| 2.25 \| 5.30 \| 30.7 \| 3.11 ω (104s−1) \| 9.14 \| 3.07 \| 3.34 \| 1.99 ¯n \| 6.69 \| 3.65 \| 0.78 \| 0.41 σQσB−1 \| 9.3×10−4 \| 3.1×10−3 \| 6.4×10−2 \| 0.20 Table 1: In the following table, the values of R, m, and ω are from the measurements presented in the articles by J. Mo et al. mo2015 and T. Li et al. li2010 . These quantities are used to calculate ¯n as in Section II as well as the relative difference between σQ and σB. Calculations were done using the following values specified by J. Mo et al. mo2015 for measurements in liquid phases: ρSilica=2.0g⋅cm−3, ρBaTiO3=4.2g⋅cm−3, ρH2O=0.998g⋅cm−3, ρAcetone=0.789g⋅cm−3, ηH2O=9.55×10−4Pas, and ηAcetone=3.17×10−4Pas and we used ω=√k/m where k is the trapping strength given in μN/m. For the measurement in the air, we used ρSilica=2.2g⋅cm−3, ρAir≈0 and ηAir=1.81×10−5Pas The papers presenting the measurements do not report any incompatibility between the above mentioned models to the data statistics. However the physical parameters that are kept free in the fitting procedures are the strength of the harmonic well $\omega$, the micro-sphere radius $R$ in the case of the measurements in liquids [21] and for the measurements in air [17] the free parameters are the strength of the harmonic well $\omega$ and the momentum relaxation time $\tau_{p}=\frac{m}{6\pi\eta R}$. We have seen that $\sigma_{B}$ is inversely proportional to $\omega$. A difference of a few percents in the spread of the micro-sphere position distribution could be absorbed by a few percent change in the well strength. Possibly more importantly, the calibration factor $K$ used to convert the balanced detector output signal to an actual position is also kept free, at least in the analysis of the measurements in liquids [21] while no details are given for the calibration of the position detector used in the measurements in the air. The factor $K$ is a scaling factor for the micro-sphere position distribution and it could as well absorb a few percent difference. As a consequence the difference between $\sigma_{Q}$ and $\sigma_{B}$ might have gone unnoticed as it could have been absorbed in fit parameters. A search for a transition around $\bar{n}\sim 1$ could start in the regime $\bar{n}\gg 1$ where all the calibration could be performed. Then, the strength of the harmonic well could be increased progressively to bring the system to the regime $\bar{n}\ll 1$. This supposes the strength of the well can be adjusted without affecting the position detector calibration. It also requires some mean of independently calibrating the strength of the well over at least an order of magnitude. ## VI Summary The Scale Relativity principle extends the usual relativity principle to include changes of resolution-scale and leads to considering non-differentiable mechanical paths. In order to retain differential calculus as a tool, these paths maybe considered at a set resolution scale which becomes an explicit parameter. Then the development of a mechanics for paths with their non-differentiable component resulting from a Wiener process leads to a generalized newtonian relation of dynamics, which cas be rewritten as a Schrödinger equation in which $\hbar$ is replaced by $2m\mathcal{D}$ where $m$ is the mass of the micro-sphere and $\mathcal{D}$ is a diffusion constant. If the Scale Relativity principle is implemented in nature, chaotic or complex systems whose evolution can effectively be described as a Wiener stochastic process should display a dynamic structure in accordance to this _emergent quantum-like mechanics_. In particular the Brownian motion of a micro-sphere in an optical trap could reveal such a behavior. In Section III, we have seen that the Boltzmann statistics occupation of the _quantum-like_ levels of the harmonic oscillator reproduces the classical position probability density in the limit $\frac{3\pi\eta R}{m\omega}\gg 1$ with a spread scaling with $\frac{1}{\omega}$. However, when $\frac{3\pi\eta R}{m\omega}\ll 1$, the statistics of the _quantum-like_ ground state may dominate and the spread of the position probability density then scales with $\frac{1}{\sqrt{\omega}}$. In Section IV we have seen that the dissipative drag force and other effects may affect the details of the transition between the two regimes but are not expected to prevent distinguishing them observationally. The observation of the change of behavior between the two regimes would be indicative that the Scale Relativity principle is actually implemented in nature. In Section V we have seen that modern experiments [21; 17] used to record the motion of a micro-sphere in an optical trap have a sensitivity that allows probing the domain where the transition between classical and quantum-like regimes may occur. While these experiments did not report any departure from the classically expected behavior, I argued that the relatively small relative difference in the spread of the micro-sphere position probability density could have been absorbed in the calibration factor of the position measurement and in the strength of the optical well, which are both fit parameters in the reported data analysis. Observational evidences for the type of _macroscopic stochastic quantization_ discussed in this paper would have major implications for many research fields such as astrophysical structure formation, geology, meteorology, biology, ecology, sociology or economy as the assumed underlying dynamics may in some cases have to be revisited. ###### Acknowledgements. The author is grateful to Mark Raizen for his clarifications and encouragements. 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1605.02162	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 148062, "num_imgs": 4, "llama3_tokens_count": 49637 }	[ "content_image/1605.02162/x1.png", "content_image/1605.02162/x23.png", "content_image/1605.02162/x25.png", "content_image/1605.02162/x36.png" ]	# Point counting on curves using a gonality preserving lift Wouter Castryck and Jan Tuitman ###### Abstract † [FOOTNOTE:†][ENDFOOTNOTE] † [FOOTNOTE:†][ENDFOOTNOTE] We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using $p$-adic cohomology. ## 1 Introduction This article is about efficiently lifting algebraic curves over finite fields to characteristic zero, in a genus and gonality preserving way, with an application to $p$-adic point counting. Throughout, our curves are always understood to be geometrically irreducible, but not necessarily non-singular and/or complete. By the genus of a curve we mean its geometric genus, unless otherwise stated. As for the gonality of a curve over a field $k$, we make a distinction between two notions: by its _$k$-gonality_ we mean the minimal degree of a non-constant $k$-rational map to the projective line, while by its _geometric gonality_ we mean the $\bar{k}$-gonality, where $\bar{k}$ denotes an algebraic closure of $k$. We also make a notational distinction between projective, affine or toric (= affine minus coordinate hyperplanes) $n$-space in characteristic zero, in which case we write $\operatorname{\mathbf{P}}^{n},\operatorname{\mathbf{A}}^{n},\operatorname{ \mathbf{T}}^{n}$, and their finite characteristic counterparts, where we opt for $\operatorname{\mathbb{P}}^{n},\operatorname{\mathbb{A}}^{n},\operatorname{ \mathbb{T}}^{n}$. Apart from that we avoid reference to the base field, which should always be clear from the context. Similarly we write $\operatorname{\mathbf{Q}}$ for the field of rational numbers and $\operatorname{\mathbb{F}}_{q}$ for the finite field with $q$ elements, where $q$ is a power of a prime number $p$. For each such $q$ we fix a degree $\log_{p}q$ extension $K\supset\operatorname{\mathbf{Q}}$ in which $p$ is inert, and let $\mathcal{O}_{K}$ denote its ring of integers. We then identify $\operatorname{\mathbb{F}}_{q}$ with the residue field $\mathcal{O}_{K}/(p)$. Our lifting problem is as follows: Problem 1.: _Given a curve $\overline{C}$ over $\operatorname{\mathbb{F}}_{q}$, find an efficient algorithmic way of producing a polynomial $f\in\mathcal{O}_{K}[x,y]$ such that_ 1. _its reduction mod_ $p$ _defines a curve that is birationally equivalent to_ $\overline{C}$_,_ 2. _the curve_ $C\subset\operatorname{\mathbf{A}}^{2}$ _it defines has the same genus as_ $\overline{C}$_,_ 3. _its degree in_ $y$ _equals the_ $\operatorname{\mathbb{F}}_{q}$_-gonality of_ $\overline{C}$_._ Note that these conditions imply that the $K$-gonality of $C$ equals the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$, because the gonality cannot increase under reduction mod $p$; see e.g. [21, Thm. 2.5]. We are unaware of whether an $f$ satisfying (i-iii) exists in general. Grothendieck’s existence theorem [36] implies that in theory one can achieve (i) and (ii) over the ring of integers $\operatorname{\mathbf{Z}}_{q}$ of the $p$-adic completion $\operatorname{\mathbf{Q}}_{q}$ of $K$, but, firstly, it is not clear that we can always take $f$ to be defined over $\mathcal{O}_{K}$ and, secondly, we do not know whether it is always possible to incorporate (iii), let alone in an effective way. To give a concrete open case, we did not succeed in dealing with Problem 1 for curves of genus four having $\operatorname{\mathbb{F}}_{q}$-gonality five, which can only exist if $q\leq 7$. (However, as we will see, among all curves of genus at most five, the only cases that we cannot handle are pathological examples of the foregoing kind.) We are intentionally vague about what it means to be _given_ a curve $\overline{C}$ over $\operatorname{\mathbb{F}}_{q}$. It could mean that we are considering the affine plane curve defined by a given absolutely irreducible polynomial $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$. Or it could mean that we are considering the affine/projective curve defined by a given more general system of equations over $\operatorname{\mathbb{F}}_{q}$. In all cases we will ignore the cost of computing the genus $g$ of $\overline{C}$. Moreover, in case $g=0$ we assume that it is easy to realize $\overline{C}$ as a plane conic (using the anticanonical embedding) and if $g=1$ we ignore the cost of finding a plane Weierstrass model. By the Hasse-Weil bound every genus one curve over $\operatorname{\mathbb{F}}_{q}$ is elliptic, so this is indeed possible. If $g\geq 2$ then we assume that one can easily decide whether $\overline{C}$ is hyperelliptic or not (note that over finite fields, curves are hyperelliptic iff they are geometrically hyperelliptic, so there is no ambiguity here). If it is then we suppose that it is easy to find a generalized Weierstrass model. If not then it is assumed that one can effectively compute a canonical embedding \[\kappa:\overline{C}\hookrightarrow\operatorname{\mathbb{P}}^{g-1}\] along with a minimal set of generators for the ideal of its image. The latter will usually be our starting point. Most of the foregoing tasks are tantamount to computing certain Riemann-Roch spaces. There is extensive literature on this functionality, which has been implemented in several computer algebra packages, such as Magma [7] and Macaulay2 [28]. The idea is then to use the output polynomial $f$ as input to a recent algorithm due to the second author [53, 54] for computing the Hasse-Weil zeta function of $\overline{C}$. This algorithm uses $p$-adic cohomology, which it represents through the map $\pi:C\rightarrow\operatorname{\mathbf{P}}^{1}:(x,y)\mapsto x$. The algorithm only works if $C$ and $\pi$ have appropriate reduction modulo $p$, in a rather subtle sense for the precise description of which we refer to [54, Ass. 1]. This condition is needed to be able to apply a comparison theorem between the (relative) $p$-adic cohomology of $\overline{C}$ and the (relative) de Rham cohomology of $C\otimes\operatorname{\mathbf{Q}}_{q}$, which is where the actual computations are done. For such a theorem to hold, by dimension arguments it is necessary that $C$ and $\overline{C}$ have the same genus, whence our condition (ii). This may be insufficient, in which case $f$ will be rejected, but for $p>2$ our experiments show that this is rarely a concern as soon as $q$ is sufficiently large. Moreover, in many cases below, our construction leaves enough freedom to retry in the event of a failure. The algorithm from [53, 54] has a running time that is sextic in $\deg\pi$, which equals the degree in $y$ of $f$, so it is important to keep this value within reason. Because the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$ is an innate lower bound, it is natural to try to meet this value, whence our condition (iii). At the benefit of other parameters affecting the complexity, one could imagine it being useful to allow input polynomials whose degree in $y$ exceeds the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$, but in all cases that we studied the best performance results were indeed obtained using a gonality-preserving lift. At the same time, looking for such a lift is a theoretically neat problem. _Remark 2_.: For the purpose of point counting, it is natural to wonder why we lift to $\mathcal{O}_{K}$, and not to the ring $\operatorname{\mathbf{Z}}_{q}$, which is a priori easier. In fact, most computations in the algorithm from [53, 54] are carried out to some finite $p$-adic precision $N$, so it would even be sufficient to lift to $\mathcal{O}_{K}/(p^{N})=\operatorname{\mathbf{Z}}_{q}/(p^{N})$. A first reason for lifting to $\mathcal{O}_{K}$ is simply that this turns out to be possible in the cases that we studied, without additional difficulties. A second more practical reason is that at the start of the algorithm from [53, 54] some integral bases have to be computed in the function field of the curve. Over a number field $K$ this is standard and implemented in Magma, but to finite $p$-adic precision it is not clear how to do this, and in particular no implementation is available. Therefore, the integral bases are currently computed to exact precision, and we need $f$ to be defined over $\mathcal{O}_{K}$. ContributionsAs explained in Section 2 the cases where $\overline{C}$ is rational, elliptic or hyperelliptic are straightforward. In this article we give a recipe for tackling Problem 1 in the case of curves of $\operatorname{\mathbb{F}}_{q}$-gonality $3$ and $4$. Because of their practical relevance, our focus lies on curves having genus at most five, which is large enough for the main trigonal and tetragonal phenomena to be present. The details can be found in Section 3; more precisely in Sections 3.1, 3.2 and 3.3 we attack Problem 1 for curves of genus three, four and five, respectively, where we restrict ourselves to finite fields $\operatorname{\mathbb{F}}_{q}$ having odd characteristic. Each of these sections is organized in a stand-alone way, as follows: * In a first part we classify curves by their $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma$ and solve Problem 1 in its basic version (except for some pathological cases such as pentagonal curves in genus four or hexagonal curves in genus five, which are irrelevant for point counting because these can only exist over extremely small fields). If the reader is interested in such a basic solution only, he/she can skip the other parts, which are more technical. * Next, in an optimization part we take into account the fact that the actual input to the algorithm from [53, 54] must be monic when considered as a polynomial in $y$. This is easily achieved: if we write \[f=f_{0}(x)y^{\gamma}+f_{1}(x)y^{\gamma-1}+\dots+f_{\gamma-1}(x)y+f_{\gamma}(x),\] then the birational transformation $y\gets y/f_{0}(x)$ gives \[y^{\gamma}+f_{1}(x)y^{\gamma-1}+\dots+f_{\gamma-1}(x)f_{0}(x)^{\gamma-2}y+f_{ \gamma}(x)f_{0}(x)^{\gamma-1},\] (1) which still satisfies (i), (ii) and (iii). But one sees that the degree in $x$ inflates, and this affects the running time. We discuss how our basic solution to Problem 1 can be enhanced such that (1) becomes a more compact expression. * We have implemented the algorithms from this paper in the computer algebra system Magma. The resulting package is called goodmodels and can be found at the webpage http://perswww.kuleuven.be/jan_tuitman. In a third part we report on this implementation and on how it performs in composition with the algorithm from [53, 54] for computing Hasse-Weil zeta functions. We give concrete runtimes, memory usage and failure rates, but avoid a detailed complexity analysis, because in any case the lifting step is heavily dominated by the point counting step. All computations were carried out with Magma v2.22 on a single Intel Core i7-3770 CPU running at 3.40 GHz. The code used to generate the tables with running times, memory usage and failure rates can be found in the subdirectory ./profiling of goodmodels. As we will see, the case of trigonal curves of genus five provides a natural transition to the study of general curves of $\operatorname{\mathbb{F}}_{q}$-gonality $3$ and $4$. These are discussed in Section 4, albeit in a more sketchy way. ConsequencesThe main consequences of our work are that * computing Hasse-Weil zeta functions using $p$-adic cohomology has now become practical on virtually all curves of genus at most five over finite fields $\operatorname{\mathbb{F}}_{q}$ of (small) odd characteristic, * the same conclusion for curves of $\operatorname{\mathbb{F}}_{q}$-gonality at most four looms around the corner, even though some hurdles remain, as explained in Section 4, * we have a better understanding of which $\operatorname{\mathbb{F}}_{q}$-gonalities can occur for curves of genus at most five, see the end of Section 2.1 for a summarizing table. We stress that the general genus five curve, let alone the general tetragonal curve of any given genus, cannot be tackled using any of the previous Kedlaya-style point counting algorithms, that were designed to deal with elliptic curves [43], hyperelliptic curves [19, 29, 31, 35, 37], superelliptic curves [26, 40], $C_{ab}$ curves [12, 20, 55] and nondegenerate curves [11, 52], in increasing order of generality. We refer to [13] for a discussion of which classes of curves do admit a nondegenerate model. A reference problem ($\dagger$)At sporadic places in this article, we refer to a paper that develops its theory over $\operatorname{\mathbf{C}}$ only, while in fact we need it over other fields, such as $\overline{\operatorname{\mathbb{F}}}_{q}$. This concern mainly applies to the theory of genus five curves due to Arbarello, Cornalba, Griffiths and Harris [2, VI.§4.F]. We are convinced that most of the time this is not an issue (the more because we rule out even characteristic) but we did not sift every one of these references to the bottom to double-check this: we content ourselves with the fact that things work well in practice. In our concluding Section 4 on trigonal and tetragonal curves, the field characteristic becomes a more serious issue, for instance in the Lie algebra method developed by de Graaf, Harrison, Pílniková and Schicho [18]. More comments on this will be given there. Each time we cite a $\operatorname{\mathbf{C}}$-only (or characteristic zero only) reference whose statement(s) we carry over to finite characteristic without having verified the details, we will indicate this using the dagger symbol $\dagger$. AcknowledgementsWe would like to thank Arnaud Beauville, Tom De Medts, Jeroen Demeyer, Steve Donnelly and Josef Schicho for answering several of our questions. A large part of this paper was prepared while the first author was affiliated with the University of Ghent. The second author is a postdoctoral research fellow of the Research Foundation Flanders (FWO). Further support for this research was received from the project G093913N of the Research Foundation Flanders (FWO) and from the European Commission through the European Research Council under the FP7/2007-2013 programme with ERC Grant Agreement 615722 MOTMELSUM. ## 2 Background ### First facts on the gonality Let $k$ be a field and let $C$ be a curve over $k$. The geometric gonality $\gamma_{\text{geom}}$ of $C$ is a classical invariant. It is $1$ if and only if the genus of $C$ equals $g=0$, while for curves of genus $g\geq 1$, by Brill-Noether theory $\gamma_{\text{geom}}$ lies in the range \[2,\dots,\lceil g/2\rceil+1.\] For a generic curve the upper bound $\lceil g/2\rceil+1$ is met [15], but in fact each of the foregoing values can occur: inside the moduli space of curves of genus $g\geq 2$ the corresponding locus has dimension $\min\{2g-5+2\gamma_{\text{geom}},3g-3\}$; see [1, §8]${}^{\dagger}$. From a practical point of view, determining the geometric gonality of a given curve is usually a non-trivial computational task, although in theory it can be computed using so-called scrollar syzygies [44]. In the arithmetic (= non-geometric) case the gonality has seen much less study, even for classical fields such as the reals [16]. Of course $\gamma_{\text{geom}}$ is always less than or equal to the $k$-gonality $\gamma$, but the inequality may be strict. In particular the Brill-Noether upper bound $\lceil g/2\rceil+1$ is no longer valid. For curves of genus $g=1$ over certain fields $\gamma$ can even be arbitrarily large [14]. As for the other genera, using the canonical or anticanonical linear system one finds * if $g=0$ then $\gamma\leq 2$, * if $g\geq 2$ then $\gamma\leq 2g-2$. These bounds can be met. We refer to [41, Prop. 1.1] and the references therein for precise statements, along with some additional first facts. If $k=K$ is a number field then the notion of $K$-gonality has enjoyed more attention, both from a computational [21, 22] and a theoretical [41] point of view, especially in the case where $C$ is a modular curve. This is due to potential applications towards effective versions of the uniform boundedness conjecture; see [49] for an overview. In the non-modular case not much literature seems available, but our rash guess would be that almost all (in any honest sense) curves of genus $g\geq 2$ over $K$ meet the upper bound $\gamma\leq 2g-2$. This is distantly supported by the Franchetta conjecture; see again [41, Prop. 1.1] and the references therein for a more extended discussion. Over finite fields $k=\operatorname{\mathbb{F}}_{q}$ the notion has attracted the attention of coding theorists in the context of Goppa codes [51]. They proved the following result: Lemma 3.: _If the $\overline{C}$ is a curve over a finite field $\operatorname{\mathbb{F}}_{q}$ then its $\operatorname{\mathbb{F}}_{q}$-gonality is at most $g+1$. Moreover, if equality holds then $g\leq 10$ and $q\leq 31$._ Proof.: See [51, §4.2]. ∎ In [51, §4.2] it is stated as an open problem to find tighter bounds for the $\operatorname{\mathbb{F}}_{q}$-gonality. In fact we expect the sharpest possible upper bound to be $\lceil g/2\rceil+1+\varepsilon$ for some small $\varepsilon$; maybe $\varepsilon\leq 1$ is sufficient as soon as $q$ is large enough. A byproduct of this paper is a better understanding of which $\operatorname{\mathbb{F}}_{q}$-gonalities can occur for curves of genus at most five, in the cases where $q$ is odd (the cases where $q$ is even should be analyzable in a similar way). The following table summarizes this. \begin{tabular}{c\|c\|c\|c\|c} $g$ & Brill-Noether & possible $\operatorname{\mathbb{F}}_{q}$-gonalities & possible $\operatorname{\mathbb{F}}_{q}$-gonalities & $B$ \\ & upper bound & (union over all odd $q$) & (for a given odd $q>B$) & \\ \hline $0$ & $1$ & $1$ & 1 & 1 \\ $1$ & $2$ & $2$ & 2 & 1 \\ $2$ & $2$ & $2$ & 2 & 1 \\ $3$ & $3$ & $2,3,4$ & $2,3$ & $29$ \\ $4$ & $3$ & $2,3,4,5$ & $2,3,4$ & $7$ \\ $5$ & $4$ & $2,3,4,5,6^{?}$ & $2,3,4,5$ & $3$ \\ \end{tabular} For background we refer to Section 2.3 (for $g\leq 2$), Lemma 7 (for $g=3$), Lemma 11 (for $g=4$), and Lemma 18, Remark 19 and Remark 20 (for $g=5$). The question mark indicates that over $\operatorname{\mathbb{F}}_{3}$ there might exist curves of genus $g=5$ having $\operatorname{\mathbb{F}}_{3}$-gonality $6$, but there also might not exist such curves, see Remark 20. ### Baker’s bound Throughout a large part of this paper we will use the convenient language of Newton polygons. Let x <figure><img src="content_image/1605.02162/x1.png"><figcaption></figcaption></figure> \[f=\sum_{(i,j)\in\operatorname{\mathbf{Z}}_{\geq 0}^{2}}c_{i,j}x^{i}y^{j}\in k[ x,y]\] be an irreducible polynomial over a field $k$. Then its Newton polygon $\Delta(f)$ is defined as $\operatorname{conv}\left\{\,\left.(i,j)\in\operatorname{\mathbf{Z}}_{\geq 0}^{ 2}\,\right\|\,c_{i,j}\neq 0\,\right\}\subset\operatorname{\mathbf{R}}^{2}$. Note that $\Delta(f)$ lies in the first quadrant and meets the coordinate axes in at least one point each, by the irreducibility of $f$. Let $C$ be the affine curve that is cut out by $f$. Then one has the following bounds on the genus and the gonality of $C$, purely in terms of the combinatorics of $\Delta(f)$. GenusThe genus of $C$ is at most _the number of points in the interior_ of $\Delta(f)$ having integer coordinates: this is Baker’s theorem. See [5, Thm. 2.4] for an elementary proof and [17, §10.5] for a more conceptual version (using adjunction theory on toric surfaces). If one fixes the Newton polygon then Baker’s bound on the genus is generically attained, i.e. meeting the bound is a non-empty Zariski-open condition; this result is essentially due to Khovanskii [38]. An explicit sufficient generic condition is that $f$ is nondegenerate with respect to its Newton polygon [11, Prop. 2.3, Cor. 2.8]. GonalityThe $k$-gonality is at most the _lattice width_$\operatorname{lw}(\Delta(f))$ of $\Delta(f)$. By definition, the lattice width is the minimal height $d$ of a horizontal strip \[\left\{\left.\,(a,b)\in\operatorname{\mathbf{R}}^{2}\,\right\|\,0\leq b\leq d\,\right\}\] inside which $\Delta(f)$ can be mapped using a unimodular transformation, i.e. an affine transformation of $\operatorname{\mathbf{R}}^{2}$ with linear part in $\operatorname{GL}_{2}(\operatorname{\mathbf{Z}})$ and translation part in $\operatorname{\mathbf{Z}}^{2}$. This is discussed in [8, §2], but briefly the argument goes as follows. By applying the same transformation to the exponents, which is a $k$-rational birational change of variables, our polynomial $f$ can be transformed along with its Newton polygon. When orienting $f$ in this way one obtains $\deg_{y}f=\operatorname{lw}(\Delta(f))$, and the gonality bound follows by considering the $k$-rational map $(x,y)\mapsto x$. If a unimodular transformation can be used to transform $\Delta(f)$ into 2.4/2 - 0.2 ($2\Upsilon$) or 2.4/2 - 0.2 ($d\Sigma$) for $d\geq 2$, then the _geometric_ gonality enjoys the sharper bound $\operatorname{lw}(\Delta(f))-1$ (amounting to $3$ resp. $d-1$); see [8, Thm. 3]. If one fixes the Newton polygon then the sharpest applicable foregoing upper bound on the geometric gonality, i.e. * $\operatorname{lw}(\Delta(f))-1$ in the exceptional cases $2\Upsilon$, $d\Sigma$ ($d\geq 2$), * $\operatorname{lw}(\Delta(f))$ in the non-exceptional cases, is generically met, and again nondegeneracy is a sufficient condition [10, Cor. 6.2]. In fact, the slightly weaker condition of meeting Baker’s genus bound is already sufficient [10, §4]. _Remark 4_.: The results from [10] are presented in characteristic zero only, but [10, Cor. 6.2] holds in finite characteristic too, as can be seen as follows. Assume for simplicity that $\Delta(f)$ is not of the form $2\Upsilon$ or $d\Sigma$ for some $d\geq 2$, these cases are easy to deal with separately. Suppose that $C$ meets Baker’s bound but that the gonality of $C$ is strictly less than $\operatorname{lw}(\Delta(f))$, say realized by a map $\pi:C\rightarrow\operatorname{\mathbb{P}}^{1}$. We split this map in the usual way into a purely inseparable and a separable part \[C\stackrel{{ F_{q}}}{{\longrightarrow}}C^{F_{q}}\stackrel{{ \pi_{s}}}{{\longrightarrow}}\operatorname{\mathbb{P}}^{1},\] where $F_{q}$ denotes an appropriate Frobenius power and $C^{F_{q}}$ is the curve defined by $f^{F_{q}}$, the polynomial obtained by applying $F_{q}$ to each coefficient of $f$. Note that $\Delta(f)=\Delta(f^{F_{q}})$, so one sees that $C^{F_{q}}$ also meets Baker’s bound because Frobenius preserves the genus [30, Prop. IV.2.5]. Clearly $\deg\pi_{s}<\operatorname{lw}(\Delta(f^{F_{q}}))$. Now the crucial ingredient in the proof of [10, Cor. 6.2] is a theorem due to Serrano on the possibility of extending morphisms from curves to ambient surfaces, which assumes $\operatorname{char}k=0$. However as Serrano points out [47, Rmk. 3.12] his theorem also holds in finite characteristic, provided that the morphism is separable, the ambient surface $S$ is rational, and $h^{0}(\mathcal{O}_{S}(C))$ is large enough compared to the degree of the morphism to be extended. The reader can verify that these conditions are satisfied when applying the proof of [10, Thm. 6.1] to $\pi_{s}$, leading to the conclusion that it is necessarily of the form $(x,y)\mapsto x^{a}y^{b}$ for some pair of coprime integers $a,b$. This contradicts that $\deg\pi_{s}<\operatorname{lw}(\Delta(f^{F_{q}}))$. Summing up in the non-geometric case, if we are not in the exceptional cases $2\Upsilon,d\Sigma$ ($d\geq 2$) then meeting Baker’s bound is sufficient for the $k$-gonality to equal $\operatorname{lw}(\Delta(f))$. In the exceptional cases the $k$-gonality is either $\operatorname{lw}(\Delta(f))$ or $\operatorname{lw}(\Delta(f))-1$. This yields a large class of defining polynomials $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$ for which finding an $f\in\mathcal{O}_{K}[x,y]$ satisfying (i), (ii) and (iii) is easy. Indeed, by semi-continuity the genus cannot increase under reduction modulo $p$. Therefore if $\overline{f}$ attains Baker’s upper bound on the genus, then it suffices to pick any $f\in\mathcal{O}_{K}[x,y]$ that reduces to $\overline{f}$ mod $p$, in such a way that $\Delta(f)=\Delta(\overline{f})$: the corresponding curve $C/K$ necessarily attains Baker’s upper bound, too. If moreover we are not in the exceptional cases $2\Upsilon$ and $d\Sigma$ ($d\geq 2$), then from the foregoing discussion we know that both the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$ and the $K$-gonality of $C$ are equal to $\operatorname{lw}(\Delta(\overline{f}))=\operatorname{lw}(\Delta(f))$. A unimodular transformation then ensures that $\deg_{y}f=\operatorname{lw}(\Delta(f))$ as desired; such a transformation is computationally easy to find [24]. It is therefore justifiable to say that conditions (i), (ii) and (iii) are easy to deal with for almost all polynomials $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$. But be cautious: this does not mean that almost all _curves_$\overline{C}/\operatorname{\mathbb{F}}_{q}$ are defined by such a polynomial. In terms of moduli, the locus of curves for which this is true has dimension $2g+1$, except if $g=7$ where it is $16$; see [13, Thm. 12.1]. Recall that the moduli space of curves of genus $g$ has dimension $3g-3$, so as soon as $g\geq 5$ the defining polynomial $\overline{f}$ of a plane model of a generic curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ of genus $g$ can never attain Baker’s bound. For such curves, the foregoing discussion becomes _counterproductive_: if we take a naive coefficient-wise lift $f\in\mathcal{O}_{K}[x,y]$ of $\overline{f}$, then it is very likely to satisfy Baker’s bound, causing an increase of genus. This shows that $f$ has to be constructed with more care, which is somehow the main point of this article. ### Preliminary discussion We will attack Problem 1 in the cases where the genus $g$ of $\overline{C}$ is at most five (in Section 3) or the $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma$ of $\overline{C}$ is at most four (in Section 4), where we recall our overall assumption that $q$ is odd. In this section we quickly discuss the cases where $g$ and/or $\gamma$ are at most $2$. _Remark 5_.: Note that for the purpose of computing the Hasse-Weil zeta function using the algorithm from [53, 54], the characteristic $p$ of $\operatorname{\mathbb{F}}_{q}$ should moreover not be too large: this restriction is common to all $p$-adic point counting algorithms. For the lifting methods described in the current paper, the size of $p$ does not play a role. If $\overline{C}$ is a curve of genus $g=0$ then we can assume that $\overline{C}=\mathbb{P}^{1}$, because every plane conic carries at least one $\operatorname{\mathbb{F}}_{q}$-point, and projection from that point gives an isomorphism to the line. In particular $\gamma=1$ if and only if $g=0$, in which case Problem 1 can be addressed by simply outputting $f=y$. Next, if $g=1$ then we can assume that $\overline{C}$ is defined by a polynomial $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$ in Weierstrass form, i.e. $\overline{f}=y^{2}-\overline{h}(x)$ for some squarefree cubic $\overline{h}(x)\in\operatorname{\mathbb{F}}_{q}[x]$. In this case $\gamma=2$, and any $f\in\mathcal{O}_{K}[x,y]$ for which $\Delta(f)=\Delta(\overline{f})$ will address Problem 1 (for instance because Baker’s bound is attained, or because a non-zero discriminant must lift to a non-zero discriminant). Finally, if $g\geq 2$ then $\overline{C}$ is geometrically hyperelliptic if and only if $\kappa$ realizes $\overline{C}$ as a degree $2$ cover of a curve of genus zero [30, IV.5.2-3]. By the foregoing discussion the latter is isomorphic to $\operatorname{\mathbb{P}}^{1}$, and therefore every geometrically hyperelliptic curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ admits an $\operatorname{\mathbb{F}}_{q}$-rational degree $2$ map to $\operatorname{\mathbb{P}}^{1}$. In particular, one can unambiguously talk about hyperelliptic curves over $\operatorname{\mathbb{F}}_{q}$. In this case it is standard how to produce a defining polynomial $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$ that is in Weierstrass form, i.e. $\overline{f}=y^{2}-\overline{h}(x)$ for some squarefree $\overline{h}(x)\in\operatorname{\mathbb{F}}_{q}[x]$. Then again any $f\in\mathcal{O}_{K}[x,y]$ for which $\Delta(f)=\Delta(\overline{f})$ will address Problem 1. _Remark 6_.: Let $g^{1}_{d}$ be a complete base-point free $\operatorname{\mathbb{F}}_{q}$-rational linear pencil of degree $d$ on a non-singular projective curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$. Then from standard arguments in Galois cohomology (that are specific to finite fields) it follows that this $g^{1}_{d}$ automatically contains an $\operatorname{\mathbb{F}}_{q}$-rational effective divisor, which can be used to construct an $\operatorname{\mathbb{F}}_{q}$-rational map to $\operatorname{\mathbb{P}}^{1}$ of degree $d$. See for instance the proof of [27, Lem. 6.5.3]. This gives another way of seeing that a geometrically hyperelliptic curve over $\operatorname{\mathbb{F}}_{q}$ is automatically $\operatorname{\mathbb{F}}_{q}$-hyperelliptic, because the hyperelliptic pencil $g^{1}_{2}$ is unique, hence indeed defined over $\operatorname{\mathbb{F}}_{q}$. The advantage of this argument is that it is more flexible: for instance it also shows that a geometrically trigonal curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ of genus $g\geq 5$ always admits an $\operatorname{\mathbb{F}}_{q}$-rational degree $3$ map to $\operatorname{\mathbb{P}}^{1}$, again because the $g^{1}_{3}$ on such a curve is unique. So we can unambiguously talk about trigonal curves from genus five on. Summing up, throughout the paper, it suffices to consider curves of $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma>2$, so that the canonical map $\kappa:\overline{C}\rightarrow\operatorname{\mathbb{P}}^{g-1}$ is an embedding. In particular we have $g\geq 3$. From the $p$-adic point counting viewpoint, all omitted cases are covered by the algorithms of Satoh [43] and Kedlaya [29, 37]. ## 3 Curves of low genus ### Curves of genus three #### 3.1.1 Lifting curves of genus three Solving Problem 1 in genus three in its basic version is not hard, so we consider this as a warm-up discussion. We first analyze which $\operatorname{\mathbb{F}}_{q}$-gonalities can occur: Lemma 7.: _Let $\overline{C}/\operatorname{\mathbb{F}}_{q}$ be a non-hyperelliptic curve of genus $3$ and $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma$, and assume that $q$ is odd. If $\#\overline{C}(\operatorname{\mathbb{F}}_{q})=0$ then $\gamma=4$, while if $\#\overline{C}(\operatorname{\mathbb{F}}_{q})>0$ (which is guaranteed if $q>29$) then $\gamma=3$._ Proof.: Using the canonical embedding we can assume that $\overline{C}$ is a smooth plane quartic. It is classical that such curves have geometric gonality $3$, and that each gonal map arises as projection from a point on the curve. For a proof see [47, Prop. 3.13], where things are formulated in characteristic zero, but the same argument works in positive characteristic; alternatively one can consult [33]. In particular if there is no $\operatorname{\mathbb{F}}_{q}$-point then there is no rational gonal map and $\gamma>3$. But then a degree $4$ map can be found by projection from an $\operatorname{\mathbb{F}}_{q}$-point outside the curve. By [34, Thm. 3(2)] there exist pointless non-hyperelliptic curves of genus three over $\operatorname{\mathbb{F}}_{q}$ if and only if $q\leq 23$ or $q=29$. ∎ We can now address Problem 1 as follows. As in the proof we assume that $\overline{C}$ is given as a smooth quartic in $\operatorname{\mathbb{P}}^{2}$. First suppose that $\#\overline{C}(\operatorname{\mathbb{F}}_{q})=0$. Because this is possible for $q\leq 29$ only, the occurrence of this event can be verified exhaustively. In this case the Newton polygon of the defining polynomial $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$ of the affine part of $\overline{C}$ equals: 2.4/2 - 0.2 ($\Delta_{3}^{0,0}$) In particular Baker’s bound is attained, and a naive Newton polygon preserving lift $f\in\mathcal{O}_{K}[x,y]$ automatically addresses (i), (ii) and (iii). If $\#\overline{C}(\operatorname{\mathbb{F}}_{q})>0$ then one picks a random $\operatorname{\mathbb{F}}_{q}$-point $P$ (which can be found quickly) and one applies a projective transformation that maps $P$ to $(0:1:0)$. After doing so the Newton polygon of $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$ becomes contained in (and typically equals): 2/2 - 0.2 ($\Delta_{3}^{1,0}$) Again Baker’s bound is attained, and a naive Newton polygon preserving lift $f\in\mathcal{O}_{K}[x,y]$ satisfies (i), (ii) and (iii). It is important to transform the curve _before_ lifting to characteristic $0$. Indeed, if one would immediately lift our input quartic to a curve $C\subset\operatorname{\mathbf{P}}^{2}$ then it is highly likely that $C(K)=\emptyset$, and therefore that the $K$-gonality equals $4$ (by the same proof as above). This type of reasoning plays an important role throughout the paper, often in a more subtle way than here. _Remark 8_ (purely notational).: The indices $i,j$ in $\Delta_{3}^{i,j}$ refer to the multiplicities of intersection of $\overline{C}$ with the line at infinity at the coordinate points $(0:1:0)$ and $(1:0:0)$, assuming that it is defined by a polynomial having Newton polygon $\Delta_{3}^{i,j}$. Note that $\Delta_{3}^{0,0}$ is just another way of writing $3\Sigma$. Algorithm 9.: Lifting curves of genus $3$: basic solution Input: non-hyperelliptic genus $3$ curve $\overline{C}$ over $\operatorname{\mathbb{F}}_{q}$ Output: lift $f\in\mathcal{O}_{K}[x,y]$ satisfying (i), (ii), (iii) that is supported $\bullet$ on $\Delta_{3}^{0,0}$ if $\overline{C}(\operatorname{\mathbb{F}}_{q})=\emptyset$, or else $\bullet$ on $\Delta_{3}^{2,0}$ 1 : $\overline{C}\leftarrow\text{CanonicalImage}(\overline{C})$ in $\operatorname{\mathbb{P}}^{2}=\operatorname{Proj}\operatorname{\mathbb{F}}_{q} [X,Y,Z]$ 2 : if$q>29$ or $\overline{C}(\operatorname{\mathbb{F}}_{q})\neq\emptyset$ (verified exhaustively) then 3 : $P:=\text{Random}(\overline{C}(\operatorname{\mathbb{F}}_{q}))$ 4 : apply automorphism of $\operatorname{\mathbb{P}}^{2}$ transforming $T_{P}(\overline{C})$ into $Z=0$ 5 : apply automorphism of $\operatorname{\mathbb{P}}^{2}$ transforming and $P$ into $(0:1:0)$ 6 : return NaiveLift(Dehomogenization${}_{Z}$(DefiningPolynomial($\overline{C}$))) #### 3.1.2 Optimizations For point counting purposes we can of course assume that $q>29$, so that $\gamma=3$. By applying (1) to a polynomial with Newton polygon $\Delta_{3}^{1,0}$ one ends up with a polynomial that is monic in $y$ and that has degree $4+(\gamma-1)=6$ in $x$. This can be improved: in addition to mapping $P$ to $(0:1:0)$, we can have its tangent line $T_{P}(\overline{C})$ sent to the line at infinity. If we then lift $\overline{f}$ to $\mathcal{O}_{K}[x,y]$ we find an $f$ whose Newton polygon is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{3}^{\text{2,0}}$) In particular $f$ is monic (up to a scalar) and $\deg_{x}f\leq 4$. We can in fact achieve $\deg_{x}f=3$ in all cases of practical interest. Indeed, with an asymptotic chance of $1/2$ our tangent line $T_{P}(\overline{C})$ intersects $\overline{C}$ in two other rational points. The above construction leaves enough freedom to position one of those points $Q$ at $(1:0:0)$. The resulting lift $f$ then becomes contained in (and typically equals) 2/2 - 0.2 ($\Delta_{3}^{\text{2,1}}$) In the case of failure we retry with another $P$. If $q>59$ (say) then there are enough $\operatorname{\mathbb{F}}_{q}$-points $P\in\overline{C}$ for this approach to work with near certainty, although there might exist sporadic counterexamples well beyond that point. _Remark 10_ (non-generic optimizations).: For large values of $q$ one might want to pursue a further compactification of the Newton polygon. Namely, if one manages to choose $P\in\overline{C}(\operatorname{\mathbb{F}}_{q})$ such that it is an ordinary flex or such that $T_{P}(\overline{C})$ is a bitangent, then $T_{P}(\overline{C})$ meets $\overline{C}$ in a unique other point $Q$, which is necessarily defined over $\operatorname{\mathbb{F}}_{q}$. By proceeding as before one respectively ends up inside the first and second polygon below. If one manages to let $P\in\overline{C}(\operatorname{\mathbb{F}}_{q})$ be a non-ordinary flex, i.e. a hyperflex, then positioning it at $(0:1:0)$ results in a polygon of the third form: 2/2 - 0.2 ($\Delta_{3}^{\text{3,1}}$) 2/2 - 0.2 ($\Delta_{3}^{\text{2,2}}$) 2/2 - 0.2 ($\Delta_{3}^{\text{4,0}}$) Heuristically, as $q\rightarrow\infty$ we expect to be able to realize the first two polygons with probablities $1-1/e$ and $1-1/\sqrt{e}$, respectively; more background can be found in an arXiv version of our paper (1605.02162v2). In contrast the hyperflex case $\Delta_{3}^{\text{4,0}}$ is very exceptional, but we included it in the discussion because it corresponds to the well-known class of $C_{3,4}$ curves: even though $\deg_{x}f=4$ here, the corresponding point count is slightly faster. #### 3.1.3 Implementation We now report on timings, memory usage and failure rates of our implementation of the algorithms in this section for various values of $p$ and $q=p^{n}$. The first column in each table contains the time used to compute the lift to characteristic $0$ averaged over $1000$ random examples. Then the second column gives the time used by the point counting code pcc from [53, 54] averaged over $10$ different random examples. Next, the third column contains the total memory used in the computation. Finally, the last column gives the number of examples out of the $1000$ where we did not find a lift satisfying [54, Ass. 1], which each time turned out to be $0$, i.e. we always found a good lift. \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $p$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & 0.2 & $0.2$ & $32$ & 0 \\ $67$ & 0.2 & $0.6$ & $32$ & 0 \\ $521$ & 0.2 & $4.2$ & $64$ & 0 \\ $4099$ & 0.2 & $41$ & $165$ & 0 \\ $32771$ & 0.2 & $590$ & $1124$ & 0 \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $0.4$ & $2.4$ & $64$ & 0 \\ $7^{5}$ & $0.4$ & $6.6$ & $64$ & 0 \\ $17^{5}$ & $0.4$ & $12$ & $76$ & 0 \\ $37^{5}$ & $0.4$ & $26$ & $124$ & 0 \\ $79^{5}$ & $0.4$ & $66$ & $241$ & 0 \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $0.5$ & $15$ & $76$ & 0 \\ $7^{10}$ & $0.6$ & $40$ & $118$ & 0 \\ $17^{10}$ & $0.7$ & $82$ & $241$ & 0 \\ $37^{10}$ & $0.7$ & $181$ & $403$ & 0 \\ $79^{10}$ & $0.8$ & $473$ & $831$ & 0 \\ \end{tabular} Alternatively, without using the methods from this section, we can just make any plane quartic monic using (1), then lift naively to characteristic $0$ and try to use this lift as input for pcc. This way, we obtain the following three tables. \begin{tabular}{r\|\|r\|r\|r} & time & space & fails \\ $p$ & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & $0.4$ & $32$ & 225 \\ $67$ & $1.3$ & $32$ & 52 \\ $521$ & $8.7$ & $76$ & 5 \\ $4099$ & $83$ & $307$ & 1 \\ $32771$ & $1153$ & $2086$ & 0 \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r} & time & space & fails \\ $q$ & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $6.1$ & $32$ & $13$ \\ $7^{5}$ & $14$ & $32$ & $0$ \\ $17^{5}$ & $32$ & $80$ & $0$ \\ $37^{5}$ & $71$ & $156$ & $0$ \\ $79^{5}$ & $161$ & $288$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r} & time & space & fails \\ $q$ & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $42$ & $76$ & $0$ \\ $7^{10}$ & $94$ & $124$ & $0$ \\ $17^{10}$ & $248$ & $320$ & $0$ \\ $37^{10}$ & $524$ & $589$ & $0$ \\ $79^{10}$ & $1296$ & $1311$ & $0$ \\ \end{tabular} Comparing the different tables, we see that the approach described in this section saves a factor of about $3$ in runtime and a factor of about $2$ in memory usage. Moreover, for small fields the naive lift of a plane quartic sometimes does not satisfy [54, Ass. 1], while this never seems to be the case for the lift constructed using our methods. ### Curves of genus four #### 3.2.1 Lifting curves of genus four By [30, Ex. IV.5.2.2] the ideal of a canonical model $\overline{C}\subset\operatorname{\mathbb{P}}^{3}=\text{Proj}\,\operatorname{ \mathbb{F}}_{q}[X,Y,Z,W]$ of a non-hyperelliptic genus $g=4$ curve is generated by a cubic $\overline{S}_{3}$ and a unique quadric $\overline{S}_{2}$. Since $q$ is assumed odd, the latter can be written as \[\begin{pmatrix}X&Y&Z&W\end{pmatrix}\cdot\overline{M}\cdot\begin{pmatrix}X&Y&Z& W\end{pmatrix}^{t},\qquad\overline{M}\in\operatorname{\mathbb{F}}_{q}^{4\times 4 },\ \overline{M}^{t}=\overline{M}.\] Let $\chi_{2}:\operatorname{\mathbb{F}}_{q}\rightarrow\{0,\pm 1\}$ denote the quadratic character on $\operatorname{\mathbb{F}}_{q}$. Then $\chi_{2}(\det\overline{M})$ is an invariant of $\overline{C}$, which is called the discriminant. If we let $S_{2},S_{3}\in\mathcal{O}_{K}[X,Y,Z,W]$ be homogeneous polynomials that reduce to $\overline{S}_{2}$ and $\overline{S}_{3}$ modulo $p$, then by [30, Ex. IV.5.2.2] these define a genus $4$ curve $C\subset\operatorname{\mathbf{P}}^{3}$ over $K$, thereby addressing (i) and (ii). However as mentioned in Section 2.1 we expect the $K$-gonality of $C$ to be typically $2g-2=6$. This exceeds the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$: Lemma 11.: _Let $\overline{C}/\operatorname{\mathbb{F}}_{q}$ be a non-hyperelliptic curve of genus $4$ and $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma$, and assume that $q$ is odd. If the discriminant of $\overline{C}$ is $0$ or $1$ then $\gamma=3$. If it is $-1$ and $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})>0$ (which is guaranteed if $q>7$) then $\gamma=4$. Finally, if it is $-1$ and $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})=0$ then $\gamma=5$._ Proof.: By [30, Ex. IV.5.5.2] our curve carries one or two geometric $g^{1}_{3}$’s, depending on whether the quadric $\overline{S}_{2}$ is singular (discriminant $0$) or not. In the former case the quadric is a cone, and the $g^{1}_{3}$ corresponds to projection from the top. This is automatically defined over $\operatorname{\mathbb{F}}_{q}$. In the latter case the quadric is $\operatorname{\mathbb{F}}_{q^{2}}$-isomorphic to the hyperboloid $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}\subset \operatorname{\mathbb{P}}^{3}$ and the $g^{1}_{3}$’s correspond to the two rulings of the latter. If the isomorphism can be defined over $\operatorname{\mathbb{F}}_{q}$ (discriminant $1$) then the $g^{1}_{3}$’s are $\operatorname{\mathbb{F}}_{q}$-rational. In the other case (discriminant $-1$) the smallest field of definition is $\operatorname{\mathbb{F}}_{q^{2}}$. So we can assume that the discriminant of $\overline{C}$ is $-1$, and therefore that $\gamma>3$. Now suppose that $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})>0$, which is guaranteed if $q>7$ by [34, Thm. 2]. If there is an $\operatorname{\mathbb{F}}_{q}$-point then let $\overline{\ell}$ be the tangent line to $\overline{C}$ at it. In the other case we can find two conjugate $\operatorname{\mathbb{F}}_{q^{2}}$-points, and we let $\overline{\ell}$ be the line connecting both. In both cases $\overline{\ell}$ is defined over $\operatorname{\mathbb{F}}_{q}$, and the pencil of planes through $\overline{\ell}$ cuts out a $g^{1}_{4}$, as wanted. The argument can be reversed: if there exists a $g^{1}_{4}$ containing an effective $\operatorname{\mathbb{F}}_{q}$-rational divisor $D$, then by Riemann-Roch we find that $\|K-D\|$ is non-empty. In particular there exists an effective $\operatorname{\mathbb{F}}_{q}$-rational divisor of degree $\deg(K-D)=2$ on $\overline{C}$, and $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})>0$. So if $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})=0$ then $\gamma>4$. Now note that $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{5}})>0$ by the Weil bound. So $\overline{C}$ carries an effective divisor $D$ of degree $5$. The linear system $\|K-D\|$ must be empty, for otherwise there would exist an $\operatorname{\mathbb{F}}_{q}$-point on $\overline{C}$. But then Riemann-Roch implies that $\dim\|D\|=1$, i.e. our curve carries an $\operatorname{\mathbb{F}}_{q}$-rational $g^{1}_{5}$. ∎ _Remark 12_.: An example of a genus four curve $\overline{C}/\operatorname{\mathbb{F}}_{3}$ having $\operatorname{\mathbb{F}}_{3}$-gonality five can be found in an arXiv version of our paper (1605.02162v2). To address Problem 1 in the non-hyperelliptic genus $4$ case we make a case-by-case analysis. $\chi_{2}(\det\overline{M}_{2})=0$In this case $\overline{S}_{2}$ is a cone over a conic. A linear change of variables takes $\overline{S}_{2}$ to the form $ZW-X^{2}$, which we note is one of the standard realizations inside $\operatorname{\mathbb{P}}^{3}$ of the weighted projective plane $\operatorname{\mathbb{P}}(1,2,1)$. It is classical how to find such a linear change of variables (diagonalization, essentially). Projecting from $(0:0:0:1)$ on the $XYZ$-plane amounts to eliminating the variable $W$, to obtain \[Z^{3}\overline{S}_{3}(X,Y,Z,\frac{X^{2}}{Z})=\overline{S}_{3}(XZ,YZ,Z^{2},X^{2 }).\] (2) After dehomogenizing with respect to $Z$, renaming $X\gets x$ and $Y\gets y$ and rescaling if needed, we obtain an affine equation $\overline{f}=y^{3}+\overline{f}_{2}(x)y^{2}+\overline{f}_{4}(x)y+\overline{f}_ {6}(x)$, with $\overline{f}_{i}\in\operatorname{\mathbb{F}}_{q}[x]$ of degree at most $i$. Its Newton polygon is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{4,0}^{0}$) So Baker’s bound is attained and we take for $f\in\mathcal{O}_{K}[x,y]$ a naive coefficient-wise lift. $\chi_{2}(\det\overline{M}_{2})=1$In this case $\overline{S}_{2}$ is a hyperboloid. A linear change of variables takes $\overline{S}_{2}$ to the standard form $XY-ZW$, which we note is the image of $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ in $\operatorname{\mathbb{P}}^{3}$ under the Segre embedding. Projection from $(0:0:0:1)$ on the $XYZ$-plane amounts to eliminating the variable $W$, to obtain \[Z^{3}\overline{S}_{3}(X,Y,Z,\frac{XY}{Z})=\overline{S}_{3}(XZ,YZ,Z^{2},XY).\] After dehomogenizing with respect to $Z$ and renaming $X\gets x$ and $Y\gets y$ we obtain an affine equation $\overline{f}=\overline{f}_{0}(x)y^{3}+\overline{f}_{1}(x)y^{2}+\overline{f}_{2 }(x)y+\overline{f}_{3}(x)$ with all $\overline{f}_{i}\in\operatorname{\mathbb{F}}_{q}[x]$ of degree at most $3$. Its Newton polygon is contained in (and typically equals) 2/2 - 0.2 ($\Delta_{4,1}^{0}$) So Baker’s bound is attained and we can take for $f\in\mathcal{O}_{K}[x,y]$ a coefficient-wise lift of $\overline{f}$. $\chi_{2}(\det\overline{M}_{2})=-1$This is our first case where in general no plane model can be found for which Baker’s bound is attained [13, §6]. If $\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})=\emptyset$, or in other words if $\gamma=5$, then unfortunately we do not know how to address Problem 1. We therefore assume that $\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})\neq\emptyset$ and hence that $\gamma=4$. This is guaranteed if $q>7$, so for point counting purposes this is amply sufficient. We follow the proof of Lemma 11: by exhaustive search we find a point $P\in\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})$ along with its Galois conjugate $P^{\prime}$ and consider the line $\overline{\ell}$ connecting both (tangent line if $P=P^{\prime}$). This line is defined over $\operatorname{\mathbb{F}}_{q}$, so that modulo a projective transformation we can assume that $\overline{\ell}:X=Z=0$. When plugging in $X=Z=0$ in $\overline{S}_{2}$ we find a non-zero quadratic expression in $Y$ and $W$. Indeed: $\overline{S}_{2}$ cannot vanish identically on $\overline{\ell}$ because no three points of $\overline{S}_{2}(\operatorname{\mathbb{F}}_{q})$ are collinear. Because $\overline{C}$ intersects $\overline{\ell}$ in two points (counting multiplicities) we find that \[\overline{S}_{3}(0,Y,0,W)=(\overline{a}Y+\overline{b}W)\overline{S}_{2}(0,Y,0,W)\] for certain $\overline{a},\overline{b}\in\operatorname{\mathbb{F}}_{q}$ that are possibly zero. Lift $\overline{S}_{2}$ coefficient-wise to a homogenous quadric $S_{2}\in\mathcal{O}_{K}[X,Y,Z,W]$ and let $a,b\in\mathcal{O}_{K}$ reduce to $\overline{a},\overline{b}$ mod $p$. We now construct $S_{3}\in\mathcal{O}_{K}[X,Y,Z,W]$ as follows: for the coefficients at $Y^{3},Y^{2}W,YW^{2},W^{3}$ we make the unique choice for which \[S_{3}(0,Y,0,W)=(aY+bW)S_{2}(0,Y,0,W),\] while the other coefficients are randomly chosen lifts of the corresponding coefficients of $\overline{S}_{3}$. Then the genus $4$ curve $C\subset\operatorname{\mathbf{P}}^{3}$ defined by $S_{2}$ and $S_{3}$ is of gonality $4$. Indeed, it is constructed such that the line $\ell:X=Z=0$ intersects the curve in two points (possibly over a quadratic extension), and the pencil of planes through this line cuts out a $g^{1}_{4}$. Now we project our lift $C\subset\operatorname{\mathbf{P}}^{3}$ from $(0:0:0:1)$ to a curve in $\operatorname{\mathbf{P}}^{2}$. This amounts to eliminating $W$ from $S_{2}$ and $S_{3}$. By dehomogenizing the resulting sextic with respect to $Z$, and by renaming $X\gets x$ and $Y\gets y$ we end up with a polynomial $f\in\mathcal{O}_{K}[x,y]$ whose Newton polygon is contained in (and typically equals): 2.4/2 - 0.2 ($\Delta_{4,-1}^{6}$) Geometrically, what happens is that the points of $C$ on $\ell$ are both mapped to $(0:1:0)$ under projection from $(0:0:0:1)$, creating a singularity there, which in terms of the Newton polygon results in $6\Sigma$ with its top chopped off. The polynomial $f$ satisfies (i), (ii) and (iii) from Problem 1. Note that Baker’s bound is usually _not_ attained here: it gives an upper bound of $9$, while $C$ has genus $4$. So it is crucial to lift the equations to $\mathcal{O}_{K}$_before_ projecting on the plane. Algorithm 13.: Lifting curves of genus $4$: basic solution Input: non-hyperelliptic genus $4$ curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ of $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma\leq 4$ Output: lift $f\in\mathcal{O}_{K}[x,y]$ satisfying (i), (ii), (iii) that is supported $\bullet$ on $\Delta_{4,0}^{0}$ if the discriminant is $0$, or else $\bullet$ on $\Delta_{4,1}^{0}$ if the discriminant is $1$, or else $\bullet$ on $\Delta_{4,-1}^{6}$ 01 : $\overline{C}\leftarrow\text{CanonicalImage}(\overline{C})$ in $\operatorname{\mathbb{P}}^{3}=\operatorname{Proj}\operatorname{\mathbb{F}}_{q} [X,Y,Z,W]$ 02 : $\overline{S}_{2}\leftarrow\text{unique quadric in Ideal}(\overline{C})$; $\overline{M}_{2}\leftarrow\text{Matrix}(\overline{S}_{2})$; $\chi\leftarrow\chi_{2}(\det\overline{M}_{2})$ 03 : $\overline{S}_{3}\leftarrow\text{cubic that along with }\overline{S}_{2}\text{ generates Ideal}(\overline{C})$ 04 : if$\chi=0$then 05 : apply automorphism of $\operatorname{\mathbb{P}}^{3}$ transforming $\overline{S}_{2}=0$ into $ZW-X^{2}=0$ 06 : return NaiveLift(Dehomogenization${}_{Z}$($\overline{S}_{3}(XZ,YZ,Z^{2},X^{2})$)) 07 : else if$\chi=1$then 08 : apply automorphism of $\operatorname{\mathbb{P}}^{3}$ transforming $\overline{S}_{2}=0$ into $XY-ZW=0$ 09 : return NaiveLift(Dehomogenization${}_{Z}$($\overline{S}_{3}(XZ,YZ,Z^{2},XY)$)) 10 : else 11 : $P:=\text{Random}(\overline{C}(\operatorname{\mathbb{F}}_{q^{2}}))$; $P^{\prime}:=\text{Conjugate}(P)$ 12 : $\overline{\ell}\leftarrow\text{line through $P$ and $P^{\prime}$ (tangent line if $P=P^{\prime}$)}$ 13 : apply automorphism of $\operatorname{\mathbb{P}}^{3}$ transforming $\overline{\ell}$ into $X=Z=0$ 14 : $S_{2}\leftarrow\text{NaiveLift}(\overline{S}_{2})$ 15 : $S_{3}\leftarrow$ lift of $\overline{S}_{3}$ satisfying $S_{3}(0,Y,0,W)=(aY+bW)S_{2}(0,Y,0,W)$ for $a,b\in\mathcal{O}_{K}$ 16 : return Dehomogenization${}_{Z}$(res${}_{W}$($S_{2},S_{3}$)) #### 3.2.2 Optimizations $\chi_{2}(\det\overline{M}_{2})=0$By applying (1) to a polynomial with Newton polygon $\Delta_{4,0}^{0}$ one ends up with a polynomial that is monic in $y$ and that has degree $6$ in $x$. This can be improved as soon as $\overline{C}(\operatorname{\mathbb{F}}_{q})\neq 0$, which is guaranteed if $q>49$ by [34, Thm. 2]. Namely we can view (2) as the defining equation of a smooth curve in the weighted projective plane $\operatorname{\mathbb{P}}(1,2,1)$. Using an automorphism of the latter we can position a given $\operatorname{\mathbb{F}}_{q}$-rational point $P$ at $(1:0:0)$ and the corresponding tangent line at $X=0$, in order to end up with a Newton polygon that is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{4,0}^{1}$) See Remark 14 below for how to do this in practice. So we find $\deg_{x}f=4$, which is optimal because the $g^{1}_{3}$ is unique in the case of a singular $\overline{S}_{2}$. There is a caveat here, in that the tangent line at $P$ might exceptionally be vertical, i.e. $P$ might be a ramification point of our degree $3$ map $(x,y)\mapsto x$. In this case it is impossible to position this line at $X=0$, but in practice one can simply retry with another $P$. But in fact having a vertical tangent line is an even slightly better situation, as explained in Remark 15 below. _Remark 14_.: The automorphisms of $\operatorname{\mathbb{P}}(1,2,1)$ can be applied directly to $\overline{f}$. They correspond to * substituting $y\leftarrow\overline{a}y+\overline{b}x^{2}+\overline{c}x+\overline{d}$ and $x\leftarrow\overline{a}^{\prime}x+\overline{b}^{\prime}$ in $\overline{f}$ for some $\overline{a},\overline{a}^{\prime}\in\operatorname{\mathbb{F}}_{q}^{\ast}$ and $\overline{b},\overline{b}^{\prime},\overline{c},\overline{d}\in\operatorname{ \mathbb{F}}_{q}$, * exchanging the line at infinity for the $y$-axis by replacing $\overline{f}$ by $x^{6}\overline{f}(x^{-1},x^{-2}y)$, or to a composition of both. For instance imagine that an affine point $P=(\overline{a},\overline{b})$ was found with a non-vertical tangent line. Then $\overline{f}\leftarrow\overline{f}(x+\overline{a},y+\overline{b})$ translates this point to the origin, at which the tangent line becomes of the form $y=\overline{c}x$. Substituting $\overline{f}\leftarrow\overline{f}(x,y+\overline{c}x)$ positions this line horizontally, and finally replacing $\overline{f}$ by $x^{6}\overline{f}(x^{-1},x^{-2}y)$ results in a polynomial with Newton polygon contained in $\Delta_{4,0}^{1}$. _Remark 15_ (non-generic optimizations).: If $P$ has a vertical tangent line then positioning it at $(1:0:0)$ results in a Newton polygon that is contained in (and typically equals) the first polygon below: 2/2 - 0.2 ($\Delta_{4,0}^{2}$) 2/2 - 0.2 ($\Delta_{4,0}^{3}$) Even though $\deg_{x}f=5$ here, this results in a slightly faster point count. Such a $P$ will exist if and only if the ramification scheme of $(x,y)\mapsto x$ has an $\operatorname{\mathbb{F}}_{q}$-rational point. Following the same heuristic as in Remark 10 we expect that this works in about $1-1/e$ of the cases. If there exists a point of ramification index $3$ then one can even end up inside the second polygon. This event is highly exceptional, but we include it in our discussion because this corresponds to the well-known class of $C_{3,5}$ curves. $\chi_{2}(\det\overline{M}_{2})=1$By applying (1) to a polynomial with Newton polygon $\Delta_{4,1}^{0}$ one ends up with a polynomial that is monic in $y$ and that has degree $3+(\gamma-1)3=9$ in $x$. This can be improved as soon as $\overline{C}(\operatorname{\mathbb{F}}_{q})\neq 0$, which is guaranteed if $q>49$ by [34, Thm. 2]. Assume as before that $\overline{S}_{2}$ is in the standard form $XY-ZW$. So it is the image of the Segre embedding \[\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1} \hookrightarrow\operatorname{\mathbb{P}}^{3}:((X_{0}:Z_{0}),(Y_{0}:W_{0})) \mapsto(X_{0}W_{0}:Y_{0}Z_{0}:Z_{0}W_{0}:X_{0}Y_{0}).\] (3) That is: we can view $\overline{C}$ as the curve in $\mathbb{P}^{1}\times\mathbb{P}^{1}$ defined by the bihomogeneous polynomial \[\overline{S}_{3}(X_{0}W_{0},Y_{0}Z_{0},Z_{0}W_{0},X_{0}Y_{0})\] of bidegree $(3,3)$. Remark that if we dehomogenize with respect to both $Z_{0}$ and $W_{0}$ and rename $X_{0}\gets x$ and $Y_{0}\gets y$ then we get the polynomial $\overline{f}$ from before. Now if our curve has a rational point $P$, by applying an appropriate projective transformation in each component we can arrange that $P=((1:0),(1:0))$. If we then dehomogenize we end up with a Newton polygon that is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{4,1}^{1}$) So Baker’s bound is attained and we take for $f\in\mathcal{O}_{K}[x,y]$ a naive coefficient-wise lift. Now applying (1) typically results in a polynomial of degree $3+(\gamma-1)2=7$ in $x$. _Remark 16_.: The automorphisms of $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ can again be applied directly to $\overline{f}$. They correspond to * substituting $y\leftarrow\overline{a}y+\overline{b}$ and $x\leftarrow\overline{a}^{\prime}x+\overline{b}^{\prime}$ in $\overline{f}$ for some $\overline{a},\overline{a}^{\prime}\in\operatorname{\mathbb{F}}_{q}^{\ast}$ and $\overline{b},\overline{b}^{\prime}\in\operatorname{\mathbb{F}}_{q}$, * exchanging the $x$-axis for the horizontal line at infinity by replacing $\overline{f}$ by $y^{3}\overline{f}(x,y^{-1})$, * exchanging the $y$-axis for the vertical line at infinity by replacing $\overline{f}$ by $x^{3}\overline{f}(x^{-1},y)$, or to a composition of these. For instance imagine that an affine point $P=(\overline{a},\overline{b})$ was found, then $\overline{f}\leftarrow\overline{f}(x+\overline{a},y+\overline{b})$ translates this point to the origin, and subsequently replacing $\overline{f}$ by $x^{3}y^{3}\overline{f}(x^{-1},y^{-1})$ results in a polynomial with Newton polygon contained in $\Delta_{4,1}^{1}$. _Remark 17_ (non-generic optimizations).: If one manages to let $P$ be a point with a horizontal tangent line, i.e. if $P$ is a ramification point of the projection map from $\overline{C}$ onto the second component of $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$, then the Newton polygon even becomes contained in (and typically equals): 2/2 - 0.2 ($\Delta_{4,1}^{2}$) This eventually results in a polynomial $f\in\mathcal{O}_{K}[x,y]$ of degree $3+(\gamma-1)1=5$ in $x$. As in the discriminant $0$ case, we heuristically expect the probability of success to be about $1-1/e$. However, it is also fine to find a ramification point of the projection of $\overline{C}$ onto the first component of $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$, because we can change the role of $(X_{0},Z_{0})$ and $(Y_{0},W_{0})$ if wanted. Assuming independence of events, the percentage of non-hyperelliptic genus $4$ curves with discriminant $1$ that admit a Newton polygon of the form $\Delta_{4,1}^{2}$ should be approximately $1-1/e^{2}$. $\chi_{2}(\det\overline{M}_{2})=-1$By applying (1) to a polynomial with Newton polygon $\Delta_{4,-1}^{6}$ we end up with a polynomial that is monic in $y$ and that has degree $3+(\gamma-1)2=9$. This can be improved as soon as $\overline{C}(\operatorname{\mathbb{F}}_{q})\neq 0$, which is guaranteed if $q>49$ by [34, Thm. 2]. In this case we redo the construction with $\overline{\ell}$ the tangent line to a point $P\in\overline{C}(\operatorname{\mathbb{F}}_{q})$. As before we apply a projective transformation to obtain $\overline{\ell}:X=Z=0$, but in addition we make sure that $P=(0:0:0:1)$. This implies that $\overline{S}_{2}(0,Y,0,W)=Y^{2}$, possibly after multiplication by a scalar. We now proceed as before, to find lifts $S_{2},S_{3}\in\mathcal{O}_{K}[X,Y,Z,W]$ that cut out a genus $4$ curve $C\subset\operatorname{\mathbf{P}}^{3}$, still satisfying the property of containing $(0:0:0:1)$ with corresponding tangent line $\ell:X=Z=0$. If we then project from $(0:0:0:1)$ we end up with a quintic in $\operatorname{\mathbf{P}}^{2}$, rather than a sextic. The quintic still passes through the point $(0:1:0)$, which is now non-singular: otherwise the pencil of lines through that point would cut out a $K$-rational $g^{1}_{3}$. We can therefore apply a projective transformation over $K$ that maps the corresponding tangent line to infinity, while keeping the point at $(0:1:0)$. After having done so, we dehomogenize to find a polynomial $f\in\mathcal{O}_{K}[x,y]$ whose Newton polygon is contained in (and typically equals) 2.4/2 - 0.2 ($\Delta_{4,-1}^{5}$) It still satisfies (i), (ii) and (iii), while here $\deg_{x}f\leq 5$. #### 3.2.3 Implementation The tables below contain timings, memory usage and failure rates for $\chi_{2}=0,1,-1$ and various values of $p$ and $q=p^{n}$. For the precise meaning of the various entries in the table see Section 3.1.3. $\mathbf{\chi_{2}=0}$ \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $p$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & $0.01$ & $0.3$ & $32$ & $159$ \\ $67$ & $0.01$ & $1.4$ & $32$ & $2$ \\ $521$ & $0.01$ & $13$ & $73$ & $2$ \\ $4099$ & $0.01$ & $189$ & $323$ & $0$ \\ $32771$ & $0.01$ & $2848$ & $2396$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $0.04$ & $6.6$ & $64$ & $2$ \\ $7^{5}$ & $0.05$ & $13$ & $73$ & $0$ \\ $17^{5}$ & $0.1$ & $32$ & $118$ & $0$ \\ $37^{5}$ & $0.1$ & $73$ & $197$ & $0$ \\ $79^{5}$ & $0.1$ & $183$ & $371$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $0.3$ & $34$ & $112$ & $0$ \\ $7^{10}$ & $0.4$ & $76$ & $156$ & $0$ \\ $17^{10}$ & $0.6$ & $205$ & $320$ & $0$ \\ $37^{10}$ & $0.7$ & $537$ & $653$ & $0$ \\ $79^{10}$ & $0.9$ & $1392$ & $1410$ & $0$ \\ \end{tabular} $\mathbf{\chi_{2}=1}$ \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $p$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & $0.01$ & $0.4$ & $32$ & $169$ \\ $67$ & $0.02$ & $1.8$ & $32$ & $1$ \\ $521$ & $0.02$ & $14$ & $76$ & $0$ \\ $4099$ & $0.02$ & $230$ & $508$ & $0$ \\ $32771$ & $0.02$ & $2614$ & $3616$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $0.1$ & $7.5$ & $64$ & $0$ \\ $7^{5}$ & $0.1$ & $16$ & $112$ & $0$ \\ $17^{5}$ & $0.2$ & $41$ & $197$ & $0$ \\ $37^{5}$ & $0.2$ & $94$ & $320$ & $0$ \\ $79^{5}$ & $0.2$ & $241$ & $589$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $0.7$ & $41$ & $150$ & $0$ \\ $7^{10}$ & $1.2$ & $102$ & $320$ & $0$ \\ $17^{10}$ & $2.1$ & $276$ & $556$ & $0$ \\ $37^{10}$ & $2.8$ & $736$ & $1070$ & $0$ \\ $79^{10}$ & $3.9$ & $1904$ & $2016$ & $0$ \\ \end{tabular} $\mathbf{\chi_{2}=-1}$ \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $p$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & $0.06$ & $2.4$ & $73$ & $0$ \\ $67$ & $0.02$ & $4.3$ & $73$ & $0$ \\ $521$ & $0.02$ & $32$ & $124$ & $0$ \\ $4099$ & $0.03$ & $503$ & $815$ & $0$ \\ $32771$ & $0.02$ & $5958$ & $6064$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $0.15$ & $20$ & $76$ & $0$ \\ $7^{5}$ & $0.3$ & $46$ & $156$ & $0$ \\ $17^{5}$ & $0.4$ & $108$ & $241$ & $0$ \\ $37^{5}$ & $0.6$ & $243$ & $403$ & $0$ \\ $79^{5}$ & $0.8$ & $570$ & $749$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $1.3$ & $130$ & $273$ & $0$ \\ $7^{10}$ & $2.8$ & $312$ & $416$ & $0$ \\ $17^{10}$ & $5.0$ & $815$ & $813$ & $0$ \\ $37^{10}$ & $6.5$ & $1939$ & $1463$ & $0$ \\ $79^{10}$ & $8.4$ & $4609$ & $2942$ & $0$ \\ \end{tabular} Contrary to the genus $3$ case, we see that for very small $p$ or $q=p^{n}$, sometimes we do not find a lift satisfying [54, Ass. 1]. However, in these cases we can usually compute the zeta function by counting points naively, so not much is lost here in practice. Note that the point counting is considerably slower for $\chi_{2}=-1$ than for $\chi_{2}=0,1$ which is due to the map from the curve to $\operatorname{\mathbb{P}}^{1}$ having degree $4$ instead of $3$ in this case. ### Curves of genus five #### 3.3.1 Lifting curves of genus five By Petri’s theorem [42] a minimal set of generators for the ideal of a canonical model \[\overline{C}\subset\operatorname{\mathbb{P}}^{4}=\text{Proj}\,\operatorname{ \mathbb{F}}_{q}[X,Y,Z,W,V]\] of a non-hyperelliptic genus $5$ curve consists of * three quadrics $\overline{S}_{2,1},\overline{S}_{2,2},\overline{S}_{2,3}$ and two cubics $\overline{S}_{3,1},\overline{S}_{3,2}$ in the trigonal case, * just three quadrics $\overline{S}_{2,1},\overline{S}_{2,2},\overline{S}_{2,3}$ in the non-trigonal case. So given such a minimal set of generators, it is straightforward to decide trigonality. We denote the space of quadrics in the ideal of $\overline{C}$ by $\mathcal{I}_{2}(\overline{C})$. Then in both settings $\mathcal{I}_{2}(\overline{C})$ is a three-dimensional $\operatorname{\mathbb{F}}_{q}$-vector space of which $\overline{S}_{2,1},\overline{S}_{2,2},\overline{S}_{2,3}$ form a basis. Trigonal caseHere Petri’s theorem moreover tells us that $\mathcal{I}_{2}(\overline{C})$ cuts out a smooth ir- reducible surface $\overline{S}$ that is a rational normal surface scroll of type $(1,2)$. This means that up to a linear change of variables, it is the image $\overline{S}(1,2)$ of \[\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1} \hookrightarrow\operatorname{\mathbb{P}}^{4}:((s:t),(u:v))\mapsto(vst:ut:vt^{2 }:us:vs^{2}),\] i.e. it is the ruled surface obtained by simultaneously pa- rameterizing a line in the $YW$-plane (called the directrix) and a conic in the $XZV$-plane, each time drawing the rule through the points under consideration (each of these rules intersects our trigonal curve in three points, counting multiplicities). In other words, modulo a linear change of variables the space $\mathcal{I}_{2}(\overline{C})$ admits the basis \[X^{2}-ZV,\qquad XY-ZW,\qquad XW-YV.\] (4) Note that these are (up to sign) the $2\times 2$ minors of \[\left(\begin{array}[]{cc}X&V\\ Z&X\\ \end{array}\right\|\hskip-2.845276pt\left.\begin{array}[]{c}W\\ Y\\ \end{array}\right).\] It is not trivial to _find_ such a linear change of variables. A general method using Lie algebras for rewriting Severi-Brauer surfaces in standard form was developed by de Graaf, Harrison, Pílniková and Schicho [18], and a Magma function ParametrizeScroll for carrying out this procedure in the case of rational normal surface scrolls was written by Schicho. Unfortunately this was intended to work over fields of characteristic zero only, and indeed the function always seems to crash when invoked over fields of characteristic three; see also Remark 25 below. We do not know how fundamental this flaw is, or to what extent it is an artefact of the implementation, but to resolve this issue we have implemented an ad hoc method that is specific to scrolls of type $(1,2)$. It can be found in convertscroll.m; more background on the underlying reasoning can be read in an arXiv version of this paper (1605.02162v2). Once our quadrics $\overline{S}_{2,1},\overline{S}_{2,2},\overline{S}_{2,3}$ are given by (4) we project from the line $X=Y=Z=0$, which amounts to eliminating the variables $V$ and $W$, in order to obtain the polynomials \[\overline{S}_{3,i}^{\text{pr}}=Z^{3}\overline{S}_{3,i}(X,Y,Z,\frac{X^{2}}{Z}, \frac{XY}{Z})=\overline{S}_{3,i}(XZ,YZ,Z^{2},X^{2},XY)\] for $i=1,2$. Dehomogenizing with respect to $Z$ and renaming $X\gets x$ and $Y\gets y$ we obtain two polynomials $\overline{f}_{1},\overline{f}_{2}\in\operatorname{\mathbb{F}}_{q}[x,y]$, whose zero loci intersect in the curve defined by $\overline{f}=\gcd(\overline{f}_{1},\overline{f}_{2})$. The Newton polygon of $\overline{f}$ is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{5,\text{trig}}^{0,0}$) Note that in particular $\overline{f}$ attains Baker’s bound, and a naive Newton polygon preserving lift $f\in\mathcal{O}_{K}[x,y]$ satisfies (i), (ii) and (iii). An alternative (namely, toric) viewpoint on our construction of $\overline{f}$, along with more background on the claims above, is given in Section 4.1. Non-trigonal caseIn the non-trigonal case, let us write the quadrics as \[\overline{S}_{2,i}=\begin{pmatrix}X&Y&Z&W&V\end{pmatrix}\cdot\overline{M}_{i} \cdot\begin{pmatrix}X&Y&Z&W&V\end{pmatrix}^{t},\qquad\overline{M}_{i}\in \operatorname{\mathbb{F}}_{q}^{5\times 5},\ \overline{M}_{i}^{t}=\overline{M}_ {i}.\] The curve $\mathfrak{D}(\overline{C})$ in $\operatorname{\mathbb{P}}^{2}=\text{Proj}\,\operatorname{\mathbb{F}}_{q}[ \lambda_{1},\lambda_{2},\lambda_{3}]$ defined by \[\det(\lambda_{1}\overline{M}_{1}+\lambda_{2}\overline{M}_{2}+\lambda_{3} \overline{M}_{3})=0\] parameterizes the singular members of $\mathcal{I}_{2}(\overline{C})$. It is a possibly reducible curve called the discriminant curve of $\overline{C}$, known to be of degree $5$ and having at most nodes as singularities [2]${}^{\dagger}$. The non-singular points correspond to quadrics of rank $4$, while the nodes correspond to quadrics of rank $3$. For a point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$, let us denote by $\overline{M}_{P}$ the corresponding $(5\times 5)$-matrix and by $\overline{S}_{P}$ the corresponding quadric, both of which are well-defined up to a scalar. We define \[\chi:\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})\rightarrow\{0, \pm 1\}:P\mapsto\left\{\begin{array}[]{ll}\chi_{2}(\operatorname{pdet}( \overline{M}_{P}))&\text{if $P$ is non-singular,}\\ 0&\text{if $P$ is singular,}\\ \end{array}\right.\] where $\operatorname{pdet}$ denotes the pseudo-determinant, i.e. the product of the non-zero eigenvalues. If we let $S_{2,i}\in\mathcal{O}_{K}[X,Y,Z,W,V]$ be homogeneous polynomials that reduce to $\overline{S}_{2,i}$ modulo $p$, then by [30, Ex. IV.5.5.3] these define a genus $5$ curve $C\subset\operatorname{\mathbf{P}}^{4}$ over $K$, thereby addressing (i) and (ii). But as mentioned in Section 2.1 we expect the $K$-gonality of $C$ to be typically $2g-2=8$, which exceeds the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$: Lemma 18.: _Let $\overline{C}/\operatorname{\mathbb{F}}_{q}$ be a non-hyperelliptic non-trigonal curve of genus $5$ and $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma$, and assume that $q$ is odd. If there is a point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ for which $\chi(P)\in\{0,1\}$ then $\gamma=4$. If there does not exist such a point and $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{3}})>0$ (which is guaranteed if $q>3$) then $\gamma=5$. If there does not exist such a point and $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{3}})=0$ then $\gamma=6$._ Proof.: By [2, VI.Ex. F]${}^{\dagger}$ the geometric $g^{1}_{4}$’s are in correspondence with the singular quadrics containing $\overline{C}$. More precisely: * Each rank $4$ quadric is a cone over $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$. By taking its span with the top, each line on $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ gives rise to a plane intersecting the curve in $4$ points. By varying the line we obtain two $g^{1}_{4}$’s, one for each ruling of $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$. * Each rank $3$ quadric is a cone with a $1$-dimensional top over a conic. By taking its span with the top, every point of the conic gives rise to a plane intersecting the curve in $4$ points. By varying the point we obtain a $g^{1}_{4}$. There are no other geometric $g^{1}_{4}$’s. Over $\operatorname{\mathbb{F}}_{q}$, we see that there exists a rational $g^{1}_{4}$ precisely * when there is a rank $4$ quadric that is defined over $\operatorname{\mathbb{F}}_{q}$, such that the base of the corresponding cone is $\operatorname{\mathbb{F}}_{q}$-isomorphic to $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$, or * when there is a rank $3$ quadric that is defined over $\operatorname{\mathbb{F}}_{q}$. In terms of the discriminant, this amounts to the existence of a $P\in\mathfrak{D}(\overline{C})$ for which $\chi(P)\in\{0,1\}$. So let us assume that $\gamma>4$. If $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{3}})>0$, which by the Serre-Weil bound is guaranteed for $q>3$, then there exists an effective $\operatorname{\mathbb{F}}_{q}$-rational degree $3$ divisor $D$ on $\overline{C}$. Because our curve is non-trigonal we find $\dim\|D\|=0$, so by the Riemann-Roch theorem we have that $\dim\|K-D\|=1$, and because $\deg(K-D)=5$ we conclude that there exists a rational $g^{1}_{5}$ on $\overline{C}$. (Remark: geometrically, this $g^{1}_{5}$ is cut out by the pencil of hyperplanes through the plane spanned by the support of $D$, taking into account multiplicities.) The argument can be reversed: if there exists a $g^{1}_{5}\ni D$ for some $\operatorname{\mathbb{F}}_{q}$-rational divisor $D$ on $\overline{C}$, then Riemann-Roch implies that $\|K-D\|$ is non-empty, yielding an effective divisor of degree $3$, and in particular $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{3}})>0$. So it remains to prove that if $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{3}})=0$ then there exists a rational $g^{1}_{6}$. We make a case distinction: * If $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})>0$ then there exists a rational effective divisor $D$ of degree $2$, and Riemann-Roch implies that $\dim\|K-D\|=2$, yielding the requested rational $g^{1}_{6}$ (even a $g^{2}_{6}$, in fact). * If $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{2}})=0$ then at least $\#\overline{C}(\operatorname{\mathbb{F}}_{q^{6}})>0$ by the Weil bound, so there exists a rational effective divisor $D$ of degree $6$. Then $K-D$ is of degree $2$ and by our assumption $\|K-D\|$ is empty. But then Riemann-Roch asserts that $\dim\|D\|=1$, and we have our rational $g^{1}_{6}$. This ends the proof. ∎ _Remark 19_.: If $q$ is large enough then it is very likely that $\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ will contain a point $P$ with $\chi(P)\in\{0,1\}$, and therefore that $\gamma=4$; a more precise discussion is given below. There do however exist counterexamples for every value of $q$, as is shown by a construction explained in an arXiv version of this paper (1605.02162v2). _Remark 20_.: We do not know whether gonality $6$ actually occurs or not. For this one needs to verify the existence of a non-trigonal genus five curve over $\operatorname{\mathbb{F}}_{3}$ which is pointless over $\operatorname{\mathbb{F}}_{27}$ and whose discriminant curve has no $\operatorname{\mathbb{F}}_{3}$-rational points $P$ for which $\chi(P)\in\{0,1\}$. We ran a naive brute-force search for such curves, but did not manage to find one. If $q$ is large enough and $\mathfrak{D}(\overline{C})$ has at least one (geometrically) irreducible component that is defined over $\operatorname{\mathbb{F}}_{q}$, then a point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ with $\chi(P)\in\{0,1\}$ exists and therefore $\overline{C}$ has $\operatorname{\mathbb{F}}_{q}$-gonality $4$. To state a precise bound on $q$, let us analyze the (generic) setting where $\mathfrak{D}(\overline{C})$ is a non-singular plane quintic. In this case the ‘good’ points $P$ are in a natural correspondence with pairs of $\operatorname{\mathbb{F}}_{q}$-points on an unramified double cover of $\mathfrak{D}(\overline{C})$; we refer to [4, §2(c)] and the references therein for more background. By Riemann-Hurwitz this cover is of genus $11$, for which the lower Serre-Weil bound is positive from $q>467$ on. The presence of singularities or of absolutely irreducible $\operatorname{\mathbb{F}}_{q}$-components of lower degree can be studied in a similar way and leads to smaller bounds. There are two possible ways in which $\mathfrak{D}(\overline{C})$ does _not_ have an absolutely irreducible $\operatorname{\mathbb{F}}_{q}$-component: either it could decompose into two conjugate lines over $\operatorname{\mathbb{F}}_{q^{2}}$ and three conjugate lines over $\operatorname{\mathbb{F}}_{q^{3}}$, or it could decompose into five conjugate lines over $\operatorname{\mathbb{F}}_{q^{5}}$. But in the former case the $\operatorname{\mathbb{F}}_{q}$-rational point $P$ of intersection of the two $\operatorname{\mathbb{F}}_{q^{2}}$-lines satisfies $\chi(P)=0$, so here too our curve $\overline{C}$ has $\operatorname{\mathbb{F}}_{q}$-gonality $4$. Thus the only remaining case is that of five conjugate lines over $\operatorname{\mathbb{F}}_{q^{5}}$, which can occur for every value of $q$. Let us now address Problem 1. First assume that $\gamma=4$, i.e. that there exists a point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ with $\chi(P)\in\{0,1\}$. This can be decided quickly: if $q\leq 467$ then one can proceed by exhaustive search, while if $q>467$ it is sufficient to verify whether or not $\mathfrak{D}(\overline{C})$ decomposes into five conjugate lines. To _find_ such a point, we first look for $\operatorname{\mathbb{F}}_{q}$-rational singularities of $\mathfrak{D}(\overline{C})$: these are exactly the points $P$ for which $\chi(P)=0$. If no such singularities exist then we look for a point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ for which $\chi(P)=1$ by trial and error. Once our point has been found, we proceed as follows. $\chi(P)=0$In this case $P$ corresponds to a rank $3$ quadric, which using a linear change of variables we can assume to be in the standard form $\overline{S}=ZW-X^{2}$. Choose homogeneous <figure><img src="content_image/1605.02162/x23.png"><figcaption></figcaption></figure> quadratic polynomials \[\overline{S}_{2},\overline{S}_{2}^{\prime}\in\operatorname{\mathbb{F}}_{q}[X,Y ,Z,W,V]\] that along with $\overline{S}$ form a basis of $\mathcal{I}_{2}(\overline{C})$. (In practice one can usually take $\overline{S}_{2}=\overline{S}_{2,1}$ and $\overline{S}^{\prime}_{2}=\overline{S}_{2,2}$.) Let $S_{2},S_{2}^{\prime}\in\mathcal{O}_{K}[X,Y,Z,W,V]$ be quadrics that reduce to $\overline{S}_{2},\overline{S}_{2}^{\prime}$ modulo $p$. Along with \[S=ZW-X^{2}\in\mathcal{O}_{K}[X,Y,Z,W,V]\] these cut out a canonical genus $5$ curve $C\subset\operatorname{\mathbf{P}}^{4}$. We view the quadric defined by $S$ as a cone over the weighted projective plane $\operatorname{\mathbf{P}}(1,2,1)$ with top $(0:0:0:0:1)$. Our curve is then an intersection of two quadrics inside this cone, and by projecting from the top we obtain a curve $C^{\mathrm{pr}}$ in $\operatorname{\mathbf{P}}(1,2,1)$. In terms of equations this amounts to eliminating $V$ from $S_{2}$ and $S_{2}^{\prime}$ by taking the resultant $S_{2}^{\mathrm{pr}}:=\text{res}_{V}(S_{2},S_{2}^{\prime})$, which is a homogeneous quartic. Now as in (2) we further eliminate the variable $W$ to end up with $S_{2}^{\mathrm{pr}}(XZ,YZ,Z^{2},X^{2})$. After dehomogenizing with respect to $Z$, renaming $X\gets x$ and $Y\gets y$ and rescaling if needed, we obtain an affine equation $f=y^{4}+f_{2}(x)y^{3}+f_{4}(x)y^{2}+f_{6}(x)y+f_{8}(x)$, with $f_{i}\in\mathcal{O}_{K}[x]$ of degree at most $i$. Its Newton polygon is contained in (and typically equals): 2.4/2 - 0.2 ($\Delta_{5,0}^{0}$) Note that Baker’s genus bound reads $9$, so this exceeds the geometric genus by $4$. Thus it was important to lift $\overline{S}_{2},\overline{S}_{2}^{\prime}$ before projecting. $\chi(P)=1$In this case $P$ corresponds to a rank $4$ quadric whose pseudo-determinant is a square. Using a linear change of variables we can assume it to be in the standard form $\overline{S}=XY-ZW$, which is a cone over $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ with top $(0:0:0:0:1)$. Choose homogeneous quadratic polynomials \[\overline{S}_{2},\overline{S}_{2}^{\prime}\in\operatorname{\mathbb{F}}_{q}[X,Y ,Z,W,V]\] that along with $\overline{S}$ form a basis of $\mathcal{I}_{2}(\overline{C})$. (In practice one can usually take $\overline{S}_{2}=\overline{S}_{2,1}$ and $\overline{S}^{\prime}_{2}=\overline{S}_{2,2}$.) Let $S_{2},S_{2}^{\prime}\in\mathcal{O}_{K}[X,Y,Z,W,V]$ be quadrics that reduce to $\overline{S}_{2},\overline{S}_{2}^{\prime}$ modulo $p$. Along with \[S=XY-ZW\in\mathcal{O}_{K}[X,Y,Z,W,V]\] these cut out a canonical genus $5$ curve $C\subset\operatorname{\mathbf{P}}^{4}$, which can be viewed as an intersection of two quadrics inside a cone over $\operatorname{\mathbf{P}}^{1}\times\operatorname{\mathbf{P}}^{1}$ with top $(0:0:0:0:1)$. We first project from <figure><img src="content_image/1605.02162/x25.png"><figcaption></figcaption></figure> this top, to obtain a curve $C^{\mathrm{pr}}$ in $\operatorname{\mathbf{P}}^{1}\times\operatorname{\mathbf{P}}^{1}$. In terms of equations, this amounts to eliminating $V$ from $S_{2}$ and $S_{2}^{\prime}$ by taking the resultant $S_{2}^{\mathrm{pr}}:=\text{res}_{V}(S_{2},S_{2}^{\prime})$, which is a homogeneous quartic. As in the discussion following (3), we conclude that $C^{\mathrm{pr}}$ is defined by the bihomogeneous polynomial \[S_{2}^{\mathrm{pr}}(X_{0}W_{0},Y_{0}Z_{0},Z_{0}W_{0},X_{0}Y_{0})\] (5) of bidegree $(4,4)$. Let $f\in\mathcal{O}_{K}[x,y]$ be the polynomial obtained from (5) by dehomogenizing with respect to $Z_{0}$ and $W_{0}$ and by renaming $X_{0}\gets x$ and $Y_{0}\gets y$. Then the Newton polygon of $f$ is contained in (and typically equals): 2.4/2 - 0.2 ($\Delta_{5,1}^{0}$) In particular $\deg_{y}f=4$, as wanted. Here again Baker’s bound reads $9$, which exceeds the geometric genus by $4$. $\forall P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q}):\chi(P)=-1$This case is very rare, so we will be rather sketchy here. If $\gamma=6$ then we do not know how to address Problem 1, which for point counting purposes is not an issue because this could only occur when $q=3$. If $\gamma=5$ then one can try to address Problem 1 by following the proof of Lemma 18, similar to the way we treated the $\chi(\det\overline{M}_{2})=-1$ case in genus four. For instance this works as follows if $\overline{C}(\operatorname{\mathbb{F}}_{q})$ has at least three non-collinear points, which is guaranteed as soon as $\#\overline{C}(\operatorname{\mathbb{F}}_{q})\geq 4$, which in turn is guaranteed if $q>101$ by the Serre-Weil bound. Apply a transformation of $\operatorname{\mathbb{P}}^{4}$ to position these points at $(0:1:0:0:0)$, $(0:0:0:1:0)$ and $(0:0:0:0:1)$, so that the plane they span is $X=Z=0$. This implies that the defining quadrics have no terms in $Y^{2}$, $W^{2}$ and $V^{2}$, a property which is of course easily preserved when lifting to $\mathcal{O}_{K}[X,Y,Z,W,V]$, resulting in a curve $C\subset\operatorname{\mathbf{P}}^{4}$ again passing through $(0:1:0:0:0)$, $(0:0:0:1:0)$ and $(0:0:0:0:1)$. Eliminating $W$ and $V$, which geometrically amounts to projecting from the line $X=Y=Z=0$, results in a sextic in $\operatorname{\mathbf{P}}^{2}=\operatorname{Proj}K[X,Y,Z]$ passing through $(0:1:0)$ in a non-singular way (otherwise the pencil of lines through that point would cut out a $K$-rational $g^{1}_{4}$). We can therefore apply a projective transformation that maps the corresponding tangent line to infinity, while keeping the point at $(0:1:0)$. Then by dehomogenizing with respect to $Z$ and renaming $X\gets x$ and $Y\gets y$ we end up with a polynomial $f\in\mathcal{O}_{K}[x,y]$ whose Newton polygon is contained in (and typically equals): 2.8/2 - 0.2 ($\Delta^{5}_{5}$) We omit a further discussion. Algorithm 21.: Lifting curves of genus $5$: basic solution Input: non-hyperelliptic genus $5$ curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ of $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma\leq 5$ or of $\operatorname{\mathbb{F}}_{q}$-gonality $\gamma=5$ and $\#\overline{C}(\operatorname{\mathbb{F}}_{q})\geq 4$ Output: lift $f\in\mathcal{O}_{K}[x,y]$ satisfying (i), (ii), (iii) that is supported $\bullet$ on $\Delta_{5,\text{trig}}^{0,0}$ if $\overline{C}$ is trigonal, or else $\bullet$ on $\Delta_{5,0}^{0}$ if $\exists P\in\mathfrak{D}(\overline{C}):\chi(P)=0$, or else $\bullet$ on $\Delta_{5,1}^{0}$ if $\exists P\in\mathfrak{D}(\overline{C}):\chi(P)=1$, or else $\bullet$ on $\Delta_{5}^{5}$ 01 : $\overline{C}\leftarrow\text{CanonicalImage}(\overline{C})$ in $\operatorname{\mathbb{P}}^{4}=\operatorname{Proj}\operatorname{\mathbb{F}}_{q} [X,Y,Z,W,V]$ 02 : if$\text{Ideal}(\overline{C})$ is generated by quadrics then 03 : $\overline{S}_{2,1},\overline{S}_{2,2},\overline{S}_{2,3}\leftarrow\text{ quadrics that generate $\text{Ideal}(\overline{C})$}$ 04 : $\overline{M}_{i}\leftarrow\text{Matrix}(\overline{S}_{2,i})$ ($i=1,2,3$) 05 : $\mathfrak{D}(\overline{C})\leftarrow$ curve in $\operatorname{\mathbb{P}}^{2}=\operatorname{Proj}\operatorname{\mathbb{F}}_{q} [\lambda_{1},\lambda_{2},\lambda_{3}]$ defined by $\det(\lambda_{1}\overline{M}_{1}+\lambda_{2}\overline{M}_{2}+\lambda_{3} \overline{M}_{3})$ 06 : if$q\leq 467$ and $\forall P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q}):\chi(P)=-1$ (verified exhaustively) 07 : or$q>467$ and $\mathfrak{D}(\overline{C})$ decomposes into five conjugate lines then 08 : goodpoints $\leftarrow$ false 09 : else 10 : goodpoints $\leftarrow$ true 11 : if goodpoints then 12 : if$\mathfrak{D}(\overline{C})$ has $\operatorname{\mathbb{F}}_{q}$-rational singular point $P$then 13 : $\overline{S}_{2},\overline{S}_{2}^{\prime}\leftarrow\text{quadrics such that } \langle\overline{S}_{P},\overline{S}_{2},\overline{S}_{2}^{\prime}\rangle_{ \operatorname{\mathbb{F}}_{q}}=\langle\overline{S}_{2,1},\overline{S}_{2,2}, \overline{S}_{2,3}\rangle_{\operatorname{\mathbb{F}}_{q}}$ 14 : apply automorphism of $\operatorname{\mathbb{P}}^{4}$ transforming $\overline{S}_{P}$ into $WZ-X^{2}$ 15 : $S_{2}\leftarrow\text{NaiveLift}(\overline{S}_{2})$; $S_{2}^{\prime}\leftarrow\text{NaiveLift}(\overline{S}_{2}^{\prime})$; $S_{2}^{\text{pr}}\leftarrow\text{res}_{V}(S_{2},S_{2}^{\prime})$ 16 : return Dehomogenization${}_{Z}(S_{2}^{\text{pr}}(XZ,YZ,Z^{2},X^{2}))$ 17 : else 18 : repeat$P\leftarrow\text{Random}(\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_ {q}))$until$\chi(P)=1$ 19 : $\overline{S}_{2},\overline{S}_{2}^{\prime}\leftarrow\text{quadrics such that } \langle\overline{S}_{P},\overline{S}_{2},\overline{S}_{2}^{\prime}\rangle_{ \operatorname{\mathbb{F}}_{q}}=\langle\overline{S}_{2,1},\overline{S}_{2,2}, \overline{S}_{2,3}\rangle_{\operatorname{\mathbb{F}}_{q}}$ 20 : apply automorphism of $\operatorname{\mathbb{P}}^{4}$ transforming $\overline{S}_{P}$ into $XY-ZW$ 21 : $S_{2}\leftarrow\text{NaiveLift}(\overline{S}_{2})$; $S_{2}^{\prime}\leftarrow\text{NaiveLift}(\overline{S}_{2}^{\prime})$; $S_{2}^{\text{pr}}\leftarrow\text{res}_{V}(S_{2},S_{2}^{\prime})$ 22 : return Dehomogenization${}_{Z}(S_{2}^{\text{pr}}(XZ,YZ,Z^{2},XY))$ 23 : else 24 : $P_{1},P_{2},P_{3}\leftarrow$ distinct random points of $\overline{C}(\operatorname{\mathbb{F}}_{q})$ 25 : apply automorphism of $\operatorname{\mathbb{P}}^{4}$ sending $P_{1}$, $P_{2}$, $P_{3}$ to $(0:1:0:0:0)$, 25$(0:0:0:1:0)$, $(0:0:0:0:1)$ 26 : $S_{2,i}\leftarrow\text{NaiveLift}(\overline{S}_{2,i})$$(i=1,2,3)$ 27 : $C^{\text{pr}}\leftarrow\text{res}_{W,V}(S_{2,1},S_{2,2},S_{2,3})$ 28 : apply automorphism of $\operatorname{\mathbf{P}}^{2}$ transforming $T_{(0:1:0)}(C^{\text{pr}})$ into $Z=0$ 29 : return Dehomogenization${}_{Z}(C^{\text{pr}})$ 30 : else 31 : apply automorphism of $\operatorname{\mathbb{P}}^{4}$ transforming space of quadrics in $\text{Ideal}(\overline{C})$ to 31$\langle X^{2}-ZV,XY-ZW,XW-YV\rangle_{\operatorname{\mathbb{F}}_{q}}$ 32 : $\overline{S}_{3,1},\overline{S}_{3,2}\leftarrow\text{cubics that along with quadrics generate $\text{Ideal}(\overline{C})$}$ 33 : $\overline{f}_{i}\leftarrow\text{Dehomogenization}_{Z}(\overline{S}_{3,i}(XZ,YZ ,Z^{2},X^{2},XY))$ ($i=1,2$) 34 : return NaiveLift($\gcd(\overline{f}_{1},\overline{f}_{2})$) #### 3.3.2 Optimizations Trigonal caseBy applying (1) to a polynomial with Newton polygon $\Delta_{5,\text{trig}}^{0,0}$ we end up with a polynomial $f\in\mathcal{O}_{K}[x,y]$ that is monic in $y$ and that has degree $5+(\gamma-1)2=9$ in $x$. This can be improved as soon as our curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ has a rational point $P$, which is guaranteed if $q>89$ by the Serre-Weil bound (probably this bound is not optimal). The treatment below is very similar to the genus four case where $\chi_{2}(\det\overline{M}_{2})=0$, as elaborated in Section 3.2.2. The role of $\operatorname{\mathbb{P}}(1,2,1)$ is now played by our scroll $\overline{S}(1,2)$. Recall that the latter is a ruled surface spanned by a line (the directrix) and a conic that are being parameterized simultaneously. Using an automorphism of $\overline{S}(1,2)$ we can position $P$ at the point at infinity of the spanning conic, in such a way that the curve and the conic meet at $P$ with multiplicity at least two. This results in a Newton polygon that is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{5,\text{trig}}^{0,1}$) See Remark 22 below for how this can be done in practice. Here an application of (1) typically results in $\deg_{x}f=3+(\gamma-1)2=7$. There are two caveats here: our curve might exceptionally be tangent at $P$ to a rule of the scroll, in which case it is impossible to make it tangent to the conic at that point. Or worse: our point $P$ might lie on the directrix, in which case it is just impossible to move it to the spanning conic. In these cases one can most likely just retry with another $P$. But in fact these two situations are better, as explained in Remark 23 below. _Remark 22_.: The automorphisms of $\overline{S}(1,2)$ can be applied directly to $\overline{f}$. They correspond to * substituting $y\leftarrow\overline{a}y+\overline{b}x+\overline{c}$ and $x\leftarrow\overline{a}^{\prime}x+\overline{b}^{\prime}$ in $\overline{f}$ for some $\overline{a},\overline{a}^{\prime}\in\operatorname{\mathbb{F}}_{q}^{\ast}$ and $\overline{b},\overline{b}^{\prime},\overline{c}\in\operatorname{\mathbb{F}}_{q}$, * exchanging the rule at infinity for the $y$-axis by replacing $\overline{f}$ by $x^{5}\overline{f}(x^{-1},x^{-1}y)$, or to a composition of both. For instance imagine that an affine point $P=(\overline{a},\overline{b})$ was found with a non-vertical tangent line. Then $\overline{f}\leftarrow\overline{f}(x+\overline{a},y+\overline{b})$ translates this point to the origin, at which the tangent line becomes of the form $y=\overline{c}x$. Substituting $\overline{f}\leftarrow\overline{f}(x,y+\overline{c}x)$ positions this line horizontally, and finally replacing $\overline{f}$ by $x^{5}\overline{f}(x^{-1},x^{-1}y)$ results in a polynomial with Newton polygon contained in $\Delta_{5,\text{trig}}^{0,1}$. _Remark 23_ (non-generic optimizations).: As for the first caveat, if $\overline{C}$ turns out to be tangent at $P$ to one of the rules of the scroll then moving $P$ to the point at infinity of the spanning conic results in a Newton polygon that is contained in (and typically equals): 2/2 - 0.2 ($\Delta_{5,\text{trig}}^{0,2}$) Even though this yields $\deg_{x}f=4+(\gamma-1)2=8$, the corresponding point count is slightly faster. Such a $P$ will exist if and only if the ramification scheme of $(x,y)\mapsto x$ has an $\operatorname{\mathbb{F}}_{q}$-rational point. Following the heuristics from Remark 10 we expect that this works in about $1-1/e$ of the cases. As for the second caveat, if $P$ is a point on the directrix of the scroll, we can move it to its point at infinity. This results in a Newton polygon that is contained in (and typically equals) the left polygon below. 2/2 - 0.2 ($\Delta_{5,\text{trig}}^{1,0}$) 2/2 - 0.2 ($\Delta_{5,\text{trig}}^{1,1}$) This again gives us $\deg_{x}f=5+(\gamma-1)1=7$, but here too the corresponding point count is faster. As explained in an arXiv version of our paper (1605.02162v2), the probability of being able to realize this polygon is about $1/2$, and one can even end up inside the right polygon with a probability of about $3/8$, yielding $\deg_{x}f=4+(\gamma-1)1=6$. Non-trigonal caseFor point counting purposes it is advantageous to give preference to the case $\chi(P)=0$, i.e. to use a singular point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ if it exists. Some optimizations over the corresponding discussion in Section 3.3.2 are possible, for instance generically one can replace $\Delta_{5,0}^{0}$ with the left polygon below: 2.4/2 - 0.2 ($\Delta_{5,0}^{1}$) 2.4/2 - 0.2 ($\Delta_{5,0}^{2}$) With an estimated probability of about $1-(3/8)^{\rho}$ one can even end up inside the right polygon. Here $10\geq\rho\geq 1$ denotes the number of singular points $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$. We will spend a few more words on this in Remark 24 below, after having discussed the $\chi(P)=1$ case. However usually such a singular $\operatorname{\mathbb{F}}_{q}$-point $P$ does not exist, i.e. $\rho=0$. More precisely we expect that the proportion of curves for which $\mathfrak{D}(\overline{C})$ is a smooth plane quintic tends to $1$ as $q\rightarrow\infty$. Indeed, in terms of moduli the locus of (non-hyperelliptic, non-trigonal) genus five curves having a singular point on its discriminant curve has codimension one; see [50, 25]${}^{\dagger}$. For this reason we will focus our attention on the case $\chi(P)=1$, and leave it to the interested reader to elaborate the remaining details. As for the case $\chi(P)=1$, note that by applying (1) to a polynomial with Newton polygon $\Delta_{5,1}^{0}$ one ends up with a polynomial that is monic in $y$ and that has degree $4+(\gamma-1)4=16$ in $x$. With near certainty this can be reduced to $10$, as we will explain now. The idea is to exploit the fact that in practice the discriminant curve $\mathfrak{D}(\overline{C})$ contains enough $\operatorname{\mathbb{F}}_{q}$-rational points for there to be considerable freedom in choosing a $P$ for which $\chi(P)=1$. We want to select a suited such $P$, by which we mean the following. As before, assume that an automorphism of $\operatorname{\mathbb{P}}^{4}$ has been applied such that $\overline{S}_{P}=\overline{S}=XY-ZW$ and let $\overline{S}_{2},\ \overline{S}_{2}^{\prime}\in\operatorname{\mathbb{F}}_{q}[X ,Y,Z,W,V]$ be quadrics that along with $\overline{S}$ cut out our curve $\overline{C}$. Now suppose that we would have projected $\overline{C}$ from the point $(0:0:0:0:1)$_before_ lifting to characteristic $0$. Then we would have ended up with a curve $\overline{C}^{\text{pr}}$ in \[\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}:\overline{S}= 0\quad\text{in }\operatorname{\mathbb{P}}^{3}=\operatorname{Proj}\operatorname {\mathbb{F}}_{q}[X,Y,Z,W].\] This curve has arithmetic genus $9$, because in fact that is what Baker’s bound measures. Since the excess in genus is $9-5=4$ we typically expect there to be $4$ nodes. Our point $P$ is ‘suited’ as soon as one of the singular points $Q$ of $\overline{C}^{\text{pr}}$ is $\operatorname{\mathbb{F}}_{q}$-rational. If $P$ is not suited, i.e. if there is no such $\operatorname{\mathbb{F}}_{q}$-rational singularity, then we retry with another $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ for which $\chi(P)=1$. Heuristically we estimate the probability of success to be about $5/8$. In particular if there are enough candidates for $P$ available, we should end up being successful very quickly with overwhelming probability. Given such a singular point $Q\in\overline{C}^{\text{pr}}(\operatorname{\mathbb{F}}_{q})\subset \operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ we can move it to the point $((1:0),(1:0))$, similar to what we did in the genus $4$ case where $\chi_{2}(\det\overline{M}_{2})=1$. In terms of the coordinates $X,Y,Z,W$ of the ambient space $\operatorname{\mathbb{P}}^{3}$ this means moving the point to $(0:0:0:1)$. Let’s say this amounts to the change of variables \[\begin{pmatrix}X\\ Y\\ Z\\ W\\ \end{pmatrix}\gets A\cdot\begin{pmatrix}X\\ Y\\ Z\\ W\\ \end{pmatrix}\] where $A\in\operatorname{\mathbb{F}}_{q}^{4\times 4}$. Then we can apply the change of variables \[\begin{pmatrix}X\\ Y\\ Z\\ W\\ V\\ \end{pmatrix}\leftarrow\begin{pmatrix}A&0\\ 0&1\\ \end{pmatrix}\cdot\begin{pmatrix}X\\ Y\\ Z\\ W\\ V\\ \end{pmatrix}\] directly to the defining polynomials $\overline{S},\overline{S}_{1},\overline{S}_{2}$ of $\overline{C}$ to obtain the curve $\overline{C}_{\text{tr}}$ cut out by \[\overline{S}=XY-ZW,\ \overline{S}_{2,\text{tr}},\ \overline{S}_{2^{\prime}, \text{tr}}\in\operatorname{\mathbb{F}}_{q}[X,Y,Z,W,V].\] Indeed the transformation affects $\overline{S}$ at most through multiplication by a non-zero scalar. If we would now project from $(0:0:0:0:1)$ as before, we would end up with a curve $\overline{C}_{\text{tr}}^{\text{pr}}\subset\operatorname{\mathbb{P}}^{1}\times \operatorname{\mathbb{P}}^{1}$ having a singularity at $((1:0),(1:0))$, which is at $(0:0:0:1)$ in the coordinates $X,Y,Z,W$. Recall that inside $\operatorname{\mathbb{P}}^{4}$ we view $\overline{S}$ as the defining equation of a cone over $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ with top $(0:0:0:0:1)$. The fact that the projected curve has a singularity at $(0:0:0:1)$ implies that the line $X=Y=Z=0$ meets the curve at least twice, counting multiplicities (these points of intersection need not be $\operatorname{\mathbb{F}}_{q}$-rational). Thus after multiplying $\overline{S}_{2,\text{tr}}$ by a scalar if needed we find that \[\overline{S}_{2,\text{tr}}(0,0,0,W,V)=\overline{S}^{\prime}_{2,\text{tr}}(0,0, 0,W,V)=\overline{a}W^{2}+\overline{b}WV+\overline{c}V^{2}\] for some $\overline{a},\overline{b},\overline{c}\in\operatorname{\mathbb{F}}_{q}$. Now lift $\overline{S}_{2,\text{tr}}$ and $\overline{S}_{2^{\prime},\text{tr}}$ in a consistent way, in order to obtain quadrics $S_{2},S_{2}^{\prime}\in\mathcal{O}_{K}[X,Y,Z,W,V]$ satisfying \[S_{2}(0,0,0,W,V)=S_{2}^{\prime}(0,0,0,W,V)=aW^{2}+bWV+cV^{2}\] for elements $a,b,c\in\mathcal{O}_{K}$ that reduce to $\overline{a},\overline{b},\overline{c}$ modulo $p$. If we then proceed as before, we end up with a curve $C^{\text{pr}}$ in $\operatorname{\mathbf{P}}^{1}\times\operatorname{\mathbf{P}}^{1}$ having a singularity at $((1:0),(1:0))$. This eventually results in a defining polynomial $f\in\mathcal{O}_{K}[x,y]$ whose Newton polygon is contained in (and typically equals): 2.4/2 - 0.2 ($\Delta_{5,1}^{2}$) Applying (1) to $f$ results in a polynomial having degree at most $4+(\gamma-1)2=10$ in $x$, as announced. _Remark 24_.: The same ideas apply to the case $\chi(P)=0$, with the role of $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$ replaced by $\operatorname{\mathbb{P}}(1,2,1)$. If the projection $\overline{C}^{\text{pr}}$ of $\overline{C}$ to $\operatorname{\mathbb{P}}(1,2,1)$ has an $\operatorname{\mathbb{F}}_{q}$-rational singular point, then it can be arranged that the resulting curve $C^{\text{pr}}\subset\operatorname{\mathbf{P}}(1,2,1)$ has a singularity at $(1:0:0)$, eventually yielding a polynomial $f\in\mathcal{O}_{K}[x,y]$ whose Newton polygon is contained in $\Delta^{2}_{5,0}$. As in the $\chi(P)=1$ case we expect that the probability that this works out for a given $P$ is about $5/8$. But unlike the $\chi(P)=1$ case there is not much freedom to retry in the case of failure: we have $\rho$ chances only. This explains our expected probability of $1-(3/8)^{\rho}$ to be able to realize $\Delta^{2}_{5,0}$. If the foregoing fails every time then we can play the same game with a non-singular $\operatorname{\mathbb{F}}_{q}$-rational point $Q$ on $\overline{C}^{\text{pr}}$ (guaranteed to exist if $q>89$ because then $\overline{C}$ has an $\operatorname{\mathbb{F}}_{q}$-rational point by the Serre-Weil bound). The result is a curve $C^{\text{pr}}\subset\operatorname{\mathbf{P}}(1,2,1)$ containing the point $(1:0:0)$. We can then use an automorphism of $\operatorname{\mathbf{P}}(1,2,1)$ to make $C^{\text{pr}}$ tangent to $X=0$ at that point (unless the tangent line is vertical, in which case we simply retry with another $Q$). This is done similarly to the way we handled the case $\chi_{2}(\det\overline{M}_{2})=0$ in Section 3.2.2: see in particular Remark 14. In this way one ends up in $\Delta^{1}_{5,0}$. #### 3.3.3 Implementation The tables below contain timings, memory usage and failure rates for the trigonal and non-trigonal case and various values of $p$ and $q=p^{n}$. For the precise meaning of the various entries in the tables see Section 3.1.3. Trigonal \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $p$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & $0.02$ & $0.6$ & $96$ & $206$ \\ $67$ & $0.02$ & $2.4$ & $96$ & $45$ \\ $521$ & $0.02$ & $23$ & $112$ & $4$ \\ $4099$ & $0.02$ & $358$ & $548$ & $1$ \\ $32771$ & $0.02$ & $4977$ & $3982$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $0.1$ & $17$ & $108$ & $6$ \\ $7^{5}$ & $0.1$ & $33$ & $150$ & $0$ \\ $17^{5}$ & $0.2$ & $76$ & $556$ & $0$ \\ $37^{5}$ & $0.2$ & $186$ & $1070$ & $0$ \\ $79^{5}$ & $0.3$ & $452$ & $1716$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $1.2$ & $82$ & $188$ & $0$ \\ $7^{10}$ & $2.0$ & $214$ & $621$ & $0$ \\ $17^{10}$ & $3.6$ & $587$ & $1366$ & $0$ \\ $37^{10}$ & $4.5$ & $1584$ & $2453$ & $0$ \\ $79^{10}$ & $6.3$ & $4039$ & $4176$ & $0$ \\ \end{tabular} Non-trigonal \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $p$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $11$ & $0.1$ & $2.0$ & $64$ & $14$ \\ $67$ & $0.1$ & $7.2$ & $76$ & $0$ \\ $521$ & $0.2$ & $65$ & $165$ & $0$ \\ $4099$ & $0.2$ & $1326$ & $1326$ & $0$ \\ $32771$ & $0.2$ & $21974$ & $10329$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{5}$ & $2.5$ & $59$ & $229$ & $0$ \\ $7^{5}$ & $5.3$ & $114$ & $352$ & $0$ \\ $17^{5}$ & $10$ & $261$ & $556$ & $0$ \\ $37^{5}$ & $14$ & $662$ & $919$ & $0$ \\ $79^{5}$ & $19$ & $1552$ & $1494$ & $0$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|r\|r} & time & time & space & fails \\ $q$ & lift(s) & pcc(s) & (Mb) & /1000 \\ \hline \hline $3^{10}$ & $16$ & $504$ & $780$ & $0$ \\ $7^{10}$ & $40$ & $1191$ & $1304$ & $0$ \\ $17^{10}$ & $89$ & $2946$ & $2231$ & $0$ \\ $37^{10}$ & $128$ & $7032$ & $3679$ & $0$ \\ $79^{10}$ & $193$ & $15729$ & $6267$ & $0$ \\ \end{tabular} ## 4 Curves of low gonality ### Trigonal curves Recall from Remark 6 that from genus five on a curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ is trigonal iff it is geometrically trigonal. It is known [42] that a minimal set of generators for the ideal of a canonical model $\overline{C}\subset\operatorname{\mathbb{P}}^{g-1}=\operatorname{Proj} \operatorname{\mathbb{F}}_{q}[X_{1},X_{2},\dots,X_{g}]$ of a non-hyperelliptic curve of genus $g\geq 4$ over $\operatorname{\mathbb{F}}_{q}$ consists of * $(g-2)(g-3)/2$ quadrics \[\overline{S}_{2,1},\overline{S}_{2,2},\dots,\overline{S}_{2,(g-2)(g-3)/2}\] and $g-3$ cubics \[\overline{S}_{3,1},\overline{S}_{3,2},\dots,\overline{S}_{3,g-3}\] if $\overline{C}$ is trigonal or $\operatorname{\mathbb{F}}_{q}$-isomorphic to a smooth curve in $\operatorname{\mathbb{P}}^{2}$ of degree five, * just $(g-2)(g-3)/2$ quadrics in the other cases. So given such a minimal set of generators, it is straightforward to decide trigonality, unless $g=6$ in which case one might want to check whether $\overline{C}$ is isomorphic to a smooth plane quintic or not. See Remark 27 below for how to do this. From now on assume that we are given a trigonal curve $\overline{C}/\operatorname{\mathbb{F}}_{q}$ in the above canonical form. Then the quadrics $\overline{S}_{2,i}$ spanning $\mathcal{I}_{2}(\overline{C})$ are known to define a rational normal surface scroll $\overline{S}$ of type $(a,b)$, where $a,b$ are non-negative integers satisfying \[a\leq b,\qquad a+b=g-2,\qquad b\leq(2g-2)/3,\] (6) called the Maroni invariants¹ of $\overline{C}$. This means that up to a linear change of variables, it is the image $\overline{S}(a,b)$ of [FOOTNOTE:1][ENDFOOTNOTE] \[\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1} \hookrightarrow\operatorname{\mathbb{P}}^{g-1}:((s:t),(u:v))\mapsto(ut^{a}:ut^ {a-1}s:\dots:us^{a}:vt^{b}:vt^{b-1}s:\dots:vs^{b}),\] i.e. it is the ruled surface obtained by simultaneously parameterizing * a rational normal curve of degree $a$ in the $\operatorname{\mathbb{P}}^{a}$ corresponding to $X_{1},X_{2},\dots,X_{a+1}$, and * a rational normal curve of degree $b$ in the $\operatorname{\mathbb{P}}^{b}$ corresponding to $X^{\prime}_{1},X^{\prime}_{2},\dots,X^{\prime}_{b+1}$, where $X^{\prime}_{i}$ denotes the variable $X_{a+1+i}$, each time drawing the rule through the points under consideration (each of these rules intersects our trigonal curve in three points, counting multiplicities). As a consequence, modulo a linear change of variables, the space $\mathcal{I}_{2}(\overline{C})$ admits the $2\times 2$ minors of \[\left(\begin{array}[]{cccc}X_{1}&X_{2}&\dots&X_{a}\\ X_{2}&X_{3}&\dots&X_{a+1}\\ \end{array}\right\|\hskip-2.845276pt\left.\begin{array}[]{cccc}X^{\prime}_{1}&X ^{\prime}_{2}&\dots&X^{\prime}_{b}\\ X^{\prime}_{2}&X^{\prime}_{3}&\dots&X^{\prime}_{b+1}\\ \end{array}\right)\] (7) as a basis, for some $a,b$ satisfying (6). We assume that we have a function ConvertScroll at our disposal that upon input of $\mathcal{I}_{2}(\overline{C})$ and a pair $(a,b)$ satisfying (6), either _finds_ such a linear change of variables, or outputs ‘wrong type’ in case the surface cut out by $\mathcal{I}_{2}(\overline{C})$ is not a scroll of type $(a,b)$. _Remark 25_.: If $g=5$ then $(1,2)$ is the only pair of integers satisfying (6), and one can use our ad hoc method from mentioned in Section 3.3.1 to find the requested linear change of variables as above. For higher genus we have written an experimental version of ConvertScroll in Magma, which can be found in the file convertscroll.m. It blindly relies on Schicho’s function ParametrizeScroll, which implements the Lie algebra method from [18]. Unfortunately the latter is only guaranteed to work in characteristic zero, and indeed one runs into trouble when naively applying ParametrizeScroll over finite fields of very small characteristic; empirically however, we found that $p>g$ suffices for a slight modification of ParametrizeScroll to work consistently. We remark that it is an easy linear algebra problem to verify the correctness of the output, in case it is returned. In any case further research is needed to turn this into a more rigorous step. _Remark 26_.: If ‘wrong type’ is returned then one retries with another pair $(a,b)$ satisfying (6). From a moduli theoretic point of view [48]${}^{\dagger}$ the most likely case is $a=b=(g-2)/2$ if $g$ is even, and $a+1=b=(g-1)/2$ if $g$ is odd, so it is wise to try that pair first, and then to let $a$ decrease gradually. According to [45]${}^{\dagger}$ the Lie algebra method implicitly computes the Maroni invariants, so it should in fact be possible to get rid of this trial-and-error part; recall that we just use the function ConvertScroll as a black box. _Remark 27_ ($g=6$).: If ‘wrong type’ is returned on input $(2,2)$ as well as on input $(1,3)$, then we are in the smooth plane quintic case and therefore $\overline{C}$ is not trigonal. Here $\mathcal{I}_{2}(\overline{C})$ cuts out a Veronese surface in $\operatorname{\mathbb{P}}^{5}$, rather than a scroll. We will revisit this case at the end of the section. Once our quadrics $\overline{S}_{2,i}$ are given by the minors of (7), we restrict our curve $\overline{C}$ to the embedded torus \[\operatorname{\mathbb{T}}^{2}\hookrightarrow\operatorname{\mathbb{P}}^{g-1}:(x ,y)\mapsto(y:xy:\dots:x^{a}y:1:x:\dots:x^{b})\] by simply substituting \[X_{1}\gets y,\ X_{2}\gets xy,\ \dots,\ X_{a+1}\gets x^{a}y\quad \text{and}\quad X^{\prime}_{1}\gets 1,\ X^{\prime}_{2}\gets x,\ \dots,\ X^{\prime}_{b+1}\gets x^{b}.\] This makes the quadrics vanish identically, while the cubics become \[\overline{f}_{1},\overline{f}_{2},\dots,\overline{f}_{g-3}\in\operatorname{ \mathbb{F}}_{q}[x,y].\] The ideal generated by these polynomials is principal, i.e. of the form $(\overline{f})$, where the Newton polygon of $\overline{f}=\gcd(\overline{f}_{1},\overline{f}_{2},\dots,\overline{f}_{g-3})$ is contained in (and typically equals): The correctness of these claims follows for instance from [9, §3]. Note that in particular $\overline{f}$ attains Baker’s bound, so a naive Newton polygon preserving lift $f\in\mathcal{O}_{K}[x,y]$ satisfies (i), (ii) and (iii). _Remark 28_.: It should be clear that the above is a generalization of the corresponding method from Section 3.3.1, where we dealt with trigonal curves of genus five. But the method also generalizes the genus four cases $\chi_{2}(\det\overline{M}_{2})=0$ and $\chi_{2}(\det\overline{M}_{2})=1$ from Section 3.2.1, where the scrolls are $\overline{S}(0,2)=\operatorname{\mathbb{P}}(1,2,1)$ and $\overline{S}(1,1)=\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}} ^{1}$, respectively. _Remark 29_.: Here too one could try to compress the Newton polygon by clipping off boundary points, similar to what we did in Section 3.3.2. But as the genus grows the resulting speed-ups become less and less significant, and we omit a further discussion. ExampleLet us carry out the foregoing procedure for the curve defined by \[(x^{3}+x+1)y^{3}+42(2x^{4}+x^{3}+3x^{2}+3x+1)y^{2}+(x+1)(x^{4}+2x^{2}+x+1)y+42 (x^{2}+1)=0\] over $\operatorname{\mathbb{F}}_{43}$. This is the reduction mod $43$ of the modular curve $X_{0}^{+}(164)$, or rather an affine model of it, whose equation we took from [32]. It is of genus $6$, while we note that Baker’s bound reads $7$, so it is not met here. Using the intrinsic CanonicalMap one computes that \[\left\{\begin{array}[]{l}X_{1}^{2}X_{2}+42X_{1}^{2}X_{5}+40X_{1}^{2}X_{6}+40X_ {1}X_{2}X_{6}+X_{1}X_{3}^{2}+2X_{1}X_{3}X_{6}+42X_{1}X_{4}^{2}+40X_{1}X_{4}X_{ 5}+X_{1}X_{4}X_{6}\\ \ \qquad\qquad+6X_{1}X_{5}X_{6}+7X_{1}X_{6}^{2}+42X_{2}X_{3}^{2}+2X_{2}X_{3}X_ {6}+41X_{2}X_{6}^{2}+42X_{3}^{3}+40X_{3}X_{6}^{2}+2X_{4}^{2}X_{5}+4X_{4}^{2}X_ {6}\\ \ \qquad\qquad+4X_{4}X_{5}X_{6}+X_{4}X_{6}^{2}+38X_{5}X_{6}^{2}+39X_{6}^{3}\\ X_{1}^{2}X_{3}+42X_{1}^{2}X_{6}+39X_{1}X_{2}X_{6}+X_{1}X_{3}^{2}+38X_{1}X_{3}X _{6}+42X_{1}X_{4}X_{5}+X_{1}X_{5}X_{6}+7X_{1}X_{6}^{2}+X_{2}X_{3}^{2}\\ \ \qquad\qquad+41X_{2}X_{3}X_{6}+8X_{2}X_{6}^{2}+42X_{3}^{2}X_{6}+4X_{3}X_{6}^ {2}+X_{4}^{2}X_{6}+5X_{4}X_{5}X_{6}+X_{4}X_{6}^{2}+40X_{5}X_{6}^{2}+37X_{6}^{3 }\\ 42X_{1}^{2}X_{6}+X_{1}X_{2}X_{3}+42X_{1}X_{2}X_{6}+39X_{1}X_{3}X_{6}+42X_{1}X_ {4}X_{5}+42X_{1}X_{5}X_{6}+6X_{1}X_{6}^{2}+X_{2}X_{3}^{2}+39X_{2}X_{3}X_{6}\\ \ \qquad\qquad+7X_{2}X_{6}^{2}+X_{3}^{3}+42X_{3}^{2}X_{6}+5X_{3}X_{6}^{2}+42X_ {4}^{2}X_{6}+5X_{4}X_{5}X_{6}+41X_{4}X_{6}^{2}+X_{5}X_{6}^{2}+36X_{6}^{3}\\ 42X_{1}X_{3}+42X_{1}X_{5}+X_{2}^{2}+X_{2}X_{6}+X_{3}X_{6}+42X_{4}^{2}+42X_{4}X _{6}+X_{5}X_{6}\\ 42X_{1}X_{5}+X_{2}X_{4}+X_{2}X_{6}+42X_{4}^{2}+42X_{4}X_{6}+X_{5}X_{6}\\ 42X_{1}X_{6}+X_{3}X_{4}+X_{3}X_{6}+42X_{4}X_{5}+X_{6}^{2}\\ 42X_{1}X_{6}+X_{2}X_{5}+42X_{4}X_{5}+X_{6}^{2}\\ 42X_{2}X_{6}+X_{3}X_{5}\\ 42X_{4}X_{6}+X_{5}^{2}+42X_{6}^{2}\\ \end{array}\right.\] is a minimal set of generators for the ideal $\mathcal{I}(\overline{C})$ of a canonical model $\overline{C}\subset\operatorname{\mathbb{P}}^{5}$. We are clearly in the trigonal case, so the six quadrics must cut out a rational normal surface scroll. According to (6) the type of the latter is either $(1,3)$ or $(2,2)$. Following Remark 26 we first try $(2,2)$, so we search for a linear change of variables taking $\mathcal{I}_{2}(\overline{C})$ to the space of quadrics spanned by the $2\times 2$ minors of \[\left(\begin{array}[]{cc}X_{1}&X_{2}\\ X_{2}&X_{3}\\ \end{array}\right\|\hskip-2.845276pt\left.\begin{array}[]{cc}X_{4}&X_{5}\\ X_{5}&X_{6}\\ \end{array}\right).\] Our experimental version of the function ConvertScroll turns out to work here, and the type $(2,2)$ was a correct guess: the change of variables returned by Magma reads \[\begin{pmatrix}X_{1}\\ X_{2}\\ X_{3}\\ X_{4}\\ X_{5}\\ X_{6}\\ \end{pmatrix}\leftarrow\begin{pmatrix}40&3&42&0&30&33\\ 0&12&35&40&42&2\\ 0&9&4&30&29&42\\ 20&37&5&2&8&22\\ 22&19&11&28&32&14\\ 38&29&16&21&33&36\\ \end{pmatrix}\cdot\begin{pmatrix}X_{1}\\ X_{2}\\ X_{3}\\ X_{4}\\ X_{5}\\ X_{6}\\ \end{pmatrix}.\] Applying this transformation to our generators of $\mathcal{I}(\overline{C})$ and then substituting \[X_{1}\gets y,\ \ X_{2}\gets xy,\ \ X_{3}\gets x^{2}y,\ \ X_{4} \gets 1,\ \ X_{5}\gets x,\ \ X_{6}\gets x^{2}\] annihilates the quadrics, while the cubics become \[6(x+27)(x+32)\overline{f},\ \ 39(x+13)(x+20)\overline{f},\ \ 2(x+13)^{2} \overline{f}\] respectively, where \[\begin{array}[]{rcl}\overline{f}&=&\hskip-5.690551ptx^{4}y^{3}+8x^{4}y^{2}+31x ^{4}y+29x^{4}+37x^{3}y^{3}+23x^{3}y^{2}+16x^{3}y+x^{3}+12x^{2}y^{3}+18x^{2}y^{ 2}\\ &&\qquad\qquad\quad+12x^{2}y+25x^{2}+10xy^{3}+7xy^{2}+30xy+11x+13y^{3}+36y^{2} +3y+2.\\ \end{array}\] For this polynomial Baker’s bound is attained, so a naive lift to $f\in\mathcal{O}_{K}[x,y]$ satisfies (i), (ii), (iii). After making $f$ monic using (1) it can be fed to the algorithm from [53, 54] to find the numerator \[43^{6}T^{12}+43^{5}\cdot 8T^{11}+43^{4}\cdot 154T^{10}+43^{3} \cdot 1032T^{9}+43^{2}\cdot 9911T^{8}+43\cdot 62496T^{7}\\ +444940T^{6}+62496T^{5}+9911T^{4}+1032T^{3}+154T^{2}+8T+1\] of the zeta function $Z_{\overline{C}/\operatorname{\mathbb{F}}_{43}}(T)$ in a couple of seconds. Point counting timingsDespite the lack of a well-working function ConvertScroll, we can tell how the point counting algorithm from [53, 54] should perform in composition with the above method, by simply assuming that $\overline{C}$ is _given_ as the genus $g$ curve defined by a suitably generic polynomial $\overline{f}\in\operatorname{\mathbb{F}}_{q}[x,y]$ supported on $\operatorname{conv}\{(0,0),(2b+2-a,0),(2a+2-b,3),(0,3)\}$. Then we can immediately lift to $\mathcal{O}_{K}[x,y]$. The tables below give point counting timings and memory usage for randomly chosen such polynomials in genera $g=6,7$, where for the sake of conciseness we restrict to the generic Maroni invariants $a=\lfloor(g-2)/2\rfloor$ and $b=\lceil(g-2)/2\rceil$; the other Maroni invariants give rise to faster point counts. $\mathbf{g=6}$ \begin{tabular}{r\|\|r\|r} & time & space \\ $p$ & pcc(s) & (Mb) \\ \hline \hline $11$ & $0.9$ & $32$ \\ $67$ & $6.0$ & $32$ \\ $521$ & $70$ & $118$ \\ $4099$ & $769$ & $824$ \\ $32771$ & $8863$ & $6829$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r} & time & space \\ $q$ & pcc(s) & (Mb) \\ \hline \hline $3^{5}$ & $33$ & $76$ \\ $7^{5}$ & $64$ & $80$ \\ $17^{5}$ & $176$ & $197$ \\ $37^{5}$ & $415$ & $371$ \\ $79^{5}$ & $1035$ & $791$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r} & time & space \\ $q$ & pcc(s) & (Mb) \\ \hline \hline $3^{10}$ & $183$ & $188$ \\ $7^{10}$ & $503$ & $320$ \\ $17^{10}$ & $1490$ & $749$ \\ $37^{10}$ & $3970$ & $1663$ \\ $79^{10}$ & $10945$ & $3716$ \\ \end{tabular} $\mathbf{g=7}$ \begin{tabular}{r\|\|r\|r\|} & time & space \\ $p$ & pcc(s) & (Mb) \\ \hline \hline $11$ & $1.5$ & $32$ \\ $67$ & $6.5$ & $32$ \\ $521$ & $88$ & $118$ \\ $4099$ & $955$ & $857$ \\ $32771$ & $13279$ & $6983$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|} & time & space \\ $q$ & pcc(s) & (Mb) \\ \hline \hline $3^{5}$ & $43$ & $76$ \\ $7^{5}$ & $91$ & $118$ \\ $17^{5}$ & $257$ & $241$ \\ $37^{5}$ & $602$ & $460$ \\ $79^{5}$ & $1561$ & $983$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r\|} & time & space \\ $q$ & pcc(s) & (Mb) \\ \hline \hline $3^{10}$ & $283$ & $197$ \\ $7^{10}$ & $777$ & $371$ \\ $17^{10}$ & $2384$ & $919$ \\ $37^{10}$ & $6706$ & $2212$ \\ $79^{10}$ & $18321$ & $4682$ \\ \end{tabular} Smooth plane quinticsWe end this section with a brief discussion of the genus $6$ case where our canonical curve $\overline{C}\subset\operatorname{\mathbb{P}}^{5}$ is $\operatorname{\mathbb{F}}_{q}$-isomorphic to a smooth plane quintic. Such curves are never trigonal: using a variant of Lemma 7 one verifies that the $\operatorname{\mathbb{F}}_{q}$-gonality is $4$ if and only if $\#\overline{C}(\operatorname{\mathbb{F}}_{q})>0$, which is guaranteed if $q>137$ by the Serre-Weil bound. In the other cases it is $5$. Nevertheless from the point of view of the canonical embedding, smooth plane quintics behave ‘as if they were trigonal’, which is why we include them here. (The appropriate unifying statement reads that trigonal curves and smooth plane quintics are exactly the curves having Clifford index $1$.) Here our main task towards tackling Problem 1 is to find a linear change of variables transforming the space $\mathcal{I}_{2}(\overline{C})$ into \[\langle X_{2}^{2}-X_{1}X_{4},X_{2}X_{3}-X_{1}X_{5},X_{3}^{2}-X_{1}X_{6},X_{3}X _{4}-X_{2}X_{5},X_{3}X_{5}-X_{2}X_{6},X_{5}^{2}-X_{4}X_{6}\rangle_{ \operatorname{\mathbb{F}}_{q}}\] whose zero locus is the Veronese surface in ‘standard form’, i.e. the closure of the image of \[\operatorname{\mathbb{T}}^{2}\hookrightarrow\operatorname{\mathbb{P}}^{5}:(x,y )\mapsto(x^{2}:xy:x:y^{2}:y:1).\] In order to achieve this, we simply assume that we have a function ConvertVeronese at our disposal. One could again try to use Schicho’s function ParametrizeScroll for this, but here too we expect problems because of the characteristic being finite (although we did not carry out the experiment). Once this standard form is attained, an easy substitution \[X_{1}\gets x^{2},\ X_{2}\gets xy,\ X_{3}\gets x,\ X_{4} \gets y^{2},\ X_{5}\gets y,\ X_{6}\gets 1\] makes the quadrics vanish identically, while the cubics have a gcd whose homogenization defines the desired smooth plane quintic. From here one proceeds as in the smooth plane quartic case described in Section 3.1.1. ### Tetragonal curves We conclude this article with some thoughts on how the foregoing material can be adapted to the tetragonal case. A full elaboration of the steps below (or even a rigorous verification of some corresponding claims) lies beyond our current scope. In particular we have not implemented anything of what follows. The main aim of this section is twofold: to illustrate how our treatment of non-trigonal curves of genus five from Section 3.3.1 naturally fits within a larger framework, and to propose a track for future research, involving mathematics that was developed mainly by Schreyer in [46, §6]${}^{\dagger}$ and Schicho, Schreyer and Weimann in [44, §5]. Let \[\overline{C}\subset\operatorname{\mathbb{P}}^{g-1}=\operatorname{Proj} \overline{R},\qquad\overline{R}=\operatorname{\mathbb{F}}_{q}[X_{1},X_{2}, \dots,X_{g}]\] be the canonical model of a genus $g\geq 5$ curve that is non-hyperelliptic, non-trigonal, and not isomorphic to a smooth plane quintic, so that a minimal set of generators of $\mathcal{I}(\overline{C})\subset\overline{R}$ consists of $\beta_{12}:=(g-2)(g-3)/2$ quadrics \[\overline{S}_{2,1},\overline{S}_{2,2},\dots,\overline{S}_{2,\beta_{12}}.\] The notation $\beta_{12}$ refers to the corresponding entry in the graded Betti table of the homogeneous coordinate ring of $\overline{C}$, to which we will make a brief reference at the end of this section. Assume that the $\operatorname{\mathbb{F}}_{q}$-gonality of $\overline{C}$ is four, and consider a corresponding $\operatorname{\mathbb{F}}_{q}$-rational <figure><img src="content_image/1605.02162/x36.png"><figcaption></figcaption></figure> map $\pi:\overline{C}\rightarrow\operatorname{\mathbb{P}}^{1}$. We note that unlike the trigonal case this map may not be uniquely determined modulo automorphisms of $\operatorname{\mathbb{P}}^{1}$, even for $g$ arbitrarily large. The linear spans of the fibers of $\pi$ form a one-dimensional family of planes in $\operatorname{\mathbb{P}}^{g-1}$ that cut out a rational normal _threefold_ scroll $\overline{S}$. Similar to before, up to a linear change of variables, such a scroll is obtained by simultaneously parameterizing * a rational normal curve of degree $a$ in the $\operatorname{\mathbb{P}}^{a}$ corresponding to $X_{1},X_{2},\dots,X_{a+1}$, * a rational normal curve of degree $b$ in the $\operatorname{\mathbb{P}}^{b}$ corresponding to $X^{\prime}_{1},X^{\prime}_{2},\dots,X^{\prime}_{b+1}$, where $X^{\prime}_{i}$ denotes the variable $X_{a+1+i}$, and * a rational normal curve of degree $c$ in the $\operatorname{\mathbb{P}}^{c}$ corresponding to $X^{\prime\prime}_{1},X^{\prime\prime}_{2},\dots,X^{\prime\prime}_{c+1}$, where $X^{\prime\prime}_{i}$ denotes the variable $X_{a+b+2+i}$, each time taking the plane connecting the points under consideration (each of these planes intersects our trigonal curve in four points, counting multiplicities). Again this concerns a determinantal variety, defined by the $2\times 2$ minors of \[\left(\begin{array}[]{cccc}X_{1}&X_{2}&\dots&X_{a}\\ X_{2}&X_{3}&\dots&X_{a+1}\\ \end{array}\right\|\hskip-2.845276pt\left.\begin{array}[]{cccc}X^{\prime}_{1}&X ^{\prime}_{2}&\dots&X^{\prime}_{b}\\ X^{\prime}_{2}&X^{\prime}_{3}&\dots&X^{\prime}_{b+1}\\ \end{array}\right.\hskip-2.845276pt\left\|\begin{array}[]{cccc}X^{\prime\prime} _{1}&X^{\prime\prime}_{2}&\dots&X^{\prime\prime}_{c}\\ X^{\prime\prime}_{2}&X^{\prime\prime}_{3}&\dots&X^{\prime\prime}_{c+1}\\ \end{array}\right).\] (8) Alternatively our scroll can be thought of as the Zariski closure of the image of \[\operatorname{\mathbb{T}}^{3}\hookrightarrow\operatorname{\mathbb{P}}^{g-1}:(x ,y,z)\mapsto(z:xz:\dots:x^{a}z:y:xy:\dots:x^{b}y:1:x:\dots:x^{c}),\] or if one prefers, as the toric threefold associated to the polytope $(\Delta_{(a,b,c)}).$ Let us denote this ‘standard’ scroll in $\operatorname{\mathbb{P}}^{g-1}$ by $\overline{S}(a,b,c)$. The non-negative integers $(a,b,c)$ are called the scrollar invariants of $\overline{C}$ with respect to $\pi$ and can be chosen to satisfy \[a\leq b\leq c,\qquad a+b+c=g-3,\qquad c\leq(2g-2)/4,\] (9) where the last inequality follows from Riemann-Roch. Inside the scroll $\overline{S}$ our curve $\overline{C}$ arises as a complete intersection of two hypersurfaces $\overline{Y}$ and $\overline{Z}$ that are ‘quadratic’. More precisely the Picard group of $\overline{S}$ is generated by the class $[\overline{H}]$ of a hyperplane section and the class $[\overline{\Pi}]$ of a ruling (i.e. of the linear span of a fiber of $\pi$), and $\overline{Y}$ and $\overline{Z}$ can be chosen such that \[\overline{Y}\in 2[\overline{H}]-b_{1}[\overline{\Pi}],\qquad\overline{Z}\in 2[ \overline{H}]-b_{2}[\overline{\Pi}]\] for non-negative integers $b_{1}\geq b_{2}$ satisfying $b_{1}+b_{2}=g-5$. These integers are invariants of the curve, that is, they do not depend on the choice of $\pi$. If $b_{2}<b_{1}$ then also the surface $\overline{Y}$ is uniquely determined by $\overline{C}$. This is automatic when $g$ is even. Let us now assume that $\overline{S}$ is given in the standard form $\overline{S}(a,b,c)$, which we consider along with the embedded torus $\operatorname{\mathbb{T}}^{3}$. Then for $\overline{Y}$ to be in the class $2[\overline{H}]-b_{1}[\overline{\Pi}]$ it means that $\overline{Y}\cap\operatorname{\mathbb{T}}^{3}$ is defined by an irreducible polynomial $\overline{f}_{\overline{Y}}\in\operatorname{\mathbb{F}}_{q}[x,y,z]$ whose support is contained in $(\Delta_{(a,b,c),b_{1}}).$ or more precisely² in [FOOTNOTE:2][ENDFOOTNOTE] \[\operatorname{conv}\{(0,0,0),(2c-b_{1},0,0),(0,2,0),(2b-b_{1},2,0),(0,0,2),(2a -b_{1},0,2)\}\cap\operatorname{\mathbf{R}}_{\geq 0}^{3}.\] In other words this is the polytope obtained from $2\Delta_{(a,b,c)}$ by shifting its right-most face leftwards over a distance $b_{1}$. Moreover $b_{1}$ is the maximal integer for which this containment holds. The same applies to $\overline{Z}$, leading to a polynomial $\overline{f}_{\overline{Z}}\in\operatorname{\mathbb{F}}_{q}[x,y,z]$ whose support is contained in $\Delta_{(a,b,c),b_{2}}$, which is the polytope obtained from $2\Delta_{a,b,c}$ by shifting the right-most face inwards over a distance $b_{2}$. The main observation of this section is that $\overline{f}_{\overline{Y}},\overline{f}_{\overline{Z}}\in\operatorname{ \mathbb{F}}_{q}[x,y,z]$ is a pair of polynomials meeting a version of Baker’s bound for complete intersections, again due to Khovanskii [39]${}^{\dagger}$. In the case of two trivariate polynomials supported on polytopes $\Delta_{1}$ and $\Delta_{2}$ the bound reads \[g\leq\#\left(\text{interior points of $\Delta_{1}+\Delta_{2}$}\right)-\#\left( \text{interior points of $\Delta_{1}$}\right)-\#\left(\text{interior points of $\Delta_{2}$}\right).\] In our case where $\Delta_{1}=\Delta_{(a,b,c),b_{1}}$ and $\Delta_{2}=\Delta_{(a,b,c),b_{2}}$, this indeed evaluates to $g-0-0=g$. Thus the strategy would be similar: lift these polynomials in a Newton polytope preserving way to polynomials $f_{Y},f_{Z}\in\mathcal{O}_{K}[x,y,z]$. These then again cut out a genus $g$ curve in $\operatorname{\mathbf{T}}^{3}$, and a polynomial $f\in\mathcal{O}_{K}[x,y]$ satisfying (i)-(iii) can be found by taking the resultant of $f_{Y}$ and $f_{Z}$ with respect to $z$ (or with respect to $y$). Genus $5$ curves revisitedLet us revisit our treatment of tetragonal curves of genus five $\overline{C}\subset\operatorname{\mathbb{P}}^{4}=\operatorname{Proj} \operatorname{\mathbb{F}}_{q}[X,Y,Z,W,V]$ from Section 3.3.1. 1. Our first step was to look for a point $P\in\mathfrak{D}(\overline{C})(\operatorname{\mathbb{F}}_{q})$ for which $\chi(P)=0$ or $\chi(P)=1$. The corresponding quadrics were described as cones over $\operatorname{\mathbb{P}}(1,2,1)$ and $\operatorname{\mathbb{P}}^{1}\times\operatorname{\mathbb{P}}^{1}$, respectively. But in the current language these are just rational normal threefold scrolls of type $(0,0,2)$ resp. $(0,1,1)$. Note that this shows that the scroll $\overline{S}$ may indeed depend on the choice of $\pi$. 2. For ease of exposition let us restrict to the case $\chi(P)=1$. Then the second step was to transform the quadric into $XY-ZW$, whose zero locus is the Zariski closure of \[\operatorname{\mathbb{T}}^{3}\hookrightarrow\operatorname{\mathbb{P}}^{4}:(x,y ,z)\mapsto(1:xy:x:y:z),\] i.e. the transformation takes the scroll $\overline{S}(0,1,1)$ into ‘standard form’. 3. The other quadrics $\overline{S}_{2},\overline{S}_{2}^{\prime}$ are instances of the surfaces $\overline{Y}$ and $\overline{Z}$. They are both in the class $2[\overline{H}]$, i.e. $b_{1}=b_{2}=0$. Viewing $\overline{Y}$ and $\overline{Z}$ inside the torus $\operatorname{\mathbb{T}}^{3}$ amounts to evaluating them at $(1,xy,x,y,z)$, resulting in polynomials that are supported on as predicted. With the present approach we naively lift these polynomials to $f_{Y},f_{Z}\in\mathcal{O}_{K}[x,y,z]$. In Section 3.3.1 we applied this naive lift directly to $\overline{S}_{2},\overline{S}_{2}^{\prime}$, which was fine there, but in higher genus it is more convenient to work in $\operatorname{\mathbb{T}}^{3}$, since $\overline{Y},\overline{Z}\subset\overline{S}$ will no longer be cut out by a single quadratic hypersurface of $\operatorname{\mathbb{P}}^{g-1}$. 4. The last step was to project this lifted curve from $(0:0:0:0:1)$, which in our case amounts to taking the resultant of $f_{Y},f_{Z}$ with respect to $z$. General recipeIf we want to turn the above into a rigorous recipe for lifting tetragonal curves, three questions show up naturally. We share some brief first thoughts, but further research is needed regarding each of these. 1. How do we decide whether the input curve has $\operatorname{\mathbb{F}}_{q}$-gonality $4$ or not, and how do we extract from $\mathcal{I}_{2}(\overline{C})$ the equations of a corresponding rational normal threefold scroll $\overline{S}$? In genus five we used the discriminant curve for this, but in general the desired information should be traceable from (the first few steps of) a minimal free resolution \[\overline{R}(-4)^{\beta_{34}}\oplus\overline{R}(-5)^{\beta_{35}}\to \overline{R}(-3)^{\beta_{23}}\oplus\overline{R}(-4)^{\beta_{24}}\to \overline{R}(-2)^{\beta_{12}}\rightarrow\overline{R}\rightarrow\faktor{ \overline{R}}{(\overline{S}_{2,1},\dots,\overline{S}_{2,\beta_{12}})}\] of the homogeneous coordinate ring of $\overline{C}$ as a graded $\overline{R}$-module, thanks to a proven part of Green’s canonical syzygy conjecture [44, Thm. 2.5], namely that $\beta_{24}\neq 0$ if and only if $\overline{C}$ is $\overline{\operatorname{\mathbb{F}}}_{q}$-tetragonal or $\operatorname{\mathbb{F}}_{q}$-isomorphic to a smooth plane sextic, which in turn holds if and only if $\overline{C}$ has Clifford index $2$. (The dimensions $\beta_{ij}$ are usually gathered in the so-called graded Betti table of $\overline{C}$, and in general Green’s conjecture predicts that the Clifford index equals the number of leading zeroes on the cubic strand, i.e. the minimal $i$ for which $\beta_{i,i+2}\neq 0$.) If $g\geq 7$ then a sufficiently generic geometrically tetragonal curve satisfies $\beta_{24}=g-4$. This is what Schicho, Schreyer and Weimann [44, Ex. 4.2] refer to as the _goneric_ case; see also [23, Thm. 0.3]${}^{\dagger}$. It implies that our curve admits a unique $g^{1}_{4}$, hence it is $\operatorname{\mathbb{F}}_{q}$-tetragonal, and that the ideal of the corresponding scroll $\overline{S}$ can be computed as the annihilator of the cokernel of the map \[\overline{R}(-5)^{\beta_{35}}\rightarrow\overline{R}(-4)^{\beta_{24}}.\] See [44, Prop. 4.11]. In the non-goneric cases one has $\beta_{24}=(g-1)(g-4)/2$ and a finer analysis is needed. Some further useful statements can be found in [44] and [29]${}^{\dagger}$. 2. How do we find the type $(a,b,c)$ of the scroll $\overline{S}$, along with a linear change of variables taking it into the standard form $\overline{S}(a,b,c)$ cut out by the minors of (8)? We encountered an analogous hurdle in the trigonal case. Here too it would be natural to try the Lie algebra method from [18], but as mentioned this was designed to work over fields of characteristic zero, and it is not clear to us how easily the method carries over to small finite characteristic. 3. How do we find the invariants $b_{1},b_{2}$ along with hypersurfaces $\overline{Y}\in 2[\overline{H}]-b_{1}[\overline{\Pi}]$ and $\overline{Z}\in 2[\overline{H}]-b_{2}[\overline{\Pi}]$ that inside $\overline{S}(a,b,c)$ cut out our curve $\overline{C}$? By evaluating the generators of $\mathcal{I}(\overline{C})$ in $(z,xz,\dots,x^{a}z,y,xy,\dots,x^{b}y,1,x,\dots,x^{c})$ one easily finds a set of generators for the ideal of $\overline{C}\cap\operatorname{\mathbb{T}}^{3}$. The challenge is now to replace this set by two polynomials that are supported on polytopes of the form \[\Delta_{(a,b,c),b_{1}}\quad\text{and}\quad\Delta_{(a,b,c),b_{2}},\] with $b_{1},b_{2}$ satisfying $b_{1}+b_{2}=g-5$. Here our approach would be to use a Euclidean type of algorithm to find generators whose Newton polytopes are as small as possible. Point counting timingsWe have not implemented anything of the foregoing recipe, but we can predict how its output should perform in composition with the point counting algorithm from [53, 54], by simply starting from a sufficiently generic pair of polynomials $\overline{f}_{\overline{Y}},\overline{f}_{\overline{Z}}\in\operatorname{ \mathbb{F}}_{q}[x,y,z]$ that are supported on $\Delta_{(a,b,c),b_{1}}$ and $\Delta_{(a,b,c),b_{2}}$ for non-negative integers $a,b,c$ satisfying (9) and $b_{1}+b_{2}=g-5$. Then one can naively lift to $\mathcal{O}_{K}[x,y,z]$, take the resultant with respect to $z$, make the outcome monic using (1), and feed the result to the point counting algorithm. The tables below contain point counting timings and memory usage for randomly chosen such pairs in genera $g=6,7$. For the sake of conciseness it makes sense to restrict to the case where the scrollar invariants $a,b,c$ and the tetragonal invariants $b_{1},b_{2}$ are as balanced as possible, meaning that $c-a\leq 1$ and $b_{1}-b_{2}\leq 1$, because this is the generic case [3, 6]${}^{\dagger}$. We expect the other cases to run faster. $\mathbf{g=6}$ \begin{tabular}{r\|\|r\|r} & time & space \\ $p$ & pcc(s) & (Mb) \\ \hline \hline $11$ & $8.5$ & $32$ \\ $67$ & $34.7$ & $64$ \\ $521$ & $445$ & $379$ \\ $4099$ & $4748$ & $2504$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r} & time & space \\ $q$ & pcc(s) & (Mb) \\ \hline \hline $3^{5}$ & $266$ & $214$ \\ $7^{5}$ & $549$ & $325$ \\ $3^{10}$ & $2750$ & $6072$ \\ $7^{10}$ & $6407$ & $9814$ \\ \end{tabular} $\mathbf{g=7}$ \begin{tabular}{r\|\|r\|r} & time & space \\ $p$ & pcc(s) & (Mb) \\ \hline \hline $11$ & $11$ & $32$ \\ $67$ & $46$ & $80$ \\ $521$ & $445$ & $347$ \\ $4099$ & $4350$ & $2441$ \\ \end{tabular} \begin{tabular}{r\|\|r\|r} & time & space \\ $q$ & pcc(s) & (Mb) \\ \hline \hline $3^{5}$ & $254$ & $156$ \\ $7^{5}$ & $550$ & $241$ \\ $3^{10}$ & $2347$ & $3606$ \\ $7^{10}$ & $5819$ & $5724$ \\ \end{tabular} ## References * [1] E. Arbarello, M. 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1201.4951	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 21161, "num_imgs": 3, "llama3_tokens_count": 5831 }	[ "content_image/1201.4951/x1.png", "content_image/1201.4951/x2.png", "content_image/1201.4951/x3.png" ]	# Interference between magnetic field and cavity modes in an extended Josephson junction V. Humbert Laboratoire de Physique des Solides, UMR8502-CNRS, University Paris-Sud, 91405 Orsay Cedex, France. M. Aprili aprili@lps.u-psud.fr Laboratoire de Physique des Solides, UMR8502-CNRS, University Paris-Sud, 91405 Orsay Cedex, France. J. Hammer [ Institute for Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany. February 22, 2024 ###### Abstract An extended Josephson junction consists of two superconducting electrodes that are separated by an insulator and it is therefore also a microwave cavity. The superconducting phase difference across the junction determines the supercurrent as well as its spatial distribution. Both, an external magnetic field and a resonant cavity intrafield produce a spatial modification of the superconducting phase along the junction. The interplay between these two effects leads to interference in the critical current of the junction and allows us to continuously tune the coupling strength between the first cavity mode and the Josephson phase from $1$ to $-0.5$ . This enables static and dynamic control over the junction in the ultra-strong coupling regime. pacs: Current address: ]Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany. A Josephson junction can be described as a two level system, at sufficiently low temperature, due to the non-linearity of the Josephson coupling. The strong coupling of a Josephson junction to an on-chip microwave superconducting resonator with small losses (i.e. quality factor higher than $10^{4}$) has led to the emergence of the new field of circuit quantum electrodynamics (CQED) [1]. We note that an extended Josephson junction in which the two superconductors are coupled through an insulating barrier, is at the same time a non-linear Josephson oscillator $and$ a microwave cavity [2]. Neglecting the Josephson effect, the eigenfrequencies of the electromagnetic modes are given by $\nu_{n}=k_{n}\cdot c_{s}/2\pi$, with $k_{n}=n\cdot\pi/L$ where L is the junction length (see Fig. 1(b)) and $c_{s}$ the Swihart velocity [3]. As the Josephson current and the microwave field are both localized in the insulator, extended junctions intrinsically form a microwave cavity enclosing a material resonance, the superconducting oscillator. Therefore, not only, the Josephson plasma frequency and the mode frequencies can be made, at low temperature, much larger than damping as required for strong coupling, but more importantly, the vacuum Rabi frequency, i.e. the photon exchange rate between the microwave cavity and the Josephson oscillator, can be as large as a fraction of the first cavity mode eigenfrequency, so that in fact the system is in the ultra-strong coupling limit [4]. This can be seen from the Hamiltonian of the junction [5], $H=H_{J}+H_{c}+H_{int}$ where $H_{J}$ and $H_{c}$ are the Josephson and the cavity Hamiltonian, respectively, while $H_{int}=-(h\nu_{p})^{2}/h\nu_{n}g_{n}N_{c}\varphi_{0}$ describes the interaction between the cavity modes and the Josephson phase. Here $\nu_{p}$ is the Josephson plasma frequency, $N_{c}$ the cavity photon number and $\varphi_{0}$ the macroscopic phase difference across the junction [6]. The interaction thus provides an intrinsically non-linear coupling which is formally equivalent to radiation-pressure interaction in optomechanics [7], [8]. The coupling constant $g_{n}$ is given by \[\begin{split} g_{n}=2\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dx}{L }&\sin[k_{n}x]^{2}\cos[k_{H}x]=\frac{\sin[\frac{\pi\phi}{\phi_{0} }]}{\frac{\pi\phi}{\phi_{0}}}\cdot h_{n}(\phi)\\ \textmd{with~{} }& h_{n}(\phi)=1+\frac{(\phi/\phi_{0 })^{2}\cos(n\,\pi)}{n^{2}-(\phi/\phi_{0})^{2}}\end{split}\] (1) where $k_{H}=2\pi\phi/(\phi_{0}L)$, with $\phi$ the magnetic flux in the junction and $\phi_{0}=2e/h$ the flux quantum [5]. At zero applied magnetic field $g_{1}=1$, therefore extended Josephson junctions provide an appealing system to investigate ultra-strong coupling between the superconducting phase and photons [9] with small aging factors from the electromagnetic environment. This regime is difficult to achieve in optical cavities [10], but recently has been obtained in a solid-state semiconductor system [11]. Furthermore it is worth noting, that the coupling is statically and dynamically tunable. In fact, $g_{1}$ can be continuously changed from 1 to -0.5 by the external magnetic field. In this paper we report on the magnetic field dependence of the coupling strength between the Josephson phase and the first cavity mode. Here $g_{1}$ is obtained by measuring the Josephson critical current as a function of the applied magnetic field for a microwave radiation frequency either resonant or non-resonant with the first cavity mode. Thus the magnetic field dependence of $g_{1}$, corresponds physically to the interference in the Josephson critical current between the phase differences produced by the intracavity and the magnetic fields. The critical current of a planar rectangular Josephson junction shows a Fraunhofer pattern as a function of the applied magnetic field [12] (see Fig. 1(c)). This originates from the phase difference, $\varphi_{H}$, created by the magnetic flux through the junction [13]. If self-screening of the applied magnetic field (i. e. $\lambda_{L}>>L$ where $\lambda_{L}$ is the Josephson penetration depth) is neglected, the phase difference accumulated along the junction is $\varphi_{H}$=$k_{H}$x. Here $d$ is magnetic penetration depth [13]. Therefore, the critical current, $I_{c}$ through the junction is given by [13]$I_{c}=I_{c0}\lvert\frac{\sin(\pi\phi/\phi_{0})}{\pi\phi/\phi_{0}}\rvert$. In presence of the n-th resonant mode due to microwave excitation, the phase difference produced by the electromagnetic field, $\varphi_{RF}$, has to be added to $\varphi_{H}$. The total phase difference is $\varphi$=$\varphi_{H}$+$\varphi_{RF}$+$\varphi_{0}$ where $\varphi_{RF}=a_{n}\cdot Re(e^{i2\pi\nu_{n}t})\sin(k_{n}x)$ is obtained by integrating the second Josephson equation with $a_{n}=2eV_{RF}/(\hbar\nu_{n})$. The critical current through the junction after integration of the first Josephson equation over time and space becomes [5] \[I_{c}=I_{c0}\,\lvert\frac{\sin(\pi\phi/\phi_{0})}{\pi\phi/\phi_{0}}\rvert\cdot \left(1-\frac{a_{n}^{2}}{4}\cdot h_{n}(\phi)\right)\] (2) Therefore the intracavity field of each mode contributes differently to the diffraction pattern. The second term in eq. 2 gives the magnetic field dependence of the coupling strength to the n-th resonant mode, $g_{n}$, and it accounts for the interference between $\varphi_{H}$ and $\varphi_{RF}$. For simplicity we consider only the first mode, resulting in a magnetic field dependend deviation of the critical current according to \[\Delta I_{c}=\left(\frac{2eV_{RF}}{\hbar\nu_{1}}\right)^{2}\,\lvert\frac{\sin( \pi\phi/\phi_{0})}{\pi\phi/\phi_{0}}\rvert\cdot\frac{1-2(\phi/\phi_{0})^{2}}{4 (1-(\phi/\phi_{0})^{2})}\] (3) Let us make a few remarks about eq. 3. First, since the phase-intrafield coupling is not linear, $\Delta I_{c}$ as a function of the magnetic flux is not equivalent to a normalization of the critical current and/or of magnetic quantum flux in the junction. Second $\Delta I_{c}$ changes sign at $0.7\,\phi_{0}$, meaning that the overall effect of microwave radiation is to decrease the critical current through the junction for $\phi<0.7\phi_{0}$, while it is to increase $I_{c}$ for $\phi>0.7\phi_{0}$. This is contrary to the common belief that microwave fields always reduce the Josephson critical current in the adiabatic approximation, i.e. when phase-photon coupled dynamics is not taken into account [14]. Finally from the Josephson current-phase relation $I=I_{c}\sin(\varphi)$ we observe that there is a small flux range just above $\phi_{0}$ in which the macroscopic phase difference through the junction changes from $\pi$ to $0$ under resonant microwave irradiation at frequency $\nu_{1}$. This is because close to $\phi_{0}$, the effect of the interference term $\Delta I_{c}$ is equivalent to a small shift in the Fraunhofer pattern. In long Superconductor/Normal/Superconductor Josephson junctions microwave induced changes in the current-phase relation have been proposed [15] and observed [16], based on a completely different mechanism; namely the microwave pumping produces a strong out-of-equilibrium quasiparticle distribution in the Andreev bound states in the normal metal. <figure><img src="content_image/1201.4951/x1.png"><figcaption>Figure 1: (a) Hysteretic current-voltage characteristic (red line) of extendedJosephson junction, taken at 600 mK. The first Fiske resonance at V=15μV (blueline) is taken at ϕ=0.7ϕ0. (b) Sketch of extended Josephson junction. Electricfield distribution of the first resonant mode k1 is indicated in red. (c)Fraunhofer pattern of Ic (data: red line, theory: red dashed line), and firstFiske resonance Ic1 (blue line) as a function of applied (in plane) magneticfield with theoretical curve (blue dashed line) given by eq. 4.</figcaption></figure> _Experiment._ We used superconductor/insulator/ferromagnet/superconductor (SIFS) Josephson junctions consisting of $\rm Nb(150nm)/Al_{2}O_{3}/PdNi(d_{F})/\-Nb(50nm)$ in a cross strip geometry (see Fig. 1(b)). The ferromagnetic thin layer reduces the Josephson coupling and the phase relaxation time at the working temperature of 600mK. The weak ferromagnet PdNi contains $10\%$ of Ni, has a Curie temperature of around 150 K, and its thickness $d_{F}$ varies between $50\,\rm{\AA}$ and $100\,\rm{\AA}$[17]. We also fabricated non-ferromagnetic (SIS) junctions without the PdNi layer, in order to verify that the thin ferromagnetic layer has no other effect than reducing the critical current. The fabrication details are given elsewhere [17]. The critical current of ferromagnetic junctions is between 10 $\mu A$ and $130\,\mu A$, the normal resistance $R_{n}\sim 0.2\,\Omega$, and the critical temperature around $8.2\,$K. The quasiparticle resistance of the junctions was measured to be 29 $\Omega$. The junction area is $0.7\times 0.7\,\rm mm^{2}$, the capacitance, $C$, is $30$ nF [14] making phase dynamics underdamped [13]. The magnetic field $H$ is applied in the $y$-direction. A $\mu$-metal shield ensures a negligible residual magnetic field in the one-shot ${}^{3}He$ cryostat. <figure><img src="content_image/1201.4951/x2.png"><figcaption>Figure 2: (a) Josephson spectroscopy of first cavity mode: Resonance ofnormalized critical current as function of microwave frequency (red dots) atν1=7.18 GHz with quality factor Qc=250 from Lorenzian fit (blue line). (b)Suppression of critical current Ic as function of injected microwave power.Blue line corresponds to theoretical fit. (c) Deviation of the criticalcurrent ΔIc for resonant (ν=7.185 GHz) and non-resonant (ν=6.950 GHz)microwave excitation. The blue line is obtained from theoretical predictions.</figcaption></figure> In Fig. 1(a) we show the current-voltage ($IV$) characteristic measured at zero applied magnetic field. The data follows a hysteretic $IV$ characteristic with the retrapping current practically zero, as expected for strongly underdamped Josephson junctions. The resonance at $V_{1}=15\mu$V (blue line) is the first Fiske step [18]. When a finite DC-voltage appears across the junction, the cavity modes are resonantly excited at $V_{n}^{DC}=\frac{h}{2e}\nu_{n}$, and mix with the AC-Josephson current giving rise to finite DC-resonances [18]. The first Fiske step shown in Fig. 1(a) has been recorded separately for an applied magnetic field of $\phi=0.7\,\phi_{0}$ which maximizes the step amplitude. From $V_{1}=15\mu$V we obtain a Swihart velocity $\tilde{c}=0.037c$. In Fig. 1(a) we present only the first Fiske step but higher order steps (not shown) are also observed [14]. We verify that the resonance at $V_{1}=15\mu$V corresponds to the first Fiske resonance by measuring its magnetic field dependence, $I_{c1}(\phi)$, as reported in Fig. 1(c) (blue line). For the current amplitude of the first Fiske step one obtains from theory [19] \[I_{c1}(\phi)=b\cdot I_{c0}\left(\frac{4(\phi/\phi_{0})}{2(\phi/\phi_{0})+1} \frac{\sin(\pi(\phi/\phi_{0}-0.5))}{\pi(\phi/\phi_{0}-0.5)}\right)^{2}\] (4) as experimentally observed. Equation 4 is plotted in Fig.1(c) as a blue dashed line. Here the numerical constant $b=0.275$ is in the limit of high cavity quality factor $Q_{c}$. In Fig. 1(c) the critical current is also shown as a function of the applied magnetic field. $I_{c}$ follows the Fraunhofer pattern (red dashed line) as described above. Smaller secondary maxima indicate a larger current density in the center of the junction. We now focus on the effect of the intracavity field on the Josephson switching current, i.e. on the maximum superconducting current in the junction (at zero voltage bias), before it switches to the dissipative state, which corresponds to a finite voltage across the junction. In our junctions at 600mK the switching current represents $I_{c}$ within one per cent [20]. The variation of the critical current as function of the microwave frequency for a fixed microwave injected power of $-15$ dBm is shown in Fig. 2(a). We report the microwave power provided by the source and not the actual power arriving at the sample. The coupling constant is obtained below. As microwaves are absorbed only at $\nu=\nu_{n}$, the critical current is suppressed only at resonance. This allows a fine spectroscopy of the cavity modes. We observed the first mode at $7.18$ GHz as expected from the value measured for the first Fiske step ($15\mu$V correspond to $7.5$ GHz). From the Lorenzian fit (see Fig. 2(a)), we obtain the cavity quality factor $Q_{c}=250$. The quality factor is limited by dissipation. If dissipation is only due to quasiparticle tunneling, $Q_{c}$ would be given by $\omega_{1}R_{qp}C$ (about $4\cdot 10^{4}$ in our junctions), where $R_{qp}$ is the tunneling quasiparticle resistance. Nevertheless, it has been shown [21] that at high frequency, surface losses in the electrodes and dielectric losses in the insulator are more important than quasiparticle tunneling and they both substantially reduce $Q_{c}$. The microwave induced suppression of the critical current at resonance is given by $1-(\alpha a_{1})^{2}$[22], where $\alpha$ is the coupling constant between the microwave line and the Josephson junction. In Fig. 2(b) we present $I_{c}$ versus microwave power at resonance, i.e. for $\nu=7.18$ GHz. From the theoretical fit (blue line) we get $\alpha=5\cdot 10^{-5}$. This shows that the Josephson junction is very weakly coupled to the microwave circuit. We then sweep the magnetic field and for each value of the applied field, we measure the difference in the critical current $\Delta I_{c}$ , by substracting $I_{c}$ at two microwave frequencies $6.950$ GHz and 7.185GHz for $-15$ dBm, corresponding to $-235$ MHz and $0$ MHz detuning from the first cavity mode, respectively. Note that $235$ MHz is much larger than the cavity bandwidth. $\Delta I_{c}$ as a function of the magnetic flux in the junction is reported in Fig. 2(c) (red line). Here $\Delta I_{c}$ is the interference term between the intracavity and the magnetic field, described by eq. 3, which is also shown in Fig.2 (c) (blue line). Equation 3 accounts well for the experimental data. In particular, we observe the change in sign of $\Delta I_{c}$ at $0.7\,\phi_{0}$ as predicted. The jump at about $\phi_{0}$ corresponds to the microwave induced $\pi$-$0$ transition in the macroscopic phase difference. <figure><img src="content_image/1201.4951/x3.png"><figcaption>Figure 3: (a) Colormap showing normalized deviation of critical current1−ΔIc/Ic as a function of microwave frequency and applied magnetic field. (b)Variation of the microwave resonant frequency for applied magnetic field,deduced from (a). The red line corresponds to the theoretically expectedvariation of ν1 as a function of an applied magnetic field.</figcaption></figure> The change in sign of $\Delta I_{c}$ can be seen more clearly in Fig. 3(a) where we plot the normalized critical current as a function of the microwave frequency and the applied magnetic field. The microwave power is $-15$ dBm. Blue (red) corresponds to an increase (decrease) of the critical current. We observe that the frequency of the first resonance is slightly reduced by about $25$ MHz, when $\Delta I_{c}$ changes sign, i.e. for $\phi>0.7\,\phi_{0}$. The resonance frequency at $\phi=0$ and $\phi=0.9\,\phi_{0}$ are marked by a dashed and a dotted line respectively in Fig.3(a). This magnetic field induced shift in the resonance frequency, $\Delta\nu_{1}$, is smaller than, but comparable to the cavity bandwidth and it explains the difference between data and theory in Fig. 2(c) for $0.7<(\phi/\phi_{0})<1$. In fact, in this field range because of $\Delta\nu_{1}$, the value of $\Delta I_{c}$ measured as the difference in the critical current at two microwave frequencies $6.950$ GHz and $7.185$ GHz is underestimated, as can be seen in Fig.2(c). In Fig.3(b) we present the value of $\Delta\nu_{1}$ obtained from the data in Fig.3(a) as a function of the applied magnetic field. We verified that $\Delta\nu_{1}$ is independent on the microwave power. Due to the Josephson coupling the dispersion of the electromagnetic waves in the junction is not linear, the cavity resonance frequencies become [2]: $\nu_{n}=\sqrt{\nu_{p}^{2}+(k_{n}c_{s})^{2}}$, where $\nu_{p}=1/2\pi\sqrt{I_{c}/C\phi_{0}}$. The red line accounts for $\Delta\nu_{1}$ when the magnetic field dependence of the plasma frequency is taken into account. As clear from Fig. 3(b), the correction to $\nu_{1}$ expected from the Josephson coupling is too small to explain the experimental changes observed in $\nu_{1}$. Therefore the shift in the resonance frequency $\nu_{1}$ is likely related to the non-linear dependence of the kinetic inductance of the electrodes on the magnetic field induced screening. This is of course a small correction but measurable because of the high quality factor of the cavity. Thus it is interesting to point out that the measurement of $\nu_{1}$ is a very effective way to directly determine the kinetic inductance at finite frequency (GHz regime) of complex superconducting based multilayers. In conclusion we have observed that the phase difference produced by an applied magnetic field together with the phase difference caused by the cavity intrafield due to microwave radiation, interfere in extended Josephson junctions. This interference changes the Fraunhofer pattern of the critical current. Moreover we have noticed that these types of junctions allow us to enter in the ultra-strong coupling limit between the Josephson oscillator and the cavity eigenmodes. The coupling strength is magnetic field dependent and it follows exactly the interference term measured with and without microwave radiation. ###### Acknowledgements. We thank J. Gabelli, R. Gross, B. Reulet and A. Ustinov and for valuable discussions. We are indebted to C.H.L. Quay for a critical reading of the manuscript. ## References * (1) A. Walraff _et al._ Nature 431, 162-167 (2004) * (2) K.K. Likharev, Rev. Mod. Phys. 51, 101 (1979) * (3) J.C. Swihart, J. of Appl. Phys. 32, 461 (1961) * (4) P. Nataf and C. Ciuti arXiv:1106.1159 * (5) M.V. Fistul and A.V. Ustinov, Phys. Rev. B 75, 214506 (2007) * (6) In our junctions the ratio $R=\nu_{p}/\nu_{1}$ is about 0.05, however, it is technically possible to achieve $R$ of the order of the unity. * (7) S. Groblacher _et al._ Nature 460, 724 (2009) * (8) F. Marquardt _et al._, Phys. Rev. Lett. 99, 093902 (2007) * (9) T. Niemczyk _et al._ Nature Physics 6, 772 (2010) * (10) H. Walther, B.T.H. Varcoe, BG. Englert and T. Becker Rep. Prog. Phys. B 69, 1325 (2006) * (11) G. Gunter _et al._ Nature 458, 178 (2009) * (12) R. C. Jaklevic, J. Lambe, J. E. Mercereau, and A. H. Silver Phys. Rev. 140, A1628 (1965) * (13) A. Barone and G. Paterno, _Physics and Applications of the Josephson Effect_ (Wiley, New York, 1982) * (14) J. Hammer, M. Aprili and I. Petkovic , Phys. Rev. Lett. 107, 017001 (2011) * (15) Pauli Virtanen _et al._, Phys. Rev. Lett. 104, 247003 (2010) * (16) M. Fuechsle _et al._, Phys. Rev. Lett. 102, 127001 (2009) * (17) T. Kontos, M. Aprili, J. Lesueur, F. Genêt, B. Stephanidis, and R. Boursier, Phys. Rev. Lett. 89, 137007 (2002). * (18) D.D. Coon and M.D. Fiske, Phys. Rev. 432, A744 (1965) * (19) I.O. Kulik, Sov. Phys. Tech. Phys. 12, 111 (1967) * (20) I. Petkovic and M. Aprili, Phys. Rev. Lett. 102, 157003 (2009) * (21) R. F. Broom and P. Wolf, Phys. Rev. B 16, 3100 (1977) * (22) S. Shapiro, Phys. Rev. Lett. 11, 80 (1963)
1910.07420	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 85909, "num_imgs": 9, "llama3_tokens_count": 24508 }	[ "content_image/1910.07420/codeModel_1.png", "content_image/1910.07420/x1.png", "content_image/1910.07420/conv_deconv_arch.png", "content_image/1910.07420/arch.png", "content_image/1910.07420/x2.png", "content_image/1910.07420/fig2.jpg", "content_image/1910.07420/schemes.png", "content_image/1910.07420/fig4.jpg", "content_image/1910.07420/thresholds2.png" ]	# Machine Learning for Error Correction with Natural Redundancy Pulakesh Upadhyaya, Anxiao (Andrew) Jiang, This work was supported in part by NSF Grant CCF-1718886. Parts of this paper were presented at the 2017 Allerton Conference on Communication, Control, and Computing, 2017 Information Theory and Applications (ITA) Workshop, and 2019 IEEE International Conference on Communications (ICC). The authors are with the Department of Computer Science and Engineering, Texas A&M University, College Station, TX 77840, USA (e-mail: pulakesh@tamu.edu; ajiang@cse.tamu.edu). This paper was submitted to IEEE Journal on Selected Areas in Information Theory (special issue on Deep Learning: Mathematical Foundations and Applications to Information Science). ###### Abstract The persistent storage of big data requires advanced error correction schemes. The classical approach is to use error correcting codes (ECCs). This work studies an alternative approach, which uses the redundancy inherent in data itself for error correction. This type of redundancy, called Natural Redundancy (NR), is abundant in many types of uncompressed or even compressed files. The complex structures of Natural Redundancy, however, require machine learning techniques. In this paper, we study two fundamental approaches to use Natural Redundancy for error correction. The first approach, called Representation-Oblivious, requires no prior knowledge on how data are represented or compressed in files. It uses deep learning to detect file types accurately, and then mine Natural Redundancy for soft decoding. The second approach, called Representation-Aware, assumes that such knowledge is known and uses it for error correction. Furthermore, both approaches combine the decoding based on NR and ECCs. Both experimental results and analysis show that such an integrated scheme can substantially improve the error correction performance. Machine learning, deep learning, LDPC codes, natural redundancy. ## I Introduction A large amount of data is generated on the Internet everyday, and the feasibility of storing useful data permanently has become a key concern. The most effective classic approach to improve data reliability is to add external redundancy to data using Error Correcting Codes (ECCs). We call such redundancy _artificial redundancy_. However, over time, errors accumulate in storage systems and can exceed the decoding threshold of ECCs. To ensure permanent reliability of data, many techniques have been explored to improve the error correction capabilities in long-term storage systems. Recent progress in machine learning has offered an opportunity to employ novel techniques to improve data reliability. One such approach is to use Natural Redundancy in data for error correction. By _Natural Redundancy_ (NR), we refer to the redundancy that is inherent in data, which is not artificially added by ECCs. It is abundant in many types of uncompressed or even compressed files. For instance, consider the English language. When LZW (Lempel-Ziv-Welch) coding is used with a fixed dictionary of $2^{20}$ patterns (larger than many LZW codes in practical systems), the language can be compressed to 2.94 bits/character. State-of-the art compression algorithms (e.g., syllable-based Burrows-Wheeler Transform) can further reduce it to 2 bits/character [26]. However, even with such advanced compression techniques, the result is still far from Shannon’s estimation of 1.34 bits/character, which is an upper-bound for the entropy of printed English [47]. For images, residual redundancy can also be abundant after compression, as made evident by recent inpainting techniques of deep learning [56]. Such abundant Natural Redundancy can be an excellent resource for error correction. There are two fundamental ways to utilize Natural Redundancy in an information system. The first way is enhanced _data compression_, which often uses deep learning to remove redundancy further than before [30, 41]. It is a new and active research area, and compression ratios higher than classic compression algorithms have been achieved in some cases (e.g., for high distortion regimes). The second way, which is the focus of this paper, is to use Natural Redundancy for _error correction_. That is, a new decoder is designed to mine the Natural Redundancy in data, and utilize it for error correction. The decoder can be further combined with ECC’s decoder for better performance. A strong motivation for this method is that modern storage systems already store a massive amount of data, which would be very costly to reprocess. The Natural Redundancy (NR) based decoder does not require systems to examine or modify any existing data. It only requires an enhancement to the decoding algorithm itself. Therefore, it is compatible with storage systems and convenient to use. In this paper, we study two fundamental approaches to use Natural Redundancy for error correction. The first approach, called _Representation-Oblivious_, requires no prior knowledge on how data are represented or compressed in files. It uses deep learning to detect file types accurately, and then mine Natural Redundancy for soft decoding. The second approach, called _Representation-Aware_, assumes that such knowledge is known and uses it for error correction. Furthermore, both approaches combine the decoding based on NR and ECCs. The Representation-Oblivious approach is useful for many storage systems where error correction is a low-level function. In those systems, such as hard drives or solid-state drives (SSDs), the controllers for error correction often have no access to information such as file types or compression schemes. Deep learning is a very useful tool for learning the complex patterns in data from scratch. And deep learning based classifiers are also suitable for decoding such data with Natural Redundancy. The Representation-Aware approach is useful for storage systems where error correction is a higher-level function. With knowledge on how data are represented, better error correction performance can be achieved with suitable machine learning techniques. This paper studies NR-based error correction for several types of data of common file types, including HTML files, JPEG files, PDF files, LaTex files and language-based texts. It presents new deep learning techniques for mining Natural Redundancy, and presents both soft-decoding and hard-decoding algorithms based on NR that can be combined with LDPC codes. It presents both experimental results and theoretical analysis for measuring the amount of Natural Redundancy mined for error correction, and the results show that NR-based decoding can substantially improve the error correction performance. (For instance, the Representation-Aware scheme can improve the decoding threshold for erasures of LDPC codes by a factor of five, when the channel’s erasure rate is as high as $30\%$.) Furthermore, we also analyze the computational complexity of using Natural Redundancy for error correction versus for data compression. The rest of the paper is organized as follows. In Section II, we review related works. In Section III, we present the Representation-Oblivious scheme, and combine it with LDPC codes to achieve enhanced error-correction performance. In Section IV, we present a Representation-Aware scheme for language-based texts, and analyze the performance of two approaches for combining NR-based decoders with LDPC decoders: a _sequential_ decoding scheme and an _iterative_ decoding scheme. In Section V, we study the computational complexity of using NR for error correction versus for data compression. In Section VI, we present the conclusions. ## II Related Work In this section, we review related works, including joint-source channel coding (JSCC), denoising, recent results on NR-based error correction, and deep learning for information theory. The idea of using the leftover redundancy at a source encoder to improve the performance of ECCs has been studied within the field of joint source-channel coding (JSCC) [4, 16, 17, 18, 20, 24, 39, 40, 42]. However, few works have considered the Representation-Oblivious scheme. Furthermore, not many works have considered JSCC specifically for language-based sources. Related to JSCC, denoising is also an interesting and well studied technique [1, 6, 9, 11, 31, 37, 38, 44, 57, 55]. A denoiser can use the statistics and features of input data to reduce its noise level for further processing. However, how to combine denoisers with the recent progress in LDPC codes and machine learning has remained under-explored. In recent works (including results from the authors of this work), machine learning and algorithmic techniques have been used to exploit NR to correct errors in data [21, 22, 23, 27, 33, 48, 49, 53, 54]. This work studies the Representation-Oblivious scheme for the first time, and also presents new theoretical analysis for the Representation-Aware scheme. In parallel, there have been numerous recent works on using deep learning for information theory [19, 43], especially for wireless and optical communications. They mainly focus on using deep learning to model complex channels, to design codes, and to approximate or improve decoding algorithms [3, 8, 12, 25, 35]. In contrast to those works, this paper focuses on using machine learning for _data_ with complex structures (instead of for complex channels), and on exploring error correction for such complex data. These two different directions are complementary in a communication or storage system, and can be integrated. ## III Representation-Oblivious NR-decoding In this section, we study the _Representation-Oblivious_ scheme for Natural Redundancy (NR) based decoding. In this scheme, _no prior information_ on the data is needed, including how data are represented or compressed, which file type (e.g. HTML, JPEG, etc.) they belong to, or how meta-data are appended to payload bits. This scheme has the benefit of having only minimal requirements on practical storage systems such as hard drives and SSDs. Controllers of storage systems can read out blocks of data and perform error correction (aided by NR-decoding) as usual, without having to access file systems for additional information on the data. However, the task is also challenging. For example, without knowing the data compression algorithm, we cannot use its codebook to find patterns in the data. The patterns in data are highly complex, and vary greatly for different file types. (For instance, bit patterns in HTML files and JPEG files are very distinct from each other.) To address the challenges of this new error correction paradigm, we use deep learning to perform error correction in three consecutive steps: (1) detect the file type of the given block of noisy bits; (2) perform NR-based soft decoding for the block of noisy bits; (3) use the NR-based soft-decoding results to improve the performance of ECC decoding. Our coding scheme for Representation-Oblivious error correction using NR is illustrated in Fig. 1. When files are stored, each file is partitioned into segments of $k$ bits, and each file segment is encoded by a systematic $(n,k)$ ECC into a codeword of $n$ bits. Then each ECC codeword passes through a noisy channel, which models the errors in a storage device. During decoding, first, a deep neural network (DNN) uses the $k$ noisy information bits to recognize the file type (e.g. HTML, LaTeX, PDF or JPEG) of the file segment. Then, a second DNN for that file type performs soft decoding on the $k$ noisy information bits based on Natural Redundancy, and outputs $k$ probabilities, where for $i=1,2,\cdots,k$, the $i$-th output is the probability for the $i$-th information bit to be 1. The $k$ probabilities are given as additional information to the ECC’s decoder. The ECC decoder then performs its decoding and outputs the final result. (In our experiments, the ECC is a systematic LDPC code, and the $k$ probabilities are combined with the initial LLRs (log-likelihood ratios) for information bits to obtain their updated LLRs. The LDPC code then runs its belief-propagation (BP) decoding algorithm.) In the following, we present the detailed designs. <figure><img src="content_image/1910.07420/codeModel_1.png"><figcaption>Fig. 1: Encoding and decoding scheme for a noisy file segment of an initiallyunknown file type. The k-bit file segment is encoded by a systematic (n,k) ECCinto an n-bit codeword. The codeword is transmitted through a channel to get anoisy codeword. Two neural networks use NR to decode the k noisy informationbits: the first network determines the file type of the file segment, and thena corresponding neural network for that file type performs soft decoding forthe k noisy information bits. The soft decoding result and the noisy codewordare both given to the ECC decoder for further error correction.</figcaption></figure> ### _File Type Recognition using Deep Learning_ We present here a _Deep Neural Network_ (DNN) for file type recognition. The DNN takes a noisy file segment of $k$ bits, $(y_{1},y_{2},\cdots,y_{k})$, as input, and outputs one of $T$ file types (e.g., HTML, LaTeX, PDF or JPEG). The errors in the file segment come from a binary-symmetric channel (BSC) of bit-error rate (BER) $p$. We first introduce the architecture of the DNN and its training method. We then present the experimental results, which show that it achieves high accuracy for file type recognition. #### Iii-A1 DNN Architecture and Training <figure><img src="content_image/1910.07420/x1.png"><figcaption>Fig. 2: Architecture of the CNN (convolutional neural network) for File TypeRecognition. Its input is a noisy file segment of 4095 bits, and its outputcorresponds to T=4 candidate file types (HTML, LaTex, PDF and JPEG). The CNNuses _ReLU_ and _sigmoid_ as the activation function of its convolutionallayers and output layer, respectively. It uses cross entropy as its lossfunction. Its optimizer is chosen to be an Ada Delta Optimizer.</figcaption></figure> Our DNN architecture is shown in Fig. 2. It is a Convolutional Neural Network (CNN) that takes the $k$ bits of a noisy file segment as input. In our experiments, we let $k=4095$. (The LDPC code we use is a $(4376,4095)$ code designed by MacKay [34], which can tolerate BER of 0.2%. Both the code length and the BER are in the typical range of parameters for storage systems.) The CNN has $T$ outputs that correspond to the $T$ possible file types, namely, the $T$ classification results. The output with the highest value leads to the selection of the corresponding file type. In our experiments, we consider four file types: HTML, LaTeX, PDF and JPEG. So $T=4$. Note that HTML and LaTeX files are both text sequences but have different file structures; PDF files contain both texts and images; and JPEG files are images. In the following, we will present DNNs and experiments using these parameters for the convenience of presentation. Note that the designs can be extended to other file-segment lengths and more file types. A large dataset has been used to train and test the CNN. For each of the $T=4$ file types, 24,000 noiseless file segments are used for training data, 4,000 noiseless file segments are used for validation data, and 4,800 noiseless file segments are used for test data. During training and testing, random errors of BER $p$ are added to each file segment, where each file segment uses an independently generated error pattern. #### Iii-A2 Experimental Performance Bit Error \| Overall \| HTML \| JPEG \| PDF \| LaTeX ---\|---\|---\|---\|---\|--- Rate \| Test \| Test \| Test \| Test \| Test (BER) \| Accuracy \| Accuracy \| Accuracy \| Accuracy \| Accuracy 0.2% \| 99.61% \| 99.98% \| 99.52% \| 99.17% \| 99.77% 0.4% \| 99.69% \| 99.96% \| 99.60% \| 99.25% \| 99.96% 0.6% \| 99.60% \| 99.94% \| 99.48% \| 99.06% \| 99.90% 0.8% \| 99.69% \| 99.98% \| 99.50% \| 99.35% \| 99.92% 1.2% \| 99.66% \| 99.96% \| 99.23% \| 99.48% \| 99.96% 1.6% \| 99.58% \| 99.96% \| 99.60% \| 98.83% \| 99.92% TABLE I: Bit error rate (BER) vs Test Accuracy for File Type Recognition (FTR). Here the “overall test accuracy” is for all 4 types of files together. The last four columns show the test accuracy for each individual type of files. (Their average value is the overall test accuracy.) The $(4376,4095)$ LDPC code used in our experiments can correct errors of BER up to 0.2% _by itself_. (That is, when it is used in the conventional way without the extra help of Natural Redundancy, it has a decoding threshold of 0.2%.) Our goal is to use the Natural Redundancy in file segments to correct errors of substantially higher BERs. So we have selected the target BER $p$ with substantially higher values, ranging from $0.2\%$ to $1.6\%$. We then train the CNN with the given target BER $p$. We measure the performance of the CNN by the _accuracy_ of file type recognition (FTR), which is defined as the fraction of file segments whose file types are recognized correctly. The test performance is shown in Table I. It can be seen that file types can be recognized by the CNN with high accuracy: for all BERs, the accuracy is close to 1. We can also examine the accuracy for recognizing each file type, and see if there is variance in performance from file type to file type. The results are shown in the last four columns of Table I. It can be seen that overall, the accuracy is constantly high for all file types. The CNN’s performance compares favorably with existing results on FTR, which has been studied previously for applications such as disk recovery. The work [7] considered a classification method for a pair of file types using Fisher’s linear discriminant and longest common subsequence methods. The accuracy ranges between $87\%$ and $99\%$ depending on which pair of file types are considered. The work [15] introduced an NLP (natural language processing) based method, where unigram and bigram counts of bytes and other statistics are used to generate feature representation, which is then followed by support vector machine (SVM) for classification of various file types. The classification accuracy varies from $17.4\%$ for JPEG files, $62.5\%$ for PDF files to $94.8\%$ for HTML files. The work [2] used PCA (principal component analysis) and a feed-forward auto-associative unsupervised neural network for feature extraction, and a three layer multi-layer perceptron network for classification. The classification accuracy is $98.33\%$ for six file types while considering entire files instead of file segments. Our deep-learning based method can be seen to achieve high performance, without the need to train separate modules for feature extraction and classification. The CNN has _robust_ performance because it works well not only for the BER it is trained for, but also for other BERs in the considered range. (For example, a CNN trained for $BER=1.2\%$ also works well for other BERs in the range $[0.2\%,2.0\%]$.) For succinctness we skip the details. The robustness of the overall error correction performance for different BERs will be presented in Subsection C. ### _Soft NR-decoding by Deep Neural Networks_ In this subsection, we study how to design DNNs that can perform soft decoding on noisy file segments. For each of the $T$ file types, we will design and train a different DNN, because different types of files have different types of Natural Redundancy. Given a file type, we will design a DNN whose input is a noisy file segment of $k$ bits $Y=(y_{1},y_{2},\cdots,y_{k})$. As before, the errors in the noisy file segment come from a binary-symmetric channel (BSC) of bit-error rate (BER) $p$. The output of the DNN is a vector $Q=(q_{1},q_{2},\cdots,q_{k})$, where for $i=1,2,\cdots,k$, the real-valued output $q_{i}\in[0,1]$ represents the DNN’s belief that for the $i$-th bit in the file segment, the probability that its correct value should be 1 is $q_{i}$. In other words, if we use $X=(x_{1},x_{2},\cdots,x_{k})$ to denote an error-free file segment, and let it pass through a BSC of BER $p$ to obtain a noisy file segment $Y=(y_{1},y_{2},\cdots,y_{k})$, then $q_{i}$ is the DNN’s estimation for $Pr\{x_{i}=1~{}\|~{}Y,p\}$. Note that the $k$ bits are not independent of each other because of the Natural Redundancy in them. So $Pr\{x_{i}=1~{}\|~{}Y,p\}$ depends on not only $y_{i}$ and $p$, but also the overall value of $Y$. The goal of the DNN is to learn the Natural Redundancy in file segments, and use it to make the probability estimation $q_{i}$ be as close to the true probability $Pr\{x_{i}=1~{}\|~{}Y,p\}$ as possible, for each $i$ and for each possible value $Y$ of the noisy file segment. To train the DNN, our optimization objective is to minimize the loss function \[L=\frac{1}{k}\sum\limits_{i=1}^{k}[x_{i}\log_{2}{q_{i}}+(1-x_{i})\log_{2}(1-q_ {i})],\] which measures the cross-entropy between $(x_{1},x_{2},\cdots,x_{k})$ and $(q_{1},q_{2},\cdots,q_{k})$, over all samples in the training dataset. The architecture of the DNN is presented in Fig. 3. It is related to auto-encoders, which are good choices for various applications related to denoising [52, 32]. The DNN model consists of $L$ convolutional layers followed by $L$ deconvolutional layers. (Deconvolutional layers may be seen as reverse operations of convolutional layers. Interested readers can refer to [10] for more details.) The $L$ convolutional layers have one-dimensional filters of size $s_{1},s_{2},\cdots,s_{L},$ respectively, and the number of feature maps at the output of each layer is $m_{1},m_{2},\cdots,m_{L},$ respectively. The filter sizes and the number of feature maps for deconvolutional layers change in the reverse order. <figure><img src="content_image/1910.07420/conv_deconv_arch.png"><figcaption>Fig. 3: General architecture of deep neural networks (DNNs) for NR-based softdecoding of noisy file segments. It consists of L convolutional layersfollowed by L deconvolutional layers. The activation function for the lastlayer is relu, and is sigmoid for the other layers. It uses cross-entropy asthe loss function, and uses the Adam optimizer.</figcaption></figure> <figure><img src="content_image/1910.07420/arch.png"><figcaption>Fig. 4: Hyper-parameters of optimized DNN models for NR-based soft decoding,for T=4 file types and different bit error rates. Here L is the number ofconvolutional/deconvolutional layers, s=(s1,s2,⋯,sL) represents the filtersizes, and m=(m1,m2,⋯,mL) represents the numbers of feature maps.</figcaption></figure> We optimize the hyper-parameters of DNNs (including filter sizes, number of feature maps, etc.) for each file type. Their performance is robust: an optimized DNN usually performs soft decoding well for a wide range of BERs. However, the performance can be slightly improved further if the hyper-parameters are also optimized based on BERs. Such optimization results are presented in Fig. 4. (Here, for PDF and JPEG files, the hyper-parameters are optimized based on two sub-ranges of BERs.) We will present their decoding performance (when combined with ECC decoding) and robustness in the next subsection. ### _Combine Soft NR-decoding with Soft LDPC-decoding_ In this subsection, we present a scheme that combines the soft NR-decoding, which applies deep learning to noisy file segments of different file types, with soft LDPC-decoding. The experimental results confirm that the scheme substantially improves the reliability of different types of files. We adopt a _robust scheme_ here: the DNNs for file-type recognition and for soft decoding have been trained with a constant BER $p_{DNN}$, but they are used for a wide range of BERs $p$ for the BSC channel. (For example, the DNNs may be trained just for $p_{DNN}=1.2\%$, but are used for any BER $p$ from 0.2% to 1.6% in experiments here.) We choose this robust scheme because when DNNs are designed, the future BER in data can be highly unpredictable. Given a noisy systematic LDPC codeword, we first use a DNN to recognize its file type based on its $k$ noisy information bits. Then a second DNN for that file type is used to do soft decoding for the $k$ noisy information bits, and output $k$ probabilities: for $i=1,2,\cdots,k$, the $i$-th output $q_{i}$ represents the estimated probability for the $i$-th information bit to be 1. Those $k$ probabilities can be readily turned into LLRs (log-likelihood ratios) for the information bits using the formula \[LLR_{i}^{DNN}=\log(\frac{1-q_{i}}{q_{i}})\] For $i=1,2,\cdots,n$, let $LLR_{i}^{channel}$ be the LLR for the $i$-th codeword bit (with $1\leq i\leq k$ for information bits, and $k+1\leq i\leq n$ for parity-check bits) derived for the binary-symmetric channel, which is either $\log(\frac{1-p}{p})$ (if the received codeword bit is 0) or $\log(\frac{p}{1-p})$ (if the received codeword bit is 1). Then we let the _initial LLR_ for the $i$-th codeword bit be \[LLR_{i}^{init}=LLR_{i}^{channel}+LLR_{i}^{DNN}\] for $1\leq i\leq k$, and \[LLR_{i}^{init}=LLR_{i}^{channel}\] for $k+1\leq i\leq n$. We then perform belief-propagation (BP) decoding using the initial LLRs, and obtain the final result. Note that there is a positive – although very small – chance that the file type will be recognized incorrectly. In that case, the incorrect soft-decoding DNN will be used, which is accounted for in the overall decoding performance for fair evaluation. We measure the performance of the error correction scheme by the percentage of codewords that are decoded correctly, which we call _Decoding Success Rate_. (Let us call the scheme the _NR-LDPC decoder_, since it combines decoding based on Natural Redundancy and the LDPC code.) We focus on BERs that are beyond the decoding threshold of the LDPC code, because NR becomes helpful in such cases. Note that the $(4376,4095)$ LDPC code used in our experiments has a decoding threshold of $BER=0.2\%$. In our experiments, we focus on BERs $p$ that are not only beyond the decoding threshold, but also can be significantly larger: $p\in[0.2\%,~{}1.6\%]$. The experimental results for $p_{DNN}=1.0\%$ are presented in Fig. 5 (a). Here the $x$-axis is the channel error probability $p$, and the $y$-axis is the _Decoding Success Rate_. (For each $p$, 1000 file segments with independent random error patterns have been used in experiments.) The curve for “ldpc” is the performance of the LDPC decoder alone, and the curve for “nr-lpdc” is for the NR-LDPC decoder. It can be seen that the NR-LDPC decoder achieves significantly higher performance. For example, as $p=0.6\%$, the decoding success rate of the NR-LDPC decoder is approximately 4 times as high as the LDPC decoder. The figure also shows the performance for each of the 4 file types. (The 4 curves are labelled by “html”, “latex”, “pdf”, “jpeg”, respectively. Their average value becomes the curve for “nr–ldpc”.) It shows that the error correction performance for HTML and LaTex files are significantly better than for PDF and JPEG files. It is probably because the former two mainly consist of languages, for which the soft-decoding DNNs are better at finding their patterns and mining their natural redundancy, while PDF is a mixture of languages and images and JPEG is image only. It is interesting to notice that even for JPEG files, when $p>0.6\%$, the NR-LDPC decoder again performs better than the LDPC decoder, which means the DNNs can extract Natural Redundancy from images, too. Fig. 5 (b) to Fig. 5 (d) show the performance for $p_{DNN}$ = 1.2%, 1.4% and 1.6%, respectively. The NR-LDPC decoder performs equally well in those cases, which proves the value of Natural Redundancy for decoding. <figure><img src="content_image/1910.07420/x2.png"><figcaption>Fig. 5: Decoding success rate vs BER for (a) pDNN = 1.0% , (b) pDNN = 1.2%,(c) pDNN = 1.4%, (d) pDNN = 1.6%.</figcaption></figure> In summary, although no prior information is known on data representation, deep learning can recognize file types with high accuracy, and perform soft decoding effectively. When combined with ECCs, it can improve the error correction performance substantially. It is expected that with future improvements in deep learning, more natural redundancy can be mined from data to improve the reliability of storage systems even further. ## IV Representation-Aware NR-decoding The previous section studies _Representation-Oblivious_ schemes. In this section, we study schemes that are _Representation-Aware_: how the source data is mapped to bits in files is known. This is also a highly useful scenario, especially when error correction is performed at a high level in computer systems, or when the controller in storage devices perform their own compression schemes. In this work, we focus on language-based data, which form an important part of big data. In particular, we focus on the English language compressed by LZW algorithms. The results can be generalized to more languages and other sequential compression algorithms, such as Huffman codes [27]_etc._ In this section, we first present an NR-based hard-decoding algorithm for languages, and analyze its performance. We then study two important cases for combining NR-decoding with ECC decoding: the _sequential_ decoding scheme, and the _iterative_ decoding scheme. For both cases, we study how NR-decoding improves the decoding thresholds of LDPC codes. Both the experimental results and the theoretical analysis show the ability of NR decoding to enhance the reliability of storage systems. ### _NR-decoder for Languages_ Consider English texts compressed by an LZW (Lempel-Ziv-Welch) algorithm that uses a fixed dictionary of size $2^{\ell}$. In our experiments, we use $\ell=20$, which gives a dictionary of $2^{20}$ patterns (larger than many practical LZW codes). The dictionary has $2^{\ell}$ text strings (called patterns) of variable lengths, where every pattern is encoded as an $\ell$-bit codeword. Given a text to compress, the LZW algorithm scans it and partitions it into patterns, and maps them to codewords. For example, if we compress _“Flash memory is an $\cdots$”_, _“Flash m”_ gets mapped to a 20-bit codeword, _“emory i”_ gets mapped to another codeword and so on. The LZW code has been constructed using the Wikipedia corpus. It can compress English texts to 2.94 bits/character, which is substantially higher than the rate of 4.59 bits/character achieved by the commonly used character-level Huffman codes. The fixed dictionary of the LZW code also makes it easy to use in practice. In this section, we focus on bit-erasure channels. For long LZW-compressed texts with erasures, to make the NR-decoding efficient, we present a decoding algorithm based on sliding-windows of variable lengths as follows. #### Iv-A1 Baseline Algorithm Let $n_{min}$ and $n_{max}$ be two integers, where $n_{min}<n_{max}$ and let $\ell$ be the length of LZW codewords. We first use a sliding-window of $n_{min}\ell$ bits to scan the compressed text (where every such window contains exactly $n_{min}$ LZW codewords of size $\ell$), and obtain candidate solutions for each window based on the validity of words. (Specifically, if the bits in the window contain $t$ erasures, there are $2^{t}$ possible solutions, each of which can be mapped back to a text string. If all the whole words in the text string are valid words, the solution is considered a candidate solution.) We then increase the size of the window to $(n_{min}+1)\ell$, $(n_{min}+2)\ell$, $\cdots$, $n_{max}\ell$, and do decoding for each size in the following dynamic programming approach. Consider a window of $k\ell$ bits that contains $k$ LZW-codewords $C_{1}$, $C_{2}$, $\cdots$, $C_{k}$. Let $S_{1}\subseteq\{0,1\}^{(k-1)\ell}$ be the set of candidate solutions for the sub-window that contains the LZW-codewords $C_{1}$, $C_{2}$, $\cdots$, $C_{k-1}$; and let $S_{2}\subseteq\{0,1\}^{(k-1)\ell}$ be the set of candidate solutions for the sub-window that contains the LZW-codewords $C_{2}$, $C_{3}$, $\cdots$, $C_{k}$. (Both $S_{1}$ and $S_{2}$ have been obtained in the previous round of decoding.) We now obtain the set of candidate solutions for the current window, which contains $C_{1}$, $C_{2}$, $\cdots$, $C_{k}$, this way. A bit sequence $(b_{1},b_{2},\cdots,b_{k\ell})$ is in $S$ only if it satisfies two conditions: (1) its first $(k-1)\ell$ bits are a solution in $S_{1}$, and its last $(k-1)\ell$ bits are a solution in $S_{2}$; (2) the decompressed text corresponding to it contains no invalid words (except on the boundaries). This way, potential solutions filtered by smaller windows will not enter solutions for larger windows, making decoding more efficient. As a final step, an erased bit is decoded this way: if _any_ of the windows of size $n_{max}\ell$ containing it (note that there are up to $2n_{max}-1$ such windows) can recover its value, decode it to that value; otherwise it remains as an erasure. #### Iv-A2 Phrase and Word Length Filter To make the above decoding algorithm more efficient, we also use phrases (such as “information theory”, “flash memory”) and features such as word/phrase lengths. If a solution for a window contains a valid word or phrase that is particularly long, we may remove other candidate solutions that contain only short words. That is because long words and phrases are very rare: their density among bit sequences of the same length decreases exponentially fast as the length increases [21]. So if they appear, the chance that they are the correct solution is high based on Bayes’ rule. The thresholds for such word/phrase lengths can be set sufficiently high such that the probability of making a decoding error is sufficiently small. #### Iv-A3 Co-location Filter We also enhance the decoding performance by using the _co-location_ relationship. Co-location means that certain pairs of words/phrases appear unusually frequently in the same context (because they are closely associated), such as “dog” and “bark”, or “information theory” and “channel capacity”. If two words/phrases with the co-location relationship are detected among candidate solutions for two windows close to each other, we may keep them as candidate solutions and remove other less likely solutions. The reason for this approach is similar to that for long words/phrases. The co-location relationship can appear in multiple places in a text, and therefore help decoding in non-trivial ways. For example, for the text in Fig. 6 (a), the words/phrases that have the co-location relationship with the phrase “flash memory” are shown in Fig. 6 (b). How to find words/phrases with the co-location relationship from a corpus of training texts is a well-known technique in Natural Language Processing (NLP) [36]. So we skip its details here. <figure><img src="content_image/1910.07420/fig2.jpg"><figcaption>Fig. 6: Co-location relationship between words and phrases. (a) A sampleparagraph from Wikipedia (part of which was omitted to save space). (b)Phrases in it that have the co-location relationship with “flash memory”.</figcaption></figure> We present the above decoding algorithm’s performance for the binary erasure channel (BEC). The output of the NR-decoder has both erasures and errors (which will be further decoded by ECC later on). Let $\epsilon\in[0,1]$ be the raw bit-erasure rate (RBER) of BEC. After NR-decoding, for an originally erased bit, let $\delta\in[0,1]$ denote the probability that it remains as an erasure, and let $\rho\in[0,1-\delta]$ denote the probability that it is decoded to 0 or 1 incorrectly. Then the amount of noise after NR-decoding can be measured by the entropy of the noise (erasures and errors) per bit: \[E_{NR}(\epsilon)\triangleq\epsilon(\delta+(1-\delta)H(\frac{\rho}{1-\delta})),\] where $H(p)=-p\log p-(1-p)\log(1-p)$ is the entropy function. Some typical values of $E_{NR}(\epsilon)$ are shown in Table II. The reduction in noise by NR-decoding is $\frac{\epsilon-E_{NR}(\epsilon)}{\epsilon}$. The table shows that noise is reduced very effectively (from 88.0% to 91.6%) for the LZW compressed data (without any help from ECC), for RBER from $5\%$ to $30\%$, which is a wide range for storage systems. RBER ϵ \| 0.05 \| 0.10 \| 0.15 ---\|---\|---\|--- δ \| 8.22×10−2 \| 8.67×10−2 \| 9.19×10−2 ρ \| 9.18×10−5 \| 1.83×10−4 \| 1.82×10−4 ENR(ϵ) \| 4.18×10−3 \| 8.92×10−3 \| 1.42×10−2 Noise \| 91.6% \| 91.1% \| 90.6% reduction \| \| \| RBER ϵ \| 0.20 \| 0.25 \| 0.30 ---\|---\|---\|--- δ \| 9.76×10−2 \| 1.05×10−1 \| 1.12×10−1 ρ \| 3.61×10−4 \| 4.48×10−4 \| 7.11×10−4 ENR(ϵ) \| 2.04×10−2 \| 2.76×10−2 \| 3.60×10−2 Noise \| 89.8% \| 89.0% \| 88.0% reduction \| \| \| TABLE II: Noise reduction by NR-based language decoder for different erasure rates ϵ. RBER ϵ \| 0.20 \| 0.25 \| 0.30 ---\|---\|---\|--- δ \| 9.76×10−2 \| 1.05×10−1 \| 1.12×10−1 ρ \| 3.61×10−4 \| 4.48×10−4 \| 7.11×10−4 ENR(ϵ) \| 2.04×10−2 \| 2.76×10−2 \| 3.60×10−2 Noise \| 89.8% \| 89.0% \| 88.0% reduction \| \| \| TABLE II: Noise reduction by NR-based language decoder for different erasure rates ϵ. Suppose that the LZW-codewords, seen as information bits, are protected by a systematic ECC. The NR-decoder can work collaboratively with the ECC decoder to maximize the number of correctable erasures. We now study two important cases for combining NR decoding with ECC decoding: the _sequential_ decoding scheme, and the _iterative_ decoding scheme. ### _Sequential Decoding by NR and LDPC code_ <figure><img src="content_image/1910.07420/schemes.png"><figcaption>Fig. 7: Two schemes for combining NR-decoding with LDPC-decoding. (a) Asequential decoding scheme by NR and LDPC code. (b) An iterative decodingscheme by NR and LDPC code.</figcaption></figure> This subsection discusses the combination of NR-decoder with LDPC codes. We protect compressed text as information bits by a systematic LDPC code of rate $R$. The NR-decoder studied here generalizes the one presented in the previous subsection: it decodes the information bits by NR, and possibly the parity check bits as well using their relations with the information bits. The decoding process is a concatenation of two decoders: (1) first, the NR-decoder corrects erasures and outputs a partially corrected codeword; (2) then, the LDPC decoder takes that codeword as input (where the erasure and error probabilities result from the NR-decoding), and uses belief propagation (BP) for decoding. (See Fig. 7 (a) for an illustration.) We present a theoretical analysis for the decoding performance, and show that the NR-decoder can substantially improve the performance of LDPC codes. Consider a binary-erasure channel (BEC) with erasure probability $\epsilon_{0}$. Let us call the non-erased bits _fixed bits_. Assume that after NR-decoding, a non-fixed bit (i.e., erasure) remains as an erasure with probability $p_{0}(\epsilon_{0})\in[0,1]$, becomes an error (0 or 1) with probability $(1-p_{0}(\epsilon_{0}))\gamma_{0}(\epsilon_{0})\in[0,1-p_{0}(\epsilon_{0})]$, and is decoded correctly (as 0 or 1) with probability $(1-p_{0}(\epsilon_{0}))(1-\gamma_{0}(\epsilon_{0}))$. (In general, $p_{0}(\epsilon_{0})$ and $\gamma_{0}(\epsilon_{0})$ can be functions of $\epsilon_{0}$. Note that if the NR-decoder decodes only information bits, and an erasure in the information bits remains as an erasure with probability $p_{0}(\epsilon_{0})^{\prime}$, then $p_{0}(\epsilon_{0})=Rp_{0}(\epsilon_{0})^{\prime}+(1-R)$. Also note that the LDPC decoder needs to decode all bits with both errors and erasures.) #### Iv-B1 Decoding Algorithm We design the following iterative LDPC decoding algorithm, which generalizes both the peeling decoder for BEC and the Gallager B decoder for BSC [46]: Algorithm$\!$ 1.: _Generalized LDPC decoding algorithm._ _1) Let $\pi\in[1,d_{v}-1]$ and $\tau\in[1,d_{v}-1]$ be two integer parameters;_ _2) In each iteration, for a variable node $v$ that is an erasure, if $\pi$ or more non-erased message bits come from $d_{v}-1$ check nodes and they all have the same value, set $v$ to that bit value;_ _3) If $v$ is not a fixed bit and not an erasure (but possibly an error) in this iteration, change $v$ to the opposite bit value if $\tau$ or more non-erased message bits come from $d_{v}-1$ check nodes and they all have that opposite value. (The updated value of $v$ will be sent to the remaining check node in the next iteration.)_ #### Iv-B2 Density Evolution Analysis We now analyze the density evolution for the decoding algorithm, for an infinitely long and randomly constructed LDPC code of regular degrees. For $t=0,1,2\cdots$, let $\alpha_{t}$ and $\beta_{t}$ be the fraction of codeword bits that are errors or erasures, respectively, after $t$ iterations of LDPC decoding. We have $\alpha_{0}=\epsilon_{0}(1-p_{0}(\epsilon_{0}))\gamma_{0}(\epsilon_{0})$ and $\beta_{0}=\epsilon_{0}p_{0}(\epsilon_{0})$. Let $\kappa_{0}=\epsilon_{0}(1-p_{0}(\epsilon_{0}))(1-\gamma_{0}(\epsilon_{0}))$. Theorem$\!$ 2.: _For a regular $(d_{v},d_{c})$ LDPC code with variable-node degree $d_{v}$ and check-node degree $d_{c}$, we have_ \[\alpha_{t+1}=\alpha_{0}C_{t}+\kappa_{0}D_{t}+\beta_{0}\mu_{t},\] _where_ \[C_{t}=1-(1-A_{t})^{d_{v}-1}+\sum_{i=0}^{\tau-1}{d_{v}-1\choose i}B_{t}^{i}(1-A _{t}-B_{t})^{d_{v}-i-1},\] \[D_{t}=\sum_{j=\tau}^{d_{v}-1}{d_{v}-1\choose j}A_{t}^{j}(1-A_{t}-B_{t})^{d_{v} -1-j},\] \[\mu_{t}=\sum_{m=\pi}^{d_{v}-1}{d_{v}-1\choose m}A_{t}^{m}(1-A_{t}-B_{t})^{d_{v }-1-m},\] _whose component variables are computed iteratively as_ \[A_{t}=\frac{(1-\beta_{t})^{d_{c}-1}-(1-\beta_{t}-2\alpha_{t})^{d_{c}-1}}{2},\] \[B_{t}=\frac{(1-\beta_{t})^{d_{c}-1}+(1-\beta_{t}-2\alpha_{t})^{d_{c}-1}}{2}.\] _For the LDPC code, we also have_ \[\beta_{t+1}=\beta_{0}(1-\mu_{t}-\nu_{t}),\] _where_ \[\nu_{t}=\sum_{m=\pi}^{d_{v}-1}{d_{v}-1\choose m}B_{t}^{m}(1-A_{t}-B_{t})^{d_{v }-1-m}.\] Proof:: Consider the root variable node of a computation tree. After $t$ iterations, let $A_{t}$ denote the probability that an incoming message to the root node from a neighboring check node is an error, and let $B_{t}$ denote the probability that the message is correct. Then $1-A_{t}-B_{t}$ is the probability that the message is an erasure. Let $\mu_{t}$ (respectively, $\nu_{t}$) be the probability that among the $d_{v}-1$ incoming messages from neighboring check nodes to the root node, $\pi$ or more messages are errors (respectively, correct) and the remaining messages are all erasures. In the $(t+1)$-th iteration, we can have an error in the root node in one of the following cases: 1. The root node was initially (namely, before decoding begins) an error (which has probability $\alpha_{0}$), and either of the two disjoint events happens: 1) fewer than $\tau$ check-node messages are correct and the remaining messages are all erasures, which happens with probability $\sum\limits_{i=0}^{\tau-1}{d_{v}-1\choose i}B_{t}^{i}(1-A_{t}-B_{t})^{d_{v}-i-1}$; 2) at least one check-node message is an error, which happens with probability $1-(1-A_{t})^{d_{v}-1}$. The probability that either of the two events occurs is $C_{t}=1-(1-A_{t})^{d_{v}-1}+\sum\limits_{i=0}^{\tau-1}{d_{v}-1\choose i}B_{t}^ {i}(1-A_{t}-B_{t})^{d_{v}-i-1}$. 2. The root node was initially correct (which has probability $\kappa_{0}$), but $\tau$ or more check-node messages are errors and the rest are all erasures (which happens with probability $D_{t}=\sum\limits_{j=\tau}^{d_{v}-1}{d_{v}-1\choose j}A_{t}^{j}(1-A_{t}-B_{t}) ^{d_{v}-1-j}$). 3. The root node was initially an erasure (which has probability $\beta_{0}$), and $\pi$ or more check-node messages are errors and the rest are all erasures (which happens with probability $\mu_{t}$). Therefore the error rate after $t+1$ iterations will be $\alpha_{t+1}=\alpha_{0}C_{t}+\kappa_{0}D_{t}+\beta_{0}\mu_{t}$. In the $(t+1)$-th iteration, we can correct an erasure at a root node correctly if the root node was initially an erasure, and $\pi$ or more check-node messages are correct and the rest are all erasures. This happens with probability $\beta_{0}\nu_{t}$. The root node will remain as an erasure if it is neither corrected mistakenly nor corrected correctly. So the erasure rate after $t+1$ iterations will be $\beta_{t+1}=\beta_{0}(1-\mu_{t}-\nu_{t})$. Now we need to find the values of $A_{t}$, $B_{t}$, $\mu_{t}$ and $\nu_{t}$. The incoming message from a check node to the root node is correct if out of the $d_{c}-1$ non-root variable nodes connected to the check node, an even number of nodes are errors and the rest are all correct (i.e., neither errors nor erasures). That probability is $B_{t}=\sum\limits_{k=0}^{\lfloor\frac{d_{c}-1}{2}\rfloor}{d_{c}-1\choose 2k} \alpha_{t}^{2k}(1-\alpha_{t}-\beta_{t})^{d_{c}-1-2k}=\frac{(1-\beta_{t})^{d_{c }-1}+(1-\beta_{t}-2\alpha_{t})^{d_{c}-1}}{2}$. The incoming message from a check node to the root node is an error if out of the $d_{c}-1$ non-root variable nodes connected to the check node, an odd number of nodes are errors and the rest are all correct. That probability is $A_{t}=\sum\limits_{k=1}^{\lfloor\frac{d_{c}}{2}\rfloor}{d_{c}-1\choose 2k-1} \alpha_{t}^{2k-1}(1-\alpha_{t}-\beta_{t})^{d_{c}-2k}=\frac{(1-\beta_{t})^{d_{c }-1}-(1-\beta_{t}-2\alpha_{t})^{d_{c}-1}}{2}$. The probability that $\pi$ or more neighboring check-node messages are errors and the rest are all erasures can be simplified as $\mu_{t}=\sum_{m=\pi}^{d_{v}-1}{d_{v}-1\choose m}A_{t}^{m}(1-A_{t}-B_{t})^{d_{v }-1-m}$. The probability that $\pi$ or more neighboring check-node messages are correct and the rest are all erasures can be simplified as $\nu_{t}=\sum_{m=\pi}^{d_{v}-1}{d_{v}-1\choose m}B_{t}^{m}(1-A_{t}-B_{t})^{d_{v }-1-m}$. This completes the proof. ∎ #### Iv-B3 Erasure Threshold Define _erasure threshold_$\epsilon^{}$ as the maximum erasure probability (for $\epsilon_{0}$) for which the LDPC code can decode successfully (which means the error/erasure probabilities $\alpha_{t}$ and $\beta_{t}$ both approach 0 as $t\to\infty$). Let us show how the NR decoder can substantially improve $\epsilon^{}$. Consider a regular LDPC code with $d_{v}=5$ and $d_{c}=100$, which has rate 0.95 (a typical code rate for storage systems). Without NR-decoding, the erasure threshold is $\tilde{\epsilon}^{}=0.036$. Now let $\pi=1$ and $\tau=4$. For LZW-compressed texts, when $\epsilon_{0}=0.2$, the NR-decoder in the previous subsection gives $p_{0}=0.143$ and $\gamma_{0}=0.0003$, for which the LDPC decoder has $\lim_{t\to\infty}\alpha_{t}=0$ and $\lim_{t\to\infty}\beta_{t}=0$. (The same happens for $\epsilon_{0}<0.2$.) So with NR-decoding, $\epsilon^{}\geq 0.2$, which means the improvement in erasure threshold is more than $455.6\%$. ### _Iterative Decoding by NR and LDPC code_ In this subsection, we study the decoding performance when we use _iterative decoding_ between the LDPC decoder and NR-decoder, as shown in Fig. 7 (b). (In last subsection’s study, the NR-decoder is followed by the LDPC decoder, without iterations between them.) As before, we focus on languages and systematic LDPC codes, and present a theoretical model for compressed languages as follows. Let $\mathbb{T}=(b_{0},b_{1},b_{2},\cdots)$ be a compressed text. Partition $\mathbb{T}$ into segments $S_{0},S_{1},S_{2}\cdots$, where each segment $S_{i}=(b_{il},b_{il+1},\cdots,b_{il+l-1})$ has $l$ bits. Consider erasures in the compressed text. Let $\theta\in[0,1]$, $l_{\theta}\triangleq\lfloor l\theta\rfloor$ and $p\in[0,1]$ be parameters. We assume that when a segment $S_{i}$ has at most $l_{\theta}$ erasures, the NR-decoder can decode it by checking the validity of up to $2^{l_{\theta}}$ candidate solutions (based on the validity of their corresponding words/phrases, grammar, etc.), and either determines (independently) the correct solution with probability $p$ or makes no decision with probability $1-p$. (Note that an NR-decoder does not have to check the $2^{l_{\theta}}$ candidate solutions one by one. For example, the NR-decoder introduced earlier can remove many invalid solutions early on without exhaustive search.) And this NR-decoding operation can be performed _only once_ for each segment (because if the correct solution cannot be determined by such an NR-based operation the first time, there is no guarantee that such operations in the future will find the correct solution). The parameter $l_{\theta}$ here is used to bound the computational complexity and erasure-correction capability of the NR-decoder in the worst case, and $p$ models the probability of making an error-free decision. This is a simplification of the practical NR-decoder shown in the previous subsection that makes very high-confidence – although not totally error-free – decisions. The model is suitable for compression algorithms such as LZW coding with a fixed dictionary, Huffman coding, etc., where each segment can be decompressed to a piece of text. The greater $l$ is, the better the model is. #### Iv-C1 Iteration with LDPC Decoder The compressed text $\mathbb{T}$ is protected as information bits by a systematic LDPC code. The LDPC code uses the peeling decoder for BEC (where $d_{c}-1$ incoming messages of known values at a check node determine the value of the outgoing message on the remaining edge) to correct erasures. See the decoding model in Fig. 7 (b). In each iteration, the LDPC decoder runs _one iteration_ of BP decoding, then the NR-decoder tries to correct those $l$-information-bit segments that contain at most $l_{\theta}$ erasures (if those segments were never decoded by the NR-decoder in any of the previous iterations). Let $\epsilon_{0}<1$ be the BEC’s erasure rate. Let $\epsilon_{t}^{\prime}$ and $\epsilon_{t}$ be the LDPC codeword’s erasure rate after the $t$-th iteration of the LDPC decoder and the NR-decoder, respectively. Next, we analyze the density evolution for regular $(d_{v},d_{c})$ LDPC codes of rate $R$ = $1-\frac{d_{v}}{d_{c}}$. Note that since the NR-decoder decodes only information bits, for the LDPC decoder, the information bits and parity-check bits will have different erasure rates during decoding. Furthermore, information bits consist of $l$-bit segments, while parity-check bits do not. For such an $l$-bit segment, if the NR-decoder can decode it successfully when it has no more than $l_{\theta}$ erasures, let us call the segment _lucky_; otherwise, call it _unlucky_. Lucky and unlucky segments will have different erasure rates during decoding, too. Every $l$-information-bit segment is _lucky_ with probability $p$, and _unlucky_ with probability $1-p$. A lucky segment is guaranteed to be decoded successfully by the NR-decoder once the number of erasures in it becomes less than or equal to $l_{\theta}$; and an unlucky segment can be considered as _never_ to be decoded by the NR-decoder (because such decoding will not succeed). Since whether a segment is lucky or not is independent of the party-check constraints and the LDPC-decoder, for analysis we can consider it as an inherent property of the segment (which exists even before the decoding begins). #### Iv-C2 Density Evolution Analysis Define $q_{0}=1$, $q_{t}\triangleq\frac{\epsilon_{t}}{\epsilon_{t}^{\prime}}$ and $d_{t}\triangleq\frac{\epsilon_{t}^{\prime}}{\epsilon_{t-1}}$ for $t\geq 1$. Note that decoding will end after $t$ iterations if one of these conditions occurs: (1) $\epsilon_{t}^{\prime}=0$, because all erasures are corrected by the $t$-th iteration; (2) $d_{t}=1$, because the LDPC decoder corrects no erasure in the $t$-th iteration, and nor will the NR-decoder since the input codeword is identical to its previous output. We now study density evolution before those boundary cases occur. For $t=1,2,3\cdots$ and $k=0,1,\cdots,l$, let $f_{k}(t)$ denote the probability that a lucky segment contains $k$ erasures after $t$ iterations of decoding by the NR-decoder. Lemma$\!$ 3.: \[f_{k}(1)=\begin{cases}\sum\limits_{i=0}^{l_{\theta}}{l\choose i}(\epsilon_{1}^ {\prime})^{i}(1-\epsilon_{1}^{\prime})^{l-i}\mbox{~{}~{}~{}~{}~{}~{}if~{}}k=0 \\ 0\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}~{}~{}~{}~{}if~{}}1\leq k\leq l_{\theta}\\ {l\choose k}(\epsilon_{1}^{\prime})^{k}(1-\epsilon_{1}^{\prime})^{l-k}\mbox{~{ }~{}~{}~{}~{}~{}~{}~{}~{}if~{}}l_{\theta}+1\leq k\leq l\end{cases}\] Proof:: Consider the LDPC-decoding and the NR-decoding in the first iteration. Since the initial erasure rate is $\epsilon_{0}$, the erasure rate after LDPC decoding will now be $\epsilon_{1}^{\prime}=q_{0}\epsilon_{0}(1-(1-\epsilon_{0})^{d_{c}-1})^{d_{v}-1}$ where $q_{0}=1$ by definition. The probability that an $l$-information-bit segment contains exactly $i$ erasures is given by ${l}\choose{i}$$(\epsilon_{1}^{\prime})^{i}(1-\epsilon_{1}^{\prime})^{l-i}$, which is independent of whether the segment is lucky or unlucky. Thus the probability that a _lucky_ segment contains up to $l_{\theta}$ erasures is given by $\sum_{i=0}^{l_{\theta}}$${l}\choose{i}$$(\epsilon_{1}^{\prime})^{i}(1-\epsilon_{1}^{\prime})^{l-i}$. All such segments are decoded by the NR-decoder successfully, while the remaining segments are not. That leads to the conclusion. ∎ Lemma$\!$ 4.: _The erasure rate after the first iteration of NR-decoding is_ \[\epsilon_{1}=\epsilon_{0}d_{1}((1-R)+R(1-p))+(\sum_{k=l_{\theta}+1}^{l}\frac{k }{l}f_{k}(1))Rp\] Proof:: After NR-decoding, the erasure rate of a lucky segment with $k$ erasures is $\frac{k}{l}$, and the erasure rate for unlucky segments and parity-check bits is still $\epsilon_{1}^{\prime}$. We have $d_{1}=\epsilon_{1}^{\prime}/\epsilon_{0}$. Hence the overall erasure rate after the 1st iteration of NR-decoding is $\epsilon_{1}=\epsilon_{0}d_{1}((1-R)+R(1-p))+(\sum_{k=l_{\theta}+1}^{l}\frac{k }{l}f_{k}(1))Rp$. (See Fig. 8 (b) for an illustration of the computation tree for density evolution. For comparison, we show the tree for classic BP decoding for BEC in Fig. 8 (a).) ∎ Lemma$\!$ 5.: _The erasure rate after the second iteration of LDPC-decoding is_ \[\epsilon_{2}^{\prime}=q_{0}q_{1}\epsilon_{0}(1-(1-\epsilon_{1})^{d_{c}-1})^{d_ {v}-1}.\] Proof:: We have $q_{1}=\frac{\epsilon_{1}}{\epsilon^{\prime}_{1}}$. Since the NR-decoding of the 1st iteration reduces the _overall_ erasure probability by a factor of $q_{1}$ (from $\epsilon_{1}^{\prime}$ to $\epsilon_{1}$), and the root variable node of a computation tree is chosen uniformly at random from the infinitely long and randomly constructed LDPC code, the root node in the tree for the 2nd iteration of LDPC decoding now has the erasure probability $q_{1}\epsilon_{0}$. (See Fig. 8 (b).) Hence the equation for the LDPC-decoder for the 2nd iteration will be given by $\epsilon_{2}^{\prime}=q_{0}q_{1}\epsilon_{0}(1-(1-\epsilon_{1})^{d_{c}-1})^{d_ {v}-1}$. Note that LDPC decoding is independent of NR-decoding because the parity-check constraints are independent of the bits being lucky-segment bits, unlucky-segment bits or parity-check bits. And note that $d_{2}=\frac{\epsilon_{2}^{\prime}}{\epsilon_{1}}$ is the probability that an erasure _remains as an erasure_ after the LDPC decoding. If $d_{2}=1$, no change was made by the LDPC-decoder; if $d_{2}=0$, all erasures have been corrected. In both cases, the decoding will end. ∎ Lemma$\!$ 6.: _For $t\geq 2$,_ \[f_{k}(t)=\begin{cases}f_{k}(t-1)+\sum\limits_{i=l_{\theta}+1}^{l}\sum\limits_{ j=0}^{l_{\theta}}f_{i}(t-1){i\choose j}(d_{t})^{j}(1-d_{t})^{i-j},\mbox{~{}~{} ~{}~{}~{}~{}~{}~{}~{}if~{}}k=0\\ \\ 0,\mbox{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~ {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~ {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}if~{}}1\leq k\leq l_{\theta}\\ \sum\limits_{i=k}^{l}f_{i}(t-1){i\choose k}(d_{t})^{k}(1-d_{t})^{i-k},\mbox{~{ }~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{ }~{}~{}~{}if~{}}l_{\theta}+1\leq k\leq l\end{cases}\] Proof:: Now consider the second iteration of NR-decoding. We only consider the case when $0<d_{2}<1$. A lucky segment has zero errors after the second iteration if an only if either one of the two cases happen : 1) the segment already has zero errors after the first iteration, or 2) the segment had $l_{\theta}+1$ or more errors after the first iteration and it has at most $l_{\theta}$ erasures after second iteration of the LDPC-decoding. Thus if $k=0$, \[f_{k}(2)=f_{k}(1)+\sum\limits_{i=l_{\theta}+1}^{l}\sum\limits_{j=0}^{l_{\theta }}f_{i}(1){i\choose j}(d_{2})^{j}(1-d_{2})^{i-j}\] A lucky segment cannot have $k\leq l_{\theta}$ erasures (with $k\geq 1$) after the second iteration of NR-decoding (because if so, it would have corrected those erasures). So we have $f_{k}(2)=0$ for that case. Finally, a lucky segment has $l_{\theta}+1\leq k\leq l$ erasures if and only if it had $k$ or more erasures after the first iteration of NR-decoding and it has $k$ erasures after the second iteration of LDPC-decoding. Thus \[f_{k}(2)=\sum\limits_{i=k}^{l}f_{i}(1){i\choose k}(d_{2})^{k}(1-d_{2})^{i-k} \textrm{ if $l_{\theta}+1\leq k\leq l$}\] The remaining cases can be analyzed similarly. That leads to the conclusion. ∎ We now present the analytical formulas for the density evolution of the iterative LDPC-NR decoding scheme. Its proof follows the previous lemmas. Theorem$\!$ 7.: _For $t\geq 1$,_ \[\epsilon_{t}=((1-R)+R(1-p))\epsilon_{0}(\prod\limits_{i=1}^{t}d_{t})+Rp\sum \limits_{k=l_{\theta}+1}^{l}\frac{k}{l}f_{k}(t),\] \[\epsilon^{\prime}_{t}=(\prod_{m=0}^{t-1}q_{m})\epsilon_{0}(1-(1-\epsilon_{t-1} )^{d_{c}-1})^{d_{v}-1}.\] _Proof:_: The decoding performance for the 2nd iteration of the LDPC-decoding has been analyzed in Lemma 5. The erasure rate in unlucky-segment bits and parity-check bits was decreased from $\epsilon_{1}^{\prime}$ to $\epsilon_{1}^{\prime}d_{2}$ = $\epsilon_{0}d_{1}d_{2}$ by the LDPC-decoding. Now the NR-decoder corrects those lucky segments that had more than $l_{\theta}$ erasures before the LDPC-decoding but now has at most $l_{\theta}$ erasures after the LDPC-decoding. So $\epsilon_{2}=\epsilon_{0}d_{1}d_{2}((1-R)+R(1-p))+(\sum\limits_{k=l_{\theta}+1 }^{l}\frac{k}{l}f_{k}(2))Rp$. The analysis for the following iterations is similar to the 2nd iteration. In general, since in the $i$-th iteration the NR-decoder reduces the overall erasure rate by a factor of $q_{i}$, the root variable node in the computation tree for the $t$-th iteration of LDPC decoding has the erasure probability $(\prod_{i=0}^{t-1}q_{i})\epsilon_{0}$. That leads to the conclusion. ∎ <figure><img src="content_image/1910.07420/fig4.jpg"><figcaption>Fig. 8: Comparison of the computation tree for density evolution analysis. (a)First three iterations of classic BP decoding (alone) for BEC. (b) First threeiterations of BP-decoding and NR decoding.</figcaption></figure> #### Iv-C3 Performance We now numerically show that the iterative NR-LDPC decoder can improve the decoding threshold for erasures significantly. Note that the analysis in this subsection is based on the assumption that the NR-decoder corrects erasures but does not create errors. However, all our existing NR-decoders still create errors with small probabilities (such as $10^{-4}$ in Table II) which, although small, are still non-zero due to the complexity of languages. Extending the NR-decoders here to correct both erasures and errors is beyond the scope of this paper. Therefore, the following analysis is based on the same assumption as above, and the parameters of the NR-decoder are chosen reasonably based on existing experimental evidence: let each segment have $l=120$ bits (which corresponds to 6 LZW codewords of 20 bits each); and let $l_{\theta}=30$. (Note that in the experiments of the previous two subsections, sliding windows of the same size and more erasures have been considered.) Let the LDPC code be a regular code with $d_{v}=5$ and $d_{c}=100$. Recall that $p$ is the probability that an NR-decoder can correct the erasures in a segment successfully when the segment has at most $l_{\theta}$ erasures. Based on the previous analysis, given the value of $p$, we can obtain the corresponding decoding threshold for erasures for the iterative NR-LDPC decoder. The results are shown in Fig. 9 (a). It can be seen that as $p$ increases, the decoding threshold $\epsilon^{}$ increases quickly. Note that without the NR-decoder, the decoding threshold of the LDPC code alone for erasures is $\tilde{\epsilon}^{}=0.036$. In Fig. 9 (a), the decoding threshold increases from 0.039 to 0.224, all of which are higher than $\tilde{\epsilon}^{}$. Based on Table II, it is reasonable to consider $p=0.9$. In this case, the decoding threshold is $\epsilon^{}=0.224$, which represents a $522.22\%$ increase from $\tilde{\epsilon}^{}$. We also study how quickly decoding converges in the iterative decoding scheme. The results are shown in Fig. 9 (b). Here, the BER of the BEC channel is $\epsilon_{0}=0.2$ (which is above the decoding threshold of the LDPC code alone). It can be seen that decoding converges faster as $p$ increases. In particular, when $p=0.9$, it takes only about 7 iterations for decoding to converge. <figure><img src="content_image/1910.07420/thresholds2.png"><figcaption>Fig. 9: Performance of the iterative NR-LDPC decoding. (a) Here parameter p isthe probability that the NR-decoder corrects erasures in a segment when it hasat most lθ erasures, and parameter ϵ∗ is the decoding threshold for erasuresof the iterative NR-LDPC decoder. The figure shows that ϵ∗ increases rapidlyas p increases, and it can substantially outperform the decoding threshold ofthe LDPC code alone (which is 0.036). (b) Here t is the number of iterationsof the iterative NR-LDPC decoding process, and ϵt is the overall bit erasurerate of the LDPC codeword after the t-th iteration. The bit erasure rate ofthe BEC channel is set to be ϵ0=0.2. The figure shows that the higher p is,the more quickly decoding ends.</figcaption></figure> In summary, with knowledge on how data are represented by bits, effective NR-based decoding schemes can be designed. Both sequential and iterative schemes are presented for combining NR-decoders with LDPC codes, and their performance is rigorously analyzed. The results show that the inclusion of NR-decoding can improve LDPC decoding substantially, and iterative decoding between the two decoders can further improve performance effectively. ## V Computational-Complexity Tradeoff for NR-based Coding In the Introduction section, we have mentioned that the Natural Redundancy in data can be used for both compression and error correction. How to use it suitably depends on many factors, such as available coding techniques, hardware design, etc. In this chapter, we discuss one such tradeoff of _central importance_: the computational complexity of using NR for compression or error correction. Real NR is hard to model precisely, so we explore this topic from a theoretical point of view, and consider NR in general forms. We show that certain types of redundancy are computationally efficient for compression, while others are so for error correction. Note that there exist works on analyzing the hardness of certain types of source coding schemes [28, 29, 45] and channel coding schemes [5, 13, 14, 50, 51]. In contrast, here we focus on the tradeoff between the two, and the analysis is NR-oriented. Let $B=(b_{1},b_{2},\cdots,b_{n})\in\{0,1\}^{n}$ be an $n$-bit message with NR. Define $\mathcal{V}~{}:~{}\{0,1\}^{n}\to\{0,1\}$ as a _validity function_: $B$ is a valid message if and only if $\mathcal{V}(B)=1$. The set of all valid messages of $n$ bits is $\mathcal{M}\triangleq\{B\in\{0,1\}^{n}~{}\|~{}\mathcal{V}(B)=1\}$. For simplicity, for both source and channel coding, assume that the valid messages in $\mathcal{M}$ are equally likely. First, consider source coding. Let $k~{}=~{}\lceil\log_{2}\|\mathcal{M}\|\rceil$. Define an _optimal lossless compression scheme_ to be an injective function $C_{opt}~{}:~{}\mathcal{M}\to\{0,1\}^{k}$ that compresses any valid message $B\in\mathcal{M}$ to a distinct $k$-bit vector $C_{opt}(B)$. Define the _Data Compression Problem_ as follows: Given a validity function $\mathcal{V}$, find an injective function $C_{opt}~{}:~{}\mathcal{M}\to\{0,1\}^{k}$. Next, consider channel coding. Assume that a valid message $X=(x_{1},x_{2},\cdots,x_{n})\in\mathcal{M}$ is transmitted through a binary-symmetric channel (BSC), and is received as a noisy message $Y=(y_{1},y_{2},\cdots,y_{n})\in\{0,1\}^{n}$. Maximum likelihood (ML) decoding requires us to find a message $Z=(z_{1},z_{2},\cdots,z_{n})\in\mathcal{M}$ that minimizes the Hamming distance $d_{H}(Y,Z)$. Define the _Error Correction Problem_ as follows: Given a validity function $\mathcal{V}$ and a message $Y\in\{0,1\}^{n}$, find a valid message $Z\in\mathcal{M}$ that minimizes the Hamming distance $d_{H}(Y,Z)$. Let $\mathcal{F}$ be the set of all functions from the domain $\{0,1\}^{n}$ to the codomain $\{0,1\}$. (We have $\|\mathcal{F}\|=2^{2^{n}}$.) The function $\mathcal{V}$ represents NR in data. In practice, different types of data have different _types_ of NR. Let us define the latter concept formally. For any subset $\mathcal{T}\subseteq\mathcal{F}$, let $\mathcal{T}$ be called a _type_ of validity functions (which represents a type of NR). When $\mathcal{V}$ can only be a function in $\mathcal{T}$ (instead of $\mathcal{F}$), we denote the Data Compression Problem and the Error Correction Problem by $\mathcal{P}_{dc}^{\mathcal{T}}$ and $\mathcal{P}_{ec}^{\mathcal{T}}$, respectively. The hardness of the problems $\mathcal{P}_{dc}^{\mathcal{T}}$ and $\mathcal{P}_{ec}^{\mathcal{T}}$ depends on $\mathcal{T}$. Let $S_{dc=NP,ec=P}$ denote the set of types $\mathcal{T}$ (where each type is a subset of $\mathcal{F}$) for which the data compression problem $\mathcal{P}_{dc}^{\mathcal{T}}$ is NP-hard while the error correction problem $\mathcal{P}_{ec}^{\mathcal{T}}$ is polynomial-time solvable. Similarly, let $S_{dc=P,ec=NP}$ (or $S_{dc=P,ec=P}$, $S_{dc=NP,ec=NP}$, respectively) denote the set of types $\mathcal{T}$ for which $\mathcal{P}_{dc}^{\mathcal{T}}$ is polynomial-time solvable while $\mathcal{P}_{ec}^{\mathcal{T}}$ is NP-hard (or $\mathcal{P}_{dc}^{\mathcal{T}}$ and $\mathcal{P}_{ec}^{\mathcal{T}}$ are both polynomial-time solvable, or both NP-hard, respectively). The following theorem shows that there exist validity-function types for each of those four possible cases. Theorem$\!$ 8.: _The four sets $S_{dc=NP,ec=P}$, $S_{dc=P,ec=NP}$, $S_{dc=P,ec=P}$ and $S_{dc=NP,ec=NP}$ are all non-empty._ Proof:: We first prove that $S_{dc=NP,ec=P}\neq\emptyset$, namely, there exists a validity-function type $\mathcal{T}_{NP,P}\subseteq\mathcal{F}$ that makes the data compression problem $\mathcal{P}_{dc}^{\mathcal{T}_{NP,P}}$ be NP-hard while making the error correction problem $\mathcal{P}_{ec}^{\mathcal{T}_{NP,P}}$ be polynomial-time solvable. We define a validity function $\mathcal{V}_{NP,P}~{}:~{}\{0,1\}^{n}\to\{0,1\}$ as follows, which takes $n$ binary variables $b_{1},b_{2},\cdots,b_{n}$ as its input. Let $f_{3SAT}(b_{1},b_{2},\cdots,b_{n-1})$ be a 3-SAT Boolean formula, which is in the Conjunctive Normal Form (CNF) where each clause contains 3 variables (such as $(b_{1}\vee\bar{b}_{2}\vee\bar{b}_{3})\wedge(b_{2}\lor b_{4}\lor b_{6})\wedge( \bar{b}_{2}\lor b_{3}\lor b_{5})\wedge\cdots$, where $\vee$ is the OR operation, $\wedge$ is the AND operation, and $\bar{x}$ is the NOT of the Boolean variable $x$). Define a function $f_{even}(b_{1},b_{2},\cdots,b_{n})$ as follows: $f_{even}(b_{1},b_{2},\cdots,b_{n})$ equals 1 if $\sum_{i=1}^{n}b_{i}$ is even, and equals 0 otherwise. Similarly, define a function $f_{odd}(b_{1},b_{2},\cdots,b_{n})$ as follows: $f_{odd}(b_{1},b_{2},\cdots,b_{n})$ equals 1 if $\sum_{i=1}^{n}b_{i}$ is odd, and equals 0 otherwise. Finally, define the validity function $\mathcal{V}_{NP,P}(b_{1},b_{2},\cdots,b_{n})$ as $\mathcal{V}_{NP,P}(b_{1},b_{2},\cdots,b_{n})\triangleq(f_{3SAT}(b_{1},b_{2}, \cdots,b_{n-1})\wedge f_{even}(b_{1},b_{2},\cdots,b_{n}))\lor f_{odd}(b_{1},b_ {2},\cdots,b_{n})$. (The validity-function type $\mathcal{T}_{NP,P}$ is the set of all specific forms for the function $\mathcal{V}_{NP,P}$. Note that the same holds for the types $\mathcal{T}_{P,NP}$, $\mathcal{T}_{P,P}$ and $\mathcal{T}_{NP,NP}$ to be discussed later.) Given the validity function $\mathcal{V}_{NP,P}$, we can see that the set of valid messages $\mathcal{M}$ has cardinality $\|\mathcal{M}\|=\|\{B\in\{0,1\}^{n}~{}\|~{}\mathcal{V}_{NP,P}(B)=1\}\|\geq\|\{B\in\{ 0,1\}^{n}~{}\|~{}f_{odd}(B)=1\}\|=2^{n-1}$, because all the messages whose bits have odd parity must be valid. So whether $\|\mathcal{M}\|>2^{n-1}$ or not (which means whether $k~{}=~{}\lceil\log_{2}\|\mathcal{M}\|\rceil>n-1$ or not) depends on whether the 3-SAT formula $f_{3SAT}(b_{1},b_{2},\cdots,b_{n-1})$ has a satisfying solution: if there is a satisfying solution to $b_{1},b_{2},\cdots,b_{n-1}$ that makes $f_{3SAT}(b_{1},b_{2},\cdots,b_{n-1})$ be 1, then we can let $b_{n}=\oplus_{i=1}^{n-1}b_{i}\in\{0,1\}$ (where $\oplus$ is the exclusive-OR operation), which gives us an $n$-bit message of even parity that is valid (because here $f_{3SAT}(b_{1},b_{2},\cdots,b_{n-1})\wedge f_{even}(b_{1},b_{2},\cdots,b_{n})= 1\wedge 1=1=\mathcal{V}_{NP,P}(b_{1},b_{2},\cdots,b_{n})$), so $k>n-1$ (which means $k=n$); otherwise, there is no valid message of even parity, so $k=n-1$. So determining whether $k=n$ or $n-1$ is equivalent to solving the 3-SAT Problem $f_{3SAT}(b_{1},b_{2},\cdots,b_{n-1})$, which is a known NP-complete problem. To solve the data compression problem $\mathcal{P}_{dc}^{\mathcal{T}_{NP,P}}$, it is necessary to know the value of $k$. So the data compression problem $\mathcal{P}_{dc}^{\mathcal{T}_{NP,P}}$ is NP-hard. Consider the error-correction problem $\mathcal{P}_{ec}^{\mathcal{T}_{NP,P}}$ given the same validity function $\mathcal{V}_{NP,P}$. Given an input noisy message $Y=\{0,1\}^{n}$, we compute $\mathcal{V}_{NP,P}(Y)$. If $\mathcal{V}_{NP,P}(Y)=1$, then $Y$ is valid and we let $Z=Y$ be the decoded message; otherwise, since all messages of odd parity are valid, we just need to flip any bit of $Y$ to get a valid message $Z$ of odd parity. In both cases, we have minimized the Hamming distance $d_{H}(Y,Z)$ (which is either 0 or 1). So the error correction problem $\mathcal{P}_{ec}^{\mathcal{T}_{NP,P}}$ is polynomial-time solvable. So $S_{dc=NP,ec=P}\neq\emptyset$. Next, we prove that $S_{dc=P,ec=NP}\neq\emptyset$. Let $H$ be an $r\times n$ binary matrix of rank $r<n$. Define the validity function $\mathcal{V}_{P,NP}~{}:~{}\{0,1\}^{n}\to\{0,1\}$ as follows: $\mathcal{V}_{P,NP}(b_{1},\cdots,b_{n})=1$ if and only if $H\cdot(b_{1},\cdots,b_{n})^{T}\equiv\mathbf{0}\mod 2$. (That is, the valid messages form a linear code.) Then the data compression problem $\mathcal{P}_{dc}^{\mathcal{T}_{P,NP}}$ becomes polynomial-time solvable: we can view $H$ as the parity-check matrix of an ECC, find its corresponding generator matrix and use it to compress any valid $n$-bit message into a distinct vector of $k=n-r$ bits (e.g., through Gaussian elimination). Its details are well known in coding theory, so we skip them here. The error correction problem $\mathcal{P}_{ec}^{\mathcal{T}_{P,NP}}$ is the same as the ML decoding problem of linear codes, which is known to be NP-hard [5]. So $S_{dc=P,ec=NP}\neq\emptyset$. To prove that $S_{dc=P,ec=P}\neq\emptyset$, we can let the validity function $\mathcal{V}_{P,P}(b_{1},\cdots,b_{n})=1$ for all inputs. In this case, all messages are valid, so both data compression and error correction become trivial problems. So $S_{dc=P,ec=P}\neq\emptyset$. Now we prove that $S_{dc=NP,ec=NP}\neq\emptyset$. Let $f_{3SAT}(b_{1},\cdots,b_{n})$ be a 3-SAT Boolean formula as defined before (except that here it takes $n$ bits, instead of $n-1$ bits, as input). Let function $f_{0}(b_{1},\cdots,b_{n})$ be defined this way: it equals 1 if $b_{1}=b_{2}=\cdots=b_{n}=0$, and 0 otherwise. Let the validity function be $\mathcal{V}_{NP,NP}(b_{1},\cdots,b_{n})=f_{3SAT}(b_{1},\cdots,b_{n})\lor f_{0} (b_{1},\cdots,b_{n})$. For the data compression problem $\mathcal{P}_{dc}^{\mathcal{T}_{NP,NP}}$, $k>0$ (namely, $\|\mathcal{M}\|>1$) if and only if $f_{3SAT}(b_{1},\cdots,b_{n})$ has a satisfying solution whose bits are not all zeros, which is NP-complete to determine. So $\mathcal{P}_{dc}^{\mathcal{T}_{NP,NP}}$ is NP-hard. For the error correction problem $\mathcal{P}_{ec}^{\mathcal{T}_{NP,NP}}$, let the input noisy message $Y\in\{0,1\}^{n}$ be $Y=(1,1,\cdots,1)$. Then $\min_{Z\in\mathcal{M}}d_{H}(Y,Z)<n$ if and only if $f_{3SAT}(b_{1},\cdots,b_{n})$ has a satisfying solution whose bits are not all zeros, which is NP-complete to determine. So $\mathcal{P}_{ec}^{\mathcal{T}_{NP,NP}}$ is NP-hard. So $S_{dc=NP,ec=NP}\neq\emptyset$. ∎ The above result shows a wide range of possibilities for the computational-complexity trade-off between source and channel coding. In practice, it is worthwhile to study the properties of Natural Redundancy (e.g., whether the redundancy is mainly local or global, which differs for different types of data), and choose appropriate coding schemes based on computational complexity along with other important factors. ## VI Conclusion This paper explores the use of Natural Redundancy in data for error correction. 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1512.04432	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 37536, "num_imgs": 4, "llama3_tokens_count": 13820 }	[ "content_image/1512.04432/x1.png", "content_image/1512.04432/x2.png", "content_image/1512.04432/x3.png", "content_image/1512.04432/x5.png" ]	# Escape rates for the Farey map with approximated holes C. Bonanno Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy claudio.bonanno@unipi.it I. Chouari Faculty of Science, Computational Mathematics Laboratory, University of Monastir, Monastir 5000, Tunisia ###### Abstract. We study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. To overcome this difficulties we propose here to consider approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for “shrinking” approximated holes. The results suggest that the scaling of the escape rate depends on the chosen approximation, but “converges” to the behavior found for piecewise linear approximations of the map in [10]. CB would like to thanks the organizers of the workshop “DinAmicI IV”, the meeting of the group of Italian researchers in dynamical systems (see http://www.dinamici.org/), because at the workshop he heard about [10], which has inspired this paper. IC acknowledges warm hospitality and financial support from the Dipartimento di Matematica, Università di Pisa, where part of this paper has been written. ## 1. Introduction In recent years there has been a quick growth of the number of papers dealing with statistical properties of dynamical systems with holes. The origin of these studies can be found in the paper [1], where it was posed the question about the statistical properties of the motion of a particle inside a billiard table with a small hole. For example, if $p_{n}$ is the probability that a trajectory remains on the table until time $n$, what is the decay rate of $p_{n}$? In general, they asked whether an initial distribution would converge, under suitable renormalization, to some limit distribution, a conditionally invariant measure. Much attention to these problems has been paid also by the physics community, see e.g. [2] and references therein. In a general dynamical system, the first question posed in [1] has become the following: let $X$ be the phase space of a dynamical system with $m$ a distribution of initial conditions; let $H$ be a hole in the space $X$; let $p_{n}:=m(S_{n})$, where $S_{n}$ is the set of surviving points up to time $n$; study the decay rate \[\gamma:=\lim_{n\to\infty}\,-\frac{\log(p_{n})}{n}\,,\] which is called the _escape rate_. The existence of the limit and its dependence on the initial distribution $m$ has been studied in [3], where it is shown that in the ideal case it is possible to compute the escape rate by using the transfer operator associated to the system. Let $X=[0,1]$ and $F:X\to X$ be a smooth map with a finite number of pre-images for each $x\in X$, then the _transfer operator_ associated to $F$ is defined as \[({\mathcal{P}}f)(x):=\sum_{y\in F^{-1}(x)}\,\frac{f(y)}{\|F^{\prime}(y)\|}\,.\] The operator ${\mathcal{P}}$ has spectral radius less or equal than 1, and if there exists a function $g$ such that ${\mathcal{P}}g=g$, then $d\mu(x)=g(x)dx$ is an $F$-invariant measure. We refer to [4] for more properties of the transfer operators. When there is a hole $H$ in $X$, one can consider the _transfer operator for the open system_ \[{\mathcal{P}}^{op}f:={\mathcal{P}}((1-\chi_{H})\,f)\,,\] where $\chi_{H}$ is the indicatrix function of the set $H$. Namely the transfer operator for the open system considers only pre-images of a point $x\in X$ which are not in the hole $H$. Then the escape rate is obtained by (1.1) \[\gamma=-\log(\lambda_{H})\] where $\lambda_{H}$ is the largest eigenvalue of ${\mathcal{P}}^{op}$. The escape rate has been largely studied mainly for hyperbolic systems and piecewise expanding maps of the interval. We refer to [3] for references. In this paper we are interested in studying the behavior of the escape rate as the hole shrinks. Rigorous results on this asymptotic behavior are given in [5] for piecewise expanding maps of the interval, for which it is found that if $\|H\|=\epsilon$ then $\gamma\sim const.\times\epsilon$ as $\epsilon\to 0^{+}$. Higher order corrections can be found in [6, 7]. However, it is well known that statistical properties of dynamical systems dramatically change passing from uniformly hyperbolic systems to intermittent ones, that is systems which have an indifferent fixed point. And in particular when the intermittent system preserves only one absolutely continuous invariant measure which is infinite. Intermittent systems have been introduced in the mathematical physics literature in [8] as a simple model of the phenomenon of intermittency, that is the alternation of a turbulent and a laminar phase in a fluid. As dynamical systems on the unit interval $[0,1]$, they may be represented by the family of maps $F(x)=x+x^{\alpha}$ (mod 1), with $\alpha>1$, which have a fixed point at $x=0$ with $F^{\prime}(0)=1$ and $F^{\prime}(x)-1\approx x^{\alpha-1}$ as $x\to 0^{+}$. For $\alpha\in(1,2)$ the system has a finite absolutely continuous invariant measure. For this case, the escape rate has been recently studied in [9], where it is shown that the probability $p_{n}$ may decrease polynomially, so in the definition of $\gamma$ one should divide by $log(n)$. However the results in [9] consider the case of a hole $H$ which is generated by the Markov partition of the map, and such that the indifferent fixed point is outside $H$. Hence the polynomial escape rate is a consequence of the typical slower decay of correlations found for intermittent systems. In this paper we are interested in the case of an intermittent dynamical system with infinite absolutely continuous invariant measure. This corresponds to values $\alpha\geq 2$ in the example $F(x)=x+x^{\alpha}$ (mod 1). This case has been recently studied by G. Knight and S. Munday in [10]. They study the asymptotic behavior of the escape rate for vanishing hole and prove that different behaviors are possible. Their methods are analytical and use the definition of the _dynamical zeta function_ associated to a system, and in particular the relations with the transfer operator. However, to apply these methods they have to consider piecewise linear maps and holes which are generated by the partition associated to the map. To give a flavor of their results, we consider a typical example of intermittent dynamical system on the interval $[0,1]$ with infinite measure, the _Farey map_. The Farey map is defined by (1.2) \[F(x)=\left\{\begin{array}[]{ll}\frac{x}{1-x}\,,&\mbox{if }\ 0\leq x\leq\frac{1 }{2}\\ \frac{1-x}{x}\,,&\mbox{if }\ \frac{1}{2}\leq x\leq 1\end{array}\right.\] and is studied in particular for its relations with the continued fractions expansion of real numbers (see e.g. [11]). Its graph is shown in Figure 1. <figure><img src="content_image/1512.04432/x1.png"><figcaption>Figure 1. The Farey map.</figcaption></figure> In [10], the authors consider a piecewise version of $F$, which is obtained by considering the partition $A=\{(\frac{1}{n+1},\frac{1}{n})\}$ and defining \[F_{p}(x)=\left\{\begin{array}[]{ll}2-2x\,,&\mbox{if }\ \frac{1}{2}\leq x\leq 1 \\ \frac{n+1}{n-1}\,x-\frac{1}{n(n-1)}\,,&\mbox{if }\ \frac{1}{n+1}\leq x\leq \frac{1}{n}\end{array}\right.\] Then for holes $H_{n}=[0,\frac{1}{n})$, they show that (1.3) \[\gamma\approx\frac{1}{n\,\log n}=\frac{\|H_{n}\|}{-\log\|H_{n}\|}\qquad\text{as} \quad n\to\infty\,,\] which is a slightly faster decay than those proved for expanding maps in [5]. More general behavior are found in [10] by varying the partition $A=\{(a_{n+1},a_{n})\}$ with $a_{n}\to 0^{+}$, but always choosing a hole of the form $[0,a_{n})$. The aim of this paper is to study the escape rate for the Farey map (1.2) with hole $H=[0,\epsilon)$, and its asymptotic behavior as $\|H\|=\varepsilon\to 0^{+}$. To our knowledge this is the first case where this problem is studied in such a generality for an infinite measure preserving dynamical system. We use (1.1) to compute the escape rate through the transfer operator ${\mathcal{P}}^{op}$ associated to $F$ with a hole $H$. The main problem is that due to the presence of the indifferent fixed point, the transfer operator of the Farey map doesn’t have the so-called _spectral gap_, that is a gap between the maximal eigenvalue and the essential spectrum, independently of the smoothness of functions on which the operator is applied (see [12]). The spectral gap is found for example in the study of the transfer operator of expanding maps, when restricted to the space of bounded variation functions, and is fundamental to obtain exponential decay of correlations for the map and is used also in [5] to study the asymptotic behavior of the escape rate. A good space of functions ${\mathcal{H}}$ on which to study the spectral properties of the transfer operator of the Farey map has been introduced in [13, 14]. It consists of holomorphic functions on the disc $\{\|z-\frac{1}{2}\|<\frac{1}{2}\}$ obtained as integral transform of functions on the positive real axis (see (2.2) below). So we should consider the action of ${\mathcal{P}}^{op}$ on ${\mathcal{H}}$. Unfortunately, since the indicatrix function $\chi_{H}$ is not real analytic, we have to introduce an approximation of $\chi_{H}$. Hence we study the escape rate for the Farey map (1.2) with approximated holes which contain the indifferent fixed point, and are general in the sense that we don’t require any relation between the length of the hole and the map. In Section 2 we introduce the operator ${\mathcal{P}}^{op}$ and its approximation $\tilde{\mathcal{P}}^{op}$, obtained by approximating the function $\chi_{H}$ with a real analytic function. In Section 3 we introduce the approach we use to study the spectral properties of $\tilde{\mathcal{P}}^{op}$, and show our results. The conclusions of our study are exposed in the final section. ## 2. Transfer operators ### The Farey map The transfer operator $\mathcal{P}$ associated to the map $F$ acts on functions $f:[0,1]\to\mathbb{C}$ as \[(\mathcal{P}f)(x):=\sum_{y\,:\,F(y)=x}\,\frac{f(y)}{\|F^{\prime}(y)\|}\] which using (1.2) becomes \[(\mathcal{P}f)(x)=(\mathcal{P}_{0}f+\mathcal{P}_{1}f)(x)\] with (2.1) \[(\mathcal{P}_{0}f)(x)=\frac{1}{(1+x)^{2}}\,f\Big{(}\frac{x}{1+x}\Big{)}\quad \text{and}\quad(\mathcal{P}_{1}f)(x)=\frac{1}{(1+x)^{2}}\,f\Big{(}\frac{1}{1+x }\Big{)}\,.\] The operator $\mathcal{P}$ has been studied in [14, 15] on the invariant Hilbert space ${\mathcal{H}}$ defined as (2.2) \[{\mathcal{H}}:=\left\{f:[0,1]\to\mathbb{C}\,:\,f=\mathcal{B}[\varphi]\ \text{ for some }\varphi\in L^{2}(m)\right\}\] where $\mathcal{B}[\cdot]$ denotes the generalized Borel transform (2.3) \[(\mathcal{B}[\varphi])(x):=\frac{1}{x^{2}}\int_{0}^{\infty}e^{-\frac{t}{x}}\,e ^{t}\,\varphi(t)\,dm(t)\,,\] and $L^{2}(m):=L^{2}(\mathbb{R}^{+},m)$ where $m$ is the measure on $\mathbb{R}^{+}$ \[dm(t)=te^{-t}dt.\] The space ${\mathcal{H}}$ is endowed with the inner product inherited by the inner product on $L^{2}(m)$ through the $\mathcal{B}$-transform, that is \[(f_{1},f_{2})_{{\mathcal{H}}}:=\int_{0}^{\infty}\varphi_{1}(t)\,\overline{ \varphi_{2}(t)}\,dm(t)\qquad\text{if}\quad f_{i}=\mathcal{B}[\varphi_{i}]\,.\] It is immediate to show that functions in ${\mathcal{H}}$ can be continued to holomorphic functions on the disc $\{\|z-\frac{1}{2}\|<\frac{1}{2}\}$, and moreover eigenfunctions of ${\mathcal{P}}$ in ${\mathcal{H}}$ can be continued to holomorphic function on the half-plane $\{\Re(z)>0\}$. From the computational point of view, it is more convenient to use the $\mathcal{B}$-transform to read the action of ${\mathcal{P}}$ on $L^{2}(m)$. Indeed (2.4) \[\mathcal{P}\,\mathcal{B}[\varphi]=\mathcal{B}[(M+N)\varphi]\] for all $\varphi\in L^{2}(m)$, where $M,N:L^{2}(m)\to L^{2}(m)$ are self-adjoint bounded linear operators defined by (2.5) \[(M\varphi)(t)=e^{-t}\,\varphi(t)\qquad\text{and}\qquad(N\varphi)(t)=\int_{0}^{ \infty}J_{1}\left(2\sqrt{st}\right)\,\sqrt{\frac{1}{st}}\ \varphi(s)\,dm(s)\] where $J_{q}$ denotes the Bessel function of order $q$. The spectrum of the operator ${\mathcal{P}}$ is the unit interval $[0,1]$, and it is continuous spectrum without eigenvalues on ${\mathcal{H}}$. This phenomenon is due to the presence of the indifferent fixed point at the origin. In particular $1$ is not an eigenvalue, and indeed the Farey map has an infinite absolutely continuous invariant measure with density $\frac{1}{x}$, which does not belong to ${\mathcal{H}}$. ### The Farey map with a hole The transfer operator ${\mathcal{P}}^{op}$ for the map $F$ with a hole in $H=[0,\epsilon)$ is obtained by ${\mathcal{P}}$ simply subtracting the contribution from the hole, that is \[{\mathcal{P}}^{op}f:={\mathcal{P}}((1-\chi_{H})f)={\mathcal{P}}f-{\mathcal{P}} (\chi_{H}f)\,,\] where $\chi_{H}(x)$ is the indicator function of the set $H$. First of all we notice that for $\epsilon<\frac{1}{2}$ it holds \[{\mathcal{P}}(\chi_{H}f)={\mathcal{P}}_{0}(\chi_{H}f)\,,\] since $\frac{1}{1+x}\geq\frac{1}{2}$ for all $x\in[0,1]$, hence by (2.1) \[({\mathcal{P}}(\chi_{H}f))(x)=\frac{1}{(1+x)^{2}}\,\chi_{H}\Big{(}\frac{x}{1+x }\Big{)}\,f\Big{(}\frac{x}{1+x}\Big{)}=\frac{1}{(1+x)^{2}}\,\chi_{\tilde{H}}(x )\,f\Big{(}\frac{x}{1+x}\Big{)}=\chi_{\tilde{H}}(x)\,({\mathcal{P}}_{0}f)(x)\] where $\tilde{H}=[0,\frac{\epsilon}{1-\epsilon})$. Hence (2.6) \[{\mathcal{P}}^{op}f={\mathcal{P}}f-\chi_{\tilde{H}}\,{\mathcal{P}}_{0}f\,.\] However, to study the spectral properties of the operator ${\mathcal{P}}^{op}$ on ${\mathcal{H}}$, we need an approximation of the indicator function since we are dealing with real analytic functions. We consider the family of approximations $\left\{\xi_{\mu}(x,a)\right\}_{\mu\in\mathbb{R}}$ for the indicator function of an interval $[0,a)$ given by (2.7) \[\xi_{\mu}(x,a):=\frac{1}{2}-\frac{1}{2}\text{Erf}\left(\mu\,\left(x-a\right) \right)\,,\] where Erf is the “error function” (2.8) \[\text{Erf}(x):=\frac{2}{\sqrt{\pi}}\,\int_{0}^{x}\,e^{-t^{2}}\,dt=\frac{2}{ \sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\frac{(-1)^{n}\,x^{2n+1}}{n!\,(2n+1)}\,.\] In Figure 2 we have plotted the function $\xi_{\mu}(x,a)$ for $a=0.4$ and $\mu=1,3$ and $10$, against $\chi_{[0,a)}(x)$. The functions $\xi_{\mu}(x,a)$ converge to $\chi_{[0,a)}(x)$ as $\mu\to\infty$ for all $x\not=a$, but clearly not uniformly. <figure><img src="content_image/1512.04432/x2.png"><figcaption>Figure 2. The graph of the function ξμ(x,a) for a=0.4 and μ=1,3,10, and of thecharacteristic function of [0,a].</figcaption></figure> Hence we define a family of approximated transfer operators for the Farey map with hole $[0,\epsilon)$ by using $\xi_{\mu}$ in (2.6), and let (2.9) \[(\tilde{\mathcal{P}}^{op}_{\mu}f)(x):=({\mathcal{P}}f)(x)-\xi_{\mu}\left(x, \frac{\epsilon}{1-\epsilon}\right)\,({\mathcal{P}}_{0}f)(x)\,.\] Moreover, to study the action of $\tilde{\mathcal{P}}^{op}_{\mu}$ on ${\mathcal{H}}$ as for the action of ${\mathcal{P}}$, we need the equivalent of equation (2.4) for $\tilde{\mathcal{P}}^{op}_{\mu}$. To this aim, we first recall that by definition (2.3) \[{\mathcal{B}}[\varphi(t)](x)=\frac{1}{x^{2}}\,{\mathcal{L}}[t\,\varphi(t)] \left(\frac{1}{x}\right)\] where ${\mathcal{L}}[\cdot]$ denotes the standard Laplace transform, then for all $\psi,\varphi\in L^{2}(m)$ we can write (2.10) \[{\mathcal{B}}[\psi](x)=\xi_{\mu}(x,a)\,{\mathcal{B}}[\varphi](x) \quad\Leftrightarrow\quad{\mathcal{L}}[t\,\psi(t)]\left(\frac{1}{x}\right)=\xi _{\mu}(x,a)\,{\mathcal{L}}[t\,\varphi(t)]\left(\frac{1}{x}\right)\quad\Leftrightarrow\] \[\Leftrightarrow\quad{\mathcal{L}}[t\,\psi(t)](x)=\xi_{\mu}\left( \frac{1}{x},a\right)\,{\mathcal{L}}[t\,\varphi(t)](x)\quad\Leftrightarrow\quad t \,\psi(t)=\int_{0}^{t}\,s\,\varphi(s)\,{\mathcal{L}}^{-1}\left[\xi_{\mu}\left( \frac{1}{x},a\right)\right](t-s)\,ds\] where in the last equivalence we have used the standard properties for the Laplace transform of convolution of functions. Using (2.4) and (2.10) in (2.9), we obtain (2.11) \[(\tilde{\mathcal{P}}^{op}_{\mu}\,{\mathcal{B}}[\varphi])(x)={ \mathcal{B}}[(M+N)\varphi](x)-\xi_{\mu}\left(x,\frac{\epsilon}{1-\epsilon} \right)\,{\mathcal{B}}[M\varphi](x)=\] \[={\mathcal{B}}[(M+N)\varphi](x)-{\mathcal{B}}\left[\frac{1}{t}\, \int_{0}^{t}\,s\,(M\varphi)(s)\,{\mathcal{L}}^{-1}\left[\xi_{\mu}\left(\frac{1 }{x},\frac{\epsilon}{1-\epsilon}\right)\right](t-s)\,ds\right](x)=\] \[={\mathcal{B}}[(\tilde{M}_{\mu}+N)\varphi](x)\] with $M,N$ as in (2.5), and (2.12) \[(\tilde{M}_{\mu}\varphi)(t):=(M\varphi)(t)-\frac{1}{t}\,\int_{0}^{t}\,s\,(M \varphi)(s)\,{\mathcal{L}}^{-1}\left[\xi_{\mu}\left(\frac{1}{x},\frac{\epsilon }{1-\epsilon}\right)\right](t-s)\,ds\,.\] Theorem 2.1.: _For $\xi_{\mu}(x,a)$ as in (2.7), the operator $\tilde{M}_{\mu}$ is given by_ (2.13) \[(\tilde{M}_{\mu}\varphi)(t) =\frac{1}{2}\Big{(}1-\mathrm{Erf}\Big{(}\frac{\mu\,\epsilon}{1- \epsilon}\Big{)}\Big{)}\,(M\varphi)(t)+\] \[+\frac{1}{\sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\sum_{k=1}^{2n+1}\, \left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}}{n!\,(k-1)!\,(2n+1)}\, \left(\frac{\epsilon}{1-\epsilon}\right)^{2n+1-k}\ \frac{1}{t}\,\int_{0}^{t}\, s(M\varphi)(s)\,(t-s)^{k-1}\,ds\] _and it is a bounded operator $\tilde{M}_{\mu}:L^{2}(m)\to L^{2}(m)$._ ## 3. The matrix approach and the numerical results As shown in [14], the Hilbert space $L^{2}(m)$ admits a complete orthogonal system $\{e_{\nu}\}_{\nu\geq 0}$ given by the Laguerre polynomials defined as (3.1) \[e_{\nu}(t):=\sum_{m=0}^{\nu}\,\left(\begin{array}[]{c}\nu+1\\ \nu-m\end{array}\right)\,\frac{(-t)^{m}}{m!}\] which satisfy \[(e_{\nu},e_{\nu})=\frac{\Gamma(\nu+2)}{\nu!}=\nu+1\] for all $\nu\geq 0$. Hence, using (2.11), we can study the action of $\tilde{\mathcal{P}}^{op}_{\mu}$ on ${\mathcal{H}}$ by the action on $L^{2}(m)$ of an infinite matrix representing the operators $\tilde{P}^{op}_{\mu}:=\tilde{M}_{\mu}+N$, defined in (2.5) and (2.13), for the basis $\{e_{\nu}\}_{\nu\geq 0}$. That is for any $\phi\in L^{2}(m)$, we can write \[\phi(t)=\sum_{\nu=0}^{\infty}\phi_{\nu}e_{\nu}(t)\quad\text{with}\quad\phi_{ \nu}=\frac{1}{\nu+1}\,(\phi,e_{\nu})\] hence $\phi$ is an eigenfunction of $\tilde{P}^{op}_{\mu}$ with eigenvalue $\lambda$ if and only if \[(\tilde{P}^{op}_{\mu}\phi,e_{j})=\lambda\,(\phi,e_{j})=\lambda\,(j+1)\,\phi_{j }\qquad\forall\;j\geq 0\] Using the notation $c_{j\nu}^{\mu}:=(\tilde{P}^{op}_{\mu}e_{\nu},e_{j})$ we obtain that (3.2) \[\tilde{P}^{op}_{\mu}\phi=\lambda\phi\quad\Leftrightarrow\quad C_{\mu}\phi= \lambda D\phi\quad\Leftrightarrow\quad A_{\mu}\phi=\lambda\phi\] where $C_{\mu}$ and $D$ are given by \[C_{r}=(c_{j\nu}^{\mu})_{j,\nu\geq 0}\quad\text{and}\quad D=\text{diag}(j+1)_{j \geq 0}\] and $A_{\mu}$ is the infinite matrix (3.3) \[A_{\mu}=(a_{j\nu}^{\mu})_{j,\nu\geq 0}\qquad\text{with}\quad a_{j\nu}^{\mu}= \frac{c_{j\nu}^{\mu}}{j+1}\,.\] We have then to compute the terms $c_{j\nu}^{\mu}:=(\tilde{P}^{op}_{r}e_{\nu},e_{j})$. From [16, Prop. 3.1] we have \[\frac{1}{j+1}\,(Me_{\nu},e_{j})=\left(\begin{array}[]{c}\nu+j+1\\ \nu\end{array}\right)\,\frac{1}{2^{\nu+j+2}}\] \[\frac{1}{j+1}\,(Ne_{\nu},e_{j})=\sum_{\ell=0}^{\nu}(-1)^{\ell}\, \left(\begin{array}[]{c}\nu+1\\ \nu-\ell\end{array}\right)\,\left(\begin{array}[]{c}\ell+j+1\\ l\end{array}\right)\,\frac{1}{2^{\ell+j+2}}\,.\] Hence (3.4) \[a^{\mu}_{j\nu}=\Big{(}1-\mathrm{Erf}\Big{(}\frac{\mu\,\epsilon}{ 1-\epsilon}\Big{)}\Big{)}\,\left(\begin{array}[]{c}\nu+j+1\\ \nu\end{array}\right)\,\frac{1}{2^{\nu+j+3}}+\sum_{\ell=0}^{\nu}(-1)^{\ell}\, \left(\begin{array}[]{c}\nu+1\\ \nu-\ell\end{array}\right)\,\left(\begin{array}[]{c}\ell+j+1\\ l\end{array}\right)\,\frac{1}{2^{\ell+j+2}}+\] \[+\frac{1}{(j+1)\sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\sum_{k=1}^{2n+1 }\,\left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}}{n!\,(k-1)!\,(2n+1)}\, \left(\frac{\epsilon}{1-\epsilon}\right)^{2n+1-k}\ \Big{(}\frac{1}{t}\,\int_{0 }^{t}\,s(Me_{\nu})(s)\,(t-s)^{k-1}\,ds\ ,\ e_{j}\Big{)}\] where in the last summation we have used the $L^{2}$ convergence of the series. Concerning the last terms we use [17, eq. 7.415 p. 810] to write \[\frac{1}{t}\,\int_{0}^{t}\,s(Me_{\nu})(s)\,(t-s)^{k-1}\,ds=t^{k}\,\int_{0}^{1} \,s\,(1-s)^{k-1}\,e^{-st}\,e_{\nu}(st)ds=\frac{\nu+1}{k(k+1)}\,t^{k}\,{}_{{}_{ 1}}F_{{}_{1}}(\nu+2,k+2,-t)\] where ${}_{{}_{1}}F_{{}_{1}}$ is the standard confluent hypergeometric function. Moreover, by (3.1) and [17, eq. 7.621(4) p. 822] \[\Big{(}t^{k}\,{}_{{}_{1}}F_{{}_{1}}(\nu+2,k+2,-t)\ ,\ e_{j}(t) \Big{)} =\sum_{m=0}^{j}\,\left(\begin{array}[]{c}j+1\\ j-m\end{array}\right)\,\frac{(-1)^{m}}{m!}\,\int_{0}^{\infty}\,t^{k+m+1}\,e^{- t}\,{}_{{}_{1}}F_{{}_{1}}(\nu+2,k+2,-t)\,dt=\] \[=\sum_{m=0}^{j}\,\left(\begin{array}[]{c}j+1\\ j-m\end{array}\right)\,\frac{(-1)^{m}\,(k+m+1)!}{m!}\,{}_{{}_{2}}F_{{}_{1}}( \nu+2,k+m+2;k+2;-1)\] where ${}_{{}_{2}}F_{{}_{1}}$ is the hypergeometric function. Using the previous equations in (3.4), we get the following explicit expression for the general term $a^{\mu}_{j\nu}$ of the matrix $A_{\mu}$ defined in (3.2) and (3.3), which represents the transfer operator $\tilde{\mathcal{P}}^{op}_{\mu}$ on $L^{2}(m)$ (3.5) \[a^{\mu}_{j\nu}=\Big{(}1-\mathrm{Erf}\Big{(}\frac{\mu\,\epsilon}{ 1-\epsilon}\Big{)}\Big{)}\,\left(\begin{array}[]{c}\nu+j+1\\ \nu\end{array}\right)\,\frac{1}{2^{\nu+j+3}}+\sum_{\ell=0}^{\nu}(-1)^{\ell}\, \left(\begin{array}[]{c}\nu+1\\ \nu-\ell\end{array}\right)\,\left(\begin{array}[]{c}\ell+j+1\\ l\end{array}\right)\,\frac{1}{2^{\ell+j+2}}+\] \[+\frac{\nu+1}{(j+1)\sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\sum_{k=1}^{ 2n+1}\,\sum_{m=0}^{j}\,\left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\left(\begin{array}[]{c}j+1\\ j-m\end{array}\right)\left(\begin{array}[]{c}k+m+1\\ m\end{array}\right)\frac{(-1)^{n+k+m-1}\,\mu^{2n+1}}{n!\,(2n+1)}\,\left(\frac{ \epsilon}{1-\epsilon}\right)^{2n+1-k}\cdot\] \[\cdot{}_{{}_{2}}F_{{}_{1}}(\nu+2,k+m+2;k+2;-1)\] We recall that the parameter $\mu$ determines the level of approximation of the real hole $\tilde{H}=[0,\frac{\epsilon}{1-\epsilon})$ used in (2.9). We need to comment on this approximation in order to compare our numerical results with results by previous papers, and in particular those in [10]. To estimate the scaling of the escape rate $\gamma$ defined in (1.1), as the hole $H=[0,\epsilon)$, and hence the hole $\tilde{H}=[0,\frac{\epsilon}{1-\epsilon})$, shrinks to the empty set, we compute the principal eigenvalue of north-west corner approximations of the matrix $A_{\mu}$ as $\epsilon$ decreases to 0. We remark that as $\epsilon\to 0^{+}$, our approximated hole, that is the function $\xi_{\mu}(x,\frac{\epsilon}{1-\epsilon})$, converges to the function \[\xi_{\mu}(x,0)=\frac{1}{2}-\frac{1}{2}\mathrm{Erf}(\mu\,x)\,,\] which, as $\mu\to+\infty$, converges to zero for all $x>0$. However, if we observe that the appropriate quantity with which the escape rate should be compared is the measure of the hole, we have to look at the integral of $\xi_{\mu}(x,0)$ on $[0,1]$, and more precisely on $\mathbb{R}^{+}$ since our approximated holes have effect on the whole real positive axis. Unfortunately the integral shows slow convergence to 0 as $\mu$ diverges. For $\mu=7$, the integral is of order $10^{-2}$, which implies a significant perturbation on the transfer operator. Moreover, a value of $\mu$ greater than 4 in (3.5) implies that, to have good numerical results, one should consider too many terms in the series on $n$. For this reason, we have changed our point of view, keeping in mind that what is important is to have a sequence of approximated holes with vanishing measure on the real positive axis. This can be obtained for small values of $\mu$, by letting $\epsilon$ diverge to $-\infty$ in $\xi_{\mu}(x,\frac{\epsilon}{1-\epsilon})$. In Figure 3 we show the behavior of $\xi_{\mu}(x,\frac{\epsilon}{1-\epsilon})$ on $[0,1]$ for $\epsilon=0.1,0,-1,-5,-20$ for $\mu=1$ on the left, and for $\mu=2$ on the right. We notice that the functions decrease on the positive real axis more rapidly for bigger $\mu$, only the first three cases are non-negligible for $\mu=2$, and if we measure their integral on $\mathbb{R}^{+}$ we have that \[\lim_{\epsilon\to-\infty}\int_{0}^{+\infty}\,\xi_{\mu}\Big{(}x,\frac{\epsilon} {1-\epsilon}\Big{)}\,dx\approx\left\{\begin{array}[]{ll}0.025&\text{for }\ \mu =1\\ 2\times 10^{-4}&\text{for }\ \mu=2\\ 10^{-5}&\text{for }\ \mu=2.5\\ 5\times 10^{-7}&\text{for }\ \mu=3\\ 10^{-8}&\text{for }\ \mu=3.5\end{array}\right.\] and the integral is already $\approx 10^{-6}$ for $\mu=3$ and $\epsilon=-20$, and $\approx 10^{-7}$ for $\mu=3.5$ and $\epsilon=-10$. <figure><img src="content_image/1512.04432/x3.png"><figcaption></figcaption></figure> Hence we have computed the principal eigenvalue $\lambda_{\mu}(\epsilon)$ of north-west corner approximations of the matrix $A_{\mu}$ as $\epsilon$ decreases to $-\infty$, and the principal eigenvalue $\lambda_{\infty}$ of the same approximations of the matrix associated to the transfer operator of the Farey map without hole. To find the scaling of the escape rate, we have plotted $\gamma_{\mu}(\epsilon):=-\log(\lambda_{\mu}(\epsilon)/\lambda_{\infty})$ against $M_{\mu}(\epsilon):=\int_{0}^{+\infty}\,\xi_{\mu}\Big{(}x,\frac{\epsilon}{1- \epsilon}\Big{)}\,dx$. The results are shown in Figure 4. The solid lines are the identity and the function $f(t)=\frac{t}{-\log t}$. The dotted lines are the plots of the points $(M_{\mu}(\epsilon),\gamma_{\mu}(\epsilon))$ for $\mu=1,2,2.5,3$ and 3.5 from the biggest to the lowest. Notice that for $\mu=1$ the dots stop far from the origin because $M_{1}(-\infty)\approx 0.025$. Figure 4 shows that the scaling of the escape rate for the Farey map with shrinking approximated holes is dependent on the shape and the goodness of the approximation. However, as the approximation gets better, that is for $\mu$ big in our case, we find a scaling \[\gamma_{\mu}(\epsilon)\approx\frac{M_{\mu}(\epsilon)}{-\log M_{\mu}(\epsilon)} \qquad\text{as}\quad M_{\mu}(\epsilon)\to 0^{+}\,,\] which is consistent with (1.3), the theoretical result for the Markov approximation of the Farey map studied in [10] with holes generated by the Markov partition. <figure><img src="content_image/1512.04432/x5.png"><figcaption>Figure 4. The solid lines are the identity and the function f(t)=t−logt. Thedotted lines are the points (Mμ(ϵ),γμ(ϵ)) for μ=1,2,2.5,3 and 3.5 from thebiggest to the lowest.</figcaption></figure> ## 4. Conclusions In this paper we have studied the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. To our knowledge this is the first time this problem is studied for general holes, since previous results only considered piecewise linear approximations of the map with holes generated by the associated partition. The problem we consider poses theoretical difficulties in the application of the standard methods for the study of the escape rate, in particular the transfer operator approach, due to the properties of the transfer operator of the Farey map. For this reason, we propose to modify the standard approach to open systems by considering approximations of the hole, by means of a family of functions converging to the indicatrix function of the hole. We believe that this is the main contribution of this paper. Here we have used the family (2.7), and proved in Theorem 2.1 that the associated transfer operator has the functional analytic properties necessary to pursue the approach already used in [14, 15, 16] to study the spectral properties of the transfer operator of the Farey map. Our numerical results suggest that the behavior of the escape rate is indeed dependent on the chosen approximation of the hole, but for functions close to the indicatrix function, we find the same behavior proved in [10] for the piecewise linear approximation of the map. ## Appendix A Proof of Theorem 2.1 From (2.12), we have to compute the inverse Laplace transform of the function $\xi$. For this we use the series expansion in (2.8) and write \[={\mathcal{L}}^{-1}\left[\frac{1}{2}-\frac{1}{2}\text{Erf}\left( \mu\,\left(\frac{1}{x}-a\right)\right)\right]=\] \[= \frac{1}{2}\,\delta_{0}(t)-\frac{1}{2}\,{\mathcal{L}}^{-1}\left[ \frac{2}{\sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\frac{(-1)^{n}\,\mu^{2n+1}}{n!\,(2n +1)}\left(\frac{1}{x}-a\right)^{2n+1}\right](t)=\] \[= \frac{1}{2}\,\delta_{0}(t)-\frac{1}{\sqrt{\pi}}\,\sum_{n=0}^{ \infty}\,\frac{(-1)^{n}\,\mu^{2n+1}}{n!\,(2n+1)}\,{\mathcal{L}}^{-1}\left[\sum _{k=0}^{2n+1}\,\left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\frac{1}{x^{k}}{(-a)^{2n+1-k}}\right](t)=\] \[= \frac{1}{2}\,\delta_{0}(t)-\frac{1}{\sqrt{\pi}}\,\sum_{n=0}^{ \infty}\,\frac{(-1)^{n-1}\,(\mu a)^{2n+1}}{n!\,(2n+1)}\,\delta_{0}(t)+\] \[-\frac{1}{\sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\sum_{k=1}^{2n+1}\, \left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}\,a^{2n+1-k}}{n!\,(2n+1)}\, \frac{t^{k-1}}{(k-1)!}=\] \[= \frac{1}{2}\Big{(}1+\text{Erf}(\mu a)\Big{)}\,\delta_{0}(t)-\frac {1}{\sqrt{\pi}}\,\sum_{n=0}^{\infty}\,\sum_{k=1}^{2n+1}\,\left(\begin{array}[] {c}2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}\,a^{2n+1-k}}{n!\,(2n+1)}\, \frac{t^{k-1}}{(k-1)!}\,,\] where $\delta_{0}$ denotes the Dirac delta at the origin, and we have used the standard results \[{\mathcal{L}}^{-1}[x^{-k}](t)=\frac{t^{k-1}}{(k-1)!}\quad\text{for}\ t\in \mathbb{R}^{+}\,,\,k>0\,,\qquad{\mathcal{L}}^{-1}[1](t)=\delta_{0}(t)\,.\] Moreover, from the second to the third line, we have used that \[\int_{0}^{\infty}\,e^{-xt}\,\left[\sum_{n=0}^{\infty}\,\frac{(-1)^{n}\,\mu^{2n +1}}{n!\,(2n+1)}\left(\delta_{0}(t)+\sum_{k=1}^{2n+1}\,\left(\begin{array}[]{c }2n+1\\ k\end{array}\right)\,\frac{(-a)^{2n+1-k}}{(k-1)!}\,t^{k-1}\right)\right]\,dt=\] \[=\sum_{n=0}^{\infty}\,\frac{(-1)^{n}\,\mu^{2n+1}}{n!\,(2n+1)}\,\int_{0}^{ \infty}\,e^{-xt}\,\left(\delta_{0}(t)+\sum_{k=1}^{2n+1}\,\left(\begin{array}[] {c}2n+1\\ k\end{array}\right)\,\frac{(-a)^{2n+1-k}}{(k-1)!}\,t^{k-1}\right)\,dt\] for $x$ big enough. Hence, by the analytical continuation of the Laplace transform, the equality is verified for all $x>0$. It follows that the passage of the ${\mathcal{L}}^{-1}$ operator inside the summation, from the second to the third line, is justified. Since \[\int_{0}^{t}\,s(M\varphi)(s)\delta_{0}(t-s)\,ds=t(M\varphi)(t)\] from (2.12) we obtain the first term on the right hand side of (2.13). To finish we need to show that (A.1) \[\frac{1}{t}\,\int_{0}^{t}\,s\,(M\varphi)(s)\,\sum_{n=0}^{\infty} \,\sum_{k=1}^{2n+1}\,\left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}\,a^{2n+1-k}}{n!\,(2n+1)}\, \frac{(t-s)^{k-1}}{(k-1)!}\,ds=\] \[=\sum_{n=0}^{\infty}\,\sum_{k=1}^{2n+1}\,\left(\begin{array}[]{c} 2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}\,a^{2n+1-k}}{n!\,(2n+1)}\, \frac{1}{t}\,\int_{0}^{t}\,s\,(M\varphi)(s)\,\frac{(t-s)^{k-1}}{(k-1)!}\,ds\] First of all we recall from [18, eq. 13.2.2 p. 322 and eq. 13.6.19 p. 328] that for any $\varphi\in L^{2}(m)$ we have \[\sum_{s=0}^{2n}\,\frac{(-2n)_{s}}{(2)_{s}\,s!}\,\Big{(}\frac{t}{a}\Big{)}^{s}= {}_{{}_{1}}F_{{}_{1}}\Big{(}-2n,2;\frac{t}{a}\Big{)}=\frac{1}{2n+1}\,e_{2n} \Big{(}\frac{t}{a}\Big{)}\] where $(k)_{s}=k(k+1)\dots(k+s-1)$ is the Pochhammer symbol, ${}_{{}_{1}}F_{{}_{1}}$ is the standard confluent hypergeometric function, and $e_{\nu}$ is the Laguerre polynomial defined in (3.1). Hence \[\sum_{n=0}^{\infty}\,\sum_{k=1}^{2n+1}\,\left(\begin{array}[]{c}2n+1\\ k\end{array}\right)\,\frac{(-1)^{n+k-1}\,\mu^{2n+1}\,a^{2n+1-k}}{n!\,(2n+1)}\, \frac{(t-s)^{k-1}}{(k-1)!}=\sum_{n=0}^{\infty}\,\frac{(-1)^{n}\,\mu^{2n+1}\,a^ {2n}}{n!\,(2n+1)}\,e_{2n}\Big{(}\frac{t-s}{a}\Big{)}\] We are then reduced to study the increase of the terms \[\Big{\\|}\frac{1}{t}\,\int_{0}^{t}\,s\,(M\varphi)(s)\,e_{2n}\Big{(}\frac{t-s}{a }\Big{)}\,ds\Big{\\|}_{L^{2}(m)}^{2}=\int_{0}^{\infty}\,\frac{1}{t}\,e^{-t}\, \left(\int_{0}^{t}\,s\,e^{-s}\,\varphi(s)\,e_{2n}\Big{(}\frac{t-s}{a}\Big{)}\, ds\right)^{2}\,dt\] Standard manipulations show that \[=\\|\varphi\\|_{L^{2}(m)}^{2}\,\int_{0}^{\infty}\,e^{-u}\,\int_{0}^ {u}\,\Big{(}\frac{u-v}{u+v}\Big{)}\,e_{2n}^{2}\Big{(}\frac{v}{a}\Big{)}\,dv\ du\leq a\,\\|\varphi\\|_{L^{2}(m)}^{2}\,\int_{0}^{\infty}\,e^{-u}\,\int_{0}^{u/a }\,e_{2n}^{2}(v)\,dv\,du\,.\] By (3.1), we have \[e_{2n}^{2}(v)=\sum_{i,j=0}^{2n}\,\left(\begin{array}[]{c}2n+1\\ 2n-i\end{array}\right)\left(\begin{array}[]{c}2n+1\\ 2n-j\end{array}\right)\frac{(-1)^{i+j}}{i!\,j!}\,v^{i+j}\] hence \[\int_{0}^{\infty}\,e^{-u}\,\int_{0}^{u/a}\,e_{2n}^{2}(v)\,dv\,du= \sum_{i,j=0}^{2n}\,\left(\begin{array}[]{c}2n+1\\ 2n-i\end{array}\right)\left(\begin{array}[]{c}2n+1\\ 2n-j\end{array}\right)\frac{(-1)^{i+j}}{i!\,j!}\,\int_{0}^{\infty}\,e^{-u}\, \int_{0}^{u/a}\,v^{i+j}\,dv\,du=\] \[=\sum_{i,j=0}^{2n}\,\left(\begin{array}[]{c}2n+1\\ 2n-i\end{array}\right)\left(\begin{array}[]{c}2n+1\\ 2n-j\end{array}\right)\frac{(-1)^{i+j}}{a^{i+j+1}}\,\frac{1}{i!\,j!\,(i+j+1)} \int_{0}^{\infty}\,e^{-u}\,u^{i+j+1}\,du=\] \[=\sum_{i,j=0}^{2n}\,\left(\begin{array}[]{c}2n+1\\ 2n-i\end{array}\right)\left(\begin{array}[]{c}2n+1\\ 2n-j\end{array}\right)\frac{(-1)^{i+j}}{a^{i+j+1}}\,\left(\begin{array}[]{c}i+ j\\ i\end{array}\right)\] Using the very crude estimate $\left(\begin{array}[]{c}k\\ h\end{array}\right)\leq 2^{k}$ for all $h=0,\dots,k$, and $i+j\leq 4n$, we obtain \[\int_{0}^{\infty}\,e^{-u}\,\int_{0}^{u/a}\,e_{2n}^{2}(v)\,dv\,du\leq(2n+1)^{2} \,2^{4(2n+1)}\,\max\{a^{-s}\,:\,s=1,\dots,4n+1\}\,.\] Hence \[\sum_{n=0}^{\infty}\,\frac{(-1)^{n}\,\mu^{2n+1}\,a^{2n}}{n!\,(2n+1)}\,\Big{\\|} \frac{1}{t}\,\int_{0}^{t}\,s\,(M\varphi)(s)\,e_{2n}\Big{(}\frac{t-s}{a}\Big{)} \,ds\Big{\\|}_{L^{2}(m)}\leq\\|\varphi\\|_{L^{2}(m)}\,\sum_{n=0}^{\infty}\,\frac{ (2n+1)\,c^{2n+1}}{n!}\] where \[c=\left\{\begin{array}[]{ll}16\,\frac{\mu}{a}\,,&\text{if }\,\|a\|\leq 1\\ 16\,\mu\,a\,,&\text{if }\,\|a\|>1\end{array}\right.\] In any case we obtain total convergence in $L^{2}(m)$ for the right hand side of (A.1), hence (A.1) holds in the $L^{2}$-sense. 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1804.02887	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 27041, "num_imgs": 3, "llama3_tokens_count": 8751 }	[ "content_image/1804.02887/x1.png", "content_image/1804.02887/x2.png", "content_image/1804.02887/x3.png" ]	# Some Reduction Operations to Pairwise Compatibility Graphs Mingyu Xiao Hiroshi Nagamochi ###### Abstract A graph $G=(V,E)$ with a vertex set $V$ and an edge set $E$ is called a pairwise compatibility graph (PCG, for short) if there are a tree $T$ whose leaf set is $V$, a non-negative edge weight $w$ in $T$, and two non-negative reals $d_{\min}\leq d_{\max}$ such that $G$ has an edge $uv\in E$ if and only if the distance between $u$ and $v$ in the weighted tree $(T,w)$ is in the interval $[d_{\min},d_{\max}]$. PCG is a new graph class motivated from bioinformatics. In this paper, we give some necessary and sufficient conditions for PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete $k$-partite graph, cactus, and some other graph classes are subsets of PCG. Key words. Graph Algorithms; Pairwise Compatibility Graph; Graph Theory; Reduction School of Computer Science and Engineering, University of Electronic Science and Technology of China, China, myxiao@gmail.com Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Japan, nag@amp.i.kyoto-u.ac.jp ## 1 Introduction An unweighted simple undirected graph $G=(V,E)$ with a vertex set $V$ and an edge set $E$ is called a _pairwise compatibility graph_ (PCG, for short) if there exist a tree $T$ with edges weighted by non-negative reals and two non-negative real numbers $d_{\min}$ and $d_{\max}$ such that: the leaf set of $T$ is $V$, and two vertices $u,v\in V$ are adjacent in $G$ if and only if the distance between $u$ and $v$ in $T$ is at least $d_{\min}$ and at most $d_{\max}$. The tree $T$ is also called a _pairwise compatibility tree_ (PCT, for short) of the graph $G$. The same tree $T$ can be a PCT of more than one PCG. Figure 1 shows an edge-weighted tree $(T,w)$ and two PCGs for $(T,w)$ with $(d_{\min},d_{\max})=(5,7)$ and $(d_{\min},d_{\max})=(4,8)$ respectively. The concept of PCG was first introduced by Kearney et al. [11] to model evolutionary relationships among a set of organisms in bioinformatics. However, it is a challenging problem to construct a pairwise compatibility tree for a given graph. Recognition and characterization of PCGs became interesting problems in graph theory recently. <figure><img src="content_image/1804.02887/x1.png"><figcaption>Figure 1: (a) An edge-weighted tree (T,w), (b) The PCG obtained from (T,w) and(dmin,dmax)=(5,7), and (c) The PCG obtained from (T,w) and (dmin,dmax)=(4,8).</figcaption></figure> Not every graph is a PCG. Yanhaona, Bayzid and Rahman [13] constructed the first non-PCG, which is a bipartite graph with 15 vertices. Later, an example with 8 vertices was found in [8]. This is the smallest non-PCG, since it has been checked that all graphs with at most seven vertices are PCGs [1]. Currently no polynomial-time algorithm is known to the problem of testing whether a given graph is a PCG or not. It is widely believed that recognizing PCGs is NP-hard [7, 8]. In the literature, there are several contributions to recognizing some subclasses of PCG. It is not difficult to see that every tree is a PCG [9]. Every cycle with at most one chord has also been shown to be a PCG [14]. Other subclasses of graphs currently known as PCGs are power graphs of trees [13], threshold graphs [5], triangle-free outerplanar 3-graphs [12], a special subclasses of split matrogenic graphs [6], Dilworth 2 graphs [3, 4], the complement of a forest [9], the complement of a cycle [2] and so on. Some conditions for a graph not being a PCG have also been developed [8, 9, 13, 10]. However, there is still few known method for generating PCGs and PCTs with complicated structures. In this paper, we will give more necessary and sufficient conditions for PCG. The first one is related to cut-vertices, where we show that a graph is a PCG if and only if each biconnected component of it is a PCG. The second one is about a pair of vertices with the same neighbors, called “twins.” We will show some conditions under which we can add a copy $v^{\prime}$ of a vertex $v$ into a PCG so that $v^{\prime}$ and $v$ form twins to get another PCG. One of our results answers an open problem on “true twins” [2]. These properties provide simple reductions rules, by which we can reduce some graph into a smaller graph to check if it is a PCG and find more subclasses of PCGs as well as non-PCGs with an arbitrary large size. For examples, our results imply that complete $k$-partite graphs, cacti, and some other graphs are subclasses of PCG. ## 2 Preliminary Let a graph $G=(V,E)$ stand for an unweighted simple undirected graph with sets $V$ and $E$ of vertices and edges, respectively. An edge with end-vertices $u$ and $v$ is denoted by $uv$. For a graph $G$, let $V(G)$ and $E(G)$ denote the sets of vertices and edges in $G$, respectively, and let $N_{G}(v)$ be the set of neighbors of a vertex $v$ in $G$ and let $N_{G}[v]=N_{G}(v)\cup\{v\}$. Two vertices $u$ and $v$ in a graph $G$ are _true twins_ (resp., _false twins_) if $N_{G}[v]=N_{G}[u]$ (resp., $N_{G}(v)=N_{G}(u)$). For a subset $X\subseteq V(G)$, let $G-X$ denote the graph obtained from $G$ by removing vertices in $X$ together with all edges incident to vertices in $X$, where $G-\{v\}$ for a vertex $v$ may be written as $G-v$. Let $G[X]$ denote the graph induced by a subset $X\subseteq V(G)$, i.e., $G[X]=G-(V(G)\setminus X)$. A vertex is called a _cut-vertex_ if deleting it increases the number of connected component of the graph. A graph is _biconnected_ if it has no cut-vertex. Note that a graph consisting of a single edge is biconnected. A _biconnected component_ in a graph is a maximal biconnected subgraph. A _cactus_ is a connected graph in which any two simple cycles have at most one vertex in common. Note that each biconnected component of a cactus is either a cycle or an edge. A graph is called a _complete $k$-partite graph_ if the vertex set can be partitioned into $k$ disjoint non-empty vertex subsets such that no two vertices in the same subset are adjacent whereas any two vertices from different subsets are adjacent. A complete $k$-partite graph with $k$ subsets $V_{1},V_{2},\ldots,V_{k}$ with $\|V_{i}\|=s_{i}$ is denoted by $\mathrm{K}_{s_{1},s_{2},\dots,s_{k}}$. Let $T$ be a tree. A vertex in a tree is called an _inner vertex_ if it is incident to at least two edges and is called a _leaf_ otherwise. Let $L(T)$ denote the set of leaves in the tree $T$. An edge incident to a leaf in $T$ is called a _leaf edge_ of $T$. For a subset $X\subseteq V(T)$ of vertices, let $T\langle X\rangle$ denote a minimal subtree of $T$ subject to the condition that any two vertices $u,v\in X$ remain connected in $T\langle X\rangle$. Note that for a given subset $X$, the minimal subtree is unique. An edge-weighted graph $(G,w)$ is defined to be a pair of a graph $G$ and a non-negative weight function $w:E(G)\to\Re_{+}$. For a subgraph $G^{\prime}$ of $G$, let $w(G^{\prime})$ denote the sum $\sum_{e\in E(G^{\prime})}w(e)$ of edge weights in $G^{\prime}$. Let $(T,w)$ be an edge-weighted tree. For two vertices $u,v\in V(T)$, the _distance_$\mathrm{d}_{T,w}(u,v)$ between them is defined to be $w(T\langle\{u,v\}\rangle)$, i.e., the sum of weights of edges in the path between $u$ and $v$ in $T$. For a tuple $(T,w,d_{\min},d_{\max})$ of an edge-weighted tree $(T,w)$ and two non-negative reals $d_{\min}$ and $d_{\max}$, define $G(T,w,d_{\min},d_{\max})$ to be the simple graph $(L(T),E)$ such that, for any two distinct vertices $u,v\in L(T)$, $uv\in E$ if and only if $d_{\min}\leq\mathrm{d}_{T,w}(u,v)\leq d_{\max}$. We define $E$ to be an empty set if $\|V(T)\|=1$. Note that $G(T,w,d_{\min},d_{\max})$ is not necessarily connected. For a subset $X\subseteq V(T)$, let $w_{X}:E(T\langle X\rangle)\to\Re_{+}$ be a function such that $w_{X}(e)=w(e)$, $e\in E(T\langle X\rangle)$, where we regard $w_{X}$ as null if $\|X\|\leq 1$. A graph $G$ is called a _pairwise compatibility graph_ (PCG, for short) if there exists a tuple $(T,w,d_{\min},d_{\max})$ such that $G$ is isomorphic to the graph $G(T,w,d_{\min},d_{\max})$, where we call such a tuple a _pairwise compatibility representation_ (PCR, for short) of $G$, and call a tree $T$ in a PCR of $G$ a _pairwise compatibility tree_ (PCT, for short) of $G$. We call $d_{\min}$ and $d_{\max}$ the _lower and upper bounds_ of a PCR. ## 3 Some Structures on PCR We start to review the following property, which has been frequently used in literature. The correctness of it immediately follows from the definition of PCG. Lemma 1: _Let $(T,w,d_{\min},d_{\max})$ be a PCR of a graph $G$. For any subset $X\subseteq V(G)$, the tuple $(T\langle X\rangle,w_{X},d_{\min},d_{\max})$ is a PCR of the induced graph $G[X]$._ A PCR $(T,w,d_{\min},d_{\max})$ of a PCG is called _non-singular_ if $T$ contains at least three vertices, $0<d_{\min}<d_{\max}$, and $w(e)>0$ holds for all edges $e\in E(T)$. Lemma 2: _Let $G$ be a PCG with at least two vertices. Then $G$ admits a non-singular PCR. Given a PCR of $G$, a non-singular PCR of $G$ can be constructed in linear time._ Proof. Let $G$ be a PCG with $\|V(G)\|\geq 2$ and $(T,w,d_{\min},d_{\max})$ be an arbitrary PCR of $G$. We will construct a non-singular PCR of $G$ by four steps below. First, if there is a non-leaf edge $e$ such that $w(e)=0$, we can shrink it by identifying the two end-vertices of it. The resulting graph is still a tree, a leaf in the original is still a leaf in this tree, and the distance between any two vertices in the tree remains unchanged. So the new tree is still a PCT of the graph $G$. Now we assume that the edge weight of any non-leaf edge in the tree is positive. By assumption of $\|V(G)\|\geq 2$, it holds that $\|V(T)\|\geq\|L(T)\|=\|V(G)\|\geq 2$. Next if $\|V(T)\|=2$, then we subdivide the unique edge $uv$ in $T$ with a new inner vertex $v^{}$ so that $w^{\prime}(uv^{})+w^{\prime}(v^{}v)=w(uv)$ in the new tree $T^{\prime}$ obtained by subdividing the edge $uv$. It is easy to see that the new tuple $(T^{\prime},w^{\prime},d_{\min},d_{\max})$ is still a PCR of $G$, and $\|V(T^{\prime})\|\geq 3$. In the following we assume that a PCT has at least three vertices. In a PCR $(T,w,d_{\min},d_{\max})$ with $\|V(T)\|\geq 3$, each path between two leaves contains exactly two leaf edges. As for the third step, if $w(e)=0$ for some leaf edge $e\in E(T)$ or $d_{\min}=0$, then we can change all leaf edge weights and $d_{\min}$ positive if necessary, by increasing the weight of each leaf edge by a positive real $\delta>0$ and increasing each of $d_{\min}$ and $d_{\max}$ by $2\delta$. The resulting tuple is a PCR of the same graph $G$. Now all of edge weights, $d_{\min}$ and $d_{\max}$ are positive. Finally, if the lower and upper bounds are same, i.e., $d_{\min}=d_{\max}$, then we augment the upper bound $d_{\max}$ to $d^{\prime}_{\max}:=d_{\max}+\varepsilon$ by choosing a sufficiently small positive real $\varepsilon$ so that every two leaves $u$ and $v$ in $T$ such that $\mathrm{d}_{T,w}(u,v)>d_{\max}$ still satisfies $\mathrm{d}_{T,w}(u,v)>d_{\max}+\varepsilon~{}(=d^{\prime}_{\max})$. Obviously the resulting tuple with $d^{\prime}_{\max}$ is a PCR of the same graph $G$ and satisfies $d_{\min}\neq d^{\prime}_{\max}$. After executing the above four steps, we can get a non-singular PCR of the graph $G$. Furthermore, all the four steps can be done in linear time. A PCR $(T,w,d_{\min},d_{\max})$ of a PCG is called _normalized_ if $0<d_{\min}<1$, $d_{\max}=1$, $w(e)>0$ holds for all edges $e\in E(T)$, and $w(e)>1/4$ holds for all leaf edges $e$ in $T$. We have the following lemmas. Lemma 3: _Let $G$ be a PCG with at least two vertices. Then there is a positive constant $c_{G}$ with $1/2<c_{G}<1$ such that for any real $\alpha$ with $c_{G}<\alpha<1$, $G$ admits a normalized PCR with $(d_{min},d_{max})=(\alpha,1)$. Given a PCR of $G$, such a normalized PCR of $G$ can be constructed in linear time._ Proof.* By Lemma 2, we know that a non-singular PCR $(T,w,d_{\min},d_{\max})$ of $G$ can be constructed in linear time. For $c_{G}=\frac{d_{\min}+d_{\max}}{d_{\max}+d_{\max}}$, where $1/2<c_{G}<1$, let $\alpha$ be any real such that $c_{G}<\alpha<1$. To prove the lemma, it suffices to show that a normalized PCR $(T,w^{\prime},\alpha,1)$ can be constructed in linear time. Let $\delta$ be the positive real such that $\frac{d_{\min}+\delta}{d_{\max}+\delta}=\alpha$, where $\delta>d_{\max}$ holds. We increase the weight of each leaf edge in $T$ by $\delta/2$, which increases the weight of each path between two leaves in $T$ by $\delta$. We scale the weight in the tuple so that the lower and upper bounds become $\alpha$ and 1; i.e., we divide by $d_{\max}+\delta$ the weight of each edge in $T$ and each of $d_{\min}+\delta$ and $d_{\max}+\delta$. This results in a tuple $(T,w^{\prime},\alpha,1)$ of $G$ such that $w^{\prime}(e)\geq(\delta/2)/(d_{\max}+\delta)>1/4$ for each leaf edge $e$ in $T$. Most of our arguments are based on normalized PCR, since it will be helpful for us to simplify some proofs. ## 4 Properties on Induced Subgraphs of PCGs In this section, we derive some sufficient conditions for induced subgraphs of a PCG to remain PCGs, and show how to reduce a PCG to smaller PCGs or construct a larger PCG (resp., non-PCG) from a given PCG (resp., non-PCG). For this, we first review the case when an induced subgraph of a PCG $G$ is a connected component of the graph. Components. It is known that a graph is a PCG if and only if each connected component of it is a PCG. The only if part trivially follows from Lemma 1. The if part is also easy to see: choose an inner vertex from the PCT of a PCR of each connected component of $G$, where we assume that $d_{max}=1$ for all PCRs, and join the inner vertices to a new vertex with an edge weighted by a positive real $>1$ to get a single tree whose leaf set is $V(G)$. We easily see that the resulting tree is a PCT for a PCR to $G$, showing that $G$ is a PCG. It would be natural to consider similar properties on 2-edge-connected components (resp., biconnected components) of graphs with bridges (resp., cut-vertices). In fact, we show that the above property also holds for biconnected components. Lemma 4: _Let a graph $G$ consist of biconnected components $B_{i}$, $i=1,2,\ldots,p$. Then $G$ is a PCG if and only if each biconnected component $B_{i}$ of $G$ is a PCG._ Proof. The only if part trivially follows from Lemma 1. To show the if part, it suffices to consider the case where $G$ consists of two PCG graphs $G_{1}$ and $G_{2}$ such that $\|V(G_{1})\cap V(G_{2})\|=1$. Let $v^{}\in V(G_{1})\cap V(G_{2})$. By Lemma 3, we see that, for a real $\alpha>0$, each PCG $G_{i}$$(i=1,2)$ admits a normalized PCR $(T_{i},w_{i},d_{\min}=\alpha,d_{\max}=1)$, as illustrated in Figure 2(a). Since they are normalized, it holds that $w_{i}(e)>1/4$ for each leaf edge $e$ in $T_{1}$ and $T_{2}$. Now we join the two PCRs by replacing the leaf edge $u_{i}v^{}$ in $T_{i}$$(i=1,2)$ with a new inner vertex $v^{\prime}$ and three edges $u_{1}v^{\prime}$, $u_{2}v^{\prime}$ and $v^{\prime}v^{}$ setting their weights by $w(u_{1}v^{}):=w_{1}(u_{1}v^{})$, $w(u_{2}v^{}):=w_{2}(u_{2}v^{})$ and $w(v^{\prime}v^{}):=0$, respectively. See Figure 2(b) for an illustration of the operation. Let $(T,w)$ denote the resulting edge-weighted tree, and let $G^{\prime}$ be the graph $G(T,w,\alpha,1)$. We will show that $G^{\prime}$ is isomorphic to the graph $G$. Since $w_{i}(e)>1/4$ for each leaf edge $e$ in $T_{i}$ with $i=1$ and $2$, we see that $\mathrm{d}_{T,w}(u,v)>4\cdot(1/4)=1=d_{\max}$ for any pair of vertices $u\in L(T_{1})-\{v^{}\}$ and $v\in L(T_{2})-\{v^{}\}$. This implies that $uv\not\in E(G^{\prime})$. Obviously for each $i=1,2$ and any pair $\{u,v\}\subseteq V(T_{i})$, it holds that $uv\in E(G^{\prime})$ if and only if $uv\in E(G_{i})$. Therefore $G^{\prime}$ is isomorphic to $G$, and $G$ is a PCG. <figure><img src="content_image/1804.02887/x2.png"><figcaption>Figure 2: (a) A normalized PCR (Ti,wi,dmin=α,dmax=1) for each i=1,2, (b) Theweighted tree (T,w) obtained from (Ti,wi), i=1,2 by joining edges u1v∗ andu2v∗ with a new inner vertex v′, where w(u1v′)=w1(u1v∗), w(u2v′)=w2(u2v∗) andw(v′v∗)=0</figcaption></figure> Lemma 4 is a powerful tool to construct PCGs. We can use it to ‘join’ small PCGs into a large PCG to find new subclasses of PCGs. An edge or a single cycle has been shown to be a PCG [14], and a cactus is a graph with each biconnected component being a cycle or an edge. By simply applying Lemma 4, we see the next. Lemma 5: _Every cactus is a PCG._ A special case of cacti (where each biconnected component is a cycle) was shown to be a subclass of PCG [14]. However, by using Lemma 4, we can greatly simplify the proofs [14]. Furthermore, Lemma 4 can be used to construct PCGs of more complicated structures. Twins. Since twins have similar structures, we are interested to know wether PCG remains close under the operation of adding a twin of a vertex. This problem has been considered by Calamoneri et al. [2]. They found that this property holds for false twins and raised the case for true twins as an interesting open problem. We will answer their question by exploring the property of true twins. For false twins, the following lemma has been proven [2]. We show that this can be proven by using normalized PCR. Lemma 6: _Let $G$ be a graph with false twins $v_{1}$ and $v_{2}$. Then $G$ is a PCG if and only if $G-v_{1}$ is a PCG._ Proof. The only if part trivially follows from Lemma 1. We show the if part assuming that $G^{\prime}=G-v_{1}$ is a PCG. By Lemma 3, we know that there is a normalized PCR $(T^{\prime},w^{\prime},\alpha>0,1)$ of $G^{\prime}=G-v_{1}$. We replace the leaf edge $v^{\prime}v_{2}$ in $T^{\prime}$ with a new leaf $v_{1}$ and a new inner vertex $v^{\prime\prime}$ and three edges $v^{\prime}v^{\prime\prime}$, $v^{\prime\prime}v_{2}$ and $v^{\prime\prime}v_{1}$, setting their weights by $w(v^{\prime}v_{2}):=w^{\prime}(v^{\prime}v_{2})$ and $w(v^{\prime\prime}v_{2}):=w(v^{\prime\prime}v_{1}):=0$. Let $(T,w)$ denote the resulting edge-weighted tree. Since $\mathrm{d}_{T,w}(v_{1},v_{2})=0<\alpha$, $v_{1}v_{2}$ is not an edge in the graph $G(T,w,\alpha,1)$. For any other leaf $v\in L(T)$, it holds $\mathrm{d}_{T,w}(v,v_{1})=\mathrm{d}_{T^{\prime},w^{\prime}}(v,v_{2})$; and for any leaves $u,v\in L(T)-\{v_{1},v_{2}\}$, it holds $\mathrm{d}_{T,w}(u,v)=\mathrm{d}_{T^{\prime},w^{\prime}}(u,v)$. Therefore $(T,w,\alpha,1)$ is a PCR of $G$. Lemma 6 can also be used to construct PCGs. Based on Lemma 4, we can construct large PCGs having cut-vertices. By using Lemma 6, we can increase the connectivity of PCGs. For example, for each cut-vertex in a PCG, we can add a false twin of it to the graph to get another PCG. Lemma 6 also implies the following result. Lemma 7: _Every complete $k$-partite graph is a PCG._ Note that for a complete $k$-partite graph, if we iteratively delete a vertex in a pair of false twins as long as false twins exist, finally we will get a clique of $k$ vertices. It is trivial that a clique is a PCG. By Lemma 6, we know that any complete $k$-partite graph is a PCG. In fact, complete $k$-partite graphs contain many interesting graphs. For examples, $\mathrm{K}_{1,2,2}$ is a $5$-wheel, $\mathrm{K}_{2,2,2}$ is an octahedron, $\mathrm{K}_{1,2,4}$ is a $(4,3)$-fan, $\mathrm{K}_{2,2,5}$ is a $(4,5)$-cone, $\mathrm{K}_{4,4,4}$ is a circulate graph $\mathrm{Ci}_{12}(1,2,4,5)$, and so on. Some of them have been shown to be PCGs by using different techniques in the literature. Next, we consider true twins. In fact, the statement in Lemma 6 for true twins is no longer correct because there is an example of a non-PCG $G$ such that deleting a vertex in true twins results in a PCG. The graph $G$ in Figure 3(a) has only seven vertices. This is a PCG since it has been proved that any graph with at most seven vertices is a PCG [1]. The graph $G^{\prime}$ in Figure 3(b) is obtained from the graph $G$ by a copy $v^{\prime}$ of vertex $v$ so that $v$ and $v^{\prime}$ form true twins in $G^{\prime}$. The graph $G^{\prime}$ has been shown to be a non-PCG [8]. <figure><img src="content_image/1804.02887/x3.png"><figcaption>Figure 3: (a) A graph G with seven vertices, and (b) A graph G′ obtained fromG by adding a vertex v′ so that v and v′ are true twins in G′, (c) A PCT Tobtained from T′ with a new leaf edge c0v1</figcaption></figure> We show that a non-PCG remains to be a non-PCG after removing one of three true twins. Lemma 8: _Let $G$ be a graph with three true twins $v_{1},v_{2}$ and $v_{3}$, i.e., $N_{G}[v_{1}]=N_{G}[v_{2}]=N_{G}[v_{3}]$. Then $G$ is a PCG if and only if $G-v_{1}$ is a PCG._ Proof. The only if part trivially follows from Lemma 1. We show the if part assuming that $G^{\prime}=G-v_{1}$ is a PCG. Let $(T^{\prime},w^{\prime},d^{\prime}_{\min},d^{\prime}_{\max})$ be a PCR of $G^{\prime}$, where $\|V(T^{\prime})\|\geq\|L(G)\|\geq 2$. We will construct a PCR $(T,w,d^{\prime}_{\min},d^{\prime}_{\max})$ of $G$. Let $c_{0}$ be the middle point of the path between $v_{2}$ and $v_{3}$ in $T^{\prime}$, i.e., $c_{0}$ is an inner vertex or an interior point on an edge such that $d_{T^{\prime},w^{\prime}}(v_{2},c_{0})=d_{T^{\prime},w^{\prime}}(c_{0},v_{3})$. We add $v_{1}$ to $T^{\prime}$ as a new leaf creating a new edge between $v_{1}$ and $c_{0}$ in $T^{\prime}$ to construct a tree $T$ with $L(T)=V(G)$. We set the edge weight $w(v_{1}c_{0}):={1\over 2}d_{T^{\prime},w^{\prime}}(v_{2},v_{3})$. If $c_{0}$ is an interior point on an edge $u_{1}u_{2}$ in $T^{\prime}$, then we subdivide $u_{1}u_{2}$ into two edges $u_{1}c_{0}$ and $c_{0}u_{2}$ setting their weights so that $w(u_{1}c_{0})+w(c_{0}u_{2})=w^{\prime}(u_{1}u_{2})$ and $c_{0}$ is still the middle point of the path between $v_{2}$ and $v_{3}$ in $T$. For all other edges $e$ in $T^{\prime}$, we set $w(e):=w^{\prime}(e)$. Note that $d_{T,w}(v_{1},c_{0})=d_{T,w}(v_{2},c_{0})=d_{T,w}(v_{3},c_{0})={1\over 2}d_{T, w}(v_{2},v_{3})={1\over 2}d_{T^{\prime},w^{\prime}}(v_{2},v_{3})$. To prove that $(T,w,d^{\prime}_{\min},d^{\prime}_{\max})$ is a PCR of $G$, it suffices to prove that for each vertex $u\in V(G)\setminus\{v_{1}\}$, $d_{T,w}(v_{1},u)$ is equal to $d_{T,w}(v_{i},u)$ for $i=2$ or $3$, which implies that $v_{1}u\in E(G)$ if and only if $v_{i}u\in E(G^{\prime})$. Recall that $v_{2}u\in E(G^{\prime})$ if and only if $v_{3}u\in E(G^{\prime})$ by assumption of $N_{G}[v_{2}]=N_{G}[v_{3}]$. Let $u\in V(G)\setminus\{v_{1}\}$, where we assume without loss of generality that $d_{T,w}(v_{2},u)\leq d_{T,w}(v_{3},u)$, which means that the path between $u$ and $v_{3}$ passes through $c_{0}$ in $T^{\prime}$, as illustrated in Figure 2(c). Hence $d_{T,w}(v_{3},u)=d_{T,w}(v_{1},u)$ holds, as required. Lemma 8 implies that a PCG with true twins $u_{1}$ and $u_{2}$ can be augment to a larger PCG with any number of new vertices $u_{2},\ldots,u_{k}$ so that every two vertices $u_{i}$ and $u_{j}$, $1\leq i<j\leq k$ form true twins. ## 5 Conclusions In this paper, we have introduced some reduction rules on PCGs. By using these rules, we can find more subclasses of PCG and simplify some arguments in previous papers. Also the reduction rules can be used to find a class of non-PCGs by constructing lager non-PCGs from a given non-PCG in a similar way. All graphs with at most seven vertices are known to be PCGs, and a non-PCG with eight vertices has been found. To find all non-PCGs with $n=8$ vertices, the reduction rules can be used to eliminate graphs with false twins or cut-vertices from the class of simple graphs with $n=8$ vertices, because such graphs are reduced to graphs with at most seven vertics which are all PCGs. It is interesting to find more reduction rules on PCG. ## References * [1] T. Calamoneri, D. Frascaria, B. Sinaimeri, All graphs with at most seven vertices are pairwise compatibility graphs, The Computer Journal, 56(7) (2013) 882–886 * [2] T. Calamoneri, E. Montefusco, R. Petreschi, B. Sinaimeri, Exploring pairwise compatibility graphs, Theoret. Comput. Sci., 468 (2013) 23–36 * [3] T. Calamoneri, R. Petreschi, On pairwise compatibility graphs having Dilworth number two, Theoret. Comput. Sci., 524 (2014) 34–40 * [4] T. Calamoneri, R. Petreschi, On pairwise compatibility graphs having Dilworth number $k$, Theoret. Comput. Sci., 547 (2014) 82–89 * [5] T. Calamoneri, R. Petreschi, B. Sinaimeri, On relaxing the constraints in pairwise compatibility graphs, in WALCOM: Algorithms and Computation, Md. S. Rahman and S.-i. Nakano, eds., LNCS 7157, Springer, Berlin, Heidelberg, 2012, 124–135 * [6] T. Calamoneri, R. Petreschi, B. Sinaimeri, On the pairwise compatibility property of some superclasses of threshold graphs, Discrete Math. Algorithms Appl., 5(2) (2013), article 360002 * [7] T. Calamoneri, B. Sinaimeri, Pairwise compatibility graphs: A survey, SIAM Review 58(3)(2016) 445–460 * [8] S. Durocher, D. Mondal, Md. S. Rahman, On graphs that are not PCGs, Theoret. Comput. Sci., 571 (2015) 78–87 * [9] Md. I. Hossain, S. A. Salma, Md. S. Rahman, D. Mondal: A necessary condition and a sufficient condition for pairwise compatibility graphs, J. Graph Algorithms Appl. 21(3) (2017) 341–352 * [10] S. Mehnaz, Md. S. Rahman, Pairwise compatibility graphs revisited, in Proceedings of the 2013 International Conference on Informatics, Electronics Vision (ICIEV), 2013, 1–6 * [11] P. E. Kearney, J. I. Munro, D. Phillips, Efficient generation of uniform samples from phylogenetic trees, in: Algorithms in Bioinformatics, G. Benson and R. Page, eds., LNCS 2812, Springer, Berlin, Heidelberg, 2003, 177–189 * [12] S. A. Salma, Md. S. Rahman, Md. I. Hossain, Triangle-free outerplanar 3-graphs are pairwise compatibility graphs, J. Graph Algorithms Appl., 17 (2013) 81–102 * [13] N. M. Yanhaona, M. S. Bayzid, M. S. Rahman, Discovering pairwise compatibility graphs. J. Appl. Math. Comput., 30 (2010) 479–503 * [14] M. N. Yanhaona, K. S. M. T. Hossain, Md. S. Rahman, Pairwise compatibility graphs, J. Appl. Math. Comput., 30 (2009) 479–503
1708.02076	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 29543, "num_imgs": 4, "llama3_tokens_count": 8911 }	[ "content_image/1708.02076/x1.png", "content_image/1708.02076/x2.png", "content_image/1708.02076/x3.png", "content_image/1708.02076/x4.png" ]	# Importance of initial and final state effects for azimuthal correlations in p+Pb collisions Moritz Greif greif@th.physik.uni-frankfurt.de Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Carsten Greiner Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Björn Schenke Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA Sören Schlichting Department of Physics, University of Washington, Seattle, WA 98195-1560, USA Zhe Xu Department of Physics, Tsinghua University and Collaborative Innovation Center of Quantum Matter, Beijing 100084, China February 23, 2024 ###### Abstract We investigate the relative importance of initial and final state effects on azimuthal correlations of gluons in low and high multiplicity p+Pb collisions. To achieve this, we couple Yang-Mills dynamics of pre-equilibrium gluon fields (IP-GLASMA) to a perturbative QCD based parton cascade for the final state evolution (BAMPS) on an event-by-event basis. We find that signatures of both the initial state correlations and final state interactions are seen in azimuthal correlation observables, such as $v_{2}\left\{2PC\right\}(p_{T})$, their strength depending on the event multiplicity and transverse momentum. Initial state correlations dominate $v_{2}\left\{2PC\right\}(p_{T})$ in low multiplicity events for transverse momenta $p_{T}>2~{}{\rm GeV}$. While final state interactions are dominant in high multiplicity events, initial state correlations affect $v_{2}\left\{2PC\right\}(p_{T})$ for $p_{T}>2~{}{\rm GeV}$ as well as the pT integrated $v_{2}\left\{2PC\right\}$. Introduction.The measured azimuthal momentum anisotropies of produced particles in heavy ion collisions are well described in the framework of event-by-event hydrodynamics. In this picture a fluctuating initial geometry, dominated by fluctuating nucleon positions in the incoming nuclei, is converted into anisotropic momentum space distributions by the pressure driven final state evolution. Hydrodynamic simulations agree well with a wide range of experimental observables from the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory and the Large Hadron Collider (LHC) at CERN Heinz and Snellings (2013); Gale _et al._ (2013); Derradi de Souza _et al._ (2016); Song _et al._ (2017). Measurements in smaller collision systems such as p+p and p+A Dusling _et al._ (2016), in particular those of anisotropies in multi-particle correlation functions, have shown very similar features as those in heavy ion collisions. While calculations within the hydrodynamic framework have been quite successful in describing observables in these small collision systems, alternative explanations relying entirely on intrinsic momentum correlations of the produced particles can also reproduce many features of the experimental data. This includes two and more particle azimuthal correlations and their $p_{T}$ dependence Dusling _et al._ (2016); Schlichting and Tribedy (2016); Dusling _et al._ (7) and mass splitting of identified particle $v_{n}$Schenke _et al._ (2016). Apart from the existence of alternative explanations, the applicability of hydrodynamics becomes increasingly doubtful as the system size decreases and gradients increase. Some recent studies argue that hydrodynamics should be applicable in systems of sizes down to $\sim 0.15\,{\rm fm}$Romatschke (2017), but off-equilibrium corrections to particle distribution functions for momenta $p_{T}\gtrsim 0.5\,{\rm GeV}$ can be significant Mäntysaari _et al._ (2017), which limits at least the quantitative reliability of the framework. So far all calculations of multi-particle correlations in small collision systems have studied either only intrinsic momentum correlations or purely final state driven effects. In this letter we present the first study where both effects are combined into a single framework to assess their relative importance. We compute initial state gluon Wigner-distributions from the Impact Parameter dependent Glasma model (IP-Glasma) Schenke _et al._ (11, 12) and via sampling of individual gluons feed them into the partonic transport simulation ’Boltzmann approach to multiparton scatterings’ (BAMPS) Xu and Greiner (2005). The initial gluon distributions Krasnitz and Venugopalan (2000, 2001) from the IP-Glasma model are anisotropic in momentum space Lappi _et al._ (2010); Schenke _et al._ (2015, 2016), thus contain the intrinsic momentum space correlations of the color glass condensate (CGC) picture Gelis _et al._ (2010); Gelis and Schenke (2016). Final state interactions mediated by perturbative quantum chromo dynamic (pQCD) cross sections are then simulated microscopically in BAMPS. We analyze the time evolution of the momentum space anisotropy of the partonic plasma by simulating events in two different multiplicity classes to understand how final state interactions modify initial state momentum correlations and whether signals of the latter can survive to affect final observables. Initial state & Phase-space distribution.Based on the IP-Glasma model, including event-by-event fluctuations of the proton’s geometrical structure Mäntysaari and Schenke (2016), we calculate the solution to the classical Yang-Mills equations of motion up to $\tau_{0}=0.2\,{\rm fm}/c$ following the standard procedures described in Schenke _et al._ (11, 12). Event-by-event we extract the Wigner distribution $\frac{\mathrm{d}N_{g}}{\mathrm{d}y\mathrm{d}^{2}\mathbf{x}_{T}\mathrm{d}^{2}{ \mathbf{p}_{T}}}$ in hyperbolic phase-space coordinates $x^{\mu}=(\tau\cosh\eta_{s},\mathbf{x}_{T},\tau\sinh\eta_{s}),\;$$p^{\mu}=(\|\mathbf{p}_{T}\|\cosh y,{\mathbf{p}_{T}},\|\mathbf{p}_{T}\|\sinh y),\;$ by evaluating equal time correlation functions in Coulomb gauge and projecting them onto the transversely polarized mode functions $\xi^{(\lambda)}_{{\mathbf{p}_{T}}}(\tau)$ of the free theory Berges _et al._ (2014), according to \[\frac{\mathrm{d}N_{g}}{\mathrm{d}y\mathrm{d}^{2}\mathbf{x}_{T} \mathrm{d}^{2}{\mathbf{p}_{T}}} =\frac{1}{(2\pi)^{2}}\sum_{\lambda=1,2}\sum_{a=1}^{N_{c}^{2}-1} \tau^{2}~{}g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\] \[\times\int\mathrm{d}^{2}\mathbf{s} \Big{(}\xi^{(\lambda)}_{{\mathbf{p}_{T}},\mu}(\tau)i \overleftrightarrow{\partial_{\tau}}A_{\mu^{\prime}}^{a}(\mathbf{x}_{T}+ \mathbf{s}/2)\Big{)}\] \[\Big{(}A_{\nu^{\prime}}^{a}(\mathbf{x}_{T}-\mathbf{s}/2)i \overleftrightarrow{\partial_{\tau}}\xi^{(\lambda)}_{{\mathbf{p}_{T}},\nu}( \tau)\Big{)}e^{-i{\mathbf{p}_{T}}\cdot\mathbf{s}}\;.\] (1) Even though the position and momentum dependent Wigner distribution includes all relevant information about the initial state coordinate space eccentricity as well as the initial state momentum space anisotropies, it suffers from the deficiency that it is not necessarily positive semi-definite. To warrant a probabilistic interpretation of a quasi-particle distribution entering the subsequent Boltzmann transport simulation, it is necessary to perform a smearing of the Wigner distribution over phase space volumes $\sigma_{x}\sigma_{p}\geq\hbar/2$. Accounting for the boost-invariant nature of the classical Yang-Mills fields the single particle distribution function $f^{g}_{0}$, which will enter the subsequent parton cascade, is obtained by performing the Gaussian smearing \[f^{g}_{0}(\mathbf{x}_{T},\eta_{s},\mathbf{p}_{\perp},y)=\frac{(2 \pi)^{3}}{2(N_{c}^{2}-1)}\frac{\delta(y-\eta_{s})}{\|\mathbf{p}_{T}\|\tau}\] (2) \[\qquad\times\int\frac{\mathrm{d}^{2}\mathbf{x}_{T}^{\prime} \mathrm{d}^{2}{\mathbf{p}_{T}}^{\prime}}{(2\pi)^{2}}e^{-\frac{(\mathbf{x}_{T}- \mathbf{x}_{T}^{\prime})^{2}}{2\sigma_{x}^{2}}}e^{-\frac{({\mathbf{p}_{T}}-{ \mathbf{p}_{T}}^{\prime})^{2}}{2\sigma_{p}^{2}}}\frac{\text{d}N_{g}}{\text{d}y \text{d}^{2}\mathbf{x}_{T}^{\prime}\text{d}^{2}{\mathbf{p}_{T}}^{\prime}}\;,\] with $\sigma_{x}=0.197~{}\rm{fm}$ and $\sigma_{p}=1~{}\rm{GeV}$ chosen to achieve a reasonable compromise between spatial and momentum resolution. Final state interactions.Even though the classical Yang-Mills evolution includes re-scattering effects at early times, the semi-classical description of the dynamics becomes inapplicable after a relatively short time when quantum effects become important and the subsequent dynamics is more appropriately described in terms of weakly interacting quasi-particles Baier _et al._ (2001); Berges _et al._ (2014); Kurkela and Zhu (2015). We simulate the dynamics within $0.2~{}\rm{fm/c}<\tau<2.0~{}\rm{fm/c}$, with a 3+1-dimensional Boltzmann approach to multi-parton scatterings (BAMPS), which, starting from the initial phase-space density of gluons in Eq. (2), solves the relativistic Boltzmann equation \[p^{\mu}\frac{\partial}{\partial x^{\mu}}f^{i}(x,p)=\sum\limits_{j=g,q, \overline{q}}C_{ij}(x,p),\] (3) for the phase-space distribution function $f^{i}(x,p)$ of massless on-shell quarks, anti-quarks and gluons by Monte-Carlo techniques Xu and Greiner (2005, 2007); Xu _et al._ (2015)¹. The collision integrals $C_{ij}$ include $2\leftrightarrow 2$ and $2\leftrightarrow 3$ interactions, based on perturbative QCD matrix elements (using a fixed strong coupling constant $\alpha_{s}=0.3$) where internal propagators are regulated by a dynamically computed screening mass $m_{D}^{2}\sim\alpha_{s}\int d^{3}pf^{i}(x,p)/p$ (see, e.g., Refs. Fochler _et al._ (2013); Uphoff _et al._ (2015)). Inelastic $2\leftrightarrow 3$ interactions are simulated based on the improved Gunion-Bertsch matrix elements Fochler _et al._ (2013), and the Landau-Pomeranchuk-Migdal (LPM) effect is treated effectively, based on a dynamically determined mean free path Uphoff _et al._ (2015). [FOOTNOTE:1][ENDFOOTNOTE] Since in practice the Monte-Carlo implementation is based on individual particles, propagating along straight lines between scattering events, one needs to supply a list of particle positions $x^{\mu}_{\rm Init}$ and momenta $p^{\mu}_{\rm Init}$ as initial condition for BAMPS. For every event we sample a collection of individual gluons from the momentum distribution $f^{g}_{0}(\mathbf{x}_{T},\eta_{s},{\mathbf{p}_{T}},y)$ of the IP-Glasma model, such that the overall number of gluons is given by the integral of the distribution. Since according to Eq. (2) the initial momentum rapidity $y$ is equal to the coordinate space rapidity $\eta_{s}$, which we sample uniformly between $-2<\eta_{s}<2$ from the boost invariant distribution, the initial position and momentum vectors of each particle are given by $x^{\mu}_{\rm Init}=\left(\tau_{0}\cosh(\eta_{s}),\mathbf{x}_{T},\tau_{0}\sinh( \eta_{s})\right)$ and $p^{\mu}_{\rm Init}=\left(\|\mathbf{p}_{T}\|\cosh(\eta_{s}),{\mathbf{p}_{T}},\| \mathbf{p}_{T}\|\sinh(\eta_{s})\right)$. We have checked explicitly, that the energy density $(T^{\tau\tau})$ and flow coefficients $(v_{2})$ extracted from the sampled particle ensemble agree well with the corresponding quantities extracted directly from the IP-Glasma distribution. Even though the IP-Glasma initial condition is boost invariant, the BAMPS calculation is performed in 3+1 dimensional Minkowski space. We will therefore extract all observables at $\|y\|<0.5$ for different lab times $t$, where $y=\log[(E+p_{z})/(E-p_{z})]/2$, noting that at midrapidity $\|y\|\approx\|\eta_{s}\|\approx 0$ such that the lab time $t\approx\tau$. Evolution of azimuthal anisotropies.We investigate the evolution of the azimuthal momentum space anisotropy characterized by the Fourier harmonics $v_{n}\{2PC\}$ of the two-particle correlation function. We follow the experimental analysis Chatrchyan _et al._ (2013) in decomposing the (normalized) two-particle correlation function for $N_{\text{trig}}$ trigger particles in a momentum range given by $p_{T}^{\rm ref}$ and $N_{\text{assoc}}$ particles in a momentum bin around $p_{T}$, in Fourier harmonics w.r.t. the relative azimuthal angle $\Delta\varphi_{p_{T}}$: \[\frac{2\pi}{N_{\text{trig}}N_{\text{assoc}}}\frac{\text{d}N^{ \text{pair}}}{\text{d}\Delta\varphi_{p_{T}}} (p_{T},p_{T}^{\text{ref}})=\] \[1+\sum\limits_{n} 2V_{n\Delta}(p_{T},p_{T}^{\text{ref}})\cos(n\Delta\varphi_{p_{T} }).\] (4) The two particle $v_{2}\{2PC\}$ is obtained as Chatrchyan _et al._ (2013) \[v_{n}\{2PC\}(p_{T})=\frac{V_{n\Delta}(p_{T},p_{T}^{\text{ref}})} {\sqrt{V_{n\Delta}(p_{T}^{\text{ref}},p_{T}^{\text{ref}})}},\] (5) with the reference momentum range chosen as $0~{}\mathrm{GeV}<p_{T}^{\text{ref}}<8~{}\mathrm{GeV}$ by default². Since in our model the double-inclusive spectrum in each event is given by the product of single inclusive spectra, we follow Lappi (2015); Schenke _et al._ (2015) and directly compute [FOOTNOTE:2][ENDFOOTNOTE] \[V_{n\Delta}(p_{T},p_{T}^{\text{ref}})=\left\langle\text{Re}\frac {b_{n}(p_{T})b_{n}^{}(p_{T}^{\text{ref}})}{b_{0}(p_{T})b_{0}^{}(p_{T}^{\text {ref}})}\right\rangle_{\rm events}\] (6) where in each event $b_{n}(p_{T})=\int\frac{\mathrm{d}\phi_{{p_{T}}}}{2\pi}\frac{\mathrm{d}N_{g}}{ \mathrm{d}^{2}{\mathbf{p}_{T}}}e^{in\phi_{{p_{T}}}}$ is the azimuthal Fourier coefficient of the single-inclusive spectrum. Since our model does not include correlations from back-to-back di-jet pairs, we also note that – contrary to the experimental analysis – no additional subtractions are required to eliminate such correlations. <figure><img src="content_image/1708.02076/x1.png"><figcaption>Figure 1: Gluon v2{2PC}(pT) at mid-rapidity (\|y\|<0.5) for different times inhigh multiplicity (⟨dNg/dy⟩=26, upper panel) and low multiplicity (⟨dNg/dy⟩=6,lower panel) p+Pb collisions.</figcaption></figure> Evolution of azimuthal anisotropy.Including both initial state effects and final state evolution, we analyze the time evolution of the momentum space anisotropy $v_{2}\{2PC\}(p_{T})$ for $\sqrt{s_{\rm pA}}=5.02\,{\rm TeV}$ p+Pb collisions in Fig. 1. We show $v_{2}\{2PC\}(p_{T})$ at different times, $t=0.2\,\text{(initial)},0.4,0.6,1,2\,\mathrm{fm}/c$ for low multiplicity ($0.5<\left(\mathrm{d}N_{g}/\mathrm{d}y\right)/\langle\mathrm{d}N_{g}/\mathrm{d} y\rangle<1$) and high multiplicity ($\left(\mathrm{d}N_{g}/\mathrm{d}y\right)/\langle\mathrm{d}N_{g}/\mathrm{d}y \rangle>2.5$) events. While in both cases momentum correlations lead to a sizeable initial state $v_{2}$Schenke _et al._ (2015), the subsequent dynamics is quite different: In high multiplicity events, we observe a pronounced effect of the final state interactions such that the high initial anisotropy at intermediate momenta $({p_{T}}\sim 2-5~{}\mathrm{GeV})$ is significantly reduced within the first $0.2~{}\mathrm{fm/c}$ evolution in the parton cascade, while at the same time the correlation strength at higher and lower momenta begins to increase. Subsequently, the azimuthal anisotropy increases for all ${p_{T}}$ up to maximally $5\%$. As a result, the pronounced peak at around ${p_{T}}\sim 3~{}\mathrm{GeV}$, present after the IP-Glasma stage, is washed out by the final state interactions. In contrast, for low multiplicity events modifications due to final state effects appear to be less significant, as the final curve $v_{2}(p_{T})$ closely resembles that of the IP-Glasma initial state. Only at low transverse momenta, ${p_{T}}\lesssim 2~{}\mathrm{GeV}$ the azimuthal anisotropy is increased to $2-3~{}\%$. While our results confirm the basic expectation that final state effects gain significance as the density of the medium increases in high-multiplicity events Schlichting (2016); Schlichting and Tribedy (2016), the way this is realized dynamically is in fact very interesting. We find that the average number of interactions in low-multiplicity events ($N_{\rm{scat}}=4.5\pm 1.1$) is indeed almost the same as in high-multiplicity events ($N_{\rm{scat}}=5.6\pm 1.1$). Because of the nature of the QCD cross-sections, most interactions however correspond to small momentum transfers ${\sim}m_{D}$ which itself depends on the density of the medium Arnold _et al._ (2003), such that the average momentum transfer is larger in high-multiplicity events. Hence, the average number of _large angle scatterings_, estimated according to $N_{\rm{scat}}^{\rm{large~{}angle}}=\frac{1}{N_{\rm{particles}}}\sum_{\rm{coll} }\frac{3}{2}\sin^{2}\theta^{\rm{coll}}_{\rm{c.o.m.}}$ where $\theta^{\rm{coll}}_{\rm{c.o.m.}}$ is the scattering angle in the c.o.m. frame of the partonic interaction³, is in fact significantly larger in high-multiplicity events $(N_{\rm{scat}}^{\rm{large~{}angle}}=1\pm 0.18)$ as compared to low-multiplicity events $(N_{\rm{scat}}^{\rm{large~{}angle}}=0.53\pm 0.14)$. [FOOTNOTE:3][ENDFOOTNOTE] Initial state vs. final state effects.In order to further disentangle the effects of initial state momentum correlations and final state response to geometry, we performed an additional set of simulations (henceforth labeled rand. azimuth) where the azimuthal angle of the transverse momentum ${\mathbf{p}_{T}}$ of each gluon is randomized ($0<\varphi_{p_{T}}<2\pi$) before the evolution in the parton cascade. Our results are compactly summarized in Fig. 2, where we compare the azimuthal anisotropy $v_{2}\{2PC\}(p_{T})$ in the different scenarios. By construction no initial state momentum correlations are present in the rand. azimuth case – shown as open gray symbols – and the initial state $v_{2}$ vanishes identically at $t=0.2~{}\mathrm{fm/c}$. However, over the course of the kinetic evolution a $v_{2}(p_{T})$ of $\sim 4\%$ at ${p_{T}}\sim 2~{}\mathrm{GeV}$ in high multiplicity events and $\lesssim 3\%$ at ${p_{T}}~{}\sim 1~{}\mathrm{GeV}$ in low multiplicity events is built up by $t=2.0~{}\mathrm{fm/c}$. Nevertheless, for momenta above ${p_{T}}\sim 2.0~{}\mathrm{GeV}$ (low multiplicity) and ${p_{T}}\sim 4.0~{}\mathrm{GeV}$ (high multiplicity), the purely final state $v_{2}$ in the rand. azimuth scenario remains significantly below the initial state + final state $v_{2}$ of the full calculation, indicating the importance of initial state momentum correlations. <figure><img src="content_image/1708.02076/x2.png"><figcaption>Figure 2: Comparison of initial and final two-particle v2(pT) for high (upperpanel) and low (lower panel) multiplicity √spA=5.02 TeV p+Pb events. Eventsincluding initial state momentum correlations (filled symbols) are compared tothe same events where the initial momenta were randomized in azimuth (rand.azimuth, open symbols).</figcaption></figure> <figure><img src="content_image/1708.02076/x3.png"><figcaption>Figure 3: Comparison of initial and final v2(pT) with respect to theeccentricity plane for high (upper panel) and low (lower panel) multiplicity√spA=5.02 TeV p+Pb events. Events including initial state momentumcorrelations (filled symbols) are compared to the same events where theinitial momenta were randomized in azimuth (rand. azimuth, open symbols).</figcaption></figure> Despite the fact that initial state correlations have a significant impact on $v_{2}\{2PC\}$, we find that the additional $v_{2}\{2PC\}$ built up in the parton cascade can be attributed to the response to the initial geometry. In order to demonstrate this feature more clearly, we have also computed the azimuthal anisotropy $v_{2}\{\text{ecc.~{}plane}\}$ w.r.t to the coordinate eccentricity plane – obtained by replacing the reference momentum vector $b_{n}(p_{T}^{\text{ref}})$ in Eq. (6) with the coordinate eccentricity vector $e_{n}=\int d^{2}\mathbf{x}_{T}~{}T^{\tau\tau}(\mathbf{x}_{T})~{}\|\mathbf{x}_{T }\|^{n}~{}e^{in\phi_{{x_{T}}}}$, where $\phi_{{x_{T}}}$ is the azimuthal angle in space. Our results in Fig. 3 show that the initial anisotropy with respect to the geometric eccentricity plane vanishes, as the initial momentum space anisotropy is uncorrelated with the event geometry Schenke _et al._ (2015). In contrast, during the kinetic evolution a clear correlation with the initial state geometry is built up. The magnitude of this final state generated $v_{2}\{\text{ecc. plane}\}$ depends only weakly on the presence or absence of initial state momentum correlations. While the comparison of the results for $v_{2}\{\text{ecc. plane}\}$ (Fig. 3) with $v_{2}\{2PC\}$ (Fig. 2) indicates that in the rand. azimuth case, the observed $v_{2}\{2PC\}$ can almost entirely be attributed to a geometric response, this is clearly not the case for the more realistic scenario including initial state correlations. Even though the effects of initial state momentum correlations are more apparent in low-multiplicity events, quantitative differences remain also in high-multiplicity events, as can also be observed from Fig. 4, where we study the time-evolution of the ${p_{T}}$ integrated $v_{2}\{2PC\}$. While in the rand. azimuth case, the $v_{2}\{2PC\}$ is built up slowly as a function of time in response to the initial state geometry, a qualitatively different behavior emerges in the more realistic case including initial state correlations. In this case, large angle scatterings at early times begin to destroy initial state momentum correlations leading to an initial decrease of $v_{2}\{2PC\}$ as a function of time. This happens because the directions of the initial state anisotropy and the eccentricity responsible for generating the final state $v_{2}$ are generally uncorrelated. Subsequently, between $t\sim 0.5-1~{}\mathrm{fm/c}$ the response to the initial state geometry sets in, leading again to an increase of $v_{2}\{2PC\}$. Overall, we find that the relative effect of initial state correlations on the final $v_{2}\{2PC\}$ is on the order of $25-50\%$, being larger for low multiplicity events. <figure><img src="content_image/1708.02076/x4.png"><figcaption>Figure 4: Evolution of the pT integrated azimuthal anisotropy v2{2PC} for highand low multiplicity p+Pb events.</figcaption></figure> Conclusions.The observation of long range rapidity correlations with characteristic structures in azimuthal angle in small systems has challenged our understanding of the space-time evolution of high-energy nuclear collisions. Despite the fact that several phenomenological works have attempted to explain various aspects of the experimental data, it remained unclear to what extent observed correlations should be attributed to initial state or final state effects. Based on a weak-coupling picture of the space-time dynamics, we developed a new framework including both initial state momentum correlations and final state interactions. By matching classical Yang-Mills dynamics (IP-GLASMA) to an effective kinetic description (BAMPS) on an event-by-event basis, we showed that the relative importance of initial and final state effects in p+Pb collisions at LHC energies depends on the event multiplicity as well as the transverse momenta under consideration. Especially at low multiplicity, the initial state correlations are very important for integrated as well as differential $v_{2}$, and need to be taken into account in a quantitative theoretical description. We also note that multi-particle correlations of more than two particles can provide additional insight into the nature of the observed correlations. Since final state induced correlations emerge in response to the global event-geometry, these naturally produce $m$-particle correlations (with $m>2$) of similar strength. Conversely, for initial state correlations the existence of pronounced multi-particle correlations is not a priori obvious. However, it was shown recently in an Abelian model that initial state effects can generate similar 4-, 6-, and 8- particle correlations Dusling _et al._ (32). Explicit studies of multi-particle correlations beyond $m=2$ within our framework are numerically very intensive and will be left for future work. Our results indicate that a differential study of azimuthal correlations across a large range of multiplicities and transverse momenta, can provide new insights into properties of the initial state and the early time non-equilibrium dynamics of high-energy collisions. In this context, it would also be interesting to include jet-like correlations at higher momenta, to achieve a fully comprehensive framework of multi-particle correlations. Acknowledgements.M.G. is grateful to Tsinghua University in Beijing for their hospitality and acknowledges the support from the “Helmhotz Graduate School for Heavy Ion research”. This work was supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse. S.S. acknowledges support by the U.S. Department of Energy (DOE) under Grant No. DE-FG02-97ER41014. B.P.S. is supported under DOE Contract No. DE-SC0012704. Z.X. was supported by the National Natural Science Foundation of China under Grants No. 11575092 and No. 11335005, and the Major State Basic Research Development Program in China under Grants No. 2014CB845400 and No. 2015CB856903. 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1409.6987	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23406, "num_imgs": 0, "llama3_tokens_count": 7679 }	[]	# Josephson relation for disordered superfluids Cord A. Müller Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany Institut Non Linéaire de Nice, CNRS and Université Nice–Sophia Antipolis, 06560 Valbonne, France February 24, 2024 ###### Abstract The Josephson sum rule relates the superfluid density to the condensate order parameter, via the infrared residue of the single-particle Green’s function. We establish an effective Josephson relation for disordered condensates valid upon ensemble averaging. This relation has the merit to show explicitly how superfluidity links to the coherent density, i.e., the density of particles with zero momentum. Detailed agreement is reached with perturbation theory for weak disorder. Introduction:—Bose-Einstein condensation (BEC) and superfluidity are certainly linked, and yet this link is difficult to state with precision in situations that involve, e.g., strong interactions, low dimensions, external potentials or temperatures close to critical. Josephson Josephson (1966) has derived a relation between the superfluid mass density $\rho_{\text{s}}$ and the BEC order parameter $\psi$ that provides such a link: \[\frac{m\|\psi\|^{2}}{\rho_{\text{s}}}=-\lim_{{\bm{k}}\to 0}\frac{{\bm{k}}^{2}}{m }G({\bm{k}},0).\] (1) Here, $m$ is the mass of the individual bosonic particle, and $G({\bm{k}},0)$ is the single-particle Green’s function at momentum ${\bm{k}}$ and zero (Matsubara) frequency. Because the Green function can be written as a frequency integral over its imaginary part, the spectral function, this relation is also referred to as the Josephson sum rule Baym (1969); Ueda (2010). Only within mean-field theory, neglecting quantum and thermal fluctuations, one finds that $\rho_{\text{s}}=m\|\psi\|^{2}$ (see eq. (22) below), and there is no need for subtle distinctions between condensate and superfluid. But especially under critical conditions, the Josephson relation is precious because it connects the scaling properties of condensate and superfluid order parameters through the Josephson (hyper-)scaling law Josephson (1966); Holzmann and Baym (2003, 2007). Because of its conceptual and practical importance, the Josephson relation has been re-derived over the years using various methods Baym (1969); Griffin (1984); Holzmann and Baym (2007); Ueda (2010); Dawson _et al._ (2012). These derivations all make use of translational invariance and thus are only valid, strictly speaking, in clean systems. Although the Josephson scaling law has been occasionally applied Crowell _et al._ (1997); Balibar and Caupin (2008) (and questioned Reppy (1992)) in disordered systems, it is not immediately clear how to read the relation (1) in that case. Indeed, the BEC order parameter $\psi({\bm{r}})$ acquires a spatial dependence on each realization of disorder, and also the Green function is no longer diagonal in momentum. Since one can take $\rho_{\text{s}}$ to be a self-averaging quantity in a bulk system of linear size $L$, one may be tempted to think that (1) should hold under the ensemble average, noted by the overbar $\overline{{(\dots)}}$: \[\frac{m\overline{{\|\psi\|^{2}}}}{\rho_{\text{s}}}=-\lim_{{\bm{k}}\to 0}\frac{{ \bm{k}}^{2}}{m}\overline{{G({\bm{k}},0)}}.\] (2) If this were true, the Josephson relation would constrain the ratio of superfluid density to the average condensate density ¹, [FOOTNOTE:1][ENDFOOTNOTE] \[\overline{{\|\psi\|^{2}}}=L^{-d}\int\mathrm{d}{\bm{r}}\|\psi({\bm{r}})\|^{2}=:n_{ \text{c}}.\] (3) The purpose of this paper is to show that this is _not_ the case. In the following, the correct Josephson relation is first stated and briefly discussed, then derived, and finally analytically checked in the simplest accessible regime of low temperatures, weak interactions, and weak disorder. Inhomogeneous Josephson relation:—Our main result is the following Josephson relation for inhomogeneous systems valid upon ensemble averaging: \[\frac{m\left\|\overline{{\psi}}\right\|^{2}}{\rho_{\text{s}}}=-\lim_{{\bm{k}}\to 0 }\frac{{\bm{k}}^{2}}{m}\overline{{G({\bm{k}},0)}}.\] (4) Here, instead of the average condensate density (3), it is the _coherent density_ \[\left\|\overline{{\psi}}\right\|^{2}=\left\|L^{-d}\int\mathrm{d}{\bm{r}}\psi({\bm {r}})\right\|^{2}=:n_{\text{coh}},\] (5) of condensed particles with ${\bm{k}}=0$, that is linked with the peculiar long-range, phase-coherent transport properties that we call superfluid stiffness. The coherent density can be defined equivalently by $n_{\text{coh}}=\lim_{\|{\bm{r}}\|\to\infty}\overline{{\bigl{\langle}\hat{\psi}^{ \dagger}({\bm{r}})\hat{\psi}(0)\bigr{\rangle}}}$ as the component with off-diagonal long range order of the _ensemble-averaged_ one-body density matrix. As recognized already by Penrose and Onsager Penrose and Onsager (1956), in systems that are not fully translation invariant, the condensate properly speaking comprises all particles in the maximally populated eigenmode $\psi({\bm{r}})$Astrakharchik and Krutitsky (2011); Müller and Gaul (2012) and thus contains the coherent component with ${\bm{k}}=0$ plus the “glassy” component with ${\bm{k}}\neq 0$Yukalov and Graham (2007); Krumnow and Pelster (2011). Qualitatively, this strong link between superfluid and coherent density may not surprise much, and indeed it has been observed in numerical calculations Pilati _et al._ (2009); Carleo _et al._ (2013) that superfluid and coherent fractions vanish together at (one and the same) superfluid-insulator critical point, as implied by a finite right-hand side of (4) at criticality. The preference of (4) over (2) is also consistent with the view that the insulating Bose glass close to the transition is a collection of locally condensed puddles with finite mean density (3), which fail to connect phase-coherently over the full system size Yamamoto _et al._ (2008); Falco _et al._ (20, 21); Diallo _et al._ (2014). However, to our knowledge a quantitative statement such as (4) has not been put on record before. Moreover, recent numerical results in $d=2$Astrakharchik _et al._ (2013) seem to suggest that superfluid and coherent density do not vanish together. Therefore, we believe it worthwhile to derive (4) by a microscopic calculation and check its validity in an analytically tractable limit. Derivation:—We consider a single-component Bose-condensed fluid with repulsive interactions in its kinematic ground state, at inverse temperature $\beta$, confined to a $d$-dimensional volume of linear size $L$ and subject to an external one-body potential $V({\bm{r}})$. The total average density $n=L^{-d}\int\mathrm{d}{\bm{r}}\bigl{\langle}\hat{\psi}^{\dagger}({\bm{r}})\hat {\psi}({\bm{r}})\bigr{\rangle}$ is fixed by the chemical potential $\mu$ and splits into the sum of the condensate density (3) and the non-condensed density. The latter comprises quantum depleted and, at $T>0$, thermally excited particles. The condensate is described by a scalar, stationary BEC order parameter $\psi({\bm{r}})$. Such an order parameter may be defined as the macroscopically populated eigenmode of the one-body density matrix Penrose and Onsager (1956). In the U(1) symmetry-breaking picture of BEC Hohenberg and Martin (1965), one rather defines $\psi({\bm{r}})=\bigl{\langle}\hat{\psi}({\bm{r}})\bigr{\rangle}$ as the expectation value of the bosonic field operator; we use the latter definition for its technical simplicity. Following Baym Baym (1969) (see also Ueda (2010)), we calculate via linear response how much adding a particle with momentum ${\bm{k}}$ changes the order parameter on the one hand, and the current density on the other. We assume that the external potential is an ergodic random process, and reach translation invariance by ensemble-averaging. Comparing the changes in order parameter and current density then leads to the generalized Josephson relation (4). To this end, let \[\delta\hat{H}_{\bm{k}}=\delta\,\hat{a}_{\bm{k}}^{\dagger}=\frac{\delta}{L^{d/2 }}\int\mathrm{d}{\bm{r}}e^{-i{\bm{k}}\cdot{\bm{r}}}\hat{\psi}^{\dagger}({\bm{r }})\] (6) be the small perturbation ($\|\delta\|\ll\mu$) that adds a particle with momentum ${\bm{k}}$ to the system ². The linear response of the condensate amplitude on average is [FOOTNOTE:2][ENDFOOTNOTE] \[\overline{{\delta\psi({\bm{r}})}}=-\int_{0}^{\beta}\mathrm{d}\tau\overline{{ \bigl{\langle}\hat{\psi}({\bm{r}},\tau)\delta\hat{H}_{\bm{k}}(0)\bigr{\rangle} }}=\frac{\delta}{L^{d/2}}e^{i{\bm{k}}{\bm{r}}}\overline{{G({\bm{k}},0)}},\] (7) which brings about the zero-frequency component of the ensemble-averaged Matsubara-Green function (8) (If the ensemble average were not taken at this stage, one would face a Green function that is not diagonal in ${\bm{k}}$, which would compromise the following derivation.) The average condensate amplitude, \[\overline{{\psi({\bm{r}})}}+\overline{{\delta\psi({\bm{r}})}}=:\overline{{\psi ({\bm{r}})}}\left[1+i\,\delta\varphi\right],\] (9) is changed by a pure phase factor when \[\delta\varphi=-i\frac{\overline{{\delta\psi({\bm{r}})}}}{\overline{{\psi({\bm{ r}})}}}=\frac{-i\delta}{L^{d/2}\overline{{\psi({\bm{r}})}}}e^{i{\bm{k}}\cdot{ \bm{r}}}\overline{{G({\bm{k}},0)}}\] (10) is real, which can be realized by choosing the phase of $\delta$ appropriately and in the limit ${\bm{k}}\to 0$ (this is the step where taking the limit is required). This phase’s gradient then induces on average the superfluid mass current \[m\,\overline{{\delta{\bm{j}}({\bm{r}})}}=\left.\frac{\rho_{\text{s}}}{m}\bm{ \nabla}\delta\varphi\right\|_{{\bm{k}}\to 0}=\delta\frac{\rho_{\text{s}}{\bm{k} }e^{i{\bm{k}}\cdot{\bm{r}}}}{L^{d/2}m\overline{{\psi({\bm{r}})}}}\left. \overline{{G({\bm{k}},0)}}\right\|_{{\bm{k}}\to 0}.\] (11) Now we calculate the current directly via linear response, \[\delta{\bm{j}}({\bm{r}})=-\int_{0}^{\beta}\mathrm{d}\tau\bigl{\langle}\hat{\bm {j}}({\bm{r}},\tau)\delta\hat{H}_{\bm{k}}(0)\bigr{\rangle}.\] (12) Yet, even for a perturbation as simple as (6), this is in general impossible, for one cannot compute the full time-dependence of the current in the presence of interactions. But we can invoke particle number conservation, as expressed by the continuity equation, in imaginary time: \[i\partial_{\tau}\hat{n}({\bm{r}},\tau)+\nabla\cdot\hat{\bm{j}}({\bm{r}},\tau)=0.\] (13) (Its proof is elementary: Given the Hamiltonian $\hat{H}[\hat{\psi},\hat{\psi}^{\dagger}]=\hat{K}+\hat{U}$ with kinetic energy $\hat{K}=\frac{1}{2m}\int\mathrm{d}{\bm{r}}\bm{\nabla}\hat{\psi}^{\dagger}\cdot \bm{\nabla}\hat{\psi}$, and an interaction $\hat{U}=U[\hat{n}]$ that is a functional of the density only, (13) is equivalent to the equation of motion $\partial_{\tau}\hat{n}=[\hat{K},\hat{n}]$.) In the momentum representation, the continuity equation (13) becomes \[\partial_{\tau}\hat{n}_{\bm{p}}(\tau)+{\bm{p}}\cdot\hat{\bm{j}}_{\bm{p}}(\tau) =0,\] (14) and thus permits to replace the longitudinal current by the density variation according to $\|{\bm{p}}\|\hat{j}_{\bm{p}}^{\parallel}=-\partial_{\tau}\hat{n}_{\bm{p}}$. This allows us to evaluate the Matsubara-time integral, \[\delta j^{\parallel}_{\bm{p}}=\|{\bm{p}}\|^{-1}\int_{0}^{\beta}\mathrm{d}\tau \bigl{\langle}\partial_{\tau}\hat{n}_{\bm{p}}(\tau)\delta\hat{H}_{\bm{k}}(0) \bigr{\rangle}=-\|{\bm{p}}\|^{-1}\bigl{\langle}[\hat{n}_{\bm{p}},\delta\hat{H}_{ \bm{k}}]\bigr{\rangle}\] (15) and we are left with the simple equal-time commutator \[[\hat{n}_{\bm{p}},\delta\hat{H}_{\bm{k}}]=\delta\,\hat{a}^{\dagger}_{{\bm{k}}- {\bm{p}}}.\] (16) Thus we find after ensemble-averaging \[\overline{{\delta j^{\parallel}({\bm{r}})}}=-\frac{\delta}{L^{d/2}\|{\bm{k}}\|} \overline{{\psi^{}({\bm{r}})}}e^{i{\bm{k}}\cdot{\bm{r}}}.\] (17) Comparing this result with (11), whose leading contribution in the limit ${\bm{k}}\to 0$ is also purely longitudinal, then establishes (4). We remark that the zero-frequency Green function appearing here contains the full dynamical single-particle correlations and can in general not be reduced to the equal-time momentum distribution that enters, for instance, the one-body density matrix Pitaevskii and Stringari (2003). Consistency check in perturbation theory:—Exact analytical results are hard to obtain, but we can evaluate the factors entering (4) perturbatively for weak disorder using inhomogeneous quadratic Bogoliubov theory Gaul and Müller (2011); Müller and Gaul (2012) and check whether they match. First, the coherent density is given by eq. (11) in Müller and Gaul (2012), \[n_{\text{coh}}=\left\|\overline{{\psi}}\right\|^{2}=n_{\text{c}}[1-V_{2}+O(V^{3} )].\] (18) It is thus smaller than the total condensate density, eq. (3), by a factor that is determined by the glassy fraction Yukalov and Graham (2007) \[V_{2}:=\sum_{\bm{p}}\frac{\overline{{\|V_{\bm{p}}\|^{2}}}}{(\epsilon^{0}_{{\bm{p }}}+2gn_{\text{c}})^{2}}\] (19) with $\epsilon^{0}_{{\bm{p}}}={\bm{p}}^{2}/2m$ the free dispersion. Furthermore, using eqs. (18)–(20) of Müller and Gaul (2012), the single-particle Green’s function can be expressed in terms of quasiparticle normal and anomalous Green’s functions, \[\overline{{G({\bm{k}},0)}}=\sum_{{\bm{p}},{\bm{q}}} \Big{[}\overline{{(u_{{\bm{k}}{\bm{p}}}u^{}_{{\bm{k}}{\bm{q}}}+v _{{\bm{k}},-{\bm{p}}}v^{}_{{\bm{k}},-{\bm{q}}})G_{{\bm{p}}{\bm{q}}}(0)}}\] (20) \[\quad-\overline{{(v_{{\bm{k}}{\bm{p}}}u^{}_{{\bm{k}}{\bm{q}}}+u_ {{\bm{k}},-{\bm{p}}}v^{}_{{\bm{k}},-{\bm{q}}})F_{{\bm{p}}{\bm{q}}}(0)}}\Big{]}.\] The matrix coefficients $u_{{\bm{k}}{\bm{p}}}$ and $v_{{\bm{k}}{\bm{p}}}$ generalize the usual Bogoliubov factors $u_{\bm{k}},v_{\bm{k}}$ to the case where the condensate, or Bogoliubov vacuum, is inhomogeneous. They encode the condensate deformation by the external potential $V({\bm{r}})$ on the mean-field level. All these factors can be Taylor-expanded to the desired order in $V$ (see Sec. 3.4 in Müller and Gaul (2012) and Sec. III.B. in Gaul and Müller (2011)). To zeroth order in $V$, for the clean system, one has \[\overline{{G^{(0)}({\bm{k}},0)}}=-(u_{\bm{k}}^{2}+v_{\bm{k}}^{2})\epsilon_{{ \bm{k}}}^{-1}\] (21) where $\epsilon_{{\bm{k}}}=[\epsilon^{0}_{{\bm{k}}}(\epsilon^{0}_{{\bm{k}}}+2gn_{ \text{c}})]^{1/2}$ is the Bogoliubov dispersion. Multiplication by ${\bm{k}}^{2}$ and taking the limit ${\bm{k}}\to 0$ as required by (4) selects the most divergent contribution, which reduces the number of terms quite substantially. (21) diverges like $1/(2a_{\bm{k}}^{2}\epsilon_{{\bm{k}}})\sim 1/2\epsilon^{0}_{{\bm{k}}}=m/{\bm{k }}^{2}$, such that from (4) one finds \[\rho_{\text{s}}=m\left\|\psi\right\|^{2}=mn_{\text{c}}=:\rho_{\text{c}}.\] (22) As expected, in a clean system and to the quadratic order of the Bogoliubov Hamiltonian considered, the whole condensate is superfluid. At order $V^{2}$ in disorder strength, two types of contributions survive in (20): (i) products like \[u_{{\bm{k}}{\bm{p}}}^{(2)}u_{{\bm{k}}{\bm{q}}}^{(0)}G^{(0)}_{{\bm{p}}{\bm{q}}} (0)\propto V_{2}\delta_{{\bm{k}}{\bm{p}}}\delta_{{\bm{k}}{\bm{q}}}u_{\bm{k}}(u _{\bm{k}}-2v_{\bm{k}})\epsilon_{{\bm{k}}}^{-1}\] (23) with the clean, normal propagator $G_{{\bm{p}}{\bm{q}}}^{(0)}(0)=-\delta_{{\bm{p}}{\bm{q}}}\epsilon_{{\bm{p}}}^{-1}$, but no anomalous terms since $F^{(0)}=0$, and (ii) products like \[u_{{\bm{k}}{\bm{p}}}^{(0)}u_{{\bm{k}}{\bm{q}}}^{(0)}G^{(2)}_{{\bm{p}}{\bm{q}}} (0)\propto\delta_{{\bm{k}}{\bm{p}}}\delta_{{\bm{k}}{\bm{q}}}u_{\bm{k}}^{2}G^{( 2)}_{\bm{k}}(0)\] (24) and similar with $u_{\bm{k}}v_{\bm{k}}F_{\bm{k}}^{(2)}(0)$. Mixed terms of the type $u^{(1)}u^{(0)}G^{(1)}$ and the like do not survive the limit ${\bm{k}}\to 0$. Type (i) terms yield, after taking the limit ${\bm{k}}\to 0$, a correction $(1-V_{2})$ on the right-hand side of (4) that cancels exactly the same factor introduced on the left hand side by the coherent fraction (18). Type (ii) terms after a bit of algebra finally yield a correction of the form \[\lim_{{\bm{k}}\to 0}\sum_{{\bm{p}}}\frac{({\bm{k}}\cdot{\bm{p}})^{2}}{\epsilon ^{0}_{{\bm{k}}}\epsilon^{0}_{{\bm{p}}}}\frac{\overline{{\|V_{\bm{p}}\|^{2}}}}{( \epsilon^{0}_{{\bm{p}}}+2gn_{\text{c}})^{2}}=\frac{4m^{2}}{d}V_{2}.\] (25) All in all, (4) predicts to order $V^{2}$ the correction \[\rho_{\text{s}}=\rho_{\text{c}}\left(1-\frac{4}{d}V_{2}\right),\] (26) which is already well documented in the literature, see eq. (12) in Huang and Meng (1992), eq. (19) in Giorgini _et al._ (1994), eq. (20) in Lopatin and Vinokur (2002), and eq. (6) in Astrakharchik _et al._ (2013). This then explicitly validates the inhomogeneous Josephson relation (4) to order $V^{2}$ and at the same time rules out (2). Note, though, that one cannot obtain a temperature dependence from (20) with the quadratic quasiparticle Hamiltonian of Gaul and Müller (2011); Müller and Gaul (2012) that contains only elastic impurity scattering. In order to recover Landau’s celebrated finite-temperature superfluid depletion Landau and Lifshitz (1980) microscopically, one would have to introduce interactions between the quasiparticles. A different method of calculating the superfluid density is to compute the normal density $\rho_{\text{n}}=\rho_{\text{c}}-\rho_{\text{s}}$ directly from the transverse current-current correlation Landau and Lifshitz (1980); Baym (1969). Inhomogeneous Bogoliubov theory Gaul and Müller (2011); Müller and Gaul (2012) then predicts, at $T=0$, \[\rho_{\text{n}}=\frac{1}{4n_{\text{c}}}\sum_{{\bm{p}},{\bm{q}}}\left.\frac{p_{ z}q_{z}}{a_{\bm{p}}a_{\bm{q}}}\overline{{{n_{\text{c}}}_{{\bm{k}}-{\bm{p}}}{n_ {\text{c}}}_{{\bm{q}}-{\bm{k}}}[F_{{\bm{p}}{\bm{q}}}(0)-G_{{\bm{p}}{\bm{q}}}(0 )]}}\right\|_{{\bm{k}}={\bm{k}}_{\perp}\to 0}.\] (27) Here, ${n_{\text{c}}}_{\bm{k}}=L^{-d}\int\mathrm{d}re^{-i{\bm{k}}{\bm{r}}}n_{c}({\bm{ r}})$ are the Fourier components of the deformed condensate density, and ${\bm{k}}_{\perp}$ lies in the $xy$ plane transverse to the $z$ axis. In the clean case, to zeroth order in $V$, the condensate is homogeneous, ${n_{\text{c}}}_{\bm{q}}=n_{\text{c}}\delta_{{\bm{q}},0}$, and since $k_{z}=0$, the normal density vanishes. To order $V^{2}$, only a single type of term survives the limit ${\bm{k}}_{\perp}\to 0$, namely $n^{(1)}n^{(1)}G^{(0)}$. Using eq. (11) of Gaul and Müller (2011), this expression evaluates rather immediately to $(4/d)\rho_{\text{c}}V_{2}$ and thus agrees with (26). Clearly, to this order it is much simpler to evaluate (27) than to find $\rho_{\text{s}}$ from the Josephson relation, since there are no common terms that cancel, like on the two sides of (4), and only the clean quasiparticle propagator $G_{{\bm{p}}{\bm{q}}}^{(0)}(0)$ enters together with the condensate deformation. Lastly, we remark that this approach can be generalized to finite temperature and thus permits to derive disorder corrections to Landau’s superfluid depletion [32]. Summary:—A Josephson-type relation has been established for disordered Bose fluids between the superfluid density, the infrared residue of the single-particle Green’s function and the coherent density, i.e., density of condensed particles with zero momentum. 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0706.0987	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 35119, "num_imgs": 3, "llama3_tokens_count": 8939 }	[ "content_image/0706.0987/x1.png", "content_image/0706.0987/x2.png", "content_image/0706.0987/x3.png" ]	# Dynamical constraints on the component masses of the cataclysmic variable WZ Sge Danny Steeghs,¹² Steve B. Howell,³ , Christian Knigge,⁴ Boris T. Gänsicke,² Edward M.Sion,⁵ , William F.Welsh,⁶ [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:5][ENDFOOTNOTE] [FOOTNOTE:6][ENDFOOTNOTE] ###### Abstract We present phase-resolved spectroscopy of the short period cataclysmic variable WZ Sge obtained with the _Hubble Space Telescope_. We were able to resolve the orbital motion of a number of absorption lines that likely probe the environment near the accreting white dwarf. The radial velocities derived from simultaneous fits to 13 absorption lines indicate an orbital velocity semi-amplitude of $K_{UV}=47\pm 3$ km/s. However, we find that the phase zero is offset from the white dwarf ephemeris by +0.1. Our offset and velocity amplitude are very similar to constraints derived from optical emission lines from the quiescent accretion disk, despite the fact that we are probing material much closer to the primary. If we associate the UV amplitude with $K_{1}$, our dynamical constraints together with the $K_{2}$ estimates from Steeghs et al. (2001) and the known binary inclination of $i=77\pm 2$ imply 0.88$<M_{1}<$1.53$M_{\odot}$, $0.078<M_{2}<0.13M_{\odot}$ and $0.075<q=M_{2}/M_{1}<0.101$. If we interpret the mean velocity of the UV lines ($-16\pm 4$ km/s) as being due to the gravitational red-shift caused in the high-$g$ environment near the white dwarf, we find $v_{grav}=56\pm 5$km/s which provides an independent estimate on the mass of the primary of $M_{1}=0.85\pm 0.04M_{\odot}$ when coupled with a mass-radius relation. Our primary mass estimates are in excellent agreement and are also self-consistent with spectrophotometric fits to the UV fluxes despite the observed phase offset. It is at this point unclear what causes the observed phase-offset in the UV spectra and by how much it distorts the radial velocity signature from the underlying white dwarf. Subject headings:binaries:general – novae,cataclysmic variables – stars:individual(WZ Sge) – white dwarfs † [FOOTNOTE:†][ENDFOOTNOTE] ## 1. Introduction WZ Sge is a crucial and well-studied member of the cataclysmic variables (CVs), harboring a white dwarf primary accreting via Roche lobe overflow from a low mass companion. Its long outburst recurrence time ($\sim$30 yrs), large outburst amplitude ($\sim$7 mags), short orbital period (82 mins) and low mass accretion rate have long been associated with the picture of an evolved CV close to its period minimum (see Kato et al. 2001 and Patterson et al. 2002 for reviews). With a distance of only 43.5 pc (Thorstensen 2003, Harrison et al. 2004), it is the closest known cataclysmic variable and reaches V$\sim$8 near its outburst peak. Most of its life, WZ Sge sits near V$\sim$15 corresponding to a rather modest M${}_{V}\sim$12. Its proximity makes WZ Sge a rewarding object of study even if its characteristics among the growing family of short orbital period CVs are no longer that extreme or unusual. The rare, large amplitude outbursts and the absence of regular dwarf nova outbursts have now been seen in a small group of WZ Sge-like systems (Patterson et al. 2005a; Templeton et al. 2006), making these binaries particularly interesting from the point of view of understanding the disk instability mechanism that is thought to drive outbursts in CVs, X-ray transients and FU Orionis stars. Attempts at determining the component masses and thereby verify the evolutionary status of WZ Sge have long been frustrated without the detection of direct spectroscopic radial velocity signatures tracing either of the two stars. Published mass values for the white dwarf have ranged from $0.45M_{\odot}$ (Smak 1993) to $\sim 1.1M_{\odot}$ (Gilliland, Suntzeff & Kemper 1986). Spruit & Rutten (1998) used the broad disk emission lines as well as the bright spot dynamics and find primary masses from $0.9M_{\odot}$ to as high as $1.2M_{\odot}$ depending on the assumptions concerning the nature of the hotspot tail. Indirect estimates using the dynamics of the emission lines may not reflect the true dynamical motion of the white dwarf itself as was illustrated by significant phase offsets between the motion in the disk lines and that expected for the primary. The donor star also proved elusive despite the low mass accretion rate during quiescence implying an intrinsically faint and low mass object (e.g. Howell et al. 2004). However, during the 2001 outburst activity, Steeghs et al. (2001) were able to detect sharp emission features originating from the irradiated hemisphere of the donor, thus providing the first direct trace of its orbital dynamics corresponding to mass limits of $M_{2}<0.11M_{\odot}$ and $M_{1}>0.7M_{\odot}$. If we can determine the mass of the white dwarf star directly by measuring the radial velocities of its photospheric absorption lines, then WZ Sge offers parameter solutions of a double-lined spectroscopic binary. As part of our monitoring of the cooling of the white dwarf in response to the July 2001 superoutburst (Sion et al. 2003; Long et al. 2004), five HST orbits were dedicated to phase-resolved spectroscopy to attempt a mass determination of the WZ Sge white dwarf. In this Letter we report the results of this spectroscopy. ## 2. Observations and Data Reduction WZ Sge was observed on five consecutive orbits with the _Hubble Space Telescope_ during 2004 July 10-11. We used STIS and the E140M echelle grating with the 0.2” x 0.2” aperture. This setup yielded a spectral resolution of $\sim$90,000 and covered the range of 1150-1710Å. Each orbit consisted of approximately 2800 total seconds of on-source observation. We refer to Godon et al. (2006) for a discussion of the observations as well as the time-averaged properties of WZ Sge derived from these data. For the purposes of our phase-resolved analysis, we divided each of the five orbits of STIS time-tagged spectroscopy into four summed datasets giving us 20 individual spectra to work with. Orbit one was slightly shorter than the other four, and we divided the spectra into four 575 sec bins. The final four orbits were all equal in length and we divided them into 16 spectra of 712 sec each. These spectra provided full-phase coverage across the 82 min binary orbit. All spectra were transfered into a helio-centric frame by applying both velocity and time-system corrections. The orbital phases were then calculated using the photometric bright spot ephemeris of Patterson et al. (1998b) shifted by -0.046 in orbital phase in order to make phase zero coincident with conjunction of the donor star (Steeghs et al. 2001). The uncertainty on this ephemeris is limited by the phase zero correction which is good to 0.03 binary orbits. The absolute wavelength calibration of STIS is expected to be good to less than a pixel provided the target is well centered in the slit. The slit acquisition precision should also be good enough for a relatively bright source such as WZ Sge and cause at most a 1 pixel shift in the zero-point. For our E140M echelle data this amounts to a potential zero-point shift of up to 3km/s. We verified this by fitting the position of the geocoronal Ly-$\alpha$ emission and found that its position was only 1 km/s off from the nominal velocity of the Earth at the time of the observations. We therefore consider the absolute wavelength scale to be accurate to better than 3 km/s. The average spectrum of WZ Sge as derived from our 20 STIS spectra is presented in Figure 1. Here we have removed the orbital velocities of the white dwarf absorption lines from the individual spectra before averaging using the radial velocity amplitudes derived in this paper. The spectrum shows a rich absorption line spectrum plus broad emission features from the accretion flow (see also Godon et al. 2006 for line identifications). <figure><img src="content_image/0706.0987/x1.png"><figcaption>Figure 1.— Average HST spectrum of WZ Sge obtained in July 2004. The orbitalvelocities of the narrow absorption lines were removed from each individualspectrum before averaging using the radial velocity amplitude derived in thisLetter. The thirteen spectral lines that were used in our final radialvelocity fits are marked by vertical ticks.</figcaption></figure> ## 3. Analysis In order to search for radial velocity signatures in the rich absorption line spectrum that is associated with the primary white dwarf, we first re-binned the data to a constant velocity-scale wavelength grid, and subsequently normalized the continuum. The continuum shape was derived from a 2nd order spline-fit to line free regions of the spectrum. To boost the signal to noise further, the normalized spectra were then phase-folded into variance weighted bins to provide 10 phase bins across the binary orbit. We investigated the time dependent properties of a range of spectral lines using a variety of techniques, including by-eye centroid estimates, formal Gaussian fits to individual lines and cross-correlation techniques. We found significant radial velocity changes between the individual spectra with a semi-amplitude of approximately 50 km/s and close in phase to what is expected for the white dwarf. The formal Gaussian fits to the line profiles provided the most reliable and stable velocities, and we thus choose this as our preferred method. A large number of strong absorption lines are present in the spectra, although only those lines with unblended line cores were suitable for Gaussian fitting. For this reason, we are not including the strong carbon blends in our analysis. When performing the Gaussian fits, the width was fixed to a value estimated from the average spectrum leaving the line center and strength as the two free parameters. This fixed width value was then revised by shifting out the fitted velocities and re-fitting for the width using the average derived after shifting the individual bins. Starting with the strongest lines, we found that individual spectral lines displayed significant and periodic radial velocities as a function of the binary orbit. Even more encouraging, the phasing and amplitudes derived from different lines were in agreement. It thus appeared that these spectral lines are all tracking a common radial velocity curve. Since these features are expected to be formed close to the photosphere of the white dwarf, they appear to be promising tracers of its orbital motion. In order to improve the signal to noise of our radial velocities, we grouped lines together in order to perform simultaneous fits to multiple lines. The free parameters in these fits involving more than one line were the strength of each line as well as a _single_ common velocity offset for the set of lines fitted, relative to their respective rest wavelengths. In all our Gaussian fits, formal fitting errors on the fit parameters were calculated based on the variances of each individual data point used in the fit. We tried different groups of lines in order to check how stable the derived velocities were to the specific sets of lines used and found that all the lines seem to share the same orbital kinematics. This led us to select a set of 13 absorption lines in the spectrum of WZ Sge that were well resolved and strong enough to be fitted with Gaussians in the individual spectra. We list the wavelengths of the 13 features used in these joint fits as well as the widths of the employed Gaussians in Table 1 and the resulting radial velocity measurements in Table 2. To further test the reliability of our fits, we also re-binned the data at various levels to test the robustness of our velocity values to various levels of binning. We found that the derived radial velocity variations were only weakly sensitive to the binning of the data and the variations were within the statistical uncertainties associated with the fits. Some level of binning is preferred to avoid fitting to very low S/N spectra and for the actual velocity values reported in this study, we worked with individual spectra binned to a constant velocity dispersion of 20 km/s/pixel. In Figure 2, we present the derived radial velocities as a function of the orbital phase of the binary, together with the best fit sinusoidal radial velocity curve. These fits were with fixed period and the zero-phase, the amplitude and the mean velocity as free parameters. For comparison, we have plotted the results from fitting to just two strong spectral lines as well as our final fit that involves the 13 spectral lines listed in Table 1. All other trial fits to several groups of lines show the same kinematics, with amplitudes varying between 44-51 km/s and 0.10-0.14 phase offsets relative to the binary ephemeris. It thus appears that despite the fact that the origin of all the observed lines is not fully understood (Long et al. 2004), they do share the same orbital kinematics. Of particular interest are the strong Si IV doublet lines near 1400Å (Fig 2) that track the Si II and other lines perfectly even though Si IV requires temperatures in excess of 25,000K while the photospheric temperature of the WD is close to 15,000K. However, zero phase is consistently and significantly offset from what is expected for material co-moving with the white dwarf’s center of mass by $\sim 0.1$. Our formal phase offset uncertainty is only 0.02 and the uncertainty in the orbital ephemeris is less than 0.03. We comment that the error bars plotted in Fig.2 reflect the formal fitting errors for our joint Gaussian fits. When comparing different group of lines, some systematic residuals from a sinusoidal radial velocity are observed. However, these additional systematic effects are modest as we typically achieve goodness of fit values close to $\chi^{2}_{\nu}=1$. For our final joint fit to 13 lines for example, we find a formal $\chi^{2}_{\nu}=1.4$. We conclude that we have detected sinusoidal velocity shifts in a large number of absorption lines that are expected to be formed very close to the primary. They should be the most reliable tracers we currently have for the orbital dynamics of the primary, but the $\sim 0.1$ phase offset suggests that these lines appear to be contaminated by line of sight components other than the underlying white dwarf photosphere. From our joint fit, we derive a mean velocity of $-16\pm 3$ km/s, a radial velocity amplitude of $47\pm 3$ km/s and a phase zero offset of $+0.12\pm 0.02$ with the errors corresponding to formal $1\sigma$ uncertainties on the fit parameters. Given that the absolute wavelength scale is accurate to 3 km/s, we added this additional uncertainty in quadrature to the formal least-squares error of the mean velocity which brings its final uncertainty to 4 km/s. Ion \| rest wavelength \| FWHM (km/s) ---\|---\|--- Si II \| 1260.4221 \| 500 Si II \| 1264.7377 \| 500 Si II \| 1265.0020 \| 500 O I \| 1302.1680 \| 175 Si II \| 1309.2760 \| 175 Si IV \| 1393.7546 \| 250 Si IV \| 1402.7697 \| 250 N II \| 1492.6254 \| 250 N II \| 1492.8200 \| 250 N II \| 1494.6751 \| 250 Si II \| 1526.7066 \| 250 Si II \| 1533.4310 \| 250 AlII \| 1670.7870 \| 250 Table 1Lines included in dynamical fits Phase \| velocity (km/s) \| σ ---\|---\|--- 0.010 \| -00.09 \| 12.1 0.140 \| -23.70 \| 13.5 0.190 \| -23.32 \| 10.7 0.258 \| -58.39 \| 9.4 0.370 \| -60.61 \| 9.2 0.507 \| -58.27 \| 7.3 0.601 \| -42.26 \| 9.9 0.704 \| 10.09 \| 6.3 0.819 \| 29.29 \| 12.8 0.865 \| 11.69 \| 9.8 Table 2Radial velocities <figure><img src="content_image/0706.0987/x2.png"><figcaption>Figure 2.— Radial velocity measurements of the photospheric absorptionfeatures from the accreting white dwarf. Top panel are the velocities derivedfrom fitting to the two strong Si IV features only whereas the bottom panelshows our final velocities derived from simultaneous fits to thirteen spectrallines. Least-squares sine fits are overplotted.</figcaption></figure> ## 4. Dynamical constraints on the component masses We were able to resolve the orbital motion of the UV absorption line system in WZ Sge using our high-resolution STIS time-series data. Unfortunately, the radial velocity curve displays a significant phase offset with the radial velocity curve expected for the primary and we must therefore be careful with associating our derived semi-amplitude with the orbital velocity of the white dwarf: $K_{UV}=47\pm 3$ km/s. The bright-spot eclipse has always served as a reliable clock for the system (Patterson et al. 1998). It is difficult to see how the accurate bright spot ephemeris combined with the correction to true orbital phase zero as provided by the detection of the mass donor star (Steeghs et al. 2001) could lead to a significant error in the calculation of the orbital phases and we estimated that our shifted ephemeris should be good to 0.03 cycles. Also while the STIS absolute timing accuracy can drift somewhat, it is not expected to approach the 8 minutes of clock error we would need to undo the 0.1 phase shift we observe. Interestingly enough, our amplitude _and_ phase offset is consistent with previous $K_{1}$ estimates reported by Gilliland et al. (1986), Spruit & Rutten (1998), Mason et al. (2000) and Steeghs et al. (2001). Those estimates relied on optical emission from the quiescent accretion disc and is based on the kinematics of the disk gas several to several tens of white dwarf radii out. The emission region we are studying in the UV on the other hand originates very close to the white dwarf surface. It appears that despite our use of UV lines formed much closer to the primary, a very similar radial velocity behavior is observed. In this and the next section, we consider the implications our UV constraints would have on the system parameters of WZ Sge. We will revisit the reliability of these constraints given the observed phase offset in the discussion section. If we consider our amplitude estimate from the UV spectroscopy as an estimate for $K_{1}$ and combine it with the conservative donor star velocity estimates ($493<K_{2}<585$ km/s) of Steeghs et al. (2001), we derive mass limits for the primary ranging between $M_{1}\sin^{3}{i}=0.83M_{\odot}$ and $M_{1}\sin^{3}{i}=1.38M_{\odot}$. Similarly, it would constrain the donor star to the range $0.074M_{\odot}<M_{2}\sin^{3}{i}<0.12M_{\odot}$. Only our dynamical constraints on $K_{1}$ and $K_{2}$ have so far entered these mass estimates. Additional constraints are provided by the prominent bright spot eclipses, even though the primary itself is never eclipsed. This limits the binary inclination range considerably since the lack of a white dwarf eclipse sets an upper limit to the allowed inclination, whereas the prominent bright spot eclipse indicates the inclination needs to be large enough to obscure the outer disc. Several inclination estimates have been reported in the literature. Skidmore et al. (2002) modeled the infrared eclipse lightcurve in quiescence and finds $i=75.9$, Spruit & Rutten (1998) quote $i=77\pm 2$. For $i=75-79$, our formal mass limits correspond to $0.88M_{\odot}<M_{1}<1.53M_{\odot}$ and $0.078M_{\odot}<M_{2}<0.13M_{\odot}$. ## 5. M${}_{1}$ from the Gravitational Redshift Apart from the radial velocity amplitude, the second insight provided by the observed absorption line velocities is the recovered mean velocity of $-16\pm 4$ km/s. Previous studies of the optical emission lines using both the broad disk emission (Spruit & Rutten 1998) as well as the narrow emission component from the mass donor star (Steeghs et al. 2001) indicate a systemic binary velocity of $\gamma=-72\pm 2$ km/s relative to a heliocentric velocity frame. The UV absorption lines are thus shifted by $+56\pm 5$ km/s. This would be expected for lines formed near the primary due to the gravitational redshift introduced in the high-$g$ environment of the white dwarf (Eddington 1924; Greenstein & Trimble 1967; Sion et al. 1994). The gravitational redshift is proportional to $M/R$ and can thus provide a completely independent determination of the primary mass given a mass-radius relation. Assuming that the lines are formed at or close to the surface of the primary and using Eggleton’s zero-temperature mass radius relation as quoted in Verbunt & Rappaport (1988) implies $M_{1}=0.84\pm 0.04M_{\odot}$. The above mass radius relation is expected to be relevant for the WD masses considered here, but does not consider the possible effects of the internal chemical composition of the white dwarf and the presence of substantial surface layers of H and He. We have no direct constraints on the internal composition of the white dwarf in WZ Sge, but stellar and binary evolution models tell us that a CO dominated WD is expected with relatively thin and low mass H/He shells. Regular nova eruptions keep the shells thin and of low mass fraction and for the purpose of this paper we do not attempt to constrain such composition issues using our redshift measurement. In Figure 3, we illustrate the possible effects of WD composition and temperature on the expected mass-radius relation, and thus the inferred mass. We used the models of Panei, Althaus & Benvenuto (2000) to plot mass-radius curves for various compositions. These provide models for He,C and O dominated white dwarfs with possible H and He layers for a range of temperatures. Given that WZ Sge’s white dwarf was close to 15,000K at the time of our observations (Godon et al. 2006), we selected relevant models at 15,000K and plot a family of model curves in Fig.3. It can thus be seen that temperature and composition effects are relatively modest in comparison to our formal error on $v_{grav}$. If we take C or O dominated models at 15,000K and allow for the presence of modest H and He layers the inferred mass is pushed up slightly, but $M_{1}=0.85\pm 0.04M_{\odot}$ encompasses the relevant models within $v_{grav}$=56$\pm$5 km/s. We comment that the mean red-shift observed reflects the $M/R$ at the radius where the lines are formed, which may not necessarily be the canonical white dwarf surface. If the lines are instead formed away from the surface at larger radii we would underestimate the white dwarf mass since we overestimate its radius. There are therefore potential systematic effect at play that will affect our inferred mass from the gravitational red-shift which are not reflected in the 5% formal error quoted. The key purpose in this paper is to provide a complementary estimate for the mass of the white dwarf, that can be compared to the masses inferred from the orbital velocities. Our primary mass/radius estimates based on the gravitational redshift interpretation of the mean velocity are in excellent agreement with recent radius estimates derived from spectrophotometric fits to FUSE and HST spectra. Given our knowledge of the distance to WZ Sge, such fits directly constrain the size of the emitting region and achieve self-consistent solutions between the required log $g$ and the radius for a white dwarf mass close to $0.9M_{\odot}$ (Long et al. 2003; Long et al. 2004; Godon et al. 2006). <figure><img src="content_image/0706.0987/x3.png"><figcaption>Figure 3.— Mass-radius constraints for the primary based on our gravitationalredshift measurement. The labeled thick dashed line represent the Vgrav=56km/scurve with additional dashed curves for the ±1σ constraints. The zero-temperature Hamada-Salpeter mass-radius relation is shown as the solid curve.Additional models from Panei et al. (2000) show the dependence on WDcomposition and temperature. Thin solid curves are for carbon compositioncomparing a He layer only (lower curve) with a He and a H layer at 15,000 K.The dot-dashed curves are the same for oxygen composition. Finally, the dottedcurve is a O model with a He layer at 5,000K.</figcaption></figure> ## 6. Discussion A robust determination of the white dwarf radial velocity amplitude, $K_{1}$, is the key to reliable system parameter estimates for WZ Sge. WZ Sge has always played an important role in our understanding of CV evolution, but despite its bright outbursts and proximity there is still considerable uncertainty about the mass and nature of its companion star. Our HST observations did reveal diagnostic UV absorption lines that display stable orbital kinematics, but suffer from the same phase offset as previous estimates using the accretion disk emission lines in the optical. It is at this point unclear what causes the observed phase-offset in the UV spectra and by how much it distorts the radial velocity signature from the underlying white dwarf. The observed Si IV lines also demand formation temperatures well above that of the underlying photoshpere. We must therefore be cautious with our derived system parameter estimates. The observed radial velocity amplitude implies a primary mass of at least $0.83M_{\odot}$ and a binary mass ratio $0.075<q<0.10$ if we ignore the phase offset and assume we are indeed measuring $K_{1}$. In addition to our estimate for $K_{1}$, we derived an independent estimate for the mass of the primary if one interprets the mean velocity as corresponding to the gravitational redshift near the white dwarf surface. This implied $M_{1}=0.85\pm 0.04M_{\odot}$. If we wish to make the system parameters consistent with both our $K_{1}=47\pm 3$ as well as $M_{1}=0.85\pm 0.04M_{\odot}$ from the redshift, we require $q=0.092\pm 0.008$, $M_{2}=0.078\pm 0.06M_{\odot}$ and $K_{2}=510\pm 15$ km/s. Despite the phase offset, there is excellent agreement between our dynamical mass estimates, the gravitational redshift inferred mass as well as the log $g$ and emitting radius constraints from fits to the UV fluxes given the known distance to WZ Sge. A white dwarf mass of $0.85M_{\odot}$ holds an important implication for the accretion heating of the white dwarf. Quasi-static evolutionary models with compressional heating, boundary layer irradiation and time variable accretion by Godon et al. (2006) require additional heating mechanisms (e.g. ongoing accretion after the outburst, boundary layer irradiation among others) to bring the evolutionary models in agreement with the observed cooling for a primary mass of $0.8-0.9M_{\odot}$. Although the various published estimates for the white dwarf mass in WZ Sge tended to converge near $0.9M_{\odot}$, in line with our estimates, it has long been thought that the mass donor star was much less massive and accordingly that the mass ratio was very small. Indeed, WZ Sge was considered one of the prime candidates for being an evolved CV that has whittled down its donor star and has evolved through its orbital period minimum (Ciardi et al. 1998; Patterson 1998a). Depending on the precise value for the period minimum, a matter that has not yet been settled, a post-period bounce system at the period of WZ Sge would have a donor mass of $0.04-0.06M_{\odot}$ (Kolb & Baraffe 1998; Howell 2001). The first solid example of such a system has only recently been discovered in the sample of SDSS CVs (Littlefair et al. 2006). We now have a growing sample of CVs accreting at low rates, with periods near the period minimum. Several of these display rare, large amplitude dwarf nova outbursts in contrast with the more prolific outburst behaviour at longer orbital periods and higher mass transfer rates. There is still a debate ongoing whether the long outburst recurrence times of these systems require an unusually low viscosity during the quiescent phases (Smak 1993), or if the inner disk can be stabilised or depleted (Hameury et al. 1997; Matthews et al. 2007). Given the intrinsic faintness of the short period systems, and the fact that a significant sample of such systems has only been inventarised very recently, their long-term variability has not been that well characterised. But it is clear that WZ Sge is part of a class of evolved CVs and that system parameter studies of these systems can provide important clues concerning our understanding of CV evolution and the nature of the disk instability process. The mass ratio implied by our constraints is rather high, resulting in a correspondingly large donor mass estimate close to what is expected for a main-sequence object filling its Roche-lobe at the orbital period of WZ Sge ($M_{2_{ZAMS}}=0.08M_{\odot}$). This is in contrast with previous constraints on the nature of the mass donor that imply a low luminosity object. Searches for signatures of the mass donor in the red and infrared have so far failed. Indeed, this has been one of the motivations for considering WZ Sge as one of the prime candidates for harboring a degenerate secondary (Howell et al. 2004, Patterson et al. 2001) as the inferred brightness constraints are not consistent with a $0.08M_{\odot}$ main-sequence object. An alternative method for estimating the mass ratio of CVs uses the precession periods of the accretion disks as a dynamical probe through the observed superhump periodicities observed in many systems (Patterson et al. 2005b, Knigge 2006). This method has revealed a remarkable correlation between the observed superhump properties and the mass ratio of the binary. Although WZ Sge is considered a calibrator for this relation, with a cited mass ratio of $q=0.05\pm 0.015$, we point out that we do not have a reliable independent determination of this value. However, it is clear that our implied mass ratio of $\sim 0.09$ if we combine $K_{1}$=47 km/s with $M_{1}=0.85M_{\odot}$ is substantially larger than the trend suggested by the superhump behaviour. It is unclear whether our UV results are significantly affected by a component that distorts the radial velocity curve causing us to over-estimate $K_{1}$ and thereby over-estimate the donor star mass and the binary mass ratio considerably. Intervening gas that is not in Keplerian rotation with the primary could spoil the diagnostics provided by the UV lines, although the exact driver behind such a component is unclear and the spectrum and its features are quite well described by WD models. Since we know the orientation of the binary fairly well thanks to the stable bright spot eclipses and the detection of the mass donor during outburst, the relative phase offset is significant at the $3-4\sigma$ level. The inferred donor mass for our favored system parameters ($M_{2}=0.078\pm 0.06M_{\odot}$) corresponds to an $\sim$L2 type star. If we take the Knigge (2006) donor sequence, we find $M_{2}\sim 0.073M_{\odot}$ for a system still evolving towards its period minimum and $M_{2}\sim 0.044M_{\odot}$ for a post period minimum system at the orbital period of WZ Sge. Our $M_{2}$ value is right near the brown dwarf treshold and would suggest that WZ Sge has not bounced back yet from its period minimum. However, an object of that mass should have been detected during previous near-infrared studies of WZ Sge given the expected $M_{K}\sim 10.4$ and the $M_{K}>12$ constraint from Howell (2004). The absence of any signatures of the mass donor might suggest a massive, but sub-luminous and evolved object instead of an object near its main-sequence. We could speculate how the low mass donor star may have adjusted to its long history of mass loss, but at this point have no direct evidence for abnormal abundances in the accretion flow for example that might be expected for such an evolved and/or stripped object. Despite our exploitation of high-resolution HST observations of the glowing white dwarf in WZ Sge, we must still keep an open mind about the system parameters of this important cataclysmic variable. The nature of the observed phase offset and its impact on the absorption line radial velocities need to be clarified. It is clear that additional high resolution UV spectroscopy of the system as it settles into its long quiescent phase could provide useful constraints on the environment near the white dwarf photosphere. DS acknowledges a Smithsonian Astrophysical Observatory Clay Fellowship as well as support through a PPARC/STFC Advanced Fellowship. The research reported here was based on observations made with the NASA/ESA Hubble Space Telescope. Support for this work was provided by NASA through grant number GO-09459. from the Space Telescope Science Institute. 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0906.1734	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 7670, "num_imgs": 2, "llama3_tokens_count": 2159 }	[ "content_image/0906.1734/x1.png", "content_image/0906.1734/x2.png" ]	# Finite lifetime effects on the photon production from a quark-gluon plasma F. Michler${}^{1}$ B. Schenke${}^{2}$ C. Greiner${}^{1}$ ###### Abstract We use the real-time Keldysh formalism to investigate finite lifetime effects on the photon emission from a quark-gluon plasma (QGP). We provide an ansatz which eliminates the divergent contribution from the vacuum polarization and renders the photon spectrum UV-finite if the time evolution of the QGP is described in a suitable manner. ${}^{1}$Institut für Theoretische Physik Johann Wolfgang Goethe - Universität Frankfurt Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany, ${}^{2}$Department of Physics, McGill University, Montreal, Quebec, H3A 2T8, Canada The quark-gluon plasma (QGP) created during heavy-ion collisions can only be accessed indirectly via experimental signatures such as hard and electromagnetic probes. Besides the role of memory effects in time evolution (see _e.g._[1, 2]) it is also of particular interest within the framework of non-equilibrium quantum field theory how these probes are affected by the finite lifetime of the QGP itself. For photons, this question has first been addressed by Boyanovsky _et al._[3, 4]. The main result has been the contribution of first order processes which are kinematically forbidden in equilibrium. The spectra from these processes were found to decay algebraically for $k>1.5$ GeV and thus dominate over higher order equilibrium contributions in that range. Two major problems arose in the above investigations, namely the divergent contribution from the vacuum polarization and the non-integrability of the remaining contributions in the ultraviolet domain [5]. The topic was then touched on by Fraga _et al._[6] where it was claimed that the divergent vacuum contribution is unphysical and thus requires an appropriate renormalization technique. It was concluded that the ansatz used in [5] is inadequate as it produces the mentioned problems. We provide an ansatz that eliminates the divergent contribution from the vacuum polarization. For the scenario of a heavy-ion collision, it also renders the resulting photon spectrum UV-finite if the time evolution is described in a suitable manner. The photon production rate from a homogeneous but non-stationary emitting system reads [2, 7]: \[k\frac{d^{7}n(t)}{d^{4}xd^{3}k}=\frac{1}{(2\pi)^{3}}\mbox{Re}\left\{\int_{- \infty}^{t}du\mbox{ }i\Pi^{<}_{T}(\vec{k},t,u)e^{ik(t-u)}\right\}\] (1) As in [5], we use the one-loop approximation for the photon self energy $i\Pi^{<}_{T}(\vec{k},t,u)$ including the processes of first order in $\alpha_{e}$. We model the finite lifetime of the QGP via time dependent occupation numbers of the quarks \[n_{F}(E)\to n_{F}(E,t)=f(t)n_{F}(E)\mbox{ ,}\] (2) and couple the time evolution to the interaction vertices by replacing the occupation numbers and the number of holes in $i\Pi^{<}_{T}(\vec{k},t,u)$ by their geometric mean from the two times $t$ and $u$. If we now decompose $i\Pi^{<}_{T}(\vec{k},t,u)$ into the vacuum polarization and the medium contribution \[i\Pi^{<}_{T}(\vec{k},t,u)=i\Pi^{<}_{T,0}(\vec{k},t-u)+i\Pi^{<}_{T,M}(\vec{k},t ,u)\,,\\] (3) and insert (3) into (1), we see that the vacuum polarization is evaluated on-shell and thus does not contribute. Therefore, the divergence associated with it does not show up in the photon yield. As the occupation numbers are time dependent, the medium part of the photon self energy is evaluated off-shell which makes the contribution of first order processes possible. For our numerical investigations, we consider a QGP with a temperature of $T=0.3$ GeV. It is ’switched on’ over a timescale of $\tau_{F}=1$ fm/c which is modeled by three different switching functions $f(t)$. \[f_{1}(t) = \Theta(t)\] \[f_{2}(t) = \Theta(-t)e^{t/t_{F}}+\Theta(t)\] \[f_{3}(t) = \frac{1}{2}\Theta(-t)e^{t/t_{F}}+\Theta(t)\left[1-\frac{1}{2}e^{- t/t_{F}}\right]\] Figure 1 shows that the resulting photon spectrum decays as $\sim 1/k^{3}$ for all switching functions $f(t)$ and is thus not integrable. <figure><img src="content_image/0906.1734/x1.png"><figcaption>Figure 1: The photon spectrum decays as ∼1/k3 independently from f(t).</figcaption></figure> This behaviour can be understood as follows. The dominant contributions to the photon yield are Bremsstrahlung and a negative contribution from Pauli Blocking of the pair creation process. Both of them behave as $\sim 1/k^{3}$ for large $k$ independently of $f(t)$[8]. If we turn from an instantaneous to a smoother switching, the Bremsstrahlung contribution essentially halves in value whereas the Pauli Blocking contribution is left unchanged [8]. This accounts for the slightly steeper decay in this case. The problem with the UV-integrability can be circumvented if the QGP is also switched off again as to mimique a heavy-ion collision. The photon yield is assumed to be observed at $t\rightarrow\infty$. We adopt a creation and hadronization time of $\tau_{F}=\tau_{H}=1$ fm/c and a lifetime of $\tau_{L}=4$ fm/c. Again different switching functions $g(t)$ are compared. \[g_{1}(t) = f_{1}(t)\cdot\Theta(\tau_{L}-t)\] \[g_{2}(t) = f_{2}(t)\cdot\left[\Theta(\tau_{L}-t)+\Theta(t-\tau_{L})e^{-(t- \tau_{L})/t_{H}}\right]\] \[g_{3}(t) = f_{2}(t)\cdot\left[\Theta(\tau_{L}-t)\left(1-\frac{1}{2}e^{(t- \tau_{L})/t_{H}}\right)+\frac{1}{2}\Theta(t-\tau_{L})e^{-(t-\tau_{L})/t_{H}}\right]\] In this case, the negative Pauli Blocking contribution vanishes [8]. One can furthermore infer from fig. (2) that the resulting photon spectrum strongly depends on the considered switching function $g(t)$. For an instantaneous switching, it still decays as $\sim 1/k^{3}$ for large $k$. But if we consider an exponential ($g_{2}(t)$) or an even smoother switching ($g_{3}(t)$) which represents a more realistic scenario, the spectrum is suppressed to $\sim 1/k^{5}$ and $\sim 1/k^{7}$, respectively, and thus rendered UV-finite. <figure><img src="content_image/0906.1734/x2.png"><figcaption>Figure 2: The decay of the photon spectrum is highly sensitive to g(t).</figcaption></figure> In summary, we have presented an ansatz which eliminates the divergent photon yield from the vacuum polarization and partially solves the problem with the UV-integrability. As the next step, we have to revisit it in a way that it also leads to UV-finite photon spectra for the general case. We also will consider the treatment of possible infrared singularities. F. M. gratefully acknowledges financial support by the Helmholtz Research School for Quark Matter Studies (H-QM) and by the Helmholtz Graduate School for Hadron and Ion Research (HGS-HIRe for FAIR). B. S. gratefully acknowledges a Richard H. Tomlinson grant by McGill University and support by the Natural Sciences and Engineering Research Council of Canada. This work was (financially) supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Ökonomischer Exzellenz) launched by the State of Hesse. ## References * [1]B. Schenke and C. Greiner, _Phys. Rev._C73, 034909 (2005). * [2]F. Michler, B. Schenke, and C. Greiner, arXiv:0905.2930, 2009. * [3]S.-Y. Wang and D. Boyanovsky, _Phys. Rev._D63, 051702 (2001). * [4]S.-Y. Wang, D. Boyanovsky, and K.-W. Ng, _Nucl. Phys._A699, 819 (2002). * [5]D. Boyanovsky and H. J. de Vega, _Phys. Rev._D68, 065018 (2003). * [6]E. Fraga, F. Gelis, and D. Schiff, _Phys. Rev._D71, 085015 (2005). * [7]J. Serreau, _JHEP_05, 078 (2004). * [8]F. Michler, work in progress.
1906.05960	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 65377, "num_imgs": 7, "llama3_tokens_count": 20795 }	[ "content_image/1906.05960/null1.png", "content_image/1906.05960/alpha1.png", "content_image/1906.05960/alpha2.png", "content_image/1906.05960/RMstab.png", "content_image/1906.05960/ourstab1.png", "content_image/1906.05960/ourstab2.png", "content_image/1906.05960/searchdestab.png" ]	# Global analysis of a predator-prey model with variable predator search rate Ben Dalziel¹, Enrique Thomann², Jan Medlock³ , and Patrick De Leenheer⁴ [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] ###### Abstract We consider a modified Rosenzweig-MacArthur predator-prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant. A complete analysis of the global behavior of the model is presented, and shows that the model exhibits a dichotomy similar to the classical Rosenzweig-MacArthur model: either the coexistence steady state is globally stable; or it is unstable, and then a unique, globally stable limit cycle exists. We discuss the similarities, but also important differences between our model and the Rosenweig-MacArthur model. The main differences are that: 1. The paradox of enrichment which always occurs in the Rosenzweig-MacArthur model, does not always occur here, and 2. Even when the paradox of enrichment occurs, predators can adapt by lowering their search rate, and effectively stabilize the system. ## 1 Introduction Predator-prey interactions are among the most common in many ecological systems, and have received considerable attention. A prototype model that captures this is: \[{\dot{N}} = rN\left(1-\frac{N}{K}\right)-f(N)P\] \[{\dot{P}} = P\left(ef(N)-m\right)\] Here, $N$ and $P$ denote the prey and predator density respectively, each expressed as numbers per unit area. In the absence of the predator, the prey is assumed to grow logistically, characterized by the positive parameters $r$ and $K$ representing the prey’s maximal per capita growth rate, and carrying capacity respectively. The prey is consumed by the predator at a rate $f(N)$ per unit of predator density, and is assumed to depend on the prey density. The choice of the functional form for this rate function $f(N)$ -which is commonly known as the functional response- has important implications for the model behavior. In this paper, we propose a specific functional response that incorporates particular predator behavior, that will be explained below. The positive parameters $m$ and $e$ are the predator’s mortality rate, and the conversion efficiency of prey into predator respectively. The parameter $e$ represents the number (or density) of predators obtained, per consumed prey (or density of prey). Since we shall assume throughout this paper that $e$ is a constant, we can scale it out by setting ${\bar{N}}=N$, ${\bar{P}}=P/e$ and ${\bar{f}}(N)=ef(N)$. In these transformed variables, and after dropping the bars, the model takes the following form: \[{\dot{N}} = rN\left(1-\frac{N}{K}\right)-f(N)P\] (1) \[{\dot{P}} = P\left(f(N)-m\right)\] (2) A common choice for $f(N)$ is the Holling type II functional response: \[f_{II}(N)=\frac{sN}{shN+1},\] (3) where $s$ and $h$ are positive constants representing the predator’s search (or attack) rate, and the handling time respectively. The main qualitative features of this functional are that it is zero when $N$ equals zero, is increasing, saturates for large prey densities at $1/h$, and is concave (the second derivative of $f_{II}(N)$ is negative for all $N\geq 0$). The latter property implies that although the per-predator consumption rate increases with prey density $N$, it is attenuated (i.e., it slows down) for larger values of $N$. Using a Holling type II functional response in $(\ref{s1})-(\ref{s2})$ yields the Rosenzweig-MacArthur model [8], which is one of the benchmark predator-prey models in ecology. To understand the main motivation for this paper, it is useful to review a mechanistic derivation of the Holling type II functional [4, 2, 1] here: Consider a sufficiently long window of time $T$ during which an average predator catches $M$ prey in a landscape where the prey density is fixed at $N$. Then the functional response equals $M/T$: \[f_{II}(N)=\frac{M}{T}.\] Let $s$ be the search rate, i.e. the area searched by the average predator per unit of time. If $h$ is the time spent handling a single prey, then the average predator will spend a total amount of $T-Mh$ units of time searching for prey, during which the predator covers an area of $s(T-Mh)$. The average predator therefore catches a total of $Ns(T-mh)$ prey, and thus: \[M=Ns(T-Mh).\] Dividing by $T$, and solving for $f_{II}(N)=M/T$ yields: \[f_{II}(N)=\frac{M}{T}=\frac{sN}{shN+1},\] which is Holling’s type II functional response. Next we offer a conceptual framework to determine the value of $s$ in practice. Imagine that a predator moves in a plane at a constant velocity, meaning that its direction and magnitude $v$ are fixed. It seems plausible that field biologists can determine relatively accurate estimates of $v$. Suppose that at any fixed time, the predator is centered in a disk of radius $r$, and is capable to instantaneously search this disk for prey. Assume now that the predator moves for a period of time $T$ through the plane at the constant velocity $v$. The area searched by the predator in this time interval $[0,T]$ is equal to: $(2r)(vT)+\pi r^{2}$ (the sum of the area of a rectangle of length $vT$ and width $2r$, and the area of two half-disks with radius $r$). Thus, the search rate during this time interval equals: \[2rv+\frac{\pi r^{2}}{T}.\] Letting $T\to+\infty$, we obtain the predator’s search rate: \[s=2vr.\] (4) Clearly, one can make different assumptions on how the predator moves (e.g. by allowing deterministic or random changes to the direction of movement and/or speed $v$; or assume diffusive movement etc), and these will lead to different values of $s$, related to the measurable characteristics of the predator’s movement pattern. However, the expression obtained above is obviously a reasonable upper bound of the actual search rate, if we use the predator’s largest possible speed, and largest possible radius it can search at any given time, two quantities that are likely well-documented for many predators. The main purpose of this paper is to investigate the implications on the model behavior when the assumption that the search rate $s$ is constant, is relaxed. It seems plausible that when predators survey the environment they operate in, and sense the prey density, they may adapt their search rate based on the perceived prey density. We shall focus on a case where predators always increase their search rate when they perceive higher prey densities. Moreover, we assume that when the prey is absent, predators cease to search, and that the search rate is limited by a maximally achievable search rate, perhaps due to physiological limitations of the predators and/or physical constraints imposed by the environment. Specifically, we shall consider: \[s(N)=\frac{aN}{N+g},\] (5) where $a$ and $g$ are positive constants. The parameter $a$ is the maximally achievable search rate, and $g$ is the half-saturation constant, which corresponds to the prey density at which the search rate is equal to half of the maximal value $a$. For all $N>0$, an increase in $g$ leads to a decrease in $s(N)$. In other words, increasing $g$ enables predators to decrease their search rate, a feature with important implications that will be discussed later. Also note that when setting $g=0$ in $(\ref{search})$, we recover a constant search rate, as in the Rosenzweig-MacArthur model. We can also easily generalize the conceptual framework used earlier to derive the formula $(\ref{speed})$, to the current context where the search rate is dependent on the prey density $N$. It suffices to assume that the predator makes its speed $v$ dependent on $N$. Specifically, choosing $v(N)=v_{\max}N/(N+g)$ expresses that the predator interpolates its speed nonlinearly between zero (when $N=0$), $v_{\max}/2$ (when $N=g$), and $v_{\max}$ (when $N$ becomes infinitely large). Replacing $v$ by $v(N)$ in $(\ref{speed})$, and $s$ by $s(N)$, yields $(\ref{search})$, when we set $a=2v_{\max}r$. This provides us once again with a reasonable way to parameterize the model, and let’s us determine the value of $a$ based on predator characteristics ($v_{\max}$, $r$ and $g$) that should be readily available in the literature for many predator species. Starting with Holling’s type II functional response $(\ref{hollingII})$, but replacing $s$ by the expression $s(N)$ in $(\ref{search})$, we obtain the following functional response: \[f(N)=\frac{aN^{2}}{ahN^{2}+N+g}\] (6) The main qualitative features this functional response shares with Holling’s type II functional response, is that it is smooth, zero when $N$ equals zero, increasing, and still saturates at $1/h$ for large prey densities. But the main qualitative difference is that its second derivative changes sign from positive to negative at a unique inflection point $N_{0}$. Consequently, this functional response is an example of what in the literature is known as a Holling type III functional response. In this paper, we perform a complete analysis of the global behavior of the model $(\ref{s1})-(\ref{s2})$ when the functional response $f(N)$ is given by $(\ref{functional})$, and compare it to the classical Rosenzweig-MacArthur model obtained when setting $f(N)=f_{II}(N)$ in $(\ref{s1})-(\ref{s2})$. For both models, the most interesting behavior occurs when one assumes that the systems have a steady state where both predator and prey coexist, and when $K>1/ah$ (respectively $K>1/sh$ for the Rosenzweig-MacArthur model). In this case, both models exhibit a dichotomy: Either the coexistence steady state is globally stable, or it is unstable, and then the systems have a unique globally stable limit cycle. But there are fundamental differences between the two models as well. Indeed, one of the main features of the Rosenzweig-MacArthur model is the so-called Paradox of Enrichment [9]. This paradox comes from the observation that for an increased carrying capacity $K$ for the prey (the ’enrichment’ in the paradox), the model can be destabilized, changing its behavior from a system with a globally stable coexistence steady state, to a system with a globally stable limit cycle. This leads to possibly severe fluctuations in both predator and prey that may bring either species close to extinction. For the model presented here, an increase in the carrying capacity $K$ will at first also lead to a similar destabilization phenomenon in some, but interestingly, not in all cases. If the system is destabilized, predators can adaptively lower their search rate (by increasing the model parameter $g$), which in turn lets the system regain its pre-existing behavior characterized by the globally stable coexistence steady state. Our results offer an intriguing evolutionary mechanism that may allow predator-prey systems to cope with the dangers associated to enrichment in the prey’s resource. ## 2 Preliminaries We start by showing that the model $(\ref{s1})-(\ref{s2})$ with $(\ref{functional})$ is well-posed. Lemma 1.: _All solutions of $(\ref{s1})-(\ref{s2})$ with functional response $(\ref{functional})$ remain in the non-negative orthant $\mathbb{R}^{2}_{+}$ when initiated there, exist for all times $t>0$, and remain bounded._ Proof.: For all $B>0$, consider the triangular regions \[T_{B}=\{(N,P)\in\mathbb{R}^{2}_{+}\,\|\,N+P\leq B\}.\] We claim that $T_{B}$ is forward invariant for all sufficiently large $B$. This see this, we check that the vector field of the system is inward-pointing on the boundary of each such $T_{B}$. For the boundary parts where $N=0$ or where $P=0$, this is straightforward, where in fact it holds for all $B>0$. To see why it holds when $N+P=B$, note that then \[{\dot{N}}+{\dot{P}}=rN\left(1-\frac{N}{K}\right)-m(B-N)=-\frac{r}{K}N^{2}+(r-m )N-mB,\] which is negative for all $N\geq 0$, provided that: \[(r-m)^{2}<4\frac{r}{K}mB.\] Thus, the vector field is inward-pointing on this part of the boundary of $T_{B}$, provided that $B$ is sufficiently large. ∎ Prey-nullcline: For all $N>0$, we define the prey-nulline \[P=h(N),\] (7) where \[h(N):=\frac{rN\left(1-\frac{N}{K}\right)}{f(N)}=\frac{r}{a}\left(\left(1-\frac {N}{K}\right)(ahN+1)+g\left(\frac{1}{N}-\frac{1}{K}\right)\right).\] (8) It is clear that for fixed $K>0$, the function is smooth for all $N>0$, positive for $0<N<K$, zero at $N=K$, and negative for $N>K$, and that the graph of $h(N)$ has a vertical asymptote at $N=0$. We need to understand better the qualitative properties of the graph of $h(N)$ on the interval $(0,K]$, which is why we calculate the derivatives of $h(N)$: \[h^{\prime}(N) = \frac{r}{a}\left(-2\frac{ah}{K}N+ah-\frac{1}{K}-\frac{g}{N^{2}}\right)\] (9) \[h^{\prime\prime}(N) = \frac{r}{a}\left(-2\frac{ah}{K}+2\frac{g}{N^{3}}\right)\] (10) \[h^{\prime\prime\prime}(N) = -6\frac{rg}{aN^{4}}<0,\textrm{ for all }N>0.\] (11) Case 1: $K-1/ah\leq 0$. In this case it is clear that $h^{\prime}<0$ for all $N>0$, and thus $h(N)$ is decreasing on $(0,K]$. Case 2: $K-1/ah>0$. In this case there are two possibilities: Either $h^{\prime}(N)<0$ for all $N>0$, and then $h(N)$ is decreasing on $(0,K]$ as in Case 1. Or, there exist $N_{\min}$ and $N_{\max}$ in the interval $(0,(K-1/ah)/2)$, with $N_{\min}\leq N_{\max}$ such that: \[h^{\prime}(N)=\begin{cases}<0,\textrm{ if }0<N<N_{\min}\textrm{ and if }N_{ \max}<N\leq K\\ 0,\textrm{ if }N=N_{\min}\textrm{ and if }N=N_{\max}\\ >0,\textrm{ if }N_{\min}<N<N_{\max}\end{cases}\] (12) When $N_{\min}=N_{\max}$, then $h(N)$ is decreasing on $(0,K]$. But when $N_{\min}<N_{\max}$, the function $h(N)$ has a unique local minimum at $N=N_{\min}$, and a unique local maximum at $N=N_{\max}$ in the interval $(0,K]$. Furthermore, $h(N)$ is decreasing on $(0,N_{\min})$ and on $(N_{\max},K)$, but increasing on $(N_{\min},N_{\max})$, and has a unique inflection point at $N=N_{i}$, where: \[N_{\min}<N_{i}<N_{\max},\textrm{ and }N_{i}^{3}=\frac{Kg}{ah},\] (13) and where $h^{\prime\prime}(N)$ switches from positive to negative when crossing $N=N_{i}$. In summary, the function $h(N)$ is either decreasing on $(0,K]$, or it is not. In the latter case, $h(N)$ has exactly two critical points for $N$ in $(0,K]$ (one is a local minimum, the other a local maximum for $h$), and a unique inflection point located between the two critical points. Both possibilities of Case 2 are illustrated in Figure 1. <figure><img src="content_image/1906.05960/null1.png"><figcaption>(a) Decreasing h(N)</figcaption></figure> Predator nullcline: The predator nullcline is determined by the equation $f(N)=m$, where $f(N)$ is given by $(\ref{functional})$. Since $f$ is increasing, the equation $f(N)=m$ has a unique positive solution at $N=N^{}$ if and only if $m<1/h$. For convenience we define $N^{}=+\infty$ if $m\geq 1/h$. Note that when $N^{}<+\infty$, the predator nullcline is given by the vertical line $N=N^{}$ in the phase plane $\mathbb{R}^{2}_{+}$ of the system. From the qualitative behavior of the prey and predator nullclines follows that the model has a unique coexistence steady state $E^{}=(N^{},P^{})$ with $P^{}=h(N^{})$, if and only if: \[N^{}<K.\] (14) We shall first consider the less interesting case when model $(\ref{s1})-(\ref{s2})$ with $(\ref{functional})$ has no coexistence steady state, or equivalently, when $N^{}\geq K$. The proof is omitted since it is easily obtained using standard phase plane arguments. Theorem 1.: _Assume that $N^{}\geq K$. Then system $(\ref{s1})-(\ref{s2})$ with $(\ref{functional})$ has two steady states $E_{0}=(0,0)$ and $E_{1}=(K,0)$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}=0$, converge to $E_{0}$ at $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}>0$, converge to $E_{1}$ as $t\to+\infty$. In particular, the predator always goes extinct._ This result is not surprising: It says that if the predator’s break-even density $N^{}$ equals or exceeds the prey’s carrying capacity, then the predator is doomed. Next, we turn to a more interesting scenario where the model has a unique coexistence steady state $E^{}=(N^{},P^{})$, but where the prey-nullcline is assumed to decrease on $(0,K]$: Theorem 2.: _Assume that $N^{}<K$, and that $h(N)$ is decreasing for $N$ in $(0,K]$. Then system $(\ref{s1})-(\ref{s2})$ with $(\ref{functional})$ has three steady states $E_{0}=(0,0)$, $E_{1}=(K,0)$ and a coexistence steady state $E^{}=(N^{},P^{})$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}=0$, converge to $E_{0}$ at $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}>0$ and $P_{0}=0$, converge to $E_{1}$ as $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}>0$ and $P_{0}>0$, converge to $E^{}$ as $t\to+\infty$._ Proof.: The existence of the 3 steady states $E_{0}$, $E_{1}$ and $E^{}$ is immediate. Linearization of the system yields the following Jacobian matrices at these steady states: \[J(E_{0})=\begin{pmatrix}r&0\\ 0&-m\end{pmatrix},\,J(E_{1})=\begin{pmatrix}-r&-f(K)\\ 0&f(K)-m\end{pmatrix},\textrm{ and }\] \[J(E^{})=\begin{pmatrix}r(1-2N^{}/K)-f^{\prime}(N^{})P^{}&-m\\ P^{}f^{\prime}(N^{})&0\end{pmatrix}=\begin{pmatrix}\frac{m}{r}h^{\prime}(N^{ })&-m\\ P^{}f^{\prime}(N^{})&0\end{pmatrix},\] From this follows that $E_{0}$ is a saddle, and so is $E_{1}$ because $N^{}<K$ implies that $m=f(N^{})<f(K)$ ($f$ is increasing). Finally, if $h^{\prime}(N^{})<0$, then $E^{}$ is a stable because the trace of $J(E^{})$ is negative and its determinant is positive; if $h^{\prime}(N^{})=0$, then $E^{}$ is a center. The statements regarding the convergence of solutions initiated on the boundary of $\mathbb{R}^{2}_{+}$ are obvious because the boundary is invariant. Thus, to conclude the proof, it suffices to show that every solution initiated in the interior of $\mathbb{R}^{2}_{+}$ converges to $E^{}$. Let $\omega(N_{0},P_{0})$ be the omega limit set of such a solution. By Lemma 1, $\omega(N_{0},P_{0})$ is non-empty. Standard arguments show that neither $E_{0}$, nor $E_{1}$ can belong to $\omega(N_{0},P_{0})$. We shall only prove that $E_{0}$ cannot belong to $\omega(N_{0},P_{0})$, because the argument is similar for $E_{1}$. By contradiction, suppose that $\omega(N_{0},P_{0})$ contains $E_{0}$. Then this limit set cannot be equal to the singleton $\{E_{0}\}$ because that would imply that $(N_{0},P_{0})$ belongs to the stable manifold of the saddle $E_{0}$. But this stable manifold coincides with the non-negative $P$-axis, which would contradict that $(N_{0},P_{0})$ belongs to the interior of $\mathbb{R}^{2}_{+}$. Thus, $\omega(N_{0},P_{0})$ would then also have to contain a point distinct from $E_{0}$, and then the Butler-McGehee Lemma [10] implies that $\omega(N_{0},P_{0})$ must also contain a point of the stable manifold of $E_{0}$, distinct from $E_{0}$. Thus, some point on the positive $P$-axis would be contained in $\omega(N_{0},P_{0})$, and then forward and backward invariance of omega limit sets would imply that $\omega(N_{0},P_{0})$ contains the entire positive $P$-axis, contradicting compactness of $\omega(N_{0},P_{0})$. To conclude the proof, we must show that $\omega(N_{0},P_{0})=\{E^{}\}$. To do that we invoke the Poincaré-Bendixson Theorem. If we can establish that the system does not have a nontrivial periodic solution, then the proof will be completed. To rule out periodic solutions, we shall use the Bendixson-Dulac criterion. First we note that any periodic solution must necessarily be located in the open strip $S=\{(N,P)\in\mathbb{R}^{2}_{+}\,\|\,0<N<K\textrm{ and }P>0\}$. Indeed, this follows from the fact that the non-negative $N$-axis, and the non-negative $P$-axis are forward invariant for the system, and because $dN/dt\leq-f(N)P<0$ when $N\geq K$ and $P>0$. Next, we multiply the vector field in $(\ref{s1})-(\ref{s2})$ by the function $1/(Pf(N))$ and take the divergence of the scaled vector field, to obtain: \[h^{\prime}(N)\] (15) Recall that by assumption, $h(N)$ is decreasing for $N$ in $(0,K]$. If $h^{\prime}(N)<0$ for all $N$ in $(0,K]$, then $(\ref{divergence0})$ is negative everywhere in $S$, which concludes the proof in this case. A very special case may occur where $h^{\prime}(N)$ is not negative, but only non-positive for all $N\in(0,K]$. However, in this case $h^{\prime}(N)$ will have a unique zero in this interval. This happens if and only if $N_{\min}=N_{i}=N_{\max}$ (see the earlier discussion of the prey nullcline), and then the zero of $h^{\prime}(N)$ occurs at this very value. It is clear that in this case, $(\ref{divergence0})$ is still negative almost everywhere in $S$. ∎ Theorems 1 and 2 leave us with one last case to consider, namely when a unique coexistence steady state $E^{}=(N^{},P^{})$ with $N^{}<K$ exists, and when the graph of the prey-nullcline $h(N)$ is not decreasing for $N$ in $(0,K)$, and instead has a local minimum and a local maximum with a unique inflection point sandwiched between the two critical points. The next Section will be devoted to the analysis of this case, but before we proceed, we discuss a key property regarding the location of the inflection point $N_{i}$ of the non-monotone function $h(N)$, and the unique inflection point $N_{0}$ of the function $f(N)$: Lemma 2.: _Assume that $h(N)$ is non-monotone for $N$ in $(0,K]$, and let $N_{i}$ be the unique inflection point of $h(N)$ for $N$ in $(0,K]$, and $N_{0}$ be the unique inflection point of $f(N)$ for $N\geq 0$ respectively. Then_ \[N_{0}<N_{i},\textrm{ and thus }\] \[f^{\prime\prime}(N)<0\textrm{ for all }N\geq N_{i}.\] (16) Proof.: Let’s first locate the inflection point of $f(N)$: \[f^{\prime}(N) = \frac{aN(N+2g)}{(ahN^{2}+N+g)^{2}},\] (17) \[f^{\prime\prime}(N) = 2a\frac{(N+g)(ahN^{2}+N+g)-(N^{2}+2gN)(2ahN+1)}{(ahN^{2}+N+g)^{3}}\] (18) \[= 2a\frac{-ahN^{3}-3gahN^{2}+g^{2}}{(ahN^{2}+N+g)^{3}}=:2a\frac{G( N)}{(ahN^{2}+N+g)^{3}}\] Thus, since $G^{\prime}(N)<0$ for all $N>0$, and $G(0)>0$, there exists a unique $N_{0}>0$ such that $G(N_{0})=f^{\prime\prime}(N_{0})=0$. Moreover, $f^{\prime\prime}(N)>0\,(f^{\prime\prime}(N)<0)$ for all $N<N_{0}\,(N>N_{0})$. By assumption, $h(N)$ is non-monotone in $(0,K]$, hence there exist $N_{\min}$ and $N_{\max}$ in $(0,K]$ with $N_{\min}<N_{\max}$, such that $h^{\prime}(N_{\min})=h^{\prime}(N_{\max})=0$. Then there exists $N_{i}$ in $(N_{\min},N_{\max})$ such that $h^{\prime\prime}(N_{i})=0$, and from $(\ref{hprime-signs})$ follows that $h^{\prime}(N_{i})>0$. From $(\ref{h2})$ we see that $N_{i}$ is uniquely determined by: \[N_{i}^{3}=\frac{Kg}{ah},\] (19) Then $h^{\prime}(N_{i})>0$, is equivalent to: \[-2\frac{ah}{K}N_{i}+ah-\frac{1}{K}-\frac{g}{N_{i}^{2}}>0\quad\Leftrightarrow \quad ahN_{i}^{2}>2\frac{ah}{K}N_{i}^{3}+g+\frac{N_{i}^{2}}{K},\] and using $(\ref{inflec})$ this implies that: \[ahN_{i}^{2}>3g+\frac{N_{i}^{2}}{K}.\] (20) Our goal is to show that $N_{0}<N_{i}$, or equivalently that $G(N_{i})<0$. There holds that: \[G(N_{i}) = -ahN_{i}^{3}-3gahN_{i}^{2}+g^{2}\] \[= -ahN_{i}^{3}+g(g-3ahN_{i}^{2})\] \[< -ahN_{i}^{3}+g\left(-8g-\frac{3N_{i}^{2}}{K}\right),\textrm{ by } (\ref{key})\] \[< 0,\] which concludes the proof. ∎ ## 3 Dichotomy We now investigate the most interesting case, which occurs when the system has a unique coexistence steady state $E^{}=(N^{},P^{})$, and when the prey nullcline $P=h(N)$ is not decreasing for $N$ in $(0,K]$. We have seen before that in this case the prey nullcline has two critical points for $N$ in $(0,K]$, namely a local minimum at $N=N_{\min}$ and a local maximum at $N=N_{\max}$, with a unique inflection point at $N=N_{i}$, where $N_{\min}<N_{i}<N_{\max}$. Recall that the predator nullcline is given by the vertical line $N=N^{}$. Depending on the location of $N^{}$ in comparison to the critical points $N_{\min}$ and $N_{\max}$ of the prey nullcline $P=h(N)$, we will see that the system displays two distinct dynamical behaviors. When $0<N^{}<N_{\min}$, or when $N_{\max}<N^{}<K$, the system has a unique, globally stable steady state. This case will be discussed in the next Subsection. When $N_{\min}<N^{}<N_{\max}$, the system displays a unique, globally stable limit cycle. This case will be shown in the second Subsection. ### Globally stable coexistence steady state $E^{}$ Our first main result is as follows: Theorem 3.: _Assume that $N^{}<K$, and that $h(N)$ has a local minimum at $N=N_{\min}$ and a local maximum at $N_{\max}$, where $0<N_{\min}<N_{\max}<K$. Furthermore, assume that either_ \[N_{\max}<N^{},\] (21) _or that_ \[N^{}<N_{\min}.\] (22) _Then system $(\ref{s1})-(\ref{s2})$ with $(\ref{functional})$ has three steady states $E_{0}=(0,0)$, $E_{1}=(K,0)$ and a coexistence steady state $E^{}=(N^{},P^{})$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}=0$, converge to $E_{0}$ at $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}>0$ and $P_{0}=0$, converge to $E_{1}$ as $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}>0$ and $P_{0}>0$, converge to $E^{}$ as $t\to+\infty$._ Proof.: Before we start the proof, we must introduce some new notation. Recall that we defined $N^{}$ as the solution to the equation $f(N)=m$. However, with that notation, $m$ is asumed to be fixed, but later in this proof we shall need to treat $m$ as a variable parameter. Thus, we redefine $m$ as $m^{}$. In other words, in this proof, $N^{}$ will denote the unique solution to the equation $f(N)=m^{}$. Now, fixing all parameters $r,K,a,h$ and $g$, but treating $m$ as a variable parameter, the implicit function Theorem implies that $N^{}(m)$ (the unique solution of $f(N)=m$) is a smooth map which is increasing on its domain $(0,1/h)$. It is easy to show that $\lim_{m\to 0+}N^{}(m)=0$, and $\lim_{m\to 1/h}N^{}(m)=+\infty$, which implies that the map $N^{}(m)$ is onto $(0,+\infty)$. We now turn to the proof of Theorem 3. At first, we can apply the same reasoning as the proof of Theorem 2, up to the point where the Bendixson-Dulac criterion is invoked to rule out the existence of nontrivial periodic solutions in the open strip $S=\{(N,P)\,\|\,0<N<K,\textrm{ and }P>0\}$. Since $h(N)$ is no longer decreasing for $N$ in $(0,K]$, the scaling function $1/(Pf(N))$ for the vector field used in the proof of Theorem 2 is no longer appropriate. Instead, we shall consider a different scaling function here, namely $P^{\alpha-1}/f(N)$, where the constant $\alpha$ will be determined later. Scaling the vector field in $(\ref{s1})-(\ref{s2})$ (but where $m$ is replaced by $m^{}$, for reasons discussed earlier) by this function, and then taking the divergence, yields: \[P^{\alpha}\left(h^{\prime}(N)+{\alpha}\left(\frac{f(N)-m^{}}{f(N)}\right)\right)\] (23) Our goal is to show that there exists some $\alpha$ such that this divergence has fixed sign in the strip $S$. For all $m$ in $(0,1/h)$, we define the following function for all $N$ in $(0,K]$ with $N\neq N^{}(m)$: \[\alpha(N,m)=-\frac{f(N)h^{\prime}(N)}{f(N)-m}\] Case 1: $N_{\max}<N^{}$. Recall that $N^{}(m)$ is onto $(0,+\infty)$, and thus there exists $m_{\max}<m^{}$ such that $N^{}(m_{\max})=N_{\max}$. We wish to investigate the graph of the function $\alpha(N,m_{\max})$, and claim that: 1. $\alpha(N,m_{\max})$ is continuous for $N$ in $[0,K]$, and \[\lim_{N\to 0+}\alpha(N,m_{\max})=-\frac{r}{m_{\max}},\textrm{ and }\lim_{N\to N _{\max}}\alpha(N,m_{\max})=:\alpha^{}>0.\] 2. $\alpha(N,m_{\max})$ is increasing on $[0,K]$. 3. For $m^{}>m_{\max}$, the graph of $\alpha(N,m_{\max})$ lies above the graph of $\alpha(N,m^{})$ for $N$ in $(N_{\min},N_{\max})$, but below it for $N$ in $(N^{},K]$, see Figure 2. <figure><img src="content_image/1906.05960/alpha1.png"><figcaption>Figure 2: Graph of α(N,m∗) (blue) and α(N,mmax) (green). Parameter values:r=a=h=1, K=7, g=1/7, m∗=0.8294 (and N∗=5), mmax=0.7373 (and Nmax=2.9422).</figcaption></figure> Items 1, 2 and 3, together with the fact that $\alpha(N,m^{})\leq 0$ when $N$ belongs to $(0,N_{\min}]$ or to $[N_{\max},N^{})$, and the fact that $\alpha^{}$ defined in item 1 above is positive, imply that the divergence of the scaled vector field in $(\ref{divergence})$ is negative in the strip $\{(N,P)\,\|\,0<N<K,P>0\}$ when we set $\alpha=\alpha^{}$. Proofs of the 3 items above: 1. We calculate: \[\lim_{N\to 0+}\alpha(N,m_{\max})\] \[= \frac{1}{m_{\max}}\lim_{N\to 0+}\frac{aN^{2}}{ahN^{2}+N+g}.\; \frac{r}{a}\left(-2\frac{ah}{K}N+\left(ah-\frac{1}{K}\right)-\frac{g}{N^{2}}\right)\] \[= \frac{r}{m_{\max}}\lim_{N\to 0+}\frac{1}{ahN^{2}+N+g}.\;\left(-2 \frac{ah}{K}N^{3}+\left(ah-\frac{1}{K}\right)N^{2}-g\right)\] \[= -\frac{r}{m_{\max}}\] By de L’Hopital’s rule: \[\lim_{N\to N_{\max}}\alpha(N,m_{\max})=\lim_{N\to N_{\max}}\frac{-f^{\prime}h^ {\prime}-fh^{\prime\prime}}{f^{\prime}}=\lim_{N\to N^{}}\frac{-fh^{\prime \prime}}{f^{\prime}}=:\alpha^{}>0\] Continuity of $\alpha(N,m_{\max})$ is now obvious. 2. After simplification, and using the specific expression $(\ref{functional})$ of the functional $f(N)$, we have that for $N>0$, the partial derivative of $\alpha$ with respect to $N$ is given by: \[\alpha^{\prime}(N,m_{\max}) = \frac{m_{\max}f^{\prime}h^{\prime}-fh^{\prime\prime}(f-m_{\max})} {(f-m_{\max})^{2}}\] (24) \[= \frac{r}{(f-m_{\max})^{2}(ahN^{2}+N+g)^{2}}g(N),\] where \[g(N) = 2\frac{ah}{K}a(1-m_{\max}h)N^{4}-4m_{\max}\frac{ah}{K}N^{3}+(-6m _{\max}g\frac{ah}{K}+m_{\max}(ah-\frac{1}{K}))N^{2}\] (25) \[+2g(m_{\max}(ah-\frac{1}{K})-a(1-m_{\max}h))N+m_{\max}g\] We wish to show that this function is zero for $N=N_{\max}$, and positive for any $N\neq N_{\max}$ in the interval $(0,K]$. From this, the desired result follows. First, we claim that $g(N)$ has a zero of at least second order at $N_{\max}$ (i.e. $g(N_{\max})=g^{\prime}(N_{\max})=0$). To see this, note that it follows from $(\ref{alpha-prime})$ that for $N>0$: \[rg(N)=(ahN^{2}+N+g)^{2}\left(m_{\max}f^{\prime}h^{\prime}-fh^{\prime\prime}(f- m_{\max})\right),\] from which it is clear that $g(N_{\max})=0$. Furthermore, taking the derivative with respect to $N$, yields: \[rg^{\prime}(N) = 2(ahN^{2}+N+g)(2ahN+1)\left(m_{\max}f^{\prime}h^{\prime}-fh^{ \prime\prime}(f-m_{\max})\right)\] \[+(ahN^{2}+N+g)^{2}\left(m_{\max}f^{\prime\prime}h^{\prime}-(f-m_{ \max})(2f^{\prime}h^{\prime\prime}+fh^{\prime\prime\prime})\right),\] from which also follows that $g^{\prime}(N_{\max})=0$. Thus, there exist constants $\alpha,\beta$ and $\gamma$ such that the 4th order polynomial $g(N)$ can be factored as: \[g(N)=(N-N_{\max})^{2}(\alpha N^{2}+\beta N+\gamma)\] To determine $\alpha$, $\beta$ and $\gamma$, we identify the above expression with $(\ref{quartic})$, which yields: \[\alpha = 2\frac{ah}{K}a(1-m_{\max}h)\] \[\beta = 4\frac{ah}{K}a(1-m_{\max}h)(N_{\max}-N^{}_{0})\] \[\gamma = \frac{m_{\max}g}{(N_{\max})^{2}}\] where $N^{}_{0}$ is the solution to the equation $f(N)=m_{\max}$ but for the case where $g=0$. It is easy to see that $N_{\max}=N^{}(m_{\max})>N^{}_{0}$. Since $m_{\max}$ belongs to $(0,1/h)$, there follows that $\alpha>0$, and then also that $\beta>0$. Finally, $\gamma$ is obviously positive as well. Consequently, $g(N)>0$ for all positive $N\neq N_{\max}$. 3. Observe that for all $m>0$, and as long as $N_{\max}\leq N^{}(m)$: \[\frac{\partial\alpha}{\partial m}(N,m) = -\frac{fh^{\prime}}{(m-f)^{2}}\begin{cases}<0,\textrm{ for }N \textrm{ in }(N_{\min},N_{\max})\\ >0,\textrm{ for }N\textrm{ in }(N^{}(m),K]\end{cases}\] From this follows the statement made in item 3. Case 2: If $N^{}<N_{\min}$, then there exists $m_{\min}>m^{}$ such that $N^{}(m_{\min})=N_{\min}$. This time we investigate the graph of the function $\alpha(N,m_{\min})$. We claim that: 1. $\alpha(N,m_{\min})$ is continuous for $N$ in $[0,K]$, and \[\lim_{N\to 0+}\alpha(N,m_{\min})=-\frac{r}{m_{\min}},\textrm{ and }\lim_{N\to N _{\min}}\alpha(N,m_{\min})=:\alpha^{}<0.\] 2. $\alpha(N,m_{\min})$ is increasing on $[0,K]$. 3. The graph of $\alpha(N,m_{\min})$ lies above the graph of $\alpha(N,m^{})$ for $N$ in $(0,N^{})$, but below it for $N$ in $(N_{\min},N_{\max})$, see Figure 3. The proof of these 3 items is entirely analogous to the proof given in Case 1, and therefore omitted. To conclude the proof in this case, we note that items 1, 2 and 3, together with the fact that $\alpha(N,m^{})\geq 0$ when $N$ belongs to $(N^{},N_{\min})$ or to $(N_{\max},K]$, and the fact that $\alpha^{}$ defined in item 1 above is negative, imply that the divergence of the scaled vector field in $(\ref{divergence})$ is negative in the strip $\{(N,P)\,\|\,0<N<K,P>0\}$ when we set $\alpha=\alpha^{}$. <figure><img src="content_image/1906.05960/alpha2.png"><figcaption>Figure 3: Graph of α(N,m∗) (blue) and α(N,mmax) (green). Parameter values:r=a=h=1, K=7, g=1/7, m∗=0.1220 (and N∗=1/4), mmin=0.2791 (and Nmin=0.5326).</figcaption></figure> ∎ ### Unique stable limit cycle Our second main result is as follows: Theorem 4.: _Assume that $N^{}<K$, and that $h(N)$ has a local minimum at $N=N_{\min}$ and a local maximum at $N_{\max}$, where $0<N_{\min}<N_{\max}<K$. Furthermore, assume that_ \[N_{\min}<N^{}<N_{\max},\] (26) _Then the system $(\ref{s1})-(\ref{s2})$ with $(\ref{functional})$ has three steady states $E_{0}=(0,0)$, $E_{1}=(K,0)$ and a coexistence steady state $E^{}=(N^{},P^{})$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}=0$, converge to $E_{0}$ at $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})$ such that $N_{0}>0$ and $P_{0}=0$, converge to $E_{1}$ as $t\to+\infty$. All solutions with initial condition $(N_{0},P_{0})\neq E^{}$ such that $N_{0}>0$ and $P_{0}>0$, converge to a unique limit cycle as $t\to+\infty$._ Proof.: The existence of the 3 steady states $E_{0}$, $E_{1}$ and $E^{}$ is immediate, and as in the proof of Theorem 2, a linearization argument shows that $E_{0}$ and $E_{1}$ are saddles. The Jacobian matrix at $E^{}$ is: \[J(E^{})=\begin{pmatrix}r(1-2N^{}/K)-f^{\prime}(N^{})P^{}&-m\\ P^{}f^{\prime}(N^{})&0\end{pmatrix}=\begin{pmatrix}\frac{m}{r}h^{\prime}(N^{ })&-m\\ P^{}f^{\prime}(N^{})&0\end{pmatrix},\] which has positive trace, from which follows that $E^{}$ is unstable. To show the existence of a unique, stable limit cycle, we apply Theorem 4.2 in [5]. That result is proved under the assumption that $f(N)$ has a simple zero at $N=0$ (i.e. $f(0)=0$, and $f^{\prime}(0)\neq 0$, see assumption (H3) in [5]), which is not satisfied for the functional $f(N)$ in $(\ref{functional})$ used here. Indeed, the $f(N)$ used here has a zero of order two at $N=0$ (i.e. $f(0)=f^{\prime}(0)=0$ and $f^{\prime\prime}(0)\neq 0$). However, the simplicity of the zero of $f(N)$ at $N=0$ is never used in the proof of Theorem 4.2 in [5]. Finally, the main condition imposed in [5] to establish the existence of a unique periodic solution, is condition (4.18) in that paper. This condition states that a specific function, stated below in $(\ref{unique})$, must be sign-definite for all $N$ in $[0,K]$. However, in our model, this function is only defined for $N$ in $(0,K]$ (this is due precisely to the fact that $f(N)$ has a zero of order two at $N=0$, as pointed out above). But again, this does not create significant problems. Instead, it suffices to check that this function is sign-definite for $N$ in $(0,K]$, and this will suffice to establish the existence of a unique and stable limit cycle. The condition we need to verify is as follows: \[mf^{\prime}h^{\prime}-f(f-m)h^{\prime\prime}\geq 0,\textrm{ for all }0<N\leq K.\] (27) To verify that his condition holds, we shall divide the interval $(0,K]$ into three subintervals, and prove the validity of $(\ref{unique})$ on each subinterval. 1. $0<N\leq N_{i}$: For fixed parameters $r,K,a,h$ and $g$, and once again treating $m$ as a variable parameter, the implicit function Theorem implies that $N^{}(m)$ (the unique solution of $f(N)=m$) is a smooth, and increasing function defined for $m$ in $(0,1/h)$. Recall also that $\lim_{m\to 0+}N^{}(m)=0$ and $\lim_{m\to 1/h}N^{}(m)=+\infty$, which implies that the map $N^{}(m)$ is onto $(0,+\infty)$. Let $m_{\min}<m^{}<m_{\max}$ be the 3 values of $m$ where the function $N^{}(m)$ equals $N_{\min}<N^{}<N_{\max}$ respectively. Using the specific expression $(\ref{functional})$ for the functional $f(N)$, the function appearing on the left hand side of the inequality in $(\ref{unique})$ is: \[mf^{\prime}h^{\prime}-f(f-m)h^{\prime\prime}=\frac{r}{(ahN^{2}+N+g)^{2}}g(N,m),\] (28) where $g(N,m)$ was already defined in $(\ref{quartic})$ (but only for the case that $m=m_{\max}$) as follows: \[g(N,m) = 2\frac{ah}{K}a(1-mh)N^{4}-4m\frac{ah}{K}N^{3}+(-6mg\frac{ah}{K}+ m(ah-\frac{1}{K}))N^{2}\] \[+2g(m(ah-\frac{1}{K})-a(1-mh))N+mg\] Our goal is to show that \[g(N,m^{})\geq 0,\textrm{ for all }0\leq N\leq N_{i}.\] (29) First, note that $g(N,m)$ is linear in $m$, and recalling $(\ref{h2})$ we can re-write $g(N,m)$ as follows: \[g(N,m) = \left(2\frac{a^{2}h}{K}N^{4}-2agN\right)+ms(N)\] (30) \[= -\frac{a^{2}}{r}N^{4}\frac{\partial^{2}h}{\partial N^{2}}(N)+ms(N),\] where \[s(N)=-2\frac{(ah)^{2}}{K}N^{4}-4\frac{ah}{K}N^{3}+(-6g\frac{ah}{K}+ah-\frac{1} {K})N^{2}+2g(2ah-\frac{1}{K})N+g\] We have established in item 2 of the proof of both cases of Theorem 3 that \[g(N,m_{\min})\geq 0,\;\textrm{for all }N\geq 0.\] (31) Now, for every $m$ in $(0,1/h)$, and $N\geq 0$, there holds: \[\frac{\partial g}{\partial m}=s(N).\] Consequently, using $(\ref{aux1})$ and $(\ref{aux2})$, we have that for all $m$ in $(0,1/h)$, and $N$ in $[0,K]$: \[\frac{\partial g}{\partial m}=s(N)=\frac{\partial g}{\partial m}(N,m_{\min}) \geq\frac{a^{2}}{rm_{\min}}N^{4}\frac{\partial^{2}h}{\partial N^{2}}(N)\] (32) But $\partial^{2}h/\partial N^{2}(N)\geq 0$ for $N$ in $(0,N_{i}]$, and thus $(\ref{help})$ and $(\ref{aux2})$ imply that: \[g(N,m)\geq 0\textrm{ for all }m\geq m_{\min}\textrm{ and }N\textrm{ in }(0,N_{ i}].\] In particular, $(\ref{goal})$ holds. 2. $N_{i}\leq N\leq N_{\max}$: We distinguish 2 cases, depending on the relative location of $N^{}$ and $N_{i}$: Case 1: $N_{i}<N^{}$. In this case we divide the interval $[N_{i},N_{\max}]$ into two further subintervals: $N_{i}\leq N\leq N^{}$: To establish that $(\ref{unique})$ holds when $N$ belongs to this interval, we first evaluate the function in the right-endpoint $N^{}$, and see that the function is positive there. Next, we calculate the derivative of this function: \[mf^{\prime\prime}h^{\prime}-(f-m)(2f^{\prime}h^{\prime\prime}+fh^{\prime\prime \prime})\] By inspection it follows that this derivative is negative when $N$ belongs to the interval $[N_{i},N^{}]$ (here, we have used Lemma 2 which implies that $f^{\prime\prime}(N)<0$ when $N\geq N_{i}$). Consequently, the function $mf^{\prime}h^{\prime}-f(f-m)h^{\prime\prime}$ is decreasing on the interval $[N_{i},N^{}]$, and as it is positive in the right-endpoint, the function is positive in the entire interval. * $N^{}\leq N\leq N_{\max}$: It is immediately clear that $(\ref{unique})$ holds when $N$ belongs to the interval $[N^{},N_{\max}]$ by inspection of the signs of the various factors and terms in the function $mf^{\prime}h^{\prime}-f(f-m)h^{\prime\prime}$, given the fact that $N_{i}<N^{}\leq N$ when $N$ belongs to this interval, whence $h^{\prime\prime}(N)\leq 0$. Case 2: $N^{}\leq N_{i}$. In this case, $(\ref{unique})$ is easily seen to hold on the interval $[N_{i},N_{\max}]$, using the same argument as in the second item of Case 1 above. 3. $N_{\max}\leq N\leq K$: We first evaluate the function $mf^{\prime}h^{\prime}-f(f-m)h^{\prime\prime}$ in the left-endpoint $N_{\max}$, and see that the function is positive there. The derivative \[mf^{\prime\prime}h^{\prime}-(f-m)(2f^{\prime}h^{\prime\prime}+fh^{\prime\prime \prime})\] of this function is positive on the interval $[N_{\max},N^{}]$ (here, we have used Lemma 2 which implies that $f^{\prime\prime}(N)<0$ when $N\geq N_{\max}$). Consequently, the function $mf^{\prime}h^{\prime}-f(f-m)h^{\prime\prime}$ is increasing on the interval $[N_{\max},K]$, and as it is positive in the left-endpoint $N_{\max}$, the function is positive in the entire interval. ∎ Hopf bifurcations are supercritical: Theorem 3 and 4 suggest that when $N^{}$ coincides with either $N_{\min}$ (where $h(N)$ achieves a local minimum), or with $N_{\max}$ (where $h(N)$ achieves a local maximum), then a Hopf bifurcation occurs. The Jacobian matrix at the coexistence steady state $E^{}=(N^{},P^{})$ is: \[J(E^{})=\begin{pmatrix}\frac{m}{r}h^{\prime}(N^{})&-m\\ P^{}f^{\prime}(N^{})&0\end{pmatrix},\] and clearly shows that $E^{}$ is a center when $N^{}=N_{\min}$ or $N^{}=N_{\max}$, and also reveals the switch in stability of $E^{}$ when $N^{}$ crosses either $N_{\min}$ or $N_{\max}$: $E^{}$ is a stable spiral when $h^{\prime}(N^{})<0$, and an unstable spiral when $h^{\prime}(N^{})>0$. Moreover, using $N^{}$ as a bifurcation parameter, it is clear that the eigenvalues of $J(E^{})$ cross the imaginary axis transversally when $N^{}$ crosses either $N_{\min}$ or $N_{\max}$. Indeed, the sum of both eigenvalues (which is twice the real part of each eigenvalue) equals $(m/r)h^{\prime}(N^{})$, and the derivative with respect to $N^{}$ of this expression is $(m/r)h^{\prime\prime}(N^{})$, which is positive when $N^{}=N_{\min}$, and negative when $N^{}=N_{\max}$. To determine the nature of the Hopf bifurcation (sub- or supercritical), we determine the sign of the following quantity [3]: \[\Omega(N^{})=h^{\prime\prime}(N^{})\left(2f^{\prime}(N^{})-\frac{f(N^{})f^ {\prime\prime}(N^{})}{f^{\prime}(N^{})}\right)+h^{\prime\prime\prime}(N^{}) f(N^{})\] in the cases where $N^{}=N_{\min}$, and $N^{}=N_{\max}$. When $\Omega(N^{})<0$, the Hopf bifurcation is supercritical, and when $\Omega(N^{})>0$ it is subcritical [3]. We will see that in both cases, $N^{}=N_{\min}$ and $N^{}=N_{\max}$, the Hopf bifurcation is supercritical. Indeed, suppressing a straightforward algebraic calculation using the derivatives $(\ref{h1})$, $(\ref{h2})$ and $(\ref{h3})$ of $h(N)$, and the derivatives $(\ref{f1})$ and $(\ref{f2})$ of $f(N)$, yields that: \[\Omega(N^{})=-\frac{2rf(N^{})}{aN^{}(N^{}+2g)}\left(2\frac{ah}{K}(N^{}+3g )+\frac{3g}{(N^{})^{2}}\right),\] which is clearly negative when $N^{}=N_{\min}$ or $N^{}=N_{\max}$. Consequently, we can generalize the conclusion of Theorem 3, to also include the cases when $N^{}=N_{\min}$, and $N^{}=N_{\max}$: Corollary 1.: _Theorem 3 remains valid if $(\ref{more})$ and $(\ref{less})$ are respectively replaced by_ \[N_{\max}\leq N^{},\textrm{ and }N^{}\leq N_{\min}.\] ## 4 Comparison to the Rosenzweig-MacArthur model Here we shall compare the dynamics of the model studied in this paper, to the Rosenzweig-MacArthur model [8], obtained by setting $f(N)=f_{II}(N)$ (see $(\ref{hollingII})$) in $(\ref{s1})-(\ref{s2})$. But first we offer some historical perspective. Despite the central role of the Rosenzweig-MacArthur model in ecology and mathematical biology, more than 20 years (25 to be precise) has passed between its initial proposal in [8], and a complete and rigorous analysis of its dynamics. The difficulty seems to have been to establish the proof of uniqueness of the limit cycle, which was first announced in [6]. According to [3] however, the proof in [6] contained a flaw, which was fixed later in [7]. A concise analysis of the dynamics of the Rosenzweig-MacArthur model can be found in [1], and is summarized next. First, the Rosenzweig-MacArthur model also always has the extinction steady state $E_{0}=(0,0)$ and the prey-only steady state $E_{1}=(K,0)$, just like the model presented here. The prey nullcline of the Rosenzweig-MacArthur model is a segment of a parabola, given by: \[P=\frac{r}{s}\left(1-\frac{N}{K}\right)(shN+1).\] The maximum of the parabola is located in the interior of the positive orthant $\mathbb{R}^{2}_{+}$ if and only if: \[K>1/sh,\] (33) and in this case this maximum occurs at: \[{\bar{N}}_{\max}:=\frac{1}{2}(K-1/sh)\] (34) The predator nullcline is a vertical line $N=N^{}$, where $N^{}$ is the solution of $f_{II}(N)=m$. Note that $N^{}$ exists if and only if $m<1/h$, a condition which is assumed to hold henceforth. Therefore, the Rosenzweig-MacArthur model has a unique coexistence steady state $E^{}=(N^{},P^{})$ if and only if $P^{}:=(1-N^{}/K)(shN^{}+1)$ is positive, or equivalently when $N^{}<K$. The global dynamics of the Rosenzweig-MacArthur model is summarized next. Theorem 5.: _Consider system $(\ref{s1})-(\ref{s2})$ with $f(N)=f_{II}(N)$ the Holling type II functional response defined in $(\ref{hollingII})$. Assume that $(\ref{up-down})$ holds, and that there exists a unique coexistence steady state $E^{}=(N^{},P^{})$, in addition to the steady states $E_{0}=(0,0)$ and $E_{1}=(K,0)$ which always exist._ Case 1_: If ${\bar{N}}_{\max}\leq N^{}$, then $E^{}$ is globally asymptotically stable with respect to initial conditions $(N_{0},P_{0})$ in the interior of $\mathbb{R}^{2}$._ Case 2_: If $N^{}<{\bar{N}}_{\max}$, then $E^{}$ is unstable, and there exists a unique stable limit cycle which attracts all solutions with initial conditions $(N_{0},P_{0})\neq E^{}$ in the interior of $\mathbb{R}^{2}$._ Comparing this to Corollary 1, we see that the global behavior of the Rosenzweig-MacArthur model exhibits the same dichotomy as the model investigated in this paper: Either the coexistence steady state is globally stable; or it is not, and then a unique, globally stable limit cycle exists. However, a significant difference is that, depending on the location of $N^{}$ -the predator’s break-even density of prey- there is only a single threshold ${\bar{N}}_{\max}$ for $N^{}$ in the Rosenzweig-MacArthur model that separates the two distinct dynamical regimes, and the coexistence steady state is globally stable if and only if ${\bar{N}}_{\max}\leq N^{}$. In the model presented here, there are two thresholds $N_{\min}$ and $N_{\max}$ for $N^{}$, and the globally stable coexistence steady state occurs when $N^{}\leq N_{\min}$, or when $N_{\max}\leq N^{}$ according to Corollary 1. In other words, here the coexistence steady state is globally stable for all sufficiently large, but also for all sufficiently small values of the predator’s break-even density of prey $N^{}$, whereas in the Rosenzweig MacArthur model this only happens for all sufficiently large values of $N^{}$. We shall see in a moment that this phenomenon also has important implications in the context of the paradox of enrichment, first pointed out for the Rosenzweig-MacArthur model in [9]. Before proceeding to that discussion, we investigate how $N_{\min}$ and $N_{\max}$ in the model studied here, vary with the parameters $K$ and $g$. Recall that $N_{\min}$ and $N_{\max}$ are critical points for the function $h(N)$, and thus $h^{\prime}(N_{\min})=h^{\prime}(N_{\max})=0$, where $h^{\prime}(N)$ is given in $(\ref{h1})$. 1. Dependence on $K$: Fixing all model parameters, except for $K$, and assuming that $N_{\min}(K)<N_{\max}(K)$, it follows from implicit differentiation with respect to $K$ of the respective expressions $h^{\prime}(N_{\min}(K))=0$ and $h^{\prime}(N_{\max}(K))=0$, and using that $h^{\prime\prime}(N_{\min}(K))>0$ and $h^{\prime\prime}(N_{\max}(K))<0$, that: \[\frac{dN_{\min}}{dK}(K)<0,\textrm{ and }\frac{dN_{\max}}{dK}(K)>0.\] Moreover, taking limits for $K\to+\infty$ in $h^{\prime}(N_{\min}(K))=0$, and in the inequality $N_{i}=(Kg/ah)^{1/3}<N_{\max}(K)$ -see $(\ref{infl})$- we obtain that: \[\lim_{K\to+\infty}N_{\min}(K)=\left(\frac{g}{ah}\right)^{1/2}=:{\bar{N}}_{\min },\textrm{ and }\lim_{K\to+\infty}N_{\max}(K)=+\infty.\] (35) These results capture what happens when the prey’s carrying capacity $K$ is increased: the gap between the two critical points of the prey nullcline widens, and while $N_{\max}(K)$ grows unbounded, $N_{\min}(K)$ is bounded below and converges to a positive value ${\bar{N}}_{\min}$. 2. Dependence on $g$: Fixing all model parameters except for $g$, and assuming that $N_{\min}(g)<N_{\max}(g)$, implicit differentiation with respect to $g$ yields in a similar fashion that: \[\frac{dN_{\min}}{dg}(g)>0,\textrm{ and }\frac{dN_{\max}}{dg}(g)<0.\] Moreover, taking limits for $g\to 0+$ in the inequality $N_{\min}(g)<N_{i}=(Kg/ah)^{1/3}$ -see $(\ref{infl})$-, and in $h^{\prime}(N_{\max}(g))=0$, we obtain that: \[\lim_{g\to 0+}N_{\min}(g)=0,\textrm{ and }\lim_{g\to 0+}N_{\max}(g)=(K-1/(ah)) /2=:{\bar{N}}_{\max}.\] (36) In other words, the gap between the critical points of the prey nullcline also grows when $g$ is decreased. In this case, $N_{\min}(g)$ converges to zero, but $N_{\max}(g)$ is bounded above, and converges to an upper bound ${\bar{N}}_{\max}$. Note that this bound is the same as the single threshold defined in $(\ref{max})$ for the Rosenzweig-MacArthur model (when we set $a=s$). Paradox of enrichment (or lack thereof)* To see why these properties are important in the context of the paradox of enrichment, we first review this paradox for the Rosenzweig-MacArthur model. Suppose that initially, the system parameters are such that $(K-1/(sh))/2={\bar{N}}_{\max}(K)\leq N^{}<K$. By Theorem 5, the coexistence steady state $E^{}$ is globally stable. If all model parameters remain fixed, except for $K$, and if we assume that $K$ is increased to a new value $K_{new}>K$, such that $N^{}<{\bar{N}}_{\max}(K_{new})$, then the coexistence steady state is destabilized. The paradox of enrichment is precisely this destabilization phenomenon, illustrated in Figure 4. <figure><img src="content_image/1906.05960/RMstab.png"><figcaption>(a) ¯Nmax(K)<N∗</figcaption></figure> <figure><img src="content_image/1906.05960/ourstab1.png"><figcaption>(a) Nmax(K)<N∗</figcaption></figure> <figure><img src="content_image/1906.05960/ourstab2.png"><figcaption>(a) N∗<¯Nmin<Nmin(K)</figcaption></figure> Let us now investigate whether the paradox of enrichment also occurs for the model presented in this paper. According to Corollary 1, there are two distinct possible initial scenarios that correspond to having a system with a globally stable coexistence steady state: Either $N_{\max}(K)\leq N^{}<K$, or $0<N^{}\leq N_{\min}(K)$. In both cases we shall determine what happens when all model parameters -except for $K$- remain fixed, and when $K$ increases to a new value $K_{new}>K$. If initially $N_{\max}(K)\leq N^{}<K$, then by $(\ref{K-dep})$ there exist sufficiently large $K_{new}>K$ such that $N_{\min}(K_{new})<N^{}<N_{\max}(K_{new})$, which destabilizes the coexistence steady state $E^{}$, as illustrated in Figure 5. Similarly, if initially $0<N^{}\leq N_{\min}(K)$, and if also ${\bar{N}}_{\min}<N^{}$, then there exist sufficiently large $K_{new}>K$, such that $N_{\min}(K_{new})<N^{}<N_{\max}(K_{new})$, once again destabilizing the coexistence steady state $E^{}$. However, if initially $0<N^{}\leq N_{\min}(K)$, and $N^{}\leq{\bar{N}}_{\min}$, then there are no $K_{new}>K$ that can destabilize $E^{}$, as illustrated in Figure 6. . This follows from $(\ref{K-dep})$ because $N^{}\leq{\bar{N}}_{\min}<N_{\min}(K_{new})$, for all $K_{new}>K$. In other words, in this last case, the paradox of enrichment does not occur for the model studied here, which is a striking difference with the Rosenzweig-MacArthur model, where the paradox of enrichment always occurs. The role of ${\bar{N}}_{\min}$, defined in $(\ref{K-dep})$, is that it serves as a buffer: When initially $N^{}\leq{\bar{N}}_{\min}$, the system cannot be destabilized by any enrichment event in the prey’s carrying capacity. Stabilizing effect when predators decrease their search rate* We shall now discuss an important feature of the model studied here that is absent from the Rosenzweig-MacArthur model. Suppose that the system parameters are initially such that the coexistence steady state is unstable, and that a unique globally stable limit cycle exists. This may be the result of an enrichment event for the prey’s carrying capacity $K$ as described above. Our goal is to show that the predator can respond to this by modifying its behavior in a way that stabilizes the coexistence steady state. To achieve this, the predator should simply increase the value of $g$. Recall that this corresponds to a decrease in its search rate $s(N)$ in $(\ref{search})$, for every $N>0$. To see why this happens, assume that all parameters except for $g$ are fixed, and that $g$ will be increased to $g_{new}>g$. <figure><img src="content_image/1906.05960/searchdestab.png"><figcaption>(a) Nmin(g)<N∗<Nmax(g)</figcaption></figure> Thus, we assume that initially $N_{\min}(g)<N^{}<N_{\max}(g)$, implying that $E^{}$ is unstable and that the system has a unique globally stable limit cycle by Theorem 4. If $g_{new}$ is chosen sufficiently large, then we can ensure that $h^{\prime}(N)<0$ for all $N$ in $(0,K]$, effectively making the prey nullcline decreasing in $N$, as illustrated in Figure 7. It follows from Theorem 3, that in this case $E^{}$ is globally stable, which establishes our claim. We can get a better idea of how quickly this happens by considering $(\ref{g-dep})$. By increasing $g$, the gap between $N_{\min}(g)$ and $N_{\max}(g)$ shrinks, and both move towards $N^{}$. Global stability of $E^{}$ will occur for the first time, when either $N_{\min}(g)$ or $N_{\max}(g)$ collides with $N^{}$ (by Corollary 1). Destabilizing effect (or lack thereof) when predators increase their search rate To conclude we will demonstrate how an increased predator’s search rate $s(N)$, realized by decreasing the parameter $g$, may destabilize a globally stable coexistence steady state in certain cases, but not in all cases in the model investigated in this paper. The mechanism turns out to be similar to how the paradox of enrichment following an enrichment event in the prey’s carrying capacity can sometimes be avoided, as described above. Suppose that initially, $g$ is such that $E^{}$ is globally stable. According to Corollary 1, this means that either $0<N^{}\leq N_{\min}(g)$, or $N_{\max}(g)\leq N^{}<K$. If $0<N^{}\leq N_{\min}(g)$, it follows from $(\ref{g-dep})$, there exist sufficiently small $g_{new}$ such that $N_{\min}(g_{new})<N^{}<N_{\max}(g_{new})$, effectively destabilizing $E^{}$. If $N_{\max}(g)\leq N^{}<K$, and if also $N^{}<{\bar{N}}_{\max}$, then there exist sufficiently small $g_{new}$ such that $N_{\min}(g_{new})<N^{}<N_{\max}(g_{new})<{\bar{N}}_{\max}$, which again destabilizes $E^{}$. But if $N_{\max}(g)\leq N^{}<K$, and if also ${\bar{N}}_{\max}\leq N^{}$, then no matter how small $g_{new}$ is chosen, $(\ref{g-dep})$ implies that $N_{\max}(g_{new})<{\bar{N}}_{\max}\leq N^{}$, and then $E^{}$ remains globally stable. Thus, whenever ${\bar{N}}_{\max}\leq N^{}$, there are no limits to increases in the predator’s search rate $s(N)$ that can destabilize the system. The bound ${\bar{N}}_{\max}$ in $(\ref{g-dep})$ also serves as a buffer for the predator’s break-even prey density $N^{}$, in the sense that if $N^{}$ is larger than ${\bar{N}}_{\max}$, destabilization cannot occur following an increase in the predator’s search rate. As a final comment, we point out that ${\bar{N}}_{\max}$ corresponds to the prey density where the parabola of the prey nullcline in the Rosenzweig-MacArthur model achieves its maximum (when setting $a=s$). This is not surprising, because taking $g\to 0$ in our model, yields the Rosenzweig-MacArthur model, and when $N^{}$ is to the right of this maximum, Theorem 5 implies that $E^{}$ is globally stable. ## 5 Conclusions Rosenzweig-MacArthur’s predator-prey model employs a Holling type II functional response which is predicated on the assumption that the predator’s search rate is constant, and independent of the prey density. It seems plausible however that predators can modify their search rate, and instead adapt it based on the prey’s density. The goal of this paper was to examine the implications on the model behavior when replacing the constant search rate in the Rosenzweig-MacArthur model by a density-dependent search rate $s(N)=aN/(N+g)$, which effectively leads to a Holling type III functional response in the model instead. The following summarizes our findings: 1. We provided a complete global analysis of the dynamics of the model , showing that just like the Rosenzweig-MacArthur model, the model investigated here exhibits a dichotomy: Either the coexistence steady state is globally stable; or, it is unstable, and then a unique globally stable limit cycle exists (Theorems 3, 4 and Corollary 1). 2. Whereas there is a single threshold ${\bar{N}}_{\max}$ for the predator’s break-even prey density $N^{}$, that determines which of the two possible regimes occurs in the Rosenzweig-MacArthur model, the model presented here can have two thresholds, $N_{\min}<N_{\max}$. If the predator’s break-even prey density $N^{}$ is such that either $N^{}\leq N_{\min}$, or if $N_{\max}\leq N^{}$, then the model has a globally stable coexistence steady state. When $N^{}$ is sandwiched between $N_{\min}$ and $N_{\max}$, there is a unique, globally stable limit cycle. 3. Whereas the Rosenzweig-MacArthur model always exhibits the paradox of enrichment -a destabilization phenomenon that occurs for all sufficiently strong enrichment events in the prey’s carrying capacity $K$- this is not always the case for the model presented here. We identified a threshold ${\bar{N}}_{\min}=(g/ah)^{1/2}$, such that if $N^{}\leq{\bar{N}}_{\min}$, the model can never be destabilized following an enrichment of the prey’s carrying capacity. 4. In those cases where the model studied here, does exhibit destabilization following enrichment in the prey’s carrying capacity, the predator can adapt by lowering its search rate, and then the system can always be stabilized again, provided the reduction in the predator’s search rate is large enough. This offers an intriguing evolutionary explanation for how predators may have evolved to respond to enrichment events experienced by the prey. Other mechanisms that can stabilize predator-prey dynamics have been proposed, that rely on certain hypothesized movement patterns of predators and/or prey. Discrete-time, nonlinear host-parasitoid models with aggregation of parasitoids -and where parasitoid aggregation may or may not depend on prey density- were investigated in [11] and generalized in [12]. A continuous-time, 2-patch predator-prey system with a diffusive predator but static prey was considered in [13]. For a more recent review of predator-prey models that incorporate movement of predators and/or prey, as well as spatial heterogeneities in the environment, see [14]. Most of these models are quite complicated due to the fact that explicit decisions have to be made about how the two species move, and because there is a large number of possible scenarios to choose from in this context. Some of these choices are targeted to capture the movement patterns of predators and prey for very specific systems, which may not apply more generally. In contrast, the model presented here neglects explicit spatial effects. Consequently, no decisions on how the two species move have to be made at any stage in the modeling process. Despite the hypothesis of a well-mixed environment, our results indicate that a very simple mechanism -namely, the biologically reasonable assumption that predators adapt their search rate based on the perceived prey density- always exhibits stabilizing effects on the predator-prey dynamics. To conclude this paper, we point out that the choice of the search rate $s(N)=aN/(N+g)$ employed here, is very specific. It would be reasonable to ask how robust our conclusions are with respect to changes in this functional $s(N)$. Further research will be needed to answer this question. To caution against unwarranted optimism, we refer to the recent intriguing results in [3], where the dynamics of three predator-prey models with distinct functional responses was considered. All three functional responses qualitatively resembled the Holling type II functional response of the Rosenzweig-MacArthur model in the sense that $f(N)$ was assumed to be smooth, zero at $N=0$, increasing but bounded above, and concave (i.e. $f^{\prime\prime}(N)<0$ for all $N>0$). Based on these common features of the functional responses, it would be reasonable to expect that these models would exhibit the same, or at least similar behavior as the Rosenzweig-MacArthur model. Surprisingly, it was shown in [3] that this is not the case. One of the models could have two limit cycles, one stable and the other unstable, surrounding a stable coexistence steady state. This implies that this model is bi-stable, with one attractor being a steady state, and another being a stable limit cycle. It is therefore remarkable that the model presented here, which employs a specific example of a Holling type III functional response $f(N)$ that transitions from being convex to concave for increasing values of $N$, cannot exhibit more complicated behavior than the original Rosenzweig-MacArthur model. ## References [1] Smith, H.L., The Rosenzweig-MacArthur predator-prey model, downloaded from https://math.la.asu.edu/$\sim$halsmith * [2] Dawes, J.H.P., and Souza, M.O., A derivation of Holling’s type I, II and III functional responses in predator-prey systems, Journal of Theoretical Biology 327, p.11-22, 2013. * [3] Seo, G., and Wolkowicz, G.S.K., Sensitivity of the dynamics of the general Rosenzweig-MacArthur model to the mathematical form of the functional response: a bifurcation theory approach, Journal of Mathematical Biology 76, p.1873-1906, 2018. * [4] Geritz, S., and Gyllenberg, M., A mechanistic derivation of the DeAngelis-Beddington functional response, Journal of Theoretical Biology 314, p. 106-108, 2012. * [5] Kuang, Y., and Freedman, H.I., Uniqueness of Limit Cycles in Gause-Type Models of Predator-Prey Systems, Mathematical Biosciences 88, p. 67-84, 1988. * [6] Cheng, K.S., Uniqueness of a limit cycle for a predator-prey system, SIAM Journal on Mathematical Analysis 12, p. 541-548,1981. * [7] Liou, L.P., and Cheng, K.S. On the uniqueness of a limit cycle for a predator-prey system, SIAM Journal on Mathematical Analysis 19, p. 867-878, 1988. * [8] Rosenzweig, M.L., and MacArthur, R.H., Graphical representation and stability conditions of predator-prey interaction, American Naturalist 97, 209-223, 1963. * [9] Rosenzweig, M.L., The Paradox of Enrichment, Science 171, p. 385-387, 1971. * [10]Smith, H.L., and Waltman, P., The theory of the chemostat, Cambridge University Press, 1995. * [11]May, R.M., Host-Parasitoid Systems in Patchy Environments: A Phenomenological Model, Journal of Animal Ecology 47, p. 833-844, 1978. * [12] Chesson, P.L., and Murdoch, W.W., Aggregation of Risk-Relationships Among Host-Parasitoid Models, The American Naturalist 127, p. 696-715, 1986. * [13]Jansen, V.A.A., The Dynamics of Two Diffusively Coupled Predator–Prey Populations, Theoretical Population Biology 59, p. 199-131, 2001. * [14]Briggs, C.J., and Hoopes, M.F., Stabilizing effects in spatial parasitoid-host and predator-prey models: a review. Theoretical Population Biology 65, p. 299-315, 2004.
1810.10535	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 88481, "num_imgs": 19, "llama3_tokens_count": 21343 }	[ "content_image/1810.10535/AgentEnvironmentFigure.png", "content_image/1810.10535/x1.png", "content_image/1810.10535/x5.png", "content_image/1810.10535/game1_playstep_example.png", "content_image/1810.10535/selfplay_learn.png", "content_image/1810.10535/MCTS_graph.png", "content_image/1810.10535/game1_violinplot.png", "content_image/1810.10535/game1_trainning_session.png", "content_image/1810.10535/game1_predicts.png", "content_image/1810.10535/x7.png", "content_image/1810.10535/game2_violinplot.png", "content_image/1810.10535/game2_trainning_session.png", "content_image/1810.10535/game2_predicts.png", "content_image/1810.10535/SinusInterfaceMesh.png", "content_image/1810.10535/FEMLSTM_diffstrain_50.png", "content_image/1810.10535/FEMLSTM_UsTs.png", "content_image/1810.10535/DataDEMRVE.png", "content_image/1810.10535/DEMRVE_UnTnUsTs_120.png", "content_image/1810.10535/DEMRVE_CN.png" ]	# Meta-modeling game for deriving theoretical-consistent, micro-structural-based traction-separation laws via deep reinforcement learning Kun Wang WaiChing Sun Corresponding author: WaiChing Sun, Assistant Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University , 614 SW Mudd, Mail Code: 4709, New York, NY 10027 _wsun@columbia.edu_ ###### Abstract This paper presents a new meta-modeling framework to employ deep reinforcement learning (DRL) to generate mechanical constitutive models for interfaces. The constitutive models are conceptualized as information flow in directed graphs. The process of writing constitutive models are simplified as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). Thus meta-modeling can be formulated as a Markov decision process with well-defined states, actions, rules, objective functions and rewards. By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive model it generated through self-playing, in the same way AlphaGo Zero (the algorithm that outplayed the world champion in the game of Go) improves its gameplay. Our numerical examples show that this automated meta-modeling framework not only produces models which outperform existing cohesive models on benchmark traction-separation data, but is also capable of detecting hidden mechanisms among micro-structural features and incorporating them in constitutive models to improve the forward prediction accuracy, which are difficult tasks to do manually. ## 1 Introduction Constitutive responses of interfaces are important for a wide spectrum of problems that involve spatial domain with embedded strong discontinuity, such as fracture surfaces (Rice, 1968; Park et al., 2009; Wang and Sun, 2017; Bryant and Sun, 2018), slip lines (Rabczuk and Areias, 2006; Borja and Foster, 2007), joints (Elices et al., 2002) and faults (Ohnaka and Yamashita, 1989; Wang and Sun, 2018; Sun and Wong, 2018). While earlier modeling efforts, in particular those involving the modeling of cohesive zones , often solely focus on mode I kinematics, the mixed mode predictions of traction-separation law relations are critical for numerous applications, ranging from predicting damage upon impacts (Ortiz and Pandolfi, 1999), to predicting seismic events (Rudnicki, 1980). Recent work by Park et al. (2009) provide a comprehensive account of the major characteristics of traction-separation laws and conclude that, while there are differences in details, most of the traction-separation laws obey a number of universal principles, such as the indifference of any superimposed rigid-body motion, the finite work required to create new surface, the existence of characteristics length scales, and the vanishing of cohesive traction with sufficient separations. In the case where the loading history is not monotonic, constitutive responses of interfaces often become path-dependent. For instance, geomaterials, such as fault gauges, are known to exhibit rate- and state-dependent frictional responses (Paterson and Wong, 2005; Sun, 2013; Borja, 2013; Wang et al., 2016; Na and Sun, 2017). While there are phenomenological models designed to capture the path-dependent responses of the interfaces, a recent trend that gains increasing popularity is to replace the phenomenological traction-separation laws with computational homogenization procedure to capture the responses of materials with heterogeneous microstructures (cf. Moës et al. (2003); Hirschberger et al. (2008, 2009). Nevertheless, as pointed out previously in Wang and Sun (2018), the major issue of applying hierarchical multiscale coupling on interfacial problems is the increasing computational demand due to the large number of required representative elementary simulations, a trade-off that is widely known in FEM${}^{2}$(Feyel, 2003) and other homogenization-based multiscale methods, such as FEM-DEM (Sun et al., 64, 60; Fish, 2013; Sun et al., 65; Wang and Sun, 2015; Liu et al., 36; Kuhn et al., 2015; Wang and Sun, 2016; Wu et al., 2018). To overcome this computational barrier, surrogate models are often derived to replicate the homogenized responses of sub-scale simulations (Kirane and Ghosh, 2008; Verhoosel et al., 2010; Keshavarz and Ghosh, 2013; Panchal et al., 2013; Faisal et al., 2014; Liu et al., 35; Tallman et al., 2017). Nevertheless, since surrogate models are often constitutive laws hand-crafted by modelers to incorporate morphology-dependent features (Liu et al., 35), deriving, verifying and validating a surrogate model that can incorporate the essential information to yield macroscopic predictions with sufficient accuracy and robustness remain difficult and time-consuming. Data-driven models such as Le et al. (2015); Bessa et al. (2017); Versino et al. (2017); Kafka et al. (2018) and Wulfinghoff et al. (2018) attempted to overcome this issue via supervised machine learning (e.g. neural network (Lefik and Schrefler, 2002), symbolic regression model (Versino et al., 2017)) and unsupervised machine learning (e.g. dimensional reduction, feature extraction and clustering (Bessa et al., 2017; Wulfinghoff et al., 2018)). In particular, recent work by Wang and Sun (2018) attempted to resolve this issue by building a generic recurrent neural network that can easily incorporate different types of sub-scale information (e.g. porosity, fabric tensor, and relative displacement) to predict traction. This technique uses the concept of directed graph on the transfer learning approach (cf. Pan et al. (2010)) in which multiple neural networks trained to make predictions on other physical quantities (e.g. relationship between porosity and fabric tensor) are re-used to generate additional inputs for predicting traction. However, the determination of the optimal input information (in addition to the displacement jump history) and configurations of information flow that enhances the prediction accuracy still requires a time-consuming trial-and-error task (cf. Section 4.3 Wang and Sun (2018)). In this work, we introduce a general artificial intelligence approach to automate the creation and validation of traction-separation models. Unlike the previous approach in which neural networks are often used to either identify material parameters or create black-box constitutive laws, this work focuses on leveraging the capacity of a computer to improve via self-playing, a technique commonly referred as (deep) reinforcement learning in the computer science community (Sutton, 1992; Silver et al., 2016, 58). In the past two years, the functionality of algorithms automatically generated from deep reinforcement learning have achieved remarkable success. In many cases, the demonstrated capacities were thought to be impossible in the past. For instance, the algorithm trained by deep reinforcement learning created by a company called DeepMind is able to outperform human experts in Go, Chess and Atari games. The most exciting part of this achievement is that, unlike previous AI such as the IBM Deep Blue, the deep reinforcement learning does not rely on hand-crafted policy evaluation functions and is therefore applicable to different kinds of games once they are defined and implemented. This success motivates this research of proposing a meta-modeling approach where deep reinforcement learning may generate constitutive laws for (1) a given set of data, (2) a well-defined objective, and (3) a given set of universal principles. To achieve this goal, we recast the process of writing a constitutive model as a game with components suitable for deep reinforcement learning, involving a sequence of actions completely compatible with the stated rules (i.e., the law of physics). First, we define the model score, which could be any objective function suitable for a given task. For instance, this objective can be minimizing the discrepancy between calibrated experimental results and _blind predictions_ measured by a norm, or a constrained optimization problem that gives considerations on other attributes such as consistency, speed, and robustness (Wang et al., 2016). Once the score (i.e., the objective) is clearly defined, we then implement the rules, which are the universal principles of mechanics, such as material frame indifference, laws of thermodynamics. These rules are applied in an environment in which scores are sampled. In the case of traction-separation law, the environment is simply the validation process itself. Following this, we then define the action space which consists of a number of actions available for the modelers to write constitutive models. Once the action space and the model score are defined, we leverage the directed graph modeling technique to generate a state. The state at the end of each game represents a constitutive model automatically generated from the computer algorithm. In reality, the action space could be of very high dimensions such that manually deriving, implementing, verifying and validating all possible configurations are not feasible. This situation is similar of playing the games of chess and Go where the number of possible combinations of decisions or moves (each can be represented by a decision tree) remains finite but is so enormous that it is not possible to seek the optimal moves by exhausting all possibilities (Shannon, 1950). With the state, action, rule and objective defined, the most critical part is to assign reward for each action. In principle, if the action space is of very low dimension, i.e., there are not many ways to model the physical processes, then the reward for each action can be determined by exhausting all the possible model configurations. However, in the case of writing a complex traction-separation model, we cannot evaluate the quality of the model until its predictions are compared with benchmark data. Therefore, the ability to approximate the reward for each action (in our case the modeling choices) without the need to evaluate all the available options becomes crucial for the success of the meta-modeling approach. The deep reinforcement learning is therefore ideal for us to achieve this goal. We can approximate the rewards via neural networks and the Bellman expectation equation (Bellman, 1957; Dolcetta and Ishii, 1984). By repeatedly generating new constitutive laws (i.e., playing the game of writing models), the agent will use the reward obtained from each played game, in analogy with the binary game result (win/loss) at the end of a Go game, to update the action probabilities and value functions to improve the agent’s ability to write good constitutive laws. Through sufficient self-plays, the reinforcement learning algorithm then improves the modeling choices it made over time until it is ready for predictions. <figure><img src="content_image/1810.10535/AgentEnvironmentFigure.png"><figcaption>Figure 1: Scheme of the reinforcement learning algorithm in which an agentinteracts with environment and receives rewards. Through exploration, theagent then determines better actions to achieve a particular goal defined bythe reward. In our case, the reward is the score which represents the qualityof the forward prediction, the action is any possible activities required toderive a constitutive law, and the environment is the procedure that comparesthe predictions with the benchmark data.</figcaption></figure> There are a few major upshots for this approach. First, once the reinforcement learning algorithm is established, it can serve as a model generator without any human intervention. Second, since we regard the validation process as the environment component of the reinforcement learning, the performance of a resultant model is simultaneously evaluated and therefore validations are always a part of the model writing process. Third, the meta-modeling approach may easily embed any existing model generated by domain experts into the action space without re-implementing a new model. These unique capabilities enable us to have an unbiased tool to evaluate how well existing models fulfill a particular objective. Furthermore, since the model generation procedure is automated once an objective function is defined, this work may potentially eliminate the need of writing multiple incremental models for the same materials over time. Finally, this modeling approach is particularly powerful for discovering hidden physical coupling mechanisms that are otherwise too subtle to detect with human observation. The rest of the paper is organized as follows. We first review the directed graph approach that enables us to generate and utilize a decision tree to represent the modeling process (Section 2). The definition of model scores is then described in Section 3. We then provide a formal definition of a game invented to generate traction-separation laws for predictions (Section 4). This is followed by a description on how to use the reinforcement learning for the traction-separation law generation (Section 5). Two numerical experiments are then used to showcase the performance of the automated meta-modeling approach using synthetic data from microscale discrete element simulations (Section 6). The major findings are then summarized in the conclusions. ## 2 Representing traction-separation law in directed graph In this section, we introduce a building block for a simplified and extensible game that generates traction-separation laws by considering the relationships among different types of data collected from sub-scale simulations. In this game, the goal is to find a specific way to link different types of data such that a score function is maximized. Before we introduce the formal definition of the game, one necessary step is to recast the algorithm that leads to predictions from constitutive laws as a network of unidirectional information flow, i.e., a directed graph (also referred as digraph) (Sun et al., 66, 2014; Sun, 2015; Salinger et al., 2016; Wang and Sun, 2018). Recall that a digraph $D=(V,E)$ is an ordered pair of non-empty finite sets which consists of a vertex set V and an edge set E (Bang-Jensen and Gutin, 2008). Each edge connects a source vertex (tail) to a target vertex (head). Following the treatment in Sun (2015) and Wang and Sun (2018), the following rules are applied to generate the traction-separation law. 1. The traction $\boldsymbol{t}$ is placed as the only leaf of the digraph (i.e., the vertex that is not source to any other vertices). 2. The displacement jump $\boldsymbol{\delta}$ is placed as the only root of the digraph (i.e., the vertex that is not target of any other vertices). 3. There may exist isolated vertices in the digraph, i.e., some internal variables or microstructural features between $\boldsymbol{\delta}$ and $\boldsymbol{t}$ may not contribute to the final completed digraph and the corresponding constitutive model. 4. The digraph is acyclic, which means that there must be no cycle in the digraph. 5. If a vertex has sources or targets connected to it, it must be on at least one of the paths leading from $\boldsymbol{\delta}$ to $\boldsymbol{t}$. This ensures that an internal variable, once considered, is fully incorporated into the final constitutive model. In the previous published work (cf. Sun (2015); Wang and Sun (2018)), we prescribed theoretical models or, in some cases, neural network models to create linkages and enforce the hierarchy among physical quantities (e.g. porosity-permeability relation). While this treatment is convenient for software engineering and code design (Salinger et al., 2016), this approach only works if we have a prior knowledge about the relationships among the physical quantities. While one may presumably make ad hoc assumptions to complete the models, such a treatment is often at the expense of robustness. Another possible remedy is to gather all the measurement and data one may possibly obtain from observations and experiments, then find the key mechanisms that incorporate the most essential physics (e.g. the critical state plasticity for soil). This latter approach can be re-expressed as a problem in the directed graph in which we only know the elements of the vertex set but have no idea whether and how these vertices are connected, except that the traction is the leaf and the displacement jump is the root of the directed graph. Note that, in reality, the creation of a deterministic constitutive law does not only limit at determining connections among vertices (physical quantities), but also includes finding hidden vertices and appropriate edges. These actions are not modeled in this paper, but will be considered in future studies. Furthermore, while our focus in this paper is on deriving the traction-separation laws, in principle, the idea can be easily extended to other problems, such as the stress-strain relation for bulk materials, the porosity-temperature-fabric-tensor-permeability relations for porous media, among others. For demonstration purposes, we consider a constitutive law $\boldsymbol{t}(\boldsymbol{\delta},\boldsymbol{q})$ that predicts the traction vector $\boldsymbol{t}$ based on the history of the displacement jump $\boldsymbol{\delta}$ over a cohesive or cohesive-frictional surface with the normal direction vector being $\boldsymbol{n}$. $\boldsymbol{q}$ is a collection of state variables with $n$ degrees of freedom, i.e., $q_{1},q_{2},q_{3},...,q_{n}$. We use sub-scale discrete element simulations to generate synthetic data and attempt to create a traction-separation model which can replicate the constitutive responses of complex loading histories. Imposing restrictions of material frame indifference and assuming isotropic cohesive-frictional surface, the traction-separation model can be simplified to (Ortiz and Pandolfi, 1999) \[\boldsymbol{t}(\boldsymbol{\delta},\boldsymbol{q})=\boldsymbol{t}(\delta_{n}, \delta_{m},\boldsymbol{q}),\] (1) where $\delta_{n}=\boldsymbol{\delta}\cdot\boldsymbol{n}$ and $\delta_{m}=\|\boldsymbol{\delta_{m}}\|=\|\boldsymbol{\delta}-\delta_{n} \boldsymbol{n}\|$. Hence, the traction $\boldsymbol{t}$ is related to its components $t_{n}$ and $t_{m}$ that \[\boldsymbol{t}(\boldsymbol{\delta},\boldsymbol{q})=t_{n}(\delta_{n},\delta_{m} ,\boldsymbol{q})\boldsymbol{n}+t_{m}(\delta_{n},\delta_{m},\boldsymbol{q}) \frac{\boldsymbol{\delta_{m}}}{\delta_{m}}.\] (2) The internal variables in $\boldsymbol{q}$, if the cohesive surface is composed of a thin layer of granular materials, can be chosen among a large set of geometrical measures on micro-structural attributes (Sun et al., 65; Kuhn et al., 2015). In this work, we first manually select the following measures to be the intermediate vertices (the vertices that are neither the leaves nor the roots) to make forward predictions on the traction vector. * Porosity $\phi$, the ratio between the volume of the void and the total volume of a representative volume element (RVE) of the material layer. * Coordination number $CN=N_{\text{contact}}/N_{\text{particle}}$ where $N_{\text{contact}}$ is the number of particle contacts and $N_{\text{particle}}$ is the number of particles in the RVE. * Fabric tensor $\boldsymbol{A}_{f}=\frac{1}{N_{\text{contact}}}\sum_{c=1}^{N_{\text{contact}}} \boldsymbol{n}^{c}\otimes\boldsymbol{n}^{c}$, where $\boldsymbol{n}^{c}$ is the normal vector of a particle contact $c$, $c$ = 1, 2, …,$N_{\text{contact}}$ in the RVE. * Strong fabric tensor $\boldsymbol{A}_{sf}=\frac{1}{N_{\text{strongcontact}}}\sum_{c=1}^{N_{\text{ strongcontact}}}\boldsymbol{n}^{c}\otimes\boldsymbol{n}^{c}$, where $\boldsymbol{n}^{c}$ is the normal vector of a strong particle contact (having a compressive normal force greater than mean contact force) $c$, $c$ = 1, 2, …,$N_{\text{strongcontact}}$ in the RVE. All particle contacts inside the RVE can form a graph with particles as vertices and interactions as edges. Some quantitative measures of this graph of connectivity can be included in the internal variables $\boldsymbol{q}$ as additional microstructural characteristics. Here, we focus on four measures, which are computed using the software package NetworkX (Hagberg et al. (2008)), and their detailed explanations can be found in the software documentation. * $d_{a}$, degree assortativity, a scalar value between -1 and 1 measuring the similarity of connections in the graph with respect to the node degree. * $c_{t}$, transitivity coefficient, $c_{t}=3\frac{n_{triangles}}{n_{triads}}$, the fraction between the number of triangles and the number of triads present in contact graph. * $l_{sp}$, average shortest path length in the contact graph. * $\rho_{g}$, density of the graph, $\rho_{g}=\frac{2m}{n(n-1)}$, where $n$ is the total number of nodes and $m$ is the total number of edges in the graph. To sum up, in the digraph representations of traction-separation models, $\boldsymbol{\delta}$ is the root and $\boldsymbol{t}$ is the leaf, and currently we consider $\boldsymbol{q}$ to be a subset of the following set of physical quantities $\{\delta_{n,m},\ t_{n,m},\ \phi,\ CN,\ \boldsymbol{A}_{f},$$\ \boldsymbol{A}_{sf},\ d_{a},\ c_{t},\ l_{sp},\ \rho_{g}\}$. For the edges, we classify them as either ”definitions” (such as $t_{n,m}\rightarrow\boldsymbol{t}$, $\boldsymbol{\delta}\rightarrow\delta_{n,m}$) which are determined by universal principles in mechanics and should not be modified, or the ”phenomenological relations” (such as $\delta_{n,m}\rightarrow\boldsymbol{A}_{f}$, $\phi\to CN$, $l_{sp}\to t_{n,m}$) which incorporate material parameters chosen to fit experimental data. The latter category of edges provide opportunities for researchers to propose hand-crafted constitutive relations of different degrees of complexities. For example, their forms can be linear, quadratic, exponential functions or be approximated by artificial neural networks (ANNs). For illustration purposes, we consider a simple digraph of traction-separation models involving only the nodes $\{\boldsymbol{\delta},\ \boldsymbol{t},\ \delta_{n,m},\ t_{n,m},\ \phi,\ CN,\ \boldsymbol{A}_{f}\}$. Figure 2 provides examples of two admissible and two illegal digraph configurations according to the Rules 1-5. <figure><img src="content_image/1810.10535/x1.png"><figcaption>(a) The digraph is admissible.</figcaption></figure> ## 3 Score system for model evaluation and objective function A score system must be introduced to evaluate the generated directed graphs for constitutive models such that the accuracy and credibility in replicating the mechanical behavior of real-world materials can be assessed. This score system may also serve as the objective function that defines the rewards for the deep reinforcement learning agent to improve the generated digraphs and resultant constitutive laws. In this work, we define the score as a positive real-valued function of the range $[0,1]$ which depends on the measures $A_{i}$$(i=1,2,3,...,n)$ of $n$ important features of a constitutive model, \[\text{SCORE}=F(A_{1},A_{2},A_{3},...,A_{n}),\] (3) where $0\leq A_{i}\leq 1$. Some features are introduced to measure the performance of a model such as the accuracy and computation speed. Other features are introduced to enforce constraints to ensure the admissibility of a constitutive model, such as the frame indifference and the thermodynamics consistency. Suppose there are $n_{\text{pfm}}$ measures of performance features $A^{\text{pfm}}_{i}$ and $n_{\text{crit}}$ measures of critical features $A^{\text{crit}}_{i}$ in the measure system of constitutive models, the score takes the form, \[\text{SCORE}=(\prod_{j=1}^{n_{\text{crit}}}A^{\text{crit}}_{j})\cdot(\sum_{i=1 }^{n_{\text{pfm}}}w_{i}A^{\text{pfm}}_{i}),\] (4) where $w_{i}\in[0,1]$ is the weight associated with the measure $A^{\text{pfm}}_{i}$, and $\sum_{i=1}^{n_{\text{pfm}}}w_{i}=1$. In this section, two examples of measures of accuracy $A_{\text{accuracy}}$ and prediction consistency $A_{\text{consistency}}$ are presented. ### Accuracy of calibrations and forward predictions In this work, the abilities of the models to replicate calibration data and make forward predictions are considered separately. Here we introduce a cross-validation procedure in which the dataset used for training the models (e.g. identifying material parameters (e.g. Wang et al. (2016); Liu et al. (35)) or adjusting weights of neurons in recurrent neural networks (e.g. Lefik and Schrefler (2002); Wang and Sun (2018)) is mutually exclusive to the testing dataset used to evaluate the quality of blind predictions. The details of the generation of these calibration and testing data sets using frictional discrete element simulations are presented in Appendix A. Both calibration and blind prediction results are compared against the target data. The mean squared error (MSE) commonly used in statistics and also as objective function in machine learning is chosen as the error measure for each data sample $i$ in this study, i.e., \[\text{MSE}_{i}=\frac{1}{N_{\text{feature}}}\sum_{j=1}^{N_{\text{feature}}}[ \mathcal{S}_{j}(Y_{i_{j}}^{\text{data}})-\mathcal{S}_{j}(Y_{i_{j}}^{\text{ model}})]^{2},\] (5) where $Y_{i_{j}}^{\text{data}}$ and $Y_{i_{j}}^{\text{model}}$ are the values of the $j$th feature of the $i$th data sample, from target data value and predictions from constitutive models, respectively. $N_{\text{feature}}$ is the number of output features. $\mathcal{S}_{j}$ is a scaling operator (standardization, min-max scaling, …) for the output feature $\{Y_{i_{j}}\},\ i\in[1,N_{\text{data}}]$. The empirical cumulative distribution functions (eCDFs) are computed for MSE of the entire dataset $\{\text{MSE}_{i}\},\ i\in[1,N_{\text{data}}]$, for MSE of the training dataset $\{\text{MSE}_{i}\},\ i\in[1,N_{\text{traindata}}]$ and for MSE of the test dataset $\{\text{MSE}_{i}\},\ i\in[1,N_{\text{testdata}}]$, with the eCDF defined as (Kendall et al., 1946), \[F_{N}(\text{MSE})=\left\{\begin{aligned} & 0,&\text{ MSE}<\text{MSE}_{1},\\ &\frac{r}{N},&\text{MSE}_{r}\leq\text{MSE}<\text{MSE }_{r+1},\ r=1,...,N-1,\\ & 1,&\text{MSE}_{N}\leq\text{MSE},\end{aligned}\right.\] (6) where $N=N_{\text{data}}$, or $N_{\text{traindata}}$, or $N_{\text{testdata}}$, and all $\{\text{MSE}_{i}\}$ are arranged in increasing order. A measure of accuracy is proposed based on the above statistics, \[A_{\text{accuracy}}=\max(\frac{\log[\max(\varepsilon_{P\%},\varepsilon_{\text{ crit}})]}{\log\varepsilon_{\text{crit}}},0),\] (7) where $\varepsilon_{P\%}$ is the $P$th percentile (the MSE value corresponding to $P\%$ in the eCDF plot) of the eCDF on the entire, training or test dataset. $\varepsilon_{\text{crit}}\ll 1$ is the critical MSE chosen by users such that a model can be considered as ”satisfactorily accurate” when $\varepsilon_{P\%}\leq\varepsilon_{\text{crit}}$. ### Consistency of accuracy between calibrations and forward predictions For the examination of the consistency in model predictions on training data and test data, the K-sample Anderson-Darling (AD) test of goodness-of-fit (gof) is conducted to check whether the eCDFs of training and test data come from the same probability distribution, while this distribution is unspecified (Anderson and Darling, 1954; Scholz and Stephens, 1987). It is a non-parametric hypothesis test and determines whether the null hypothesis $H_{0}$ that the two eCDFs come from the same continuous distribution can be rejected or not, under a chosen significance level $\alpha_{\text{gof}}$. The method consists of calculating a normalized AD test statistic, critical values of the AD statistic that depends on the sample sizes, and a $p$-value indicating the approximated significance level at which $H_{0}$ can be rejected. If the $p$-value is smaller than the significance level $\alpha_{\text{gof}}$, the $H_{0}$ hypothesis is rejected. Otherwise there is insufficient evidence to reject $H_{0}$. In this work, we define the following binary measure for the consistency of the MSE distributions, with the significance level $\alpha_{\text{gof}}$, (8) ## 4 Game of the traction-separation law Our focus in this paper is primarily on the meta-modeling game invented for generating traction-separation models. Nevertheless, similar games can be defined for generating other types of constitutive models based on the ideas presented in this work. With the directed graph representations of traction-separation models as presented in Section 2, the process of developing a model can be recast as a game of making a sequence of decisions in generating edges between nodes in the digraphs. The player of the game can be a human or an AI agent. The game starts with an initial ”game board” of digraph with predefined nodes and no edge formed between them. Each step of the game consists of activating only one edge among all possible choices of edges in the predefined action space, following the predefined rules of the game. The game terminates when a complete and admissible digraph following the rules in Section 2 is established. The output models of the game are measured by a score system as presented in Section 3. <figure><img src="content_image/1810.10535/x5.png"><figcaption>(a) Initial configuration of the ”game board”</figcaption></figure> The game can be mathematically formalized as a Markov decision process. The human or AI agent observes the state of the game $s_{t}$ at the current step $t$ from the game environment (the directed graph that represents a constitutive model) in the form of a vector of binaries indicating the on/off status of each valid edge choices in the action space. The agent takes an action $a_{t}$ on the game environment in the form of an integer indicating the next edge to switch on in the action space. The action $a_{t}$ is sampled from a vector of probabilities $\pi(s_{t})$ of taking each valid action from the state $s_{t}$. Consequently, the state of the game becomes $s_{t+1}$ at the next step $t+1$. The agent also receives a reward $r_{t+1}$ for the action $a_{t}$ of taking the game state from $s_{t}$ to $s_{t+1}$. Each policy applied in a complete gameplay produces a particular trajectory $s_{0}$, $a_{0}$, $r_{1}$, $s_{1}$, $a_{1}$, $r_{2}$, …, $a_{t-1}$, $r_{t}$, $s_{t}$, $a_{t}$, …, $a_{T-1}$, $r_{T}$, $s_{T}$. Once a complete constitutive model is generated, the model score is evaluated. The final reward $r_{T}$ is defined as: if the current score is higher than the average score of models from a group of already played games by the agent, then the current game wins and $r_{T}=1$, otherwise, the current game loses and $r_{T}=-1$. The average score can be initialized to 0 for the first game. Note that the reward is only available at the end of the game, similar to the game of Chess and Go. $r_{T}$ is known according to the score of the generated model. The previous rewards $r_{t<T}$, however, can only be estimated. For a human agent, both rewards $r_{t<T}$ and move probabilities $\pi(s)$ come from ”intuition” gained during many constitutive modeling practices. An experienced human modeler estimates the rewards and probabilities more accurately and hence more likely generates better constitutive models. For an AI agent, $r_{t<T}$ and $\pi(s)$ are approximated by hand-crafted mathematical functions or recently neural networks as in deep reinforcement Q-learning. They are estimated based on the expected game reward of taking action $a$ from state $s$ (Q-value) $Q(s,a)$ and the value of current state $v(s)$. The above-mentioned important quantities for mathematical descriptions of the gameplays are summarized in Table 1. Moreover, the constitutive modeling game is compared side-by-side with the game of Chess more familiar to the public in Table 2, in the aspects of the board to play on, the permitted actions to execute, the criteria for wining the game, etc. Environment \| Benchmark training and test data, idealized multigraph for constitutive models ---\|--- Agent \| Human or AI State s \| A list of binaries indicating the on/off status of each valid edge choice Action a \| An integer indicating the next edge to switch on from the current game state Reward r \| Win (1) / loss (-1) according to the score of the constitutive model in Section 3 π(s,a) \| Probability of taking action a at state s v(s) \| Expected reward of state s Q-value Q(s,a) \| Expected reward from taking action a at state s Table 1: Key ingredients of the game of constitutive models in directed graph. \| Game of Chess \| Game of constitutive modeling in directed graph ---\|---\|--- Definition of game \| Make a sequence of decisions to maximize the probability to win \| Make a sequence of decisions to maximize the score of the constitutive model Game board \| 8×8 grid \| Directed graph with predefined nodes of physical quantities and edges of definition or universal principles Game state \| Configuration of chess pieces on the board \| Configuration of directed graph representing the constitutive model Game action \| Move chess pieces \| Select among modeling choices. For instance 1. What physical quantities are included? 2. How physical quantities are linked? 3. What are the edges between physical quantities? Game rule \| Restrictions on chess piece movements \| Universal principles Rules in Section 2 Specific restrictions on edge choices Game reward \| Win, draw or loss (discontinuous) \| Win or loss (discontinuous) from comparison of model scores (continuous) Reward evaluation \| Only available at the end \| Only available at the end Table 2: Comparison of the essential definitions between the game of Chess and the game of constitutive modeling in directed graph. For illustration purposes, we provide a simple game example for the digraph presented in Figure 2, which only involves the nodes $\{\boldsymbol{\delta},\ \boldsymbol{t},\ \delta_{n,m},\ \ t_{n,m},\ \phi,\ CN, \ \boldsymbol{A}_{f}\}$. Figure 3 presents the ”initial game board” and all possible edges choices in the current game definition. The configuration of the digraph, or the state of the game, can be totally described by a list of binaries for 13 edges $[\delta_{n,m}\rightarrow\phi,\ \delta_{n,m}\to CN,\ \delta_{n,m} \rightarrow\boldsymbol{A}_{f},\ \delta_{n,m}\to t_{n,m},\ \phi \to CN,\ \phi\rightarrow\boldsymbol{A}_{f},\ \phi\to t_{n,m}, \ CN\rightarrow\phi,\ CN\rightarrow\boldsymbol{A}_{f},\ CN\to t_{n,m}, \ \boldsymbol{A}_{f}\rightarrow\phi,\ \boldsymbol{A}_{f}\to CN,\ \boldsymbol{A}_{f}\to t_{n,m}]$ (The edges $\boldsymbol{\delta}\rightarrow\delta_{n,m}$ and $t_{n,m}\rightarrow\boldsymbol{t}$ are definitions and always active). The list also represents the entire action space. The action $a$ is an integer $\in[0,12]$ indicating the next edge ID to activate in the list. The legal moves at the current game state are represented by a list of 13 binaries indicating whether the corresponding edges are allowed to be activated for the next action step. The rule of the legal moves are as follows: (1) if one edge has already been selected, it is excluded from the selection of actions; (2) if an edge between two intermediate nodes has been selected, the other edge involving these two nodes but with opposite direction is also excluded (e.g., The edges $\phi\to CN$ and $CN\rightarrow\phi$ are mutually exclusive); (3) the resultant digraph must obey the rules in Section 2. Figure 4 provides a gameplay example of the constitutive modeling game in Figure 3, with mathematical representations of game states, actions and legal actions, as well as the Markov decision process. <figure><img src="content_image/1810.10535/game1_playstep_example.png"><figcaption>Figure 4: A gameplay example formalized as a Markov decision process (s0, a0,r1, s1, a1, r2, s2, a2, r3, s3, a3, r4, s4, a4, r5, s5) for the digraph gamein Figure 3. The states are lists of binaries for 13 edges [δn,m→ϕ, δn,m→CN,δn,m→Af, δn,m→tn,m, ϕ→CN, ϕ→Af, ϕ→tn,m, CN→ϕ, CN→Af, CN→tn,m, Af→ϕ, Af→CN,Af→tn,m]. The actions a are integers ∈[0,12] (the list indices start from 0)indicating the next edge ID to activate. The legal moves are lists of binariesindicating whether the edges are allowed to be activated next. Final reward r5is determined by the model score evaluated at the end of the game. r1−4 areonly estimated by ”intuitions” on whether the current policy can lead to a winor not, until r5 is known. Note that the Markov decision process leading tothe final digraph configuration s5 is not unique.</figcaption></figure> The score evaluation (Section 3) requires model calibration on training data, and forward predictions on test data. The procedure for score evaluation is as follows. Once the final digraph configuration is determined, all paths (information flows) leading from $\boldsymbol{\delta}$ to $\boldsymbol{t}$ and all predecessors for each node in the paths are identified using the graph theory (software package NetworkX). Secondly, the predecessor nodes for the terminal node $\boldsymbol{t}$ within these paths are identified. Recursively going upstream along all the information flows, the predecessors for these nodes are identified, until the final predecessor node is the start node $\boldsymbol{\delta}$ only. All the predecessor-successor node pairs can be connected by either mathematical equations frequently used in handcrafted constitutive models (linear, quadratic, exponential, power law, etc.) or artificial neural networks. In this work, we take the advantage of the flexibility of ANNs that they are universal function approximators to continuous functions of various complexity on compact subsets of $R^{n}$ (Universal approximation theorem, (Hornik et al., 1989)). Moreover, a special type of ANN, recurrent neural network (e.g., long short-term memory LSTM (Hochreiter and Schmidhuber, 1997), gated recurrent units GRU (Cho et al., 2014; Chollet et al., 2015)), can capture the function of a time series of inputs, which is ideal for replicating the path-dependent material behavior. Hence, we only focus on ANN edges, without loss of generality of the meta-modeling games. The hybridized constitutive models with both mathematical equation edges and ANN edges will be studied in a separate research. The predecessor-successor node pairs are also inputs and outputs of all ANNs involved in the constitutive model. For example, there are two paths in the final digraph $s_{5}$ in Figure 4: $\{\boldsymbol{\delta}\rightarrow\delta_{n,m}\to CN\to \boldsymbol{A}_{f}\to t_{n,m}\rightarrow\boldsymbol{t}\}$ and $\{\boldsymbol{\delta}\rightarrow\delta_{n,m}\to CN\rightarrow\phi \to t_{n,m}\rightarrow\boldsymbol{t}\}$. Then the three required ANNs are, represented as input-output pairs, $[\delta_{n,m}\to CN]$, $[CN\rightarrow\phi,\boldsymbol{A}_{f}]$ and $[\phi,\boldsymbol{A}_{f}\to t_{n,m}]$. The parameters in each ANN are calibrated with training data of the input and output features using back propagations. The final output of $\boldsymbol{t}$ is predicted by executing consecutively the ANNs following the established paths from $\boldsymbol{\delta}$ to $\boldsymbol{t}$ in the directed graph. In the numerical examples of this paper, the same neural network architecture is used for all ANNs: two hidden layers of 32 GRU neurons in each layer, and the output layer is a dense layer with linear activation function. All input and output data are preprocessed by standard scaling using mean values and standard deviations (Pedregosa et al., 2011). Each input feature considers its current value and 19 history values prior to the current loading step. Each ANN is trained for 1000 epochs using the Adam optimization algorithm (Kingma and Ba, 2014), with batch size of 256. ## 5 Deep reinforcement learning for generating constitutive laws With the game of constitutive modeling completely defined, a deep reinforcement learning (DRL) algorithm is employed as a guidance of taking actions in the game to maximize the final model score (Figure 5). This tactic is considered one of the key ideas leading to the major breakthrough in AI playing the game of Go (AlphaGo Zero) (Silver et al., 58), Chess and shogi (Alpha Zero) (Silver et al., 57) and many other games. The learning is completely free of human interventions. It does not need previous human knowledge in traction-separation model as a starter database. The AI agent simply learns to improve from a number of games it played and from the corresponding model scores and game rewards, even if the initially generated digraph configurations make very little sense for a traction-separation model. Moreover, during the self-plays and training, no human guidance is needed. <figure><img src="content_image/1810.10535/selfplay_learn.png"><figcaption>Figure 5: Self-play reinforcement learning of traction-separation law.</figcaption></figure> A (deep) neural network $f_{\theta}$ with parameters $\theta$ (weights, bias, … of the artificial neurons) takes in the current configuration of the directed graph of the constitutive law $s$ and outputs a policy vector $\boldsymbol{p}$ with each component $p_{a}=p(s,a)$ representing the probability of taking the action $a$ from state $s$, as well as a scaler $v$ estimating the expected score of the constitutive law game from state $s$, i.e., \[(\boldsymbol{p},v)=f_{\theta}(s).\] (9) These outputs from the policy/value network guide the game play from the AI agent. At each state $s$, the action to take is sampled from an action probability $\boldsymbol{\pi}(s)$. This probability is based on the policy $\boldsymbol{p}$ predicted from the neural network enhanced by a Monte Carlo Tree Search (MCTS) ((Browne et al., 2012)). The search tree is composed of nodes representing states $s$ of the game, and edges representing permitted actions $a$ from $s$. Each edge $(s,a)$ possesses a list of statistics $[N(s,a),W(s,a),Q(s,a)]$, where $N(s,a)$ is the number of visits to the edge during MCTS search, $W(s,a)$ is the total action value and $Q(s,a)=\frac{W(s,a)}{N(s,a)}$ is the mean action value. The search procedure consists of firstly a recursive selection of a sequence of optimal actions $a^{0},a^{1},a^{2},...$ leading to the corresponding child states $s^{1},s^{2},s^{3},...$, starting from the root state $s^{0}$, until a leaf node of state $s^{l}$ (that has never been encountered before in the search) is reached. The criteria for selection from a state $s$ is that the action $a$ maximizes the upper confidence bound $U(s,a)$ of the Q-value, among all valid actions. The upper bound is defined as \[U(s,a)=Q(s,a)+U_{Q}(s,a)=Q(s,a)+c_{puct}p(s,a)\frac{\sqrt{\sum_{b}N(s,b)}}{1+N (s,a)}.\] (10) where $c_{puct}$ is a parameter controlling the level of exploration. If $s^{l}$ is not a terminal state that ends the game, then its $\boldsymbol{p}(s^{l})$ and $v(s^{l})$ are predicted from the policy/value neural network $f_{\theta}(s^{l})$. The search tree is expanded and the statistics for each edge $(s^{l},a)$ is initialized to $[N(s,a)=0,W(s,a)=0,Q(s,a)=0]$. Otherwise, $v(s^{l})$ is equal to the final reward of the constitutive modeling game. Finally, $v(s^{l})$ is propagated back to the parent states $\{s^{0},s^{1},s^{2},...s^{l}\}$ and actions $\{a^{0},a^{1},a^{2},...a^{(l-1)}\}$ traversed during the seach. Their statistics are updated as \[N(s^{i},a^{i})=N(s^{i},a^{i})+1,\ W(s^{i},a^{i})=W(s^{i},a^{i})+v(s^{l}),\ Q(s ^{i},a^{i})=\frac{W(s^{i},a^{i})}{N(s^{i},a^{i})},\ \text{for all}\ i<l.\] (11) The MCTS procedure is repeated a number of times. The searches in MCTS eventually yield a vector of search probabilities $\boldsymbol{\pi}(s^{0})$ recommending actions to take from the root position $s^{0}$. $\boldsymbol{\pi}(s^{0})$ is proportional to the exponentiated visit count for each edge, i.e., \[\boldsymbol{\pi}(s^{0},a)=\frac{N(s^{0},a)^{-\tau}}{\sum_{b}N(s^{0},b)^{-\tau}},\] (12) where $\tau$ is a positive temperature parameter that also controls the level of exploration. The MCTS algorithm for the game of constitutive models is illustrated in Figure 6. <figure><img src="content_image/1810.10535/MCTS_graph.png"><figcaption>Figure 6: Monte Carlo Tree Search (MCTS) in a game of constitutive models(figure design adopted from (Silver et al., 2017b)). A sequence of actions areselected from the root state s0, each maximizing the upper confidence boundQ(s,a)+UQ(s,a). The leaf node sl is expanded and its policy probabilities andposition value are evaluated from the neural network (p(sl),v(sl))=fθ(sl). Theaction values Q in the tree are updated from the evaluation of the leaf node.Finally the search probability π(s0) for the root state s0 is returned toguide the next action in self-play.</figcaption></figure> During one episode of self-play by the AI agent, the above MCTS algorithm is executed for each state $s_{t}$ in the sequence of encountered states $\{s_{0},s_{1},s_{2},...,s_{T-1}\}$. The root node $s^{0}$ of the search tree is set to $s_{t}$ as the game progresses to the state $s_{t}$. All child nodes and their statistics constructed in the MCTS for the prior game states are preserved. The training data for the neural network consists of $(s_{t},\boldsymbol{\pi}_{t},z_{t})$ obtained from a number of full plays of the constitutive law game guided by the aforementioned reinforcement learning algorithm. $\boldsymbol{\pi}_{t}$ is the estimation of policy after performing MCTS from state $s_{t}$ and $z_{t}$ is the reward of the generated constitutive model at the end of the game $s_{T}$. The loss function to be minimized by adjusting parameters $\theta$ using back propagation is, \[\text{l}=\sum_{t}(v(s_{t})-z_{t})^{2}+\sum_{t}\boldsymbol{\pi}_{t}\log[ \boldsymbol{p}(s_{t})],\] (13) which is the combination of mean squared errors in game reward (1 or -1) and cross-entropy losses in policy probabilities. Hence, accordingly, the activation functions for the output layer are the hyperbolic tangent function $\text{tanh}(x)=\frac{e^{2x}-1}{e^{2x}+1}$ and the softmax function (Nasrabadi, 2007). The procedure of DRL guided self-plays and the following training of the network $f_{\theta}$ is iterated until the score of the generated directed graph of the constitutive model does not improve. ## 6 Numerical Experiments and Applications In this section, we present two traction-separation modeling games with different digraph complexities to demonstrate the intelligence, robustness and efficiency of the deep reinforcement learning algorithm on improving the accuracy and consistency of the generated traction-separation models through self-plays. In both examples, sub-scale discrete element simulations (DEM) are used to generate synthetic benchmark data for model calibrations and blind prediction evaluations. The procedure of database generation from a pre-consolidated representative volume element (RVE) of a frictional contact material is described in Appendix A. The third numerical example presents a multiscale finite element simulation where the previously auto-generated optimal model is used as a scale-bridging constitutive model for pre-existing interface. ### Numerical Experiment 1: Determining optimal physical relationships for traction-separation laws In the first example, our goal is to test the DRL algorithm and see whether it can determine the optimal topological relations among microstructural physical quantities of porosity $\phi$, coordination number $CN$ and fabric tensor $\boldsymbol{A}_{f}$. In Wang and Sun (2018), the authors use domain expertise, i.e., knowledge from previous literature on fabric tensor and critical state theory to deduce that the porosity and fabric tensor can be used as state variables to improve the forward prediction accuracy of the traction-separation law (cf. Fu and Dafalias (2011); Li and Dafalias (2011); Sun (2013); Wang and Sun (2016)). In this work, we do not make any assumption or introduce any interpretation to the meta-modeling computer agent. Instead, we simply make a number of physical quantities measured from discrete element simulations available as vertices in the directed graph but do not introduce any relation (edge) manually. In other words, the edge set that represents the relations of the physical quantities is self-discovered by the computer agent from the reinforcement learning without any human intervention. We document our training procedure and analyze the performance of the models generated by the meta-modeling approach. The directed graphs, states, actions, rewards and game rules of the modeling game have been defined in Sections 2 and 4, and illustrated in Figures 2, 3 and 4. The action space is of dimension 13. Through exhaustive plays of the game, the authors count 3200 possible game states, among which 591 states represent complete and admissible directed graph configurations according to the game rules. The model score is defined as: \[\text{SCORE}=0.45A^{\text{calibration}}_{\text{accuracy}}+0.45A^{\text{ prediction}}_{\text{accuracy}}+0.1A_{\text{consistency}},\] (14) where $P\%=90\%$ and $\varepsilon_{\text{crit}}=1e^{-6}$ for accuracy evaluations and $\alpha_{\text{gof}}=1\%$ for consistency evaluations. The training data for model calibration contains 50 loading cases, and the test data for forward prediction evaluation contains 150 loading cases. The DRL meta-modeling procedure contains 10 iterations of ”exploration and exploitation” of game strategies, by setting the temperature parameter $\tau$ to 1. Then an iteration of ”competitive gameplay” ($\tau=0.01$) is conducted to showcase the performance of the final trained AI agent. Each iteration consists of 20 self-plays of the game. Each game starts with a randomly initialized neural network for the policy/value predictions, and each play step require 20 MCTS simulations. Then the play steps and corresponding final game rewards are append to the set of training examples for the training of the policy/value network. Due to the randomness of the initialized neural network and the MCTS search for each play step, each run of the DRL algorithm may yield different game play results, concerning the starting game policy, the speed of improvement, and the converged optimal policy. In this numerical example, the DRL procedure is repeated 20 times and all model scores that the AI agent played during the iterations are recorded. Hence for each iteration, there are 400 gamplays for statistical analysis (20 gamplays X 20 repeated procedures). The gameplay results are presented in Figure 7. At the first DRL iteration, the AI agent only knows the rule of the game without human knowledge on neither which physical quantities are essential in predicting the tractions nor how they should be connected. The AI just plays with trial-and-error following strategies guided by random initial neural network and MCTS. This lack of gameplay knowledge can be seen from the widely spread density distribution of model scores between maximum and minimum scores, large interquartile range between 25% and 75%, and the large standard deviation (Figure 7). In the subsequent iterations, the AI plays with increasing knowledge of game play reinforced by the ultimate game rewards, and it shows intelligence in keep playing games with better outcomes. This is shown by the increase in median and average of scores, the narrowing of interquartile range and the migration of the density distribution towards higher scores. The automatic learning is very efficient. Statistically, after 5 iterations (100 games out of the total 591 possible game outcomes), the scores already concentrate around the maximum. Few bad games could be played, since the AI is still allowed to explore different game possibilities to avoid convergence to local maximum. The strength of the AI agent after 10 iterations is tested by suppressing the ”exploration plays”, and the outcome game scores show outstanding performances. Figure 8 illustrates the improvement of knowledge of traction-separation constitutive modeling by four representative digraph games played during the DRL iterations. The traction predictions from the resultant constitutive models are compared against both training data and unseen test data. In addition, five examples of blind predictions from the optimal digraph configuration (the 4th digraph in Figure 8) obtained in this game are shown in Figure 9. <figure><img src="content_image/1810.10535/game1_violinplot.png"><figcaption>(a) Violin plots of the density distribution of model scores in each DRLiteration</figcaption></figure> <figure><img src="content_image/1810.10535/game1_trainning_session.png"><figcaption>Figure 8: Knowledge of directed graphs of traction-separation models learnedby deep reinforcement learning in Numerical Experiment 1. Four representativedigraph games played during the DRL iterations and their prediction accuracyagainst training and test data are presented.</figcaption></figure> <figure><img src="content_image/1810.10535/game1_predicts.png"><figcaption>Figure 9: Five examples of blind predictions from the optimal digraphconfiguration (the 4th digraph in Figure 8) against unseen data among testdatabase of 150 loading cases.</figcaption></figure> ### Numerical Experiment 2: Data-driven discovery for enhancement of traction-separation laws In the second example, we consider another common scenario in which we attempt to convert qualitative observations into quantitative predictions with the help of the reinforcement learning algorithm as a tool for augmented intelligence. The need to interpret observations of mechanisms into predictions is one of the oldest problems in constitutive modeling (Pastor et al., 2011). For instance, the observation that yielding depends on the amount of normal traction leads to the Mohr-Coulomb yield criterion (Timoshenko, 1953). The evolution of fabric tensor has been incorporated into the hardening law and the plastic flow rule to capture the induced anisotropy and critical state of sand (Mehrabadi et al., 1982; Dafalias and Manzari, 2004; Dafalias et al., 2004). However, recent advancements on the application of graph theory as well as the experimental techniques such as micro-CT imaging have revealed many geometric measures on the grain connectivity that help explaining the onset of shear band (Tordesillas et al., 2011, 2010), coherent vortex structure (Williams and Rege, 1997) and post-bifurcation behaviors in granular materials (Sun et al., 65; Wang and Sun, 2015; Liu et al., 2018). While these discoveries of new knowledge are indeed encouraging, one cannot make use of them without investing significant efforts and time to derive, verify, and validate new constitutive laws that incorporate those new information. Hence the graph-theoretical approaches, although have found great promises on analyzing the granular assembles obtained from real or virtual experiments, have not yet made significant impacts on constitutive laws used for engineering applications. Our meta-modeling approach is capable of overcoming this bottleneck by efficiently automating some of these tasks currently undertaken by modelers. This second numerical experiment is used to demonstrate how the augmented intelligence can be used to incorporate the insights from observations into predictions without manually re-writing an existing constitutive law every time new information comes up. This example is an extension of the first numerical experiment, in which more microstructual information are considered, including the fabric of strong interactions $A_{sf}$ and four measures of grain connectivities $d_{a},\ c_{t},\ l_{sp},\ \rho_{g}$. The task of identifying their roles in constitutive models for granular materials is now simply recast as defining a new game with augmented vertex set in digraph and extended action space. The ”game board” and all possible actions for this new game are shown in Figure 10. The dimension of the action space increases from 13 to 71. A particular game rule is added to test the flexibility of the DRL algorithm in handling different types of game constraints: the strong fabric tensor $A_{sf}$ and the fabric tensor $A_{f}$, since both are geometric measures of inter-particles forces, are mutually exclusive in the final digraphs of constitutive models. The number of possible game states increases from 3200 to over 400000. The number of complete and admissible directed graph configurations increases from 591 to over 20000. The score definition is the same as Equation (14). The meta-modeling algorithm try to learn the optimal ways to incorporate the microstructual information to make better predictions only from the training database of 50 loading cases, while the gained knowledge is validated on the test database of another 150 loading cases. The parameters for the DRL meta-modeling algorithm are set as: 10 iterations of ”exploration and exploitation”, 1 iteration of ”competitive gameplay”, 30 self-plays in each iteration, and 30 MCTS simulations in each play step. <figure><img src="content_image/1810.10535/x7.png"><figcaption>(a) Initial configuration of the ”game board”</figcaption></figure> The statistics of the gameplay results from 5 separate runs of the DRL procedure are presented in Figure 11. We observe a very efficient improvement in generated traction-separation models, even though the number of legal game states in the new game has largely increased. Figure 12 exhibits four representative digraph configurations developed during DRL iterations, as well as their prediction quality on calibration data and unseen data. It can be seen that the information flows in a constitutive model are of crucial importance. Although the first and the fourth graphs both incorporate the same types of microstructual information, the difference in the ways how these information are connected results in significant difference in model scores of 0.191 and 0.915, respectively. Moreover, the DRL algorithm develops the intelligence of selecting the strong fabric tensor $A_{sf}$ over the fabric tensor $A_{f}$ in order to further improve the prediction score of the model. Five blind prediction examples of the optimal digraph configuration (the 4th digraph in Figure 12) obtained in this game are presented in Figure 13. Comparing to the numerical example 1 (Figure 9), the augmented knowledge of additional microstructural information in constitutive models lead to more accurate representations of granular materials. <figure><img src="content_image/1810.10535/game2_violinplot.png"><figcaption>(a) Violin plots of the density distribution of model scores in each DRLiteration</figcaption></figure> <figure><img src="content_image/1810.10535/game2_trainning_session.png"><figcaption>Figure 12: Knowledge of directed graphs of traction-separation models learnedby deep reinforcement learning in Numerical Experiment 2.</figcaption></figure> <figure><img src="content_image/1810.10535/game2_predicts.png"><figcaption>Figure 13: Five examples of blind predictions from the optimal digraphconfiguration (The 4th digraph in Figure 12) against unseen data among testdatabase of 150 loading cases.</figcaption></figure> ### Application: Multiscale bridging using DRL-generated traction-separation model in finite element modeling In this example, we demonstrate that the traction-separation model auto-generated by the proposed DRL meta-modeling can be applied in multiscale finite element simulations, as a surrogate model to replace the computationally expensive DEM RVEs. The example consists of a mixed-mode tension-shear test on a plane-strain granular specimen with embedded strong discontinuity. The sample is 0.1 m x 0.1 m in dimension. The bottom edge is fixed, while the top edge moves rigidly following a mixed-mode loading-unloading-reloading displacement path, as shown in Fig. 14. The sample is assumed periodic in the horizontal direction, thus periodic displacement boundary condition is applied on the lateral edges. The geometry of the pre-existing interface in the center is a sinusoid with the spatial period of 0.05 m and amplitude of 0.005 m. The elements along the interface are enhanced by assumed strain formulation to embed the strong discontinuity (Oliver et al., 2002), while the other elements are regular bulk finite elements. The optimal traction-separation model found in the Numerical Example 2 is applied in the interface. The bulk material is assumed isotropic linear elastic and the parameters are homogenized from the DEM RVE for generation of data (Young’s modulus $E=300$ MPa, Poisson’s ratio $\nu=0.24$). The specimen is initially under isotropic pressure of $10$ MPa, the same as the DEM RVEs, and the lateral pressure remains constant during the loading steps. <figure><img src="content_image/1810.10535/SinusInterfaceMesh.png"><figcaption>(a) Embedded interface of sinusoidal line</figcaption></figure> The differential stress and strain fields computed by FEM-DRL multiscale approach at selected loading steps are shown in Fig. 15. The shear strain localizes in the embedded strong discontinuity in both specimens. The global traction-displacement curves in normal ($U_{y}-T_{y}$) and shear ($U_{x}-T_{x}$) directions are compared for both FEM-DRL and FEM-DEM simulations in Fig. 16. Results show great agreement on the mechanical behavior of the specimen. Hence the optimal traction-separation model automatically developed by the DRL meta-modeling approach can be used as a surrogate model to microscale DEM RVEs in multiscale FEM simulations. <figure><img src="content_image/1810.10535/FEMLSTM_diffstrain_50.png"><figcaption>(a) Load step 50</figcaption></figure> <figure><img src="content_image/1810.10535/FEMLSTM_UsTs.png"><figcaption>(a) Embedded interface of sinusoidal line</figcaption></figure> ## 7 Conclusion This paper presents a meta-modeling approach in which we attempt to generate traction-separation laws not through explicitly writing a particular model but to provide the computer with modeling options such that it can explore on its own through self-practicing. Unlike previous deep-learning models that leverage supervised learning techniques to train neural networks that makes black-box predictions, this new approach focuses on reinforcement learning technique to discover hidden relationships among data and therefore make modeling decisions to emulate the process of writing constitutive models by human. Given the rules (frame indifference, thermodynamic laws, balance principles), we introduce an agenda-based approach where the DRL technique is used to find the optimal way to generate a forward prediction. As demonstrated in our numerical experiments, this approach can be regarded as a generalization of the previous models where neural network predictions may still embed in part of the predictions but are not necessarily completely replacing all components in the conventional models. This flexibility is the key for us to exploit the computer to make _repeated_ trial-and-errors and improve from experiments over time to generate the best outcomes, instead of spending significant human time to explore through trial-and-errors. The idea of inventing the metal-modeling game could be significant in the sense that it frees modelers from focusing on curve-fitting a physical process. Instead, as future improvements on the models can be made by expanding the action space or simply leveraging the power of computer to improve the models over time, this allows us unprecedented luxury to place our focuses on finding best cause of actions that lead to the most predictive model. In addition, this meta-modeling approach also provides the following unique benefits against the conventional hand-crafting approach and black-box neural network models. 1. Since the machine learning procedure is automated, models intended for different purposes or designed to fulfill different demands (speed, accuracy, robustness) can be automatically generated and improved over time through self-plays in the model-creation game. 2. Since the validation procedure is introduced as the reward mechanism for the agent to find the optimal models available, the resultant models are always validated at the end of the games. 3. By recasting constitutive models as directed graphs, previous models established by domain experts can be easily embedded in the proposed framework to expand action spaces efficiently and shorten the training time. 4. The metal-modeling approach is generic and reusable, which means that it can handle different situations with different data, objective functions and rules set by human without going through additional derivation, implementation, material parameter identification and validation. Hence it does not require any debugging once the the game is implemented correctly. There are also limitations of the current approaches. For instance, the demands for data could be higher than conventional modeling approach, particularly when more sophisticated and rigorous validation metric is used to assign model score and game reward. The model has not yet introduced any technique to handle noise, nor does it consider any mechanism to consider uncertainty. While these topics are of great importance, the corresponding research activities are beyond the scope of this study and will be considered in the future, should opportunities come. ## Acknowledgments The corresponding author’s work is supported by the Earth Materials and Processes program from the US Army Research Office under grant contract W911NF-15-1-0442 and W911NF-15-1-0581, the Dynamic Materials and Interactions Program from the Air Force Office of Scientific Research under grant contract FA9550-17-1-0169, the nuclear energy university program from department of energy under grant contract DE-NE0008534 as well as the Mechanics of Material program at National Science Foundation under grant contract CMMI-1462760. Meanwhile, the first author is supported by US Army Research Office under grant contract W911NF-15-1-0442 and W911NF-15-1-0581, and the 2018 Interdisciplinary Research Seed funding from Columbia University. These supports are gratefully acknowledged. The views and conclusions contained in this document are those of the authors, and should not be interpreted as representing the official policies, either expressed or implied, of the sponsors, including the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. ## Appendix A: Generation of synthetic data from discrete element modeling (DEM) The data for calibration and evaluation of prediction accuracy of the deep-reinforcement-learned traction-separation models are generated by numerical simulations on a representative volume element (RVE) representing the granular materials on a frictional surface. The open-source software YADE for DEM is used (Šmilauer et al., 2010). The discrete element particles in the RVE have radii between $1\pm 0.3$ mm with uniform distribution. The RVE has the height of $20$ mm in the normal direction of the frictional surface and is initially consolidated to isotropic pressure of 10 MPa. The Cundall’s elastic-frictional contact model ((Cundall and Strack, 1979)) is used for the inter-particle constitutive law. The material parameters are: interparticle elastic modulus $E_{eq}=1$ GPa, ratio between shear and normal stiffness $k_{s}/k_{n}=0.3$, frictional angle $\varphi=$$30\lx@arcdegree$, density $\rho=2600$$kg/m^{3}$, Cundall damping coefficient $\alpha_{damp}=0.2$. The DEM RVE is loaded in the normal $\boldsymbol{n}$ and tangential $\boldsymbol{m}$ directions of the frictional surface by displacement controls $\delta_{n}$ and $\delta_{m}$ (Figure 17(a)). The synthetic database consists of 200 numerical experiments under different loading paths. They differ from each other in the ratio of normal and tangential loading rate $\dot{\delta_{n}}/\dot{\delta_{m}}$, as well as the loading-unloading-reloading cycles, as illustrated in Figure 17(b), 17(c) and 17(d). The traction-separation curves of the experiments are recored and three examples corresponding to the paths in Figure 17 are presented in Figure 18. The microstructural information required for the intermediate nodes in the directed graphs, such as porosity, coordination number and fabric tensor, are also recored during the simulations. The open-source library NetworkX (Hagberg et al., 2008) is employed to analyze the graph of the particle interactions in the RVEs. Figure 19 presents examples of microstructural information and graph characteristics for the three example loading paths. The 200 numerical simulations in the database are shuffled. The first 50 simulations are used as ”training data” for the calibration of model parameters for the edges in the directed graphs. 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1101.1877	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 17510, "num_imgs": 4, "llama3_tokens_count": 6057 }	[ "content_image/1101.1877/b195_pcacVSm0.png", "content_image/1101.1877/x1.png", "content_image/1101.1877/fig_all_1mstar2.png", "content_image/1101.1877/RCs_4masses1.png" ]	# Renormalization constants for Wilson fermion lattice QCD with four dynamical flavours ETM Collaboration Petros Dimopoulos${}^{a}$, Roberto Frezzotti${}^{b,c}$, Gregorio Herdoiza${}^{d}$, Karl Jansen${}^{d}$, Vittorio Lubicz${}^{e}$, ${}^{c}$, Giancarlo Rossi${}^{b,c}$ ${}^{a}$Dipartimento di Fisica, Universit di Roma “La Sapienza” Piazzale A. Moro, I-00185 Rome, Italy ${}^{b}$Dipartimento di Fisica, Universit di Roma “Tor Vergata” Via della Ricerca Scientifica 1, I-00133 Rome, Italy ${}^{c}$INFN Sezione di “Roma Tor Vergata” c/o Dipartimento di Fisica, Universit di Roma “Tor Vergata” Via della Ricerca Scientifica 1, I-00133 Rome, Italy ${}^{d}$DESY, Platanenallee 6, D-15738 Zeuthen, Germany ${}^{e}$Dipartimento di Fisica, Universit di Roma Tre and INFN Via della Vasca Navale 84, I-00146 Rome, Italy E-mail: ###### Abstract: We report on an ongoing non-perturbative computation of RI-MOM scheme renormalization constants for the lattice action with four dynamical flavours currently in use by ETMC. For this goal dedicated simulations with four degenerate sea quark flavours are performed at several values of the standard and twisted quark mass parameters. We discuss a method for removing possible O(a) artifacts at all momenta and extrapolating renormalization constant estimators to the chiral limit. We give preliminary results at one lattice spacing. ## 1 Introduction Simulations including two degenerate light flavours and a non-degenerate doublet of quarks are currently being performed by the European Twisted Mass (ETM) Collaboration. The inclusion of $n_{f}=2+1+1$ flavours is a necessary step to move towards a realistic situation. Fermions are described by the maximally twisted mass lattice QCD (MtmLQCD) action [1] and gluons by the Iwasaki action [2]. While the first physical results are very encouraging [3], dedicated simulations are required to perform the non-perturbative renormalization of operators in a mass-independent scheme, where renormalization constants (RCs) are defined at zero quark mass. In the study of $n_{f}=2+1+1$ QCD ETMC is adopting the RI-MOM scheme [4]. The RCs are evaluated by extrapolating to the chiral limit the RC estimators computed in the theory with $n_{f}=4$ mass degenerate quarks for a range of mass values ¹. Here we report on the progress we made in this project. [FOOTNOTE:1][ENDFOOTNOTE] ### Action and quark mass parameters For the present study we consider the lattice action \[S_{L}=S^{\rm YM}_{Iwa}+a^{4}\sum_{x}\sum_{f=1}^{4}\bar{\chi}_{f}\left[\gamma \cdot\widetilde{\nabla}-\tfrac{a}{2}\nabla^{}\nabla+m_{0,f}+i\gamma_{5}r_{f} \mu_{f}\right]\chi_{f}(x)\] (1) where $\chi_{f}$ is a one-flavour quark field in the so-called _twisted basis_ and in this work $r_{f}$ is set to either $1$ or $-1$. Passing from the twisted to the physical quark basis² [FOOTNOTE:2][ENDFOOTNOTE] \[S_{L}=S^{\rm YM}_{Iwa}+a^{4}\sum_{x}\sum_{f=1}^{4}\bar{q}_{f}\left[\gamma\cdot \widetilde{\nabla}-i\gamma_{5}r_{f}e^{i\gamma_{5}r_{f}\theta_{0,f}}(-\tfrac{a} {2}\nabla^{}\nabla+m_{\rm cr})+M_{0,f}\right]q_{f}(x)\,.\] (3) The bare mass parameters can be rewritten as \[M_{0,f}=\sqrt{(m_{0,f}-m_{\rm cr})^{2}+\mu_{f}^{2}}\,,\quad\sin\theta_{0,f}= \frac{m_{0,f}-m_{\rm cr}}{M_{0,f}}\,,\quad\cos\theta_{0,f}=\frac{\mu_{f}}{M_{0 ,f}}\,.\] (4) Their renormalized counterparts read $M_{f}=Z_{P}\hat{M}_{f}=\sqrt{Z_{A}^{2}m_{\rm PCAC}^{2}+\mu_{f}^{2}}$ and $\tan\theta_{f}=\frac{Z_{A}m_{\rm PCAC}}{\mu_{f}}$. The parametrization in terms of $M$ and $\theta$ is convenient because the leading term of the Symanzik local effective Lagrangian involves only $M$, not $\theta$. As we will see later (see the end of section 2), this remark is at the basis of our method to obtain $O(a)$-improved RC-estimators at all scales even _out of maximal twist_. Since, for practical reasons, we work in a partially quenched setup with all four flavours having equal mass parameters, we will have to consider in our analysis four quark mass parameters: $M_{\rm sea},\theta_{\rm sea},M_{\rm val},\theta_{\rm val}$. ### RI’-MOM scheme and our setup The focus of the present study is on flavour non-singlet quark bilinear operators, $O_{\Gamma}=\bar{\chi}_{f}\Gamma\chi_{f^{\prime}}$ (or $\bar{\chi}_{f^{\prime}}\Gamma\chi_{f}$), with $\Gamma\;=\;S,P,V,A,T$, which are written in terms of $\chi$ and $\bar{\chi}$ quark fields (i.e. in the standard quark basis for untwisted Wilson fermions). RCs are named after the expression of the operators in this basis so as to match the usual notation in the literature about Wilson fermions. As convenient in lattice studies, we adopt the RI’-MOM scheme [6, 4], which is defined as follows. A first condition fixes the quark field renormalization, namely \[Z_{q}^{-1}\frac{-i}{12N(p)}\sum_{\rho}\mbox{}^{\prime}\left[\frac{\mathrm{Tr}( \gamma_{\rho}S_{f}(p)^{-1})}{\tilde{p}_{\rho}}\right]_{\tilde{p}^{2}=\mu^{2}} \,=\,1\,,\qquad\mathrm{any}f\,,\] (5) where $\tilde{p}^{2}=\sum_{\mu}\tilde{p}_{\mu}^{2}\,,\ \tilde{p}_{\mu}\equiv\tfrac{1} {a}\sin ap_{\mu}$. The sum $\sum_{\rho}^{\prime}$ only runs over the Lorentz indices for which $p_{\rho}$ is different from zero and $N(p)=\sum_{\rho}^{\prime}1$. The renormalization condition for the operators $O_{\Gamma}$ reads \[Z_{q}^{-1}Z^{(ff^{\prime})}_{\Gamma}\mathrm{Tr}\left[\Lambda^{(ff^{\prime})}_{ \Gamma}(\tilde{p},\tilde{p})P_{\Gamma}\right]_{\tilde{p}^{2}=\mu^{2}}\,=\,1\,, \qquad f\neq f^{\prime}\,.\] (6) Above $S_{f}(p)\,=\,a^{4}\sum_{x}\,e^{-ipx}\left\langle\chi_{f}(x)\bar{\chi_{f}}(0)\right\rangle$ is the $\chi_{f}$ field propagator in momentum space, while \[\Lambda^{(ff^{\prime})}_{\Gamma}(p,p)\,=\,S_{f}^{-1}(p)G_{\Gamma}^{(ff^{\prime })}(p,p)S_{f^{\prime}}^{-1}(p)\] (7) denotes the quark bilinear vertex that is obtained by “amputating” the Green function \[G^{(ff^{\prime})}_{\Gamma}(p,p)\,=\,a^{8}\sum_{x,y}\,e^{-ip(x-y)}\left\langle \chi_{f}(x)(\bar{\chi}_{f}\Gamma\chi_{f^{\prime}})(0)\bar{\chi}_{f^{\prime}}(y )\right\rangle\qquad\Gamma\;=\;S,P,V,A,T\,.\] (8) Barring cutoff effects, RCs are independent of $sign(r_{f})$. For practical reasons here we limit ourselves to $r_{f^{\prime}}=-r_{f}$ in evaluating $Z_{\Gamma}$, see eq. (6). ## 2 Strategy for RCs in the $N_{f}=4$ theory In order to extract useful information from simulations performed with twisted mass Wilson fermions one must know the twist angle, $\omega=\tfrac{\pi}{2}-\theta$, with good precision. The level of precision requested for $\omega$ depends on the observable of interest. In our case, after an exploratory study on a few $16^{3}\times 32$ lattices [7], and some tests near maximal twist on a $24^{3}\times 48$ lattice we have chosen to work _out of maximal twist_. Figure 1 illustrates the difficulties of tuning to maximal twist, i.e. setting $m_{\rm PCAC}$ to zero, in the simulation setup for RC computations, at least if the lattice spacing is not very fine. Specifically, the slope of $m_{\rm PCAC}$ vs $1/(2\kappa)$ in figure 1 suggests that near $m_{\rm PCAC}=0$ simulations are in a region with a sharpe change of the slope for $m_{\mathrm{PCAC}}$ where it is difficult to extract useful information. On the other hand figure 1 gives a more quantitative view of this problem showing results from one simulation close to the critical point (the point closest to $m_{\mathrm{PCAC}}=0$ in figure 1). It appears that due to the long fluctuations a precise measurement of the PCAC mass will require for this case a very large number of Monte Carlo trajectories. In fact, we have observed a similar feature for all the ensembles with $\|am_{\mathrm{PCAC}}\|\lesssim 0.01$ at both $\beta=1.95$ and $\beta=1.90$ (i.e. for $a\geq 0.08$ fm). In summary, working at maximal twist for the chosen range of twisted masses (see table 1) would imply a considerable fine tuning work owing to the difficulties in determining $am_{\mathrm{PCAC}}$ near $am_{\mathrm{PCAC}}=0$. To alleviate the problem one would need to increase the value of the twisted mass, $\mu_{f}$, and thus $M_{f}$. Instead, working away from _maximal twist_, one can avoid the metastable region of parameter space and measure the twist angle with good precision. This comes at the price of a moderate increase of the quark mass $M_{f}$ and of a slightly more involved analysis. In our RC-estimators cutoff effects linear in $a$ are expected to be small and can anyway be removed with controlled precision by averaging the results obtained for a given $M_{f}$ at opposite values of $\theta_{f}$. <figure><img src="content_image/1101.1877/b195_pcacVSm0.png"><figcaption></figcaption></figure> In fact, from the symmetry of the lattice action $S_{L}$ under ${\cal P}\times(\theta_{0}\to-\theta_{0})\times{\cal D}_{d}\times(M_{0}\to-M_{0})$[1, 8, 12] it follows that the $O(a^{2k+1})$ artifacts occurring in the vacuum expectation values of (multi)local operators $O$ that are invariant under ${\cal P}\times(\theta_{0}\to-\theta_{0})$ are quantities that change sign upon changing the sign of $\theta_{0}$ (or $\theta$). Hence $O(a^{2k+1})$ cutoff effects vanish in $\theta$-averages: $~{}\frac{1}{2}\Big{[}\langle O\rangle\|_{\hat{M},\theta}+\langle O\rangle\|_{ \hat{M},-\theta}\Big{]}$. The same is true for operator form factors invariant under ${\cal P}\times(\theta_{0}\to-\theta_{0})$ and, in particular, for our RC-estimators at all values of $M_{f}$ and $\tilde{p}^{2}$. ## 3 Current analysis and preliminary results ensemble \| aμsea \| amseaPCAC \| aMsea0 \| θsea \| aμval \| amvalPCAC ---\|---\|---\|---\|---\|---\|--- 1m \| 0.0085 \| -0.04125(13) \| 0.03288(10) \| -1.3093(8) \| [0.0085,…, 0.0298] \| -0.0216(2) 1p \| 0.0085 \| +0.04249(13) \| 0.03380(10) \| 1.3166(7) \| [0.0085,…, 0.0298] \| +0.01947(19) 3m \| 0.0180 \| -0.0160(2) \| 0.02182(9) \| -0.601(6) \| [0.0060,…, 0.0298] \| -0.0160(2) 3p \| 0.0180 \| +0.0163(2) \| 0.02195(9) \| 0.610(6) \| [0.0060,…, 0.0298] \| +0.0162(2) 2m \| 0.0085 \| -0.02091(16) \| 0.01821(11) \| -1.085(3) \| [0.0085,…, 0.0298] \| -0.0213(2) 2p \| 0.0085 \| +0.0191(2) \| 0.01696(16) \| 1.046(6) \| [0.0085,…, 0.0298] \| +0.01909(18) 4m \| 0.0085 \| -0.01459(13) \| 0.01409(8) \| -0.923(4) \| [0.0060,…, 0.0298] \| -0.01459(13) 4p \| 0.0085 \| +0.0151(2) \| 0.01441(14) \| 0.940(7) \| [0.0060,…, 0.0298] \| +0.0151(2) Table 1: Mass parameters of the ensembles analysed for this contribution. From the formulae in sect. 1.1 it follows that in the valence sector we have 0.013≲aMval≲0.033 and 0.4≲\|θval\|≲1.2 (θval/mvalPCAC>0). Here we detail the analysis procedure we followed in order to obtain _very preliminary_ results on the RCs of interest. Indeed, at this stage our main goal was checking the feasibility of the project. In particular, the analysis procedure is not yet the optimal one, for instance concerning the order of the various steps, and some refinements, such as the subtraction of the known cutoff effects at O($a^{2}g^{2}$) [9], are still omitted. While these improvements will be included in the final analysis, the present work shows that the strategy advocated in section 2 allows to extract the RCs of the quark field and quark bilinear operators with a $\sim 1\%$ level precision by means of stable simulations at a lattice spacing ($a\sim 0.08$ fm) which is among the coarsest ones explored in the study of $n_{f}=2+1+1$ QCD by ETMC. In practice, for a sequence of $M_{\mathrm{sea}}$-values, we produced for each $M_{\mathrm{sea}}$ two ensembles with opposite values of $\theta_{\mathrm{sea}}$. We label them as Ep/m, where E$\,=1,2\ldots$ and p/m refers to sign($\theta_{\mathrm{sea}}$). On each ensemble Ep/m, with $(M_{\mathrm{sea}}^{\mathtt{E}\mathrm{p/m}},\theta_{\mathrm{sea}}^{\mathtt{E} \mathrm{p/m}})$ we compute the RC-estimators for several values of the valence mass parameters $(M_{\rm val},\theta_{\rm val})$ and $\tilde{p}^{2}$ (all corresponding to “democratic” momenta $p$, in the sense specified in [10]), as summarized in table 1. Then we proceed in various steps as follows. <figure><img src="content_image/1101.1877/x1.png"><figcaption></figcaption></figure> Valence chiral limit.: A fit of RC-estimators linear in $(M^{PS}_{\rm val})^{2}$ turns out to be numerically adequate (see fig. 2). For $\Gamma=P$ (see fig. 2) or, due to O($a^{2}$) terms, $\Gamma=S$, we have also kept into account the contribution $\propto(M^{PS}_{\rm val})^{-2}$ coming from the Goldstone boson pole. $O(a^{2}\tilde{p}^{2})$ discretization errors.: We applied two different methods, following [10]. In the first method (“M1”), after bringing, via the known [11] perturbative evolution the RC-estimators to a common renormalization scale ($\tilde{p}^{2}_{\mathrm{M1}}=1/a^{2}$), we remove the remaining $O(a^{2}\tilde{p}^{2})$ discretization errors by a linear fit in $\tilde{p}^{2}$. The second method (“M2”) consists in simply taking the value of the RCs estimators at a high momentum point fixed in physical units. We chose $\tilde{p}^{2}_{\mathrm{M2}}\,=\,12.2~{}\mathrm{GeV}^{2}$. The two approaches yield RC results differing only by cutoff effects. <figure><img src="content_image/1101.1877/fig_all_1mstar2.png"><figcaption></figcaption></figure> Removal of $O(a)$ artifacts.: It is achieved by $\theta$-average (see section 2) of the RCs estimators, \[Z_{\Gamma}(M_{\mathrm{sea}}^{\mathtt{E}},\|\theta_{\mathrm{sea}}^{\mathtt{E}}\|) \,=\,\frac{1}{2}\Big{[}\langle Z_{\Gamma}(\hat{M}_{\mathrm{sea}}^{\mathtt{E} \mathrm{p}},\theta_{\mathrm{sea}}^{\mathtt{E}\mathrm{p}};\theta_{\mathrm{val; eff}}^{\mathrm{E}\mathrm{p}})\rangle+\langle Z_{\Gamma}(\hat{M}_{\mathrm{sea}} ^{\mathtt{E}\mathrm{m}},\theta_{\mathrm{sea}}^{\mathtt{E}\mathrm{m}};\theta_{ \mathrm{val;eff}}^{\mathrm{E}\mathrm{m}})\rangle\Big{]}\] (9) where $\theta_{\mathrm{val;eff}}^{\mathrm{E}\mathrm{p(m)}}$ parameterizes the dominating O($a$) effects in RC-estimators that (in the present analysis) arise from employing $M_{\rm val;Ep(m)}^{\rm PS}$ in the valence chiral extrapolation. Sea chiral limit.: The quantities $Z_{\Gamma}(M_{\mathrm{sea}}^{\mathtt{E}},\|\theta_{\mathrm{sea}}^{\mathtt{E}}\|)$ are extrapolated to $M_{\rm sea}=0$ by using the fit Ansatz \[Z_{\Gamma}(M_{\mathrm{sea}},\theta_{\mathrm{sea}})\,=\,Z_{\Gamma}\;+\;A\,M^{2} _{\mathrm{sea}}\;+\;B\,M^{2}_{\mathrm{sea}}\,\cos(2\theta_{\mathrm{sea}})\,.\] (10) This Ansatz can be justified by an analysis à la Symanzik of the lattice artifacts in $Z_{\Gamma}(M_{\mathrm{sea}},\theta_{\mathrm{sea}})$ up to O($M_{\rm sea}^{2}$) and neglecting chiral spontaneous symmetry breaking effects [12]. <figure><img src="content_image/1101.1877/RCs_4masses1.png"><figcaption></figcaption></figure> The first, very preliminary results of this analysis are summarized in table 2. Method \| ZA \| ZV \| ZP(1/a) \| ZS(1/a) \| ZP/ZS \| ZT(1/a) \| Zq(1/a) ---\|---\|---\|---\|---\|---\|---\|--- M1 \| 0.761(08) \| 0.630(05) \| 0.438(08) \| 0.614(09) \| 0.716(21) \| 0.753(07) \| 0.767(06) M2 \| 0.771(03) \| 0.674(03) \| 0.496(04) \| 0.647(03) \| 0.767(08) \| 0.768(03) \| 0.813(02) Table 2: Preliminary RC results at β=1.95 from the analysis of section 3\. We also get ZV(WI)=0.612(1). ## 4 Conclusions and outlook We have described our strategy to compute O($a$) improved operator RCs for the $N_{f}=4$ lattice action currently used by ETMC. We have shown that the method advocated in this work provides very encouraging results at one lattice spacing ($a\sim 0.08$ fm) that is among the coarsest simulated in the study of QCD with $n_{f}=2+1+1$ dynamical flavours. In particular, the observed dependences of RCs on valence and sea quark masses are mild and quite in line with our experience [10] in $n_{f}=2$ QCD. 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1803.10093	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 128455, "num_imgs": 7, "llama3_tokens_count": 41371 }	[ "content_image/1803.10093/x1.png", "content_image/1803.10093/x5.png", "content_image/1803.10093/x9.png", "content_image/1803.10093/x13.png", "content_image/1803.10093/x17.png", "content_image/1803.10093/x21.png", "content_image/1803.10093/x23.png" ]	# Search for a heavy resonance decaying into a boson and a or boson in $2\ell 2\PQq$ final states at $\sqrt{s}=13\TeV$ February 21, 2024 ###### Abstract A search has been performed for heavy resonances decaying to $\PZ\PZ$ or $\PZ\PW$ in $2\ell 2\PQq$ final states, with two charged leptons ($\ell=\Pe,\Pgm$) produced by the decay of a boson, and two quarks produced by the decay of a or boson. The analysis is sensitive to resonances with masses in the range from 400 to 4500. Two categories are defined based on the merged or resolved reconstruction of the hadronically decaying vector boson, optimized for high- and low-mass resonances, respectively. The search is based on data collected during 2016 by the CMS experiment at the LHC in proton-proton collisions with a center-of-mass energy of $\sqrt{s}=13\TeV$, corresponding to an integrated luminosity of 35.9. No excess is observed in the data above the standard model background expectation. Upper limits on the production cross section of heavy, narrow spin-1 and spin-2 resonances are derived as a function of the resonance mass, and exclusion limits on the production of bosons and bulk graviton particles are calculated in the framework of the heavy vector triplet model and warped extra dimensions, respectively. B2G-17-013 $HeadURL:svn+ssh://svn.cern.ch/reps/tdr2/papers/B2G-17-013/trunk/B2G-17-013.tex$$Id:B2G-17-013.tex4758232018-09-2214:00:46Zjpazzini$ B2G-17-013 ## 0.1 Introduction The validity of the standard model (SM) of particle physics is corroborated by a wide set of precise experimental results with an impressive level of accuracy. Nonetheless, there are several open points where the SM fails to provide an explanation, either for experimental observations, as in the case of the presence of dark matter in the universe, or for theoretical questions, such as the omission of gravity from the SM, and the hierarchy problem. Several SM extensions addressing the open questions of the SM predict the presence of new heavy particles with an enhanced branching fraction for decays into pairs of vector bosons. The existence of heavy spin-2 gravitons () is predicted in the Randall–Sundrum model with warped extra spatial dimensions (WED) [1, 2, 3]. In the bulk scenario [4, 5], the main free parameters are the mass of the first Kaluza–Klein graviton excitation (the bulk graviton mass), and the ratio $\widetilde{\kappa}\equiv\kappa/\overline{M}_{\text{Pl}}$, where $\kappa$ is a curvature parameter of the WED metric and $\overline{M}_{\text{Pl}}\equiv\Mpl/\sqrt{8\pi}$ is the reduced Planck mass. The introduction of a spin-1 triplet of and bosons is described in the heavy vector triplet (HVT) model [6], which generalizes a large number of explicit models in terms of a small set of parameters: $c_{\mathrm{H}}$, controlling the interactions of the triplet with the SM vector and Higgs bosons; $c_{\mathrm{F}}$, which describes the direct interaction with fermions; and $g_{\mathrm{V}}$, which represents the overall strength of the new vector boson triplet interactions. A variety of searches for heavy resonances decaying to two vector bosons have been carried out in the past. The most recent results from the CERN LHC [7, 8, 9, 10, 11], with no evidence of signal, have provided stringent upper limits on signal cross sections in these models. This paper reports on the results of a search for heavy, narrow resonances (collectively indicated as $\cmsSymbolFace{X}$) decaying into $2\ell 2\cPq$ final states, with two charged leptons ($\ell=\Pe,\Pgm$) produced by the leptonic decay of a boson and a pair of quarks produced from the hadronic decay of a vector boson ($\cmsSymbolFace{V}$ = or ). In the narrow-width assumption, the width of the heavy resonance is taken to be small in comparison to the experimental resolution. Two complementary search strategies are defined to span the mass range $400<m_{\cmsSymbolFace{X}}<4500\GeV$, where $m_{\cmsSymbolFace{X}}$ is the mass of the heavy resonance. The first strategy, referred to as the “high-mass analysis”, is optimized for the range $850<m_{\cmsSymbolFace{X}}<4500\GeV$ by selecting events where the vector bosons have a large Lorentz boost, resulting in the collimation of their decay products. The high-mass analysis uses dedicated leptonic reconstruction and identification techniques to reconstruct leptons in close proximity to each other in order to retain high signal efficiency, as well as jet substructure techniques to identify the hadronic decay of the or boson into a pair of quarks contained in a single merged reconstructed jet. For lower resonance masses, the quarks produced by the hadronic decay of the $\cmsSymbolFace{V}$ boson may be sufficiently separated to be reconstructed as two single narrow jets (dijet). A second strategy, referred to as the “low-mass analysis”, is therefore defined in this regime, exploiting dijet reconstruction in addition to the reconstruction of merged jets to retain signal efficiency in the range $400<m_{\cmsSymbolFace{X}}<850\GeV$ for those events in which no merged $\cmsSymbolFace{V}$ candidate is found. To increase the signal sensitivity, in the low-mass analysis a categorization based on the flavor of the jets is used, to exploit the relatively large decay branching fraction of the boson to pairs of b quarks. This paper is organized as follows: in Section 0.2, a description of the data and simulated samples used in the analysis is provided; Section 0.3 briefly describes the CMS detector; Section 0.4 provides a description of the event reconstruction; in Section 0.5, the event selection is discussed; Section 0.6 contains the description of the signal and describes the estimation of the SM background; the systematic uncertainties affecting the analysis are presented in Section 0.7; and the results of the search for heavy spin-1 and spin-2 resonances are presented in Section 0.8. Finally, results are summarized in Section 0.9. ## 0.2 Data and simulated samples This analysis uses data collected by the CMS detector during proton-proton () collisions at the LHC at $\sqrt{s}=13\TeV$, corresponding to an integrated luminosity of 35.9. The events were selected online by criteria that require the presence of at least one electron or muon; these criteria are described in Sec. 0.5. Simulated signal samples are used in the analysis to optimize the search for the potential production of heavy spin-1 or spin-2 resonances. For this purpose, signal samples are generated according to the HVT and WED scenarios, respectively. For both scenarios, the samples are generated at leading order (LO) in QCD with the 2.2.2 generator [12]. Two HVT models are considered as benchmarks, “model A” and “model B”, with different values of the three defining parameters described earlier: for “model A”, $g_{\mathrm{V}}=1$, $c_{\mathrm{H}}=-0.556$, and $c_{\mathrm{F}}=-1.316$, while for “model B”, $g_{\mathrm{V}}=3$, $c_{\mathrm{H}}=-0.976$, and $c_{\mathrm{F}}=1.024$. Different resonance mass hypotheses are considered in the range from 400 to 4500. The resonance width is predicted to be between 0.4 and 2.3for a candidate in HVT model A, and between 14 and 64for HVT model B, depending on the mass hypothesis [6]; in the WED model with $\widetilde{\kappa}=0.1$, the bulk graviton signal width is predicted to range from 3.6 to 54 [13]. Since the resonance width is small in comparison with the experimental resolution, for simplicity, the width is taken to be 1in the simulation. In the case of the spin-1 , the resonance is forced to decay into one and one boson; additionally, the boson is then forced to decay to a pair of electrons, muons, or tau leptons, while the boson is forced to decay into a pair of quarks. The generated spin-2 bulk graviton is instead forced to decay into two bosons, one decaying leptonically into any pair of charged leptons, and the other boson decaying hadronically into a pair of quarks. Several SM processes yielding final states with charged leptons and jets are sources of background events for the analysis, and corresponding Monte Carlo (MC) simulated samples have been generated and used in the analysis. The SM production of a boson in association with quarks or gluons in the final state ($\PZ+\text{jets}$) represents the dominant background process for the analysis, having topological similarities to the signal because of the presence of a pair of charged leptons and jets. However, since the quark- and gluon-induced jets are not associated with the decay of a vector boson, the jet mass spectrum is characterized by a smooth distribution and the distribution of the $2\ell+\text{jet}$ system invariant mass falls exponentially, in contrast with the peaking distribution expected from the signal in both the jet and $2\ell+\text{jet}$ mass spectra. The $\PZ+\text{jets}$ MC samples are produced with at next-to-leading order (NLO), using the FxFx merging scheme [14] between the jets from matrix element calculations and parton showers, and normalized to the next-to-NLO cross section computed using v3.1 [15]. Another important source of SM background arises from processes leading to top quark production. Simulated samples describing the production of top quark pairs are generated with at LO, with the MLM matching scheme [16]. Single top quark production is also considered; $s$- and $t$-channel single top quark samples are produced in the four-flavor scheme using and v2 [17, 18, 19, 20], respectively, while $\PQt\PW$ production is simulated at NLO with in the five-flavor scheme [21]. Additional top quark background processes, such as the associated production of a or boson with pair-produced top quarks, and the production of $\PQt\PQq\PZ$, are also considered in the analysis and produced at NLO with . The SM diboson production of $\cmsSymbolFace{V}\cmsSymbolFace{V}$ is an irreducible source of background for the analysis, since the jet mass spectrum will contain a peak from the hadronic decay of and bosons, like the expected jet mass spectrum for the signal; however, this process produces a smoothly falling $2\ell+\text{jet}$ invariant mass distribution. The SM production of pairs of vector bosons ($\PW\PW$, $\PW\PZ$, and $\PZ\PZ$) is simulated at NLO with . For all the simulated samples used in the analysis, the simulation of parton showering and hadronization is described by interfacing the event generators with 8.212 [22] with the CUETP8M1 [23] tune, while the parton distribution functions (PDFs) of the colliding protons are given by the NNPDF 3.0 [24] PDF set. Additional $\Pp\Pp$ interactions occurring in the same or nearby bunch crossings (pileup) are added to the event simulation, with a frequency distribution adjusted to match that observed in data. All samples are processed through a simulation of the CMS detector using [25], and reconstructed using the same algorithms as those for the data collected. ## 0.3 The CMS detector The central feature of the CMS apparatus is a superconducting solenoid of 6m internal diameter, providing a magnetic field of 3.8T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. The silicon tracker covers the pseudorapidity range $\abs{\eta}<2.5$, while the ECAL and HCAL cover the range $\abs{\eta}<3.0$. Forward calorimeters extend the coverage provided by the barrel and endcap detectors to $\abs{\eta}<5.2$. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid, with detection planes made using three technologies: drift tubes, cathode strip chambers, and resistive-plate chambers. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [26]. ## 0.4 Event reconstruction The event reconstruction is performed globally using a particle-flow (PF) algorithm [27], which reconstructs and identifies each individual particle with an optimized combination of information from the various elements of the CMS detector. The reconstructed vertex with the largest value of summed physics-object $\pt^{2}$ is taken to be the primary $\Pp\Pp$ interaction vertex. The physics objects chosen are those that have been defined using information from the tracking detector. These objects include jets, the associated missing transverse momentum, which was taken as the negative vector sum of the transverse momentum () of those jets, and charged leptons. In the silicon tracker, isolated charged particles with $\pt=100\GeV$ and $\abs{\eta}<1.4$ have track resolutions of 2.8% in and 10 (30)in the transverse (longitudinal) impact parameter [28]. The energy of charged hadrons is determined from a combination of their momenta measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. The energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies. Electrons are required to be within the range $\abs{\eta}<2.5$ covered by the silicon tracker, and are reconstructed from a combination of the deposited energy of the ECAL clusters associated with the track reconstructed from the measurements determined by the inner tracker, and the energy sum of all photons spatially compatible with being bremsstrahlung from the electron track. The identification of electrons is based on selection criteria relying on the direction and momentum of the track in the inner tracker, its compatibility with the primary vertex of the event [27], and on observables sensitive to the shape of energy deposits along the electron trajectory. The momentum resolution for electrons with $\pt\approx 45\GeV$ from $\PZ\to\Pe\Pe$ decays ranges from 1.7% to 4.5% [29]. It is generally better in the barrel region than in the endcaps, and also depends on the amount of bremsstrahlung emitted by the electron as it traverses the material in front of the ECAL. Muons are reconstructed in the entire CMS muon system acceptance region of $\abs{\eta}<2.4$ by combining in a global fit the information provided by the measurements in the silicon tracker and the muon spectrometer. Candidate muons are selected using criteria based on the degree of compatibility of the inner track, which is reconstructed using the silicon tracker only, and the track reconstructed using the combination of the hits in both the tracker and spectrometer. Further reconstruction requirements include the compatibility of the trajectory with the primary vertex of the event, and the number of hits observed in the tracker and muon systems. The relative resolution achieved is 1.3–2.0% for muons with $20<\pt<100\GeV$ in the barrel and better than 6% in the endcaps. The resolution in the barrel is better than 10% for muons with up to 1 [30]. Both electrons and muons are required to be isolated from hadronic activity and other leptons in the event. An isolation variable is defined as the scalar sum of the of charged hadrons originating from the primary vertex, plus the scalar sums of the transverse momenta for neutral hadrons and photons, in a cone of $\Delta R=\sqrt{\smash[b]{(\Delta\eta)^{2}+(\Delta\phi)^{2}}}<0.3\,(0.4)$ around the electron (muon) direction corrected to account for the contribution from neutral candidates originating from pileup, where $\phi$ is the azimuthal angle in radians. In the high-mass analysis, a specific muon isolation requirement is implemented to retain signal efficiency for high resonance masses, where the large boson boost may result in extremely close pairs of muons. For this reason, muon candidates in the high-mass analysis are retained if they pass an isolation requirement based on the sum of reconstructed of all tracks within $\Delta R<0.3$ from the muon trajectory, ignoring tracks associated with other reconstructed muons. Hadron jets are clustered from particles reconstructed by the PF algorithm using the infrared- and collinear-safe anti-algorithm [31, 32] with distance parameters of 0.4 (AK4 jets) and 0.8 (AK8 jets). The jet momentum is determined as the vectorial sum of all constituent particle momenta. Contamination from pileup is suppressed using charged hadron subtraction (CHS) which removes from the list of PF candidates all charged particles originating from vertices other than the primary interaction vertex of the event. The residual contribution from neutral and charged particles originating from pileup vertices is removed by means of an event-by-event jet-area-based correction to the jet four-momentum. Identification requirements, based on the estimation of the energy fraction carried by the different types of PF candidates clustered into a jet, along with the multiplicity of the PF candidates, are used to remove jets originating from calorimetric noise. Corrections to the jet energy are derived from the simulation, and are confirmed with in situ measurements with the energy balance of dijet, multijet, $\text{photon}+\text{jet}$, and leptonically decaying $\PZ+\text{jet}$ events [33]. A jet grooming technique is used for AK8 jets in this analysis to help identify and discriminate between jets from boosted hadronic $\cmsSymbolFace{V}$ decays, which we refer to as “merged jets”, and jets from quarks and gluons. The AK8 jets are groomed by means of the modified mass drop tagger algorithm [34], also known as the soft drop algorithm, with angular exponent $\beta=0$, soft cutoff threshold $z_{\text{cut}}<0.1$, and characteristic radius $R_{0}=0.8$ [35]. The soft drop algorithm does not fully reject contributions from the underlying event and pileup. The mass of the AK8 jet ($m_{\mathrm{j}}$) is therefore defined as the invariant mass associated to the four-momentum of the soft drop jet, after the application of the pileup mitigation corrections provided by the pileup per particle identification (PUPPI) algorithm [36]. Discrimination between AK8 jets originating from vector boson decays and those originating from gluons and quarks is also achieved by the $N$-subjettiness jet substructure variable [37]. This observable exploits the distribution of the jet constituents found in the proximity of the subjet axes to determine if the jet can be effectively subdivided into a number $N$ of subjets. The generic $N$-subjettiness variable $\tau_{N}$ is defined as the -weighted sum of the angular distance of all the $k$ jet constituents from the closest subjet: \[\tau_{N}=\frac{1}{d_{0}}\sum_{k}p_{\mathrm{T},k}\min(\Delta R_{1,k},\Delta R_{ 2,k},\dots,\Delta R_{N,k}).\] (1) The normalization factor $d_{0}$ is defined as $d_{0}=\sum_{k}p_{\mathrm{T},k}R_{0}$, with $R_{0}$ the clustering parameter of the original jet. In this analysis, which aims to select $\cmsSymbolFace{V}\to\cPq\cPq^{(\prime)}$ decays, the variable that best discriminates V boson jets from those from quarks and gluons is the ratio of the 2-subjettiness to the 1-subjettiness: $\tau_{21}=\tau_{2}/\tau_{1}$. The $\tau_{21}$ observable is calculated for the jet before the grooming procedure, and includes the PUPPI algorithm corrections for pileup mitigation. For the identification of jets originating from the hadronization of bottom quarks, the combined secondary vertex (CSVv2) algorithm [38, 39] is used, either directly on the AK4 jets or on the AK8 soft drop subjets with CHS pileup mitigation applied. Only AK4 and AK8 jets reconstructed centrally in the detector acceptance, within $\abs{\eta}<2.4$, are considered in the analysis. ## 0.5 Event selection Events are selected online by requiring the reconstruction at trigger [40] level of at least one charged lepton. For the high-mass analysis, thresholds of 115 (50)are used for electrons (muons). No isolation requirements are applied at trigger level, to retain efficiency for high-mass signals, where the large boost expected for the leptonically decaying boson will cause the two charged leptons to be collimated in the detector. For the low-mass analysis a larger separation between the leptons is expected because of the lower of the boson, and isolation requirements are included in the trigger selection, allowing the use of lower lepton thresholds. The online selection for the low-mass analysis requires at least one electron with $\pt>25\GeV$ and $\abs{\eta}<2.1$ passing tight identification and isolation requirements, or at least one muon with $\pt>24\GeV$ and $\abs{\eta}<2.4$, subject to loose identification and isolation requirements, using the variables described in Ref. [40]. To reconstruct the boson candidate, at least two well-identified leptons with opposite charge and the same flavor are required to be present in the event. The leading lepton in the event is required to pass more stringent selection requirements than the online thresholds to avoid inefficiencies induced by the trigger selections. In the high-mass analysis, the leading (subleading) lepton is required to have $\pt>135\,(35)\GeV$ for electrons, and $\pt>55\,(20)\GeV$ for muons. Loose isolation and identification requirements are applied to the leptons to retain high signal efficiency. For electrons, we use a set of requirements that have been observed to have an efficiency of about 90% for both low and high mass points. For muons, as the CMS standard requirements [41] only have an efficiency of about 65% for close muons, we instead use a dedicated selection where one of the two muons is allowed to be identified only in the tracker. The isolation variable is calculated removing the contribution of the other muon if it falls within the isolation cone, therefore recovering a signal efficiency of about 90% for high mass resonances. For the low-mass analysis, the leading (subleading) lepton is required to have larger than $40\,(30)\GeV$ and to fall in the range $\abs{\eta}<2.1\,(2.4)$. The selection of the boson candidate relies on the invariant mass of the dilepton pair, $m_{\ell\ell}$. This is required to satisfy $70<m_{\ell\ell}<110\GeV$, except for the low-mass analysis in the resolved category (discussed below) where the requirement is $76<m_{\ell\ell}<106\GeV$ to enhance the sensitivity to the signal by reducing the nonresonant contribution in the sample with tagged jets. Different strategies are used in the low- and high-mass analyses to identify and reconstruct the hadronically decaying $\cmsSymbolFace{V}$ boson, as described below, to cope with the different $\cmsSymbolFace{V}$ boson boost regimes expected for low- and high-mass signal candidates. In the high-mass analysis a merged jet is required in the event, and its mass $m_{\mathrm{j}}$ is used to select the hadronically decaying or . The signal is expected to be almost fully contained in the mass range $65<m_{\mathrm{j}}<105\GeV$, which is thus defined as the signal region (SR). In order to select candidate signal events, where a heavy massive particle decays into a pair of boosted vector bosons, both the dilepton pair and the leading jet selected in an event are required to have $\pt>200\GeV$; this is motivated by the spectrum of the $\cmsSymbolFace{V}$ bosons observed in simulation. Events are divided into categories depending on the flavor of the charged leptons ( or ) and the value of the jet $\tau_{21}$ variable. As the signal is expected to have lower values of $\tau_{21}$, two different purity categories are defined: events with $\tau_{21}<0.35$ are defined as the high-purity (HP) category, while events with $0.35<\tau_{21}<0.75$ fall into a low-purity (LP) category, used to retain some sensitivity to signal although a larger amount of background is expected with respect to the HP category. The $\tau_{21}>0.75$ region is expected to be dominated by the background, and is therefore not used in the high-mass analysis. In total, four exclusive categories (from the two purity and two lepton flavor categories) are defined for the high-mass analysis. In the low-mass analysis, events are divided into two categories depending on whether the two quarks from the hadronic $\cmsSymbolFace{V}$ decay merge into a single reconstructed jet or can be resolved as two distinct jets. In the merged category, merged jets with $\pt>200\GeV$ and $\tau_{21}<0.40$ are selected. The choice of a looser $\tau_{21}$ selection with respect to the cutoff applied in the HP category of the high-mass analysis is driven by the higher expected signal efficiency for merged events, which are selected in the low-mass analysis using only one $\tau_{21}$ category. As in the high-mass analysis, the jet mass is required to be in the range $65<m_{\mathrm{j}}<105\GeV$ for the jet to be considered a candidate or boson, which is also defined as the SR for the merged low-mass analysis. The resolved category contains events that do not contain a merged $\cmsSymbolFace{V}$ candidate, but instead two AK4 jets, both with $\pt>30\GeV$ that form a dijet candidate with invariant mass $m_{\mathrm{jj}}>30\GeV$ and $\pt>100\GeV$. In both the merged and resolved cases, the selection is determined by comparing the spectrum of simulated signal events with the expected background. Both the merged and resolved categories are further split into two tag categories. Events in the merged tagged category are required to have at least one subjet satisfying a tagging requirement corresponding to $\approx$65% efficiency for b quark identification and $\approx$1% light-flavor jet mistag rate; events not passing this requirement are placed in the merged untagged category. For the resolved tagged category, events are required to have at least one jet satisfying the same tagging requirement used in the merged category; a looser tag selection is instead required for the other jet, with $\approx$80% efficiency and $\approx$10% light-flavor jet mistag rate. Events failing these requirements fall in the so-called resolved untagged category. An arbitration procedure is used to select the dijet candidate in case of events containing more than two selected narrow jets: first, if a dijet passing the tagging requirements is selected in the event, the candidate in the tag category is chosen; then the dijet candidate closest in mass to the boson mass is selected as the candidate $\cmsSymbolFace{V}$ boson. The signal region for the low-mass resolved category accepts events in the dijet mass range $65<m_{\mathrm{jj}}<110\GeV$. Eight categories are defined in the low-mass analysis, based on the lepton flavor, the -tag category, and the merged or resolved reconstruction of the hadronically decaying $\cmsSymbolFace{V}$ candidate. The $\tau_{21}$ and merged jet distributions of the $\cmsSymbolFace{V}$ candidate for events selected in the merged category of the low-mass analysis are shown in Fig. 1, where the $m_{\mathrm{j}}$ and $m_{\mathrm{jj}}$ distributions for events with a $\cmsSymbolFace{V}$ candidate are also shown for the merged and resolved low-mass analysis categories, respectively. <figure><img src="content_image/1803.10093/x1.png"><figcaption>Figure 1: Upper row: distribution of the merged \cmsSymbolFaceV candidate τ21(left), where the τ21<0.4 requirement has been removed, and the jet \pt(right)in data and simulation for events in the signal region of the low-massanalysis. Lower row: \cmsSymbolFaceV candidate mj (left) and mjj (right) indata and simulation for events in the signal regions of the low-mass search.The points show the data while the filled histograms show the backgroundcontributions. The gray band shows the statistical and systematicuncertainties in the background, while the dashed vertical region (“Higgs”)shows the expected SM Higgs boson mass range, which is excluded from thisanalysis. A 600\GeVbulk graviton signal prediction is represented by the blackdashed histogram; for visibility, the signal cross-section is increased by afactor of 5 in the merged category and 50 in the resolved category. With theexception of the jet \pt, which typically peaks at approximately half of theresonance mass, the quantities shown have minimal dependence on the mass ofthe resonance. The background normalization is derived from the final fit tothe m\cmsSymbolFaceV\PZ observable in data.</figcaption></figure> ## 0.6 Background estimation ### High-mass analysis The main source of background events in the final state of the analysis arises from the production of a leptonically decaying boson in association with quark and gluon jets. A second background source relevant for the analysis is SM diboson production, mainly $\PZ\PZ$ and $\PZ\PW$, with a leptonically decaying boson together with a or boson decaying hadronically. These diboson events are an irreducible background for the analysis, as the mass distribution of the SM $\cmsSymbolFace{V}$ jet peaks in the same region as the signal. Finally, top quark production is considered as a source of background in the analysis, despite having a much smaller contribution with respect to other SM backgrounds in the region probed by this analysis, mostly because of the boson invariant mass selection and the large boost required in the event. All SM background processes are characterized by a smoothly falling distribution of the invariant mass of the dilepton pair and the jet selected ($m_{\cmsSymbolFace{V}\PZ}$), whereas the signal is instead expected to appear as a narrow peak at a value of $m_{\cmsSymbolFace{V}\PZ}$ close to the actual value of the mass of the resonance $m_{\cmsSymbolFace{X}}$. To minimize the dependency on the accuracy of the simulation, the contribution of the dominant background, $\PZ+\text{jets}$ SM production, is estimated using data. Two signal-depleted regions are defined by selecting events with jet mass outside the $m_{\mathrm{j}}$ signal mass window defined in Section 0.5; these are the sideband (SB) regions. A lower sideband (LSB) region is defined for events with $30<m_{\mathrm{j}}<65\GeV$, close to the SR of the analysis, while a higher sideband (HSB) region contains events with $135<m_{\mathrm{j}}<300\GeV$. The region $105<m_{\mathrm{j}}<135\GeV$ is not used in the analysis, to exclude events containing the hadronic decays of a SM Higgs boson, which are targeted in other CMS analyses, such as that described in [42]. The $\PZ+\text{jets}$ background $m_{\cmsSymbolFace{V}\PZ}$ shape and normalization are obtained by extrapolation from fits to data in the SB regions. The $m_{\mathrm{j}}$ distribution for the SM background sources considered in the analysis is modeled by means of analytic functions describing the spectrum of each background in the mass region $30<m_{\mathrm{j}}<300\GeV$. In the LP category, the $m_{\mathrm{j}}$ spectrum in $\PZ+\text{jets}$ events is described by a smoothly falling exponential distribution, while a broad structure centered around the mass of the boson present in the HP category is modeled with an error function convolved with an exponential distribution, which is of particular importance for describing the behavior at large values of $m_{\mathrm{j}}$. The peaking structure of the diboson background, originating from the presence of a jet from a genuine or boson in the event, is described in both the LP and HP categories with a Gaussian distribution. The remaining component of the distribution, consisting of tails extending far from the SR, is modeled in the LP category with an exponential function, similarly to the $\PZ+\text{jets}$ case. In the HP category, the $\cmsSymbolFace{V}\cmsSymbolFace{V}$ events are mostly contained in the SR, and the small fraction of events present in the Higgs boson and LSB regions is described with an additional broad Gaussian contribution. The top quark background (, single top quark, $\PQt\PZ\PQq$, and $\ttbar\cmsSymbolFace{V}$ production) is mostly similar in shape to the $\PZ+\text{jets}$ background; in the LP category, in addition to the exponentially falling component, a Gaussian is included to model the top quark peak appearing in the HSB for $m_{\mathrm{j}}\approx 170\GeV$. The expected yield of the $\PZ+\text{jets}$ background in the SR is extracted by a fit of the $m_{\mathrm{j}}$ distribution in the SBs taking into account all background contributions. The parameters describing the $m_{\mathrm{j}}$ shape and normalization of the subdominant background processes are fixed to those extracted from the simulation. All the parameters used to describe the $\PZ+\text{jets}$ contribution are left free to float in the fit to the data SBs. Alternative functions modeling the $m_{\mathrm{j}}$ shape of the main $\PZ+\text{jets}$ background are used to evaluate the impact of the function choice on the signal normalization. The $m_{\mathrm{j}}$ distribution for expected and observed events is shown in Fig. 2. <figure><img src="content_image/1803.10093/x5.png"><figcaption>Figure 2: The mj distributions of the events in data, compared to the expectedbackground shape, for the high-mass analysis in the electron (upper) and muon(lower) channels, and for the high-purity (left) and low-purity (right)categories. The expected background shape is extracted from a fit to the datasidebands (\PZ+jets) or derived from simulation (“top quark” and“\cmsSymbolFaceV\cmsSymbolFaceV”). The dashed region around the background sumrepresents the uncertainty in the \PZ+jets distribution, while the dashedvertical region (“Higgs”) shows the expected SM Higgs boson mass range,excluded from the analysis. The bottom panels show the pull distributionbetween data and SM background expectation from the fit, where σdata is thePoisson uncertainty in the data.</figcaption></figure> To describe the shape of the $m_{\cmsSymbolFace{V}\PZ}$ variable for the $\PZ+\text{jets}$ background in the SR, the following transfer function is defined from simulation: \[\alpha(m_{\cmsSymbolFace{V}\PZ})=\frac{f_{\text{SR}}^{\text{MC},\PZ+\text{jets }}(m_{\cmsSymbolFace{V}\PZ})}{f_{\text{SB}}^{\text{MC},\PZ+\text{jets}}(m_{ \cmsSymbolFace{V}\PZ})},\] (2) where $f_{\text{SR}}^{\text{MC},\PZ+\text{jets}}(m_{\cmsSymbolFace{V}\PZ})$ and $f_{\text{SB}}^{\text{MC},\PZ+\text{jets}}(m_{\cmsSymbolFace{V}\PZ})$ are the probability density functions describing the $m_{\cmsSymbolFace{V}\PZ}$ spectrum in the SR and SBs, respectively, of the simulated $\PZ+\text{jets}$ sample. The shape of the $\PZ+\text{jets}$ background in the SR is then extracted from a simultaneous fit to data in the SBs, and to simulation in both the SR and SBs, to correct the functional form obtained from data using the $\alpha(m_{\cmsSymbolFace{V}\PZ})$ ratio. The $m_{\cmsSymbolFace{V}\PZ}$ shape is described by two-parameter exponential functions for both data and simulation. The final estimate of the background $m_{\cmsSymbolFace{V}\PZ}$ shape predicted in the SR is then given by the following relation: \[N_{\text{SR}}^{\text{pred}}(m_{\cmsSymbolFace{V}\PZ})=N_{\text{SR}}^{\PZ+\text {jets}}f_{\text{SB}}^{\text{obs},\PZ+\text{jets}}(m_{\cmsSymbolFace{V}\PZ}) \alpha(m_{\cmsSymbolFace{V}\PZ})+N_{\text{SR}}^{\text{MC},\PQt}f_{\text{SR}}^{ \text{MC},\PQt}(m_{\cmsSymbolFace{V}\PZ})+N_{\text{SR}}^{\text{MC}, \cmsSymbolFace{V}\cmsSymbolFace{V}}f_{\text{SR}}^{\text{MC},\cmsSymbolFace{V} \cmsSymbolFace{V}}(m_{\cmsSymbolFace{V}\PZ}),\] (3) where $N_{\text{SR}}^{\text{pred}}(m_{\cmsSymbolFace{V}\PZ})$ is the predicted background in the SR and $f_{\text{SB}}^{\text{obs},\PZ+\text{jets}}(m_{\cmsSymbolFace{V}\PZ})$ is the probability distribution function describing the $\PZ+\text{jets}$ background in the SBs. This is obtained from a fit of the overall background components to data in the SBs, after subtracting the subdominant top quark and $\cmsSymbolFace{V}\cmsSymbolFace{V}$ components, which are derived from simulation. The functions $f_{\text{SR}}^{\text{MC},\PQt}(m_{\cmsSymbolFace{V}\PZ})$ and $f_{\text{SR}}^{\text{MC},\cmsSymbolFace{V}\cmsSymbolFace{V}}(m_{\cmsSymbolFace {V}\PZ})$ are the probability distributions of the top quark and diboson components, respectively, also in this case fixed to the shapes derived from the simulated samples in the SR. The normalization of the $\PZ+\text{jets}$ background in the SR, $N_{\text{SR}}^{\PZ+\text{jets}}$, is provided by the result of the fit on the $m_{\mathrm{j}}$ data sidebands described above, while the normalization of the top quark and $\cmsSymbolFace{V}\cmsSymbolFace{V}$ backgrounds, $N_{\text{SR}}^{\text{MC},\PQt}$ and $N_{\text{SR}}^{\text{MC},\cmsSymbolFace{V}\cmsSymbolFace{V}}$, are fixed to the expected yields from simulation. The $\alpha(m_{\cmsSymbolFace{V}\PZ})$ function accounts for differences and correlations in the transfer process from the SB regions to the SR, and is largely unaffected by uncertainties in the overall $\PZ+\text{jets}$ cross section and distribution shapes. The final $m_{\cmsSymbolFace{V}\PZ}$ spectra in the SR are shown in Fig. 3, compared to the expected estimated background. The validity and robustness of the background estimation method is demonstrated by the agreement observed between the shape and normalization for events selected in an intermediate $m_{\mathrm{j}}$ mass region ($50<m_{\mathrm{j}}<65\GeV$), corresponding to the part of the LSB shown in Fig. 2 above 50, and the prediction made using the events in the remaining part of the LSB and the full HSB regions. <figure><img src="content_image/1803.10093/x9.png"><figcaption>Figure 3: Expected and observed distributions of the resonance candidate massm\cmsSymbolFaceV\PZ in the high-mass analysis, in the electron (upper) andmuon (lower) channels, and separately for the high-purity (left) and low-purity (right) categories. The shaded area represents the post-fit uncertaintyin the background. The bottom panels show the pull distribution between dataand post-fit SM background fit, where σdata is the Poisson uncertainty in thedata. The expected contribution from \PWprsignal candidates with massm\cmsSymbolFaceX=2000\GeV, normalized to a cross section of 100\unitfb, isalso shown.</figcaption></figure> The description of the signal $m_{\cmsSymbolFace{V}\PZ}$ shape is extracted from simulated signal samples. Several signal samples generated with resonance mass ranging from 400 to 4500in the narrow width approximation are modeled independently for each channel with a Crystal Ball (CB) function [43]. The power-law component of the CB function improves the description of the $m_{\cmsSymbolFace{V}\PZ}$ signal distribution by accounting for the small contribution from lower $m_{\cmsSymbolFace{V}\PZ}$ tails appearing for high signal masses. The resolution of the reconstructed $m_{\cmsSymbolFace{V}\PZ}$ can be extracted from the Gaussian core width of the CB function, and is estimated to be 2–3.5% in the electron channel and 3–4% in the muon channel, depending on the mass of the resonance. ### Low-mass analysis For the low-mass analysis, the $\PZ+\text{jets}$ background is characterized using simulated $\text{Drell--Yan}+\text{jets}$ events. Because of the limited number of simulated events, the $m_{\cmsSymbolFace{V}\PZ}$ distributions in the -tagged categories are susceptible to sizable statistical fluctuations, which affect the quality of the background modeling. It has been observed, however, that within simulation uncertainties, the $\PZ+\text{jets}$ mass shape is the same for events with and without -tagged jets. Therefore, the $\PZ+\text{jets}$ shape in the -tagged category is described using the $m_{\cmsSymbolFace{V}\PZ}$ shape obtained from the simulation without making any tag requirements. Sideband regions are defined depending on the mass of the hadronic $\cmsSymbolFace{V}$ boson candidate. The mass ranges $30<m_{\mathrm{j}}<65\GeV$ and $135<m_{\mathrm{j}}<180\GeV$ are used for the merged category, whereas for the resolved event selection the upper mass threshold is raised to 300to take advantage of the increased number of events in that region. In the final fit to the data, the $\PZ+\text{jets}$ background normalization in the SR is constrained by the observed yield in the SBs; this procedure is applied independently to each category. The shape predictions from the NLO $\PZ+\text{jets}$ simulation are taken as a baseline $m_{\cmsSymbolFace{V}\PZ}$ shape in the SR of every category; additionally, a family of linear correction functions: \[\text{Corr}(m_{\cmsSymbolFace{X}},s)=1+s(m_{\cmsSymbolFace{X}}-500\GeV)/(500 \GeV),\] (4) with individual members of the family defined by the slope parameter $s$, is considered. Figure 4 shows fits to the SB $m_{\cmsSymbolFace{V}\PZ}$ distributions where the slope parameter $s$, allowed to float freely, is constrained by the observed shapes in data. The two-standard-deviation uncertainties in the fitted linear correction functions, which are in the range from $2\ten{-4}$ to $6\ten{-4}\GeV^{-1}$, depending on the category, are observed to cover the residual shape differences in the SBs. In the signal region fit of each category, the SB-constrained slope parameter $s$ is treated as a $\PZ+\text{jets}$ shape systematic effect. In this way the background shape can be corrected to that observed in data. Statistical uncertainties associated with the simulated $\PZ+\text{jets}$ distributions are also taken into account in the fit. The fits in the merged $\cmsSymbolFace{V}$ categories include the peaking region of the background; Fig. 4 shows that the SB data in this particular region are described well by the fit. Dilepton backgrounds that do not contain a leptonic boson decay are estimated from data using $\Pe\Pgm$ events passing the analysis selection. This approach accounts for production, $\PW\PW+\text{jets}$, $\PZ\to\tau\tau+\text{jets}$, single top quark, and hadrons misidentified as leptons, which we collectively refer to as $\cPqt+\text{X}$. The relative yield of $\Pe\Pe$ and $\Pgm\Pgm$ events with respect to $\Pe\Pgm$ events has been estimated on a top quark–enriched control sample and shown to be consistent with expectations. Also, the $\Pe\Pgm$$m_{\cmsSymbolFace{V}\PZ}$ distribution was compared with the prediction from simulated background events with symmetric lepton flavor, and found to be in agreement. The contribution of this $\cPqt+\text{X}$ background is 2% and 20% of the total background in the untagged and tagged categories of the resolved analysis, respectively. The merged analysis has a $\cPqt+\text{X}$ contribution of 0.5% and 1% in the untagged and tagged categories, respectively. The diboson background ($\PZ\PZ$ and $\PZ\PW$, with $\PZ\to\ell\ell$) is estimated directly from simulation. The contribution from these events represents 4% and 5% of the total background in the untagged and tagged categories of the resolved analysis, respectively, while in the merged analysis it is about 14% and 16% in the untagged and tagged categories, respectively. The $m_{\cmsSymbolFace{V}\PZ}$ distributions for the signal region for the merged and resolved categories are depicted in Fig. 5. <figure><img src="content_image/1803.10093/x13.png"><figcaption>Figure 4: Sideband m\cmsSymbolFaceV\PZ distributions for the low-mass searchin the merged \cmsSymbolFaceV (upper), resolved \cmsSymbolFaceV (lower),untagged (left), and tagged (right) categories, after fitting the sidebanddata alone. The points show the data while the filled histograms show thebackground contributions. Electron and muon categories are combined. The grayband indicates the statistical and post-fit systematic uncertainties in thenormalization and shape of the background. Larger bin widths are used athigher values of m\cmsSymbolFaceV\PZ; the bin widths are indicated by thehorizontal error bars.</figcaption></figure> <figure><img src="content_image/1803.10093/x17.png"><figcaption>Figure 5: The signal region m\cmsSymbolFaceV\PZ distributions for the low-masssearch, in the merged \cmsSymbolFaceV (upper), resolved \cmsSymbolFaceV(lower), untagged (left), and tagged (right) categories, after fitting thesignal and sideband regions. Electron and muon categories are combined. A600\GeVbulk graviton signal prediction is represented by the black dashedhistogram. The gray band indicates the statistical and post-fit systematicuncertainties in the normalization and shape of the background. Larger binwidths are used at higher values of m\cmsSymbolFaceV\PZ; the bin widths areindicated by the horizontal error bars.</figcaption></figure> ## 0.7 Systematic uncertainties Several sources of systematic uncertainties influence both the normalization and shape of the backgrounds and signal distributions in the analysis. In the high-mass analysis, where the $\PZ+\text{jets}$ background component is estimated with data, the main systematic uncertainties in the predicted normalization for the $\PZ+\text{jets}$ background arise from the statistical uncertainties in the fit of the $m_{\mathrm{j}}$ sidebands in data. Another uncertainty affecting the normalization of the main background is evaluated by taking the difference between the expected $\PZ+\text{jets}$ contribution in the SR obtained by the main function used to describe the $m_{\mathrm{j}}$ spectrum, and an alternative function choice. An additional normalization uncertainty is related to the choice of the function used to describe the $m_{\mathrm{j}}$ spectrum for the subdominant top quark and $\cmsSymbolFace{V}\cmsSymbolFace{V}$ backgrounds, evaluated from simulation, and propagated to the $\PZ+\text{jets}$ normalization prediction in the SR. Overall, the $\PZ+\text{jets}$ normalization uncertainties contribute from 9 to 15%, depending on the category. The main shape uncertainties in the $\PZ+\text{jets}$ background are extracted from the covariance matrix of the fit to the $m_{\cmsSymbolFace{V}\PZ}$ data SB spectrum, convolved with the uncertainties provided by the $\alpha(m_{\cmsSymbolFace{V}\PZ})$ ratio, via the simultaneous fit procedure described in Section 0.6.1. In the low-mass analysis, to account for background shape systematic effects not explicitly evaluated, data and simulation are compared in the sideband region, and the residual shape difference is treated as an additional uncertainty, resulting in the dominant background shape systematic uncertainty of the low-mass analysis. The top quark and $\cmsSymbolFace{V}\cmsSymbolFace{V}$ background components have a systematic uncertainty in the normalization arising from the degree of knowledge of the respective process production cross sections. The value of the $\cmsSymbolFace{V}\cmsSymbolFace{V}$ production cross section, taken from a recent measurement by the CMS Collaboration [44, 45], is assigned an uncertainty of 12%. The top quark background uncertainties are estimated differently in the low- and high-mass analyses: in the low-mass analysis, where a dedicated $\Pe\Pgm$ control region is exploited to measure the $\cPqt+\text{X}$ background normalization, a 4% uncertainty is estimated by comparing the yield of $\Pe\Pgm$ events with $\Pe\Pe+\Pgm\Pgm$ data; in the high-mass analysis, where the top quark production is taken from simulation, a 5% uncertainty in the cross section is used, which is extracted from the recent CMS measurement of top quark pair production in dilepton events [46]. Uncertainties associated with the description in simulation of the trigger efficiencies, as well as the uncertainties in the efficiency for electron and muon reconstruction, identification, and isolation, are extracted from dedicated studies of events with leptonic decays, and amount to 1.5–3%, depending on the lepton flavor. The uncertainties in the lepton momentum and energy scales are taken into account, and propagated to the signal shapes and normalization, with a typical impact on the normalization of about 0.5–2%, depending on the lepton flavor. Uncertainties in the jet energy scale and resolution [47] affect both the normalization and the shape of the background and signal samples. The momenta of the reconstructed jets are varied according to the uncertainties in the jet energy scale, and the selection efficiencies and $m_{\cmsSymbolFace{V}\PZ}$ signal shapes are reevaluated using these modified samples, resulting in a change of 0.1 to 1.8%, depending on the jet selection. The impact of the jet energy resolution is also propagated, and a smaller impact is observed compared with that due to the uncertainty in the energy scale. The dominant uncertainty in the signal selection efficiency is the uncertainty in the $\cmsSymbolFace{V}$ boson identification efficiency, corresponding to 11% (23%) for the HP (LP) category in the high-mass analysis, and 6% for the merged category of the low-mass analysis [48]. The V boson identification efficiency, the groomed mass resolution of $\cmsSymbolFace{V}$ jets, and the related systematic uncertainty are measured in data and simulation in an almost pure selection of semileptonic events where boosted bosons produced in the top quark decays are separated from the combinatorial background by means of a simultaneous fit to the soft drop mass. The uncertainties in the soft drop mass scale and resolution are propagated to the groomed jet mass, and the impact on the expected selection efficiency of signal and $\cmsSymbolFace{V}\cmsSymbolFace{V}$ background is taken into account. An additional uncertainty affecting the signal normalization is included to account for the extrapolation of the uncertainties extracted from a sample at typical jet of 200to higher regimes, estimated from the differences between 8 and ++ [49] showering models, yielding an uncertainty from 2.5 to 20% depending on the category. For the high-mass analysis, the uncertainties in the V boson identification efficiency and the extrapolation are treated as anticorrelated between the low- and high-purity categories. For the low-mass analysis, one of the largest signal selection uncertainties is the uncertainty in the tagging efficiency for the tagged categories of the analysis. The tagging efficiencies and their corresponding systematic uncertainties are measured in data using samples enriched in b quark content, and their propagation to the signal region of the low-mass analysis produces an uncertainty of up to 4.3%. The uncertainties in the mistag efficiency are also considered; the uncertainties in the tagging and mistag efficiencies are treated as anticorrelated between the tagged and untagged categories. The impact of the uncertainties in the factorization and renormalization scales is propagated both to the normalization and the $m_{\cmsSymbolFace{V}\PZ}$ shapes for signal, and for the high-mass analysis to top quark and $\cmsSymbolFace{V}\cmsSymbolFace{V}$ backgrounds. The corresponding scales are varied by a factor of 2 to measure the effect, resulting in an uncertainty of 2% for the diboson background normalization and 15% for top quarks. The impact on the signal acceptance is evaluated to be 0.1–3%, depending on the resonance mass and analysis category. A systematic uncertainty associated with the choice of the set of PDFs used to generate the simulated samples is evaluated by varying the NNPDF 3.0 PDF set within its uncertainties, and its effect is propagated to both the signal and background $m_{\cmsSymbolFace{V}\PZ}$ shapes and normalization, resulting in a measured uncertainty of approximately 1%. Additional systematic uncertainties affecting the normalization of backgrounds and signal from the contributions of pileup events and the integrated luminosity [50] are also considered and are reported in Table 0.7, together with the complete list of uncertainties considered in the analysis. In the high-mass analysis, the typical total uncertainty in the background normalization is in the range 10–60%, depending on the signal mass, and it is 1–5%, depending on the category, in the low-mass analysis. \topcaption Summary of systematic uncertainties, quoted in percent, affecting the normalization of background and signal samples. Where a systematic uncertainty depends on the resonance mass (for signal) or on the category (for background), the smallest and largest values are reported in the table. In the case of a systematic uncertainty applying only to a specific background source, the source is indicated in parentheses. Systematic uncertainties too small to be considered are written as “<0.1”, while a dash (\NA) represents uncertainties not applicable in the specific analysis category. High-mass Low-mass Low-mass Merged Merged Resolved Source Background Signal Background Signal Background Signal Electron trigger and ID 2.0–3.0 2.0 2.0 Muon trigger and ID 1.5–3.0 1.5 1.5 Electron energy scale <0.1 1.0 0.8 0.1–0.5 1.3 1.2–2.5 Muon momentum scale <0.1 0.5–2.0 0.6 0.1–0.4 1.4 0.2–2.0 Jet energy scale 0.1–0.5 0.1 1.0 0.3–0.6 1.3 0.6–1.8 Jet energy resolution <0.1 <0.1 0.6 0.1 0.2 0.1–0.2 \PQbtag SF untagged \NA \NA 0.2 0.3–0.4 0.1 0.6 \PQbtag SF tagged \NA \NA 2.0 2.0–2.3 3.8 4.1–4.3 Mistag SF untagged \NA \NA 0.5 0.5–0.6 0.4 0.2–0.4 Mistag SF tagged \NA \NA 1.5 0.4–0.6 4.3 0.5–1.4 SM \cmsSymbolFaceV\PZ production 12 \NA 12 \NA 12 \NA SM t quark production 5 \NA 4 (\Pe\Pgm) \NA 4 (\Pe\Pgm) \NA \cmsSymbolFaceV identification (τ21) \NA 11–23 6 (\cmsSymbolFaceV\PZ) 6 \NA \NA \cmsSymbolFaceV identification (extrapolation) \NA 2.5–20 \NA 2.6–6.0 \NA \NA \cmsSymbolFaceV mass scale 0.5–2.5 1.0–2.0 0.2 (\cmsSymbolFaceV\PZ) 0.5–1.1 \NA \NA \cmsSymbolFaceV mass resolution 5.5 5–6 5.6 (\cmsSymbolFaceV\PZ) 5.7–6.0 \NA \NA \PZ+jets normalization 9–15 \NA \NA \NA \NA \NA Pileup 0.5–4.0 0.4 0.5 0.1–0.3 0.1 0.3–0.5 PDFs 0.3–1.5 0.5 \NA 1.5–1.6 \NA 0.3–1.1 Renorm./fact. scales 2 (\cmsSymbolFaceV\PZ), 15 (Top) 1.0–3.0 \NA 0.1–0.3 \NA 0.2–0.3 Integrated luminosity 2.5 2.5 2.5 ## 0.8 Results and interpretation Results are extracted separately for the high- and low-mass analyses from a combined maximum likelihood fit of signal and background to the m\cmsSymbolFaceV\PZ distribution, simultaneously in all the categories used in the respective analysis. An unbinned fit is performed in the high-mass analysis, while a binned fit is performed in the low-mass one; this choice is determined by the fact that in the high-mass analysis, the signal and background shapes are described with analytical functions, while in the low- mass analysis, the background shapes are described by binned histograms. The systematic uncertainties discussed in Section 0.7 are included as nuisance parameters in the maximum likelihood fit, and the background-only hypothesis is tested against the combined background and signal hypothesis [51, 52]. The largest excess of events with respect to the background-only hypothesis, with a local significance of 2.5 standard deviations, is observed in the vicinity of m\cmsSymbolFaceX≈1.2\TeV, and arises predominantly from a localized excess of events in the dimuon HP category of the high-mass analysis. The limit at 95% confidence level (\CL) on the signal cross section for the production of a heavy spin-1 or spin-2 resonance is set using the asymptotic modified frequentist method (CLs) [51, 52, 53, 54]. The results of the low- and high-mass analyses should agree for the intermediate mass range 800–900\GeV, which is accessible to both strategies with similar expected efficiencies for signal candidates. The results of the analysis are therefore presented based on the low-mass strategy up to resonance masses m\cmsSymbolFaceX≤850\GeV, and based on the high-mass analysis for m\cmsSymbolFaceX≥850\GeV. At the intermediate mass point m\cmsSymbolFaceX=850\GeV, the results of both strategies are presented, and the expected limits at 95% \CLof the low- and high-mass analyses on the signal cross sections are found to be in agreement within 3 and 6% for the \PWprand bulk graviton signal model, respectively. The observed upper limits on the resonance cross section, multiplied by the branching fraction for the decay into one \PZboson and a \PW or \PZboson, σ\PWprB(\PWpr→\PZ\PW) or σ\cPGB(\cPG→\PZ\PZ), are reported as a function of the resonance mass in Fig. 6 assuming a \PWpror \cPG produced in the narrow- width approximation, and the local p-value [55] is shown in Fig. 7. Based on the observed (expected) upper limits on the signal cross section, a \PWprsignal is excluded up to 2270 (2390)\GeVin the framework of HVT model A (gV=1), and up to 2330 (2630)\GeVfor HVT model B (gV=3); a WED bulk graviton is excluded up to masses of 925 (960)\GeVfor ˜κ=0.5. ![Observed and expected 95% ](x21.png) \| ![Observed and expected 95% ](x22.png) ---\|--- Figure 6: Observed and expected 95% \CLupper limit on σ\PWprB(\PWpr→\PZ\PW) (left) and σ\cPGB(\cPG→\PZ\PZ) (right) as a function of the resonance mass, taking into account all statistical and systematic uncertainties. The electron and muon channels and the various categories used in the analysis are combined together. The green (inner) and yellow (outer) bands represent the 68% and 95% coverage of the expected limit in the background-only hypothesis. The dashed vertical line represents the transition from the low-mass to the high-mass analysis strategy. Theoretical predictions for the signal production cross section are also shown: (left) \PWprproduced in the framework of HVT model A with gv=1 and model B with gv=3; (right) \cPG produced in the WED bulk graviton model with ˜κ=0.5. ## 0.8 Results and interpretation Results are extracted separately for the high- and low-mass analyses from a combined maximum likelihood fit of signal and background to the $m_{\cmsSymbolFace{V}\PZ}$ distribution, simultaneously in all the categories used in the respective analysis. An unbinned fit is performed in the high-mass analysis, while a binned fit is performed in the low-mass one; this choice is determined by the fact that in the high-mass analysis, the signal and background shapes are described with analytical functions, while in the low-mass analysis, the background shapes are described by binned histograms. The systematic uncertainties discussed in Section 0.7 are included as nuisance parameters in the maximum likelihood fit, and the background-only hypothesis is tested against the combined background and signal hypothesis [51, 52]. The largest excess of events with respect to the background-only hypothesis, with a local significance of 2.5 standard deviations, is observed in the vicinity of $m_{\cmsSymbolFace{X}}\approx 1.2\TeV$, and arises predominantly from a localized excess of events in the dimuon HP category of the high-mass analysis. The limit at 95% confidence level () on the signal cross section for the production of a heavy spin-1 or spin-2 resonance is set using the asymptotic modified frequentist method ($\text{CL}_{\mathrm{s}}$) [51, 52, 53, 54]. The results of the low- and high-mass analyses should agree for the intermediate mass range 800–900, which is accessible to both strategies with similar expected efficiencies for signal candidates. The results of the analysis are therefore presented based on the low-mass strategy up to resonance masses $m_{\cmsSymbolFace{X}}\leq 850\GeV$, and based on the high-mass analysis for $m_{\cmsSymbolFace{X}}\geq 850\GeV$. At the intermediate mass point $m_{\cmsSymbolFace{X}}=850\GeV$, the results of both strategies are presented, and the expected limits at 95% of the low- and high-mass analyses on the signal cross sections are found to be in agreement within 3 and 6% for the and bulk graviton signal model, respectively. The observed upper limits on the resonance cross section, multiplied by the branching fraction for the decay into one boson and a or boson, $\sigma_{\PWpr}\mathcal{B}(\PWpr\to\PZ\PW)$ or $\sigma_{\cPG}\mathcal{B}(\cPG\to\PZ\PZ)$, are reported as a function of the resonance mass in Fig. 6 assuming a or produced in the narrow-width approximation, and the local $p$-value [55] is shown in Fig. 7. Based on the observed (expected) upper limits on the signal cross section, a signal is excluded up to 2270 (2390)in the framework of HVT model A ($g_{\mathrm{V}}=1$), and up to 2330 (2630)for HVT model B ($g_{\mathrm{V}}=3$); a WED bulk graviton is excluded up to masses of 925 (960)for $\widetilde{\kappa}=0.5$. <figure><img src="content_image/1803.10093/x21.png"><figcaption>Figure 6: Observed and expected 95% \CLupper limit on σ\PWprB(\PWpr→\PZ\PW)(left) and σ\cPGB(\cPG→\PZ\PZ) (right) as a function of the resonance mass,taking into account all statistical and systematic uncertainties. The electronand muon channels and the various categories used in the analysis are combinedtogether. The green (inner) and yellow (outer) bands represent the 68% and 95%coverage of the expected limit in the background-only hypothesis. The dashedvertical line represents the transition from the low-mass to the high-massanalysis strategy. Theoretical predictions for the signal production crosssection are also shown: (left) \PWprproduced in the framework of HVT model Awith gv=1 and model B with gv=3; (right) \cPG produced in the WED bulkgraviton model with ˜κ=0.5.</figcaption></figure> <figure><img src="content_image/1803.10093/x23.png"><figcaption>Figure 7: Observed local p-values for \PWpr(left) and \cPG (right) narrowresonances as a function of the resonance mass. The dashed vertical linerepresents the transition from the low-mass to the high-mass analysisstrategy.</figcaption></figure> ## 0.9 Summary A search for a heavy resonance decaying into a boson and a or a boson in $2\ell 2\cPq$ final states has been presented. The data analyzed were collected by the CMS experiment in proton-proton collisions at $\sqrt{s}=13\TeV$ during 2016 operations at the LHC, corresponding to an integrated luminosity of 35.9. The final state of interest consists of a boson decaying leptonically into an electron or muon pair, and the decay of an additional or boson into a pair of quarks. Two analysis strategies, dedicated to the low- and high-mass regimes (below and above 850, respectively), have been used to set limits in the range of resonance mass from 400 to 4500. Depending on the resonance mass, expected upper limits of 3–3000 and 1.5–400fb have been set on the product of the cross section of a spin-1 and the $\PZ\PW$ branching fraction, and on the product of the cross section of a spin-2 graviton and the $\PZ\PZ$ branching fraction, respectively. ###### Acknowledgements. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: BMWFW and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES and CSF (Croatia); RPF (Cyprus); SENESCYT (Ecuador); MoER, ERC IUT, and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); NKFIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CINVESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS and RFBR (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI and FEDER (Spain); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU and SFFR (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie program and the European Research Council and Horizon 2020 Grant, contract No. 675440 (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the “Excellence of Science - EOS” - be.h project n. 30820817; the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Lendület (“Momentum”) Program and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence Program ÚNKP, the NKFIA research grants 123842, 123959, 124845, 124850 and 125105 (Hungary); the Council of Science and Industrial Research, India; the HOMING PLUS program of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, the Mobility Plus program of the Ministry of Science and Higher Education, the National Science Center (Poland), contracts Harmonia 2014/14/M/ST2/00428, Opus 2014/13/B/ST2/02543, 2014/15/B/ST2/03998, and 2015/19/B/ST2/02861, Sonata-bis 2012/07/E/ST2/01406; the National Priorities Research Program by Qatar National Research Fund; the Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia María de Maeztu, grant MDM-2015-0509 and the Programa Severo Ochoa del Principado de Asturias; the Thalis and Aristeia programs cofinanced by EU-ESF and the Greek NSRF; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA). ## References * [1] K. 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Dragicevic, J. Erö, A. Escalante Del Valle, M. Flechl, M. Friedl, R. Frühwirth1, V.M. Ghete, J. Grossmann, J. Hrubec, M. Jeitler1, A. König, N. Krammer, I. Krätschmer, D. Liko, T. Madlener, I. Mikulec, E. Pree, N. Rad, H. Rohringer, J. Schieck1, R. Schöfbeck, M. Spanring, D. Spitzbart, A. Taurok, W. Waltenberger, J. Wittmann, C.-E. Wulz1, M. Zarucki Institute for Nuclear Problems, Minsk, Belarus V. Chekhovsky, V. Mossolov, J. Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium E.A. De Wolf, D. Di Croce, X. Janssen, J. Lauwers, M. Pieters, M. Van De Klundert, H. Van Haevermaet, P. Van Mechelen, N. Van Remortel Vrije Universiteit Brussel, Brussel, Belgium S. Abu Zeid, F. Blekman, J. D’Hondt, I. De Bruyn, J. De Clercq, K. Deroover, G. Flouris, D. Lontkovskyi, S. Lowette, I. Marchesini, S. Moortgat, L. Moreels, Q. Python, K. Skovpen, S. Tavernier, W. Van Doninck, P. Van Mulders, I. Van Parijs Université Libre de Bruxelles, Bruxelles, Belgium D. Beghin, B. Bilin, H. Brun, B. Clerbaux, G. De Lentdecker, H. Delannoy, B. Dorney, G. Fasanella, L. Favart, R. Goldouzian, A. Grebenyuk, A.K. Kalsi, T. Lenzi, J. Luetic, T. Seva, E. Starling, C. Vander Velde, P. Vanlaer, D. Vannerom, R. Yonamine Ghent University, Ghent, Belgium T. Cornelis, D. Dobur, A. Fagot, M. Gul, I. Khvastunov2, D. Poyraz, C. Roskas, D. Trocino, M. Tytgat, W. Verbeke, B. Vermassen, M. Vit, N. Zaganidis Université Catholique de Louvain, Louvain-la-Neuve, Belgium H. Bakhshiansohi, O. Bondu, S. Brochet, G. Bruno, C. Caputo, A. Caudron, P. David, S. De Visscher, C. Delaere, M. Delcourt, B. Francois, A. Giammanco, G. Krintiras, V. Lemaitre, A. Magitteri, A. Mertens, M. Musich, K. Piotrzkowski, L. Quertenmont, A. Saggio, M. Vidal Marono, S. Wertz, J. Zobec Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil W.L. Aldá Júnior, F.L. Alves, G.A. Alves, L. Brito, G. Correia Silva, C. Hensel, A. Moraes, M.E. Pol, P. Rebello Teles Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil E. Belchior Batista Das Chagas, W. Carvalho, J. Chinellato3, E. Coelho, E.M. Da Costa, G.G. Da Silveira4, D. De Jesus Damiao, S. Fonseca De Souza, H. Malbouisson, M. Medina Jaime5, M. Melo De Almeida, C. Mora Herrera, L. Mundim, H. Nogima, L.J. Sanchez Rosas, A. Santoro, A. Sznajder, M. Thiel, E.J. Tonelli Manganote3, F. Torres Da Silva De Araujo, A. Vilela Pereira Universidade Estadual Paulista ${}^{a}$, Universidade Federal do ABC ${}^{b}$, São Paulo, Brazil S. Ahuja${}^{a}$, C.A. Bernardes${}^{a}$, L. Calligaris${}^{a}$, T.R. Fernandez Perez Tomei${}^{a}$, E.M. Gregores${}^{b}$, P.G. Mercadante${}^{b}$, S.F. Novaes${}^{a}$, Sandra S. Padula${}^{a}$, D. Romero Abad${}^{b}$, J.C. Ruiz Vargas${}^{a}$Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria A. Aleksandrov, R. Hadjiiska, P. Iaydjiev, A. Marinov, M. Misheva, M. Rodozov, M. Shopova, G. Sultanov University of Sofia, Sofia, Bulgaria A. Dimitrov, L. Litov, B. Pavlov, P. Petkov Beihang University, Beijing, China W. Fang6, X. Gao6, L. Yuan Institute of High Energy Physics, Beijing, China M. Ahmad, J.G. Bian, G.M. Chen, H.S. Chen, M. Chen, Y. Chen, C.H. Jiang, D. Leggat, H. Liao, Z. Liu, F. Romeo, S.M. Shaheen, A. Spiezia, J. Tao, C. Wang, Z. Wang, E. Yazgan, H. Zhang, J. Zhao State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing, China Y. Ban, G. Chen, J. Li, Q. Li, S. Liu, Y. Mao, S.J. Qian, D. Wang, Z. Xu Tsinghua University, Beijing, China Y. Wang Universidad de Los Andes, Bogota, Colombia C. Avila, A. Cabrera, C.A. Carrillo Montoya, L.F. Chaparro Sierra, C. Florez, C.F. González Hernández, M.A. Segura Delgado University of Split, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, Split, Croatia B. Courbon, N. Godinovic, D. Lelas, I. Puljak, P.M. Ribeiro Cipriano, T. Sculac University of Split, Faculty of Science, Split, Croatia Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, D. Ferencek, K. Kadija, B. Mesic, A. Starodumov7, T. Susa University of Cyprus, Nicosia, Cyprus M.W. Ather, A. Attikis, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A. Razis, H. Rykaczewski Charles University, Prague, Czech Republic M. Finger8, M. Finger Jr.8 Universidad San Francisco de Quito, Quito, Ecuador E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt A. Ellithi Kamel9, Y. Mohammed10, E. Salama11${}^{,}$12 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia S. Bhowmik, R.K. Dewanjee, M. Kadastik, L. Perrini, M. Raidal, C. Veelken Department of Physics, University of Helsinki, Helsinki, Finland P. Eerola, H. Kirschenmann, J. Pekkanen, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland J. Havukainen, J.K. Heikkilä, T. Järvinen, V. Karimäki, R. Kinnunen, T. Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta University of Technology, Lappeenranta, Finland T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France M. Besancon, F. Couderc, M. Dejardin, D. Denegri, J.L. Faure, F. Ferri, S. Ganjour, S. Ghosh, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry, C. Leloup, E. Locci, M. Machet, J. Malcles, G. Negro, J. Rander, A. Rosowsky, M.Ö. Sahin, M. Titov Laboratoire Leprince-Ringuet, Ecole polytechnique, CNRS/IN2P3, Université Paris-Saclay, Palaiseau, France A. Abdulsalam13, C. Amendola, I. Antropov, S. Baffioni, F. Beaudette, P. Busson, L. Cadamuro, C. Charlot, R. Granier de Cassagnac, M. Jo, I. Kucher, S. Lisniak, A. Lobanov, J. Martin Blanco, M. Nguyen, C. Ochando, G. Ortona, P. Paganini, P. Pigard, R. Salerno, J.B. Sauvan, Y. Sirois, A.G. Stahl Leiton, Y. Yilmaz, A. Zabi, A. Zghiche Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France J.-L. Agram14, J. Andrea, D. Bloch, J.-M. Brom, E.C. Chabert, C. Collard, E. Conte14, X. Coubez, F. Drouhin14, J.-C. Fontaine14, D. Gelé, U. Goerlach, M. Jansová, P. Juillot, A.-C. Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l’Institut National de Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne, France S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3, Institut de Physique Nucléaire de Lyon, Villeurbanne, France S. Beauceron, C. Bernet, G. Boudoul, N. Chanon, R. Chierici, D. Contardo, P. Depasse, H. El Mamouni, J. Fay, L. Finco, S. Gascon, M. Gouzevitch, G. Grenier, B. Ille, F. Lagarde, I.B. Laktineh, H. Lattaud, M. Lethuillier, L. Mirabito, A.L. Pequegnot, S. Perries, A. Popov15, V. Sordini, M. Vander Donckt, S. Viret, S. Zhang Georgian Technical University, Tbilisi, Georgia T. Toriashvili16 Tbilisi State University, Tbilisi, Georgia I. Bagaturia17 RWTH Aachen University, I. Physikalisches Institut, Aachen, Germany C. Autermann, L. Feld, M.K. Kiesel, K. Klein, M. Lipinski, M. Preuten, M.P. Rauch, C. Schomakers, J. Schulz, M. Teroerde, B. Wittmer, V. Zhukov15 RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany A. Albert, D. Duchardt, M. Endres, M. Erdmann, S. Erdweg, T. Esch, R. Fischer, A. Güth, T. Hebbeker, C. Heidemann, K. Hoepfner, S. Knutzen, M. Merschmeyer, A. Meyer, P. Millet, S. Mukherjee, T. Pook, M. Radziej, H. Reithler, M. Rieger, F. Scheuch, D. Teyssier, S. Thüer RWTH Aachen University, III. Physikalisches Institut B, Aachen, Germany G. Flügge, B. Kargoll, T. Kress, A. Künsken, T. Müller, A. Nehrkorn, A. Nowack, C. Pistone, O. Pooth, A. Stahl18 Deutsches Elektronen-Synchrotron, Hamburg, Germany M. Aldaya Martin, T. Arndt, C. Asawatangtrakuldee, K. Beernaert, O. Behnke, U. Behrens, A. Bermúdez Martínez, A.A. Bin Anuar, K. Borras19, V. Botta, A. Campbell, P. Connor, C. Contreras-Campana, F. Costanza, V. Danilov, A. De Wit, C. Diez Pardos, D. Domínguez Damiani, G. Eckerlin, D. Eckstein, T. Eichhorn, A. Elwood, E. Eren, E. Gallo20, J. Garay Garcia, A. Geiser, J.M. Grados Luyando, A. Grohsjean, P. Gunnellini, M. Guthoff, A. Harb, J. Hauk, M. Hempel21, H. Jung, M. Kasemann, J. Keaveney, C. Kleinwort, J. Knolle, I. Korol, D. Krücker, W. Lange, A. Lelek, T. Lenz, K. Lipka, W. Lohmann21, R. Mankel, I.-A. Melzer-Pellmann, A.B. Meyer, M. Meyer, M. Missiroli, G. Mittag, J. Mnich, A. Mussgiller, D. Pitzl, A. Raspereza, M. Savitskyi, P. Saxena, R. Shevchenko, N. Stefaniuk, H. Tholen, G.P. Van Onsem, R. Walsh, Y. Wen, K. Wichmann, C. Wissing, O. Zenaiev University of Hamburg, Hamburg, Germany R. Aggleton, S. Bein, V. Blobel, M. Centis Vignali, T. Dreyer, E. Garutti, D. Gonzalez, J. Haller, A. Hinzmann, M. Hoffmann, A. Karavdina, G. Kasieczka, R. Klanner, R. Kogler, N. Kovalchuk, S. Kurz, V. Kutzner, J. Lange, D. Marconi, J. Multhaup, M. Niedziela, D. Nowatschin, T. Peiffer, A. Perieanu, A. Reimers, C. Scharf, P. Schleper, A. Schmidt, S. Schumann, J. Schwandt, J. Sonneveld, H. Stadie, G. Steinbrück, F.M. Stober, M. Stöver, D. Troendle, E. Usai, A. Vanhoefer, B. Vormwald Institut für Experimentelle Teilchenphysik, Karlsruhe, Germany M. Akbiyik, C. Barth, M. Baselga, S. Baur, E. Butz, R. Caspart, T. Chwalek, F. Colombo, W. De Boer, A. Dierlamm, N. Faltermann, B. Freund, R. Friese, M. Giffels, M.A. Harrendorf, F. Hartmann18, S.M. Heindl, U. Husemann, F. Kassel18, S. Kudella, H. Mildner, M.U. Mozer, Th. Müller, M. Plagge, G. Quast, K. Rabbertz, M. Schröder, I. Shvetsov, G. Sieber, H.J. Simonis, R. Ulrich, S. Wayand, M. Weber, T. Weiler, S. Williamson, C. Wöhrmann, R. Wolf Institute of Nuclear and Particle Physics (INPP), NCSR Demokritos, Aghia Paraskevi, Greece G. Anagnostou, G. Daskalakis, T. Geralis, A. Kyriakis, D. Loukas, I. Topsis-Giotis National and Kapodistrian University of Athens, Athens, Greece G. Karathanasis, S. Kesisoglou, A. Panagiotou, N. Saoulidou, E. Tziaferi National Technical University of Athens, Athens, Greece K. Kousouris, I. Papakrivopoulos University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, S. Mallios, N. Manthos, I. Papadopoulos, E. Paradas, J. Strologas, F.A. Triantis, D. Tsitsonis MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University, Budapest, Hungary M. Csanad, N. Filipovic, G. Pasztor, O. Surányi, G.I. Veres22 Wigner Research Centre for Physics, Budapest, Hungary G. Bencze, C. Hajdu, D. Horvath23, Á. Hunyadi, F. Sikler, T.Á. Vámi, V. Veszpremi, G. Vesztergombi22 Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, S. Czellar, J. Karancsi24, A. Makovec, J. Molnar, Z. Szillasi Institute of Physics, University of Debrecen, Debrecen, Hungary M. Bartók22, P. Raics, Z.L. Trocsanyi, B. Ujvari Indian Institute of Science (IISc), Bangalore, India S. Choudhury, J.R. Komaragiri National Institute of Science Education and Research, Bhubaneswar, India S. Bahinipati25, P. Mal, K. Mandal, A. Nayak26, D.K. Sahoo25, S.K. Swain Panjab University, Chandigarh, India S. Bansal, S.B. Beri, V. Bhatnagar, S. Chauhan, R. Chawla, N. Dhingra, R. Gupta, A. Kaur, M. Kaur, S. Kaur, R. Kumar, P. Kumari, M. Lohan, A. Mehta, S. Sharma, J.B. Singh, G. Walia University of Delhi, Delhi, India A. Bhardwaj, B.C. Choudhary, R.B. Garg, S. Keshri, A. Kumar, Ashok Kumar, S. Malhotra, M. Naimuddin, K. Ranjan, Aashaq Shah, R. Sharma Saha Institute of Nuclear Physics, HBNI, Kolkata, India R. Bhardwaj27, R. Bhattacharya, S. Bhattacharya, U. Bhawandeep27, D. Bhowmik, S. Dey, S. Dutt27, S. Dutta, S. Ghosh, N. Majumdar, K. Mondal, S. Mukhopadhyay, S. Nandan, A. Purohit, P.K. Rout, A. Roy, S. Roy Chowdhury, S. Sarkar, M. Sharan, B. Singh, S. Thakur27 Indian Institute of Technology Madras, Madras, India P.K. Behera Bhabha Atomic Research Centre, Mumbai, India R. Chudasama, D. Dutta, V. Jha, V. Kumar, A.K. Mohanty18, P.K. Netrakanti, L.M. Pant, P. Shukla, A. Topkar Tata Institute of Fundamental Research-A, Mumbai, India T. Aziz, S. Dugad, B. Mahakud, S. Mitra, G.B. Mohanty, N. Sur, B. Sutar Tata Institute of Fundamental Research-B, Mumbai, India S. Banerjee, S. Bhattacharya, S. Chatterjee, P. Das, M. Guchait, Sa. Jain, S. Kumar, M. Maity28, G. Majumder, K. Mazumdar, N. Sahoo, T. Sarkar28, N. Wickramage29 Indian Institute of Science Education and Research (IISER), Pune, India S. Chauhan, S. Dube, V. Hegde, A. Kapoor, K. Kothekar, S. Pandey, A. Rane, S. Sharma Institute for Research in Fundamental Sciences (IPM), Tehran, Iran S. Chenarani30, E. Eskandari Tadavani, S.M. Etesami30, M. Khakzad, M. Mohammadi Najafabadi, M. Naseri, S. Paktinat Mehdiabadi31, F. Rezaei Hosseinabadi, B. Safarzadeh32, M. Zeinali University College Dublin, Dublin, Ireland M. Felcini, M. Grunewald INFN Sezione di Bari ${}^{a}$, Università di Bari ${}^{b}$, Politecnico di Bari ${}^{c}$, Bari, Italy M. Abbrescia${}^{a}$${}^{,}$${}^{b}$, C. Calabria${}^{a}$${}^{,}$${}^{b}$, A. Colaleo${}^{a}$, D. Creanza${}^{a}$${}^{,}$${}^{c}$, L. Cristella${}^{a}$${}^{,}$${}^{b}$, N. De Filippis${}^{a}$${}^{,}$${}^{c}$, M. De Palma${}^{a}$${}^{,}$${}^{b}$, A. Di Florio${}^{a}$${}^{,}$${}^{b}$, F. Errico${}^{a}$${}^{,}$${}^{b}$, L. Fiore${}^{a}$, A. Gelmi${}^{a}$${}^{,}$${}^{b}$, G. Iaselli${}^{a}$${}^{,}$${}^{c}$, S. Lezki${}^{a}$${}^{,}$${}^{b}$, G. Maggi${}^{a}$${}^{,}$${}^{c}$, M. Maggi${}^{a}$, B. Marangelli${}^{a}$${}^{,}$${}^{b}$, G. Miniello${}^{a}$${}^{,}$${}^{b}$, S. My${}^{a}$${}^{,}$${}^{b}$, S. Nuzzo${}^{a}$${}^{,}$${}^{b}$, A. Pompili${}^{a}$${}^{,}$${}^{b}$, G. Pugliese${}^{a}$${}^{,}$${}^{c}$, R. Radogna${}^{a}$, A. Ranieri${}^{a}$, G. Selvaggi${}^{a}$${}^{,}$${}^{b}$, A. Sharma${}^{a}$, L. Silvestris${}^{a}$${}^{,}$18, R. Venditti${}^{a}$, P. Verwilligen${}^{a}$, G. Zito${}^{a}$INFN Sezione di Bologna ${}^{a}$, Università di Bologna ${}^{b}$, Bologna, Italy G. Abbiendi${}^{a}$, C. Battilana${}^{a}$${}^{,}$${}^{b}$, D. Bonacorsi${}^{a}$${}^{,}$${}^{b}$, L. Borgonovi${}^{a}$${}^{,}$${}^{b}$, S. Braibant-Giacomelli${}^{a}$${}^{,}$${}^{b}$, R. Campanini${}^{a}$${}^{,}$${}^{b}$, P. Capiluppi${}^{a}$${}^{,}$${}^{b}$, A. Castro${}^{a}$${}^{,}$${}^{b}$, F.R. Cavallo${}^{a}$, S.S. Chhibra${}^{a}$${}^{,}$${}^{b}$, G. Codispoti${}^{a}$${}^{,}$${}^{b}$, M. Cuffiani${}^{a}$${}^{,}$${}^{b}$, G.M. Dallavalle${}^{a}$, F. Fabbri${}^{a}$, A. Fanfani${}^{a}$${}^{,}$${}^{b}$, D. Fasanella${}^{a}$${}^{,}$${}^{b}$, P. Giacomelli${}^{a}$, C. Grandi${}^{a}$, L. Guiducci${}^{a}$${}^{,}$${}^{b}$, S. Marcellini${}^{a}$, G. Masetti${}^{a}$, A. Montanari${}^{a}$, F.L. Navarria${}^{a}$${}^{,}$${}^{b}$, F. Odorici${}^{a}$, A. Perrotta${}^{a}$, A.M. Rossi${}^{a}$${}^{,}$${}^{b}$, T. Rovelli${}^{a}$${}^{,}$${}^{b}$, G.P. Siroli${}^{a}$${}^{,}$${}^{b}$, N. Tosi${}^{a}$INFN Sezione di Catania ${}^{a}$, Università di Catania ${}^{b}$, Catania, Italy S. Albergo${}^{a}$${}^{,}$${}^{b}$, S. Costa${}^{a}$${}^{,}$${}^{b}$, A. Di Mattia${}^{a}$, F. Giordano${}^{a}$${}^{,}$${}^{b}$, R. Potenza${}^{a}$${}^{,}$${}^{b}$, A. Tricomi${}^{a}$${}^{,}$${}^{b}$, C. Tuve${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Firenze ${}^{a}$, Università di Firenze ${}^{b}$, Firenze, Italy G. Barbagli${}^{a}$, K. Chatterjee${}^{a}$${}^{,}$${}^{b}$, V. Ciulli${}^{a}$${}^{,}$${}^{b}$, C. Civinini${}^{a}$, R. D’Alessandro${}^{a}$${}^{,}$${}^{b}$, E. Focardi${}^{a}$${}^{,}$${}^{b}$, G. Latino, P. Lenzi${}^{a}$${}^{,}$${}^{b}$, M. Meschini${}^{a}$, S. Paoletti${}^{a}$, L. Russo${}^{a}$${}^{,}$33, G. Sguazzoni${}^{a}$, D. Strom${}^{a}$, L. Viliani${}^{a}$INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, F. Fabbri, D. Piccolo, F. Primavera18 INFN Sezione di Genova ${}^{a}$, Università di Genova ${}^{b}$, Genova, Italy V. Calvelli${}^{a}$${}^{,}$${}^{b}$, F. Ferro${}^{a}$, F. Ravera${}^{a}$${}^{,}$${}^{b}$, E. Robutti${}^{a}$, S. Tosi${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Milano-Bicocca ${}^{a}$, Università di Milano-Bicocca ${}^{b}$, Milano, Italy A. Benaglia${}^{a}$, A. Beschi${}^{b}$, L. Brianza${}^{a}$${}^{,}$${}^{b}$, F. Brivio${}^{a}$${}^{,}$${}^{b}$, V. Ciriolo${}^{a}$${}^{,}$${}^{b}$${}^{,}$18, M.E. Dinardo${}^{a}$${}^{,}$${}^{b}$, S. Fiorendi${}^{a}$${}^{,}$${}^{b}$, S. Gennai${}^{a}$, A. Ghezzi${}^{a}$${}^{,}$${}^{b}$, P. Govoni${}^{a}$${}^{,}$${}^{b}$, M. Malberti${}^{a}$${}^{,}$${}^{b}$, S. Malvezzi${}^{a}$, R.A. Manzoni${}^{a}$${}^{,}$${}^{b}$, D. Menasce${}^{a}$, L. Moroni${}^{a}$, M. Paganoni${}^{a}$${}^{,}$${}^{b}$, K. Pauwels${}^{a}$${}^{,}$${}^{b}$, D. Pedrini${}^{a}$, S. Pigazzini${}^{a}$${}^{,}$${}^{b}$${}^{,}$34, S. Ragazzi${}^{a}$${}^{,}$${}^{b}$, T. Tabarelli de Fatis${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Napoli ${}^{a}$, Università di Napoli ’Federico II’ ${}^{b}$, Napoli, Italy, Università della Basilicata ${}^{c}$, Potenza, Italy, Università G. Marconi ${}^{d}$, Roma, Italy S. Buontempo${}^{a}$, N. Cavallo${}^{a}$${}^{,}$${}^{c}$, S. Di Guida${}^{a}$${}^{,}$${}^{d}$${}^{,}$18, F. Fabozzi${}^{a}$${}^{,}$${}^{c}$, F. Fienga${}^{a}$${}^{,}$${}^{b}$, G. Galati${}^{a}$${}^{,}$${}^{b}$, A.O.M. Iorio${}^{a}$${}^{,}$${}^{b}$, W.A. Khan${}^{a}$, L. Lista${}^{a}$, S. Meola${}^{a}$${}^{,}$${}^{d}$${}^{,}$18, P. Paolucci${}^{a}$${}^{,}$18, C. Sciacca${}^{a}$${}^{,}$${}^{b}$, F. Thyssen${}^{a}$, E. Voevodina${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Padova ${}^{a}$, Università di Padova ${}^{b}$, Padova, Italy, Università di Trento ${}^{c}$, Trento, Italy P. Azzi${}^{a}$, N. Bacchetta${}^{a}$, L. Benato${}^{a}$${}^{,}$${}^{b}$, D. Bisello${}^{a}$${}^{,}$${}^{b}$, A. Boletti${}^{a}$${}^{,}$${}^{b}$, R. Carlin${}^{a}$${}^{,}$${}^{b}$, A. Carvalho Antunes De Oliveira${}^{a}$${}^{,}$${}^{b}$, P. Checchia${}^{a}$, M. Dall’Osso${}^{a}$${}^{,}$${}^{b}$, P. De Castro Manzano${}^{a}$, T. Dorigo${}^{a}$, U. Dosselli${}^{a}$, F. Gasparini${}^{a}$${}^{,}$${}^{b}$, U. Gasparini${}^{a}$${}^{,}$${}^{b}$, A. Gozzelino${}^{a}$, S. Lacaprara${}^{a}$, P. Lujan, M. Margoni${}^{a}$${}^{,}$${}^{b}$, A.T. Meneguzzo${}^{a}$${}^{,}$${}^{b}$, N. Pozzobon${}^{a}$${}^{,}$${}^{b}$, P. Ronchese${}^{a}$${}^{,}$${}^{b}$, R. Rossin${}^{a}$${}^{,}$${}^{b}$, F. Simonetto${}^{a}$${}^{,}$${}^{b}$, A. Tiko, E. Torassa${}^{a}$, M. Zanetti${}^{a}$${}^{,}$${}^{b}$, P. Zotto${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Pavia ${}^{a}$, Università di Pavia ${}^{b}$, Pavia, Italy A. Braghieri${}^{a}$, A. Magnani${}^{a}$, P. Montagna${}^{a}$${}^{,}$${}^{b}$, S.P. Ratti${}^{a}$${}^{,}$${}^{b}$, V. Re${}^{a}$, M. Ressegotti${}^{a}$${}^{,}$${}^{b}$, C. Riccardi${}^{a}$${}^{,}$${}^{b}$, P. Salvini${}^{a}$, I. Vai${}^{a}$${}^{,}$${}^{b}$, P. Vitulo${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Perugia ${}^{a}$, Università di Perugia ${}^{b}$, Perugia, Italy L. Alunni Solestizi${}^{a}$${}^{,}$${}^{b}$, M. Biasini${}^{a}$${}^{,}$${}^{b}$, G.M. Bilei${}^{a}$, C. Cecchi${}^{a}$${}^{,}$${}^{b}$, D. Ciangottini${}^{a}$${}^{,}$${}^{b}$, L. Fanò${}^{a}$${}^{,}$${}^{b}$, P. Lariccia${}^{a}$${}^{,}$${}^{b}$, R. Leonardi${}^{a}$${}^{,}$${}^{b}$, E. Manoni${}^{a}$, G. Mantovani${}^{a}$${}^{,}$${}^{b}$, V. Mariani${}^{a}$${}^{,}$${}^{b}$, M. Menichelli${}^{a}$, A. Rossi${}^{a}$${}^{,}$${}^{b}$, A. Santocchia${}^{a}$${}^{,}$${}^{b}$, D. Spiga${}^{a}$INFN Sezione di Pisa ${}^{a}$, Università di Pisa ${}^{b}$, Scuola Normale Superiore di Pisa ${}^{c}$, Pisa, Italy K. Androsov${}^{a}$, P. Azzurri${}^{a}$${}^{,}$18, G. Bagliesi${}^{a}$, L. Bianchini${}^{a}$, T. Boccali${}^{a}$, L. Borrello, R. Castaldi${}^{a}$, M.A. Ciocci${}^{a}$${}^{,}$${}^{b}$, R. Dell’Orso${}^{a}$, G. Fedi${}^{a}$, L. Giannini${}^{a}$${}^{,}$${}^{c}$, A. Giassi${}^{a}$, M.T. Grippo${}^{a}$${}^{,}$33, F. Ligabue${}^{a}$${}^{,}$${}^{c}$, T. Lomtadze${}^{a}$, E. Manca${}^{a}$${}^{,}$${}^{c}$, G. Mandorli${}^{a}$${}^{,}$${}^{c}$, A. Messineo${}^{a}$${}^{,}$${}^{b}$, F. Palla${}^{a}$, A. Rizzi${}^{a}$${}^{,}$${}^{b}$, P. Spagnolo${}^{a}$, R. Tenchini${}^{a}$, G. Tonelli${}^{a}$${}^{,}$${}^{b}$, A. Venturi${}^{a}$, P.G. Verdini${}^{a}$INFN Sezione di Roma ${}^{a}$, Sapienza Università di Roma ${}^{b}$, Rome, Italy L. Barone${}^{a}$${}^{,}$${}^{b}$, F. Cavallari${}^{a}$, M. Cipriani${}^{a}$${}^{,}$${}^{b}$, N. Daci${}^{a}$, D. Del Re${}^{a}$${}^{,}$${}^{b}$, E. Di Marco${}^{a}$${}^{,}$${}^{b}$, M. Diemoz${}^{a}$, S. Gelli${}^{a}$${}^{,}$${}^{b}$, E. Longo${}^{a}$${}^{,}$${}^{b}$, B. Marzocchi${}^{a}$${}^{,}$${}^{b}$, P. Meridiani${}^{a}$, G. Organtini${}^{a}$${}^{,}$${}^{b}$, F. Pandolfi${}^{a}$, R. Paramatti${}^{a}$${}^{,}$${}^{b}$, F. Preiato${}^{a}$${}^{,}$${}^{b}$, S. Rahatlou${}^{a}$${}^{,}$${}^{b}$, C. Rovelli${}^{a}$, F. Santanastasio${}^{a}$${}^{,}$${}^{b}$INFN Sezione di Torino ${}^{a}$, Università di Torino ${}^{b}$, Torino, Italy, Università del Piemonte Orientale ${}^{c}$, Novara, Italy N. Amapane${}^{a}$${}^{,}$${}^{b}$, R. Arcidiacono${}^{a}$${}^{,}$${}^{c}$, S. Argiro${}^{a}$${}^{,}$${}^{b}$, M. Arneodo${}^{a}$${}^{,}$${}^{c}$, N. Bartosik${}^{a}$, R. Bellan${}^{a}$${}^{,}$${}^{b}$, C. Biino${}^{a}$, N. Cartiglia${}^{a}$, R. Castello${}^{a}$${}^{,}$${}^{b}$, F. Cenna${}^{a}$${}^{,}$${}^{b}$, M. Costa${}^{a}$${}^{,}$${}^{b}$, R. Covarelli${}^{a}$${}^{,}$${}^{b}$, A. Degano${}^{a}$${}^{,}$${}^{b}$, N. Demaria${}^{a}$, B. Kiani${}^{a}$${}^{,}$${}^{b}$, C. Mariotti${}^{a}$, S. Maselli${}^{a}$, E. Migliore${}^{a}$${}^{,}$${}^{b}$, V. Monaco${}^{a}$${}^{,}$${}^{b}$, E. Monteil${}^{a}$${}^{,}$${}^{b}$, M. Monteno${}^{a}$, M.M. Obertino${}^{a}$${}^{,}$${}^{b}$, L. Pacher${}^{a}$${}^{,}$${}^{b}$, N. Pastrone${}^{a}$, M. Pelliccioni${}^{a}$, G.L. Pinna Angioni${}^{a}$${}^{,}$${}^{b}$, A. Romero${}^{a}$${}^{,}$${}^{b}$, M. Ruspa${}^{a}$${}^{,}$${}^{c}$, R. Sacchi${}^{a}$${}^{,}$${}^{b}$, K. Shchelina${}^{a}$${}^{,}$${}^{b}$, V. Sola${}^{a}$, A. Solano${}^{a}$${}^{,}$${}^{b}$, A. Staiano${}^{a}$INFN Sezione di Trieste ${}^{a}$, Università di Trieste ${}^{b}$, Trieste, Italy S. Belforte${}^{a}$, M. Casarsa${}^{a}$, F. Cossutti${}^{a}$, G. Della Ricca${}^{a}$${}^{,}$${}^{b}$, A. Zanetti${}^{a}$Kyungpook National University D.H. Kim, G.N. Kim, M.S. Kim, J. Lee, S. Lee, S.W. Lee, C.S. Moon, Y.D. Oh, S. Sekmen, D.C. Son, Y.C. Yang Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea H. Kim, D.H. Moon, G. Oh Hanyang University, Seoul, Korea J.A. Brochero Cifuentes, J. Goh, T.J. Kim Korea University, Seoul, Korea S. Cho, S. Choi, Y. Go, D. Gyun, S. Ha, B. Hong, Y. Jo, Y. Kim, K. Lee, K.S. Lee, S. Lee, J. Lim, S.K. Park, Y. Roh Seoul National University, Seoul, Korea J. Almond, J. Kim, J.S. Kim, H. Lee, K. Lee, K. Nam, S.B. Oh, B.C. Radburn-Smith, S.h. Seo, U.K. Yang, H.D. Yoo, G.B. Yu University of Seoul, Seoul, Korea H. Kim, J.H. Kim, J.S.H. Lee, I.C. Park Sungkyunkwan University, Suwon, Korea Y. Choi, C. Hwang, J. Lee, I. Yu Vilnius University, Vilnius, Lithuania V. Dudenas, A. Juodagalvis, J. Vaitkus National Centre for Particle Physics, Universiti Malaya, Kuala Lumpur, Malaysia I. Ahmed, Z.A. Ibrahim, M.A.B. Md Ali35, F. Mohamad Idris36, W.A.T. Wan Abdullah, M.N. Yusli, Z. Zolkapli Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico Duran-Osuna, M. C., H. Castilla-Valdez, E. De La Cruz-Burelo, Ramirez-Sanchez, G., I. Heredia-De La Cruz37, Rabadan-Trejo, R. I., R. Lopez-Fernandez, J. Mejia Guisao, Reyes-Almanza, R, A. Sanchez-Hernandez Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, C. Oropeza Barrera, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico J. Eysermans, I. Pedraza, H.A. Salazar Ibarguen, C. Uribe Estrada Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico A. Morelos Pineda University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand S. Bheesette, P.H. Butler National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan A. Ahmad, M. Ahmad, Q. Hassan, H.R. Hoorani, A. Saddique, M.A. Shah, M. Shoaib, M. Waqas National Centre for Nuclear Research, Swierk, Poland H. Bialkowska, M. Bluj, B. Boimska, T. Frueboes, M. Górski, M. Kazana, K. Nawrocki, M. Szleper, P. Traczyk, P. Zalewski Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland K. Bunkowski, A. Byszuk38, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski, M. Misiura, M. Olszewski, A. Pyskir, M. Walczak Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal P. Bargassa, C. Beirão Da Cruz E Silva, A. Di Francesco, P. Faccioli, B. Galinhas, M. Gallinaro, J. Hollar, N. Leonardo, L. Lloret Iglesias, M.V. Nemallapudi, J. Seixas, G. Strong, O. Toldaiev, D. Vadruccio, J. Varela Joint Institute for Nuclear Research, Dubna, Russia S. Afanasiev, P. Bunin, M. Gavrilenko, I. Golutvin, I. Gorbunov, A. Kamenev, V. Karjavin, A. Lanev, A. Malakhov, V. Matveev39${}^{,}$40, P. Moisenz, V. Palichik, V. Perelygin, S. Shmatov, S. Shulha, N. Skatchkov, V. Smirnov, N. Voytishin, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St. Petersburg), Russia Y. Ivanov, V. Kim41, E. Kuznetsova42, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L. Uvarov, S. Vavilov, A. Vorobyev Institute for Nuclear Research, Moscow, Russia Yu. Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov, N. Krasnikov, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical and Experimental Physics, Moscow, Russia V. Epshteyn, V. Gavrilov, N. Lychkovskaya, V. Popov, I. Pozdnyakov, G. Safronov, A. Spiridonov, A. Stepennov, V. Stolin, M. Toms, E. Vlasov, A. Zhokin Moscow Institute of Physics and Technology, Moscow, Russia T. Aushev, A. Bylinkin40 National Research Nuclear University ’Moscow Engineering Physics Institute’ (MEPhI), Moscow, Russia M. Chadeeva43, P. Parygin, D. Philippov, S. Polikarpov, E. Popova, V. Rusinov P.N. Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin40, I. Dremin40, M. Kirakosyan40, S.V. Rusakov, A. Terkulov Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia A. Baskakov, A. Belyaev, E. Boos, V. Bunichev, M. Dubinin44, L. Dudko, A. Ershov, A. Gribushin, V. Klyukhin, O. Kodolova, I. Lokhtin, I. Miagkov, S. Obraztsov, V. Savrin, A. Snigirev Novosibirsk State University (NSU), Novosibirsk, Russia V. Blinov45, D. Shtol45, Y. Skovpen45 State Research Center of Russian Federation, Institute for High Energy Physics of NRC &quot, Kurchatov Institute&quot, , Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, D. Elumakhov, A. Godizov, V. Kachanov, A. Kalinin, D. Konstantinov, P. Mandrik, V. Petrov, R. Ryutin, A. Sobol, S. Troshin, N. Tyurin, A. Uzunian, A. Volkov National Research Tomsk Polytechnic University, Tomsk, Russia A. Babaev University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia P. Adzic46, P. Cirkovic, D. Devetak, M. Dordevic, J. Milosevic Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M. Barrio Luna, M. Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, C. Fernandez Bedoya, J.P. Fernández Ramos, J. Flix, M.C. Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M. Hernandez, M.I. Josa, D. Moran, A. Pérez-Calero Yzquierdo, J. Puerta Pelayo, I. Redondo, L. Romero, M.S. Soares, A. Triossi Universidad Autónoma de Madrid, Madrid, Spain C. Albajar, J.F. de Trocóniz Universidad de Oviedo, Oviedo, Spain J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras, I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, S. Sanchez Cruz, P. Vischia, J.M. Vizan Garcia Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M. Fernandez, P.J. Fernández Manteca, A. García Alonso, J. Garcia-Ferrero, G. Gomez, A. Lopez Virto, J. Marco, C. Martinez Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels, T. Rodrigo, A. Ruiz-Jimeno, L. Scodellaro, N. Trevisani, I. Vila, R. Vilar Cortabitarte CERN, European Organization for Nuclear Research, Geneva, Switzerland D. Abbaneo, B. Akgun, E. Auffray, P. Baillon, A.H. Ball, D. Barney, J. Bendavid, M. Bianco, A. Bocci, C. Botta, T. Camporesi, M. Cepeda, G. Cerminara, E. Chapon, Y. Chen, D. d’Enterria, A. Dabrowski, V. Daponte, A. David, M. De Gruttola, A. De Roeck, N. Deelen, M. Dobson, T. du Pree, M. Dünser, N. Dupont, A. Elliott-Peisert, P. Everaerts, F. Fallavollita47, G. Franzoni, J. Fulcher, W. Funk, D. Gigi, A. Gilbert, K. Gill, F. Glege, D. Gulhan, J. Hegeman, V. Innocente, A. Jafari, P. Janot, O. Karacheban21, J. Kieseler, V. Knünz, A. Kornmayer, M. Krammer1, C. Lange, P. Lecoq, C. Lourenço, M.T. Lucchini, L. Malgeri, M. Mannelli, A. Martelli, F. Meijers, J.A. Merlin, S. Mersi, E. Meschi, P. Milenovic48, F. Moortgat, M. Mulders, H. Neugebauer, J. Ngadiuba, S. Orfanelli, L. Orsini, F. Pantaleo18, L. Pape, E. Perez, M. Peruzzi, A. Petrilli, G. Petrucciani, A. Pfeiffer, M. Pierini, F.M. Pitters, D. Rabady, A. Racz, T. Reis, G. Rolandi49, M. Rovere, H. Sakulin, C. Schäfer, C. Schwick, M. Seidel, M. Selvaggi, A. Sharma, P. Silva, P. Sphicas50, A. Stakia, J. Steggemann, M. Stoye, M. Tosi, D. Treille, A. Tsirou, V. Veckalns51, M. Verweij, W.D. Zeuner Paul Scherrer Institut, Villigen, Switzerland W. Bertl${}^{\textrm{\textdagger}}$, L. Caminada52, K. Deiters, W. Erdmann, R. Horisberger, Q. Ingram, H.C. Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich - Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland M. Backhaus, L. Bäni, P. Berger, B. Casal, N. Chernyavskaya, G. Dissertori, M. Dittmar, M. Donegà, C. Dorfer, C. Grab, C. Heidegger, D. Hits, J. Hoss, T. Klijnsma, W. Lustermann, M. Marionneau, M.T. Meinhard, D. Meister, F. Micheli, P. Musella, F. Nessi-Tedaldi, J. Pata, F. Pauss, G. Perrin, L. Perrozzi, M. Quittnat, M. Reichmann, D. Ruini, D.A. Sanz Becerra, M. Schönenberger, L. Shchutska, V.R. Tavolaro, K. Theofilatos, M.L. Vesterbacka Olsson, R. Wallny, D.H. Zhu Universität Zürich, Zurich, Switzerland T.K. Aarrestad, C. Amsler53, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del Burgo, S. Donato, C. Galloni, T. Hreus, B. Kilminster, I. Neutelings, D. Pinna, G. Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y. Takahashi, A. Zucchetta National Central University, Chung-Li, Taiwan V. Candelise, Y.H. Chang, K.y. Cheng, T.H. Doan, Sh. Jain, R. Khurana, C.M. Kuo, W. Lin, A. Pozdnyakov, S.S. Yu National Taiwan University (NTU), Taipei, Taiwan P. Chang, Y. Chao, K.F. Chen, P.H. Chen, F. Fiori, W.-S. Hou, Y. Hsiung, Arun Kumar, Y.F. Liu, R.-S. Lu, E. Paganis, A. Psallidas, A. Steen, J.f. Tsai Chulalongkorn University, Faculty of Science, Department of Physics, Bangkok, Thailand B. Asavapibhop, K. Kovitanggoon, G. Singh, N. Srimanobhas Çukurova University, Physics Department, Science and Art Faculty, Adana, Turkey M.N. Bakirci54, A. Bat, F. Boran, S. Cerci55, S. Damarseckin, Z.S. Demiroglu, C. Dozen, I. Dumanoglu, S. Girgis, G. Gokbulut, Y. Guler, I. Hos56, E.E. Kangal57, O. Kara, A. Kayis Topaksu, U. Kiminsu, M. Oglakci, G. Onengut, K. Ozdemir58, B. Tali55, U.G. Tok, S. Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East Technical University, Physics Department, Ankara, Turkey G. Karapinar59, K. Ocalan60, M. Yalvac, M. Zeyrek Bogazici University, Istanbul, Turkey I.O. Atakisi, E. Gülmez, M. Kaya61, O. Kaya62, S. Tekten, E.A. Yetkin63 Istanbul Technical University, Istanbul, Turkey M.N. Agaras, S. Atay, A. Cakir, K. Cankocak, Y. Komurcu Institute for Scintillation Materials of National Academy of Science of Ukraine, Kharkov, Ukraine B. Grynyov National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine L. Levchuk University of Bristol, Bristol, United Kingdom F. Ball, L. Beck, J.J. Brooke, D. Burns, E. Clement, D. Cussans, O. Davignon, H. Flacher, J. Goldstein, G.P. Heath, H.F. Heath, L. Kreczko, D.M. Newbold64, S. Paramesvaran, T. Sakuma, S. Seif El Nasr-storey, D. Smith, V.J. Smith Rutherford Appleton Laboratory, Didcot, United Kingdom K.W. Bell, A. Belyaev65, C. Brew, R.M. Brown, D. Cieri, D.J.A. Cockerill, J.A. Coughlan, K. Harder, S. Harper, J. Linacre, E. Olaiya, D. Petyt, C.H. Shepherd-Themistocleous, A. Thea, I.R. Tomalin, T. Williams, W.J. Womersley Imperial College, London, United Kingdom G. Auzinger, R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock, S. Casasso, D. Colling, L. Corpe, P. Dauncey, G. Davies, M. Della Negra, R. Di Maria, Y. Haddad, G. Hall, G. Iles, T. James, M. Komm, R. Lane, C. Laner, L. Lyons, A.-M. Magnan, S. Malik, L. Mastrolorenzo, T. Matsushita, J. Nash66, A. Nikitenko7, V. Palladino, M. Pesaresi, A. Richards, A. Rose, E. Scott, C. Seez, A. Shtipliyski, T. Strebler, S. Summers, A. Tapper, K. Uchida, M. Vazquez Acosta67, T. Virdee18, N. Wardle, D. Winterbottom, J. Wright, S.C. Zenz Brunel University, Uxbridge, United Kingdom J.E. Cole, P.R. Hobson, A. Khan, P. Kyberd, A. Morton, I.D. Reid, L. Teodorescu, S. Zahid Baylor University, Waco, USA A. Borzou, K. Call, J. Dittmann, K. Hatakeyama, H. Liu, N. Pastika, C. Smith Catholic University of America, Washington DC, USA R. Bartek, A. Dominguez The University of Alabama, Tuscaloosa, USA A. Buccilli, S.I. Cooper, C. Henderson, P. Rumerio, C. West Boston University, Boston, USA D. Arcaro, A. Avetisyan, T. Bose, D. Gastler, D. Rankin, C. Richardson, J. Rohlf, L. Sulak, D. Zou Brown University, Providence, USA G. Benelli, D. Cutts, M. Hadley, J. Hakala, U. Heintz, J.M. Hogan68, K.H.M. Kwok, E. Laird, G. Landsberg, J. Lee, Z. Mao, M. Narain, J. Pazzini, S. Piperov, S. Sagir, R. Syarif, D. Yu University of California, Davis, Davis, USA R. Band, C. Brainerd, R. Breedon, D. Burns, M. Calderon De La Barca Sanchez, M. Chertok, J. Conway, R. Conway, P.T. Cox, R. Erbacher, C. Flores, G. Funk, W. Ko, R. Lander, C. Mclean, M. Mulhearn, D. Pellett, J. Pilot, S. Shalhout, M. Shi, J. Smith, D. Stolp, D. Taylor, K. Tos, M. Tripathi, Z. Wang, F. Zhang University of California, Los Angeles, USA M. Bachtis, C. Bravo, R. Cousins, A. Dasgupta, A. Florent, J. Hauser, M. Ignatenko, N. Mccoll, S. Regnard, D. Saltzberg, C. Schnaible, V. Valuev University of California, Riverside, Riverside, USA E. Bouvier, K. Burt, R. Clare, J. Ellison, J.W. Gary, S.M.A. Ghiasi Shirazi, G. Hanson, G. Karapostoli, E. Kennedy, F. Lacroix, O.R. Long, M. Olmedo Negrete, M.I. Paneva, W. Si, L. Wang, H. Wei, S. Wimpenny, B. R. Yates University of California, San Diego, La Jolla, USA J.G. Branson, S. Cittolin, M. Derdzinski, R. Gerosa, D. Gilbert, B. Hashemi, A. Holzner, D. Klein, G. Kole, V. Krutelyov, J. Letts, M. Masciovecchio, D. Olivito, S. Padhi, M. Pieri, M. Sani, V. Sharma, S. Simon, M. Tadel, A. Vartak, S. Wasserbaech69, J. Wood, F. Würthwein, A. Yagil, G. Zevi Della Porta University of California, Santa Barbara - Department of Physics, Santa Barbara, USA N. Amin, R. Bhandari, J. Bradmiller-Feld, C. Campagnari, M. Citron, A. Dishaw, V. Dutta, M. Franco Sevilla, L. Gouskos, R. Heller, J. Incandela, A. Ovcharova, H. Qu, J. Richman, D. Stuart, I. Suarez, J. Yoo California Institute of Technology, Pasadena, USA D. Anderson, A. Bornheim, J. Bunn, J.M. Lawhorn, H.B. Newman, T. Q. Nguyen, C. Pena, M. Spiropulu, J.R. Vlimant, R. Wilkinson, S. Xie, Z. Zhang, R.Y. Zhu Carnegie Mellon University, Pittsburgh, USA M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, J. Russ, M. Sun, H. Vogel, I. Vorobiev, M. Weinberg University of Colorado Boulder, Boulder, USA J.P. Cumalat, W.T. Ford, F. Jensen, A. Johnson, M. Krohn, S. Leontsinis, E. MacDonald, T. Mulholland, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, USA J. Alexander, J. Chaves, Y. Cheng, J. Chu, A. Datta, K. Mcdermott, N. Mirman, J.R. Patterson, D. Quach, A. Rinkevicius, A. Ryd, L. Skinnari, L. Soffi, S.M. Tan, Z. Tao, J. Thom, J. Tucker, P. Wittich, M. Zientek Fermi National Accelerator Laboratory, Batavia, USA S. Abdullin, M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S. Banerjee, L.A.T. Bauerdick, A. Beretvas, J. Berryhill, P.C. Bhat, G. Bolla${}^{\textrm{\textdagger}}$, K. Burkett, J.N. Butler, A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M. Cremonesi, J. Duarte, V.D. Elvira, J. Freeman, Z. Gecse, E. Gottschalk, L. Gray, D. Green, S. Grünendahl, O. Gutsche, J. Hanlon, R.M. Harris, S. Hasegawa, J. Hirschauer, Z. Hu, B. Jayatilaka, S. Jindariani, M. Johnson, U. Joshi, B. Klima, M.J. Kortelainen, B. Kreis, S. Lammel, D. Lincoln, R. Lipton, M. Liu, T. Liu, R. Lopes De Sá, J. Lykken, K. Maeshima, N. Magini, J.M. Marraffino, D. Mason, P. McBride, P. Merkel, S. Mrenna, S. Nahn, V. O’Dell, K. Pedro, O. Prokofyev, G. Rakness, L. Ristori, A. Savoy-Navarro70, B. Schneider, E. Sexton-Kennedy, A. Soha, W.J. Spalding, L. Spiegel, S. Stoynev, J. Strait, N. Strobbe, L. Taylor, S. Tkaczyk, N.V. Tran, L. Uplegger, E.W. Vaandering, C. Vernieri, M. Verzocchi, R. Vidal, M. Wang, H.A. Weber, A. Whitbeck, W. Wu University of Florida, Gainesville, USA D. Acosta, P. Avery, P. Bortignon, D. Bourilkov, A. Brinkerhoff, A. Carnes, M. Carver, D. Curry, R.D. Field, I.K. Furic, S.V. Gleyzer, B.M. Joshi, J. Konigsberg, A. Korytov, K. Kotov, P. Ma, K. Matchev, H. Mei, G. Mitselmakher, K. Shi, D. Sperka, N. Terentyev, L. Thomas, J. Wang, S. Wang, J. Yelton Florida International University, Miami, USA Y.R. Joshi, S. Linn, P. Markowitz, J.L. Rodriguez Florida State University, Tallahassee, USA A. Ackert, T. Adams, A. Askew, S. Hagopian, V. Hagopian, K.F. Johnson, T. Kolberg, G. Martinez, T. Perry, H. Prosper, A. Saha, A. Santra, V. Sharma, R. Yohay Florida Institute of Technology, Melbourne, USA M.M. Baarmand, V. Bhopatkar, S. Colafranceschi, M. Hohlmann, D. Noonan, T. Roy, F. Yumiceva University of Illinois at Chicago (UIC), Chicago, USA M.R. Adams, L. Apanasevich, D. Berry, R.R. Betts, R. Cavanaugh, X. Chen, S. Dittmer, O. Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, K. Jung, J. Kamin, I.D. Sandoval Gonzalez, M.B. Tonjes, N. Varelas, H. Wang, Z. Wu, J. Zhang The University of Iowa, Iowa City, USA B. Bilki71, W. Clarida, K. Dilsiz72, S. Durgut, R.P. Gandrajula, M. Haytmyradov, V. Khristenko, J.-P. Merlo, H. Mermerkaya73, A. Mestvirishvili, A. Moeller, J. Nachtman, H. Ogul74, Y. Onel, F. Ozok75, A. Penzo, C. Snyder, E. Tiras, J. Wetzel, K. Yi Johns Hopkins University, Baltimore, USA B. Blumenfeld, A. Cocoros, N. Eminizer, D. Fehling, L. Feng, A.V. Gritsan, W.T. Hung, P. Maksimovic, J. Roskes, U. Sarica, M. Swartz, M. Xiao, C. You The University of Kansas, Lawrence, USA A. Al-bataineh, P. Baringer, A. Bean, S. Boren, J. Bowen, J. Castle, S. Khalil, A. Kropivnitskaya, D. Majumder, W. Mcbrayer, M. Murray, C. Rogan, C. Royon, S. Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang Kansas State University, Manhattan, USA A. Ivanov, K. Kaadze, Y. Maravin, A. Modak, A. Mohammadi, L.K. Saini, N. Skhirtladze Lawrence Livermore National Laboratory, Livermore, USA F. Rebassoo, D. Wright University of Maryland, College Park, USA A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, C. Ferraioli, N.J. Hadley, S. Jabeen, G.Y. Jeng, R.G. Kellogg, J. Kunkle, A.C. Mignerey, F. Ricci-Tam, Y.H. Shin, A. Skuja, S.C. Tonwar Massachusetts Institute of Technology, Cambridge, USA D. Abercrombie, B. Allen, V. Azzolini, R. Barbieri, A. Baty, G. Bauer, R. Bi, S. Brandt, W. Busza, I.A. Cali, M. D’Alfonso, Z. Demiragli, G. Gomez Ceballos, M. Goncharov, P. Harris, D. Hsu, M. Hu, Y. Iiyama, G.M. Innocenti, M. Klute, D. Kovalskyi, Y.-J. Lee, A. Levin, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn, C. Mironov, S. Narayanan, X. Niu, C. Paus, C. Roland, G. Roland, G.S.F. Stephans, K. Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang, B. Wyslouch, S. Zhaozhong University of Minnesota, Minneapolis, USA A.C. Benvenuti, R.M. Chatterjee, A. Evans, P. Hansen, S. Kalafut, Y. Kubota, Z. Lesko, J. Mans, S. Nourbakhsh, N. Ruckstuhl, R. Rusack, J. Turkewitz, M.A. Wadud University of Mississippi, Oxford, USA J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA E. Avdeeva, K. Bloom, D.R. Claes, C. Fangmeier, F. Golf, R. Gonzalez Suarez, R. Kamalieddin, I. Kravchenko, J. Monroy, J.E. Siado, G.R. Snow, B. Stieger State University of New York at Buffalo, Buffalo, USA A. Godshalk, C. Harrington, I. Iashvili, D. Nguyen, A. Parker, S. Rappoccio, B. Roozbahani Northeastern University, Boston, USA G. Alverson, E. Barberis, C. Freer, A. Hortiangtham, A. Massironi, D.M. Morse, T. Orimoto, R. Teixeira De Lima, T. Wamorkar, B. Wang, A. Wisecarver, D. Wood Northwestern University, Evanston, USA S. Bhattacharya, O. Charaf, K.A. Hahn, N. Mucia, N. Odell, M.H. Schmitt, K. Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame, USA R. Bucci, N. Dev, M. Hildreth, K. Hurtado Anampa, C. Jessop, D.J. Karmgard, N. Kellams, K. Lannon, W. Li, N. Loukas, N. Marinelli, F. Meng, C. Mueller, Y. Musienko39, M. Planer, A. Reinsvold, R. Ruchti, P. Siddireddy, G. Smith, S. Taroni, M. Wayne, A. Wightman, M. Wolf, A. Woodard The Ohio State University, Columbus, USA J. Alimena, L. Antonelli, B. Bylsma, L.S. Durkin, S. Flowers, B. Francis, A. Hart, C. Hill, W. Ji, T.Y. Ling, W. Luo, B.L. Winer, H.W. Wulsin Princeton University, Princeton, USA S. Cooperstein, O. Driga, P. Elmer, J. Hardenbrook, P. Hebda, S. Higginbotham, A. Kalogeropoulos, D. Lange, J. Luo, D. Marlow, K. Mei, I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, J. Salfeld-Nebgen, D. Stickland, C. Tully University of Puerto Rico, Mayaguez, USA S. Malik, S. Norberg Purdue University, West Lafayette, USA A. Barker, V.E. Barnes, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada, D.H. Miller, N. Neumeister, C.C. Peng, H. Qiu, J.F. Schulte, J. Sun, F. Wang, R. Xiao, W. Xie Purdue University Northwest, Hammond, USA T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA Z. Chen, K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Guilbaud, M. Kilpatrick, W. Li, B. Michlin, B.P. Padley, J. Roberts, J. Rorie, W. Shi, Z. Tu, J. Zabel, A. Zhang University of Rochester, Rochester, USA A. Bodek, P. de Barbaro, R. Demina, Y.t. Duh, T. Ferbel, M. Galanti, A. Garcia-Bellido, J. Han, O. Hindrichs, A. Khukhunaishvili, K.H. Lo, P. Tan, M. Verzetti The Rockefeller University, New York, USA R. Ciesielski, K. Goulianos, C. Mesropian Rutgers, The State University of New Jersey, Piscataway, USA A. Agapitos, J.P. Chou, Y. Gershtein, T.A. Gómez Espinosa, E. Halkiadakis, M. Heindl, E. Hughes, S. Kaplan, R. Kunnawalkam Elayavalli, S. Kyriacou, A. Lath, R. Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D. Sheffield, S. Somalwar, R. Stone, S. Thomas, P. Thomassen, M. Walker University of Tennessee, Knoxville, USA A.G. Delannoy, J. Heideman, G. Riley, K. Rose, S. Spanier, K. Thapa Texas A&M University, College Station, USA O. Bouhali76, A. Castaneda Hernandez76, A. Celik, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R. Eusebi, J. Gilmore, T. Huang, T. Kamon77, R. Mueller, Y. Pakhotin, R. Patel, A. Perloff, L. Perniè, D. Rathjens, A. Safonov, A. Tatarinov Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, F. De Guio, P.R. Dudero, J. Faulkner, E. Gurpinar, S. Kunori, K. Lamichhane, S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang Vanderbilt University, Nashville, USA S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K. Padeken, J.D. Ruiz Alvarez, P. Sheldon, S. Tuo, J. Velkovska, Q. Xu University of Virginia, Charlottesville, USA M.W. Arenton, P. Barria, B. Cox, R. Hirosky, M. Joyce, A. Ledovskoy, H. Li, C. Neu, T. Sinthuprasith, Y. Wang, E. Wolfe, F. Xia Wayne State University, Detroit, USA R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University of Wisconsin - Madison, Madison, WI, USA M. Brodski, J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, L. Dodd, S. Duric, B. Gomber, M. Grothe, M. Herndon, A. Hervé, U. Hussain, P. Klabbers, A. Lanaro, A. Levine, K. Long, R. Loveless, V. Rekovic, T. Ruggles, A. Savin, N. Smith, W.H. Smith, N. Woods †: Deceased 1: Also at Vienna University of Technology, Vienna, Austria 2: Also at IRFU; CEA; Université Paris-Saclay, Gif-sur-Yvette, France 3: Also at Universidade Estadual de Campinas, Campinas, Brazil 4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil 5: Also at Universidade Federal de Pelotas, Pelotas, Brazil 6: Also at Université Libre de Bruxelles, Bruxelles, Belgium 7: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia 8: Also at Joint Institute for Nuclear Research, Dubna, Russia 9: Now at Cairo University, Cairo, Egypt 10: Now at Fayoum University, El-Fayoum, Egypt 11: Also at British University in Egypt, Cairo, Egypt 12: Now at Ain Shams University, Cairo, Egypt 13: Also at Department of Physics; King Abdulaziz University, Jeddah, Saudi Arabia 14: Also at Université de Haute Alsace, Mulhouse, France 15: Also at Skobeltsyn Institute of Nuclear Physics; Lomonosov Moscow State University, Moscow, Russia 16: Also at Tbilisi State University, Tbilisi, Georgia 17: Also at Ilia State University, Tbilisi, Georgia 18: Also at CERN; European Organization for Nuclear Research, Geneva, Switzerland 19: Also at RWTH Aachen University; III. Physikalisches Institut A, Aachen, Germany 20: Also at University of Hamburg, Hamburg, Germany 21: Also at Brandenburg University of Technology, Cottbus, Germany 22: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group; Eötvös Loránd University, Budapest, Hungary 23: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 24: Also at Institute of Physics; University of Debrecen, Debrecen, Hungary 25: Also at Indian Institute of Technology Bhubaneswar, Bhubaneswar, India 26: Also at Institute of Physics, Bhubaneswar, India 27: Also at Shoolini University, Solan, India 28: Also at University of Visva-Bharati, Santiniketan, India 29: Also at University of Ruhuna, Matara, Sri Lanka 30: Also at Isfahan University of Technology, Isfahan, Iran 31: Also at Yazd University, Yazd, Iran 32: Also at Plasma Physics Research Center; Science and Research Branch; Islamic Azad University, Tehran, Iran 33: Also at Università degli Studi di Siena, Siena, Italy 34: Also at INFN Sezione di Milano-Bicocca; Università di Milano-Bicocca, Milano, Italy 35: Also at International Islamic University of Malaysia, Kuala Lumpur, Malaysia 36: Also at Malaysian Nuclear Agency; MOSTI, Kajang, Malaysia 37: Also at Consejo Nacional de Ciencia y Tecnología, Mexico city, Mexico 38: Also at Warsaw University of Technology; Institute of Electronic Systems, Warsaw, Poland 39: Also at Institute for Nuclear Research, Moscow, Russia 40: Now at National Research Nuclear University ’Moscow Engineering Physics Institute’ (MEPhI), Moscow, Russia 41: Also at St. Petersburg State Polytechnical University, St. Petersburg, Russia 42: Also at University of Florida, Gainesville, USA 43: Also at P.N. Lebedev Physical Institute, Moscow, Russia 44: Also at California Institute of Technology, Pasadena, USA 45: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia 46: Also at Faculty of Physics; University of Belgrade, Belgrade, Serbia 47: Also at INFN Sezione di Pavia; Università di Pavia, Pavia, Italy 48: Also at University of Belgrade; Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia 49: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy 50: Also at National and Kapodistrian University of Athens, Athens, Greece 51: Also at Riga Technical University, Riga, Latvia 52: Also at Universität Zürich, Zurich, Switzerland 53: Also at Stefan Meyer Institute for Subatomic Physics (SMI), Vienna, Austria 54: Also at Gaziosmanpasa University, Tokat, Turkey 55: Also at Adiyaman University, Adiyaman, Turkey 56: Also at Istanbul Aydin University, Istanbul, Turkey 57: Also at Mersin University, Mersin, Turkey 58: Also at Piri Reis University, Istanbul, Turkey 59: Also at Izmir Institute of Technology, Izmir, Turkey 60: Also at Necmettin Erbakan University, Konya, Turkey 61: Also at Marmara University, Istanbul, Turkey 62: Also at Kafkas University, Kars, Turkey 63: Also at Istanbul Bilgi University, Istanbul, Turkey 64: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 65: Also at School of Physics and Astronomy; University of Southampton, Southampton, United Kingdom 66: Also at Monash University; Faculty of Science, Clayton, Australia 67: Also at Instituto de Astrofísica de Canarias, La Laguna, Spain 68: Also at Bethel University, ST. PAUL, USA 69: Also at Utah Valley University, Orem, USA 70: Also at Purdue University, West Lafayette, USA 71: Also at Beykent University, Istanbul, Turkey 72: Also at Bingol University, Bingol, Turkey 73: Also at Erzincan University, Erzincan, Turkey 74: Also at Sinop University, Sinop, Turkey 75: Also at Mimar Sinan University; Istanbul, Istanbul, Turkey 76: Also at Texas A&M University at Qatar, Doha, Qatar 77: Also at Kyungpook National University, Daegu, Korea
1009.2752	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 52932, "num_imgs": 16, "llama3_tokens_count": 18647 }	[ "content_image/1009.2752/x1.png", "content_image/1009.2752/x2.png", "content_image/1009.2752/x3.png", "content_image/1009.2752/x4.png", "content_image/1009.2752/x5.png", "content_image/1009.2752/x6.png", "content_image/1009.2752/x7.png", "content_image/1009.2752/x8.png", "content_image/1009.2752/x9.png", "content_image/1009.2752/x10.png", "content_image/1009.2752/x11.png", "content_image/1009.2752/x12.png", "content_image/1009.2752/x13.png", "content_image/1009.2752/x14.png", "content_image/1009.2752/x15.png", "content_image/1009.2752/x16.png" ]	# The Effect of Active Galactic Nuclei on the Mid-Infrared Aromatic Features Aleksandar M. Diamond-Stanic¹ , George H. Rieke¹ [FOOTNOTE:1][ENDFOOTNOTE] ###### Abstract We present Spitzer measurements of the aromatic (also known as PAH) features for 35 Seyfert galaxies from the revised Shapley–Ames sample and find that the relative strengths of the features differ significantly from those observed in star-forming galaxies. Specifically, the features at 6.2, 7.7, and 8.6 $\mu$m are suppressed relative to the 11.3 $\mu$m feature in Seyferts. Furthermore, we find an anti-correlation between the L(7.7 $\mu$m)/L(11.3 $\mu$m) ratio and the strength of the rotational H${}_{2}$ emission, which traces shocked gas. This suggests that shocks suppress the short-wavelength features by modifying the structure of the aromatic molecules or destroying the smallest grains. Most Seyfert nuclei fall on the relationship between aromatic emission and [Ne ii] emission for star-forming galaxies, indicating that aromatic-based estimates of the star-formation rate in AGN host galaxies are generally reasonable. For the outliers from this relationship, which have small L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios and strong H${}_{2}$ emission, the 11.3 $\mu$m feature still provides a valid measure of the star-formation rate. Subject headings:galaxies: active, galaxies: nuclei, galaxies: Seyfert, galaxies: ISM † [FOOTNOTE:†][ENDFOOTNOTE] ## 1. Introduction The mid-infrared (mid-IR) aromatic emission features are a universal product of star formation in moderate-to-high–metallicity galaxies (e.g., Roche et al., 1991; Lu et al., 2003; Smith et al., 62). Their molecular carriers, often assumed to be polycyclic aromatic hydrocarbons (PAHs, Leger & Puget, 1984; Allamandola et al., 1985; Tielens, 2008), radiate through IR fluorescence following vibrational excitation by a single ultraviolet (UV) photon (Tielens, 2005) and provide an indirect measurement of the UV radiation field strength, and therefore the star-formation rate (SFR), that is largely extinction independent (e.g., Peeters et al., 2004; Calzetti et al., 2007; Rieke et al., 2009). This emission is thought to originate in photo-dissociation regions where aromatic molecules are shielded from the harsh radiation field near hot stars (e.g., Povich et al., 2007). These molecules can also be destroyed by the harder radiation field associated with an active galactic nucleus (AGN, Voit, 1992; Genzel et al., 1998). Nonetheless, aromatic features are readily detected in many AGNs above IR continua boosted by hot dust, and they have been used to probe the SFR in AGN host galaxies (e.g., Schweitzer et al., 2006; Shi et al., 2007, 2009; Lutz et al., 2008). Understanding what environments destroy or modify these features is important for constraining systematic uncertainties in aromatic-based estimates of the SFR, and is a key way to probe the nature of their molecular carriers, an open issue in our understanding of the interstellar medium. Duley & Williams (1981) first suggested that vibrational modes of aromatic hydrocarbons could produce the observed features, which were subsequently identified with specific C–H and C–C bending and stretching modes (Allamandola et al., 1989). Specifically, the 6.2 and 7.7 $\mu$m features are produced by C–C stretching modes, the 8.6 $\mu$m feature by C–H in-plane bending modes, and the 11.3 and 12.7 $\mu$m features by C–H out-of-plane bending modes. While these features are commonly attributed to PAHs, we use the simpler, more general term “aromatic” to avoid making assumptions about the detailed structure of the molecules. It is worth noting, for example, that PAH spectra from laboratory measurements and quantum chemical calculations are unable to match the range of astronomical spectra without artificial enhancements of the 6.2, 7.7, and 8.6 $\mu$m feature strengths (e.g., Li & Draine, 2001). Regardless of this uncertainty associated with uniquely matching observed spectra with expectations for specific molecules, one can probe the properties of the aromatic carriers by measuring the relative strengths of the emission features, which are expected to vary as a function of charge state (e.g., Bakes et al., 2001), molecular size (e.g., Draine & Li, 2001), and molecular structure (e.g., Vermeij et al., 2002). Efforts to study variations in aromatic feature strengths outside the Milky Way have focused on star-forming galaxies (e.g., Smith et al., 62; Galliano et al., 2008; Roseboom et al., 2009; O’Dowd et al., 2009), but the AGNs included in these studies have shown evidence for suppression of shorter wavelength features (e.g., those at 6.2, 7.7, and 8.6 $\mu$m) relative to those at longer wavelengths. For example, Smith et al. (62) studied a sample of 59 galaxies from the Spitzer Infrared Nearby Galaxy Survey (SINGS, Kennicutt et al., 2003), of which 12 have Seyfert nuclei and 20 have low-ionization nuclear emission-line regions (LINERs), and they found a tendency for the systems with reduced 6–8 $\mu$m features to be associated with low-luminosity AGNs. They speculated on possible causes for this behavior, including whether the AGN can modify the aromatic grain size distribution or serve as the excitation source for aromatic emission. Similarly, O’Dowd et al. (2009) studied a sample of 92 galaxies at $z\sim 0.1$ from the Spitzer SDSS GALEX Spectroscopic Survey, including eight AGN-dominated and 20 composite systems, and found that the AGNs exhibited lower 7.7/11.3 $\mu$m ratios. They interpreted this behavior as being consistent with destruction of small aromatic molecules by shocks or X-rays associated with the AGNs, but they were unable to distinguish any differences between the AGN-dominated and composite objects, nor any strong correlation with AGN power. The physical slit size at their median redshift is 6 kpc, so there is little spatial information. In this paper we analyze the aromatic features drawing from the sample of 89 local Seyfert galaxies studied by Diamond-Stanic et al. (2009). This sample is from the revised Shapley–Ames catalog (RSA, Sandage & Tammann, 1987), and includes every galaxy brighter than $B_{T}=13$ that is known to host Seyfert activity (Maiolino & Rieke, 1995; Ho et al., 1997). The median distance of the sample is 22 Mpc, so the 3.7″ slit width of the Short-Low (SL) module of the Infrared Spectrograph (IRS, Houck et al., 2004) on the Spitzer Space Telescope (Werner et al., 2004) provides spatial information on scales of a few hundred parsecs, allowing us to isolate nuclear regions distinct from the rest of the galaxy. As a result, we are able to probe the effect of AGNs on the aromatic features more systematically than has previously been done. ## 2. Data We gathered data from the Spitzer archive taken with the IRS SL module ($\lambda=5.2$–14.5 $\mu$m, $R=64$–128) from a variety of programs (24, 86, 159, 668, 3247, 3269, 3374, 30471, 30572, 30577, 30745, 40018, and 50702), as well as dedicated data taken for this study (program 40936, PI: G. H. Rieke). We use CUBISM (Smith et al., 63) to extract one-dimensional spectra from the basic calibrated data using $3.6\arcsec\times 7.2\arcsec$ apertures oriented along the slit direction. This aperture size was chosen to isolate the nuclear component of the aromatic emission while still including a substantial fraction of the diffraction-limited beam. We use the calibration for extended sources based on the assumption that the regions producing aromatic emission are spatially extended, so the extracted spectra are in units of surface brightness. We use overlapping data in the 7.59–8.42 $\mu$m region to scale the SL2/SL3 orders to the SL1 order when offsets are apparent; these offsets are $<10$% in all cases. We then use a modified version of PAHFIT (Smith et al., 62) to determine the strength of the various aromatic features. This spectral fitting package includes aromatic features represented by Drude profiles, dust continuum emission represented by modified blackbodies at fixed temperatures, fine-structure lines and H${}_{2}$ rotational lines represented by Gaussian profiles, starlight represented by $T=5000$ K blackbody emission, and dust extinction represented by a power-law and silicate features. Because Seyfert galaxies exhibit higher-ionization emission lines, silicate dust emission, and hot-dust continuum emission, we additionally include a [Ne vi] $\lambda$7.652$~{}\mu$m emission line and a silicate emission component, both represented by Gaussian profiles, and we use temperatures of 1000, 750, 500, 350, 225, 150, and 100 K for the dust continuum emission. We show example PAHFIT decompositions for three sources exhibiting a range in continuum shape and silicate extinction in Figure 1. <figure><img src="content_image/1009.2752/x1.png"><figcaption>Figure 1.— Example PAHFIT spectral decompositions for three RSA Seyfertnuclei. The observed spectra are shown in black. The blue lines above thecontinuum correspond to the aromatic features, while red lines correspond tounresolved atomic and molecular emission lines. The total fit is shown ingreen, and the dotted line indicates the extinction profile.</figcaption></figure> ## 3. Results The data exhibit a range of signal-to-noise ratios (S/N), so we visually inspected all of the nuclear spectra to identify those that have clear detections of the relevant aromatic features and whose spectra are adequately described by the PAHFIT model. The spectra for these 35 sources are shown in Figure 2. Since many of the observations were executed with the mapping mode of IRS, it is also possible to extract spectra for off-nuclear regions in some galaxies, allowing for comparison between spectra dominated by the active nucleus and spectra dominated by H ii regions within the same galaxy. We identified off-nuclear regions that were covered by the IRS slit and had sufficient S/N to detect the relevant aromatic features in 21/35 galaxies. We show a comparison between the nuclear and off-nuclear extractions for these galaxies in Figure 5, and we compile relevant measurements in Tables 1 and 2. <figure><img src="content_image/1009.2752/x2.png"><figcaption>Figure 2.— Low-resolution 5.2–14.2 μm Spitzer/IRS nuclear spectra for the 35RSA Seyfert galaxies considered in this study.</figcaption></figure> <figure><img src="content_image/1009.2752/x3.png"><figcaption>Figure 2.— Continued</figcaption></figure> <figure><img src="content_image/1009.2752/x4.png"><figcaption>Figure 2.— Continued</figcaption></figure> <figure><img src="content_image/1009.2752/x5.png"><figcaption>Figure 5.— Nuclear and off-nuclear spectra for the 21/35 Seyfert galaxieswhere off-nuclear regions were covered by the IRS slit and had sufficient S/Nto detect the relevant aromatic features. The panel to the right of eachspectrum shows the corresponding 3.6\arcsec×7.2\arcsec extraction regionoverlaid on the central 1\arcmin×1\arcmin of an IRAC 8.0 μm image.</figcaption></figure> <figure><img src="content_image/1009.2752/x6.png"><figcaption>Figure 5.— Continued</figcaption></figure> <figure><img src="content_image/1009.2752/x7.png"><figcaption>Figure 5.— Continued</figcaption></figure> <figure><img src="content_image/1009.2752/x8.png"><figcaption>Figure 8.— The cumulative distribution of aromatic feature ratios for 35 RSASeyfert nuclei, 21 off-nuclear regions, and 27 SINGS H ii galaxies. The firstthree panels show that the 6.2, 7.7, and 8.6 μm features are systematicallyweaker relative to the 11.3 μm feature for the Seyfert nuclei than for theoff-nuclear regions or the SINGS H ii galaxies. Panel (c), for example, showsthat half of the RSA Seyfert nuclei have L(8.6 μm)/L(11.3 μm) ratios <0.5,whereas for the H ii galaxies, half have ratios <0.75. The remaining threepanels show that the ratios among the 6.2, 7.7, and 8.6 μm features show nosignificant differences between any of the samples.</figcaption></figure> <figure><img src="content_image/1009.2752/x9.png"><figcaption>Figure 9.— The cumulative distribution of the ratio of 6.2, 7.7, and 8.6 μmfeatures to the 11.3 μm feature for SINGS galaxies with Seyfert, LINER, and Hii optical classifications. This illustrates the result found by Smith et al.(2007a) that the Seyferts and LINERs have ratios that are significantly lowerthan the H ii galaxies. The apparent difference between SINGS Seyferts andLINERs is not statistically significant.</figcaption></figure> ### Aromatic Feature Ratio Distributions In Figure 8, we show the distribution of aromatic feature ratios for the 35 RSA Seyfert nuclei and 21 off-nuclear regions, as well as for 27/59 SINGS galaxies from Smith et al. (62) that have H ii nuclear classifications (i.e., those that are not Seyferts or LINERs). We find that the L(6.2 $\mu$m)/L(11.3 $\mu$m), L(7.7 $\mu$m)/L(11.3 $\mu$m), and L(8.6 $\mu$m)/L(11.3 $\mu$m) ratios are systematically lower for the Seyfert nuclei than for the off-nuclear regions or the SINGS H ii galaxies. These differences are all statistically significant with $p\leq 0.003$ based on the two-sample Kolmogorov–Smirnov (K-S) test (see Table 3), a non-parametric test that considers the maximum deviation between two cumulative distribution functions (Press et al., 1992; Wall & Jenkins, 2003). On the other hand, there are no significant differences between these feature ratios for the off-nuclear regions and the SINGS H ii galaxies, so the feature strengths in regions of star formation are consistent with being drawn from the same parent distribution. Furthermore, the ratios among the 6.2, 7.7, and 8.6 $\mu$m features show no significant differences between any of the samples. Smith et al. (62) noted that the LINERs and Seyferts in the SINGS sample were offset towards lower L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios when compared to the H ii galaxies. We illustrate this result graphically in Figure 9, which shows the distribution functions of the ratio of 6.2, 7.7, and 8.6 $\mu$m features to the 11.3 $\mu$m feature for galaxies with Seyfert, LINER, and H ii optical classifications. Both Seyferts ($p=2\times 10^{-3}$) and LINERs ($p=0.04$) have ratios that are significantly lower than the H ii galaxies. While the SINGS Seyferts have somewhat lower ratios than the LINERs, this difference is not statistically significant, and neither sample is statistically distinguishable from the RSA Seyferts. <figure><img src="content_image/1009.2752/x10.png"><figcaption>Figure 10.— The relationship between the strength of the H2 S(3) rotationalline, normalized to the strength of the aromatic features, and the L(7.7μm)/L(11.3 μm) ratio for RSA Seyfert nuclei. The sources with small L(7.7μm)/L(11.3 μm) ratios also exhibit strong H2 emission. The most extremesources with L(7.7 μm)/L(11.3 μm)<1.6 are highlighted with filled circles.</figcaption></figure> <figure><img src="content_image/1009.2752/x11.png"><figcaption>Figure 11.— Nuclear spectra for 12 additional RSA Seyfert nuclei that exhibitsmall L(7.7 μm)/L(11.3 μm) ratios and strong H2 S(3) lines, but were excludedfrom the sample due to a lack of 6.2, 7.7, or 8.6 μm aromatic featuredetections. The spectra are sorted from top to bottom by the equivalent widthof the 11.3 μm aromatic feature. The wavelengths of the 9.67 μm H2 S(3) lineand the 11.3 μm aromatic feature are marked by dotted lines.</figcaption></figure> ### Trends with H${}_{2}$ emission Inspection of Figure 2 reveals that several Seyferts with small L(7.7 $\mu$m)/L(11.3 $\mu$m) aromatic feature ratios also exhibit a strong H${}_{2}$ S(3) rotational line at 9.67 $\mu$m (e.g., NGC4501, NGC5194.) To investigate this behavior, we plot the strength of the H${}_{2}$ S(3) line, normalized to the strength of the aromatic features, as a function of the L(7.7 $\mu$m)/L(11.3 $\mu$m) ratio in Figure 10. We find a strong anti-correlation in this plot such that sources with the smallest L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios also have the strongest H${}_{2}$ emission. The Spearman’s $\rho$ rank correlation coefficient is $-0.78$ with a probability $p=3\times 10^{-8}$ of no correlation, while Kendall’s $\tau$ is $-0.62$ with $p=1\times 10^{-7}$; these non-parametric tests consider the agreement between the ranks of quantities in pairs of measurements (Press et al., 1992; Wall & Jenkins, 2003), with coefficient values ranging from -1 (perfect disagreement) to 1 (perfect agreement). Roussel et al. (2007) found that H${}_{2}$ rotational lines scale tightly with the aromatic features for SINGS $\textrm{H}~{}\textsc{ii}$ galaxies, but that Seyferts and LINERs often exhibit excess H${}_{2}$ emission, which they attribute to shocks. We explore the hypothesis that shocks cause both the excess H${}_{2}$ emission and the anomalous aromatic ratios for AGNs in Section 4.4. Among the sources excluded from the above analysis due to a lack of 6.2, 7.7, or 8.6 $\mu$m aromatic feature detections, there are a significant number with clearly detected 11.3 $\mu$m features and H${}_{2}$ S(3) lines. In Figure 11, we show the nuclear spectra for a dozen of these sources, sorted by the equivalent width of the 11.3 $\mu$m feature. These spectra exhibit the small L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios and strong H${}_{2}$ S(3) lines characteristic of sources in the top-left of Figure 10. Due to uncertainties associated with estimating the strength of weak, broad features and determining robust upper limits (e.g., proper continuum placement), we do not include any of these sources in our subsequent analysis. However, their behavior is consistent with that in Figure 10 and supports the reality of the trend between aromatic feature characteristics and H${}_{2}$ line strength. ### Evidence for extinction of aromatic features The sources with the largest L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios, NGC4945 and NGC3079, also have the strongest silicate absorption features. This suggests that the 11.3 $\mu$m feature is being significantly attenuated, consistent with previous results for starburst and luminous infrared galaxies (e.g., Brandl et al., 2006; Pereira-Santaella et al., 2010), and implies that a significant fraction of the silicate-absorbing material is extended relative to the regions that produce the aromatic features. Although the aromatic feature measurements in PAHFIT are corrected for extinction, in cases as extreme as these two galaxies the resulting feature strengths are highly uncertain. For all other galaxies in our sample, the inferred extinctions are $<50\%$ for all features. ## 4. Discussion The result that Seyfert galaxies exhibit weak 6.2, 7.7, and 8.6 $\mu$m aromatic features relative to the 11.3 $\mu$m feature could be explained by radiative or mechanical processing of the molecular carriers by the active nucleus. Here we explore the relevant physical and chemical effects that could modify the observed feature strengths. ### Ionization Balance Previous experimental (e.g., Szczepanski & Vala, 1993; Hudgins & Allamandola, 1995) and theoretical (e.g., DeFrees et al., 1993; Langhoff, 1996) work on PAHs has shown that the C–C stretching modes that produce the 6.2 and 7.7 $\mu$m features, as well as the C–H in-plane bending modes that produce the 8.6 $\mu$m feature, are more efficiently excited in ionized molecules. The ratios of these features to the 11.3 $\mu$m feature, which is produced by C–H out-of-plane bending modes, are lower for neutral molecules (see Figure 1 of Allamandola et al., 1999). The fraction of ionized aromatic molecules is set by the balance between ionization and recombination, which depends on the UV radiation field density ($G_{0}$), the gas temperature ($T)$, and the electron density ($n_{e}$) according to $G_{0}~{}T^{1/2}/n_{e}$(Bakes & Tielens, 1994). Galliano et al. (2008) argued that the variations in aromatic feature ratios for a heterogeneous sample of 50 objects (including Galactic regions, Magellanic H ii regions, and galaxies, as well as spatially resolved regions within seven of those objects) are controlled by this ionization balance. Similar to our results, they found that the relative strengths of 6.2, 7.7, and 8.6 $\mu$m features showed little variation, while the ratios between these features and the 11.3 $\mu$m feature varied by an order of magnitude. This hypothesis is also supported by observations of Galactic reflection nebulae by Joblin et al. (1996) and Bregman & Temi (2005), who found decreasing L(8.6 $\mu$m)/L(11.3 $\mu$m) and L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios as a function of distance from the ionizing source, consistent with an increasing neutral fraction. To compare with model expectations for ionized and neutral aromatic molecules, we plot L(11.3 $\mu$m)/L(7.7 $\mu$m) v. L(6.2 $\mu$m)/L(7.7 $\mu$m) for the Seyfert nuclei, off-nuclear sources, and SINGS H ii galaxies in Figure 12. This can be compared with Figure 16 of Draine & Li (2001) and Figure 5 of O’Dowd et al. (2009), although we use a condensed plot range. We find that a number of Seyferts lie beyond the range of model predictions, even for completely neutral aromatic molecules; these are the 6/35 Seyferts with L(11.3 $\mu$m)/L(7.7 $\mu$m$)>0.6$: NGC5194, NGC4501, NGC4639, NGC1433, NGC2639, and NGC5005. While such L(11.3 $\mu$m)/L(7.7 $\mu$m) ratios could be produced by large ($>200$ C atoms) neutral molecules, they would be expected to have L(6.2 $\mu$m)/L(7.7 $\mu$m$)<0.25$, which is inconsistent with the data. Similar extreme aromatic band strengths were observed by Reach et al. (2000) for the quiescent molecular cloud SMC B1 No. 1, and Li & Draine (2002) were unable to reproduce the observed band ratios even with completely neutral grains. The above comparison is for a single Milky Way–based model, and laboratory studies have found larger L(11.3 $\mu$m)/L(7.7 $\mu$m) ratios for neutral PAHs, but it does illustrate the difficulty in explaining our results for Seyfert galaxies in terms of a low ionized fraction. Furthermore, under the assumption that aromatic features are produced by star formation (see Section 4.5), the temperatures and densities of the aromatic-emitting regions should be typical of PDRs, whereas the UV radiation field would likely be enhanced by the AGN. This implies that the ionized fraction would be higher, not lower. Thus, ionization balance arguments appear unable to explain the behavior of the aromatic features around AGNs. <figure><img src="content_image/1009.2752/x12.png"><figcaption>Figure 12.— The relative strengths of the 6.2, 7.7, and 11.3 μm features forRSA Seyfert nuclei, off-nuclear regions, and SINGS H ii galaxies compared withmodel predictions from Draine & Li (2001) for neutral and ionized PAHs. Thedashed lines correspond to predictions for completely neutral and completelyionized molecules; the permitted region of the diagram is bounded by these twolines. The arrows illustrate the effects of increasing grain size andincreasing ionization on the aromatic feature ratios. The Seyferts highlightedas filled circles in Figure 10 all lie beyond the range of model predictions,even for completely neutral molecules.</figcaption></figure> ### Grain Size Smaller aromatic molecules contribute preferentially to the shorter-wavelength features (e.g., Schutte et al., 1993), but they are subject to photodestruction by the UV radiation field and collisional destruction by shocks. Based on laboratory studies, Jochims et al. (1994) found a critical size of 30–40 C atoms, below which PAHs would mainly be photodissociated, while Allain et al. (1996) suggested a larger critical value of 50 C atoms based on their models. Le Page et al. (2003) agreed that small PAHs with $15$–20 C atoms or fewer would be destroyed in most environments, but their models indicated that PAHs in the 20–30 C atom range may survive, albeit with most of their peripheral H atoms stripped away, while larger PAHs would survive with their H atoms intact. Micelotta et al. (42) found that PAHs with 50 C atoms would not survive in shocks with velocities greater than 100 km s${}^{-1}$, while PAHs with 200 C atoms would be destroyed by shocks with velocities above 125 km s${}^{-1}$. Destruction of the smallest molecules is expected to result in the 6.2 and 7.7 $\mu$m features being suppressed relative to the 11.3 $\mu$m feature, as well as the 6.2 $\mu$m feature being suppressed relative to the 7.7 $\mu$m feature (e.g., Draine & Li, 2001; Galliano et al., 2008). The former effect is clearly seen in Figure 8, but the latter is not. Thus, the hypothesis that small-grain destruction can explain the observed ratios is only tenable if the molecules that produce the 6.2, 7.7, and 8.6 $\mu$m features are destroyed with similar efficiency, which is inconsistent with existing models. ### Hydrogenation and Molecular Structure The level of hydrogenation of the aromatic molecules will affect the number of C–H bonds and therefore the relative strength of the C–H and C–C vibrational modes. An increase in the C–H/C–C ratio was proposed by Reach et al. (2000) to explain the large L(11.3 $\mu$m)/L(7.7 $\mu$m) ratio observed in SMC B1 No. 1, although Draine & Li (2001) and Li & Draine (2002) point out that PAHs with $>30$ C atoms are already expected to be fully hydrogenated. Some range in C–H/C–C ratios, even for fully hydrogenated molecules, is facilitated by the structure of the C skeleton, which can be compact with more C–C bonds or open with more C–H bonds (e.g., pericondensed PAHs v. catacondensed PAHs, Tielens, 2005). The structure also affects the number of adjacent C–H groups per aromatic ring, and therefore the relative strengths of the 11.3 $\mu$m feature, which is produced by solo C–H bonds, and the 12.7 $\mu$m feature, which is produced by C–H multiplets (e.g., Hony et al., 2001). For example, based on the large L(12.7 $\mu$m)/L(11.3 $\mu$m) ratio for SMC B1 No. 1, Vermeij et al. (2002) argued for a compact structure with a higher incidence of C–H multiplets. To investigate such behavior, we plot the L(12.7 $\mu$m)/L(11.3 $\mu$m) ratios for Seyfert nuclei, off-nuclear regions, and SINGS H ii galaxies in Figure 13. The Seyfert nuclei exhibit significantly smaller ratios ($p\leq 0.001$), while the ratios for off-nuclear regions and SINGS H ii galaxies are not distinguishable. This implies that the aromatic molecules in Seyfert nuclei may have fewer C–H multiplets. Thus a scenario where AGN processing or environment results in open, uneven molecular structures with higher C–H/C–C ratios and fewer adjacent C–H groups could qualitatively explain the observed 6–13 $\mu$m aromatic spectra. <figure><img src="content_image/1009.2752/x13.png"><figcaption>Figure 13.— The cumulative distribution of L(12.7 μm)/L(11.3 μm) ratios forRSA Seyfert nuclei, off-nuclear regions, and SINGS H ii galaxies. The resultthat Seyfert nuclei exhibit significantly smaller ratios suggests aromaticmolecules that have fewer adjacent C–H groups.</figcaption></figure> ### The role of AGN-driven shocks As presented in Section 3.2 and Figure 10, the Seyfert galaxies with the smallest L(7.7 $\mu$m)/L(11.3 $\mu$m) aromatic feature ratios also exhibit the strongest H${}_{2}$ S(3) emission, which probes hot molecular gas (upper level temperature 2500 K). The incidence of this excess H${}_{2}$ emission does not scale with AGN luminosity, indicating that shock excitation is more important than X-ray heating (e.g., Roussel et al., 2007). A connection between shock-heated, H${}_{2}$-emitting gas and small L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios was found by Ogle et al. (2007) for the radio galaxy 3C 326 and by Guillard et al. (2010) for Stephan’s Quintet, a compact group of interacting galaxies exhibiting a large-scale shock (e.g., Appleton et al., 2006; Cluver et al., 2010). Similarly, Kaneda et al. (2008) found strong H${}_{2}$ emission and small L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios in a sample of local elliptical galaxies, many of which host low-luminosity AGNs. More recently, Vega et al. (2010) affirmed this result for a sample of four early-type galaxies classified as LINERs, and they argued that shock processing of aromatic molecules may be responsible for the observed behavior. As discussed above, the Seyferts and LINERs in the SINGS sample also exhibit smaller L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios (Smith et al., 62) and stronger H${}_{2}$ emission (Roussel et al., 2007) than do the H ii galaxies. <figure><img src="content_image/1009.2752/x14.png"><figcaption>Figure 14.— The relationship between the aromatic features and the [Ne ii] and[O iv] emission lines. The filled circles correspond to the RSA Seyfertsdefined in Figure 10 that have the smallest L(7.7 μm)/L(11.3 μm) ratios. Thestrong correlation with [Ne ii], which traces star formation, and the weakcorrelation with [O iv], which traces AGN activity, implies that the aromaticfeatures are primarily associated with star formation. Most of the Seyfertnuclei lie on the relationship between aromatic feature and [Ne ii] emissionfor H ii galaxies; the only outliers are among the sources highlighted withfilled circles, which have extreme aromatic feature ratios (see Figures 10 and12). The aromatic feature and [Ne ii] emission values are in surfacebrightness units (W m−2 sr−1), while the [O iv] values, taken from Diamond-Stanic et al. (2009), are in flux units (W m−2).</figcaption></figure> Shocks are expected to have profound impacts on interstellar dust via shattering in grain-grain collisions and sputtering in ion-grain collisions (e.g., Jones et al., 1994, 1996). Aromatic features are nonetheless observed in the shocked environments of supernova remnants (e.g., Tappe et al., 2006; Reach et al., 2006) and galactic winds (e.g., Tacconi-Garman et al., 2005; Engelbracht et al., 2006). The observed emission may come from entrained clumps that are not fully exposed to the shock or the hot, post-shock gas (Micelotta et al., 42, 43). Micelotta et al. (42) study the processing of small carbon grains ($N_{C}\leq 200$, corresponding to aromatic molecules) by interstellar shocks and find that their molecular structure is severely denatured for shock velocities of 75–100 km s${}^{-1}$ and they are completely destroyed when $v\geq 125$ km s${}^{-1}$. The effect of this shock processing on the observed aromatic feature ratios is not known. A possibility that could explain the association of modified aromatic feature ratios with strong H${}_{2}$ emission is that shocks may leave open, uneven structures in the surviving aromatic molecules. We note that AGN-driven shocks, if responsible for the observed behavior, do not strongly suppress the 11.3 $\mu$m feature (see Section 4.6). <figure><img src="content_image/1009.2752/x15.png"><figcaption>Figure 15.— The relationship between aromatic feature and [Ne ii] emission asa function of the L(7.7 μm)/L(11.3 μm) ratio. The filled circles correspond tothe RSA Seyferts defined in Figure 10 that have the smallest L(7.7 μm)/L(11.3μm) ratios. This confirms that the sources with suppressed aromatic features,relative to [Ne ii], have the smallest L(7.7 μm)/L(11.3 μm) ratios.</figcaption></figure> <figure><img src="content_image/1009.2752/x16.png"><figcaption>Figure 16.— The relationship between [Ne ii] emission and the 7.7 and 11.3 μmaromatic features. The filled circles correspond to the RSA Seyferts definedin Figure 10 that have the smallest L(7.7 μm)/L(11.3 μm) ratios. While the 7.7μm feature can be strongly suppressed, the 11.3 μm feature is still a robusttracer of the SFR. The solid line in the bottom panel corresponds to themedian ratio L[Ne \textscii]/L11.3=0.12, and the dotted lines correspond tofactors of two above and below this median value. Scatter in this ratio isexpected because [Ne ii] traces somewhat younger stellar populations than dothe aromatic features. All values are in surface brightness units (W m−2sr−1).</figcaption></figure> ### Could the aromatic features be excited by the AGN? Smith et al. (62) speculated that the AGNs could directly excite aromatic emission. If this were the case, SFRs estimated from aromatic features would be overestimated due to this AGN contribution. To investigate the relationship between star formation rate, AGN luminosity, and aromatic feature strength, we plot the fluxes of the [Ne ii] and [O iv] emission lines versus those of the 7.7 $\mu$m and 11.3 $\mu$m aromatic features in Figure 14. The [Ne ii] line has an ionization potential of 21 eV and is a reasonable tracer of the SFR (e.g., Ho & Keto, 2007), while the [O iv] line has an ionization potential of 55 eV and traces the AGN intrinsic luminosity (e.g., Meléndez et al., 2008; Diamond-Stanic et al., 2009; Rigby et al., 2009). Figure 14 shows the strong correlation between [Ne ii] and aromatic feature strength for RSA Seyferts (Spearman’s $\rho=0.93$), which matches the relationship for SINGS H ii galaxies, and it shows the weak correspondence between [O iv] and aromatic feature strength (Spearman’s $\rho=0.39$). This confirms that the aromatic features are primarily tracing star-formation activity. The Seyferts that are outliers in the [Ne ii]–aromatic feature relationship have weak aromatic features, and we show in Figure 15 that these correspond to the sources with the smallest L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios. There are no examples with stronger aromatic features as might be expected if the AGN were exciting additional emission. The three obvious outliers, NGC2639, NGC4501, and NGC5194, all have $[\textrm{O}~{}\textsc{iv}]/[\textrm{Ne}~{}\textsc{ii}]<0.25$, implying that the AGN contribution to [Ne ii] is $<10$% (e.g., Sturm et al., 2002). We note that the incidence of modified aromatic spectra does not show a dependence on AGN luminosity, confirming the results of Baum et al. (2010), who found no correlation between the L(6.2 $\mu$m)/L(11.3 $\mu$m) ratio and [Ne v] luminosity. ### Use of Aromatic Features to Determine SFRs Several studies (e.g., Schweitzer et al., 2006; Lutz et al., 2008; Shi et al., 2009) have used the 6.2 and 7.7 $\mu$m aromatic features to measure the SFRs in AGN host galaxies. The result that some AGNs exhibit suppressed short-wavelength aromatic features (e.g., the outliers in Figure 15) suggests that such SFR measurements may be underestimated. To determine whether the 11.3 $\mu$m feature is robust to such effects, we plot separately the relationships between [Ne ii] and the 7.7 and 11.3 $\mu$m features in Figure 16. We find that almost all of the RSA Seyferts, including those with anomalously high L${}_{\scriptsize{[\textrm{Ne}~{}\textsc{ii}]}}$/L${}_{7.7}$ values, are within a factor of two of the median value L${}_{\scriptsize{[\textrm{Ne}~{}\textsc{ii}]}}$/L${}_{11.3}=0.12$. Scatter in this ratio is expected as a function of the age of the stellar population because 21 eV photons from young stars ($<10$ Myr) are required to produce [Ne ii], while somewhat older stars can produce 6–13.6 eV UV photons that excite aromatic emission (e.g., Peeters et al., 2004; Díaz-Santos et al., 2010; Pereira-Santaella et al., 2010). Silicate absorption will tend to increase the observed ratio, but this is only a significant effect for sources like NGC4945 and NGC3079 (see Section 3.3). While SFR estimates based on the 11.3 $\mu$m feature are still subject to the uncertainties that apply to H ii galaxies (e.g., Smith et al., 62), such measurements for AGN hosts appear to be robust to the effects of AGN- and shock-processing of aromatic molecules. ## 5. Conclusions We have shown that the relative strengths of the mid-IR aromatic features for Seyfert galaxies differ significantly from those for star-forming galaxies, with the 6.2, 7.7, and 8.6 $\mu$m features being suppressed relative to the 11.3 $\mu$m feature in Seyferts. The sources with the smallest L(7.7 $\mu$m)/L(11.3 $\mu$m) aromatic feature ratios also exhibit the strongest H${}_{2}$ S(3) rotational lines, which likely trace shocked gas (see Figure 10). We explore the relevant physical and chemical effects that could produce the observed aromatic spectra. An enhanced fraction of neutral aromatic molecules could produce qualitatively similar behavior, but the observed ratios lie beyond model predictions for completely neutral molecules and the presence of an AGN would be expected to increase the level of ionization rather than reduce it. Destruction of the smallest aromatic molecules could explain the suppression of shorter wavelength features, but the expected variations in the relative strengths of the 6.2, 7.7, and 8.6 $\mu$m features are not seen. A modification of the molecular structure that enhances the C–H/C–C ratio could reproduce the observed behavior, and an open C skeleton with fewer adjacent C–H groups would furthermore explain the reduced strength of the 12.7 $\mu$m feature. Given the connection between strong H${}_{2}$ emission and modified aromatic ratios, we speculate that shock processing could produce such structures. Finally, we show that the aromatic features correlate well with [Ne ii] (i.e., star formation) but not with [O iv] (i.e., AGN luminosity), indicating that AGN excitation of aromatic emission is not significant and that aromatic-based estimates of the SFR are generally reasonable. There are a few outliers with strong H${}_{2}$ emission, small L(7.7 $\mu$m)/L(11.3 $\mu$m) ratios, and small aromatic/[Ne ii] ratios, but for these sources the 11.3 $\mu$m feature is still a reasonably robust tracer of the SFR. We acknowledge useful discussions with and assistance from Anthony Jones, Yong Shi, Amelia Stutz, Alexander Tielens, Jonathan Trump, and Gregory Walth. We thank the anonymous referee for helpful suggestions that have improved the manuscript. This work was supported by contract 1255094 from Caltech/JPL to the University of Arizona. _Facilities:_Spitzer ## References * Allain et al. (1996) Allain, T., Leach, S., & Sedlmayr, E. 1996, A&A, 305, 602 * Allamandola et al. 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W., et al. 2004, ApJS, 154, 1 NAME \| 6.2 μm \| 7.7 μmaaConsists of sub-features at 7.42, 7.60, and 7.85 μm. \| 8.6 μm \| 11.3 μmbbConsists of sub-features at 11.23 and 11.33 μm. \| 12.7 μmccConsists of sub-features at 12.62 and 12.69 μm. \| [Ne ii] \| H2 S(3) ---\|---\|---\|---\|---\|---\|---\|--- IC3639 \| 1.15±0.05e-06 \| 5.19±0.79e-06 \| 3.41±0.54e-07 \| 1.38±0.05e-06 \| 4.94±0.54e-07 \| 3.11±0.05e-07 \| 4.45±0.79e-08 NGC1058 \| 1.47±0.10e-07 \| 4.81±0.64e-07 \| 9.52±0.69e-08 \| 1.91±0.06e-07 \| 9.12±0.88e-08 \| 9.00±1.09e-09 \| 5.22±1.28e-09 NGC1097 \| 9.24±0.31e-07 \| 3.69±0.19e-06 \| 6.90±0.22e-07 \| 1.48±0.03e-06 \| 8.07±0.21e-07 \| 1.54±0.02e-07 \| 8.36±0.68e-08 NGC1241 \| 5.38±0.83e-07 \| 1.95±0.44e-06 \| 3.50±0.44e-07 \| 5.86±0.71e-07 \| 2.67±0.35e-07 \| 7.89±0.48e-08 \| 3.65±1.61e-08 NGC1365 \| 4.77±0.07e-06 \| 1.98±0.05e-05 \| 3.10±0.05e-06 \| 3.85±0.04e-06 \| 3.00±0.06e-06 \| 5.12±0.05e-07 \| 1.11±0.07e-07 NGC1433 \| 3.58±0.25e-07 \| 8.93±1.79e-07 \| 2.50±0.18e-07 \| 6.08±0.15e-07 \| 2.66±0.23e-07 \| 4.96±0.21e-08 \| 3.34±0.62e-08 NGC1566 \| 8.12±0.25e-07 \| 3.16±0.14e-06 \| 5.14±0.20e-07 \| 1.06±0.01e-06 \| 4.87±0.18e-07 \| 7.46±0.20e-08 \| 5.01±0.32e-08 NGC2273 \| 2.54±0.03e-06 \| 9.23±0.14e-06 \| 1.45±0.02e-06 \| 2.69±0.03e-06 \| 1.08±0.02e-06 \| 3.20±0.03e-07 \| 1.04±0.04e-07 NGC2639 \| 9.29±0.97e-08 \| 2.13±0.61e-07 \| 4.19±0.66e-08 \| 1.34±0.05e-07 \| 8.02±0.71e-08 \| 7.55±0.09e-08 \| 1.47±0.13e-08 NGC2992 \| 1.55±0.05e-06 \| 6.82±0.35e-06 \| 8.24±0.36e-07 \| 1.56±0.03e-06 \| 7.80±0.43e-07 \| 3.49±0.04e-07 \| 5.99±0.52e-08 NGC3079 \| 1.82±0.03e-05 \| 7.32±0.08e-05 \| 1.19±0.03e-05 \| 1.15±0.03e-05 \| 6.68±0.07e-06 \| 9.66±0.10e-07 \| 5.04±0.26e-07 NGC3185 \| 1.11±0.05e-06 \| 3.51±0.26e-06 \| 6.99±0.40e-07 \| 1.00±0.04e-06 \| 4.57±0.34e-07 \| 7.56±0.36e-08 \| 4.19±0.98e-08 NGC3227 \| 2.91±0.03e-06 \| 9.99±0.15e-06 \| 1.24±0.01e-06 \| 3.24±0.03e-06 \| 1.35±0.01e-06 \| 5.17±0.05e-07 \| 2.03±0.03e-07 NGC3735 \| 5.92±0.42e-07 \| 3.39±0.34e-06 \| 5.61±0.28e-07 \| 7.59±0.30e-07 \| 4.47±0.38e-07 \| 5.84±0.37e-08 \| 2.44±0.69e-08 NGC4051 \| 8.84±0.54e-07 \| 4.42±0.31e-06 \| 3.96±0.32e-07 \| 1.12±0.03e-06 \| 5.67±0.52e-07 \| 1.04±0.06e-07 \| 8.09±0.77e-08 NGC4258 \| 3.40±0.07e-07 \| 1.45±0.07e-06 \| 1.36±0.04e-07 \| 3.38±0.03e-07 \| 1.60±0.06e-07 \| 7.43±0.07e-08 \| 6.58±0.14e-08 NGC4501 \| 1.09±0.05e-07 \| 3.51±0.35e-07 \| 8.02±0.37e-08 \| 3.05±0.04e-07 \| 1.50±0.03e-07 \| 4.15±0.04e-08 \| 6.37±0.25e-08 NGC4639 \| 9.15±0.56e-08 \| 2.33±0.30e-07 \| 3.47±0.35e-08 \| 1.67±0.03e-07 \| 6.65±0.52e-08 \| 9.47±0.57e-09 \| 9.22±0.69e-09 NGC4945 \| 7.99±0.08e-05 \| 3.80±0.04e-04 \| 3.91±0.04e-05 \| 2.51±0.03e-05 \| 2.81±0.03e-05 \| 5.59±0.06e-06 \| 3.24±0.71e-08 NGC5005 \| 1.40±0.06e-06 \| 5.39±0.31e-06 \| 1.20±0.04e-06 \| 3.38±0.05e-06 \| 1.26±0.03e-06 \| 3.27±0.04e-07 \| 4.13±0.22e-07 NGC5033 \| 6.04±0.40e-07 \| 2.25±0.25e-06 \| 4.07±0.21e-07 \| 1.10±0.03e-06 \| 5.77±0.19e-07 \| 1.23±0.03e-07 \| 4.00±0.57e-08 NGC5135 \| 5.65±0.06e-06 \| 2.07±0.02e-05 \| 3.62±0.04e-06 \| 4.76±0.05e-06 \| 2.58±0.03e-06 \| 8.04±0.08e-07 \| 1.22±0.04e-07 NGC5194 \| 2.57±0.32e-07 \| 6.23±1.25e-07 \| 1.12±0.19e-07 \| 6.82±0.16e-07 \| 2.85±0.17e-07 \| 1.68±0.02e-07 \| 1.32±0.09e-07 NGC5395 \| 6.69±0.43e-08 \| 2.45±0.34e-07 \| 4.14±0.38e-08 \| 1.22±0.05e-07 \| 5.88±0.45e-08 \| 4.44±0.50e-09 \| 6.12±1.16e-09 NGC5427 \| 2.23±0.06e-07 \| 1.12±0.06e-06 \| 1.49±0.05e-07 \| 2.68±0.04e-07 \| 1.43±0.08e-07 \| 4.03±0.06e-08 \| 1.32±0.11e-08 NGC5643 \| 1.10±0.09e-06 \| 4.32±0.87e-06 \| 4.27±1.17e-07 \| 1.93±0.09e-06 \| 9.62±1.02e-07 \| 1.66±0.10e-07 \| 7.28±2.05e-08 NGC5728 \| 9.91±0.23e-07 \| 5.03±0.17e-06 \| 7.19±0.17e-07 \| 2.10±0.02e-06 \| 8.19±0.17e-07 \| 2.18±0.07e-07 \| 1.55±0.07e-07 NGC6221 \| 9.07±0.09e-06 \| 3.02±0.03e-05 \| 4.92±0.05e-06 \| 6.50±0.07e-06 \| 3.88±0.05e-06 \| 1.77±0.02e-06 \| 1.62±0.09e-07 NGC6951 \| 1.87±0.05e-06 \| 6.39±0.26e-06 \| 1.30±0.04e-06 \| 2.04±0.06e-06 \| 9.54±0.29e-07 \| 2.44±0.04e-07 \| 8.03±1.14e-08 NGC7130 \| 2.71±0.05e-06 \| 1.08±0.03e-05 \| 1.71±0.05e-06 \| 2.48±0.06e-06 \| 1.37±0.06e-06 \| 5.21±0.07e-07 \| 6.38±1.11e-08 NGC7314 \| 1.71±0.23e-07 \| 1.20±0.17e-06 \| 1.09±0.17e-07 \| 2.57±0.11e-07 \| 1.33±0.14e-07 \| 6.41±0.15e-08 \| 2.50±0.36e-08 NGC7469 \| 8.23±0.08e-06 \| 2.93±0.06e-05 \| 4.28±0.04e-06 \| 5.41±0.05e-06 \| 3.29±0.07e-06 \| 1.16±0.01e-06 \| 1.32±0.05e-07 NGC7496 \| 2.18±0.05e-06 \| 6.48±0.25e-06 \| 1.15±0.03e-06 \| 1.42±0.03e-06 \| 7.30±0.32e-07 \| 2.77±0.04e-07 \| 6.14±0.93e-08 NGC7582 \| 2.26±0.05e-06 \| 8.55±0.26e-06 \| 2.58±0.04e-06 \| 4.10±0.04e-06 \| 1.55±0.03e-06 \| 3.60±0.04e-07 \| 1.37±0.10e-07 NGC7590 \| 3.01±0.57e-07 \| 1.19±0.31e-06 \| 2.72±0.36e-07 \| 4.49±0.48e-07 \| 1.95±0.29e-07 \| 2.72±0.30e-08 \| 2.18±1.10e-08 Note. – Measurements are in units of W m−2 sr−1. Table 1Nuclear Measurements NAME \| RA \| Dec \| 6.2 μm \| 7.7 μmaaConsists of sub-features at 7.42, 7.60, and 7.85 μm. \| 8.6 μm \| 11.3 μmbbConsists of sub-features at 11.23 and 11.33 μm. \| 12.7 μmccConsists of sub-features at 12.62 and 12.69 μm. ---\|---\|---\|---\|---\|---\|---\|--- IC3639 \| 12:40:53.13 \| −36:45:10.3 \| 4.24±0.32e-07 \| 1.63±0.22e-06 \| 1.94±0.28e-07 \| 2.84±0.20e-07 \| 1.63±0.32e-07 NGC1097 \| 02:46:19.06 \| −30:16:20.0 \| 4.26±0.07e-06 \| 1.39±0.04e-05 \| 2.63±0.05e-06 \| 2.86±0.05e-06 \| 1.62±0.03e-06 NGC1365 \| 03:33:36.71 \| −36:08:18.0 \| 8.53±0.09e-06 \| 3.37±0.04e-05 \| 6.68±0.08e-06 \| 6.24±0.07e-06 \| 4.15±0.05e-06 NGC1566 \| 04:20:02.13 \| −54:56:37.1 \| 2.69±0.18e-07 \| 8.33±1.14e-07 \| 1.16±0.13e-07 \| 1.33±0.10e-07 \| 7.92±1.39e-08 NGC2992 \| 09:45:42.07 \| −14:19:29.4 \| 1.05±0.10e-06 \| 3.59±0.75e-06 \| 6.48±0.84e-07 \| 9.31±1.03e-07 \| 6.07±0.47e-07 NGC3079 \| 10:01:57.49 \| +55:40:58.5 \| 2.61±0.10e-06 \| 9.02±0.30e-06 \| 1.58±0.09e-06 \| 1.80±0.08e-06 \| 9.90±0.27e-07 NGC3227 \| 10:23:30.87 \| +19:51:43.1 \| 1.04±0.08e-07 \| 4.36±0.43e-07 \| 7.23±0.46e-08 \| 1.51±0.07e-07 \| 7.30±0.49e-08 NGC4258 \| 12:18:59.31 \| +47:18:24.8 \| 4.44±0.05e-07 \| 1.44±0.02e-06 \| 2.60±0.03e-07 \| 2.87±0.03e-07 \| 1.51±0.04e-07 NGC4501 \| 12:32:00.42 \| +14:25:25.2 \| 2.97±0.08e-07 \| 1.09±0.03e-06 \| 1.92±0.05e-07 \| 2.43±0.06e-07 \| 1.33±0.04e-07 NGC4945 \| 13:05:28.26 \| −49:27:39.6 \| 2.24±0.06e-06 \| 7.69±0.11e-06 \| 1.61±0.07e-06 \| 1.72±0.06e-06 \| 9.13±0.19e-07 NGC5005 \| 13:10:56.89 \| +37:03:24.8 \| 4.87±0.65e-07 \| 1.98±0.50e-06 \| 3.28±0.32e-07 \| 4.59±0.25e-07 \| 2.71±0.34e-07 NGC5033 \| 13:13:27.87 \| +36:35:25.6 \| 9.14±0.28e-07 \| 3.47±0.15e-06 \| 6.73±0.34e-07 \| 7.55±0.23e-07 \| 4.72±0.21e-07 NGC5135 \| 13:25:44.60 \| −29:50:08.6 \| 1.90±0.13e-07 \| 8.02±0.97e-07 \| 1.31±0.09e-07 \| 2.24±0.07e-07 \| 1.27±0.25e-07 NGC5194 \| 13:29:50.36 \| +47:11:36.0 \| 7.64±0.25e-07 \| 2.82±0.11e-06 \| 4.08±0.24e-07 \| 5.82±0.19e-07 \| 3.60±0.20e-07 NGC5395 \| 13:58:38.82 \| +37:25:38.2 \| 2.49±0.04e-07 \| 8.21±0.31e-07 \| 1.47±0.04e-07 \| 1.61±0.03e-07 \| 8.41±0.56e-08 NGC5427 \| 14:03:26.11 \| −06:01:43.2 \| 2.64±0.07e-07 \| 1.02±0.05e-06 \| 1.87±0.05e-07 \| 2.24±0.04e-07 \| 1.24±0.07e-07 NGC6221 \| 16:52:46.03 \| −59:13:08.8 \| 1.05±0.04e-06 \| 3.04±0.19e-06 \| 6.22±0.25e-07 \| 1.21±0.04e-06 \| 6.46±0.31e-07 NGC7130 \| 21:48:19.38 \| −34:56:56.1 \| 1.31±0.03e-06 \| 4.35±0.13e-06 \| 8.07±0.25e-07 \| 1.03±0.03e-06 \| 5.14±0.14e-07 NGC7314 \| 22:35:46.89 \| −26:03:13.7 \| 1.11±0.13e-07 \| 4.55±1.01e-07 \| 8.01±1.30e-08 \| 1.00±0.10e-07 \| 5.26±1.40e-08 NGC7582 \| 23:18:22.64 \| −42:21:57.7 \| 4.14±0.29e-07 \| 1.82±0.19e-06 \| 2.29±0.24e-07 \| 3.36±0.18e-07 \| 1.76±0.31e-07 NGC7590 \| 23:18:55.05 \| −42:14:28.0 \| 5.19±0.53e-07 \| 1.78±0.37e-06 \| 2.81±0.29e-07 \| 3.58±0.22e-07 \| 1.79±0.29e-07 Note. – Measurements are in units of W m−2 sr−1. Table 2Off-Nuclear Measurements ratio \| Seyferts v. SINGS \| Seyferts v. off nuclear \| SINGS v. off nuclear ---\|---\|---\|--- 6/11 \| 5×10−4 \| 5×10−5 \| 0.682 7/11 \| 0.003 \| 0.001 \| 0.074 8/11 \| 9×10−4 \| 2×10−4 \| 0.063 6/7 \| 0.230 \| 0.447 \| 0.888 6/8 \| 0.489 \| 0.347 \| 0.689 7/8 \| 0.108 \| 0.303 \| 0.374 Note. – Values correspond to probabilities from two-sample K-S tests. Table 3Statistical Tests
1904.11121	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 9443, "num_imgs": 0, "llama3_tokens_count": 1793 }	[]	introduction Most popular tools for building machine learning (ML) algorithms (such as TensorFlow tensorflow, Caffe2 caffe2, and PyTorch pytorch) are essentially numerical optimization engines bolted on top of a compute engine. In the ideal case, to use such a tool, the programmer specifies a model in the form of a loss function, and the tool generates a numerical optimization procedure (typically, gradient decent) by automatically differentiating the model. This optimization procedure is then run on top of the system’s compute engine. Parameter Servers for Machine Learning. Most widely-used ML algorithm-generation tools (such as TensorFlow tensorflow) use a so-called parameter server Smola:2010, Li:2014 as the underlying distributed compute engine for executing automatically-generated algorithms. A parameter server is essentially a distributed key-value store. In a parameter server, a set of machines are designated as servers. Servers typically store fragments of the model, addressable via a unique key. The remainder of the machines are designated as workers. Workers repeatedly load a small portion of the training data, request some relevant part of the model, use the data to update the part of the model, and then push the updated model to the servers. In the common case that the numerical optimization procedure to minimize the loss function is generated automatically and the end user is more skilled in mathematics than in writing Big Data codes, it is desirable for the compute engine to run the numerical optimization procedure well, no matter how large or small the data set and model, and no matter how many machines are used to run the computation, or the amount of resources (CPU and RAM) available on each machine. To learn a model twice as quickly, just requisition twice the machines (and hence twice the CPUs/GPUs) and run the same code. This is the “one implementation, any model/data size and compute hardware” ideal. Unfortunately, this is not the reality for today’s distributed deep learning tools. It is true that a parameter server can be used to implement any distributed computation—just as a tool such as MPI gropp1996high can be used to implement any distributed computation—but the problem is that getting many distributed computations to work on top of a parameter server will typically require a large amount of human effort. For anything other than relatively simple computations, a programmer must manually break the computation into discrete units and then make choices about which servers serve which data, which workers run which computations, which machines workers and servers are housed on, how to synchronize the servers and workers, etc. The result is that in practice, parameter servers such as TensorFlow are most often used to execute computations on a single machine. For distributed learning, parameter servers are typically used for running asynchronous, stochastic gradient descent algorithms bottou2010large in the simple case where the model is small enough to be easily broadcast in its entirety to each worker. The various workers each have a small subset of the data, which they use to update the model, and then asynchronously (or synchronously) send their updates back to the parameter server. This is the so-called data parallel implementation. In practice data parallel implementations are preferred because they are easy. Each processing unit performs the same computation over different subsets of the data. In contrast, in many situations, a model parallel implementation—where various workers work on different parts of a huge model—would be desirable. Depending upon the application, the model may be so large that it cannot fit into the RAM of any particular machine and so model parallelism is required. Or, the model may be complicated and difficult to compute, and as we will argue, data parallelism is limited in its ability to speed convergence of distributed gradient descent algorithms. Model parallelism can help. Unfortunately, various claims both in the folklore and in research papers, parameter servers do not support easy model parallelism, where “easy” model parallelism would insulate a programmer from the details of the distributed implementation, the same way in which SQL insulates a database programmer from the details of how a distributed query is executed. In the ideal case, the gradient descent algorithm automatically generated by the ML platform’s front end would automatically run well on any model/data size and compute hardware. However, this is far from the reality of the situation. Adapting RDBMS Technology for ML. In contrast, the goal of “one implementation, any model/data size and compute hardware” coincides almost perfectly to the relational database management system (RDBMS) concept of data independence, which has been a guiding principle of RDBMS design since their inception in the 1970’s. To support data independence, databases are fundamentally based upon the declarative programming paradigm codd1970relational: the programmer (or algorithm generator) specifies what he/she/it wants, and not how to compute it, and the computation is automatically optimized to match the data and hardware. We will argue that this makes an RDBMS a great platform for distributed machine learning. Besides the principle of data independence, RDBMSs have many other characteristics to recommend their use as a replacement for a parameter server as the compute platform for distributed ML computations. Databases are fast, robust, distributed query optimization is now well-understood and highly effective chaudhuri1998overview. It is not an accident that competing distributed compute platforms such as Spark zaharia2010spark (which now promotes the use of relational-style DataFrames Armbrust:2015 and DataSets datasets interfaces) are beginning to look more like a parallel RDBMSs. Our central thesis in this paper is that it makes little sense to reinvent the wheel—that ideas such as declarative programming and data independence have made RDBMSs so successful make a lot of sense for distributed machine learning as well. We will show that it is possible to express deep learning computations in a few lines of SQL, and have those computations automatically parallelized in a truly model-parallel way. We will argue that there is nothing unique about modern ML computations that make them particularly amenable to the key-value style of programming required by a parameter server. In fact, such computations look at lot like the classical analytics queries that have been successfully evaluated by distributed RDBMSs for the past two decades. Hence, tools such as TensorFlow should be using RDBMS-like backends, rather than those based upon parameter serves. However, there are a couple of reasons that a modern RDBMS’ cannot be used out-of-the-box as a platform for most large-scale machine learning algorithms. Crucially, such systems lack sufficient support for recursion. In deep learning, for example, it is necessary to “loop” through the layers of a deep neural network, and then “loop” backwards through the network to propagate error and update the model. All of this is repeated within another “loop” that repeatedly runs the forward-backward passes. Such “looping” could be expressed declaratively via recursive dependencies among tables, but RDBMS support for recursion is typically limited (if it exists at all) to computing fixed-points over sets–specifically with an eye towards computing transitive closures aho1979universality. And crucially, even if one adds sufficient support for recursion to an RDBMS, there is the problem that the “query” plan for a typical deep-learning computation may run to tens of thousands of operators, which no existing RDBMS optimizer is going to be able to handle. Our Contributions. In this paper, we consider exactly how database support for recursion should be enhanced to handle large-scale machine learning computations, and then address the question of how a typical query optimization framework can be modified so that it can handle the massive query plans that result from complicated recursive computations. We implement our ideas on top of SimSQL, which is a prototype distributed RDBMS that is specifically designed to handle large-scale statistical computation. For two distributed deep learning problems (a feed-forward neural network and an implementation of Word2Vec word2vec:NIPS2013, mikolov2013efficient) we show that declarative SimSQL codes of the kind that could be auto-generated by an ML algorithm-generation tool outperform corresponding TensorFlow parameter server implementations, and scale to huge model sizes that are beyond the capabilities of the implementations that ship with TensorFlow. This is despite the fact that SimSQL has the performance handicap of being a Java-based prototype RDBMS implemented on top of MapReduce. A high-performance RDBMS would likely do even better. We also show that a distributed LDA implemented on top of SimSQL generally scales better and outperforms the same implementation on top of TensorFlow’s parameter server (as well as Apache Spark). Taken together, these results call into question the current practice of developing state-of-the-art machine learning platforms on top of a distributed parameter server, and instead suggest that a more RDBMS-like backend is perhaps a more suitable choice.
1002.3143	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 60761, "num_imgs": 14, "llama3_tokens_count": 18110 }	[ "content_image/1002.3143/fig1.jpg", "content_image/1002.3143/fig2a.jpg", "content_image/1002.3143/fig3a.jpg", "content_image/1002.3143/fig4a.jpg", "content_image/1002.3143/fig5.jpg", "content_image/1002.3143/fig6.jpg", "content_image/1002.3143/fig7a.jpg", "content_image/1002.3143/fig8a1.jpg", "content_image/1002.3143/fig9a.jpg", "content_image/1002.3143/fig10.jpg", "content_image/1002.3143/fig11a.jpg", "content_image/1002.3143/fig12a.jpg", "content_image/1002.3143/fig13a.jpg", "content_image/1002.3143/fig14a.jpg" ]	# Quark pair creation in color electric fields and effects of magnetic fields Naoto Tanji¹ [FOOTNOTE:][ENDFOOTNOTE] _Institute of physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan_ ###### Abstract The time evolution of a system where a uniform and classical SU(3) color electric field and quantum fields of quarks are dynamically coupled with each other is studied focusing on non-perturbative pair creation and its back reaction. We characterize the color direction of the electric field in a gauge invariant way, and investigate its dependence. Momentum distributions of created quarks show plasma oscillation as well as quantum effects such as the Pauli blocking and interference. Pressure of the system is also calculated, and we show that pair creation moderates degree of anisotropy of pressure. Furthermore, enhancement of pair creation and induction of chiral charge under a color magnetic field which is parallel to the electric field are discussed. UT-Komaba/10-1 ## 1 Introduction Study of non-perturbative pair creation from a classical electric field, which is known as the Schwinger mechanism [1], has a long history and wide range of applications (see Ref. [2] for a recent review). One of those applications can be found in studies of relativistic heavy-ion collisions, where the Schwinger mechanism has been used as a mechanism of matter formation from a color flux tube [3]. The color flux-tube model assumes that a strong color electric field is formed in a beam direction just after two nuclei collide and pass through each other [4, 5]. Formation of longitudinal color electric fields is also predicted in the framework of color glass condensate [6, 7]. Therefore, particle production due to the Schwinger mechanism attracts renewed interest [8, 9, 10, 11, 12]. Under these circumstances, getting an understanding of how an initial electric field and created particles evolve in time is of prime importance. To properly describe the time evolution, calculating vacuum persistence probability or pair creation probability, which were first derived by Schwinger, is not sufficient [13], and an electric field should be treated as a dynamical variable rather than a background field controlled by hand, i.e. back reaction should be taken into account. There have been considerable numbers of studies treating back reaction; the ones based on a kinetic theory [3, 14, 15] and the others on quantum field theory [13, 16, 17, 18, 19]. To our knowledge, however, field theoretical treatment of the back reaction problem under a _color_ electric field has been lacking. Therefore, in this paper we investigate the pair creation of quarks under a color electric field incorporating back reaction. In studies of physics under non-Abelian electromagnetic fields, SU(2) theory has been often used for simplicity. In the case of SU(3), however, a new feature arises: anisotropy in color space. It has been shown that an SU(3) color electric field has two independent directions and it is characterized by two gauge invariant parameters: one of them is determined by its field strength and the other is related with the color direction of the field [20, 21]. More generally, an SU($N_{c}$) color vector has $(N_{c}-1)$-independent directions in color space, and physical contents can generally depend on a color direction of an electric field [22]. In this paper, we deal with SU(3) color electric fields and examine the color direction dependence. Not only new features which arise in non-Abelian fields, we also analyze phenomena whose essence is common to the Abelian case. Collective motion of created particles which couples to an electric field shows plasma oscillation. During this evolution, several phenomena are observed: suppression of pair creation or annihilation of the particles due to the Pauli blocking, damping of the electric field, and rapid oscillations in the momentum distribution of the created particles due to interference. We shall give an analysis of these phenomena to advance an understanding of physics in pair creation. We take a uniform color electric field as an initial state. Pressure of this initial state is quite anisotropic: the longitudinal pressure is negative and the transverse pressure is positive. Therefore, if local thermalization is achieved starting from the flux-tube initial condition, isotropization of pressure should be needed during the time evolution. However, the full understanding of a thermalization process in heavy-ion collisions has not been obtained. In this paper, we examine the role of pair creation for the isotropization of pressure as a first step to understand a mechanism of thermalization in heavy-ion collisions. One of remarkable differences of the color flux tube given by the color glass condensate from that in the original flux-tube model is the existence of a longitudinal color magnetic field in addition to an electric field [7]. It has been shown that a longitudinal magnetic field enhances pair creation of fermions and speeds up the decay of an electric field in the previous paper [13]. We extend it to the quark pair creation under a longitudinal color electric and magnetic field. Furthermore, we study induction of chiral charge due to pair creation under a magnetic field. Since the chiral anomaly is a semi-classical effect where the quantum aspect of a gauge field is unnecessary, we can also apply our framework to study the chiral anomaly due to pair creation. The relation between pair creation and the chiral anomaly has been also studied in Refs. [24, 25]. Emergence of a nonzero chirality in heavy-ion collisions attracts interest in the context of the chiral magnetic effect [23]. The remainder of this paper is organized as follows. In the next section, we shall explain the Abelianization of a color electromagnetic field, and introduce the parameter characterizing the color direction of the field. Although this formalism is essentially the same as that given in Ref. [26], we make the existence of color direction dependence clearer with the help of the method in Refs. [20, 21]. In Section 3, we introduce time-dependent particle picture to describe the time evolution of the system. Then, we shall show our numerical results in Section 4. Time evolution of momentum distribution functions of created quarks, color current density, electric field strength and pressure of the system are displayed and discussed. Color direction dependence of the results is also examined there. In Section 5, effects of a longitudinal magnetic field, i.e. enhancement of pair creation and induction of chiral charge, are discussed. ## 2 General framework Quark pair creation incorporated with back reaction is described by the following Lagrangian density \[\mathcal{L}=\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{1}{4}F_{ \mu\nu}^{a}F^{a\mu\nu},\] (1) where $\psi$ is a quark field and color indices $i\ (i=1,2,\cdots,N_{c})$ are omitted. We assume for simplicity that each flavor has the same mass $m$, and flavor indices are also omitted. The number of flavor is set to be $N_{f}=3$ throughout this paper. The covariant derivative and the field strengths are defined in terms of a background gauge field $A_{\mu}^{a}$ as \[D_{\mu}=\partial_{\mu}+igT^{a}A_{\mu}^{a},\] (2) \[F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{ a}-gf^{abc}A_{\mu}^{b}A_{\nu}^{c},\] (3) where $T^{a}$ is the generator in the fundamental representation of gauge group SU($N_{c}$), and $f^{abc}$ is the anti-symmetric structure constant ($a,b,c=1,2,\cdots,N_{c}^{2}-1$). The equations of motion for $A_{\mu}^{a}$ and $\psi$ now read \[D_{\mu}F^{a\mu\nu}=g\bar{\psi}\gamma^{\nu}T^{a}\psi,\] (4) \[\left(i\gamma^{\mu}D_{\mu}-m\right)\psi=0.\] (5) Because we treat the gauge field $A_{\mu}^{a}$ as a classical background field, the charge current operator $g\bar{\psi}\gamma^{\nu}T^{a}\psi$ in Eq.(4) is replaced by its expectation value $g\langle\bar{\psi}\gamma^{\nu}T^{a}\psi\rangle$ in the following. These coupled equations govern the pair creation and its back reaction. We restrict the background field to spatially homogeneous and Abelian-like one, which is expressed as \[F_{\mu\nu}^{a}=\bar{F}_{\mu\nu}n^{a},\] (6) where $\bar{F}_{\mu\nu}$ is an Abelianized field strength and is independent of the space coordinates. $n^{a}$ is a constant vector indicating a color orientation of the electromagnetic field, and is normalized so that $n^{a}n^{a}=1$. This field strength is given by the gauge field \[A_{\mu}^{a}=\bar{A}_{\mu}n^{a}.\] (7) The relation between $\bar{F}_{\mu\nu}$ and $\bar{A}_{\mu}$ is just the same as that in the Abelian case: \[\bar{F}_{\mu\nu}=\partial_{\mu}\bar{A}_{\nu}-\partial_{\nu}\bar{A}_{\mu}.\] (8) Under the Abelianized gauge field (7), the covariant derivative (2) reads $D_{\mu}=\partial_{\mu}+ign^{a}T^{a}\bar{A}_{\mu}$. Because $\left(n^{a}T^{a}\right)_{ij}$ is an $N_{c}\times N_{c}$ hermitian matrix, we can diagonalize it by a unitary transformation²(global gauge transformation): $Un^{a}T^{a}U^{\dagger}=\mathrm{diag}(w_{1},\cdots,w_{N_{c}})$. Then, the equations of motion (4) and (5) can be rewritten as [FOOTNOTE:2][ENDFOOTNOTE] \[\partial_{\mu}\bar{F}^{\mu\nu}=\sum_{i=1}^{N_{c}}w_{i}g\langle \bar{\psi}^{i}\gamma^{\nu}\psi^{i}\rangle,\] (9) \[\left[i\gamma^{\mu}(\partial_{\mu}+iw_{i}g\bar{A}_{\mu})-m\right] \psi^{i}=0\hskip 20.0pt(i=1,2,\cdots,N_{c}).\] (10) These equations have the same form with those in the Abelian theory, except the presence of the weight vector $w_{i}$. Hence, we can solve the coupled equations (9) and (10) as in the Abelian case [13]. Each quark field $\psi^{i}$ couples with the background field $\bar{A}_{\mu}$ via a coupling constant $w_{i}g$. <figure><img src="content_image/1002.3143/fig1.jpg"><figcaption>Figure 1: A graphical representation of Eqs.(12). The triangle of dotted linescorresponds to θ=0 and is an usual weight diagram for a fundamentalrepresentation of SU(3). Each wi is given by the x coordinate of each vertexof the θ-rotated triangle.</figcaption></figure> In the case of SU(3), a diagonalized $n^{a}T^{a}$ may be expressed as \[Un^{a}T^{a}U^{\dagger}=T^{3}\cos\theta-T^{8}\sin\theta,\] (11) because $T^{3}=\mathrm{diag}(1/2,-1/2,0)$ and $T^{8}=\mathrm{diag}(1/2\sqrt{3},1/2\sqrt{3},-1/\sqrt{3})$ are diagonal. (We represent $T^{a}$ as $T^{a}=\frac{1}{2}\lambda_{a}$ where $\lambda_{a}$ is the Gell-Mann matrix.) Then, $w_{i}$ are expressed in terms of $\theta$ as follows \[w_{1}=\frac{1}{\sqrt{3}}\cos(\theta+\frac{\pi}{6}),\ w_{2}=\frac{1}{\sqrt{3}} \cos(\theta+\frac{5\pi}{6}),\ w_{3}=\frac{1}{\sqrt{3}}\cos(\theta+\frac{3\pi}{ 2}).\] (12) The angle $\theta$ is related with the Casimir (gauge) invariant [20, 21]. In our parametrization³, the relation is [FOOTNOTE:3][ENDFOOTNOTE] \[\sin^{2}3\theta=3C_{2},\] (13) where $C_{2}=[d^{abc}n^{a}n^{b}n^{c}]^{2}$ is the second Casimir invariant for SU(3) and characterizes a direction of the color electromagnetic field in a gauge invariant way. Fig.1 is a graphical representation of Eqs.(12). Each $w_{i}$ is given by the $x$ coordinate of each vertex of the rotated triangle (weight diagram). Owing to the symmetry of the diagram, it is sufficient to take the angle $\theta$ restricted to $0\leq\theta\leq\frac{\pi}{6}$. ## 3 Canonical quantization in background fields To describe the time evolution of the system, we introduce an instantaneous particle picture and quantize the field using that particle picture. Of course, a definition of particle in the presence of a pair-creating background is rather ambiguous. Thus, the instantaneous particle picture should be regarded as our working definition to describe the system evolution. To calculate field quantities such as current and energy density, we need to identify and subtract the contribution from the Dirac sea at each time. We can do this just by the normal ordering in terms of the instantaneous particle basis, and then energy conservation is automatically guaranteed [13]. Therefore, one can interpret defining a particle picture at each time as a means to properly regularize field quantities, and can consider particle number at intermediate time as a byproduct. Nevertheless, we will show time evolution of a particle number or momentum distribution function defined through the instantaneous particle picture in the following, because it behaves in a physically reasonable manner, and helps us understand the dynamics of this system. First, we consider only the electric fields. We will introduce the magnetic fields in Section 5. We set an initial state with no electric field and switch it on at time $t=0$. An initial field strength is $E_{0}$ and its direction is along the $z$-axis. The gauge $\bar{A}_{0}=0$ is chosen so that \[\bar{A}_{3}(t)=0\ \ (t\leq 0),\] \[\frac{d\bar{A}_{3}}{dt}\bigg{\|}_{t=0}=-E_{0},\] (14) and $\bar{A}_{1}=\bar{A}_{2}=0$. After switching on, $\bar{A}_{3}(t)$ evolves according to Eqs.(9) and (10). A quantized quark field may be expanded as \[\psi^{i}(x)=\sum_{s=\uparrow,\downarrow}\int\!d^{3}p\left[{}_{+}\!\psi^{i\, \mathrm{in}}_{\mathbf{p}s}(x)a^{i\,\mathrm{in}}_{\mathbf{p},s}+{}_{-}\!\psi^{i \,\mathrm{in}}_{\mathbf{p}s}(x)b^{i\,\mathrm{in}\dagger}_{-\mathbf{p},s}\right],\] (15) where $a^{i\,\mathrm{in}}_{\mathbf{p},s}$ [$b^{i\,\mathrm{in}}_{\mathbf{p},s}$] is the annihilation operator of a particle [antiparticle] with momentum $\mathbf{p}$ and spin-$s$ satisfying the anti-commutation relation $\{a^{i\,\mathrm{in}}_{\mathbf{p},s},a^{j\,\mathrm{in}\dagger}_{\mathbf{q},s^{ \prime}}\}=\{b^{i\,\mathrm{in}}_{\mathbf{p},s},b^{j\,\mathrm{in}\dagger}_{ \mathbf{q},s^{\prime}}\}=\delta_{ij}\delta_{ss^{\prime}}\delta^{3}(\mathbf{p}- \mathbf{q})$, and ${}_{\pm}\!\psi^{i\,\mathrm{in}}_{\mathbf{p}s}(x)$ are classical solutions of the Dirac equation (10). The superscript ‘in’ distinguishes the initial condition for ${}_{\pm}\!\psi^{i\,\mathrm{in}}_{\mathbf{p}s}(x)$: at $t<0$, ${}_{+}\!\psi^{i\,\mathrm{in}}_{\mathbf{p}s}(x)$ [${}_{-}\!\psi^{i\,\mathrm{in}}_{\mathbf{p}s}(x)$] is identical to the positive [negative] energy solution of the free Dirac equation. We set the state to be in-vacuum $\|0,\mathrm{in}\rangle$, where no particle exists initially and which is defined by $a^{i\,\mathrm{in}}_{\mathbf{p},s}\|0,\mathrm{in}\rangle=b^{i\,\mathrm{in}}_{ \mathbf{p},s}\|0,\mathrm{in}\rangle=0$. At $t>0$, ${}_{\pm}\!\psi^{i\,\mathrm{in}}_{\mathbf{p}s}(x)$ evolve under influence of the electric field and become superposition of a positive and negative energy (frequency) state. To describe the pair creation dynamically, we introduce a time-dependent particle picture by decomposing the field operator $\psi^{i}(x)$ into positive and negative frequency instantaneously: \[\psi^{i}(t_{0},\mathbf{x})=\sum_{s=\uparrow,\downarrow}\int\!d^{3}p\left[{}_{+ }\!\psi^{i\,(t_{0})}_{\mathbf{p}s}(x)a^{i}_{\mathbf{p},s}(t_{0})+{}_{-}\!\psi^ {i\,(t_{0})}_{\mathbf{p}s}(x)b^{i\,\dagger}_{-\mathbf{p},s}(t_{0})\right],\] (16) where ${}_{+}\!\psi^{i\,(t_{0})}_{\mathbf{p}s}(x)$ [${}_{-}\!\psi^{i\,(t_{0})}_{\mathbf{p}s}(x)$] is a positive [negative] frequency solution of the Dirac equation under the pure gauge $\bar{A}_{3}=\bar{A}_{3}(t=t_{0})$. Instantaneous particle picture is defined by $a^{i}_{\mathbf{p},s}(t)$ and $b^{i}_{\mathbf{p},s}(t)$. Of course, $a^{i}_{\mathbf{p},s}(t)$ and $b^{i}_{\mathbf{p},s}(t)$ agree with $a^{i\,\mathrm{in}}_{\mathbf{p},s}$ and $b^{i\,\mathrm{in}}_{\mathbf{p},s}$ at $t<0$, respectively. The particle picture at time $t$ and that of the in-state are related by the time-dependent Bogoliubov transformation: \[\begin{matrix}a^{i}_{\mathbf{p}s}(t)=\alpha^{i}_{\mathbf{p}s}(t)a^{i\,\mathrm{ in}}_{\mathbf{p}s}+\beta^{i}_{\mathbf{p}s}(t)b^{i\,\mathrm{in}\dagger}_{- \mathbf{p}s},\\ b^{i\,\dagger}_{-\mathbf{p}s}(t)=\alpha^{i\,}_{\mathbf{p}s}(t)b^{i\,\mathrm{ in}\dagger}_{-\mathbf{p}s}-\beta^{i\,}_{\mathbf{p}s}(t)a^{i\,\mathrm{in}}_{ \mathbf{p}s},\end{matrix}\] (17) of which coefficients satisfy $\|\alpha^{i}_{\mathbf{p}s}(t)\|^{2}+\|\beta^{i}_{\mathbf{p}s}(t)\|^{2}=1$ and are given by \[\begin{matrix}\alpha^{i}_{\mathbf{p}s}(t)\delta^{3}(\mathbf{p}-\mathbf{q}+w_{i }g\bar{A}_{3}(t))={\int}\!d^{3}x{}_{+}\!\psi^{i\,(t)\dagger}_{ \mathbf{p}s}(x){}_{+}\!\psi^{i\,\mathrm{in}}_{\mathbf{q}s}(x),\\ \beta^{i}_{\mathbf{p}s}(t)\delta^{3}(\mathbf{p}-\mathbf{q}+w_{i}g\bar{A}_{3}(t ))={\int}\!d^{3}x{}_{+}\!\psi^{i\,(t)\dagger}_{\mathbf{p}s}(x){}_ {-}\!\psi^{i\,\mathrm{in}}_{\mathbf{q}s}(x).\end{matrix}\] (18) A quark pair distribution function is defined by \[f^{i}_{\mathbf{p}s}(t)=\langle 0,\mathrm{in}\|a^{i\,\dagger}_{\mathbf{p}s}(t)a^ {i}_{\mathbf{p}s}(t)\|0,\mathrm{in}\rangle\frac{(2\pi)^{3}}{V}=\langle 0, \mathrm{in}\|b^{i\,\dagger}_{-\mathbf{p}s}(t)b^{i}_{-\mathbf{p}s}(t)\|0,\mathrm{ in}\rangle\frac{(2\pi)^{3}}{V},\] (19) where $V$ is the volume of the system. With the help of Eqs.(17), we can rewrite $f^{i}_{\mathbf{p}s}(t)$ in terms of the Bogoliubov coefficients: \[f^{i}_{\mathbf{p}s}(t)=\|\beta^{i}_{\mathbf{p}s}(t)\|^{2}.\] (20) The expectation of the charge current operator, which is regularized by the normal ordering is \[\begin{split} j_{z}(t)&=\sum_{i=1,2,3}\langle 0, \mathrm{in}\|:w_{i}g\bar{\psi}^{i}\gamma_{3}\psi^{i}:\|0,\mathrm{in}\rangle\\ &=2N_{f}\sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow}w_{i}g\int\! \frac{d^{3}p}{(2\pi)^{3}}\frac{p_{z}}{\omega_{p}}f^{i}_{\mathbf{p}s}(t)+2N_{f} \sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow}w_{i}g\int\!\frac{d^{3}p}{(2\pi)^{3} }\frac{m_{\mathrm{T}}}{\omega_{p}}g^{i}_{\mathbf{p}s}(t),\end{split}\] (21) where the transverse mass $m_{\mathrm{T}}=\sqrt{m^{2}+p_{x}^{2}+p_{y}^{2}}$ and the anomalous distribution \[g^{i}_{\mathbf{p}s}(t)=\frac{1}{2}\langle 0,\mathrm{in}\|\left[e^{-2i\omega_{p} t}b^{i}_{-\mathbf{p}s}a^{i}_{\mathbf{p}s}+e^{2i\omega_{p}t}a^{i\,\dagger}_{ \mathbf{p}s}b^{i\,\dagger}_{-\mathbf{p}s}\right]\|0,\mathrm{in}\rangle\frac{(2 \pi)^{3}}{V}=\mathrm{Re}\left[e^{-2i\omega_{p}t}\alpha^{i}_{\mathbf{p}s}(t) \beta^{i}_{\mathbf{p}s}(t)\right]\] (22) are introduced.⁴ The first term of the last expression of Eq.(21) is a conduction current, which is caused by movement of real particles and the second is a polarization current, which is generated by variation of electric dipole density. [FOOTNOTE:4][ENDFOOTNOTE] ## 4 Pair creation under electric fields ### Plasma oscillation and Pauli blocking We have solved the coupled equation (9) and (10) numerically with the help of Eqs.(20), (21) and (22). The results are shown in Figs.2, 3 and 4. The time evolution of longitudinal momentum distributions with fixed transverse momentum is exhibited in Fig.2 and that of transverse momentum distributions with fixed longitudinal momentum is in Fig.3. Fig.4 shows the time evolution of color current density, electric field strength and quark number density. The parameters are set to be $a=\frac{m^{2}}{2gE_{0}}=0.01$ or $a=0$, and $g=1,\theta=0$. To concentrate our attention at first on general features which are common to the Abelian case, only the distributions of “blue” quarks, whose effective coupling to the background is $\frac{1}{2}g$ when $\theta=0$, are presented. Hereafter, all figures are shown in the dimension-less unit scaled by $\sqrt{gE_{0}}$. <figure><img src="content_image/1002.3143/fig2a.jpg"><figcaption>Figure 2: Longitudinal momentum distributions of “blue” quarks. g=1,pT=0,θ=0.</figcaption></figure> <figure><img src="content_image/1002.3143/fig3a.jpg"><figcaption>Figure 3: Transverse momentum distribution of “blue” quarks.g=1,pz/√gE0=1.0,θ=0</figcaption></figure> After the switch-on of the electric field, particles are created with approximately 0 longitudinal momenta. Their occupation number is approximately $\exp\left(-\pi m_{\text{T}}^{2}/\|w_{i}\|gE_{0}\right)$, which accords with the form expected from the semi-classical tunneling calculation [27], and the transverse momentum distribution exhibits a Gaussian-like form.⁵ In particular, particles in zero mode ($m=0,p_{\text{T}}=0$) are created as many as possible under the restriction of Pauli’s exclusion principal $f_{\mathbf{p},s}^{i}(t)\leq 1$, so that their occupation is exactly equal to one. Therefore, the longitudinal momentum distribution with $a=0$ and $p_{\text{T}}=0$ [Fig.2(b)] shows squarish form. [FOOTNOTE:5][ENDFOOTNOTE] After created, particles [anti-particles] are accelerated to the direction of $w_{i}g\mathbf{E}$ [$-w_{i}g\mathbf{E}$]. They generate charge current in the direction of $n^{a}$, which decreases the electric field according to the “Maxwell’s” equation (9). Then, the direction of the electric field is flipped at some time and the particles start to be decelerated. Repeats of this process result in oscillating behavior of the longitudinal momentum distribution, the charge current and the electric field, which is known as plasma oscillation. Other than plasma oscillation, which is a classical dynamics, also the Pauli blocking, which is a quantum effect, plays a role in the time evolution of the longitudinal momentum distributions. Because of Pauli’s exclusion principal, a particle blocks the subsequent pair creation at the point where it locates in phase space. In particular, if a value of a distribution exceeds $1/2$, not only pair creation is suppressed but also pair annihilation occurs (see the second term of the right hand side of Eq.(32)). Therefore, the distributions get dented when particles cross the line of $p_{z}=0$ in momentum space. (Notice that pair creation and annihilation can happen only in the vicinity of the line of $p_{z}=0$ because we now take only a classical gauge field.) The effect of the Pauli blocking is the most conspicuous in distributions with $m_{\mathrm{T}}=0$. In this case, because the occupation of the created particles is equal to one, particles are totally annihilated when they cross the line of $p_{z}=0$ [See Fig.2(b)]. Collecting the facts above, the momentum distribution function can be approximated by the following equation: \[f^{i}_{\mathbf{p}s}(t)\simeq\exp\left(-\frac{\pi m_{\text{T}}^{2}}{\|w_{i}\|gE_{ 0}}\right)\theta\left(p_{z}(-w_{i}g\bar{A}_{3}-p_{z})\right).\] (23) This equation roughly replicates the distributions obtained by the numerical calculation. Using this empirical and analytic expression of the distribution, we analyze the numerical results and study the parameter dependence of current and particle number density. Equation (23) is exactly correct for the zero mode with $m_{\text{T}}=0$. In contrast, it loses its accuracy for higher $m_{\text{T}}$ modes. However, it does not matter for our purpose because pair creation of those modes is strongly suppressed compared with low $m_{\text{T}}$ modes. Although Eq.(23) is neither an exact one nor obtained by some systematic approximation, it reproduces the numerical results with sufficient accuracy for a rough analysis and is useful because of its simple structure. <figure><img src="content_image/1002.3143/fig4a.jpg"><figcaption>Figure 4: Time evolution of the color current density, the electric field andquark number density. g=1,θ=0.</figcaption></figure> Using Eq.(23), the momentum integrations in the particle number density and the conduction current can be done and we obtain \[N(t)\simeq\frac{4N_{f}}{(2\pi)^{3}}E_{0}\sum_{i=1,2,3}(w_{i}g)^{ 2}e^{-\frac{\pi m^{2}}{\|w_{i}\|gE_{0}}}\left\|\bar{A}_{3}(t)\right\|\] (24) \[j_{z}^{\text{cond}}(t)\simeq-\frac{4N_{f}}{(2\pi)^{3}}E_{0}\sum_ {i=1,2,3}\|w_{i}g\|^{3}e^{-\frac{\pi m^{2}}{\|w_{i}\|gE_{0}}}\bar{A}_{3}(t).\] (25) In the derivation of Eq.(25), the approximation $p_{z}/\omega_{p}\simeq\text{sgn}(p_{z})\ (m^{2}\ll gE_{0})$ has been used, which makes Eq.(25) overestimate. Equation (25) can explain the plasma oscillation obtained by the numerical calculation. Neglecting the polarization current and using Eq.(25), we can solve the Maxwell equation $\frac{d^{2}\bar{A}_{3}}{dt^{2}}=j_{z}$ and obtain the oscillating electric field \[\bar{A}_{3}(t)=-\frac{E_{0}}{\Omega}\sin\Omega t\] (26) \[E_{z}(t)=E_{0}\cos\Omega t,\] (27) where its frequency is \[\Omega=\sqrt{\frac{4N_{f}}{(2\pi)^{3}}E_{0}\sum_{i}\|w_{i}g\|^{3}e^ {-\frac{\pi m^{2}}{\|w_{i}\|gE_{0}}}}.\] (28) Let us introduce the time $t_{c}$ when the electric field first reduces its strength to zero. This $t_{c}$ gives a typical time scale of variation of the electric field. In the present approximation, $t_{c}$ is given as follows \[t_{c}=\frac{1}{4}\frac{2\pi}{\Omega}=\frac{\pi}{2}\sqrt{\frac{(2 \pi)^{3}}{4N_{f}E_{0}\sum_{i}\|w_{i}g\|^{3}e^{-\frac{\pi m^{2}}{\|w_{i}\|gE_{0}}}}}.\] (29) This gives a smaller value of $t_{c}$ than the result of the numerical calculations because the current (25) is an overestimate. However, their discrepancy is less than factor 2 and Eq.(29) correctly describes the parameter dependence of $t_{c}$ obtained by the numerical calculations. Equation (29) tells us that the stronger the electric field or the larger the coupling constant, the shorter time the electric field takes to vanish. Substituting Eq.(26) into Eq.(24) leads the oscillating behavior of $N(t)$ while the actual $N(t)$ [Fig.4(c)] does not show oscillation. This discrepancy is because of the incorrectness of the approximation (23) at $t>t_{c}$. For explanation, suppose particles having a positive charge $w_{i}g>0$. They get positive momentum at first by acceleration of the electric field, and after $t_{c}$ they go into the negative momentum area in the momentum space because the direction of the field is flipped at $t=t_{c}$. The modeled expression (23) does not describe particles plunging into the negative momentum area. If the particles are those in the zero mode $m_{\text{T}}=0$, they totally disappear when they cross the line of $p_{z}=0$ due to the Pauli blocking and thus Eq.(23) is correct at all time. If $m_{\text{T}}\neq 0$, however, they do not totally disappear and the simple approximation (23) becomes incorrect after $t_{c}$. That is why Eq.(24) is not correct at $t>t_{c}$ and the actual $N(t)$ shows saturation rather than oscillation. Under a strong magnetic field, however, $N(t)$ does oscillate in time [Fig.13(c)] and the present approximation becomes correct because only the lowest $m_{\text{T}}$ mode mainly contributes (See Section 5). ### Polarization current and damping of the electric field The current density (21) consists of two parts: the conduction current and the polarization current. As discussed in the previous subsection, the conduction current is associated with collective motion of particles, i.e. plasma oscillation. In contrast, the polarization current is related with microscopic processes of pair creation. It is induced at an instant when pair creation happens. In a classical view, when a particle pair is created and becomes on-shell, a distance between them is nonzero except the case that their transverse mass is zero. Therefore, the pair creation process can be interpreted as creation of an electric dipole, so that it generates the polarization current. (See Ref. [13] for more detailed argument.) Because of its origin, the polarization current is expected to be induced in the same direction with the electric field. In other words, $E_{z}$ and $j_{z}^{\text{pol}}$ have the same sign. Therefore, the polarization current _always_ reduces the electric field strength through the Maxwell equation $\frac{dE_{z}}{dt}=-j_{z}$. That is, the polarization current causes damping of the electric field. <figure><img src="content_image/1002.3143/fig5.jpg"><figcaption>Figure 5: Comparison between the polarization and conduction current.a=0.01,g=1,θ=0. The electric field is also plotted for reference.</figcaption></figure> Nevertheless, the polarization current obtained by our numerical calculation shows irregular behavior [Fig.5] rather than simple one expected from above argument based on a classical view. This is because the system undergoes a complicated evolution due to plasma oscillation and the Pauli blocking, and furthermore the anomalous distribution, which is included in the integrand of the polarization current (21), is sensitive to phases of the Bogoliubov coefficients. What is especially remarkable in the plot of the polarization current is a peak just after switching the electric field on. Because the electric field is turned on suddenly, the vacuum experiences prompt polarization. That is why the large polarization current is induced at an early time. Although the naive expectation based on a classical view does not hold in this dynamic system, damping of electric fields actually happen. However, it is slight and hard to be recognized in Fig.4(b) because the polarization current is far smaller than the conduction current (except the time just after the field is switched on) in our parametrization. If behavior in longer time is calculated, damping of the field would be observed. Indeed, damping behavior can be recognized in the result under a magnetic field which is parallel to an electric field [Fig.12(b)], because a magnetic field speeds up the time evolution of the system (see Section 5). Besides it, the damping would be quickened by strong coupling. Before closing this subsection, let us emphasize that a zero transverse mass mode does not contribute to polarization current. It is evident from the explicit expression of a polarization current, which contains the factor $m_{\text{T}}$ (the second term of Eq.(21)). In a classical view, the distance between a massless pair is zero when they are created, so that its electric dipole moment is also zero. That is why no polarization current is induced by zero modes. ### Interference One may notice that the longitudinal distribution shows rapid oscillations after $t_{c}$ [Fig.2]. Fig.6 shows time slices of the longitudinal distribution with $a=0.01,g=1$ and $\theta=0$, whose whole picture is Fig.2(a). Before $t_{c}$ ($\sim 15/\sqrt{gE_{0}}$ in this case), the distribution is smooth. After $t_{c}$, particles created before $t_{c}$ start to cross the line of $p_{z}=0$. Then, the distribution begins to show rapid oscillations. These oscillation have been observed also in earlier works where back reaction is taken into account [16, 17]. These can be interpreted as interference between particles created before $t_{c}$ and those created after $t_{c}$. <figure><img src="content_image/1002.3143/fig6.jpg"><figcaption>Figure 6: Time slices of the longitudinal momentum distribution in Fig.2(a).</figcaption></figure> Because the problem with back reaction cannot be treated analytically, we deal with pair creation under a constant electric field to explain the interference noted above. The problem without back reaction can be solved analytically and an explicit expression for distributions is available [13]. Under a constant electric field, however, particles are accelerated to one direction, so that there is no event that two distributions overlap in momentum space nor interfere. Therefore, we suppose an initial state with a distribution $f_{0}(\mathbf{p})$ instead of a vacuum with no particle $\|0,\text{in}\rangle$. (In this subsection, we omit spin and color indices for simplicity.) This initial distribution would join a distribution of particles created from an electric field, and they would interfere. That initial state is expressed by a two-mode squeezed state as \[\|f_{0}\rangle=\prod_{\mathbf{p}}\mathcal{N}_{\mathbf{p}}^{-1/2} \exp\left[\frac{(2\pi)^{3}}{V}F(\mathbf{p})a^{\text{in}\,\dagger}_{\mathbf{p}} b^{\text{in}\,\dagger}_{-\mathbf{p}}\right]\|0,\text{in}\rangle,\] (30) where $\mathcal{N}_{\mathbf{p}}$ and $F(\mathbf{p})$ are related with the initial distribution $f_{0}(\mathbf{p})$ as follows \[\|F(\mathbf{p})\|^{2}=\frac{f_{0}(\mathbf{p})}{1-f_{0}(\mathbf{p})} ,\hskip 10.0pt\mathcal{N}_{\mathbf{p}}=\frac{1}{1-f_{0}(\mathbf{p})}\] (31) Notice that a phase of $F(\mathbf{p})$ is irrelevant to $f_{0}(\mathbf{p})$. Because now $f_{0}(\mathbf{p})$ is given by hand, there is no criterion to decide a phase of $F(\mathbf{p})$. However, if $f_{0}(\mathbf{p})$ is a distribution of particles created from an electric field, a phase of $F(\mathbf{p})$ is automatically determined by its time history of evolution. Letting this system evolve under the electric field which is given by the gauge $A^{\mu}=\left(0,\mathbf{A}(t)\right)$, we obtain the distribution function at time $t$: \[\tilde{f}_{\mathbf{p}}(t) =\langle f_{0}\|a^{\dagger}_{\mathbf{p}}(t)a_{\mathbf{p}}(t)\|f_{0} \rangle\frac{(2\pi)^{3}}{V}\] \[=f_{0}\left(\mathbf{p}+w_{i}g\mathbf{A}(t)\right)+\left\{1-2f_{0} \left(\mathbf{p}+w_{i}g\mathbf{A}(t)\right)\right\}f_{\mathbf{p}}(t)\] (32) \[+2\left\{1-f_{0}\left(\mathbf{p}+w_{i}g\mathbf{A}(t)\right) \right\}\text{Re}\left[\alpha_{\mathbf{p}}(t)\beta^{}_{\mathbf{p}}(t)F( \mathbf{p}+w_{i}g\mathbf{A}(t))\right].\] The first term of the right hand side represents the initial particles accelerated by the field. $f_{\mathbf{p}}(t)$ in the second term is the distribution of particles created from the field, which is defined by Eq.(19). The factor $\left\{1-2f_{0}\left(\mathbf{p}+w_{i}g\mathbf{A}(t)\right)\right\}$ expresses the effect of the Pauli blocking: initial particles suppress the subsequent pair creation, and pair annihilation occurs if the initial occupation exceeds $1/2$. The third term describes interference between the initial particles and those created from the field.⁶ Now $\alpha_{\mathbf{p}}(t)$ and $\beta_{\mathbf{p}}(t)$ are the Bogoliubov coefficients giving the distribution $f_{\mathbf{p}}(t)$. Although the distributions $f_{\mathbf{p}}(t)$ and $f_{0}(\mathbf{p})$ are independent of the phases of the Bogoliubov coefficients or $F(\mathbf{p})$, the third term is sensitive to those phases. Because of this term, the distribution shows rapid oscillations when the two distributions overlap in momentum space. [FOOTNOTE:6][ENDFOOTNOTE] <figure><img src="content_image/1002.3143/fig7a.jpg"><figcaption>Figure 7: Distribution functions under a constant electric field with theinitial distribution f0(p)=12e−(p/√eE0+5)2. a=0.01,e=1,pT=0.</figcaption></figure> As an illustration, we plot in Fig.7 the distribution (32) with the initial distribution $f_{0}(\mathbf{p})=\frac{1}{2}e^{-(p/\sqrt{eE_{0}}+5)^{2}}$ under a constant electric field which is switched on at $t=0$. For simplicity, results in Abelian theory, which is obtained by replacing $w_{i}g$ with $e$ in (32), are shown. The parameters are chosen as $a=0.01,e=1$. For reference, we show in Fig.7(a) the result in which the interference term is neglected. We can see clearly the effect of the Pauli blocking. In the presence of the third term, the distributions show oscillations of which contour is along classical trajectories of particles [Fig.7(b),(d)]. What is remarkable is Fig.7(c), in which a phase factor of $F(\mathbf{p})$ is set to be $e^{ip_{z}^{2}/eE_{0}}$. In this case, oscillations seen in Fig.7(b) disappears. That is to say, the factor $e^{ip_{z}^{2}/eE_{0}}$ cancels the phases of $\alpha_{\mathbf{p}}$ and $\beta^{}_{\mathbf{p}}$. This is reasonable because the phase factors of $\alpha_{\mathbf{p}}$ and $\beta_{\mathbf{p}}$ are expected to be $e^{iS_{cl}}$ and $e^{-iS_{cl}}$, respectively, in which $S_{cl}$ is the classical action of a particle under the constant electric field: \[S_{cl}\approx-\frac{1}{2}\frac{p_{z}^{2}}{eE_{0}}\hskip 20.0pt(p _{z}\to\infty).\] (33) Hence, if one want to reproduce the situation in which a distribution of particles created from the electric field interfere with a distribution of particles which are _also_ created from the electric field, it is natural to assume $F(\mathbf{p})$ has the same phase factor with $\alpha_{\mathbf{p}}\beta^{}_{\mathbf{p}}$, i.e. $e^{-ip_{z}^{2}/eE_{0}}$. A distribution in such case is plotted in Fig.7(d). Rapid oscillations similar to those seen in the distribution with back reaction are obtained. Note that like as the polarization current, the interference term has no contribution from a zero transverse mass mode. That is because the occupation $n_{\mathbf{p},s}(t)=\|\beta_{\mathbf{p}s}(t)\|^{2}$ of a zero mode takes the maximum value 1, and there is the constraint $\|\alpha_{\mathbf{p}s}\|^{2}+\|\beta_{\mathbf{p}s}\|^{2}=1$, so that $\alpha_{\mathbf{p}}(t)\beta^{}_{\mathbf{p}}(t)$ is 0 for this mode. These rapid oscillations are a key ingredient for seeming irreversibility of time evolution [28]. Because of this interference as well as a polarization current, effective dissipation of energy from the electric field to the quantum fields happens. For example, the number density in Fig.4(c) shows saturation, and there is seemingly an arrow of time, although the equations of motion (4) and (5) hold time-reversal symmetry. In contrast, if there is no polarization current nor the interference term, effective dissipation would not happen and evolution of the system would be periodic in time. We shall demonstrate it by studying pair creation under a strong magnetic field in Section 5.1 (see Fig.13(c)). <figure><img src="content_image/1002.3143/fig8a1.jpg"><figcaption>Figure 8: Color direction dependence of the longitudinal momentumdistributions. a=0.01,g=1,pT=0.</figcaption></figure> ### Color direction dependence As shown in Section 2, direction of a color electric field in color space can be characterized by the Casimir invariant $C_{2}$ or the angle $\theta$ in a gauge invariant way. Therefore, physical values may generally depend on $C_{2}$ or $\theta$. Casimir dependence of transverse distribution⁷ under a constant color electric field has been studied in Ref. [29]. However, as far as the author knows, color direction dependence incorporated with back reaction has not been investigated. [FOOTNOTE:7][ENDFOOTNOTE] Fig.8 shows the color direction dependence of the longitudinal momentum distributions ($a=0.01,g=1,p_{\text{T}}=0$). We can understand the behavior of these distributions based on the values of the effective coupling $w_{i}g$. For example, when $\theta=0$, the distributions of “blue” and “red” quarks are symmetric in momentum space because $w_{1}=-w_{2}$, and “green” quarks are not at all created since $w_{3}=0$. Concerning the momentum distributions, the result greatly depends on the color direction of the electric field. This is a matter of course because the distributions of unconfined quarks are not color singlet object. <figure><img src="content_image/1002.3143/fig9a.jpg"><figcaption>Figure 9: Color direction dependence of the current density, the electricfield and the quark number density. a=0.01,g=1.</figcaption></figure> Color direction dependence of the charge current density, the field strength and the quark number density is shown in Fig.9. Unlike the momentum distributions, the color direction dependence is very small for these quantities. This is because the current and the particle number are obtained by summing up the colors, so that they depend on $\theta$ only through the elementary symmetric polynomials of $w_{i}$. The relations between $\theta$ and the elementary symmetric polynomials of $w_{i}$ are \[w_{1}+w_{2}+w_{3}=0\] (34) \[w_{1}^{2}+w_{2}^{2}+w_{3}^{2}=\frac{1}{2}\] (35) \[w_{1}w_{2}w_{3}=-\frac{1}{12\sqrt{3}}\sin 3\theta.\] (36) The first equation is the trace of Eq.(11), the second is the trace of the square of Eq.(11) and the third is the determinant of Eq.(11). Because $w_{1}+w_{2}+w_{3}$ and $w_{1}^{2}+w_{2}^{2}+w_{3}^{2}$ are independent of $\theta$ and only $w_{1}w_{2}w_{3}$ has the dependence on $\theta$, the field quantities depend on the color direction parameter $\theta$ very weakly. As an illustration, see Eqs.(24) and (25). Because our attention is now on the regime where pair creation strongly happens, i.e. $m^{2}/gE_{0}\ll 1$, the factor $\exp\left(-\frac{\pi m^{2}}{\|w_{i}\|gE_{0}}\right)$ is nearly equals to 1 and its $\theta$-dependence is negligible. Therefore, leading dependence on $\theta$ comes from the factor $(w_{i}g)^{2}$ or $\|w_{i}g\|^{3}$. However, the factor $(w_{i}g)^{2}$ brings no $\theta$-dependence because of Eq.(35). That is why color direction dependence of the number density is very small. In the case of the current density, although the factor $\sum_{i}\|w_{i}g\|^{3}$ does bring $\theta$-dependence, one can show that the $\theta$-dependence is numerically small by substituting Eq.(12) explicitly into $\sum_{i}\|w_{i}g\|^{3}$. It varies between $0.24g^{3}$ and $0.25g^{3}$. ### Pressure The initial state with the longitudinal electric field $\mathbf{E}^{a}=(0,0,E)n^{a}$ is quite anisotropic: longitudinal pressure is $P_{\text{L}}=-E^{2}/2$ and transverse pressure is $P_{\text{T}}=E^{2}/2$. In contrast, pressure is locally isotropic in a locally thermalized quark-gluon plasma phase. Hence, if longitudinal color electric fields are formed in the initial stage of heavy-ion collisions and if local thermalization is realized in the later stage, isotropization of pressure must be achieved during the evolution of the system. In this subsection, we examine how the initial anisotropic pressure evolves under the influences of pair creation and its back reaction. Pressure generated by quarks is given by the expectation of the symmetric energy-momentum tensor: \[\begin{split}\langle 0,\text{in}\|:\Theta^{\mu\nu}:\|0,\text{in} \rangle&=-\frac{i}{4}\langle 0,\text{in}\|:\bar{\psi}\left(\gamma^ {\mu}\overleftrightarrow{\partial}^{\nu}+\gamma^{\nu}\overleftrightarrow{ \partial}^{\mu}\right)\psi:\|0,\text{in}\rangle\\ &=\text{diag}\left(\mathcal{E},P_{\text{T}},P_{\text{T}},P_{\text {L}}\right),\end{split}\] (37) which is regularized by the normal ordering. Notice that $\langle\Theta^{\mu\nu}\rangle$ is diagonal because we are in the center of mass frame. Inserting Eq.(16) into Eq.(37) and using Eq.(17), one can express $\mathcal{E}$ (energy density; Fig.10), $P_{\text{L}}$ and $P_{\text{T}}$ in terms of the momentum distribution $f^{i}_{\mathbf{p}s}(t)$ and the anomalous distribution $g^{i}_{\mathbf{p}s}(t)$: \[\mathcal{E}=2N_{f}\sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow}\int \!\frac{d^{3}p}{(2\pi)^{3}}\omega_{p}f^{i}_{\mathbf{p}s}(t)\] (38) \[P_{\text{L}}=2N_{f}\sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow} \int\!\frac{d^{3}p}{(2\pi)^{3}}\left[\frac{p_{\text{L}}^{2}}{\omega_{p}}f^{i}_ {\mathbf{p}s}(t)+\frac{p_{\text{L}}m_{\text{T}}}{\omega_{p}}g^{i}_{\mathbf{p}s }(t)\right]\] (39) \[P_{\text{T}}=2N_{f}\sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow} \int\!\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2}\left[\frac{p_{\text{T}}^{2}}{\omega _{p}}f^{i}_{\mathbf{p}s}(t)-\frac{p_{\text{L}}p_{\text{T}}^{2}}{m_{\text{T}} \omega_{p}}g^{i}_{\mathbf{p}s}(t)\right].\] (40) The terms containing $f^{i}_{\mathbf{p}s}(t)$ are contributions from real particles, which are common to classical theory. What is peculiar to our quantum field theoretical calculation is the presence of the terms including $g^{i}_{\mathbf{p}s}(t)$, which exhibit pressure generated by pair creation processes. One can confirm that the relation \[\begin{split}\langle 0,\text{in}\|:\Theta^{\mu}_{\ \mu}:\|0,\text{ in}\rangle&=\mathcal{E}-2P_{\text{T}}-P_{\text{L}}\\ &=2N_{f}\sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow}\int\!\frac{d^{ 3}p}{(2\pi)^{3}}\left[\frac{m^{2}}{\omega_{p}}f^{i}_{\mathbf{p}s}(t)-\frac{p_{ \text{L}}m^{2}}{\omega_{p}m_{\text{T}}}g^{i}_{\mathbf{p}s}(t)\right]\\ &=m\langle 0,\text{in}\|:\bar{\psi}\psi:\|0,\text{in}\rangle\end{split}\] (41) holds.⁸ [FOOTNOTE:8][ENDFOOTNOTE] The result of our numerical calculation is shown in Fig.11. The parameters are set to be $a=0,g=1$ and $\theta=0$. Due to pair creation and subsequent acceleration by the electric field, the particles generate positive pressure in the longitudinal direction, whereas transverse pressure by the particles is relatively small because the particles are accelerated only to the longitudinal direction. As pressure by the particles increases, pressure of the electric field is weakened owing to back reaction. The negative longitudinal pressure at the initial time is compensated by pressure generated by the particles. Although full isotropization is not achieved because this system is collision-less plasma, degree of anisotropy is moderated due to pair creation. <figure><img src="content_image/1002.3143/fig10.jpg"><figcaption>Figure 10: Energy balance.</figcaption></figure> <figure><img src="content_image/1002.3143/fig11a.jpg"><figcaption>Figure 11: Time evolution of pressure. a=0.01,g=1,θ=0. L and T denotelongitudinal and transverse, respectively.</figcaption></figure> ## 5 Effects of magnetic fields In the color glass condensate framework, it has been shown that longitudinal color magnetic fields are produced in addition to color electric fields just after heavy-ion collisions [6, 7]. In this section, we study the effects of a longitudinal color magnetic field on pair creation. Let us consider the situation such that both electric and magnetic fields are along the $z$-axis in configuration space and along $n^{a}$ in color space. This setup realizes a gauge field configuration with nonzero topological charge $F_{\mu\nu}^{a}\tilde{F}^{a\,\mu\nu}\neq 0$ which is predicted by the color glass condensate. Because this system is spatially uniform, induced current is also uniform, so that the magnetic field is not changed by back reaction. Therefore, the magnetic field acts as only a catalyst in this situation. To treat back reaction to a magnetic field, a non-uniform field must be considered. However, it is beyond the scope of our present paper. ### Enhancement of pair creation Under a longitudinal magnetic field, pair creation is enhanced, so that the time evolution of the system becomes faster due to the enhanced back reaction [13]. This phenomenon is caused by appearance of Landau levels and spin-magnetic field interaction. Due to interaction with the magnetic field, transverse momentum of created particles is discretized into Landau levels, and degeneracy between modes with spin parallel and antiparallel to the magnetic field, which are denoted by $\uparrow$ and $\downarrow$ each in the following, is broken: \[p_{\text{T}}^{2}\longrightarrow 2n_{s}\|w_{i}\|gB,\] (42) where \[n_{s}=\left\{\begin{array}[]{lll}n&&\text{for $s=\uparrow$\ }\\ n+1&\raisebox{6.0pt}[0.0pt][0.0pt]{$(n=0,1,\cdots)$}&\text{for $s=\downarrow$. }\end{array}\right.\] (43) The expressions of field quantities under the longitudinal magnetic field are available from those under no magnetic field by the replacements (42) and \[\begin{split} m_{\text{T}}^{2}&\longrightarrow m_{ \text{T},s}^{2}=m^{2}+2n_{s}\|w_{i}\|gB\\ \omega_{p}&\longrightarrow\omega_{\mathbf{p},s}= \sqrt{p_{z}^{2}+m_{\text{T},s}^{2}}\\ \int\!\frac{d^{2}p_{\text{T}}}{(2\pi)^{2}}& \longrightarrow\frac{\|w_{i}\|gB}{2\pi}\sum_{n=0}^{\infty}\ ,\end{split}\] (44) where $\mathbf{p}$ in $\omega_{\mathbf{p},s}$ is the abbreviated expression for $(p_{z},n)$. Notice that the transverse mass with $n_{s}=0$ is independent of $B$, while those with higher $n_{s}$ depend on $B$. Therefore, a strong magnetic field makes particles in higher modes “heavy” and suppresses their pair creation. In contrast, creation of $(n_{s}=0)$-particles is not at all suppressed. Not only the creation of the lowest mode is not suppressed, the magnetic field enhances field quantities such as current and total particle number. That is because the number of modes degenerating in a unit transverse area is proportional to $B$. For example, the charge current (21) is replaced by \[j_{z}(t)=2N_{f}\sum_{i=1,2,3}\sum_{s=\uparrow,\downarrow}w_{i}g\frac{\|w_{i}\|gB }{2\pi}\sum_{n}\int\!\frac{dp_{z}}{2\pi}\left[\frac{p_{z}}{\omega_{\mathbf{p}, s}}f^{i}_{\mathbf{p}s}(t)+\frac{m_{\mathrm{T},s}}{\omega_{\mathbf{p},s}}g^{i}_ {\mathbf{p}s}(t)\right].\] (45) In a strong magnetic field $gB\gtrsim gE\gg m^{2}$, this may be approximated as follows \[j_{z}(t)\simeq 2N_{f}\sum_{i=1,2,3}w_{i}g\frac{\|w_{i}\|gB}{2\pi}\int\!\frac{dp_ {z}}{2\pi}\left[\frac{p_{z}}{\omega_{p_{z},n=0,\uparrow}}f^{i}_{p_{z},n=0, \uparrow}(t)+\frac{m}{\omega_{p_{z},n=0,\uparrow}}g^{i}_{p_{z},n=0,\uparrow}(t )\right],\] (46) which is proportional to $B$. <figure><img src="content_image/1002.3143/fig12a.jpg"><figcaption>Figure 12: Magnetic field dependence. a=0.01,g=1,θ=0.</figcaption></figure> <figure><img src="content_image/1002.3143/fig13a.jpg"><figcaption>Figure 13: Mass dependence under the magnetic field. B=E0,g=1,θ=0.</figcaption></figure> Fig.12 presents the magnetic field dependence of the current, the electric field and the particle number density ($a=0.01,g=1,\theta=0$). The color current density and the quark number density are indeed enhanced by the magnetic field, and as a consequence the frequency of the plasma oscillation increases. We can estimate $t_{c}$, which is a typical time scale of back reaction, in the same way as that in Section 4.1, and obtain \[t_{c}=\frac{\pi}{2}\sqrt{\frac{(2\pi)^{3}}{4N_{f}E_{0}\sum_{i}\|w _{i}g\|^{3}e^{-\frac{\pi m^{2}}{\|w_{i}\|gE_{0}}}}\frac{\tanh\pi\frac{B}{E_{0}}}{ \pi\frac{B}{E_{0}}}}.\] (47) This is a decreasing function of $B$. Because only the lowest $m_{\text{T}}$ mode mainly contributes under a strong magnetic field⁹, the results in the massless case ($a=0$) and those with nonzero mass ($a=0.01$) show drastic difference [Fig.13], whereas in the absence of a magnetic field, there is no considerable difference between them [Fig.4]. The particle number density is especially remarkable: Its time evolution becomes nearly periodic in the massless case. Similar behavior of a particle number has been obtained by Iwazaki [25] in a different way. Emergence of this periodic behavior is consistent with our contention that the polarization current and the interference of distributions are the source of apparent irreversibility of time, because as noted in Sections 4.2 and 4.3, the zero mode contributes to neither the polarization current nor the interference term. [FOOTNOTE:9][ENDFOOTNOTE] ### Chiral anomaly The magnetic field has another effect: induction of chiral charge. Because the lowest Landau level ($n=0$) is occupied only by particles with spin parallel to the magnetic field ($s=\uparrow$), the balance between spin-$\uparrow$ and $\downarrow$ is broken and thus chiral charge is induced. We consider the chiral charge density given by the following equation \[\mathcal{Q}_{5}=\langle 0,\text{in}\|:\bar{\psi}\gamma_{0}\gamma_{5}\psi:\|0, \text{in}\rangle.\] (48) In the chiral limit ($m=0$), the chiral charge equals the difference between the number of right-handed particles and that of left-handed particles. Substituting Eq.(16) into Eq.(48) and subtracting a divergent part by the normal ordering, we can express the chiral charge density in terms of the distribution functions: \[\mathcal{Q}_{5}(t)=2N_{f}\sum_{i=1,2,3}\frac{w_{i}gB}{2\pi}\int\!\frac{dp_{z}} {2\pi}\left[\frac{p_{z}}{\omega_{p_{z},n=0,\uparrow}}f^{i}_{p_{z},n=0,\uparrow }(t)+\frac{m}{\omega_{p_{z},n=0,\uparrow}}g^{i}_{p_{z},n=0,\uparrow}(t)\right].\] (49) Note that only the lowest mode contributes to $\mathcal{Q}_{5}$. Contribution from the higher modes is totally canceled out because they are occupied with both the spin-$\uparrow$ and $\downarrow$ modes. When $m\neq 0$, the anomalous distribution also contributes to the chiral charge. The time evolution of the chiral charge obeys the Adler-Bell-Jackiw anomaly equation [30, 31]. The expectation of the anomaly equation extended to the QCD case now reads¹⁰ [FOOTNOTE:10][ENDFOOTNOTE] \[\frac{d}{dt}\mathcal{Q}_{5}(t)=\frac{N_{f}g^{2}}{4\pi^{2}}E(t)B+2m\bar{ \mathcal{Q}}_{5}(t),\] (50) where the term $2m\bar{\mathcal{Q}}_{5}$ describes explicit breaking of the chiral symmetry due to the mass term, and \[\begin{split}\bar{\mathcal{Q}}_{5}(t)&=\langle 0, \text{in}\|:\bar{\psi}i\gamma_{5}\psi:\|0,\text{in}\rangle\\ &=2N_{f}\sum_{i=1,2,3}\frac{w_{i}gB}{2\pi}\int\!\frac{dp_{z}}{2 \pi}\mathrm{Im}\left[e^{-2i\omega_{p}t}\alpha^{i}_{p_{z},n=0,\uparrow}(t)\beta ^{i}_{p_{z},n=0,\uparrow}(t)\right].\end{split}\] (51) <figure><img src="content_image/1002.3143/fig14a.jpg"><figcaption>Figure 14: Time evolution of the chiral charge. g=1,B=E0,θ=0.</figcaption></figure> In Fig.14, each term of the time integral of Eq.(50) is plotted. Equation (50) is indeed satisfied. Due to plasma oscillation, the chiral charge density shows oscillation. As noted in the previous subsection, time evolution is nearly periodic in the chiral limit [Fig.14(a)]. This curve of $\mathcal{Q}_{5}$ becomes exactly a sine one in the limit of a strong magnetic field $B\gg E_{0}$[25]. In contrast, if $m\neq 0$, the term $\bar{\mathcal{Q}}_{5}$ grows, and the periodicity seen in $\mathcal{Q}_{5}$ is destroyed. ## 6 Summary and discussion We have studied the time evolution of the system where quantum quark fields and a classical color electric field are dynamically coupled with each other. By the procedure of Abelianization, we have reduced the equations of motion to the same form as in the Abelian case. Effective coupling strength of each quark with the Abelianized electromagnetic field depends on the direction of the field in color space. Therefore, the longitudinal momentum distributions of created quarks greatly depend on the color direction of the gauge field [Fig.8]. However, we have found that the color current density and total particle density are rather insensitive to the color orientation of the field since those quantities are obtained after summing up all the color components [Fig.9]. We have obtained the oscillating behavior of the longitudinal momentum distributions, and estimated its typical time scale $t_{c}$ [Eq.(29)] using the empirical analytic expression for the quark distributions. This $t_{c}$ also gives a time scale of net particle production. The increase of the quark number density is concentrated at initial time and gets saturated after $t_{c}$ [Fig.4(c)]. Let us roughly estimate $t_{c}$ with the parameters motivated by the color glass condensate; $m\simeq 0,N_{f}=3$ and $gE_{0}\simeq Q_{s}^{2}\sim 1\text{GeV}^{2}$, where $Q_{s}$ is a saturation scale. Then Eq.(29) yields \[t_{c}\sim 3/g\hskip 5.0pt\text{fm}.\] (52) Note in passing that the $\theta$ dependence of $t_{c}$ is also small, 2% effect, at most, in the present case. This $t_{c}$ is not drastically small compared with the typical time scale of the initial stage of heavy-ion collisions, such as thermalization time expected from hydrodynamic calculations $\lesssim 1$ fm (e.g. [32]). However, this estimation incorporates only the effect of quark pair creation. If gluon pair creation is taken into account, $t_{c}$ would become much smaller. Pair creation of gluons is stronger than that of quarks because (i) gluons are boson, so that they are Bose-enhanced rather than Pauli-blocked, and (ii) effective coupling of gluons to an electric field is larger than that of quarks.¹¹ However, applying the method in the present paper to gluon pair creation, we encounter a difficulty of infrared divergence. Furthermore, under a magnetic field, one particle energy of a gluon in the lowest Landau level becomes imaginary and instability occurs, which is known as the Nielsen–Olesen instability [33, 34]. Thus, we need to resolve these difficulties in order to treat both the quark pair and gluon pair creations in our framework in a unified manner. [FOOTNOTE:11][ENDFOOTNOTE] In addition to the plasma oscillation, we have pointed out that the response of vacuum to the electric field involves several phenomena such as the Pauli blocking, damping of the electric field and rapid oscillations in the quark distribution. In particular, it has been revealed that the cause of the rapid oscillations is the interference between the particles. Because of this interference, the distribution becomes sensitive to the phase of the Bogoliubov coefficients. Furthermore, we have found that apparent irreversibility of time evolution is induced by nonzero transverse mass modes through the polarization current and the interference. Also the time evolution of pressure of the system has been calculated. We have shown that the initial anisotropy in pressure is moderated by pair creation and back reaction even though our treatment is a mean field one. If effects of collisions among particles are taken into account, in other words, if quantum gauge fields are introduced, the system would be more isotropized. It is hoped to investigate these effects, as well as effects of non-uniformity of electromagnetic fields. Finally, we have discussed the effects of a magnetic field on quark pair creation. We have shown that a magnetic field enhances pair creation because of the emergence of the Landau level and the spin-magnetic field interaction. As a result, the longitudinal magnetic field speeds up the decay of the electric field. This mechanism may have significance in the context of heavy-ion collisions. ## Acknowledgments The author would like to thank Professors T. Matsui and H. Fujii for enlightening discussions and comments on the manuscript. He also acknowledges Professor A. 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1611.00495	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 20292, "num_imgs": 0, "llama3_tokens_count": 6837 }	[]	# Probing lepton flavour (universality) violation at NA62 and future kaon experiments Lewis C. Tunstall ¹${}^{,1}$ Andreas Crivellin ${}^{2}$ Giancarlo D’Ambrosio${}^{3}$ and Martin Hoferichter${}^{4}$ ${}^{1}$Albert Einstein Centre for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH–3012 Bern, Switzerland ${}^{2}$Paul Scherrer Institut, CH–5232 Villigen PSI, Switzerland ${}^{3}$INFN-Sezione di Napoli, Via Cintia, I–80126 Napoli, Italy ${}^{4}$Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA tunstall@itp.unibe.ch [FOOTNOTE:0][ENDFOOTNOTE] ###### Abstract Recent results from the LHC’s first run have revealed intriguing departures from lepton flavour universality in the semi-leptonic decays of $B$-mesons. We discuss the complementary role that rare kaon decays can provide in testing new physics explanations of these flavour anomalies. In the framework of minimal flavour violation, we relate the chiral low-energy constants involved in $K\to\pi\ell\ell^{\prime}$ and $K\to\ell\ell^{\prime}$ ($\ell=\mu\mbox{ or }e$) with the new physics Wilson coefficients of the $b\to s$ effective Hamiltonian. We comment on the determination of these low-energy constants at NA62 and future kaon experiments, as well as the required improvements in sensitivity necessary to test the $B$-physics anomalies in the kaon sector. ## 1 Introduction With the completion of Run 1 at the LHC, we find ourselves with several indirect hints that physics beyond the Standard Model (SM) may be lurking in the semi-leptonic decays of $B$-mesons. Among the recent experimental anomalies, two at LHCb have received considerable attention: * a $2.6\sigma$ signal of lepton flavour universality violation (LFUV) in the measured [1] branching fractions of $B\to K\ell^{+}\ell^{-}$ decays $(\ell=\mu\mbox{ or }e)$; * the measured value [2, 3] of the angular observable $P_{5}^{\prime}$ [4] in the decay $B\to K^{}\mu^{+}\mu^{-}$ deviates from the SM prediction at the $2$–$3\sigma$ level [5, 6, 7]. Taken separately, each deviation is at most a $3\sigma$ effect; however, when combined with other $b\to s$ transitions, global fits [8, 9] indicate that (a) new physics (NP) is preferred over the SM by $4$–$5\sigma$, and (b) the effect is in the $\mu\mu$ modes only. Expressed in terms of the effective Hamiltonian for $b\to s$ transitions [9] \[{\cal H}_{\text{eff}}^{\|\Delta B\|=1}=-\frac{4G_{F}}{\sqrt{2}}V_{tb}V_{ts}^{} \sum_{i}C_{i}^{B}(\mu)Q_{i}^{B}(\mu)+\text{h.c.}\,,\] (1) the potential NP can be interpreted as contributions to the Wilson coefficients $C_{9,10}^{B}$ of the semi-leptonic operators \[Q_{9}^{B}=\frac{e^{2}}{32\pi^{2}}\left[\bar{s}\gamma^{\mu}(1- \gamma_{5})b\right]\sum_{\ell=e,\mu}\left[\bar{\ell}\gamma_{\mu}\ell\right] \quad\mbox{and}\quad Q_{10}^{B}=\frac{e^{2}}{32\pi^{2}}\left[\bar{s}\gamma^{ \mu}(1-\gamma_{5})b\right]\sum_{\ell=e,\mu}\left[\bar{\ell}\gamma_{\mu}\gamma_ {5}\ell\right]\,.\] (2) Here we discuss the complementary role that rare kaon decays can provide in testing NP explanations of the $B$-physics anomalies. Our analysis [10] is based on the observation that at low energy scales $\mu\ll m_{t,b,c}$, the strangeness-changing transitions are described in terms of the effective Lagrangian [11] \[{\cal L}_{\text{eff}}^{\|\Delta S\|=1}=-\frac{G_{F}}{\sqrt{2}}V_{ud}V_{us}^{} \sum_{i}C_{i}(\mu)Q_{i}(\mu)+\text{h.c.}\,,\] (3) which contains semi-leptonic operators \[Q_{7V}=\left[\bar{s}\gamma^{\mu}(1-\gamma_{5})d\right]\sum_{\ell =e,\mu}\left[\bar{\ell}\gamma_{\mu}\ell\right]\quad\mbox{and}\quad Q_{7A}= \left[\bar{s}\gamma^{\mu}(1-\gamma_{5})d\right]\sum_{\ell=e,\mu}\left[\bar{ \ell}\gamma_{\mu}\gamma_{5}\ell\right]\,,\] (4) that are the $s\to d$ analogues of the $b\to s$ operators $Q_{9,10}^{B}$ in (2). In the framework of minimal flavour violation (MFV), the Wilson coefficients of the two sectors _are correlated_, and we use this feature to convert knowledge of $C_{7V,7A}$ into bounds on $C_{9,10}^{B}$. The quality of the bounds is hampered by non-perturbative effects from QCD, which are parametrised in terms of the low-energy constants (LECs) arising in the 3-flavour chiral expansion. In Section 2 we focus on the experimental determination of the LECs involved in $K^{\pm}\to\pi^{\pm}\ell^{+}\ell^{-}$ and $K_{L}\to\ell^{+}\ell^{-}$, and comment on how measurements at NA62 and future kaon experiments may improve the resulting limits on $C_{9,10}^{B}$. A similar strategy is adopted in Section 3 to obtain bounds on lepton flavour violation (LFV) in the $B$-meson sector, while concluding remarks are given in Section 4. For the analysis of LFUV and LFV in other kaon decays not discussed in this article, we refer the reader to [10]. ## 2 Kaon probes of lepton flavour universality violation ### $K^{\pm}\to\pi^{\pm}\ell^{+}\ell^{-}$ At low energies, the dominant contribution to $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ is due to single virtual-photon exchange. The resulting amplitude involves a vector form factor $V_{+}(z)$ which up to $O(p^{6})$ in the chiral expansion, can be decomposed in the general form [11] \[V_{+}(z)=a_{+}+b_{+}z+V_{+}^{\pi\pi}(z)\,,\qquad z=q^{2}/m_{K}^{2}\,.\] (5) Here the LECs $a_{+}$ and $b_{+}$ parametrise the polynomial part, while the rescattering contribution $V_{+}^{\pi\pi}$ can be determined from fits to $K\to\pi\pi$ and $K\to\pi\pi\pi$ data. Chiral symmetry alone does not constrain the values of the LECs,¹ so instead, we consider the differential decay rate $d\Gamma/dz\propto\|V_{+}(z)\|^{2}$ as a means to extract $a_{+}$ and $b_{+}$ from experiment. The resulting fit to the decay spectra from all available high-statistics experiments is given in Table 1. [FOOTNOTE:1][ENDFOOTNOTE] Channel \| a+ \| b+ \| Reference ---\|---\|---\|--- ee \| −0.587±0.010 \| −0.655±0.044 \| E865 [14] ee \| −0.578±0.016 \| −0.779±0.066 \| NA48/2 [15] μμ \| −0.575±0.039 \| −0.813±0.145 \| NA48/2 [16] Table 1: Fitted values of coefficients in the vector form factor (5). Now for the crucial point: if lepton flavour universality applies, the coefficients $a_{+}$ and $b_{+}$ have to be equal for the $ee$ and $\mu\mu$ channels, which within errors is indeed the case. Since the SM interactions are lepton flavour universal, deviations from zero in differences like $a_{+}^{\mu\mu}-a_{+}^{ee}$ would then be a sign of NP, and the corresponding effect would be necessarily short-distance. To convert the allowed range on $a_{+}^{\text{NP}}$ into a corresponding range in the Wilson coefficients $C_{7V}^{\ell\ell}$, we make use of the $O(p^{2})$ chiral realization of the $SU(3)_{L}$ current \[\bar{s}\gamma^{\mu}(1-\gamma_{5})d\leftrightarrow iF_{\pi}^{2}(U\partial^{\mu} U^{\dagger})_{23}\,,\qquad U=U(\pi,K,\eta)\,,\] (6) to obtain \[a_{+}^{\text{NP}}=\frac{2\pi\sqrt{2}}{\alpha}V_{ud}V_{us}^{}C_{7V}^{\text{NP} }\,.\] (7) Contributions due to NP in $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ can then be probed by considering the _difference_ between the two channels \[C_{7V}^{\mu\mu}-C_{7V}^{ee}=\alpha\frac{a_{+}^{\mu\mu}-a_{+}^{ee}}{2\pi\sqrt{2 }V_{ud}V_{us}^{}}\,.\] (8) If the framework of MFV, this can be converted into a constraint on the NP contribution to $C_{9}^{B}$: \[C_{9}^{B,\mu\mu}-C_{9}^{B,ee}=-\frac{a_{+}^{\mu\mu}-a_{+}^{ee}}{\sqrt{2}V_{td} V_{ts}^{}}\approx-19\pm 79\,,\] (9) where we have averaged over the two electron experiments listed in Table 1. Evidently, the determination of $a_{+}^{\mu\mu}-a_{+}^{ee}$ needs to be improved by an $O(10)$ factor in order to probe the parameter space relevant for the $B$-anomalies, whose explanation involves Wilson coefficients $C_{9,10}^{B}=O(1)$ [9]. Improvements of this size may be possible at NA62, especially for the experimentally cleaner dimuon mode which currently has the larger uncertainty. ### $K_{L}\to\ell^{+}\ell^{-}$ With an eye towards future kaon experiments involving $K_{L}$ beams² (e.g. a side programme like KLEVER at NA62 [17]), we also consider $K_{L}\to\ell^{+}\ell^{-}$ decays as another potential probe of LFUV. These decays are complementary to $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ since they provide the means to constrain NP effects due to axial-vector interactions. [FOOTNOTE:2][ENDFOOTNOTE] The dominant long-distance contribution is due to $K_{L}\to\gamma^{}\gamma^{}\to\ell^{+}\ell^{-}$, where the dispersive component of the decay amplitude involves a counterterm $\chi$ that is decomposed in long- and short-distance parts $\chi(\mu)=\chi_{\gamma\gamma}(\mu)+\chi_{\text{SD}}$. The SM prediction for $\chi_{\text{SD}}$ is known [11], but $\chi_{\gamma\gamma}$ depends on two LECs whose values are not fixed by chiral symmetry. Nevertheless, we invoke the same argument applied to $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$, and observe that if lepton flavour universality holds, then the SM values of $\chi$ must be equal in both the $ee$ and $\mu\mu$ channels. Then, using the chiral realization (6) of the $SU(3)_{L}$ current, one obtains an analogous relation to (7) for the NP Wilson coefficient: \[C_{7A}^{\text{NP}}=-\frac{\alpha}{F_{K}G_{F}V_{ud}V_{us}^{}}\bigg{(}\frac{2 \Gamma_{\gamma\gamma}}{\pi m_{K}^{3}}\bigg{)}^{1/2}\chi_{\text{NP}}\,,\qquad \Gamma_{\gamma\gamma}=\Gamma(K_{L}\to\gamma\gamma)\,.\] (10) The final step is to observe that within the framework of MFV, the difference \[C_{7A}^{\mu\mu}-C_{7A}^{ee}\] (11) is directly related to the Wilson coefficients of the $B$-physics sector: \[C_{10}^{B,\mu\mu}-C_{10}^{B,ee}=2.6\bigg{(}\frac{3.5\times 10^{- 4}}{V_{td}V_{ts}^{}}\bigg{)}\big{(}\chi^{\mu\mu}-\chi^{ee}\big{)}\,.\] (12) Clearly, the quality of the bounds on $C_{10}^{B,\ell\ell}$ depends on the precision with which $\chi^{\ell\ell}$ can be determined. The present situation is as follows: $\chi$ can be determined (up to a two-fold ambiguity) from the measured $K_{L}\to\ell^{+}\ell^{-}$ rates, with the resulting values shown in Table 2.³ Although Solution 2 for the $ee$ channel is easily excluded, the current data are not sufficiently precise to distinguish among the remaining Solutions. Moreover, Solution 1 for the $ee$ channel carries a large uncertainty which needs to be improved in order to test LFUV in the interesting regions of parameter space. [FOOTNOTE:3][ENDFOOTNOTE] Channel \| χ (Solution 1) \| χ (Solution 2) ---\|---\|--- ee \| 5.1+15.4−10.3 \| −(57.5+15.4−10.3) μμ \| 3.75±0.20 \| 1.52±0.20 Table 2: Values of the contact term χ(Mρ) extracted from the measured KL→e+e− and KL→μ+μ− rates. To gain an idea of the improvement in precision required, suppose the uncertainty in $\Gamma(K_{L}\to\ell^{+}\ell^{-})$ could be reduced by a factor of $10$ and that the central value remained unchanged. In this idealised scenario, Solution 2 for the $\mu\mu$ channel would be strongly disfavored, given that LFUV (if present at all) should manifest itself as a small effect. Substituting the resulting difference $\chi^{\mu\mu}-\chi^{ee}\sim 1.3\pm 1.3$ into (12) would then yield the bound $C_{10}^{B,\mu\mu}-C_{10}^{B,ee}\approx 3.5\pm 3.5$. We thus find that the improvement required to obtain competitive bounds on $C_{10}^{B}$ is of similar magnitude to what we found in the analysis of $C_{9}^{B}$ in $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$. We note that, contrary to the $K_{L}$ decay, short-distance contributions to the $K_{S}\to\ell^{+}\ell^{-}$ decay width do not interfere with any long-distance physics. In the expression for the decay width [11] \[\Gamma(K_{S}\to\ell^{+}\ell^{-})=\frac{m_{K}}{8\pi}\beta_{\ell}\left[\beta_{ \ell}^{2}\|B\|^{2}+\|C\|^{2}\right]\,,\] (13) short-distance effects contribute to the second term [19], which in the notation of [11] reads \[\text{Im}\,C_{\mathrm{SD}}=-2\sqrt{2}F_{K}m_{\ell}\,\text{Im}\,\{V_{ud}V_{us}^ {}C_{7A}\}=2\sqrt{2}F_{K}m_{\ell}\,y_{7A}\,\text{Im}\,\{V_{td}V_{ts}^{}\}\,.\] (14) Recently, LHCb have strengthened their 2012 bound [21] on the $K_{S}\to\mu^{+}\mu^{-}$ rate by roughly a factor of two: $\mathrm{BR}(K_{S}\to\mu^{+}\mu^{-})<6.9(5.8)\times 10^{-9}$ at the 95% (90%) confidence level [22]. With improvements expected in the near future, LHCb will start probing the theoretically interesting region $\mathrm{BR}(K_{S}\to\mu^{+}\mu^{-})\approx 10^{-11}$ [19] where meaningful bounds on NP contributions to the coefficient $y_{7A}$ can be derived. ## 3 Lepton flavour violating decays NA62 is projected to improve existing limits on LFV in the kaon sector by an order of magnitude or more [23]. Here we adopt a similar approach to our analysis of LFUV in Section 2, and use MFV to convert limits on the LFV Wilson coefficients $C_{7V,7A}^{\mu e}$ of the kaon sector into bounds on the corresponding $b\to s$ transitions. The analysis is simplified by the fact that LFV is absent in the SM⁴ so the decay amplitudes can be expressed directly in terms of $C_{7V,7A}^{\mu e}$ and quark operators based on the chiral realization (6). [FOOTNOTE:4][ENDFOOTNOTE] In the context of LFV searches at NA62, the mode of interest to us is $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$, whose branching fraction takes the form \[\text{Br}\big{[}K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}\big{]} =0.027\big{\{}\|C_{7V}^{\mu e}\|^{2}+\|C_{7A}^{\mu e}\|^{2}\big{\}}\,.\] (15) As a point of comparison, the same combination of Wilson coefficients enters in the $K_{L}\to\mu^{\pm}e^{\mp}$ decay, with branching fraction \[\text{Br}\big{[}K_{L}\to\mu^{\pm}e^{\mp}\big{]} =2.6\big{\{}\|C_{7V}^{\mu e}\|^{2}+\|C_{7A}^{\mu e}\|^{2}\big{\}}\,.\] (16) Based on (15) and (16), the present experimental limits [23] can be used to constrain the term $(\|C_{7V}^{\mu e}\|^{2}+\|C_{7A}^{\mu e}\|^{2})^{1/2}$. The resulting limits are given in the first line of Table 3, where the limit from $K_{L}\to\mu^{\pm}e^{\mp}$ decays is an order of magnitude more stringent than the one from $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$. Although the projected limits from NA62 [23] will also fall short by a factor of $4$, it would be interesting to examine whether removal of the GigaTracker could produce a sufficient increase in statistics to become competitive with the $K_{L}$ limits.⁵ [FOOTNOTE:5][ENDFOOTNOTE] As in the case of LFUV, we use MFV to obtain limits on the $B$-physics coefficients. These are shown in the bottom line of Table 3, where in the case of the $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$ decay, the resulting constraints are slightly better than (9), but of similar magnitude. The strongest constraint is obtained from the limit on $K_{L}\to\mu e$. \| KL→μ±e∓ \| K+→π+μ±e∓ \| K+→π+μ±e∓ (NA62 projection) ---\|---\|---\|--- (\|Cμe7V\|2+\|Cμe7A\|2)1/2 \| <1.3×10−6 \| <2.2×10−5 \| <5.1×10−6 (\|CB,μe9\|2+\|CB,μe10\|2)1/2 \| <0.71 \| <12 \| <2.7 Table 3: Limits on LFV Wilson coefficients from kaon decays. The last line shows the corresponding limits in the B-system assuming MFV, while the rightmost column refers to the projected limit from NA62 [23]. ## 4 Remarks and future prospects Rare kaon decays offer a potential probe into NP explanations of the flavour anomalies observed by LHCb in semi-leptonic $B$-decays. In the framework of MFV, we have discussed how limits on LFUV and LFV at kaon experiments can be converted into bounds on the Wilson coefficients $C_{9,10}^{B}$ of the $b\to s$ effective Hamiltonian. In this respect, we have focused on the $K^{+}$ decay modes we consider to be of most relevance to the NA62 programme (with $K_{L}$ modes included as a point of comparison): * $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ and $K_{L}\to\ell^{+}\ell^{-}$ as a means to constrain LFUV due to vector and axial-vector interactions. Although the LECs involved in these modes are poorly constrained from theory, they can be determined via precise experimental measurements. In particular, bounds on the short-distance NP effects can be obtained by considering _differences_ between the $ee$ and $\mu\mu$ parameters. For this method to obtain meaningful bounds on $C_{9,10}^{B}$, we found that order-of-magnitude reductions on the LECs’ current uncertainties are required. At NA62, this will require a measurement of the $K^{+}\to\pi^{+}\ell^{+}\ell^{-}$ spectrum which significantly reduces the uncertainties of the NA48/2 data. * the LFV modes $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$ and $K_{L}\to\mu^{\pm}e^{\mp}$. In these decays, the amplitude factorises into short- and long-distance components, so strong bounds can be obtained on the former. NA62’s projected limits on $K^{+}\to\pi^{+}\mu^{\pm}e^{\mp}$ will fall short of the existing E871 bound on $K_{L}\to\mu^{\pm}e^{\mp}$ [24], so it would be worthwhile considering if the experiment can be adapted to boost the sample of $K^{+}$ decays involving three charged particles. We note that in order to convert bounds in the kaon sector into those in $B$-physics, we have worked within the framework of MFV. In general, the potential NP may not satisfy MFV, so from this point of view there are three logical possibilities which can be tested at NA62: 1. if the NP explanations for the $B$-meson anomalies are consistent with MFV, then one should see a signal at the sensitivities discussed in Sections 2 and 3; 2. the experimental searches at a sensitivity expected from MFV turn out negative. In this case, one could immediately infer that any NP scenario explaining the $B$-anomalies would require departures from MFV; 3. a signal is observed at current or slightly improved sensitivity. 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1001.3254	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 16014, "num_imgs": 1, "llama3_tokens_count": 5015 }	[ "content_image/1001.3254/x1.png" ]	# Isospin breaking effects in the dynamical generation of the $X(3872)$ D. Gamermann J. Nieves E. Oset E. Ruiz Arriola ###### Abstract We have studied isospin breaking effects in the X(3872) resonance and found a natural explanation for the branching fraction of the X decaying to $J/\psi$ with two and three pions being close to unit. Within our framework the X(3872) is a dynamically generated resonance in coupled channels. We also study the relationship between the couplings of the resonance to the coupled channels with its wave function, which further helps us to understand the isospin structure of the resonance. Keywords:$X(3872)$, wave functions, isospin structure : 14.40.Pq 13.25.Gv ## 1 Introduction The $X(3872)$ resonance has been first observed by the Belle collaboration [1] and later confirmed by other four experiments the CDFII, D0 and BaBar collaborations [2; 3; 4]. It has first been seen in the decay channel $J/\psi\pi^{+}\pi^{-}$ and the analysis of the $\pi\pi$ spectrum indicates that this meson pair comes from a $\rho$ meson. There has been no observation of a charge partner for the $X(3872)$ state, which indicates an isoscalar nature for it. Later on the $X(3872)$ has been observed in another two decay channels [5]: The channel $J/\psi\gamma$ that fixes the C-parity of the state as positive and the channel $J/\psi\pi^{+}\pi^{-}\pi^{0}$ where the three pion spectrum shows a peak for an $\omega$ resonance. The measured branching fraction \[\frac{{\cal B}(X\to J/\psi\pi^{+}\pi^{-}\pi^{0})}{{\cal B }(X\to J/\psi\pi^{+}\pi^{-})} = 1.0\pm 0.4\pm 0.3\textrm{ ,}\] (1) raises a problem, since it seems to indicate a huge isospin violation in the decays of the $X(3872)$, since the case of $\rho$ production (two pions) has isospin 1 while the $\omega$ has isospin 0. A careful analysis of the data on this state has been done in [6], concluding that its quantum numbers must correspond to $J^{P}=1^{++}$ or $J^{P}=2^{-+}$. From the theoretical point of view the $X(3872)$ mass is very close to the $D^{0}\bar{D}^{0}$ threshold and it has been suggested [7; 8; 9; 10] that this state is a $D\bar{D}^{}$ molecule. There has been though some discussion on whether the $D^{+}D^{-}$ component is important in the structure of the state, since this component is bound by about 8 MeV, while the neutral component is bound by less than 1 MeV [11; 12]. In this work we explain how the branching fraction of Eq. (1) is naturally explained for a dominant isospin 0 nature of the $X(3872)$. We also stress the importance of taking into account the charged $D\bar{D}^{}$ components in the structure of the $X$ even though the neutral wave function extends much further in space than the charged one. This is a consequence of the fact that only the values of the wave function around the origin are important in the description of short range processes as the decays to $J/\psi$ with two or three pions. ## 2 The Decays of the $X(3872)$ into two and three Pions We follow here the approach of [12] where the $X(3872)$ is generated dynamically within coupled channels. As noted in [10; 12] only the $D^{+}D^{-}$ and $D^{0}\bar{D}^{0}$ components of the state provide sizeable couplings and therefore we consider only these two channels in the approach. The model is based on the calculation of the unitarized transition matrix (T-matrix). For the tree level interaction of the mesons, Lagrangians based on the $SU(4)$ extensions of the hidden gauge Lagrangians are used. The $SU(4)$ symmetry is broken in order to take into account the bigger mass of the charmed and hidden charmed mesons. The tree level amplitudes are used as kernel to solve a scattering equation that in an on-shell approach takes the form \[T = (1-VG)^{-1}V\] (2) where $T$ is the T-matrix, $V$ is a matrix containing the tree level amplitudes projected in s-wave and $G$ is a diagonal matrix with each diagonal element containing the two particle propagators for each channel. The loop functions in $G$ are divergent and can be regularized using dimensional regularization, which gives a subtraction constant $\alpha$ as a free parameter that can be used to adjust the pole generated in the T-matrix of Eq. (2) to its right position, fitting in this way the $X(3872)$. Close to the pole the T-matrix can be written in the separable form \[T_{ij} = \frac{g_{i}g_{j}}{s-s_{pole}}\] (3) where $s$ is the invariant mass squared, $s_{pole}$ is the pole position and $g_{k}$ are the couplings to each channel. Using Eq. (3) one can calculate the couplings of the resonance to each channel by calculating the residues of the T-matrix at the pole position. By using physical masses for the charged and neutral components of the $D$-meson doublet members, one finds a difference of at most 1.5% in the couplings of the $X$ to the charged and neutral channels. This is a tiny difference which we will neglect in what follows. In order to calculate the ratio of the decays into $J/\psi\rho$ and $J/\psi\omega$ one has to take into account the coupling first of the $X$ to a $D^{0}\bar{D}^{0}$ or $D^{+}D^{-}$ and then the coupling of the $D\bar{D}^{}$ components to $J/\psi\rho$ or $J/\psi\omega$. In the hidden gauge formalism this former coupling is equal for both cases. What differentiates the situation with the $\rho$ meson from the situation with the $\omega$ mesons is that in the case of the $\rho$ which has isospin 1 the loops with neutral and charged $D\bar{D}^{}$ mesons interfere destructively while in the case of the $\omega$ with isospin 0 the interference is constructive. As a result the ratio of the coupling squared of the $X$ to $\rho$ and $\omega$ is given by \[R_{\rho/\omega} = \frac{G_{11}-G_{22}}{G_{11}+G_{22}}=0.032\] (4) The value 0.032 was obtained in the limit of 0 binding energy with the loops calculated with dimensional regularization. In order to calculate the full ratio of the branching fractions one still has to take into account the phase space available for the $\rho$ to decay into two pions and the $\omega$ into three pions. For that we integrate the spectral function of each meson taking into account the much bigger width of the $\rho$ and the kinematical constrains. The result obtained is \[\frac{{\cal B}(X\to J/\psi\pi^{+}\pi^{-}\pi^{0})}{{\cal B }(X\to J/\psi\pi^{+}\pi^{-})} = 1.4\] (5) which is compatible with the value $1.0\pm 0.4$ from experiment [5]. One sees that although the couplings of the $X$ to the isospin 1 state $J/\psi\rho$ is suppressed in relation to the isospin 0 state $J/\psi\omega$ by a factor 30, the much bigger phase space for the $\rho$ to decay than for the $\omega$, compensates this suppression and brings the branching fraction to a value close to the observed experimentally. ## 3 the Wave Functions of the $X(3872)$ To calculate the wave functions of the $X$ resonance we start from a phenomenological potential that in momentum space reads: \[\langle\vec{p}^{\;\prime}\|V\|\vec{p}\,\rangle\equiv V(\vec{p}^{\;\prime},\vec{p}\,)= v\,\Theta(\Lambda-p)\Theta(\Lambda-p^{\prime})\] \[v =\] (6) where the cut off $\Lambda$ is fixed in order to find the resonance with the appropriate mass. To do that we write the T-matrix and the Lippmann-Schwinger equation: \[\langle\vec{p}\,\|T\|\vec{p}^{\;\prime}\rangle \equiv T(\vec{p}^{\;\prime},\vec{p}\,)=t\,\Theta(\Lambda-p)\Theta( \Lambda-p^{\prime})\] (7) \[t = (1-vG)^{-1}v\] (8) the coupled channels loop function $G$ here is a diagonal matrix given by: \[G =\] (9) The poles of the T-matrix are given by the equation det$(1-vG)=0$. The value of Lambda is chosen so that this equation is satisfied for the loop calculated at the energy where we want to have the $X$ state bound. We write then the Schorödinger equation for the potential: \[(H_{0}+V)\|\psi\rangle = E_{\alpha}\|\psi\rangle\] (10) where $H_{0}$ is the free Hamiltonian and $E_{\alpha}$ is the energy of the resonance. Projecting this equation in momentum space one obtains a set of two coupled equation for the wave function in each channel: \[\langle\vec{p}\,\|\psi_{1}\rangle = {\hat{v}}\frac{\Theta(\Lambda-p)}{E_{\alpha}-M_{1}-\frac{\vec{p}^ {\,2}}{2\mu_{1}}}\int_{k<\Lambda}d^{\hskip 0.284528pt3}\hskip-1.422638ptk\, \left(\langle\vec{k}\|\psi_{1}\rangle+\langle\vec{k}\|\psi_{2}\rangle\right)\] (11) \[\langle\vec{p}\,\|\psi_{2}\rangle = {\hat{v}}\frac{\Theta(\Lambda-p)}{E_{\alpha}-M_{2}-\frac{\vec{p}^ {\,2}}{2\mu_{2}}}\int_{k<\Lambda}d^{\hskip 0.284528pt3}\hskip-1.422638ptk\, \left(\langle\vec{k}\|\psi_{1}\rangle+\langle\vec{k}\|\psi_{2}\rangle\right)\] (12) The two integrals in the right hand side of the equations are constants that can be calculated by normalizing the wave function. The couplings of the state to the channels can again be calculated from the residues of the T-matrix: \[g_{1}^{2}=g_{2}^{2}\equiv g^{2} = \lim_{E\to E_{\alpha}}(E-E_{\alpha})t_{ij}=-\left.\left( \frac{dG_{11}}{dE}+\frac{dG_{22}}{dE}\right)^{-1}\right\|_{E=E_{\alpha}}\] (13) In the limit when $E_{B1}^{\alpha}\to 0$, we have $\left.\frac{dG_{11}}{dE}\right\|_{E=E_{\alpha}}\rightarrow\infty$ and we find \[g_{1}^{2}=g_{2}^{2} \sim \frac{\gamma_{1}}{4\pi^{2}\mu_{1}^{2}},\qquad E_{B1}^{\alpha}\to 0\] (14) where \[\gamma_{i} = \sqrt{2\mu_{i}E_{Bi}^{\alpha}}\] (15) \[E_{Bi}^{\alpha} = M_{i}-E_{\alpha}.\] (16) and the $M_{i}$ is the value of the threshold for channel $i$. As shown in [13] working out Eq. (11), Eq. (12) and Eq. (13) one obtains (17) (18) and we can recognize that the integrals in these equations are the Fourier transform of the momentum wave function for $\vec{r}=0$. So we can rewrite Eq. (17) and Eq. (18) as \[gG^{\alpha}_{11}= (2\pi)^{3/2}\psi_{1}(\vec{0}\,)= \hat{\psi_{1}}\] (19) \[gG^{\alpha}_{22}= (2\pi)^{3/2}\psi_{2}(\vec{0}\,)= \hat{\psi_{2}}\] (20) These equations show that the couplings of the the resonance to its channels are proportional to the value of the wave function at the origin (denoted as $\hat{\psi}$). Had we chosen another kind of form factor for regularize the potential in Eq. (6) we would arrive at the same equations, but with a slightly different definition of $\hat{\psi}$: \[{\hat{\psi}}_{i} = \int d^{\hskip 0.284528pt3}\hskip-1.422638ptkf(\vec{k})\langle \vec{k}\|\psi_{i}\rangle\] (21) where $f(\vec{k})$ is a form factor substituting the $\Theta(\Lambda-k)$ in Eq. (6). This new form factor has also a scale $\Lambda$ that in coordinate space represents a range about $1/\Lambda$. Since a typical cut off of 700 MeV represents in coordinate space a range of about 0.3 fm, $\hat{\psi}$ represents here a smeared value of the wave function around the origin. In Table 1 we show results for potentials with three different form factors. The wave functions for each kind of potential are slightly different but the values of $\hat{\psi}_{i}$ and the ratio of these values do not change much, since these are the quantities that carry the information about the short range physics. Form \| Λ \| gFT \| ψ1(→0)/ψ2(→0) \| ^ψ1/^ψ2 \| ^ψ1 \| ^ψ2 \| Rρ/ω ---\|---\|---\|---\|---\|---\|---\|--- Factor \| [MeV] \| [MeV] \| \| \| \| \| Sharp \| 653 \| 3202 \| 1.31 \| 1.31 \| 3.29 \| 2.50 \| 0.018 Gauss \| 731 \| 3238 \| 1.20 \| 1.29 \| 3.30 \| 2.56 \| 0.016 Lorentz \| 834 \| 3254 \| 1.17 \| 1.28 \| 3.31 \| 2.58 \| 0.015 Table 1: Comparative results for different potentials for a D0¯D∗0 binding energy of 0.1 MeV. In Figure 1 we show plots of the wave functions and probability densities for the sharp cut off potential and plots comparing the wave functions for the three different kinds of form factors. <figure><img src="content_image/1001.3254/x1.png"><figcaption>Figure 1: Neutral and charged Wave function components for a D0¯D∗0 bindingenergy of 0.1 MeV. In the upper left panel one can see the value of the wavefunctions at the origin for both channels, in the lower left panel we plot theprobability density for each channel. In the right panel we show plots of thewave functions for each channel calculated with the three different formfactors in the potential.</figcaption></figure> With regard to the isospin structure of the $X(3872)$ we can first write its wave function in terms of the charge basis and then transform it to isospin basis: \[\|X\rangle = \psi_{1}(\vec{r})\|D^{0}\bar{D}^{0}\rangle+\psi_{2}(\vec{r})\|D^{+ }D^{-}\rangle\] (22) \[\|X\rangle = \frac{\psi_{1}(\vec{r})+\psi_{2}(\vec{r})}{\sqrt{2}}\|D\bar{D}^{} _{I=0}\rangle+\frac{\psi_{1}(\vec{r})-\psi_{2}(\vec{r})}{\sqrt{2}}\|D\bar{D}^{ }_{I=1}\rangle\] (23) We see from Eq. (23) that the mixing between the two possible isospin states depends on the relative distance of the two D mesons, $\vec{r}$. But in physical processes this mixing will be given by the values of the wave function around the origin. Using the $\hat{\psi}_{i}$ values given in Table 1 one sees that for short range processes the contribution of the isospin 1 state is about 2% and the $X$ can be regarded as an almost pure isospin 0 state. ## 4 Conclusions We have studied the isospin structure of the $X(3872)$. In our model this state is generated dynamically in coupled channels and is interpreted as an s-wave bound state of $D\bar{D}^{}$. The apparent isospin violation in the branching fraction ratio of $X\to J/\psi\pi^{+}\pi^{-}$ and $X\to J/\psi\pi^{+}\pi^{-}\pi^{0}$ is naturally explained due to the much larger phase space for the $\rho$ to decay into $\pi^{+}\pi^{-}$ than the $\omega$ to decay into $\pi^{+}\pi^{-}\pi^{0}$. The mass of the $X(3872)$ is much closer to the $D^{0}\bar{D}^{0}$ threshold than to the $D^{+}D^{-}$ which is bound by around 8 MeV. As a consequence the wave function of the $X$ in the $D^{0}\bar{D}^{0}$ channel extends much further on space than in the $D^{+}D^{-}$ channel. Nevertheless physical short range processes are dominated by the couplings of the state to each channel and these couplings have a negligible isospin violation. In terms of wave functions this can be understood since the couplings reflect an averaged value of the wave function around the origin and not in the whole space. This work is partly supported by DGI and FEDER funds, under contract FIS2006-03438, FIS2008-01143/FIS and PIE-CSIC 200850I238 and the Junta de Andalucia grant no. FQM225-05. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity ”Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of EU. Work supported in part by DFG (SFB/TR 16, ”Subnuclear Structure of Matter”). ## References (1) S. K. Choi _et al._ [Belle Collaboration], Phys. Rev. Lett. 91, 262001 (2003) [arXiv:hep-ex/0309032]. * (2) D. E. Acosta _et al._ [CDF II Collaboration], Phys. Rev. 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1706.04292	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 15277, "num_imgs": 6, "llama3_tokens_count": 3785 }	[ "content_image/1706.04292/x1.png", "content_image/1706.04292/x2.png", "content_image/1706.04292/x3.png", "content_image/1706.04292/x4.png", "content_image/1706.04292/x5.png", "content_image/1706.04292/x7.png" ]	# Interference Mitigation with a Modified ASKAP Phased Array Feed on the $64\,\mathrm{m}$ Parkes Radio Telescope A.P. Chippendale CSIRO Astronomy and Space Science, PO Box 76, Epping, 1710, Australia, e-mail: Aaron.Chippendale@csiro.au G. Hellbourg Department of Astronomy, University of California, 339 Campbell Hall, Berkeley, CA 94710, USA ###### Abstract We present results from a first attempt to mitigate radio frequency interference in real-time during astronomical measurements with a phased array feed on the $64\,\mathrm{m}$ Parkes radio telescope. Suppression of up to $20\,\mathrm{dB}$ was achieved despite errors in estimating the interference spatial signature. Best results were achieved in the clean excision of a narrowband and stationary clock signal that originates from the receiver’s digital back-end system. We also contribute a method to interpolate valid beamformer weights at interference-affected channels. Correct initial beam weights are required to avoid suppressing the desired signal. ## 1 An Askap Paf at Parkes A phased array feed (PAF) is a dense array of antenna elements at the focus of a concentrator that, with digital beamforming, can produce multiple simultaneous antenna beams of high sensitivity throughout a wide field of view. We trialled interference mitigation with a Mk. II PAF system [1] built by CSIRO for the Australian Square Kilometre Array Pathfinder (ASKAP) telescope and modified by CSIRO and the Max Planck Institute for Radio Astronomy (MPIfR) for deployment on the Effelsberg $100\,\mathrm{m}$ telescope. Our experiments took place during a scientific commissioning phase on the $64\,\mathrm{m}$ Parkes telescope [2]. The ASKAP PAF [1] is a connected-element “chequerboard” array [3] operating over $0.7$–$1.8$$\,\mathrm{GHz}$. Modifications for MPIfR involved the addition of selective filters to reject broad-band radio frequency interference (RFI) [2]. The experiment described here has however used an unmodified ASKAP filter covering $0.7$–$1.2$$\,\mathrm{GHz}$. The digital beamformer can simultaneously form 36 dual-polarisation beams, each of $384\,\mathrm{MHz}$ bandwidth, from the PAF’s 188 antenna elements. The beamformer also calculates array covariance matrices (ACMs) with $1\,\mathrm{MHz}$ resolution and integrated fine filter bank (FFB) spectra of the beamformer outputs with $18.5\,\mathrm{k}\mathrm{H}\mathrm{z}$ resolution. <figure><img src="content_image/1706.04292/x1.png"><figcaption>Figure 1: Median spectrum at Parkes with and without mitigation applied to aboresight maxSNR beam.</figcaption></figure> Parkes has more challenging RFI than the radio-quiet site that the ASKAP PAF was designed for [4]. Figure 1 shows median beamformed spectra, with and without mitigation, for maximum signal-to-noise ratio (maxSNR [5]) weights over a $13\,\mathrm{min}$ observation with $18.5\,\mathrm{kHz}$ resolution. Table 1 lists dominant signals in this data. RFI scans over the $0.7$–$1.8$$\,\mathrm{GHz}$ range of the PAF, before beamforming, are given in [2]. The broader RFI status at Parkes is summarised on the observatory’s website¹. [FOOTNOTE:1][ENDFOOTNOTE] Frequency \| Bandwidth \| Service \| nopt \| Comment \| Label ---\|---\|---\|---\|---\|--- (MHz) \| \| \| \| \| 773.000 \| 30MHz \| 4G mobile telecom \| >10 \| Tower transmitters (FDD-LTE) \| A 880.000 \| 20kHz \| 4G mobile telecom \| >10 \| Tower transmitters \| B 947.500 \| 12.5MHz \| 2/3/4G mobile telecom \| >10 \| Tower transmitters (LTE/GSM) \| C 976.593 \| <18.5kHz \| Clock (CSIRO) \| 1 \| Seen at 976.609MHz \| D 1018.000 \| 500kHz \| Aeronautical radiolocation \| 4 \| Parkes VOR/DME transponder \| E \| \| \| \| \| Table 1: Dominant RFI Identification ## 2 Spatial Mitigation With Pafs Blind spatial filtering is proposed to perform RFI mitigation. The implemented approach involves (1) estimating the interfering signal subspace from the ACM, and (2) updating the beamformer weights to create spatial nulls towards the RFI. The viability of this technique has been demonstrated via simulation for full ASKAP [6] and by an experiment on ASKAP’s six-antenna Boolardy Engineering Test Array (BETA) [7]. This experiment successfully imaged a celestial source in a $1\,\mathrm{MHz}$ channel that was otherwise saturated by interference from GPS L2 over 200 times stronger than the desired signal. Here we trial spatial filtering across the full $384\,\mathrm{MHz}$ system bandwidth and evaluate its impact on $18.5\,\mathrm{kHz}$ resolution spectra. We explore the efficacy of the technique on a wider variety of RFI signals and measure its impact on spectral-line observations of a celestial source. ### Experimental method at Parkes We used FFB spectra to assess the performance of various mitigation algorithms and their parameters. Each beam was initiated with maxSNR boresight beam weights [8] with different real-time RFI mitigation algorithms and parameters. The first beam remained unprocessed for reference. Both orthogonal and oblique projectors with various RFI subspace dimension were trialled as in [7]. The subspace dimension is closely related to the number of RFI sources, their bandwidths, and their relative motion to the telescope [9]. ACMs were continuously downloaded to estimate the RFI subspace and compute the spatial filter, before updating beam weights. The system completes an ACM download and beam weights upload every $15\,\mathrm{s}$. Eight cycles of ACM downloads and weights uploads were required to apply mitigation to the full $384\,\mathrm{MHz}$ bandwidth because the ACM calculation operates on $48\,\mathrm{MHz}$ at a time. This resulted in a $120\,\mathrm{s}$ update period for any given $1\,\mathrm{MHz}$ channel. This slow rate, together with the five-fold reduction in beam feature size for the $64\,\mathrm{m}$ Parkes antenna with respect to the $12\,\mathrm{m}$ ASKAP antennas, affects the attenuation of RFI sources moving relatively to the telescope. The update rate could, however, be increased significantly by refining software. The user-defined _on-the-sky_ ACM integration time was $0.5\,\mathrm{s}$, resulting in a $2\,\mathrm{s}$ accumulation period as the ACM calculation in firmware uses every fourth sample. The $18.5\,\mathrm{kHz}$ FFB spectra were integrated for $2\,\mathrm{s}$ and were downloaded every $4.5\,\mathrm{s}$. The experiments ran over two $384\,\mathrm{MHz}$ bands centred at $891.5\,\mathrm{MHz}$ and $1340.5\,\mathrm{MHz}$. At each band, we made measurements both with the antenna stationary and tracking a celestial source at the sidereal rate of $0.25\,\mathrm{\SIUnitSymbolDegree}/\mathrm{min}$. We tracked flux calibrator PKS B$0407$$-658$[10] and a reference position offset by $+10\,\mathrm{\SIUnitSymbolArcminute}$ in right ascension. Only results of the band centred at $891.5\,\mathrm{MHz}$ are presented as they contain more fixed terrestrial sources of RFI. ## 3 Results ### Suppression and subspace dimension <figure><img src="content_image/1706.04292/x2.png"><figcaption>Figure 2: Median suppression achieved over 4min by oblique projection versusRFI subspace dimension with stationary antenna. Labels match signals to Table1.</figcaption></figure> Figure 2 shows RFI attenuation as a function of the dimension $n$ of the projected-out RFI subspace. This plots the ratio in dB of the total power, in each signal sub-band of Table 1, between a beam with no mitigation and a beam with oblique projection mitigation. Only data from when the antenna was stationary were used. The distance measuring equipment (DME) signal (E) at $1018\,\mathrm{MHz}$ achieves peak suppression at $n=4$ and the narrowband clock signal (D) does so at $n=1$. Suppression of the mobile telecommunication base-station signals (A-C) continues to rise at $n=10$. <figure><img src="content_image/1706.04292/x3.png"><figcaption>Figure 3: Suppression versus telescope motion for 4G mobile telecom (FDD-LTE)base-station signals.</figcaption></figure> Figure 3 shows the change in RFI suppression between the tracking and stationary telescope. The level of suppression is more consistent when the telescope is stationary. Error in estimating the RFI subspace also limits suppression. Performance can be improved by using reference antennas tracking RFI sources to more accurately estimate their subspace [11], reducing ACM integration time to reduce subspace smearing [9] caused by relative motion between RFI sources and the telescope, and by increasing beam weight update rates so that the mitigation is still valid when applied. ### Clean excision of a narrowband signal Expecting the current implementation to work best on stationary signals, we explored performance on a tone generated by the ASKAP digital receiver that is narrower than an $18.5\,\mathrm{kHz}$ FFB channel. The source is a $256\,\mathrm{MHz}$ FPGA clock that is multiplied by 32/27 to read out the digital receiver’s $1\,\mathrm{MHz}$ resolution oversampled coarse filter bank [12]. <figure><img src="content_image/1706.04292/x4.png"><figcaption>Figure 4: Clean removal of coarse filter bank read-out clock with negligiblesystem temperature increase.</figcaption></figure> Figure 4 shows that this signal, which appears at $976.6\,\mathrm{MHz}$ due to direct sampling in the second Nyquist zone, is neatly mitigated to the noise floor by both orthogonal and oblique projection with ${n=1}$. This plot shows the beam equivalent system-temperature-on-efficiency $T_{\text{sys}}/\eta$ measured on PKS B$0407$$-658$ assuming a flux model² fit to data in the Parkes catalogue [10]. Both mitigation algorithms change system temperature by less than $1\,\mathrm{K}$ with respect to the unmitigated spectrum. The typical system temperature here is higher than reported in [2], likely because we initialised the system with 35 day-old beam weights made at different azimuth and elevation and without adjustment for changes in system status since the weights were last calibrated. [FOOTNOTE:2][ENDFOOTNOTE] ### Interpolating correct initial weights <figure><img src="content_image/1706.04292/x5.png"><figcaption>(a) Raw.</figcaption></figure> <figure><img src="content_image/1706.04292/x7.png"><figcaption>(a) Raw.</figcaption></figure> Correct initial beam weights, that maximise the desired signal, are required for algorithms that suppress RFI and retain the desired signal. Figures 4(a) and 5(a) show that the initial maxSNR beam weights are corrupted at some frequencies by RFI. Here all weights were normalised by the weights of the port with the highest typical weight amplitude³. [FOOTNOTE:3][ENDFOOTNOTE] Figures 4(b) and 5(b) show that we can recover the weights at interference-affected channels. Amplitude is interpolated independently for each port by iteratively fitting a polynomial to its weight amplitude as a function of frequency and removing outliers that are likely to be interference-affected channels at each iteration. Before fitting a polynomial to the weight phase of a given port, we first removed a bulk linear phase (delay) estimated by taking the Fourier transform of the normalised weights with respect to frequency. We then unwrapped the phase and iteratively fit a polynomial to interference-free channels as described above. In Figure 5(b), weights corrupted by RFI are replaced with an evaluation of the fitted polynomials plus the bulk linear phase. ## 4 Conclusion We have suppressed RFI by up to $20\,\mathrm{dB}$ in real-time observations with a PAF on the Parkes telescope. Suppression could be improved by reducing the integration time and increasing the download rate of ACMs to account for dynamic RFI environments, and by correlating PAF ports against reference antennas directed at RFI sources to enhance subspace estimation. Within current limitations, best performance was achieved on a narrowband and stationary clock signal originating from the PAF back-end. This was cleanly mitigated to the noise floor with no reduction in system sensitivity. Finally, we demonstrated successful interpolation of beam weights at interference-affected channels. This is immediately useful for general PAF beamforming in addition to being necessary for mitigation algorithms that robustly preserve the desired signal. ## Acknowledgments The Parkes radio telescope is part of the ATNF, which is funded by the Commonwealth of Australia for operation as a National Facility managed by the CSIRO. MPIfR financed the PAF used in this paper and its modification for a less radio-quiet site. Dr. K. Bannister and C. Haskins supported software implementation. A. Brown improved the duty cycle of the FFB integrator. Dr. B. Indermühle supported signal identification. Dr. J. Tuthill advised on the digital receiver read-out clock. Prof. L. Staveley-Smith motivated this work. ## References * [1] G. Hampson _et al._, “ASKAP PAF ADE–advancing an L-band PAF design towards SKA,” in _2012 International Conference on Electromagnetics in Advanced Applications_, Sept 2012, pp. 807–809. * [2] A. P. Chippendale _et al._, “Testing a modified ASKAP Mark II phased array feed on the 64 m Parkes radio telescope,” in _2016 International Conference on Electromagnetics in Advanced Applications (ICEAA)_, Sept 2016, pp. 909–912. * [3] S. G. Hay and J. D. O’Sullivan, “Analysis of common-mode effects in a dual-polarized planar connected-array antenna,” _Radio Science_, vol. 43, no. 6, 2008, rS6S04. * [4] B. T. Indermüehle _et al._, “The ASKAP RFI environment as seen through BETA,” in _2016 Radio Frequency Interference (RFI)_, Oct 2016, pp. 43–48. * [5] B. D. V. Veen and K. M. Buckley, “Beamforming: a versatile approach to spatial filtering,” _IEEE ASSP Magazine_, vol. 5, no. 2, pp. 4–24, April 1988. * [6] R. A. Black _et al._, “Multi-tier interference-cancelling array processing for the ASKAP radio telescope,” in _2015 IEEE Signal Processing and Signal Processing Education Workshop (SP/SPE)_, Aug 2015, pp. 261–266. * [7] G. Hellbourg, K. Bannister, and A. Hotan, “Spatial filtering experiment with the ASKAP BETA array,” in _2016 Radio Frequency Interference (RFI)_, Oct 2016, pp. 37–42. * [8] D. McConnell _et al._, “The Australian Square Kilometre Array Pathfinder: Performance of the Boolardy Engineering Test Array,” _Proc. Astron. Soc. Aust._, vol. 33, p. e042, Sep. 2016. * [9] G. Hellbourg, “Subspace smearing and interference mitigation with array radio telescopes,” in _2015 IEEE Signal Processing and Signal Processing Education Workshop (SP/SPE)_, Aug 2015, pp. 278–282. * [10] R. E. Otrupcek and A. E. Wright (Eds). (1990) Parkes catalogue. Australia Telescope National Facility. [Online]. Available: https://www.parkes.atnf.csiro.au/observing/databases/pkscat90.html * [11] G. Hellbourg _et al._, “Reference antenna-based subspace tracking for RFI mitigation in radio astronomy,” in _2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP)_, Dec 2014, pp. 1286–1290. * [12] J. Tuthill _et al._, “Compensating for oversampling effects in polyphase channelizers: A radio astronomy application,” in _2015 IEEE Signal Processing and Signal Processing Education Workshop (SP/SPE)_, Aug 2015, pp. 255–260.
1012.2397	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 25751, "num_imgs": 4, "llama3_tokens_count": 7163 }	[ "content_image/1012.2397/x1.png", "content_image/1012.2397/x2.png", "content_image/1012.2397/x3.png", "content_image/1012.2397/x4.png" ]	# Spin-orbit driven ferromagnetic resonance: A nanoscale magnetic characterisation technique D. Fang Microelectronics Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK H. Kurebayashi Microelectronics Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK J. Wunderlich Hitachi Cambridge Laboratory, Cambridge CB3 0HE, UK Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 162 53 Praha 6, Czech Republic K. Výborný Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 162 53 Praha 6, Czech Republic L. P. Z${\rm\hat{a}}$rbo Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 162 53 Praha 6, Czech Republic R. P. Campion School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK A. Casiraghi School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK B. L. Gallagher School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK T. Jungwirth Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 162 53 Praha 6, Czech Republic School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK A. J. Ferguson ajf1006@cam.ac.uk Microelectronics Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK ###### Abstract We demonstrate a scalable new ferromagnetic resonance (FMR) technique based on the spin-orbit interaction. An alternating current drives FMR in uniform ferromagnetic structures patterned from the dilute magnetic semiconductors (Ga,Mn)As and (Ga,Mn)(As,P). This allows the direct measurement of magnetic anisotropy coefficients and damping parameters for individual nano-bars. By analysing the ferromagnetic resonance lineshape, we perform vector magnetometry on the current-induced driving field, observing contributions with symmetries of both the Dresselhaus and Rashba spin-orbit interactions. Ferromagnetic resonance (FMR) is the most common technique for exploring spin-dynamics phenomena and for the magnetic characterisation of ferromagnets.[1] However, previously developed FMR techniques, based on exciting the magnetic system by an external alternating magnetic field from a resonant cavity[2; 3; 4] or a micro-waveguide,[5; 6; 7; 8] struggle to simultaneously achieve scalability of the technique to nano-size objects, uniformity of the excitation field, and the range of available excitation frequencies. We introduce an FMR technique applicable to individual nanomagnets in which the FMR driving field is generated in the probed magnet itself. The excitation is driven by the effective field generated by an alternating electrical current passing through the ferromagnet, which results from the combined effect of the spin-orbit (SO) coupling and exchange interaction. [9; 10; 11] Our SO-FMR can be operated at tuneable frequencies and we demonstrate its sensitivity and scalability by measuring the variation of micromagnetic parameters of lithographically patterned (Ga,Mn)As and (Ga,Mn)(As,P) nano-bars. FMR induced by driving an alternating current directly through the probed sample has been previously demonstrated for specific non-uniform magnetic nano-devices such as spin-valves.[12; 13] The experiments utilised the spin-transfer torque in which spin-polarised electrical current acts on spatially varying magnetisation[14] and can be viewed as a macroscopic angular momentum transfer effect. Our SO-FMR (Figure 1a) does not require the specific samples with a non-collinear magnetisation profile. The method can be applied to a broad range of systems including uniformly polarised nanomagnets. This is because the effective field utilised in the SO-FMR does not rely on the spatial variation of the magnetisation vector but on a microscopic non-collinearity of individual electron spins due to their relativistic, SO-coupled band structure. Specifically, when an electrical current traverses through the uniformly magnetised material, the resulting non-equilibrium distribution of occupied states in the SO-coupled carrier bands yields a non-equilibrium spin polarisation.[15; 16; 17] The polarisation produces a transverse component of the internal exchange field (can be viewed as an effective magnetic field) and a torque is applied to the magnetisation vector.[9; 18] This current-induced effective field is generic to ferromagnets with SO-coupling and inversion asymmetry in their band structure. Previously it has been utilised for magnetisation switching in the ferromagnetic semiconductor (Ga,Mn)As[10] and for domain nucleation in a Pt/Co/AlO${}_{x}$ stack.[11] The micro and nano-bars employed in our SO-FMR study are patterned by electron beam lithography on 25 nm-thick films of (Ga${}_{0.94}$,Mn${}_{0.06}$)As and (Ga${}_{0.94}$,Mn${}_{0.06}$)(As${}_{0.9}$,P${}_{0.1}$), grown by low-temperature molecular beam epitaxy. The (III,Mn)V ferromagnetic semiconductors used in our study are particularly favourable systems for observing and exploring SO-FMR because of the compatibility of the material with advanced semiconductor nanofabrication techniques, because the carrier bands have strong SO-coupling, and the (III,Mn)V nanostructures have a rich phenomenology in their micromagnetic parameters. In the following text we demonstrate our scalable SO-FMR technique in lithographically patterned bars of width ranging from several $\mu$m’s to 80 nm (Figure 1b). In order to drive SO-FMR we pass a microwave-frequency current through the nano-bar. This is achieved by wire-bonding the sample between an open-circuit coplanar transmission line and a low-frequency connection which also provides a microwave ground (Figure 1c). Since the microwave excitation field originates from the material properties, only a 2-terminal nano-bar (a resistor) is required in our experiment, enabling simple and rapid sample fabrication. For detection of FMR we utilise a frequency mixing effect based on the anisotropic magnetoresistance (AMR).[5; 6; 2; 3; 4; 7] When the magnetisation precession is driven, a time-dependent change $\Delta R(t)$ in the longitudinal resistance from the equilibrium value $R$ occurs (due to the AMR). The resistance oscillates with the same frequency as the microwave current, therefore causing frequency mixing and a directly measurable dc voltage $V_{\text{dc}}$ is generated across the nano-bar. This voltage is our observable providing a direct probe of the amplitude and phase of the magnetisation precession with respect to the microwave current. We first show measurements on a 80 nm-wide nano-bar patterned in the [11̄0] direction from the (Ga,Mn)(As,P) epilayer. The magnetic field dependence of $V_{\text{dc}}$ is measured at different microwave frequencies and taken at a temperature of 6 K. The frequency of the incident current is fixed while an external dc magnetic field $\mathbf{H}_{0}$ is swept and a well-defined resonance peak appears (Figure 2a). The peak is well-fitted by the solution of the Landau-Lifshitz-Gilbert (LLG) equation, which describes the dynamics of precessional motion of the magnetisation: \[V_{\text{dc}}=V_{\text{sym}}\frac{\Delta H^{2}}{(H_{0}-H_{\text{res}})^{2}+ \Delta H^{2}}+V_{\text{asy}}\frac{\Delta H(H_{0}-\Delta H)}{(H_{0}-H_{\text{ res}})^{2}+\Delta H^{2}}\] (1) Here $H_{\text{res}}$ is the field at which resonance occurs and $\Delta H$ is the linewidth (half width at half maximum) of the FMR peak. The resonance lineshape is a combination of symmetric and anti-symmetric Lorentzian functions with amplitudes $V_{\text{sym}}$ and $V_{\text{asy}}$, respectively. Their relative contributions are determined by the phase of the driving field with respect to the current, and the direction of the driving field (see Equation 3 & 4). Figure 2b plots the frequency-dependence of the resonance field $H_{\text{res}}$. It is described by the equation for ferromagnetic resonance:[19] \[\left(\frac{\omega}{\gamma}\right)^{2}=\mu_{0}^{2}(H_{\text{res}}+H_{\text{ani }}^{{}^{\prime}})(H_{\text{res}}+H_{\text{ani}}^{{}^{\prime\prime}})\] (2) where $H_{\text{ani}}^{{}^{\prime}}$ and $H_{\text{ani}}^{{}^{\prime\prime}}$ are terms containing the demagnetisation and anisotropy energies of the ferromagnet (see Methods). A gyromagnetic constant $\gamma$ characteristic for Mn${}^{2+}$ spins of 176 GHz/T (g-factor 2) is used for the fitting. This, together with the good agreement between the observed peaks and the fitted results from the LLG equation, confirms that we observe the coherent precession of Mn spins. The FMR linewidth ($\Delta H=\Delta H_{\text{inhomo}}+\alpha\omega/\gamma$) describes the damping in the ferromagnetic system. The broadband nature of our setup allows us to determine the inhomogeneous (2.5 mT) and frequency-dependent contributions to the damping (Figure 2c) that correspond to Gilbert-damping constant $\alpha=$ 0.023. Using a vector field cryostat we also perform the SO-FMR measurements for different orientations of the external magnetic field. In Figure 2d we present the data from an in-plane scan of the magnetic field showing that there is a strong uniaxial anisotropy perpendicular to the bar direction. By analysing the peak positions (Figure 2e) using Equation 2 we quantify the anisotropy fields and find $\mu_{0}H_{2\parallel}=-180$ mT (uniaxial) and $\mu_{0}H_{4\parallel}=68$ mT (biaxial). We now demonstrate that SO-FMR can be applied to comparative investigations of nano-bars where the anisotropies differ from bulk values. The effect of strain-relaxation, due to the lithographic patterning, on the magnetic anisotropy of (Ga,Mn)As nano-bars has previously been studied by electrical transport[20; 21; 22] and optically-detected FMR.[8] We first compare the effect of strain-relaxation between 500 nm bars under compressive ((Ga,Mn)As) and tensile ((Ga,Mn)(As,P)) growth strain. The in-plane anisotropies are studied; although (Ga,Mn)(As,P) is out-of-plane magnetised[23], the applied field $\mathbf{H}_{0}$ brings the magnetisation into plane. In (Ga,Mn)As we observe an additional uniaxial contribution to the anisotropy ($\mu_{0}H_{U}=32$ mT) along the bar (Figure 3a & c) with a similar magnitude to previous reports.[20; 22; 8] By contrast in the (Ga,Mn)(As,P) nano-bar (Figure 3b & c) the sign of the uniaxial anisotropy ($\mu_{0}H_{U}=-50.1$ mT) has reversed and the easy axis is now perpendicular to the bar. This can be understood in terms of the sign of the strain relaxation: these materials become magnetically easier in the direction of most compressive (least tensile) strain. So when the tensile strain of the (Ga,Mn)(As,P) nano-bar relaxes, it introduces an easy axis perpendicular to the bar (Figure 3d). Furthermore we measure (Ga,Mn)(As,P) bars of different widths and observe a decrease in the strain-relaxation induced anisotropy from the 80 nm bar ($\mu_{0}H_{U}=-270$ mT) to the 500 nm bar ($\mu_{0}H_{U}=-50.1$ mT), and almost no effect of strain-relaxation in the $4~{}\mu$m bar ($\mu_{0}H_{U}=-10.5$ mT). As well as being able to determine the patterning-induced change in anisotropy, we also compare the damping among the nano-bars of different sizes. The frequency-dependent term (related to damping) increases for decreasing bar width: $\alpha=0.004$ (4 $\mu$m-wide), 0.006 (500 nm) and 0.023 (80 nm). The significantly higher value of Gilbert damping at 80 nm compared with the 500 nm and 4 $\mu$m bars may be due to damage during the etching process. The frequency-independent term is relevant in the case of strain relaxation as it indicates the inhomogeneity of anisotropy fields within the bar itself. The intermediate case of 500 nm shows greater inhomogeneity $\Delta H_{\text{inhomo}}=9.9$ mT than the 4 $\mu$m bar $\Delta H_{\text{inhomo}}=5.4$ mT, explained by the increased variation in local anisotropy. By contrast, for 80 nm bar reduces to $\Delta H_{\text{inhomo}}=2.5$ mT, indicative of a high degree of strain-relaxation. To characterise SO-FMR we must understand the direction and amplitude of the effective field $\mathbf{h}_{\text{eff}}$ that drives magnetisation precession. Similar to the experiments on STT-FMR in spin-valves[12; 13] we are able to perform vector magnetometry on the driving field from the angle dependence of the amplitude of the FMR peak. For a vector driving field $\mathbf{h}_{\text{eff}}(t)=(h_{x},h_{y},h_{z})e^{i\omega t}$ in-phase with the microwave current $\mathbf{I}(t)=(I_{x},0,0)e^{i\omega t}$, the amplitudes of the two components of the FMR peak are: \[V_{\text{sym}}(\theta) = \frac{I\Delta R}{2}A_{\text{sym}}\sin(2\theta)h_{z}\] (3) \[V_{\text{asy}}(\theta) = \frac{I\Delta R}{2}A_{\text{asy}}\sin(2\theta)(h_{x}\sin\theta+h_ {y}\cos\theta)\] (4) where $\Delta R$ is the AMR coefficient of the ferromagnetic sample, $\theta$ is the angle between the applied field $\mathbf{H}_{0}$ and the current $\mathbf{I}$, and $A_{\text{sym(asy)}}$ are constants determined by the magnetic anisotropies. Hence by decomposing the resonance lineshape into $V_{\text{sym}}$ and $V_{\text{asy}}$, and by measurements of the AMR and magnetic anisotropies we are able to deduce the components of $\mathbf{h}_{\text{eff}}$. No component of $V_{\text{sym}}$ is seen to behave as $\sin(2\theta)$, indicating that the driving field $\mathbf{h}_{\text{eff}}$ is predominantly in-plane. Figure 4a shows the angle-dependence of $V_{\text{asy}}$ for a 500 nm-wide (Ga,Mn)As bar patterned in the [1$\bar{1}$0] direction. We see that $V_{\text{asy}}(\theta)$ comprises a $-\sin(2\theta)\cos(\theta)$ term, indicating that the driving field is perpendicular to I. In a [110] device (Figure 3a) the amplitude of $V_{\text{asy}}$ has the opposite sign, indicating that the driving field has reversed. For nano-bars along [100] and [010] (Figure 3b), the $V_{\text{asy}}$ curve is a superposition of $\sin(2\theta)\sin(\theta)$ and $\sin(2\theta)\cos(\theta)$ functions, showing that the driving field consists of components both parallel and perpendicular to $\mathbf{I}$. These data are most clearly seen by plotting the dependence of the magnitude and direction of the effective field on the current (nano-bar) orientation (Figure 3c). Two contributions to the driving field are observed with different symmetry, $\mathbf{h}_{\text{eff}}=\mathbf{h}_{\text{R}}+\mathbf{h}_{\text{D}}$. The fields $\mathbf{h}_{\text{R}}$ and $\mathbf{h}_{\text{D}}$ have angular dependence on $\mathbf{I}$ reminiscent of the angular dependence of Rashba and Dresselhaus SO fields in the momentum space, respectively.[24; 25] The field with Dresselhaus symmetry, as previously observed in magnetisation switching experiments,[10] is due to the diagonal elements in the strain tensor (due to the lattice mismatch between GaAs substrate and (Ga,Mn)As). Therefore $\mathbf{h}_{\text{D}}$ changes sign between the (Ga,Mn)As and (Ga,Mn)(As,P) materials (comparing Figure 4c and 4d). The Rashba symmetry field $\mathbf{h}_{\text{R}}$ can be modelled by off-diagonal elements in the strain tensor. This strain is not physically present in the crystal structure of (Ga,Mn)As epilayers. It has been introduced, however, in previous studies to model the in-plane uniaxial anisotropy present in (Ga,Mn)As and the fitted values of this effective off-diagonal strain are typically several times smaller than the diagonal, growth-induced strain.[26] This is consistent with the observed smaller magnitude of $\mathbf{h}_{\text{R}}=6.5$ $\mu$T than $\mathbf{h}_{\text{D}}=18$ $\mu$T (values given at $j=10^{5}$ Acm${}^{-2}$). Both $\mathbf{h}_{\text{D}}$ and $\mathbf{h}_{\text{R}}$ are measured to be linear in current density (Figure 4e & f). We observe a larger magnitude of $\mathbf{h}_{\text{D}}$ at a given current density in the (Ga,Mn)(As,P) nano-bars. This is explained by the larger magnitude of the growth strain and larger resistivity (larger $E$ at given $j$) of (Ga,Mn)(As,P) as compared to the (Ga,Mn)As film.[23] In conclusion, we perform variable-frequency FMR experiments on individual micro and nano-bars of uniform ferromagnetic semiconductors (Ga,Mn)As and (Ga,Mn)(As,P). The FMR is driven by a torque at microwave frequencies whose origin lies in the internal effective field (due to the SO-coupling and exchange interaction) of the probed ferromagnet. We have demonstrated the utility of our SO-FMR technique by determining the rich characteristics of magnetic anisotropy fields and damping coefficients in the studied nanoscale ferromagnetic semiconductor samples. In addition, we have performed vector magnetometry on the driving field allowing us to measure a previously unobserved contribution to the current-induced field in the studied ferromagnets with symmetry of the Rashba SO-interaction. Our work demonstrates a new scalable FMR technique which provides an unprecedented method to perform magnetic characterisation of uniform ferromagnetic nanostructures and to study the nature of the current-induced effective magnetic field in SO-coupled ferromagnets. We acknowledge fruitful discussions with Ion Garate, Allan H. MacDonald and Leonid Rokhinson and support from EU Grants FP7-214499 NAMASTE, FP7-215368 SemiSpinNet, ERC Advanced Grant, from Czech Republic Grants AV0Z10100521, KAN400100652, LC510, KJB100100802 and Preamium Academiae, DF acknowledges support from Cambridge Overseas Trusts and Hitachi Cambridge Laboratory, A.J.F. acknowledges the support of a Hitachi research fellowship. ## References * (1) Vonsovskiǐ, S. V. _Ferromagnetic Resonance_ (Pergamon, Oxford, 1966). * (2) Goennenwein, S. T. B. _et al._ Electrically detected ferromagnetic resonance. _Appl. Phys. Lett._90 (2007). * (3) Mecking, N., Gui, Y. S. & Hu, C.-M. 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B_80 (2009). <figure><img src="content_image/1012.2397/x1.png"><figcaption></figcaption></figure> Figure 1, Principle of the experiment and its setup.a, Precession of the magnetisation vector M around the total magnetic field $\mathbf{H}_{\text{tot}}$. M is subject to a damping torque $\tau^{\alpha}$ due to energy dissipation, which causes the magnetic motion to relax towards $\mathbf{H}_{\text{tot}}$. The driving torque $\tau^{\text{SO}}$ due to current-induced effective field counters the effect of damping, and leads to steady-state motion $\partial\mathbf{M}/\partial t=-\gamma\mathbf{M}\times\mathbf{H}_{\text{tot}}$. b, SEM image of a 80 nm-wide bar, patterned from the (Ga,Mn)(As,P) wafer. c, Schematic of the experimental setup. <figure><img src="content_image/1012.2397/x2.png"><figcaption></figcaption></figure> Figure 2, Spin-orbit driven ferromagnetic resonance.a,$V_{\text{dc}}$ measured at 8, 10 and 12 GHz (circles) on the 80 nm-wide device. The resonance peaks are clearly observed and can be well-described by Equation 1 (solid lines are the fitted results). The difference in the signal level at different $\omega$ is caused by the frequency-dependent attenuation of the microwave circuit. b, The resonance field $H_{\text{res}}$ as a function of the microwave frequency (black triangles). The red solid line is the fitted results to Equation 2. c, Frequency-dependence of the FMR linewidth $\Delta H$ (black squares). The data are fitted to a straight line to extract information on $\Delta H_{\text{inhomo}}$ and $\alpha$. d,$V_{\text{dc}}$ measured from in-plane rotational scans of the external field $\mathbf{H}_{0}$. The colour scale represents the magnitude of the voltage. $\varphi$ is the angle between the magnetisation vector $\mathbf{M}$ and the [100] crystalline axis. e, Angle-plot of the resonance field $H_{\text{res}}$, which is extracted by fitting to each FMR peak using Equation 1 (black circles). The red line is a fitting curve to Equation 2 to calculate the magnetic anisotropy. <figure><img src="content_image/1012.2397/x3.png"><figcaption></figcaption></figure> Figure 3, SO-FMR on devices patterned from different materials and with various sizes.a,$H_{\text{res}}(\varphi)$ measured from an in-plane rotational scan on a 500 nm-wide (Ga,Mn${}_{0.06}$)As bar (patterned along the [010] axis). The circles are measurement data, and the solid line is the fitted results to Equation 2. The black arrow marks the long axis of the nano-bar. b,$H_{\text{res}}(\varphi)$ measured on a (Ga,Mn${}_{0.06}$)(As,P${}_{0.1}$) device with identical shape and orientation. c, Comparison of the in-plane anisotropy fields $H_{i}$ between the two samples. d, Schematic of the strain relaxation in the compressively-strained (Ga,Mn)As and and tensile-strained (Ga,Mn)(As,P) nanostructures. e, Comparison of the magnetic anisotropy (in terms of the profiles of $H_{\text{res}}$) among 80, 500 and 4000 nm-wide (Ga,Mn)(As,P) bars. f, The linewidth $\Delta H$ of the FMR signals measured on the three devices. <figure><img src="content_image/1012.2397/x4.png"><figcaption></figcaption></figure> Figure 4, Characterisation of the driving field in both (Ga,Mn)As and (Ga,Mn)(As,P) devices.a–b, Amplitudes of the anti-symmetric part of the FMR signal $V_{\text{asy}}$, measured on a group of 500 nm-wide (Ga,Mn)As bars (circles), patterned along different crystalline directions. The solid lines are fitted results to Equation 4. c, Plot of the magnitude and direction of the current-induced effective field $\mathbf{h}_{\text{eff}}$ measured on the (Ga,Mn)As nano-bars, scaled for a current density $j=10^{5}$ A/cm${}^{2}$. d, Similar plot for $\mathbf{h}_{\text{eff}}$ measured on the (Ga,Mn)(As,P) devices. e–f, Current density dependence of $\mathbf{h}_{\text{D}}$ and $\mathbf{h}_{\text{R}}$ in both (Ga,Mn)As and (Ga,Mn)(As,P) nano-bars. A second horizontal scale is included for the electric field, calculated from the device resistance (values given in Methods).
1510.05581	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23708, "num_imgs": 3, "llama3_tokens_count": 6875 }	[ "content_image/1510.05581/x1.png", "content_image/1510.05581/x2.png", "content_image/1510.05581/x3.png" ]	# Quark deconfinement and the duration of short Gamma Ray Bursts Alessandro Drago${}^{\text{(a)}}$, Andrea Lavagno${}^{\text{(b)}}$, Brian Metzger${}^{\text{(c)}}$ and Giuseppe Pagliara${}^{\text{(a)}}$ ${}^{\text{(a)}}$Dip. di Fisica e Scienze della Terra dell’Università di Ferrara and INFN Sez. di Ferrara, Via Saragat 1, I-44100 Ferrara, Italy ${}^{\text{(b)}}$ Department of Applied Science and Technology, Politecnico di Torino, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Italy ${}^{\text{(c)}}$ Columbia Astrophysics Laboratory, Columbia University, New York, NY, 10027, USA ###### Abstract We propose a model for short duration gamma-ray bursts (sGRBs) based on the formation of a quark star after the merger of two neutron stars. We assume that the sGRB central engine is a proto-magnetar, which has been previously invoked to explain the plateau-like X-ray emission observed following both long and short GRBs. Here, we show that: i) a few milliseconds after the merger it is possible to form a stable and massive star made in part of quarks; ii) during the early cooling phase of the incompletely formed quark star, the flux of baryons ablated from the surface by neutrinos is large and it does not allow the outflow to achieve a bulk Lorentz factor high enough to produce a GRB; iii) after the quark burning front reaches the stellar surface, baryon ablation ceases and the jet becomes too baryon poor to produce a GRB; iv) however, between these two phases a GRB can be produced over the finite timescale required for the baryon pollution to cease; a characteristic timescale of the order of $\sim 0.1$ s naturally results from the time the conversion front needs to cover the distance between the rotational pole and the latitude of the last closed magnetic field line; v) we predict a correlation between the luminosity of the sGRB and its duration, consistent with the data; vi) our model also predicts a delay of the order of ten seconds between the time of the merger event and the sGRB, allowing for the possibility of precursor emission and implying that the jet will encounter the dense cocoon formed immediately after the merger. Quark matter, compact stars, Gamma Ray Bursts pacs: 98.70.Rz,21.65.Qr,26.60.Dd Both long duration (lGRBs) and short duration Gamma Ray Bursts (sGRBs) start with a violent “prompt” emission phase, which generally lasts a few tens of seconds in the case of lGRBs and a few tenths of a second in sGRBs. The prompt emission is in many cases followed by some form of prolonged engine activity, commonly referred to as the “Quasi-Plateau” (QP) in the case of lGRBs and “Extended Emission” (EE) in the case of sGRBs Norris and Bonnell (2006). Beyond similarities in their light curve behavior, sGRBs and lGRBs show remarkably similar spectral properties Ghirlanda et al. (2009). This led to the suggestion that a similar central engine is acting in both classes of GRBs, a sGRB being similar to a lGRB cut after $0.3(1+z)$ s Calderone et al. (2015). The progenitors of lGRBs and sGRBs, on the other hand, are believed to be quite different: the collapse of a massive star for long bursts Woosley (1993) and the merger of two neutron stars (or of a neutron star and a black hole) for the short bursts Paczynski (1986). In their original forms, both models postulated a hyper-accreting black hole as the source of the relativistic outflow powering the GRB. However, following the discovery of the prolonged emission, a new model for the engine has grown in popularity, based on the relativistic wind of a newly formed, rapidly rotating proto-magnetar Thompson et al. (2004); Metzger et al. (2011). The model was initially proposed to explain the structure of lGRBs, but more recently it has been adapted to interpret also sGRBs Metzger et al. (2008); Bucciantini et al. (2012); Rowlinson et al. (2013)¹. [FOOTNOTE:1][ENDFOOTNOTE] GRB prompt emission results from dissipation within a relativistic jet composed of electron-positron pairs, photons and a small (but non-negligible) fraction of baryons Shemi and Piran (1990). The latter plays a fundamental role by setting the bulk Lorentz factor $\Gamma$ of the jet, with values of $\Gamma\sim 10^{2}-10^{3}$ required to match the observational data in most jet emission models [12]. In the case of a proto-magnetar, the requisite baryon loading is set naturally by the rate of mass ablation from the surface by neutrino heating Metzger et al. (2011). The duration of the initial prompt phase is therefore closely connected with the cooling time of the proto-neutron star, which indeed typically lasts tens of seconds or longer. The subsequent quasi-plateau is also powered by the still rapidly rotating magnetar, but the emission properties are likely to change once the wind reaches a high magnetization (pulsar-like) state after baryon loading ceases. Model fits of QP light curve to the dipole spin-down luminosity successfully describe the data Lyons et al. (2010); Dall’Osso et al. (2011). The same modeling applied to the EE of sGRBs Rowlinson et al. (2013) generally finds acceptable fits for similar values of the initial rotation period $P\sim$ few milliseconds, but the required dipole magnetic field strength $B$ is roughly an order of magnitude larger than for lGRBs. If the magnetar model is correct, a crucial question naturally arises: what is the origin of the prompt emission for sGRBs? If broadly similar values of $P$ and $B$ are needed to describe the QP and the EE, then why is sGRB prompt emission typically two orders of magnitude shorter than in lGRBs? The cleaner environment for the jet to escape, and the larger peak temperature of the proto-magnetar (reaching $\approx$ 50 MeV Sekiguchi et al. (2011)) in NS mergers compared to core collapse, would on the contrary suggest that the sGRB prompt emission should last even longer than that of lGRBs! In this Letter, we propose that due to the large mass of the proto-magnetar formed after a neutron star merger its nature is that of a quark star and not of a neutron star Kurkela et al. (2010); Drago et al. (17) following the ”two-families” scenario of Ref.Drago et al. (17, 2015); Drago and Pagliara (19) in which light compact stars are made of hadrons while the most massive ones are quark stars. Quark stars are self-bound objects, such that neutrinos with energies of a few tens MeV are not energetic enough to ablate material from the surface of the star Haensel et al. (1991); Dai and Lu (1998). Therefore, after the complete transformation of the newly formed compact star into a quark star, no baryonic material can be ablated from its surface and the prompt emission has rapidly to terminate. We associate this brief phase of cessation of the baryonic pollution with the duration of the prompt emission in sGRBs. Below, we will show that: 1) the formation of quark matter can take place within a few milliseconds after the merger, stabilizing the massive compact star; 2) the rate of baryon ablation from the surface during the formation of the quark star (until its conversation is complete) is too high to produce prompt GRB emission; 3) the duration of the prompt emission in sGRBs can therefore be linked to the switch-off of the baryonic emission, a process which we will show is indeed expected to last a few tenths of a second. In this way the prompt phase of sGRBs will look like that of lGRBs but cut at the moment of the switch-off, satisfying the analysis of Ref. Calderone et al. (2015). <figure><img src="content_image/1510.05581/x1.png"><figcaption>Figure 1: Fractions of neutrons, protons, Lambda (and Δ-resonances in thelower panel) as a function of the density (left scale). Temperature (rightscale). They are computed for matter having entropy per baryon of S/N=3.</figcaption></figure> We start by showing that, in the newly-formed compact star created by the merger, the conditions for initiating quark deconfinement are fulfilled. In Fig. 1 we display the composition of matter at beta-equilibrium and with an entropy-per-baryon $S/N=3$. This corresponds to a temperature in the center of the merger remnant of about 50 MeV, similar to what found by the simulations of Ref.Sekiguchi et al. (2011). We have employed the EoS SFHo obtained in [22] which satisfies all existing constraints below nuclear matter saturation density $n_{0}$ and we have taken into account the possible formation of $\Delta$-resonances Drago et al. (23). We also show in Fig.1 the EoS excluding $\Delta$’s to prove that the mechanism we are describing does not depend on the details of the hadronic EoS. Importantly, note that hyperons are present already at densities of the order of $n_{0}$ (in agreement with Ref.Sekiguchi et al. (2011)) due to the high temperature of the system. Bubbles of deconfined quark matter (here described by the EoS of Ref.Weissenborn et al. (2011)) will start appearing throughout the central region of the star on the time-scale of strong interaction (the temperature is large enough that thermal nucleation can take place Di Toro et al. (2006)) and will rapidly expand following the scheme of Ref.[26]. The central region will deconfine on a time-scale of $\sim 3-4$ ms 27; Pagliara et al. (2013) since in this initial phase the burning front is strongly accelerated by hydrodynamical instabilities. This phase of rapid burning halts at a depth of a few kilometers below the stellar surface, leaving the external layers unburnt and producing in a few ms an intermediate configuration which is mechanically stable, but not yet chemically equilibrated. In Fig. 2 we show the profile of this configuration, as mass-enclosed vs radius. Numerical simulations of the merger process (e.g., Sekiguchi et al. (2011)) show that, if the mass is not too large, the merger remnant can survive longer than 10 ms (due to its rapid differential rotation) before collapsing into a black-hole. For the EoS we are using, a direct collapse will not occur for the common case of the merger of two 1.3 $M_{\odot}$ stars, even neglecting the additional stabilizing effect due to the stiffening of the EoS Bauswein et al. (2013); Bauswein (2015); Bauswein et al. (2015)². [FOOTNOTE:2][ENDFOOTNOTE] <figure><img src="content_image/1510.05581/x2.png"><figcaption>Figure 2: Density profile (green line) and mass enclosed (black line) of the”hybrid” star formed after the rapid combustion as a function of the distancefrom the center.</figcaption></figure> After the conversion of the inner region to quark matter, what follows is a process of much slower burning which, being no longer accelerated by hydrodynamical instabilities, typically lasts a few tens of seconds [26]³. The entire star has converted to quark matter only after this slower burning front has reached the remnant surface. We will show that during this phase, no relativistic outflow - and hence no prompt GRB emission - is expected from the merger remnant, similarly to what happens in lGRBs. This is because in proto-magnetar models the maximum achievable Lorentz factor of the flow is given by $\Gamma_{\rm max}\sim\dot{E}/\dot{M}c^{2}$, where $\dot{E}\sim B^{2}R^{6}(2\pi/P)^{4}/3c^{3}$ is the magnetic Poynting flux, $R$ is the stellar radius, and $\dot{M}$ is the mass loss rate due to neutrino heating Metzger et al. (2011). As long as the star maintains an external layer of baryons, nucleons can be ablated from its surface by thermal neutrinos with energies of a few MeV. [FOOTNOTE:3][ENDFOOTNOTE] <figure><img src="content_image/1510.05581/x3.png"><figcaption>Figure 3: Panel (a): total neutrino luminosity. Solid lines correspond to theluminosity associated with the rapid burning of the central area (and twodifferent values for the diffusion time). Dashed line the neutrino luminosityof the slow combustion of the external layer of the star. Panel (b): Maximumbulk Lorentz factor of the magnetar jet, Γmax, as a function of time, shownfor two values of B/P2, where B is the magnetic dipole and P the rotationperiod. The two horizontal lines bracket the range of values of Γmax requiredto produce GRB prompt emission according to conventional models. Here and inpanel (a) the arrows indicate the time t0∼13 s at which the conversion of theremnant into a quark star is completed. Panel (c): duration of the promptemission of the sGRB as a function of B/P2, shown for two values of the timeneeded for baryon cessation tc (see text).</figcaption></figure> The evolution of $\dot{M}$ is quite complicated. During the first tenth of a second it reaches values as large as $\dot{M}\sim 10^{-3}M_{\odot}s^{-1}$Dessart et al. (2009). In the following few seconds, the baryon flow is associated with the generation of protons via $\beta$-decay in the cooling process. In this way the remnant atmosphere becomes progressively more proton rich, similar to the evolution of a proto-neutron star after a supernova explosion. In our simple analysis we borrow from the existing literature the result that $\dot{M}$ remains very large for a few seconds Metzger and Fernandez (2014) and we assume that it can be approximated better and better with the formula used in the case of a proto-neutron star after a supernova explosion. In that case $\dot{M}$ is approximately given by Qian and Woosley (1996): \[\dot{M}\sim 1.2\times 10^{-9}C^{5/3}L_{\overline{\nu}_{e},51}^{5/3}\epsilon^{1 0/3}_{\overline{\nu}_{e},\mathrm{MeV}}M^{-2}_{1.4}R^{5/3}_{6}M_{\odot}s^{-1}\,,\] (1) where $L_{\overline{\nu}_{e},51}$ is the electron anti-neutrino luminosity in units of $10^{51}$ erg, $\epsilon_{\overline{\nu}_{e},\mathrm{MeV}}$ is their energy in MeV, $M_{1.4}$ is the neutron star mass in units of 1.4 $M_{\odot}$, $R_{6}$ is the radius of the star in units of 10${}^{6}$ cm, and $C\sim 2$ is a correction factor to account for additional channels of neutrino heating Qian and Woosley (1996). The energy of neutrinos from the merger remnant is typically $\approx$ 10 MeV Perego et al. (2014). The crucial ingredient in the calculation of $\dot{M}$, and hence $\Gamma_{\rm max}$, is the neutrino luminosity. This has been evaluated in Pagliara et al. (2013), accounting only for the heat deposited during the rapid burning of the central region, while [26] also evaluates the emission associated with the prolonged burning of the external layer. The contributions to the neutrino luminosity from the initial phase of prompt burning in the core, $L_{\nu}^{c}$, can be approximated in a simple way by introducing the neutrino diffusion time $\tau_{\rm diff}$. Following Ref. Pagliara et al. (2013): \[L_{\nu}^{c}\sim Q/\tau_{\mathrm{diff}}\,\mathrm{e}^{-t/\tau_{\mathrm{diff}}}\,,\] (2) where $Q\sim(2-3)\times 10^{53}$ erg is the total heat deposited by quark deconfinement during the rapid burning phase and $\tau\sim 2(3)$s for a star of mass 1.4(1.8) $M_{\odot}$, respectively. We employ a similar formula in the merger case, but accounting for the larger amount of heat deposited, $Q\sim 10^{54}$ erg (also due to the gravitational potential energy before the merger and in part to the use of a different equation of state), and $\tau_{\mathrm{diff}}\sim 3-4$ s, the latter estimated following Ref.Perego et al. (2014) (their eq. 6). Fig. 3 shows that, while the quark star is still forming, the neutrino luminosity is very large and it corresponds to a mass loss rate of $\approx 10^{-4}M_{\odot}s^{-1}$. Therefore the Lorentz factor does not reach high enough values to produce the GRB prompt emission. This stage mirrors the early evolution of the proto-magnetar in lGRB, where no relativistic jet is created during the first $\sim 10$ s after core bounce due to the high baryon load. In the case of lGRBs, after that phase the baryon load slowly reduces and a GRB lasting a few tens seconds is produced. Notice that in the case of lGRB, the mass of the proto-magnetar and its initial temperature are significantly smaller and quark deconfinement need not to take place. By contrast, in the merger case, the quark conversion is unavoidable and when the front reaches the stellar surface baryonic ablation ceases. To zeroth order, therefore, the prompt emission from the rotating magnetized merger remnant is suppressed at all epochs: the mass loss rate is too large prior to quark conversion, or too low after the conversion. In neither case can a prolonged relativistic outflow of the appropriate Lorentz factor form. In this zero-order approximation, the maximum Lorentz factor $\Gamma_{\rm max}$ jumps from values of the order of unity to, virtually, infinity. Such a sudden jump in the outflow’s Lorentz factor is clearly not physical: what is missing is a description of the period over which the most external layer of the star is converted into quarks. Even if baryon loading were to cease abruptly, a minimum time would be required to clear the jet of baryons, which we estimate to be $t_{\mathrm{d}}\sim 0.01$ s as the dynamical timescale near the base of the wind (Ref. [36], Fig. 9). However, there is a potentially more important effect that delays the time for baryon cessation. Since the star is rapidly rotating near centrifugal break-up, its shape is deformed into an ellipsoid with an equatorial radius $R_{\mathrm{eq}}$ larger than its polar radius $R_{\mathrm{p}}$. For a soft EoS, such as that we employ for the hadronic phase, we expect $R_{\mathrm{eq}}/R_{\mathrm{p}}\sim 1.2-1.4$ for a rotation rate of $\sim 1$ kHz Bejger et al. (2007). Using the results of Ref.[26], we estimate that the burning front will reach the pole and the equator at times $t_{\mathrm{\rm p}}$ and $t_{\mathrm{\rm eq}}\approx(1.2-1.4)t_{\mathrm{\rm p}}$, respectively. Since $t_{p}\sim(10-20)$ s, the quark conversion of the star will move from pole to equator over a characteristic timescale of $\Delta t\sim t_{\rm eq}-t_{\rm p}\sim$ a few seconds. However, in fact baryon mass loss from the strongly magnetized remnant is confined to a relatively narrow range of latitudes near the axis of the magnetic dipole, which is likely to be aligned with the rotation axis. The latitudinal extent of this ‘open zone’ of the magnetosphere is given by $\theta_{\rm open}\approx\left(R/2R_{\rm L}\right)\approx 0.1R_{6}(P/\rm 2ms)^{ -1}$, where $R_{\rm L}=2\pi Pc$ is the light cylinder radius. Thus, for typical values of $P\sim 2$ ms, we expect the true timescale for baryon cessation to be given by $t_{c}\approx\theta_{\rm open}(\Delta t\sim t_{\rm eq}-t_{\rm p})\sim$ a few $0.1$ s, comparable to the duration of sGRBs. 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1006.4826	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 25293, "num_imgs": 0, "llama3_tokens_count": 6800 }	[]	# Neutrinos Oscillations With Long-Base-Line Beams (Past, Present and very near Future)¹ [FOOTNOTE:1][ENDFOOTNOTE] L.STANCO INFN - Padova, www.pd.infn.it Via Marzolo, 8, Padova I-35131 Italy E-mail: luca.stanco@pd.infn.it ###### Abstract We overview the status of the studies on neutrino oscillations with accelerators at the present running experiments. Past and present results enlighten the path towards the observation of massive neutrinos and the settling of their oscillations. The very near future may still have addiction from the outcome of the on-going experiments. OPERA is chosen as a relevant example justified by the very recent results released. Neutrino; Oscillations; Tau. ## 1 Introduction In the last two decades several experiments have provided strong evidence in favor of the neutrinos oscillation hypothesis. In the so called _atmospheric sector_ the flavor conversion was first established by Super-Kamiokande[1] and further by MACRO[2] and Soudan-2[3] experiments. Further confirmation was more recently obtained by the K2K[4] and MINOS[5] long-baseline experiments. However a two fold question is still unanswered, does the oscillation scenario correspond to the simple 3-flavor expectation or not? which is related to the still unobserved direct appearance of one flavor to another, in particular to the highly expected $\nu_{\mu}\rightarrow\nu_{\tau}$ oscillation. Answer to this two-fold question is relevant mainly to proceed towards the next steps in the clarification of the leptonic sector of the particle model. After a brief reminder of the physics behind we will assay to focus on the main points which brings us to the present knowledge about neutrino mixing. The recent history provided the scenario in which the neutrino oscillation framework was settled. Still new questions opened up and these bring us directly into the future. Next we will shortly report on the present results from short-base-line (SBL) experiment, mainly the MiniBooNE[6] experiment, and the long-base-line (LBL) experiments, namely MINOS and OPERA[7]. Finally some physics expectations for the near future after a personal discussion of the very recent OPERA results[8] will be drawn. ## 2 Physics layout The issue of the lepton mixing is far from being understood and even generally described as it occurs in the quark sector. In particular the generic three questions on the reason the leptons mix themselves, the details of the way they actually mix and which are the mechanisms which underlay their mixing, arize. In 1998 a new history for neutrinos began as a sort of second life with the double discovery that (a) they oscillate[1] then owing a mass after 41 years from the initial idea of B. Pontecorvo in 1957[9] and (b) they mix themselves in a peculiar way after the void result by CHOOZ[10]. The CHOOZ experiment took data in 1997-98 at a distance of about 1 km from a nuclear power plant of two reactors in France. It aimed to observe $\nu_{e}\rightarrow\nu_{\mu}$ (actually antineutrinos) oscillations. After a collection of 2991 ${\bar{\nu}_{e}}$ candidates CHOOZ put an upper limit on the direct observation of ${\bar{\nu}_{\mu}}$ events. At that time the limit was set as $\sin\theta\lesssim 0.1$ with a systematic error of 2.7%. The low error was due to the possibility for CHOOZ to measure the backgrounds before the switching on of the reactors. In 2002 the KamLAND experiment[11] repeated the measure in a site in Japan where many reactors were present, close and far away from the detector. The distribution of the ${\bar{\nu}_{e}}$ flux coming from the reactors is displayed in Fig. 1(a), with an average distance of 150 km from the reactor. Differently from CHOOZ, KamLAND obtained a positive result in term of disappearance of ${\bar{\nu}_{e}}$ flux. The beautiful oscillation pattern is shown in Fig. 1(b). [FIGURE:S2.F1][ENDFIGURE] Mainly after KamLAND (and a rather contemporary result in the _solar_ neutrino sector by the SNO experiment[12]) the increase in the oscillation neutrino studies was extremely rapid and huge bringing to a re-interpretation of the CHOOZ result in term of oscillations of flavour eigenstates. The old idea of mixing matrix by Maki et al. in 1962[13] was revitalized, similarly to what was made by Cabibbo[14] in 1963 for the quark sector. The standard parametrization of a mixing matrix at 3 components is therefore realized via the usual 3 Euler rotations, leaving us with 3 angles, $\theta_{12}$, $\theta_{23}$, $\theta_{13}$, and a phase $\delta$. Moreover in case of a Majorana picture two more phases are present, $\alpha_{1}$ and $\alpha_{2}$. To emphasize the key point it comes out quite naturally to simply establish a similar way of mixing for quarks and leptons. Of course other more complex scenarios, where more than 3 eigenstates appear, are possible. More neutrinos states are compatible with the present knowledge of the lepton physics, in particular one or more _sterile_ neutrinos[16] may be included. This is a fundament question since it may or it may not en strength parallelism between quarks and leptons. The complete description of the formalism may be found in [strumia], while several fits have been performed to take into account the whole set of measurements. Still fundamental questions remain unanswered. The first question relates to the mass ordering of the neutrino mass eigenstates. Does the mass scale ordering of $\nu_{1}$, $\nu_{2}$, $\nu_{3}$ (as defined by the parametrization) follows the same ordering of $\nu_{e}$, $\nu_{\mu}$, $\nu_{\tau}$ ? As the measured oscillation pattern is described only in term of $\Delta m^{2}_{12}$ and $\Delta m^{2}_{13}$ the exact order is not identified yet, neither it is the absolute mass scale. Are the 3 masses just below the present neutrino mass absolute limit (less than 1 eV) or are they some order of magnitude smaller? More and more unanswered questions come up as we put a closer look to the measured quantities. For example in Table 2 the present values of the mixing matrix components for quarks, $V_{CKM}$, and for leptons, $V_{MNS}$ are compared. The underlying pattern is clearly different and we finally conclude that the lepton mixing is weird². [FOOTNOTE:2][ENDFOOTNOTE] \tbl Present values for the Neutrino Mass Mixing Matrix (a) as taken from Ref. \refcitemezzetto and the unitarity values of the VCKM (b) as extracted from Ref. \refcitePDG. Note that the very recent result by MINOS[29] sets sin22θ13<0.12. \toprule(a) sin2θ12 = 0.30±0.02 sin2θ23 = 0.50±0.07 sin22θ13 < 0.13 Δm213 = 2.40+0.12−0.11×10−3eV2 δm212 = 7.6±0.2×10−5eV2 \botrule(b) ∑i=d,s,b\|Vui\|2 = 0.9999±0.0011 ∑i=u,c,t\|Vid\|2 = 1.002±0.005 ∑i=u,c;j=d,s,b\|Vij\|2 = 2.002±0.027 \botrule Also the present knowledge of the errors is largely different in the quark and lepton sector. See e.g. Ref. \refcitemezzetto for an up-to-date report on the error measurements, to be compared with the extremely well known values of the quark mixing matrix[18]. To illustrate the importance of the size of the errors we may look at Fig. 2 taken from Ref. \refciteISS (Fig. 43), which shows the large region for the possible values of the top angle of the lepton unitarity triangle. The degenerate case, θ13=0, corresponding to the bottom horizontal line, is also still allowed by the present measurements. More and exhaustive discussions may be found in Ref. \refcitefarzan. \psfig file=ISS.eps,width=3in Figure 2: The unitarity eμ-triangles. The horizontal side, \|Ue1U∗μ1\| is normalised to one. The triangles correspond to θ13=0.15 and different values of the phase δ. Each scatter point represents a possible position of vertex as the mixing parameters pick up random values within the present uncertainty ranges: sin2θ23∈[0.36,0.61], sin2θ12∈[0.27,0.37] and sin2θ13∈[0,0.031], and δ∈[0,2π]. There are also illustrated 3 different triangles for 3 different choices of δ and θ13=8.60 case. The figure is taken from [\refciteISS]. Also the present knowledge of the errors is largely different in the quark and lepton sector. See e.g. Ref. mezzetto for an up-to-date report on the error measurements, to be compared with the extremely well known values of the quark mixing matrix[18]. To illustrate the importance of the size of the errors we may look at Fig. 2 taken from Ref. ISS (Fig. 43), which shows the large region for the possible values of the top angle of the lepton unitarity triangle. The degenerate case, $\theta_{13}=0$, corresponding to the bottom horizontal line, is also still allowed by the present measurements. More and exhaustive discussions may be found in Ref. farzan. [FIGURE:S2.F2][ENDFIGURE] In summary we may conclude that the lepton mass mixing matrix might be technically similar to the quark one even if it shows a quite different pattern and it is at present rather poor known. We like to conclude this section by using the same wording of W. Buchmüller at EPS09 conference[22]: ”Right-handed neutrinos have been found; no exotics have been found (yet)”. Therefore as a whole it follows that we have to be prepared to the unexpected! ## 3 Physics perspectives Currently the lepton scenario illustrated in the previous section is the only one which is receiving attention by experimental investigation and mostly phenomenological investigation too. Other theoretical possibilities like e.g. the NSI, Non-Standard-Interactions[23], are in our judgement not so appealing and remains at the level of generic phenomenological models. Therefore a not so long list of unknowns have to be identified and measured: the 3 mixing angles ($\theta_{12}$, $\theta_{13}$ and $\theta_{23}$), the 2 neutrino squared mass differences ($\Delta m^{2}_{12}$, $\Delta m^{2}_{13}$), the sign of one the two mass differences ($\Delta m^{2}_{23}$), a CP phase ($\delta$), the absolute neutrino mass scale and their nature (Dirac or Majorana), the total number of neutrino (are there more than 3 neutrinos ? ³), not at last forgetting the detection of the undergoing source of the oscillation. The latter question corresponds to the detection of a direct appearance signal, that is the observation of the $\nu_{\tau}$ appearance for the atmospheric oscillation (and the $\nu_{e}$ for the solar one) providing a direct measurement of the Lepton Flavor Violation (LFV) process⁴. [FOOTNOTE:3][ENDFOOTNOTE] [FOOTNOTE:4][ENDFOOTNOTE] Most of the above items may be investigated at Long-Base-Line experiments by excluding the investigation of the fundamental nature of the neutrinos and their absolute mass scale. [FIGURE:S3.F3][ENDFIGURE] The physics prospects are raveled by the ”presence” of internal puzzles in the experimental side. In particular the recent results from MiniBooNE are not able to disentangle the somewhat old and controversial result by LSND[24]. The original result from LSND (see Fig. 3) of the ${\bar{\nu}_{\mu}}-{\bar{\nu}_{e}}$ observation could not be phenomenologically arranged in the 3 neutrino standard scenario. MiniBooNE[25] looked for the oscillation in either the neutrino or the antineutrino modes. In the neutrino mode it is able to rule out the result by LSND as oscillation while observing an unexplained excess in a energy region below that of LSND. In the antineutrino mode no similar excess is observed while the ruling out of LSND is not gained. Fig. 4 (a and b) as extracted by Ref. miniboone shows the MiniBooNE results. [FIGURE:S3.F4][ENDFIGURE] As a matter of fact to the author the experimental situation is rather confused. More experimental facts are needed and the question whether the ongoing two LBL experiments MINOS and OPERA may help turns out to be fully relevant. ## 4 MINOS physics results The MINOS experiment[26] is constituted by two similar apparata, the Near and the Far detectors, made of scintillator strips and a toroidal spectrometer. This layout allows the minimization of several uncertainties like the neutrino flux from the NUMI beam and the extrapolation via Monte Carlo of the unoscillated $\nu_{\mu}$ spectrum from Near to Far sites. A very detailed analysis allows to reconstruct the energy of the interacting neutrinos (Fig. 5) and estimate the percentage of disappeared neutrinos[27]. From the later MINOS extracts the oscillation parameters in the assumption of 2 flavor oscillation mode (Fig. 6 from the analysis in [minos]). [FIGURE:S4.F5][ENDFIGURE] [FIGURE:S4.F6][ENDFIGURE] Since we will discuss in the next section the OPERA experiment it is worthwhile to outline the twofold character of the MINOS analysis, the ”rate” and the ”shape”. As OPERA will be able to deal only with ”rates”, the latter significance power has to be compared with the corresponding one by MINOS which turned out to be rather poor (Fig. 7). [FIGURE:S4.F7][ENDFIGURE] The disappearance mode can be complementary studied in MINOS with the appearance of electron $\nu$. First results reported were indicative of a possible $\nu_{e}$ appearance: 35 events from $\nu_{e}$ interactions were observed against an expected background of $27\pm 5(stat)\pm 2(sys)$, corresponding to a 1.5 excess[28]. However very recent results (released after the Conference time) with an increased statistics washed out that indication[29]. It seems that the new dedicated experiments for the $\theta_{13}$ measurement have to be waited for (see the related contributions to these proceedings). ## 5 The OPERA way We will now discuss at length the OPERA experiment since the very recent on May 31${}^{rst}$ 2010 release of new results (see next Section) corresponds to a relevant new contribution in the neutrino physics. The OPERA experiment[7] has been designed to observe the $\nu_{\tau}$ appearance in the CNGS $\nu_{\mu}$ beam[30] on an event by event basis. The $\nu_{\tau}$ signature is given by the decay topology of the short-lived $\tau$ leptons produced in the $\nu_{\tau}$ Charged Current (CC) interactions decaying to one prong (electron, muon or hadron) or three prongs hadrons. The detector is located underground in the Laboratorio Nazionale del Gran Sasso (LNGS, L’Aquila, Italy) along the path of the CNGS neutrino beam, 730 km away from the source at CERN. The beam was optimized in order to maximize the number of $\nu_{\tau}$ CC interactions at the LNGS site keeping the energy constraint to be above the $\tau$ production threshold. The result is a wide band neutrino beam with an average energy of $\sim 17$ GeV; the $\bar{\nu}_{\mu}$ contamination is 2.1%, $\nu_{e}+\bar{\nu}_{e}$ is below 1% and prompt $\nu_{\tau}$ at production is negligible. With a nominal beam intensity of $4.5\times 10^{19}$ proton-on-target (p.o.t.) per year, $\nu_{\mu}$ CC and neutral current (NC) interactions at Gran Sasso are deemed to 2900/(kton$\times$year) and 875/(kton$\times$year), respectively. By assuming the oscillation parameters $\Delta m^{2}=2.5\times 10^{-3}$ eV${}^{2}$ at full mixing 10.4 events are expected to be observed in OPERA in 5 years of data taking with a background of 0.75 events. In the two years 2008 and 2009 OPERA succeeded[31] to collect $5.30\times 10^{19}$ p.o.t. corresponding to 31,550 detected events in time with the beam, 5391 of which matched to a neutrino interaction in the OPERA target within more than 99% percent accuracy. At the CNGS energies the average $\tau$ decay length is $\sim 450~{}\mu\mbox{m}$. In order to observe it OPERA makes use of $2\times 44\mu m$ nuclear emulsions films interspaced with $1~{}\mbox{mm}$ thick lead plates which form the target mass of the OPERA detector. This technique, called Emulsion Cloud Chamber (ECC), has been used successfully by the DONUT experiment for the first direct observation[32] of the $\nu_{\tau}$. Every time a trigger in the electronic detectors is compatible with an interaction inside the target (see Fig. 8), the brick with the highest probability to contain the neutrino interaction vertex is extracted from the apparatus and exposed to X-rays for film-to-film alignment. Further the brick is unsandwiched, the emulsion films are developed and analyzed. The final sensitivities are $\sim$0.3 $\mu$m spatial resolution, $\sim$2 mrad angular resolution and $\sim$90% single track detection efficiency. [FIGURE:S5.F8][ENDFIGURE] [FIGURE:S5.F9][ENDFIGURE] ADDENDUM Very recently OPERA reported the observation of a first $\nu_{\tau}$ candidate[8]. The result is obtained by the observation of a rather clean event (Fig. 9), a possible 1-prong hadron decay of a $\tau$ lepton with $(n)\pi_{0}$ derived by the presence of some electromagnetic showers. The decay topology is consistent to be that of $\tau^{-}\rightarrow\nu_{\tau}+\rho^{-}\rightarrow\nu_{\tau}+\pi^{-}\pi_{0}$. Even if the expected number of $\nu_{\tau}$ interactions and identification in OPERA is estimated to be $0.54\pm 0.13$, well in agreement with the possible observation of 1 $\nu_{\tau}$ event, the significance of the result depends totally on the value of the background. OPERA estimates the background to be $0.018\pm 0.007$ for the _1-prong_ decay channel where the candidate has been observed. That corresponds to a probability of 1.8% to fluctuate to 1 event, which may be interpreted as a significance of 2.36 sigma’s towards the observation of a $\nu_{\tau}$ interaction (_p_-values of the null hypothesis, see Ref. [PDG]). At first sight it may be surprising to extract such level of significance from just one event. That is the power of a _clean_ experiment. It is ilustrated in Fig. 10 where the significance of the result is drafted towards the number of events observed instead of the usual integrated _luminosity_ of the data collected. The curves parametrized as function of the number of p.o.t collected by OPERA⁵ show that very few events allow to set a quite robust physics result. On top of that it is also evident that whether OPERA will be able to decrease the level of background the significance will increase it. For example, in case the estimated background be increased/decreased of a factor 2, retaining the assumed 50% nuisance, the corresponding significances will decrease/increase as 2.10 and 2.61, respectively. From another point of view the detection of a second (third) $\nu_{\tau}$ candidate, with the present level of total background proportionally updated, will increase the statistical significance from 2.01 to 2.82 (3.42). The latter consideration may demonstrate that the OPERA result be potentially much more interesting that the actual measurement by Super-Kamiokande which set a 2.4 significance in the $\nu_{\tau}$ appearance observation[34]. [FOOTNOTE:5][ENDFOOTNOTE] [FIGURE:S5.F10][ENDFIGURE] The OPERA result, at 98.2% of probability, corresponds to an extremely important evidence which can be expressed in several ways. For example, we may say that it is the first direct evidence of Lepton Flavor Violation, the theoretical unsatisfaction of the Standard Models being from now on even more evident. The observation of the transition from one flavor to the other should constraint and open new horizons to the theoretical elaborations, not forgetting the parallelism (somehow _opposite_ in term of flavor eigenstates) with the quark sector. The second important point which is left to OPERA for the near future is to answer the question about the number of oscillated $\nu_{\tau}$. That issue is well illustrated by a plot similar to the previous one (Fig. 11) where the _distance_ in terms of sigma’s from the MINOS expectation is drawn towards the number of observed events. The result is parametrized as function of number of p.o.t. From the figure we may deduce that it will take some time to disentangle any deviation from the standard oscillation scenario. However it is will be fully worthwhile to pursue it. [FIGURE:S5.F11][ENDFIGURE] ## 6 Conclusions The neutrino oscillation scenario began to be clarified in 1998 with the observation of a disappearance of atmosferic $\nu_{\mu}$, followed by the determination of similar disappearance (and indirect appearance) in the solar sector. The scenario that rose up is based on a 3-flavor oscillation which however leave out some intriguing concerns like the LSND result and the presence or not of _sterile_ neutrinos. In that context possible correlations with the similar mixing pattern of the quark sector are still at the level of theoretical exercises. The powerful results by MINOS settled a stringent measurement on the $\nu_{\mu}$ oscillation. The very recent result by OPERA, even if still at the level of _evidence_, demonstrates the action of LFV and it rules out for the time being the presence of _sterile_ neutrinos. The large numbers of experiments undergoing all over the world to search for a $\theta_{13}$ value different from zero corresponds to a lively field of physics interest (see other contributions on these proceedings). However more than usual it is necessary to outline the lesson from past, nature is not obvious and the lack of experimental confirmations about theoretical models should encourage us to be prepared on the unexpected. ## 7 Acknowledgments It is a pleasure to thanks the very warm hospitality of the organizers and the kind invitation. The presentation allowed the author to further elaborate on the very attractive field of neutrino physics. Some results and discussions reported in these proceedings were stimulated just for this occasion. Some of the statistical elaborations have been checked through by my colleague S. Dusini. Also I want to thank M. Mezzetto for a critical reading of the draft, many of his suggestions have been incorporated in the present version. Finally, all the considerations and the conclusions throughout the paper are of full responsibility of the author. ## References * [1] Y. Fukuda et al. (SK Collaboration), _Phys. Rev. Lett._81, 1562 (1998). * [2] S. P. Ahlen et al. (MACRO Collaboration), _Phys. Lett._ B 434, 451 (1998). * [3] W.W.M. Allison et al. (Soudan-2 Collaboration) _Phys. Lett._ B 449, 137 (1999). * [4] M.H. Ahn et al. (K2K Collaboration), _Phys. Rev._ D 74, 072003 (2006). * [5] D.G. Michael et al. (MINOS Collaboration), _Phys. Rev. Lett._101, 131802 (2008). * [6] A.A. Aguilar-Arevalo et al. (MiniBooNE Collaboration), _Phys. Rev. Lett._98, 231801 (2007). _Phys. Rev. Lett._103, 111801 (2009). * [7] R. Acquafredda et al. (OPERA Collaboration), _JINST_4, 04018 (2009). * [8] N. 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(MINOS Collaboration), arXiV:1006.0996. * [30] CNGS project, http://proj-cngs.web.cern.ch/proj-cngs/ * [31] L. Stanco (OPERA Collaboration), _Proceedings of the BUE-CTP International Conference on Neutrino Physics in the LHC Era_, 19 November 2009, Luxor, Egypt. * [32] K. Kodama et al. (DONUT Collaboration) _Phys. Lett._ B 504, 218 (2001). * [33] M. Guler et al., (OPERA collaboration), _An appearance experiment to search for $\nu_{\mu}\rightarrow\nu_{\tau}$ oscillations in the CNGS beam: experimental proposal_, CERN-SPSC-2000-028, LNGS P25/2000. M. Guler et al. (OPERA collaboration), _Status Report on the OPERA experiment_, CERN/SPSC 2001-025, LNGS-EXP 30/2001 add. 1/01 * [34] K. Abe et al. (SK Collaboration), _Phys. Rev. Lett._97, 171801 (2006).
1806.09232	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 60552, "num_imgs": 3, "llama3_tokens_count": 18823 }	[ "content_image/1806.09232/x1.png", "content_image/1806.09232/x2.png", "content_image/1806.09232/x3.png" ]	# Measurement compatibility in Bell nonlocality tests Tassius Temistocles Departamento de Física, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Caixa Postal 702, CEP 30123-970, Belo Horizonte, Brazil Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, CEP 13083-859, Campinas, Brazil Rafael Rabelo Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, CEP 13083-859, Campinas, Brazil Marcelo Terra Cunha tcunha@ime.unicamp.br Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, CEP 13083-859, Campinas, Brazil February 22, 2024 ###### Abstract Incompatibility of observables, or measurements, is one of the key features of quantum mechanics, related, among other concepts, to Heisenberg’s uncertainty relations and Bell nonlocality. In this manuscript we show, however, that even though incompatible measurements are necessary for the violation of any Bell inequality, some relevant Bell-like inequalities may be obtained if compatibility relations are assumed between the local measurements of one (or more) of the parties. Hence, compatibility of measurements is not necessarily a drawback and may, however, be useful for the detection of Bell nonlocality and device-independent certification of entanglement. † [FOOTNOTE:†][ENDFOOTNOTE] ## I Introduction Quantum theory is fundamentally distinct from any classical theory of physics, a fact that is well known and accepted nowadays. Even though there is no consensus regarding a physical principle that explains such departure, the distinction between quantum and classical mechanics is clear at the level of their mathematical formalisms. For instance, two of the most interesting nonclassical features present in quantum theory are entanglement and incompatibility between measurements. The fact that there are measurements, or observables, that are incompatible, which can not be jointly treated as one observable within quantum theory, is one of the key ingredients behind some of the most astonishing phenomena related to nonclassicality, such as Bell nonlocality [1; 2] and Bell-Kochen-Specker contextuality [3; 4]. Both arose from investigations regarding the completeness of quantum theory [5] and are related to stronger-than-classical correlations between outcomes of measurements performed on quantum systems. The theory of Bell nonlocality and the theory of Bell-Kochen-Specker contextuality, though, have been developed independently, in a sense, and each presents its own particular features. In this manuscript we focus our attention on the former, although the reader may notice that there will be elements of the latter. The paradigmatic example of Bell nonlocality of quantum systems takes place in a bipartite measurement scenario, where two characters, Alice and Bob, are able to choose between two possible dichotomic measurements to perform on their respective share of a previously prepared joint system. Denoting by $A_{x}\in\{\pm 1\}$ the outcome of measurement $x\in\{0,1\}$ of Alice, and by $B_{y}\in\{\pm 1\}$ the outcome of measurement $y\in\{0,1\}$ of Bob, any so-called _local hidden variable_ (LHV) theory of joint probabilities that govern the behavior of the measurement devices will lead to mean values that necessarily obey the _Clauser-Horne-Shimony-Holt_ (CHSH) [6] inequality \[\left<A_{0}(B_{0}+B_{1})+A_{1}(B_{0}-B_{1})\right>\leq 2.\] (1) If the measurements are performed on quantum systems, however, then the inequality does not need to be respected, and a violation of up to the value of $2\sqrt{2}$ may be observed [7]. This shows that quantum theory is incompatible with LHV theories, an observation first made by Bell [1]. Two necessary conditions for Bell nonlocality to manifest in quantum systems are (i) entanglement, in the state of the shared system; and (ii) incompatibility between the measurements, in each party. Curiously, neither (i) [8] nor (ii) [9] is a sufficient condition. Regarding (i), it became an important question in the field (for both fundamental and practical reasons) to identify which entangled states could ultimately lead to Bell nonlocality. On one hand, there has been an effort to obtain examples of local entangled states, those that never lead to nonlocal correlations [10; 11; 12; 13; 14]. On the other hand, several nonstandard measurement scenarios have been proposed where even local entangled states can lead to Bell nonlocality, and concepts like hidden-nonlocality [15; 16; 17; 18; 19] and activation of nonlocality [20; 21; 22; 23] have been defined. In this manuscript we propose a measurement scenario where we explicitly assume the existence of subsets of compatible measurements for (at least) one of the parties. We show how this assumption drastically changes the measurement scenario and leads to a range of Bell-like inequalities that are potential tools to improve on condition (i), leading to examples of entangled states that have not been known to be nonlocal. We present an example, in one of the simplest scenarios of this approach, where, without the compatibility assumption, the only relevant Bell inequality is CHSH [24], and, with the compatibility assumption, 26 Bell-like inequalities arise. We explicitly show that one of these inequalities reveals Bell nonlocality in two families of quantum states for regions of parameters where the CHSH inequality is not violated. Also, there is numerical evidence that, in part of this region, the states of one of such families do not violate the I${}_{3322}$ inequality [25] either. In addition to revealing the nonlocality of quantum states that would not be displayed in standard Bell scenarios, multipartite measurement scenarios with local compatible measurements are interesting for both fundamental and practical reasons: fundamentally, because these are the scenarios that are suitable for the joint study and observation of Bell nonlocality and Bell-Kochen-Specker contextuality; practically, because, in some particular cases (as in the example we present), the compatibility relations can be implemented in a device-independent manner, and so such scenarios may be useful for the implementation of stronger device-independent information processing protocols, such as device-independent certification of entanglement, as in the example we present. ## II The scenario Consider a scenario where two parties, Alice and Bob, are able to perform different measurements on their subsystems of a shared physical system. The measurements are implemented by black-boxes, of which only classical inputs and outputs are available to the users. The inputs and outputs of Alice’s box are labeled $x\in\mathcal{X}$ and $a\in\mathcal{A}$, respectively, and the inputs and outputs of Bob’s box are labeled $y\in\mathcal{Y}$ and $b\in\mathcal{B}$, respectively. Since the users do not have access to the inner workings of the measurement devices, but only to their classical inputs and outputs, the best description of the experiment is given by the joint probabilities $p(a,b\|x,y)$ of the parties to observe outputs $a$ and $b$, on the condition that inputs $x$ and $y$ are chosen, respectively. The collection of probabilities $p(a,b\|x,y)$ for all $a\in\mathcal{A}$, $b\in\mathcal{B}$, $x\in\mathcal{X}$, $y\in\mathcal{Y}$ is referred as the _behavior_ or the _empirical model_ of the box, and will be denoted $\mathsf{p}$. A behavior is said to be _no-signalling_ if the choice of input of one of the parties cannot influence the marginal probability distribution of outcomes of the other, _i.e._, it satisfies the following _no-signalling conditions_: \[p(a\|x)= \sum_{b}p(a,b\|x,y)=\sum_{b}p(a,b\|x,y^{\prime}),\] (2a) \[p(b\|y)= \sum_{a}p(a,b\|x,y)=\sum_{a}p(a,b\|x^{\prime},y).\] (2b) More restrictedly, a behavior is said to be _local_ if there exist a variable $\lambda$, a probability distribution $q(\lambda)$, and probability distributions $p(a\|x,\lambda)$ and $p(b\|y,\lambda)$ such that \[p(a,b\|x,y)=\sum_{\lambda}q(\lambda)p(a\|x,\lambda)p(b\|y,\lambda).\] (3) It can be shown that the probability distributions $p(a\|x,\lambda)$ and $p(b\|y,\lambda)$ can be made deterministic without loss of generality. If the boxes perform measurements on quantum systems, then the probabilities are given by Born’s rule: \[p(a,b\|x,y)=\mathrm{tr}\left(\rho P_{a\|x}\otimes Q_{b\|y}\right),\] (4) where $\rho$ is the density operator that describes the state of the joint system while $P_{a\|x}$ and $Q_{b\|y}$ are, in general, POVM effects. It is well known that, in Bell scenarios, the set of local behaviors is strictly contained in the set of quantum behaviors, which, in its turn, is strictly contained in the set of no-signalling behaviors. Now, consider the single black box of Bob, with inputs $y\in\mathcal{Y}$, and outputs $b\in\mathcal{B}$. Suppose, however, that some measurements are _compatible_, and that each set of compatible measurements defines a _context_, $\mathsf{y}\subset\mathcal{Y}$ – _sans serif_ types refer to labels that represent ordered tuples of the corresponding _serif_ labels, _e. g._, $\mathsf{y}=(y_{i},\dots,y_{j})$, $\mathsf{b}=(b_{k},\dots,b_{l})$. Let $\mathcal{C}=\{\mathsf{y}\}$ denote the set of possible contexts of the scenario; it is usual to represent this set by means of the _compatibility hypergraph_$G=(V,E)$, where each measurement is associated to a vertex $v\in V$ and each context associated to a hyperedge $e\in E$. In particular, in scenarios where the contexts have cardinality $2$, $G$ will take the form of a regular graph. Compatible measurements can be jointly performed, and a joint probability distribution of the outcomes can be defined. Let $\mathsf{b}$ denote the ordered outcomes of the measurements in a context $\mathsf{y}$. Then, the behavior of this single box is best described by the probabilities $p(\mathsf{b}\|\mathsf{y})$. Noting that an individual measurement can appear in more than one context, it is usual to assume that the marginal behavior of each individual measurement $y\in\mathcal{Y}$ to be well defined, regardless of the context. This leads to the so-called _no-disturbance_ conditions: \[p(b\|y)=\sum_{{\mathsf{b}/b}}p(\mathsf{b}\|\mathsf{y})=\sum_{{\mathsf{b}/b}}p( \mathsf{b}\|\mathsf{y}^{\prime}),\] (5) where $\mathsf{b}/b$ denotes all labels in $\mathsf{b}$ except $b$, for all $b$ in $\mathcal{B}$, for all $y\in\mathcal{Y}$, and for all $\mathsf{y},\mathsf{y}^{\prime}\in\mathcal{C}$ such that $y\in\mathsf{y}\cap\mathsf{y}^{\prime}$. Suppose, now, a bipartite scenario, as considered previously, but let Bob be able to perform joint measurements according to given compatibility rules that lead to a set of contexts $\mathcal{C}$. We refer to this scenario as a _Bell scenario extended with compatible measurements_, or _extended Bell scenario_, for short. Then, the joint behavior of the boxes will be given by probabilities $p(a,\mathsf{b}\|x,\mathsf{y})$, for all $a\in\mathcal{A}$, $\mathsf{b}\in\mathcal{B}^{\|\mathsf{y}\|}$, $x\in\mathcal{X}$ and $\mathsf{y}\in\mathcal{C}$. We assume the behavior to be no-signalling, and the marginal, local behavior of Bob’s box to obey the no-disturbance conditions. This measurement scenario is illustrated in the lower panel of Fig. 1. <figure><img src="content_image/1806.09232/x1.png"><figcaption>Figure 1: Upper panel: a standard Bell nonlocality scenario. In each round ofthe experiment, party A (B) chooses measurement x (y) to perform on itsrespective subsystem, obtaining outcome a (b). The experiment is described bythe conditional probabilities p(a,b\|x,y). Lower panel: a Bell nonlocalityscenario where party B is able to perform compatible measurements. Now, ineach round, party B chooses two (or more, according to the context)measurements to be jointly performed, y and y′, obtaining outcomes b and b′,respectively. The experiment is described by the conditional probabilitiesp(a,b,b′\|x,y,y′); defining y=(y,y′) and b=(b,b′), the same conditionalprobabilities can be written as p(a,b\|x,y).</figcaption></figure> We define a behavior to be local in this scenario if there are a variable $\lambda$, a probability distribution $q(\lambda)$, and probability distributions $p(a\|x,\lambda)$ and $p(\mathsf{b}\|\mathsf{y},\lambda)$ such that \[p(a,\mathsf{b}\|x,\mathsf{y})=\sum_{\lambda}q(\lambda)p(a\|x,\lambda)p(\mathsf{b }\|\mathsf{y},\lambda).\] (6) Here, $p(a\|x,\lambda)$ can be assumed to be deterministic probability distributions. This is due to the fact that the set of marginal behaviors $\mathsf{p}_{A}=\{p(a\|x)\}$ is a convex set with finitely many extremal points, _i.e._, a _polytope_, whose vertices are exactly the deterministic distributions that suffice in the definition; let $\mathcal{P}_{A}$ denote this polytope. The set of marginal behaviors $p_{B}=\{p(\mathsf{b}\|\mathsf{y})\}$ is also a polytope, since it is characterized by the intersection of a finite number of subspaces, given by the no-disturbance conditions Eq. (5), together with nonnegativity and normalization conditions of probability distributions. As a polytope, it has a finite number of extremal points; however, they are not necessarily deterministic probability distributions. Let $\mathcal{P}_{B}$ denote this polytope, the _no-disturbance polytope_ of party $B$. It is easy to see, then, that, in this definition, the joint behavior $\mathsf{p}=\{p(a,\mathsf{b}\|x,\mathsf{y})\}$ will be a convex combination of a finite set of points, so the set of $\mathsf{p}$ will also be a polytope, $\mathcal{P}_{AB}$, the _local and no-disturbance polytope_; its vertices are all possible “products” between the vertices of $\mathcal{P}_{A}$ and $\mathcal{P}_{B}$. Now, given that the set of local behaviors is a polytope whose vertices are known (provided the vertices of $\mathcal{P}_{B}$ are known), we can change its representation by means of specialized software, such as porta[26] or panda[27], and obtain the inequalities associated to its facets. Such inequalities will be Bell-like inequalities whose violation certify Bell nonlocality, in the sense that these correlations can not be explained locally, by Eq. (6). ## III Compatible measurements on quantum systems In the measurement scenarios introduced in the previous section, it is assumed that compatible measurements are performed by one or more of the parties. Thus, quantum realizations of such measurement scenarios require quantum measurements that are compatible according to the predefined contexts. In quantum theory, compatibility of measurements is usually defined in terms of _observables_. An observable is an Hermitian operator acting on the Hilbert space of the quantum system, that is related to a projective measurement by means of its spectral decomposition. Two observables $B_{y}$ and $B_{y^{\prime}}$ are said to be compatible if they commute, $\left[B_{y},B_{y^{\prime}}\right]=B_{y}B_{y^{\prime}}-B_{y^{\prime}}B_{y}=0$. This condition implies that both operators can be diagonalized in the same basis, and that a third operator that represents the joint action of them can be defined. Let $B_{y}=\sum_{b}bQ_{b\|y}$ and $B_{y^{\prime}}=\sum_{b^{\prime}}b^{\prime}R_{b^{\prime}\|y^{\prime}}$ be the spectral decompositions of observables $B_{y}$ and $B_{y^{\prime}}$, where $b$ and $b^{\prime}$ are their respective eigenvalues and $Q_{b\|y}$ and $R_{b^{\prime}\|y^{\prime}}$ are projectors onto subspaces – not necessarily one-dimensional – of the local Hilbert space of Bob’s system. A sufficient condition for the observables to commute is $\left[Q_{b\|y},R_{b^{\prime}\|y^{\prime}}\right]=0$ for all $b$ and $b^{\prime}$. We, thus, use this condition to define a pair of compatible projective measurements, as the following. Let $\left\{Q_{b\|y}\right\}$ and $\left\{R_{b^{\prime}\|y^{\prime}}\right\}$ be projective measurements labeled by $y$ and $y^{\prime}$. They are compatible if, for all $b$ and $b^{\prime}$, $\left[Q_{b\|y},R_{b^{\prime}\|y^{\prime}}\right]=0$ holds. As previously, defining $\mathsf{b}=\left(b,b^{\prime}\right)$ and $\mathsf{y}=\left(y,y^{\prime}\right)$ allows us to write the joint projective measurement as $\left\{S_{\mathsf{b}\|\mathsf{y}}\right\}$, where each projector is given by $S_{\mathsf{b}\|\mathsf{y}}=Q_{b\|y}R_{b^{\prime}\|y^{\prime}}$. This construction can be directly extended to contexts that involve more than two measurements. The same definition can be extended to POVMs, as follows. Let $\left\{Q_{b\|y}\right\}$ and $\left\{R_{b^{\prime}\|y^{\prime}}\right\}$ be POVM measurements labeled by $y$ and $y^{\prime}$, _i.e._, $Q_{b\|y}\geq 0$ for all $b$, $R_{b^{\prime}\|y^{\prime}}\geq 0$ for all $b^{\prime}$, and $\sum_{b}Q_{b\|y}=\sum_{b^{\prime}}R_{b^{\prime}\|y^{\prime}}=\mathds{1}$, the identity operator. Then, if, for all $b$ and $b^{\prime}$, $\left[Q_{b\|y},R_{b^{\prime}\|y^{\prime}}\right]=0$, we define measurements $y$ and $y^{\prime}$ to be compatible, and a joint POVM can be defined as $\left\{S_{\mathsf{b}\|\mathsf{y}}\right\}$, where each POVM element is given by $S_{\mathsf{b}\|\mathsf{y}}=Q_{b\|y}R_{b^{\prime}\|y^{\prime}}$ – note that the commutation relations imply that the product of the positive semi-definite operators is itself positive semi-definite, and it is easy to see that $\sum_{\mathsf{b}}S_{\mathsf{b}\|\mathsf{y}}=\sum_{b,b^{\prime}}Q_{b\|y}R_{b^{ \prime}\|y^{\prime}}=\mathds{1}$. Now, consider a bipartite measurement scenario where one of the parties, Bob, is able to implement compatible measurements on his subsystem, as defined in Sec. II and depicted in the lower panel of Fig. 1. Assuming the measurements are performed on a shared quantum system in state $\rho$, the joint probabilities of obtaining outcomes $a$ and $\mathsf{b}$ for respective measurements $x$ and $\mathsf{y}$ are given by \[p(a,\mathsf{b}\|x,\mathsf{y})=\mathrm{tr}\left(\rho P_{a\|x}\otimes S_{\mathsf{b }\|\mathsf{y}}\right),\] (7) where $\{P_{a\|x}\}$ and $\{S_{\mathsf{b}\|\mathsf{y}}\}$ are, in general, POVM measurements, with, as previously discussed, the elements of the later given by $S_{\mathsf{b}\|\mathsf{y}}=Q_{b\|y}R_{b^{\prime}\|y^{\prime}}$, where $\{Q_{b\|y}\}$ and $\{R_{b^{\prime}\|y^{\prime}}\}$ are POVMs associated to measurements $y$ and $y^{\prime}$, respectively, respecting $\left[Q_{b\|y},R_{b^{\prime}\|y^{\prime}}\right]=0$ for all $b$ and $b^{\prime}$. Regarding the nonlocality of quantum systems, an important question is whether scenarios with compatible measurements may activate the nonlocality of entangled states that are local in standard Bell scenarios. Let us first define locality of quantum states in standard Bell scenarios and in scenarios extended with compatibilities. Let $\rho$ be a density operator acting on $\mathcal{H}=\mathcal{H}_{d_{A}}\otimes\mathcal{H}_{d_{B}}$, where $d_{A}$ and $d_{B}$ are the dimensions of the Hilbert spaces associated to the subsystems $A$ and $B$, respectively. We define $\rho$ to be _local_ if, _for all_ POVMs $\{P_{a\|x}\}$ acting on $\mathcal{H}_{d_{A}}$ and $\{Q_{b\|y}\}$ acting on $\mathcal{H}_{d_{B}}$, there exist a variable $\lambda$ in a set $\Lambda$ and probability distributions $q(\lambda)$, $p(a\|x,\lambda)$ and $p(b\|y,\lambda)$ such that \[\mathrm{tr}\left(\rho P_{a\|x}\otimes Q_{b\|y}\right)=\int_{\Lambda}p(a\|x, \lambda)p(b\|y,\lambda)q(\lambda)d\lambda.\] (8) If, in particular, Eq. (8) holds for projective measurements, then we say $\rho$ is local with respect to projective measurements. Now, let $\mathcal{H}_{d^{\prime}_{B}}$ be a Hilbert space associated to subsystem $B$, where $d_{B}^{\prime}\geq d_{B}$, and let $\rho^{\prime}$ denote state $\rho$ trivially embedded in $\mathcal{H}=\mathcal{H}_{d_{A}}\otimes\mathcal{H}_{d^{\prime}_{B}}$. We define $\rho$ to be _local in an extended Bell scenario_ – or _extended-local_, for short – if, _for all_ POVMs $\{P_{a\|x}\}$ acting on $\mathcal{H}_{d_{A}}$, and _for all_ compatible pairs of POVMs $\{Q_{b\|y}\}$ and $\{R_{b^{\prime}\|y^{\prime}}\}$ acting on $\mathcal{H}_{d_{B}^{\prime}}$, there exist a variable $\lambda$ in a set $\Lambda$ and probability distributions $q(\lambda)$, $p(a\|x,\lambda)$, and no-disturbing probability distributions $p(b,b^{\prime}\|y,y^{\prime},\lambda)$ such that \[\mathrm{tr}\left(\rho^{\prime}P_{a\|x}\otimes Q_{b\|y}R_{b^{\prime} \|y^{\prime}}\right)\\ =\int_{\Lambda}p(a\|x,\lambda)p(b,b^{\prime}\|y,y^{\prime},\lambda) q(\lambda)d\lambda.\] (9) In Appendix A we prove the following: Theorem: If $\rho$ is extended-local, then $\rho$ is local. Since the above theorem does not guarantee the equivalence between locality and extended-locality of quantum states, it could be the case that a local quantum state $\rho$ could lead to a nonlocal behavior in a scenario with compatible measurements; in other words, a scenario with compatible measurements could activate the nonlocality of $\rho$. However, even if the converse of the theorem is true and equivalence between locality and extended-locality as properties of quantum states holds, a scenario with compatible measurements could be more economical, in terms of the number of measurements, for instance, than a standard Bell scenario to display nonlocal behavior of a quantum state. In the following section, we present, in some detail, an interesting and simple example of Bell scenario with local compatible measurements where these approaches and scenario provide an advantage for the detection of nonlocality of two important families of quantum states, as compared to similar standard Bell scenarios. ## IV Application Consider a bipartite scenario where Alice is able to choose between two dichotomic measurements to perform, so $\mathcal{A}=\{\pm 1\}$, $\mathcal{X}=\{0,1\}$, and Bob is able to perform four dichotomic measurements, $\mathcal{B}=\{\pm 1\}$ and $\mathcal{Y}=\{0,1,2,3\}$, assumed to be compatible according to the contexts $\mathcal{C}=\{\{0,1\},\{1,2\},\{2,3\},\{3,0\}\}$. In this scenario, a behavior $\mathsf{p}$ will have $64$ components; however, due to normalization, no-signalling, and no-disturbance, only $26$ components will be independent. Due to this reason, it is convenient to work with correlators instead of probabilities. Defining new random variables $A_{x}$, $B_{y}$ and $B_{\mathsf{y}}=B_{y_{1}}B_{y_{2}}$, valued on the set $\{\pm 1\}$, to represent the outcomes of the respective measurements, the “full” correlators are defined as the mean value of their product, as follows: \[\left<A_{x}B_{\mathsf{y}}\right>=p(ab_{1}b_{2}=1\|x,\mathsf{y})-p(ab_{1}b_{2}=- 1\|x,\mathsf{y}),\] (10) for all $x\in\mathcal{X}$ and $\mathsf{y}\in\mathcal{C}$, $(b_{1},b_{2})$ being the respective outcomes of measurements $(y_{1},y_{2})=\mathsf{y}$. “Marginal” correlators $\left<A_{x}\right>$, $\left<B_{y}\right>$, $\left<B_{\mathsf{y}}\right>$, $\left<A_{x}B_{y}\right>$, for all $x\in\mathcal{X}$, $y\in\mathcal{Y}$ and $\mathsf{y}\in\mathcal{C}$, are analogously defined with the corresponding marginal probability distributions. The behaviors will, then, be vectors $\vec{c}\in\mathds{R}^{d}$, where each component is a correlator. It is easy to check that there are exactly $26$ correlators in total, so $d=26$; and, given the correlators, all the $64$ probabilities can be retrieved as \[p(a,\mathsf{b}\|x,\mathsf{y})=\frac{1}{8}\left[1+a\left<A_{x} \right>+{b_{1}}\left<B_{y_{1}}\right>+{b_{2}}\left<B_{y_{2}}\right>\right.\\ +{b_{1}b_{2}}\left<B_{\mathsf{y}}\right>+{ab_{1}}\left<A_{x}B_{y_ {1}}\right>\\ +\left.{ab_{2}}\left<A_{x}B_{y_{2}}\right>+{ab_{1}b_{2}}\left<A_{ x}B_{\mathsf{y}}\right>\right].\] (11) Now, we want to characterize the facets of the local and no-disturbance polytope of the scenario, $\mathcal{P}_{AB}$. The first step is to obtain all the extremal points of Bob’s no-disturbance polytope. In all scenarios where the compatibility relations among dichotomic measurements are cyclic, it is known [28] that the extremal points of the no-disturbance polytope, up to outcome or measurement relabellings that respect the compatibility relations¹, are either of the form ² [FOOTNOTE:1][ENDFOOTNOTE] [FOOTNOTE:2][ENDFOOTNOTE] \[\left<B_{y}\right>=\pm 1\] (12a) \[\left<B_{\mathsf{y}}\right>=\prod_{y\in\mathsf{y}}\left<B_{y} \right>,\] (12b) for all $y\in\mathcal{Y}$ and $\mathsf{y}\in\mathcal{C}$, or of the form \[\left<B_{y}\right>=0,\,\forall\,y\in\mathcal{Y};\] (13a) (13b) Then, the extremal points of the local, locally no-disturbing polytope will be behaviors whose bipartite correlators are of the form \[\left<A_{x}B_{y}\right>=\left<A_{x}\right>\left<B_{y}\right>,\] (14a) \[\left<A_{x}B_{\mathsf{y}}\right>=\left<A_{x}\right>\left<B_{ \mathsf{y}}\right>,\] (14b) where $\left<A_{x}\right>\in\{\pm 1\}$ and the behavior of Bob’s box is given by either Eqs. (12) or Eqs. (13), for all $x\in\mathcal{X}$, $y\in\mathcal{Y}$ and $\mathsf{y}\in\mathcal{C}$. Having all the extremal points, we used panda to obtain the facets of the local, no-disturbance polytope. We found $26$ classes of inequalities, all of which are given in Appendix B. This result should be contrasted to the fact that, in standard bipartite Bell scenarios where no assumption regarding compatibility is made, if the number of measurements of one of the parties is $2$ and they are dichotomic, the only Bell inequality, up to rellabelings, is the CHSH inequality, as has been proven by Pironio in Ref. [24]. Actually, the compatibility relations we assume can be implemented in a tripartite Bell scenario, if we assign measurements $B_{0}$ and $B_{2}$ to one party (say, Bob${}_{0}$) and $B_{1}$ and $B_{3}$ to another (say, Bob${}_{1}$). Due to this reason, some of the inequalities we obtain are equivalent to Sliwa’s inequalities [31] (see discussion in Appendix C), the Bell inequalities that completely characterize the local polytope in a tripartite scenario where each party is able to perform two dichotomic measurements. Note, however, that, had we assumed another compatibility structure for Bob’s measurements, _e.g._, if the compatibility graph $G$ was a pentagon instead of a square, than it would not be possible to relate the scenario to any usual Bell scenario, since it would not be possible to assign subsets of measurements to two or more parties in a way that is consistent with the the assumed compatibilities ³. [FOOTNOTE:3][ENDFOOTNOTE] Among the $26$ inequalities we obtain, one has the form \[2\left<B_{0}\right>+\left<(1-B_{0})[A_{0}(B_{1}+B_{3})+A_{1}(B_{1}-B_{3})] \right>\leq 2.\] (15) Note that the term in square brackets corresponds to the left-hand side of a CHSH inequality between Alice and measurements $1$ and $3$ of Bob. To study the quantum violation of inequality Eq. (15), it is convenient to define observables \[A_{x} =P_{+\|x}-P_{-\|x},\] (16a) \[B_{y} =Q_{+\|y}-Q_{-\|y},\] (16b) where $P_{a\|x}$ and $Q_{b\|y}$ are projectors associated to outcomes $a$ and $b$ of measurements $x$ and $y$, respectively, so the correlators will be evaluated as , where $B_{\mathsf{y}}=B_{y_{1}}B_{y_{2}}$, and $[B_{y_{1}},B_{y_{2}}]=0$ for all $\mathsf{y}\in\mathcal{C}$. Inequality Eq. (15) is equivalent to the class $\#4$ of Sliwa [31]. For quantum systems, it is maximally violated up to the value $4\sqrt{2}-2$, attained by a two-qubit maximally entangled state embedded in $\mathds{C}^{2}\otimes\mathds{C}^{4}$[32]. We now show that this inequality can certify the nonlocality of bipartite quantum states that do not violate the CHSH inequality. Consider the following two-parameter family of two-qubit states \[\rho\left(\alpha,w\right)=w\left\|\psi(\alpha)\right>\!\!\left<\psi(\alpha) \right\|+(1-w)\left\|00\right>\!\!\left<00\right\|,\] (17a) where \[\left\|\psi(\alpha)\right>=\sqrt{\alpha}\left\|01\right>+\sqrt{1-\alpha}\left\|10 \right>.\] (17b) This family is known to include the two-qubit states with highest entanglement (as quantified by negativity and concurrence) that do not violate the CHSH inequality [33]. We, then, perform a seesaw optimization, embedding the states in $\mathds{C}^{2}\otimes\mathds{C}^{4}$ to impose the compatibility relations among the measurements (details in the Appendix D), and search for the lowest value of $w$ such that the inequality is violated, for each $\alpha$. The results are displayed in Fig. 2, where we also plot the critical values of $w$ as a function of $\alpha$ for the CHSH inequality, provided by means of the Horodecki criterium [34], and upper bounds on the critical values of $w$, obtained by means of a seesaw optimization, for the I${}_{3322}$ inequality [25] – a relevant Bell inequality in the scenario where Alice and Bob perform three dichotomic measurements each – , given by the expression \[-\left<A_{1}\right>-\left<A_{2}\right>-\left<B_{1}\right>-\left<B _{2}\right>-\left<A_{1}B_{1}\right>-\left<A_{2}B_{1}\right>-\left<A_{3}B_{1} \right>\\ -\left<A_{1}B_{2}\right>-\left<A_{2}B_{2}\right>+\left<A_{3}B_{2} \right>-\left<A_{1}B_{3}\right>+\left<A_{2}B_{3}\right>\leq 4.\] (18) In fact, the state $\rho\left(0.80,0.85\right)$ in family Eq. (17) was the example considered in Ref. [25] of a state that does not violate the CHSH inequality, that, however, violates I${}_{3322}$. In Ref. [35], the authors show that, for $\alpha=0.80$, inequality I${}_{3322}$ is violated for $w\gtrsim 0.837$, in excellent agreement with the value $0.838$ we obtain, corroborating with the precision of our lower bounds. <figure><img src="content_image/1806.09232/x2.png"><figcaption>Figure 2: For states ρ(α,w) defined in Eq. (17), the plot shows the criticalparameter w, as a function of α, above which inequality Eq. (15) is violated(red circles), above which the CHSH inequality is violated (black squares),and above which the I3322 inequality is violated (blue triangles). The pointsfor inequalities Eq. (15) and I3322 were obtained via a see-saw optimization,and are, hence, upper bounds on the actual critical points. The points for theCHSH are exact, obtained by means of Horodecki’s necessary and sufficientcriterium for violation of the CHSH inequality by two-qubit statesHorodecki1995 . Inequality Eq. (15) is denoted I#15 for consistency with theappendices and the data related to the red points, which are available at Ref.data .</figcaption></figure> Now, consider the following two-parameter family of two-qubit states \[\sigma\left(\alpha,w\right)=w\left\|\psi(\alpha)\right>\!\!\left<\psi(\alpha) \right\|+(1-w)\mathds{1}/4,\] (19a) where, as previously, \[\left\|\psi(\alpha)\right>=\sqrt{\alpha}\left\|01\right>+\sqrt{1-\alpha}\left\|10 \right>.\] (19b) For $\alpha=1/2$, the states obtained are locally equivalent to two-qubit Werner states, known to be entangled for $w>1/3$, and local with respect to projective measurements for $w\lesssim 0.68$[29]. The best-known bounds on the locality of the states in family Eq. (19) are provided in Ref. [13]. Applying the same methods adopted in the previous example, we were able to obtain upper bounds on the values of $w$, as a function of $\alpha$, above which states Eq. (19) violate inequality Eq. (15). Results are shown in Fig. 3, where it is clear that inequality Eq. (15) is better than the CHSH inequality to witness the nonlocality of this family of states specially in the range $0.7<\alpha<1$. Note that, compared to Fig. 2, points corresponding the the $I_{3322}$ inequality are absent, and this is due to the fact that $I_{3322}$ does not provide any advantage over the CHSH inequality to witness the nonlocality of this family of states. <figure><img src="content_image/1806.09232/x3.png"><figcaption>Figure 3: For states σ(α,w) defined in Eq. (19), the plot shows the criticalparameter w, as a function of α, above which inequality (15) is violated (redcircles), and above which the CHSH inequality is violated (black squares). Thepoints for inequality (15) were obtained via a see-saw optmization, and are,hence, upper bounds on the actual critical points. The points for the CHSH areexact, obtained by means of Horodecki’s necessary and sufficient criterium forviolation of the CHSH inequality by two-qubit states Horodecki1995 . As inFig. 2, inequality (15) is denoted I#15 for consistency with the appendicesand the data related to the red points, which are available at data .</figcaption></figure> ## V Discussion Although the simple examples we consider are sufficient evidence of the potential of the approach we introduce, it is only the first step in a direction for the study of Bell nonlocality. In principle, one could assume a plethora of more intricate local compatibility structures in scenarios with any number of parties, leading to a range of Bell-like inequalities. The specific compatibility structure we consider can be realized – with some loss of generality – in a tripartite scenario, where each party is able to perform two dichotomic measurements. An advantage of the tripartite implementation is that one does not need to _assume_ the compatibility relations; they would naturally hold due to space-like separation of the parties, implying that the test would be _device-independent_. Also, note that the locality assumption in tripartite scenarios is more restrictive than the condition we demand in our scenario, since each $p\left(\mathsf{b}\middle\|\mathsf{y},\lambda\right)$ is required to obey the nondisturbance condition (5) in the latter, as opposed to strict locality, in the former. This shows that our local and nondisturbance polytope is strictly larger than the corresponding tripartite local polytope; more explicitly, notice that any vertex whose marginal behavior $p\left(\mathsf{b}\middle\|\mathsf{y}\right)$ obeys Eqs. (13) is not tripartite-local. One interesting fact, however, that is discussed in more detail in the Appendix C, is that some of the inequalities we obtain are isomorphic to the tripartite inequalities obtained by Sliwa [31], including Inequality Eq. (15). This equivalence, together with the results presented in this manuscript, prove that there are multipartite Bell inequalities that are useful to witness the Bell nonlocality of bipartite quantum states in a subtler way than just merging parts. Two other scenarios that demand comparison are the ones obtained when we consider joint measurements of $B_{y}$ and $B_{y+1}$ (addition modulo $4$) as new measurements $B^{\prime}_{y}$. In the first case, if we consider outcomes $b^{\prime}_{y}=b_{y}b_{y+1}$, Bob will have four mutually incompatible dichotomic measurements, and Ref. [24] shows that the only relevant inequalities for describing the local polytope belong to CHSH family. Hence, our inequalities can show nonlocal behavior not revealed when such coarse grained version is considered. The specific inequality Eq. (15), for example, could never be written in such a scenario, since correlators like $\left<B_{y}\right>$ or $\left<A_{x}B_{y}\right>$ cannot be written as functions of the probabilities of the outcomes of $B^{\prime}_{y}$. In the second case, if we consider $B^{\prime\prime}_{y}=\left(B_{y},B_{y+1}\right)$, then Bob will have four mutually incompatible four-outcome measurements. Once more, Ref. [24] implies that only CHSH inequalities are relevant for such a scenario, while the extra correlations coming from each $B_{y}$ being an element of $B^{\prime\prime}_{y}$ and $B^{\prime\prime}_{y-1}$ also would make it a somehow special realization of this Bell scenario (with such additional constraints). Also, it is worth mentioning that somewhat similar scenarios have been previously considered, with different focuses and assumptions. In Ref. [18], the authors present a formalism to study nonlocality in sequential measurement scenarios, mainly focused on the proper consequences due to the causal structures underlying the sequences of measurements. In Ref. [36], the author argues, considering a bipartite scenario where one of the parties is able to perform sequential measurements, that local contextuality may lead to Bell nonlocality, although the definition of locality adopted is somehow intrincated and seems to implicitly assume local noncontextuality. ## VI Conclusion In this manuscript we present an approach to Bell nonlocality, an approach that takes into account the possibility that one (or more) of the parties is able to perform joint measurements according to given compatibility rules. We provide a precise definition of locality, or, more specifically, of local behaviors in these scenarios. Applying this definition, we completely characterize the set of local behaviors in the simplest scenario with compatible measurements, and we show how this approach leads to new, interesting Bell-like inequalities that may provide advantages over known Bell inequalities in witnessing the nonlocality of quantum states. We discuss in some detail two examples where such advantage appears; in particular, both families of states Eqs. (17) and (19) show nonlocal behavior in a scenario with compatible measurements for parameters where neither CHSH, nor $I_{3322}$, are able to witness it. In the scenario considered, the compatibility relations can be implemented in a device-independent manner, and, thus, the examples show explicitly that this approach may provide advantages over standard Bell nonlocality for device-independent certification of entanglement. ###### Acknowledgements. R. R. thanks Jean-Daniel Bancal and Marcos César de Oliveira. The authors thank the anonymous referees who contributed with suggestions that made the manuscript much better than its previous versions. This work used computing resources and assistance of the John David Rogers Computing Center (CCJDR) in the Institute of Physics “Gleb Wataghin”, University of Campinas. The authors acknowledge support from the Brazilian agencies CNPq and FAEPEX. R. R. is also supported by Grant No. 2018/07258-7, São Paulo Research Foundation (FAPESP). This work is part of the Brazilian National Institute for Science and Technology on Quantum Information. ## Appendix A Locality of quantum states in scenarios with compatible measurements In this Appendix, we prove that if a quantum state $\rho$ is local in a scenario with compatible measurements, it is local in a standard Bell scenario. For simplicity, we have assumed a scenario where compatibilities are present only in party $B$, and each context is composed of two measurements. However, the following Theorem would still hold even under general definitions, which also take into account compatibilities in party $A$ and arbitrarily-sized contexts in both parties. Before proceeding, let us introduce notation and important definitions. Let $\rho$ be a density operator acting on $\mathcal{H}=\mathcal{H}_{d_{A}}\otimes\mathcal{H}_{d_{B}}$, where $d_{A}$ and $d_{B}$ are the dimensions of the Hilbert spaces associated to the subsystems $A$ and $B$, respectively. We define $\rho$ to be _local_ if, _for all_ POVMs $\{P_{a\|x}\}$ acting on $\mathcal{H}_{d_{A}}$ and $\{Q_{b\|y}\}$ acting on $\mathcal{H}_{d_{B}}$, there exist a variable $\lambda$ in a set $\Lambda$ and probability distributions $q(\lambda)$, $p(a\|x,\lambda)$ and $p(b\|y,\lambda)$ such that \[\mathrm{tr}\left(\rho P_{a\|x}\otimes Q_{b\|y}\right)=\int_{\Lambda}p(a\|x, \lambda)p(b\|y,\lambda)q(\lambda)d\lambda.\] (20) In the particular case where all measurements are assumed to be projective, we say $\rho$ is local with respect to projective measurements. Now, let $\mathcal{H}_{d^{\prime}_{B}}$ be a Hilbert space associated to subsystem $B$, where $d_{B}^{\prime}\geq d_{B}$, and let $\rho^{\prime}$ denote state $\rho$ trivially embedded in $\mathcal{H}=\mathcal{H}_{d_{A}}\otimes\mathcal{H}_{d^{\prime}_{B}}$. We define $\rho$ to be _local in an extended Bell scenario with compatible measurements in party $B$_, or _extended-local_, for short, if, _for all_ POVMs $\{P_{a\|x}\}$ acting on $\mathcal{H}_{d_{A}}$, and _for all_ compatible pairs of POVMs $\{Q_{b\|y}\}$ and $\{R_{b^{\prime}\|y^{\prime}}\}$ acting on $\mathcal{H}_{d_{B}^{\prime}}$, there exist a variable $\lambda$ in a set $\Lambda$ and probability distributions $q(\lambda)$, $p(a\|x,\lambda)$, and a no-disturbing probability distribution $p(b,b^{\prime}\|y,y^{\prime},\lambda)$ such that \[\mathrm{tr}\left(\rho^{\prime}P_{a\|x}\otimes Q_{b\|y}R_{b^{\prime} \|y^{\prime}}\right)\\ =\int_{\Lambda}p(a\|x,\lambda)p(b,b^{\prime}\|y,y^{\prime},\lambda) q(\lambda)d\lambda.\] (21) In the particular case where all measurements are assumed to be projective, we say $\rho$ is extended-local with respect to projective measurements. Theorem: If $\rho$ is extended-local, then $\rho$ is local. Proof: Let $\rho$ be extended-local, $d^{\prime}_{B}=d_{B}$, and $R_{b^{\prime}\|y^{\prime}}=\mathds{1}/n$, where $n$ denotes the number of outcomes of the measurement, for all $b^{\prime}$ and $y^{\prime}$, and $\mathds{1}$ denotes the identity operator in $\mathcal{H}_{d_{B}}$. By assumption, for all POVMs $\{Q_{b\|y}\}$ acting on $\mathcal{H}_{d_{B}}$ Eq. (21) holds. Now, marginalising over $b^{\prime}$: \[\sum_{b^{\prime}}\mathrm{tr}\left(\rho P_{a\|x}\otimes Q_{b\|y}R_{b ^{\prime}\|y^{\prime}}\right)\\ =\int_{\Lambda}p(a\|x,\lambda)\left(\sum_{b^{\prime}}p(b,b^{\prime }\|y,y^{\prime},\lambda)\right)q(\lambda)d\lambda;\] (22) leading to \[\mathrm{tr}\left(\rho P_{a\|x}\otimes Q_{b\|y}\right)=\int_{\Lambda}p(a\|x, \lambda)p(b\|y,\lambda)q(\lambda)d\lambda;\] (23) which, by definition, holds for all POVMs $\{P_{a\|x}\}$ acting on $\mathcal{H}_{d_{A}}$ and $\{Q_{b\|y}\}$ acting on $\mathcal{H}_{d_{B}}$. Thus, $\rho$ is local. ## Appendix B Inequalities In the scenario we have considered, Alice is able to perform two dichotomic measurements, and Bob is able to perform four dichotomic measurements; Bob’s measurements, however, can be jointly performed according to compatibility rules provided by the contexts $\mathcal{C}=\{\{0,1\},\{1,2\},\{2,3\},\{3,0\}\}$. Using the representation of correlators, we have listed all extremal points of the polytope of behaviors that are local, according to the definition we provided in the main text, such that Bob’s marginal behaviors respect the no-disturbance conditions. Bob’s extremal marginal behaviors belong to either one of two distinct classes: * noncontextual behaviors: are behaviors of the form \[\left<B_{y}\right>=\pm 1\] (24a) \[\left<B_{\mathsf{y}}\right>=\prod_{{y\in\mathsf{y}}}\left<B_{y} \right>,\] (24b) for all $y\in\mathcal{Y}$ and $\mathsf{y}\in\mathcal{C}$; * contextual, no-disturbing behaviors: are behaviors of the form \[\left<B_{y}\right>=0,\,\forall\,y\in\mathcal{Y};\] (25a) (25b) Then, the extremal points of the local, locally no-disturbing polytope will be behaviors whose bipartite correlators are of the form \[\left<A_{x}B_{y}\right>=\left<A_{x}\right>\left<B_{y}\right>,\] (26a) \[\left<A_{x}B_{\mathsf{y}}\right>=\left<A_{x}\right>\left<B_{ \mathsf{y}}\right>,\] (26b) where $\left<A_{x}\right>\in\{\pm 1\}$ and the behavior of Bob’s box is given by either Eqs. (24) or Eqs. (25), for all $x\in\mathcal{X}$, $y\in\mathcal{Y}$ and $\mathsf{y}\in\mathcal{C}$. Having listed all the extremal points of the the local, no-disturbance polytope, we used the software panda[27] to change the representation of the polytope, and we obtained $26$ inequalities, up to rellabelings that respect the local compatibility rules. These inequalities are listed in Table 1. This table should be read as follows: each row represents an inequality, labeled by the number in the first column. Each column, then, has the coefficient of the correlator represented in the heading, where the measurements $x$, $y_{1}$, and $y_{2}$ are the corresponding numbers in the second row. The second to last column refers to the local bound $\beta_{L}$ of each inequality. In the last column, we list the quantum maxima $\beta_{Q}$ (exact up to the given precision) of each inequality. The maxima were upper bounded by means of the Navascués-Pironio-Acín [37] hierarchy of semidefinite programs that outerapproximate the set of quantum correlations, implemented in python with the aid of the NCPOL2SDPA[38] library. The values listed correspond to the third level of the hierarchy, and the optimizations were performed with the MOSEK[39] solver. We have also computed lower bounds on the maxima by means of a seesaw optimization, detailed in Appendix D. The lower and upper bounds on quantum maxima obtained differ by less than $5\times 10^{-4}$ for all inequalities; on average, they differ by $3\times 10^{-5}$. For the 15 inequalities that are equivalent to Sliwa’s inequalities, our results are in perfect agreement with those of [32; 40], where, among other results, quantum maxima are computed and analyzed for all Sliwa’s inequalities. As an example, consider inequality 25. It is (27) ## Appendix C Relation to Sliwa’s inequalities Note that the particular scenario we consider share some similarity with a tripartite Bell scenario, where each party is able to perform two dichotomic measurements; the correspondence becomes explicit if one considers $B_{0}$ and $B_{2}$ as the possible measurements of a second party, while $B_{1}$ and $B_{3}$ are the possible choices of a third party. This Bell scenario has been studied by Sliwa [31], who obtained 46 distinct classes of Bell inequalities by assuming full locality between the three parties. Had we considered only noncontextual marginal behaviors of Bob, Eq. (12), the inequalities obtained would be all equivalent to Sliwa’s 46 inequalities. However, by including points of the form of Eq. (13) (i) we obtain inequalities that are not equivalent to Sliwa’s; and (ii) any violation of the inequalities is a certification of bipartite nonlocality, a fact that would not be true otherwise, since stronger-than-classical (contextual) correlations in the marginal behavior of Bob could lead to violations of the inequalities obtained solely via Eq. (12). Sliwa’s inequalities are, as discussed, related to a polytope that is contained in the local, no-disturbance polytope we characterize. It would not be surprising, then, if some of the facets of the polytope were equivalent to the inequalities of Sliwa, and this is exactly what we observe. For the 15 inequalities we obtained that are equivalent to an inequality of Sliwa, we provide in the second column of Table I the number referring to the enumeration in Ref. [31]. # \| # S \| ⟨Ax⟩ \| ⟨By⟩ \| ⟨AxBy⟩ \| ⟨By1By2⟩ \| ⟨AxBy1By2⟩ \| βL \| βQ ---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| 0 \| 1 \| 0 \| 1 \| 2 \| 3 \| 00 \| 01 \| 02 \| 03 \| 10 \| 11 \| 12 \| 13 \| 01 \| 12 \| 23 \| 30 \| 001 \| 012 \| 023 \| 030 \| 101 \| 112 \| 123 \| 130 \| \| 1 \| 1 \| 1 \| 0 \| 1 \| 1 \| 0 \| 0 \| -1 \| -1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| -1 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1.000 2 \| - \| 2 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 \| 2 \| 0 \| 0 \| 1 \| 0 \| -1 \| 0 \| 0 \| 0 \| 1 \| 1 \| -1 \| -1 \| -1 \| -1 \| -1 \| 1 \| 0 \| 0 \| 4 \| 5.656 3 \| - \| 2 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 2 \| -1 \| 0 \| 0 \| 0 \| 1 \| 1 \| -1 \| 1 \| -1 \| -1 \| -1 \| -1 \| 0 \| 0 \| 4 \| 5.000 4 \| - \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 1 \| 0 \| 1 \| 0 \| 1 \| 0 \| -1 \| -1 \| 1 \| 1 \| -1 \| 0 \| -1 \| -1 \| 0 \| -1 \| 0 \| 0 \| 1 \| 4 \| 5.656 5 \| - \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| -1 \| -1 \| -1 \| 1 \| -1 \| 1 \| 0 \| -1 \| 0 \| -1 \| -1 \| 0 \| 1 \| 0 \| 4 \| 5.753 6 \| - \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 2 \| 0 \| 0 \| -1 \| 1 \| 1 \| -1 \| -1 \| -1 \| -1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 4 \| 5.000 7 \| - \| 1 \| 1 \| 1 \| 0 \| 1 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| -1 \| -1 \| 2 \| -1 \| -1 \| 0 \| 0 \| 0 \| 1 \| 1 \| -1 \| 1 \| 0 \| -1 \| -1 \| 4 \| 5.656 8 \| - \| 1 \| 1 \| 1 \| 0 \| 1 \| 0 \| 1 \| 2 \| 0 \| 1 \| 0 \| 0 \| -1 \| -1 \| 0 \| 0 \| -1 \| -1 \| -1 \| -1 \| 1 \| 0 \| -1 \| 1 \| 0 \| 1 \| 4 \| 5.753 9 \| - \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 2 \| 2 \| 1 \| 1 \| 0 \| 0 \| -1 \| -1 \| -1 \| -1 \| -1 \| -1 \| 0 \| 0 \| 1 \| 0 \| -1 \| 1 \| 0 \| 1 \| 4 \| 5.656 10 \| - \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 2 \| 1 \| 1 \| 0 \| 0 \| 1 \| -1 \| 0 \| -1 \| 1 \| 1 \| -1 \| 0 \| -1 \| 0 \| 0 \| -1 \| 0 \| -1 \| 1 \| 4 \| 5.656 11 \| 17 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| -1 \| 0 \| 2 \| 0 \| -1 \| 0 \| -2 \| 0 \| 4 \| 5.656 12 \| 6 \| 1 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| -1 \| -1 \| 1 \| 1 \| 0 \| 0 \| 0 \| -1 \| -1 \| 0 \| -1 \| 0 \| 1 \| 0 \| -1 \| 3 \| 4.656 13 \| - \| 1 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| -1 \| 0 \| 0 \| 0 \| 1 \| -1 \| -2 \| -1 \| 0 \| -1 \| 2 \| -1 \| 5 \| 7.012 14 \| - \| 1 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| -1 \| 0 \| 0 \| 0 \| 1 \| -1 \| 2 \| -1 \| 0 \| -1 \| -2 \| -1 \| 5 \| 6.656 15 \| 4 \| 0 \| 0 \| 2 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 \| 0 \| -1 \| 0 \| 0 \| 0 \| 0 \| -1 \| 0 \| 0 \| -1 \| -1 \| 0 \| 0 \| 1 \| 2 \| 3.656 16 \| 19 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 \| 2 \| 1 \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| -1 \| -1 \| -1 \| -1 \| 0 \| 0 \| 1 \| -1 \| 1 \| -1 \| 4 \| 5.782 17 \| 18 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 \| 2 \| 1 \| 0 \| 0 \| 0 \| 0 \| 2 \| -1 \| -1 \| 0 \| 0 \| 0 \| 0 \| 1 \| -1 \| 1 \| -1 \| -1 \| -1 \| 4 \| 5.753 18 \| 15 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 2 \| 2 \| 2 \| 0 \| 0 \| 0 \| 0 \| -1 \| -1 \| -1 \| -1 \| 0 \| 0 \| 0 \| 0 \| 1 \| -1 \| 1 \| -1 \| 4 \| 6.000 19 \| 14 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 1 \| 0 \| 1 \| 2 \| -1 \| 0 \| -1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| -1 \| 0 \| 0 \| -1 \| 1 \| 0 \| 4 \| 5.656 20 \| 12 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 1 \| 0 \| 1 \| 0 \| 1 \| 2 \| 1 \| -1 \| -1 \| -1 \| -1 \| 0 \| 1 \| -1 \| 0 \| -1 \| 0 \| 0 \| 1 \| 4 \| 5.656 21 \| 14 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| -1 \| 1 \| -1 \| 0 \| 0 \| 0 \| 0 \| 1 \| -1 \| -1 \| 1 \| 4 \| 5.656 22 \| 13 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| -1 \| 1 \| -1 \| 1 \| -1 \| -1 \| 1 \| 4 \| 5.656 23 \| 11 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| -1 \| -1 \| -1 \| 1 \| -1 \| 4 \| 5.656 24 \| 10 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 \| 1 \| 1 \| -1 \| 1 \| -1 \| 1 \| -1 \| 1 \| -1 \| 1 \| 0 \| -1 \| 0 \| 0 \| -1 \| 0 \| 1 \| 4 \| 4.000 25 \| 9 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 2 \| 0 \| 1 \| -1 \| -2 \| 0 \| 1 \| -1 \| 4 \| 5.656 26 \| 3 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 \| 0 \| -1 \| 0 \| 2 \| 2.828 Table 1: All 26 classes of inequalities that are facets of the local, no- disturbing polytope of the measurement scenario we introduce, displayed as coefficients of correlators. Some of the inequalities are equivalent to Sliwa’s inequalities Sliwa ; the corresponding class in Ref. Sliwa is displayed in the second column, #S. The second-to-last column displays the local bounds of the inequalities, and the last column displays their respective maximal quantum violations, exact, up to the given precision. ## Appendix D Seesaw optimization To compute lower bounds on the maximum quantum violation of the 26 inequalities we study, as well as the bounds on the critical parameters of the family of quantum states we present in the main text, we implemented variations of an optmization algorithm known as see-saw iteration, introduced by Werner and Wolf in Ref. [41]. Our implementation follows the steps described in Sec. II.B.3 of Ref. [35], with minor adjustments. For standard Bell inequalities where the parties perform dichotomic measurements, the algorithm is based on the idea that, if the quantum state and the measurements of all but one of the parties are fixed, then optmization over the measurements of the remaining party can be carried out explicitly. Consider, for clarity, a bipartite scenario; extensions to multipartite ones are straightforward. Let the operator associated to a given Bell inequality be \[\beta=\sum_{x=1}^{m_{A}}\sum_{y=1}^{m_{B}}{\sum_{a=-1}^{1}\sum_{b=-1}^{1}}c_{x ,y}^{a,b}P_{a\|x}\otimes Q_{b\|y},\] (28) where $x$ ($y$) labels the choice among the $m_{A}$ ($m_{B}$) possible measurements of party $A$ ($B$), $a$ ($b$) labels the possible outcomes, $P_{a\|x}$ ($Q_{b\|y}$) is the measurement operator associated to outcome $a$ ($b$) of measurement $x$ ($y$), and $c_{x,y}^{a,b}$ are the respective coefficients that define the inequality. Then, the quantum average value of the inequality can be written, as a function of the state $\rho$ and the measurement operators, as \[S_{P}(\rho,\left\{P_{a\|x}\right\},\left\{Q_{b\|y}\right\})=\sum_{b,y}\mathrm{tr }\left[\rho_{Q_{b\|y}}Q_{b\|y}\right],\] (29a) where \[\rho_{Q_{b\|y}}=\sum_{a,x}c_{x,y}^{a,b}\mathrm{tr}_{A}\left[\rho(P_{a\|x}\otimes \mathds{1})\right],\] (29b) where $\mathrm{tr}_{A}\left(.\right)$ denotes the partial trace over subsystem $A$. For fixed $\rho$ and $P_{a\|x}$, $S_{P}$ is a linear function of $Q_{b\|y}$. And, since $Q_{1,y}=\mathds{1}-Q_{-1\|y}$, we have \[\sum_{b=\pm 1}\mathrm{tr}\left(\rho_{Q_{b\|y}}Q_{b\|y}\right)=\\ \mathrm{tr}\left(\left(\rho_{Q_{+1\|y}}-\rho_{Q_{-1\|y}}\right)Q_{+ 1\|y}\right)+\mathrm{tr}\left(\rho_{Q_{-1\|y}}\right).\] (30) This expression can be optimized by setting $Q_{+1\|y}$ equal to the projector onto the positive subspace of $\rho_{Q_{+1\|y}}-\rho_{Q_{-1\|y}}$. This procedure can, then, be iterated, so optimization can be carried over all measurements of all parties. If desired, then the quantum state can be optimized over, in an even simpler fashion: In any step, the optimal quantum state can be taken as a pure state given by an eigenvector of $\beta$ associated to its maximal eigenvalue. Note that the first step of the seesaw algorithm already requires a choice of state and measurements, so they should be randomly generated in the beginning of the process. Although it is clear that the algorithm will converge after a sufficient number of steps, one cannot guarantee that it will converge to the global maximum of the problem. Any solution, however, is a lower bound to the optimal solution, so it is recommended to restart the algorithm with as many random “seeds” as feasible. The scenario we consider, as discussed in the previous section, is similar to a tripartite scenario where each of the three parties is able to perform two dichotomic measurements. Our implementation makes use of this similarity, assuming measurements $B_{1}$ and $B_{3}$ are implemented by a third party. On one hand, this assumption guarantees the compatibility relations assumed in the scenario; on the other hand, it leads to loss of generality. This is one more reason (despite the fact that the see-saw does not necessarily converge to the global maximum) that advocates against optimality of the bounds computed via this method. Our goal was to compute upper bounds on the critical values of $w$, as a function of $\alpha$, such that the two families of states \[\rho=w\left\|\psi(\alpha)\right>\!\!\left<\psi(\alpha)\right\|+(1-w)\left\|00 \right>\!\!\left<00\right\|,\] (31a) and \[\sigma\left(\alpha,w\right)=w\left\|\psi(\alpha)\right>\!\!\left<\psi(\alpha) \right\|+(1-w)\mathds{1}/4,\] (31b) where \[\left\|\psi(\alpha)\right>=\sqrt{\alpha}\left\|01\right>+\sqrt{1-\alpha}\left\|10 \right>,\] (31c) violates inequality $\#15$ (we have numerical evidence that this is the best inequality among the ones we listed to witness the nonlocality of such states). The code was implemented in MATLAB, with the aid of the QETLAB[42] library. In both cases, we suppose a system with local Hilbert spaces $\mathcal{H}_{A}=\mathds{C}^{2}$ and ${\mathcal{H}_{B}=\mathds{C}^{4}}$, where the states Eqs. 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1504.04544	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 24409, "num_imgs": 4, "llama3_tokens_count": 7076 }	[ "content_image/1504.04544/x1.png", "content_image/1504.04544/x2.png", "content_image/1504.04544/x3.png", "content_image/1504.04544/x4.png" ]	# Creation of X-Ray Transparency of Matter by Stimulated Elastic Forward Scattering J. Stöhr stohr@slac.stanford.edu SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA A. Scherz andreas.scherz@xfel.eu European XFEL GmbH, Albert-Einstein-Ring 19, 22761 Hamburg, Germany ###### Abstract X-ray absorption by matter has long been described by the famous Beer-Lambert law. Here we show how this fundamental law needs to be modified for high-intensity coherent x-ray pulses, now available at x-ray free electron lasers, due to the onset of stimulated elastic forward scattering. We present an analytical expression for the modified polarization-dependent Beer-Lambert law for the case of resonant core-to-valence electronic transitions and incident transform limited x-ray pulses. Upon transmission through a solid, the absorption and dichroic contrasts are found to vanish with increasing x-ray intensity, with the stimulation threshold lowered by orders of magnitude through a super-radiative coherent effect. Our results have broad implications for the study of matter with x-ray lasers. Non-linear interactions of intense electromagnetic radiation with matter have long been utilized in the microwave and optical regions to control nuclear and valence electronic transitions and have enabled breakthroughs in many fields of science, such as medical imaging, telecommunication or the creation and manipulation of novel states of matter. The natural extension of these techniques into the x-ray region had to await the availability of sufficiently bright x-ray sources in the form of x-ray free electron lasers. Over the last few years, several experiments performed with rather uncontrolled x-ray pulses of high intensity, produced through the self amplification of spontaneous emission (SASE) process [1], have revealed the presence of high intensity effects due to electronic stimulation [2] or multiple ionization [3]. Here we discuss how x-ray transmission through matter can be modified in a controlled way by stimulated scattering effects induced by transform limited x-ray pulses now available through self-seeding [4]. In contrast to stimulated _inelastic_ scattering [2, 9, 10], which requires pulses with a broad bandwidth that covers the difference between excitation and de-excitation energies or multi-color pulses with separate “pump” and “dump” functions, we consider here the conceptually simpler case of _elastic_ stimulation which exists within the energy bandwidth of the incident beam itself. In this case stimulated x-ray scattering modifies the fundamental Beer-Lambert law because of the direct link of x-ray absorption and resonant elastic scattering through the optical theorem. Of particular importance and interest are experiments that utilize _resonant_ electronic core-to-valence transitions since they exhibit large cross sections, and for solids provide elemental and chemical bonding specificity, and through their polarization dependence enable the determination of bond orientation [5] and the dichroic separation of charge and spin based phenomena [6]. Resonant x-rays are widely utilized in x-ray absorption, x-ray scattering and coherent x-ray imaging experiments [5, 6, 7, 8]. We derive the modified Beer-Lambert law by utilizing the time-dependent density matrix approach where the evolution of the resonant core-valence two-level system is governed by the optical Bloch equations [11]. An analytical solution is obtained for the case of incident transform limited x-ray pulses whose coherence time is much longer than the core hole lifetime. We apply our theory to the important case of 3d transition metal samples whose polarization dependent transmission exhibits both a charge and spin response, the latter through the x-ray magnetic circular dichroism (XMCD) effect. We find that for the prominent Co L${}_{3}$ absorption resonance at 778 eV (wavelength of 1.6 nm), stimulated decays begin to rob intensity from the dominant spontaneous Auger channel at an incident intensity of about 1 mJ/cm${}^{2}$/fs (1 TW/cm${}^{2}$), with the onset lowered by a super-radiative coherent scattering enhancement in the forward direction. At higher intensities the sample becomes increasingly transparent with the spin-based XMCD contrast disappearing sooner than the charge-based absorption contrast. We follow the formalism of reference [6] and, denoting the x-ray polarization by the labels $q\!=\!0$ for linear, $q\!=\!+$ for right and $q\!=\!-$ for left circular polarization, describe the polarization dependent x-ray response of a magnetic sample in terms of the atomic scattering length in the soft-x-ray approximation as $f^{q}(\vec{Q}\!=\!0)=r_{0}Z\!+\!{f^{\prime}}^{q}\!-\mathrm{i}{f^{\prime\prime} }^{q}$, where $r_{0}$ is the Thomson scattering length and $Z$ the atomic number. The _spontaneously_ transmitted intensity through a sample of atomic number density $\rho_{\mathrm{a}}$ and thickness $d$ is given by the Beer-Lambert law \[I^{q}_{\mathrm{trans}}=I^{q}_{0}\,\mathrm{e}^{-2\lambda{f^{\prime\prime}}^{q} \rho_{\mathrm{a}}d}\] (1) where $I^{q}_{\mathrm{trans}}$ and $I^{q}_{0}$ are the polarization dependent transmitted and incident intensities and $\sigma^{q}_{\mathrm{abs}}=2\lambda{f^{\prime\prime}}^{q}$ is the x-ray absorption cross section. Our x-ray scattering length formulation is related to the optical constants and the electric susceptibility through the complex refractive index $\tilde{n}^{q}=1-\delta^{q}+\mathrm{i}\beta^{q}\simeq 1+\frac{1}{2}({\chi^{ \prime}}^{q}+\mathrm{i}{\chi^{\prime\prime}}^{q})$, where $\delta^{q}=\rho_{\mathrm{a}}\lambda^{2}(r_{0}Z+{f^{\prime}}^{q})/2\pi$ and $\beta^{q}=\rho_{\mathrm{a}}\lambda^{2}{f^{\prime\prime}}^{q}/2\pi$. The resonant polarization dependent x-ray absorption cross section $\sigma^{q}_{\mathrm{abs}}=2\lambda{f^{\prime\prime}}^{q}$ and the differential atomic elastic scattering cross section $d\sigma^{q}_{\mathrm{scat}}/d\Omega=({f^{\prime}}^{q})^{2}+({f^{\prime\prime}} ^{q})^{2}$ have a Lorentzian lineshape and are linked by the optical theorem which may be written as, \[{f^{\prime\prime}}^{q}\!\!=\!\frac{\Gamma}{\Gamma^{q}_{x}}\frac{2 \pi}{\lambda}\!\left[({f^{\prime}}^{q})^{2}\!\!+\!({f^{\prime\prime}}^{q})^{2} \right]\!=\!\frac{\Gamma_{x}^{q}}{\Gamma}\frac{\lambda}{2\pi}\frac{(\Gamma/2)^ {2}}{(\hbar\omega\!-\!{\cal E}_{0})^{2}\!\!+\!(\Gamma/2)^{2}\!}\] (2) Here ${\cal E}_{0}$ is the resonant photon energy, $\Gamma=\Gamma^{q}_{x}+\Gamma_{\mathrm{A}}$ is the total spontaneous decay width, which in the soft x-ray region is dominated by the Auger width $\Gamma\simeq\Gamma_{\mathrm{A}}$[12]. The polarization dependent radiative transition widths $\Gamma^{q}_{x}$ consist of a radial and angular part and can be calculated by _ab initio_ methods. We have derived their values for Fe, Co and Ni metal from experimental data, and they are listed in Table 1 . \| ρa \| E0 \| λ0 \| σ+0 \| σ00 \| σ−0 \| Γ+x \| Γ0x \| Γ−x \| Γ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| [\small atomsnm3] \| [eV] \| [nm] \| [Mb] \| [Mb] \| [Mb] \| [meV] \| [meV] \| [meV] \| [eV] Fe \| 84.9 \| 707 \| 1.75 \| 8.8 \| 6.9 \| 5.0 \| 1.37 \| 1.08 \| 0.78 \| 0.36 Co \| 90.9 \| 778 \| 1.59 \| 7.9 \| 6.25 \| 4.65 \| 1.208 \| 0.96 \| 0.715 \| 0.43 Ni \| 91.4 \| 853 \| 1.45 \| 5.1 \| 4.4 \| 3.7 \| 0.675 \| 0.575 \| 0.48 \| 0.48 Table 1: Polarization dependent parameters for the L3 resonances of Fe, Co and Ni metals. Listed are the atomic number densities ρa, the resonance energies and wavelengths, and the polarization dependent (q=0,±) peak experimental cross sections σq0, assuming propagation along the magnetization direction. Γqx is the polarization dependent dipole transition width which includes the number of valence holes Nh, and Γ is the natural decay energy width [12]. The polarization dependent Lorentzian x-ray absorption cross sections $\sigma^{q}_{\mathrm{abs}}=2\lambda{f^{\prime\prime}}^{q}$ calculated with Eq. 2 and the parameters for Co in Table 1 are shown as blue curves in Fig. 1 (a). They were derived from fits of the experimental resonant cross sections by Voigt profiles as shown in Fig. 1 (b), consisting of a convolution of the natural Lorentzian lineshapes in (a) with a Gaussian of 1.4 eV FWHM to account for the band-structure broadened $d$ valence states into which the $2p_{3/2}$ core electrons are excited. <figure><img src="content_image/1504.04544/x1.png"><figcaption>Figure 1: (a) Polarization and photon energy dependent L3 absorption crosssections σqabs=2λf′′q for Co metal, calculated by use of Eq. 2 and theparameters Γ and Γqx in Table 1. (b) Comparison of the experimental dichroiccross sections (black lines) and the theoretical cross sections in (a)convoluted with a Gaussian of 1.4 eV FWHM. Note that the corresponding blueand red curves have the same areas but their peak values differ by a factor ofσq1/σq0=2.9.</figcaption></figure> For a sample of finite thickness $d$ and atomic number density $\rho_{\mathrm{a}}$, the transmitted intensity decays exponentially with the number of atoms in the beam $N_{\mathrm{a}}/A=\rho_{\mathrm{a}}d$ according to Eq. 1. Since absorption and resonant scattering are related through Eq. 2, the Beer-Lambert absorption law can also be derived by considering resonant elastic forward scattering. To do so one considers scattering by a thin atomic sheet so that the first Born approximation is valid. For a sheet thickness $\Delta\!\ll\!\lambda$ the spontaneously forward scattered fields are coherent and the transmitted field is given by, \[E^{q}_{\mathrm{trans}}=E^{q}_{0}\,\mathrm{e}^{\mathrm{i}k\Delta} \left\{1-\mathrm{i}\,\lambda\,\left[r_{0}Z+{f^{\prime}}^{q}-\mathrm{i}\,{f^{ \prime\prime}}^{q}\right]\,\rho_{\mathrm{a}}\Delta\right\}\] (3) Neglecting the non-resonant term $r_{0}Z$, the intensity transmitted through the sample with $N_{\mathrm{a}}$ atoms in the beam of cross sectional area $A$ is, \[\|E^{q}_{\mathrm{trans}}\|^{2}\!\!=\!\|E^{q}_{0}\|^{2}\!\left\{\!1\!- \!2\lambda\frac{N_{\mathrm{a}}}{A}{f^{\prime\prime}}^{q}\!\!+\!\lambda^{2} \frac{N^{2}_{\mathrm{a}}}{A^{2}}\!\left[({f^{\prime}}^{q})^{2}\!\!+\!({f^{ \prime\prime}}^{q})^{2}\right]\!\right\}\] (4) The first term is the incident intensity and the second term is the absorption loss (minus sign) in linear response. Within the Born approximation, the absorption loss arises from the destructive interference of the incident field with the coherently forward scattered field. The third term is the forward scattered gain (plus sign) due to the coherent superposition of the fields scattered by the atoms in the sheet which scales with $N^{2}_{\mathrm{a}}$. In Eq. 4 we have neglected the weak intensity $(8\pi/3)N_{\mathrm{a}}[({f^{\prime}}^{q})^{2}\!+\!({f^{\prime\prime}}^{q})^{2} ]/A$ which is _incoherently_ scattered. The _coherent_ forward scattered intensity is larger by the enhancement factor, \[{\cal G}_{\mathrm{coh}}=\frac{3}{8\pi}N_{\mathrm{a}}\frac{\lambda ^{2}}{A}\] (5) where $d\Omega_{\mathrm{coh}}=\lambda^{2}/A$ represents the solid angle of coherent forward scattering. The total field transmitted through a sample of arbitrary thickness $d=N\Delta$ is obtained by using the Darwin-Prins dynamical scattering formalism to sum Eq. 3 over $N$ thin sheets, which yields the exponential Beer-Lambert law Eq. 1[13]. Remarkably, for forward scattering, the longitudinal coherence length $\ell_{\mathrm{c}}\!\simeq\!\lambda^{2}/\Delta\lambda$ does not enter [14], and both Eq. 1 and ${\cal G}_{\mathrm{coh}}$ depend on the number of atoms $N_{\mathrm{a}}$ in the beam cross sectional area $A$, and not on the sample thickness $d$. ${\cal G}_{\mathrm{coh}}$ is thus the same for a thin film with $\rho_{\mathrm{a}}\simeq 100$ atoms/nm${}^{3}$ and typical thickness $d\!\simeq\!1/(\sigma_{\mathrm{abs}}\rho_{\mathrm{a}})\!\simeq\!20$ nm and a gas sample of the same atoms of density $\rho_{\mathrm{a}}\simeq 0.01$ atoms/nm${}^{3}$ and a much larger thickness of $d\simeq 200\,\mu$m. As the incident intensity is increased, the Kramers-Heisenberg perturbation theory (see [6]) leads to un-physical results since it does not account for population changes in the excited state. This is overcome by the density matrix formalism which directly calculates the time-dependent ground and excited state populations [11], which we shall denote $\rho_{11}(t)$ and $\rho_{22}(t)=1-\rho_{11}(t)$, respectively. The populations are obtained as the solutions of the optical Bloch equations. In the presence of stimulation, we can write the atomic scattering length as the sum of a spontaneous (subscript “0”) and stimulated non-linear (subscript “NL”) part according to, \[{f^{\prime}}^{q}=r_{0}Z+{f^{\prime}}_{0}^{q}+{f^{\prime}}_{ \mathrm{NL}}^{q},~{}~{}~{}{f^{\prime\prime}}^{q}={f^{\prime\prime}}_{0}^{q}+{f ^{\prime\prime}}_{\mathrm{NL}}^{q}\] (6) and Eq. 1 is replaced by, \[I^{q}_{\mathrm{trans}}=I^{q}_{0}\,\mathrm{e}^{-2\lambda\left({f^{\prime\prime} _{0}}^{q}+2{f^{\prime\prime}}^{q}_{\!\!\!\mathrm{NL}}\right)\rho_{\mathrm{a}}d}\] (7) The spontaneous absorption cross section $\sigma_{\mathrm{abs}}=2\lambda{f^{\prime\prime}}^{q}$ with ${f^{\prime\prime}}^{q}={f^{\prime\prime}_{0}}^{q}$ in Eq. 1 becomes the coherence time and intensity dependent expression, \[\sigma_{\mathrm{abs}}\!=\!2\lambda\left[{f^{\prime\prime}}^{q}_{0 }+2{f^{\prime\prime}}^{q}_{\mathrm{NL}}\right]=2\lambda{f^{\prime\prime}}^{q}_ {0}\left[1-2\,\rho^{q}_{22}(\tau_{\mathrm{c}})\right]\] (8) Here ${f^{\prime\prime}}^{q}_{0}$ is given by the spontaneous expression Eq. 2, and $\rho^{q}_{22}(\tau_{\mathrm{c}})$ is the excited state population obtained from the time dependent solution of the optical Bloch equations [11], by integration over the coherence time $\tau_{\mathrm{c}}$ of the incident x-rays. In general, the Bloch equations have to be solved numerically for $\rho^{q}_{22}(\tau_{\mathrm{c}})$. However, if the coherence time is much longer than the Auger decay time ($\hbar/\Gamma=1.5$ fs for Co $2p_{3/2}$), the excited state population reaches an equilibrium value (see Fig. 2) and the non-linear contribution is given by the analytical expression, \[{f^{\prime\prime}}^{q}_{\!\!\!\mathrm{NL}}\!=\!-{f^{\prime\prime} _{0}}^{q}\!\underbrace{\frac{I^{q}_{0}\Gamma^{q}_{x}\,{\cal G}_{\mathrm{coh}} \lambda^{3}/(8\pi^{2}c)}{(\hbar\omega\!-\!{\cal E}_{0})^{2}\!+(\Gamma/2)^{2}\! +I^{q}_{0}\Gamma^{q}_{x}\,{\cal G}_{\mathrm{coh}}\lambda^{3}/(4\pi^{2}c)}}_{ \mbox{$\rho^{q}_{22}(\infty)$}}\] (9) where $\rho^{q}_{22}(\infty)$ is the equilibrium excited state population in the limit $\tau_{\mathrm{c}}\rightarrow\infty$. The non-linear contribution ${f^{\prime\prime}}^{q}_{\mathrm{NL}}$ is seen to have the opposite sign of the spontaneous contribution ${f^{\prime\prime}_{0}}^{q}$. In the limit of high incident intensity we simply have $\rho^{q}_{22}(\infty)=0.5$ and $2{f^{\prime\prime}}^{q}_{\!\!\!\mathrm{NL}}=-{f^{\prime\prime}_{0}}^{q}$ and the sample becomes transparent. <figure><img src="content_image/1504.04544/x2.png"><figcaption>Figure 2: (a) Excited state population ρ022(τc) as a function of the linearlypolarized incident intensity I00 for different coherence times τc of theincident pulses for the L3 edge of a 20 nm thick Co metal film. We assumedresonance excitation, ℏω=E0, linearly polarized light and experimental peakcross sections as discussed in the text. (b) ρ022(τc) as a function of theincident fluence per coherence time of the pulse.</figcaption></figure> The increase in excited state population $\rho^{0}_{22}(\tau_{\mathrm{c}})$ for Co L${}_{3}$ excitation as a function of incident intensity and different coherence times of the incident pulse is shown in Fig. 2. It was calculated by numerical solution of the optical Bloch equations, assuming a 20 nm thick Co metal film and $\hbar\omega={\cal E}_{0}$, with the spontaneous peak cross section $\sigma^{0}_{0}$ (Fig. 1 (b)), corrected for the $\tau_{\mathrm{c}}$-dependent energy bandwidth $\Delta{\cal E}=2\hbar\sqrt{\pi\ln 4}/\tau_{\mathrm{c}}$ of the incident pulse [15]. With increasing coherence time $\tau_{\mathrm{c}}$ relative to the core hole life time, the Rabi oscillations in the excited state population are suppressed. The behavior of $\rho^{0}_{22}(\tau_{\mathrm{c}})$ for our chosen $\tau_{\mathrm{c}}$ values as a function of the total intensity per coherent pulse, in units of [mJ/cm${}^{2}$/$\tau_{\mathrm{c}}$] is shown in Fig. 2 (b). The stimulated threshold is seen to be lowest for coherent pulses in the 3-10 fs range. <figure><img src="content_image/1504.04544/x3.png"><figcaption>Figure 3: (a) Change of the effective polarization dependent absorption crosssection 2λ(f′′0q+2f′′qNL) for the L3 resonance in Co metal for three incidentintensity values, assuming alignment of the x-ray propagation direction withthe film magnetization and the long coherence time limit Eq. 9. The lowintensity cross sections shown as black curves are nearly identical to thespontaneous ones shown in red in Fig. 1. (b) Dependence of the transmissioncontrast as a function of sample thickness and incident intensity for resonantL3 excitation of a Co metal film due to charge absorption with linearlypolarized light, according to Eq. 7. (c) Same as (b) for the transmitted XMCDcontrast.</figcaption></figure> Fig. 3 (a) shows the effective polarization dependent absorption cross section for Co given by Eq. 8 for three values of the incident intensity with ${f^{\prime\prime}}^{q}_{\!\!\!\mathrm{NL}}$ calculated according to Eq. 9. In Fig. 3 (b) we illustrate the thickness dependence of the absorption contrast (linear polarization) obtained from Eq. 7 for several values of the incident intensity. At low intensity the sample transmission decreases with increasing sample thickness $d$ due to absorption. However, with increasing intensity, the transmitted intensity at large $d$ is seen to decrease considerably slower due to stimulated forward scattering. The magnetic XMCD contrast, plotted in Fig. 3 (c), first increases with thickness up to a maximum around $d\!=\!1/(\sigma_{\mathrm{abs}}\rho_{\mathrm{a}})\!=\!17$ nm, corresponding to one x-ray absorption length, before it also decreases. Fig. 4 shows the dependence of the transmitted intensity for the stimulated relative to the spontaneous case as a function of the incident intensity, calculated for Co metal with $d=20$ nm and assuming resonant excitation. Both the effective absorption cross section and the transmitted intensity reveal a strong dependence on the incident intensity, with the spin related XMCD contrast (red curve) vanishing faster than the charge related XAS contrast (black curve). <figure><img src="content_image/1504.04544/x4.png"><figcaption>Figure 4: Dependence of the transmitted intensity according to Eq. 7 and Eq.9 in the presence versus absence of stimulation for a 20 nm Co metal film as afunction of incident intensity. The black curve represents the linearpolarization or charge response [I0trans]stim/[I0trans]spon and the red curveis the transmitted XMCD difference intensity[I−trans−I+trans]stim/[I−trans−I+trans]spon. The top scale is discussed in thetext. The inset shows the relative transmitted XMCD contrast for filmthicknesses of 20 nm (red) and 100 nm (dashed orange) as a function ofincident intensity.</figcaption></figure> The inset reveals a particularly interesting thickness dependence of the transmitted XMCD intensity. For a thick sample of 100 nm, the remaining small spontaneous XMCD contrast of about 1.5%, which according to Fig. 3 (c) is greatly diminished by absorption, can actually be increased by nearly a factor of 5 upon stimulation. The stimulated onset in Fig. 4 is predicted to be orders of magnitude lower than for the stimulated effects observed before [2] for SASE pulses with an average coherence time of about 0.5 fs [16]. This is due to our assumption of self-seeded pulses which besides coherence times of $\sim 10$ fs [4] offer a jitter-free photon energy that can be resonantly tuned for maximum cross section. In addition, the elastic stimulation threshold is lowered by the coherent enhancement factor ${\cal G}_{\mathrm{coh}}$, which for L${}_{3}$ excitation of a Co film is $\sim$ 500. The dependence of the non-linear contribution on the incident intensity, given by Eq. 9, may also be expressed in terms of the number of incident photons contained in a specific volume. If the volume is chosen to be the coherence volume $V_{qk}$ per mode $qk$, then the associated number of photons $n_{qk}$ is referred to as the photon degeneracy parameter. The incident photons that stimulate electronic decays, however, need to be present during the total atomic clock decay time $\hbar/\Gamma$ which defines the sample-specific atomic decay volume $V_{\Gamma}$. The two coherence volumes are given by \[V_{qk}=\lambda^{3}\frac{\hbar\omega}{\Delta(\hbar\omega)},~{}~{} ~{}V_{\Gamma}=\lambda^{3}\frac{\hbar\omega}{2\pi^{2}\Gamma}\] (10) The number of stimulating photons $n_{\Gamma}$ in the volume $V_{\Gamma}$ is that in the well-known Kramers-Heisenberg stimulated correction term $1+n_{\Gamma}$, and it can be expressed in terms of the incident polarization dependent intensity and field amplitude $E^{q}_{0}$ as, \[n^{q}_{\Gamma}=\frac{1}{2\pi^{2}c}\frac{\lambda^{3}}{\Gamma}I^{q }_{0}=\frac{\epsilon_{0}}{\pi^{2}}\frac{\lambda^{3}}{\Gamma}\|E^{q}_{0}\|^{2}= \frac{\|E^{q}_{0}\|^{2}}{\|E_{\mathrm{ZP}}\|^{2}}\] (11) On the right we have introduced the zero-point (ZP) field $E_{\mathrm{ZP}}$ responsible for spontaneous radiative decays. For $n^{q}_{\Gamma}=1$ the spontaneous and stimulated scattering intensities become the same, and the incident field $E^{q}_{0}$ is equally effective in driving decays as the ZP field $\|E_{\mathrm{ZP}}\|^{2}=\pi^{2}\Gamma/(\epsilon_{0}\lambda^{3})$ corresponding to one virtual photon in the volume $V_{\Gamma}$. This allows us to equate the intensity scale on the bottom of Fig. 4 with the number of photons $n_{\Gamma}$ on top of the figure, and for our case the ZP field has the value $E_{\mathrm{ZP}}\simeq 4.4\times 10^{9}$ V/m. Research at SLAC was supported through the Stanford Institute for Materials and Energy Sciences which is funded by the Office of Basic Energy Sciences of the U.S. Department of Energy. We would like to thank D. Higley, J. W. Goodman, and S. K. Sinha for clarifications of longitudinal coherence effects. ## References * [1] P. Emma et al., Nat. Photonics 4, 641 (2010); H. Tanaka et al., Nat. Photonics 6, 540 (2012) * [2] N. 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1911.11414	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 23839, "num_imgs": 4, "llama3_tokens_count": 5574 }	[ "content_image/1911.11414/Fig01.jpg", "content_image/1911.11414/Fig02a.jpg", "content_image/1911.11414/Fig03a.jpg", "content_image/1911.11414/Fig04a.jpg" ]	# Effect of size, temperature and strain rate on dislocation density and deformation mechanisms in Cu nanowires P. Rohith G. Sainath sg@igcar.gov.in V.S. Srinivasan Homi Bhabha National Institute (HBNI), Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamilnadu-603102, India Materials Development and Technology Division, Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamilnadu-603102, India Scientific Information Resource Division, Resource Management & Public Awareness Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamilnadu-603102, India ###### Abstract In the present study, molecular dynamics (MD) simulations have been performed to understand the effect of nanowire size, temperature and strain rate on the variations in dislocation density and deformation mechanisms in $<$100$>$ Cu nanowires. The nanowire size has been varied in the range 1.446-43.38 nm with a constant length of 21.69 nm. Different temperatures varying from 10 K to 700 K and strain rates in the range of $5\times 10^{7}$ - $1\times 10^{9}$ s${}^{-1}$ have been considered. For all the conditions, the variations in dislocation density $(\rho)$ has been calculated as a function of strain. The results indicate that the variations in dislocation density exhibits two stages irrespective of the conditions: (i) dislocation exhaustion at small strains followed by (ii) dislocation starvation at high strains. However, with decreasing size and increasing temperature, the rate of dislocation exhaustion increases, which results in early transition from dislocation exhaustion stage to dislocation starvation stage. Similarly, with increasing strain rate, the rate of dislocation exhaustion and also the transition strain increases. keywords: Molecular Dynamics simulations; Cu nanowire; Dislocation density; Dislocation exhaustion; Dislocations starvation. † [FOOTNOTE:†][ENDFOOTNOTE] ## 1 Introduction Technological advancements have paved the way towards the development of nanocomponents in nanoelectro mechanical systems (NEMS). Nano components used in NEMS are required to withstand the complex stress state with minimal probability of failure. Hence, it is important to understand the mechanical properties such as strength and associated deformation mechanisms like dislocation density. Generally, the mechanical properties of nanowires are determined using nanoindentation tests, while tensile tests are most common in bulk materials. However, performing experiments at nanoscale involves many complications. Alternatively, molecular dynamics (MD) simulations provide great insights in determining mechanical properties and understanding the deformation behaviour of nanowires with atomic-scale resolution. Plastic deformation in bulk materials is characterized by dislocation multiplication, dislocation pile-up, dislocation cross-slip and similar processes, which leads to strain hardening/softening before final failure. However, in nanomaterials, dislocations can travel limited distances before annihilating at free surfaces/grain boundaries, thereby reducing the probability of dislocation multiplication [2; 3]. The strengthening behavior in the bulk materials with respect to grain size can be described by the well known Hall-Petch relation. A physical basis for this behavior is associated with the difficulty of dislocation movement across grain boundaries and the stress concentration arising due to dislocation pile-up [4; 5]. However, as the grain size decreases to nanoscale regime, the grain boundary volume fraction increases significantly and as a result, the grain boundary mediated processes such as GB sliding and GB rotation becomes more important [4; 5]. In particular, when the grain size is reduced to below certain critical size, the strength of materials decreases with decreasing grain size, i.e., it follows inverse Hall–Petch relation. In the absence of grain boundaries in single crystalline nanowires, the surface alone influences the strength and deformation mechanisms [6]. In nanowires, the strength increases with decreasing size, which is mainly attributed to surface effects. Contrary to nanowires, the surface effects are absent in bulk single crystals. These differences suggests that the deformation mechanisms governing the plastic deformation in nanomaterials/nanowires are quite different from their bulk counterparts. Since then, many researchers have proposed different mechanisms such as source exhaustion and source truncation, dislocation exhaustion, dislocation starvation, and weakest link theory to understand the deformation behaviour at nanoscale [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]. These mechanisms originate from either micro-structural parameters or dimensional constraints. Oh et al. [12] had performed in-situ tensile tests on Al single crystals and observed the operation of single ended dislocation sources without any multiplication mechanism. Greer et al. [7] have performed the compression tests on Au nanopillars and reported that the dislocations would leave the pillars before they multiply, leading to dislocation starvation. Once nanopillar is in dislocation-starved state, very high stresses are required to nucleate new dislocations, either at surfaces or in the interior [7]. Parthasarathy et al. [9] have reported that the reduction in nanowire size transforms the double ended Frank-Read sources into single ended sources leading to increased strength by source truncation. Volkert and Lilleodden [8] have shown that the image stresses due to surfaces and source limited behaviour results in dislocation annihilation at free surface. This leads to dislocation starvation resulting in increase strength for activation of dislocation sources at smaller sizes. Sansoz [14] had performed MD simulations on compressive loading of Cu nanopillars with initial dislocation density. It has been demonstrated that the deformation exhibits a pronounced dislocation exhaustion regime followed by source limited activation regime. El-Alwady [15] had studied the effect of sample size and initial dislocation density on deformation mechanisms. It was reported that dislocation starvation is dominant in small size nanowires. However, in relatively larger size, with increasing initial dislocation density, the dominant deformation mechanism changes from dislocation starvation to single-source mechanism and then to dislocation exhaustion and finally forest hardening [15]. Further, it was mentioned that as sample size increases, dislocation density at which the deformation mechanisms transits decreases. However, all these studies were mainly focussed on size related effects on deformation mechanisms and little has been studied about the combined influence of size, temperature and strain rate. Particularly, the influence of temperature and strain rate on the variations in dislocation density and also on deformation mechanisms such as dislocation exhaustion and dislocation starvation has not been investigated. In view of this, effect of size, temperature and strain rate on deformation mechanisms of $<$100$>$ Cu nanowire has been investigated in the present study using MD simulations. The variation of dislocation density as a function of strain has been calculated for all the scenarios. Based on the variations in dislocation density, two type of deformation mechanisms (dislocation exhaustion and dislocation starvation) have been identified. ## 2 MD Simulation details Cu nanowires of square cross-section shape oriented in $<$100$>$ direction with {001} as side surfaces were chosen for the present study. In order to study the influence of size, nanowires of different size varying from 1.446 to 43.38 nm, all having a constant length of 21.69 nm were used. Periodic boundary conditions were chosen along length direction, while other two directions were kept free to mimic an infinitely long nanowire. On these nanowires, tensile loading was simulated using MD simulations through LAMMPS package [18]. Embedded atom method (EAM) potential for FCC Cu given by Mishin and co-workers [19] has been used to describe the inter-atomic interactions. This potential has been chosen for being able to reproduce the stacking fault and twinning fault energies of Cu [20], which are important to predict the dislocation related mechanisms. Following the initial construction of nanowires, energy minimization was performed by conjugate gradient method to obtain a stable structure. Velocity verlet algorithm was used to integrate the equations of motion with a time step of 2 fs. Finally, the relaxed model was thermally equilibrated to required temperature in NVT ensemble with a Nose-Hoover thermostat. This configuration of nanowire has been taken as initial state for further tensile simulations. Following thermal equilibration, nanowires were subjected to tensile loading along axial direction at required temperature and strain rate. MD simulations have been performed at various temperatures in the range 10 -700 K and strain rates varying from $5\times 10^{7}$ to $1\times 10^{9}$ s${}^{-1}$. These strain rates are significantly higher than the experimental strain rates. However, despite the high strain rates, many studies have shown that MD simulation results are in good agreement with the experimental investigations. The average stress in loading direction has been calculated using Virial expression [21] as implemented in LAMMPS. The dislocations in present study have been identified/tracked by using Dislocation eXtraction Algorithm (DXA) developed by Stukowski [22] as implemented in OVITO [23]. The detailed procedure for tracking the dislocation lines and their Burgers vectors is provided in the paper [22]. Following the detection of dislocation lines, the total length of dislocation lines within the simulated volume can be obtained in OVITO. In the present study, we have considered all type of dislocations (Shockley partials, full dislocations, Frank partials and stair rods) while calculating the total length. Once the total length is obtained, the dislocation density has been calculated as the total dislocation length divided by simulated cell volume. This procedure has been repeated for every 250 time steps (0.5 ps). ## 3 Results and Discussion Figure 1 shows the variations in dislocation density along with flow stress as a function of strain under tensile loading of a nanowire with size (d) = 21.69 nm and strain rate of $1\times 10^{9}$ s${}^{-1}$ at 10 K. It can be seen that the nanowire undergoes elastic deformation up to a peak followed by an abrupt drop in flow stress. This abrupt drop is associated with yielding through the nucleation of 1/6$<$112$>$ Shockley partial dislocations in the nanowire. During the process of this yielding, large number of dislocations nucleate and as a result, the dislocation density reaches its maximum value (Figure 1). With increasing deformation, dislocation density gradually decreases from its maximum until a strain value of 0.56. This regime, where dislocation density gradually decreases is denoted as dislocation exhaustion stage. Interestingly, this exhaustion stage is associated with slight increase in flow stress (Figure 1). Since the deformation in nanowires is nucleation controlled [24], decrease in dislocation density indicates that the rate of exhaustion or annihilation is higher than the nucleation. Following dislocation exhaustion stage, dislocation density remains very low and constant with marginal fluctuations around a mean value (Figure 1). This low value of dislocation density indicates that the nanowire is depleted of dislocations, and therefore, this regime ($\varepsilon>0.56$) is termed as dislocation starvation stage. In starvation stage, deformation proceeds through continuous nucleation and annihilation of dislocations, which is also reflected in terms of fluctuations in dislocation density as well as flow stress (Figure 1). Further, the marginal fluctuations at low and constant value of dislocation density indicates that the rate of dislocation nucleation is almost same as the rate of dislocation annihilation. Finally, it can be seen that the dislocation density during the deformation of nanowire is in the range of $1\times 10^{16}$ - $6\times 10^{17}$ m${}^{-2}$ (Figure 1), which is few orders of magnitude higher than those observed in experiments. However, many MD simulation studies have reported such high values of dislocation density, which is attributed to high applied strain rates inherent in MD [16; 17; 25; 26; 27; 28]. <figure><img src="content_image/1911.11414/Fig01.jpg"><figcaption>Figure 1: Variations in dislocation density along with flow stress as afunction of strain at 10 K and strain rate of 1×109 s−1 for a nanowire of size(d) = 21.69 nm. The regions of dislocation exhaustion and starvation areclearly marked. In the inset figures, the green colour lines show 1/6<112>partials and the magenta lines indicate stair-rod dislocations.</figcaption></figure> In order to investigate the influence of size, temperature and strain rate on the variations in dislocation exhaustion and starvation stages, the dislocation density as a function of strain (or time) has been calculated for all the cases. Figure 2a shows the variations in dislocation density as a function of strain (or time) for Cu nanowires of different cross-section width in the range 1.446-43.38 nm and strain rate of $1\times 10^{9}$ s${}^{-1}$ at 10 K. Dislocation density is shown only up to a strain level of 1 as it remains almost constant above this level. It can be seen that for all sizes except the smallest, dislocation density exhibits two stages; dislocation exhaustion stage followed by dislocation starvation stage (Figure 2a). In the smallest nanowire, large fluctuations around low value of dislocation density suggest that there is no dislocation exhaustion stage and dislocation starvation alone dominates the deformation at all strains. Further, the transition strain at which the dislocation mechanism changes from exhaustion stage to starvation stage increases with increasing size. This indicates that the rate of dislocation exhaustion is higher in small size nanowires. The rate of dislocation exhaustion is calculated as a slope of dislocation density vs. time plot as typically shown in Figure 2a for nanowire of size 43.38 nm. Figure 2b shows the variation of dislocation exhaustion rate as a function of nanowire size. It can be clearly seen that, the rate of dislocation exhaustion ($\dot{\rho}$) decreases with increasing size (d) (Figure 2b) and follows the relation $\dot{\rho}=\rho_{d0}+ae^{-bd}$, where $\rho_{d0}$, $a$, and $b$ are constants. The high exhaustion rates or low resident time of dislocations in small size nanowires is due to many factors like lower probability of dislocation-defect interactions and high image stresses. On the contrary, the high probability of dislocation-defect interactions and low image stress results in low rate of dislocation exhaustion in large nanowires (Figure 2b). <figure><img src="content_image/1911.11414/Fig02a.jpg"><figcaption></figcaption></figure> The variations in dislocation density for a nanowire of size (d) = 10.85 nm as a function of strain (or time) at different temperatures are shown in Figure 3a at a constant strain rate of $1\times 10^{9}$ s${}^{-1}$. At all temperatures, dislocation density in nanowires clearly display dislocation exhaustion and starvation stages. Further, with increasing temperature, the maximum in dislocation density, which is observed at yielding, decreases (Figure 3a), while the rate of dislocation exhaustion increases (Figure 3b). The rate of dislocation exhaustion ($\dot{\rho}$) with temperature (T) follows the relation $\dot{\rho}=\rho_{T0}-ae^{-bT}$, where $\rho_{T0}$, $a$, and $b$ are constants. The high dislocation exhaustion rates at high temperatures results in low values of transition strain for change in deformation mechanisms from exhaustion to starvation. The low exhaustion rates at low temperatures are due to high dislocation density at yielding (Figure 3a) and low velocity of dislocations [29], which increases the probability of dislocation interactions. The high probability of dislocation interactions restricts the ease of dislocation annihilation to surfaces resulting in low exhaustion rates at low temperatures. <figure><img src="content_image/1911.11414/Fig03a.jpg"><figcaption></figcaption></figure> Figure 4a shows the variations in different dislocation mechanisms (dislocation exhaustion and starvation stages) as a function of strain at different strain rates for a nanowire of size (d) = 10.85 nm at 10 K. Due to different applied strain rates, which results in different time scales, the time axis has not shown in Figure 4a. Similar to size and temperature, it shows that the strain rate also influences the dislocation mechanisms in Cu nanowires. It can be seen that, with increasing strain rates, both maximum dislocation density at yielding and transition strain (exhaustion to starvation) increases (Figure 4a). However, unlike size and temperature cases, the variations in dislocation density show different behaviour with respect to strain and time. With respect to strain (Figure 4a), it appears that the dislocations exhaust at low rates under high strain rate conditions. However, this is not actually true when the calculations are obtained from dislocation density vs. time. This difference with respect to strain and time is due to different time scales involved at different strain rates. For example, under high strain rate condition, it takes very short time to reach the strain value of 0.5, while it takes longer time to reach the same strain value under low strain rate case. Interestingly, the results obtained from dislocation density vs. time graph show that the dislocation exhaustion rate increases with increasing strain rate as shown in Figure 4b and follows the relation $\dot{\rho}=\rho_{\dot{\varepsilon}0}-ae^{-b\dot{\varepsilon}}$, where $\rho_{\dot{\varepsilon}0}$, $a$, and $b$ are constants. Since the dislocation velocity is directly proportional to strain rate [30], high exhaustion rates are expected under high strain rate conditions. <figure><img src="content_image/1911.11414/Fig04a.jpg"><figcaption></figcaption></figure> The dislocation density influences the different properties of materials, out of which the most prominent being the mechanical properties. In bulk materials increasing the dislocation density increases the yield strength, which results in work hardening [30]. Similarly, the variations in dislocation density in the nanowires has many consequences on strength, ductility and deformation mechanisms [17; 31]. For example, it has been shown that the yield stress is very sensitive to initial dislocation density [31]. In a crystal with low initial dislocation density, the nucleation and growth of twins along with strain hardening has been reported, whereas in crystal with high initial dislocation density, the deformation proceeds by dislocation multiplication and motion without any twins and exhibit no strain hardening [31]. Further, it has been shown that the nanowires with high dislocation density display large ductility as compared to nanowires with low dislocation density[17]. This has been attributed to high dislocation-dislocation interactions in nanowires containing high dislocation density. ## 4 Conclusions MD simulations were used to understand the variations in dislocation density as a function of strain for different nanowire sizes, temperatures and strain rates. The results indicate that, irrespective of temperature and strain rate, the dislocation density in all the nanowires except with d = 1.446, show two stage behaviour; dislocation exhaustion stage at small strains followed by dislocation starvation stage at large strains. However, small size nanowires with d $<$ 3.615 nm exhibit only dislocation starvation at all strains. Further, in all the cases, the dislocation density attains its maximum immediately after yielding. During dislocation exhaustion, it has been observed that the rate of dislocation exhaustion strongly depends on nanowire size, temperature and strain rate. The large size nanowires show lower exhaustion rates as compared to smaller ones, i.e., resident time of dislocations within the nanowire increases with increasing size. The lower exhaustion rates in large size nanowires are due to high probability of dislocation-defect interactions along with low image stress. As a result of low exhaustion rates, large size nanowires show higher transition strain (strain to change in dislocation mechanisms from exhaustion to starvation) as compared to small size nanowires. Similarly, in nanowire of particular size, the dislocation exhaustion rates increases with increasing temperature and strain rate. Correspondingly, the transition strain decreases with increasing temperature and decreasing strain rate. The lower exhaustion rates at low temperatures and low strain rates are mainly due to low dislocation velocities, which increases the probability of dislocation interactions with existing defects or dislocations and thus restricting the ease of dislocation annihilation to surfaces. ## References * (1) * (2) J.J. Gilmann, Appl. Micromech. Flow Solids, (1953) 185. * (3) R. Krahne, L. Manna, G. Morello, A. Figuerola, C. George, S. Deka, Nano Sci. Tech., Springer, New York, 2013. * (4) L. Zhang, C. Lu, K. Tieu, Comput. Mater. Sci. 118 (2016) 180. * (5) L. 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1709.03391	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 42364, "num_imgs": 3, "llama3_tokens_count": 14852 }	[ "content_image/1709.03391/x1.png", "content_image/1709.03391/x2.png", "content_image/1709.03391/x3.png" ]	# The driven oscillator, with friction T.B. Smith School of Physical Sciences The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK ###### Abstract This paper develops further the semi-classical theory of an harmonic oscillator acted on by a Gaussian white noise force discussed in [25] (arXiv:1508.02379 [quant-ph]). Here I add to that theory the effects of Brownian damping (friction). This requires an adaption of the original formalism and complicates the algebra somewhat. Albeit semi-classical, the theory can be used to model quantum expectations and probabilities. Among several examples, I consider some implications for the canonical phase operator. PACS: 03.65.-w; 03.65.Ge; 05.40.-a; 42.50.Lc ## 1 Introduction In the classical theory of Brownian motion in phase space a particle is subjected both to a white noise external force and to a damping force proportional to its velocity [8, 14]. To generate a semiclassical theory from this for an oscillator we make the association $(p,q)\rightarrow(\hat{p},\hat{q})$ where $[\hat{p},\hat{q}]=-{\rm i}\hbar$, but the damping term makes the transition to quantum dynamics awkward. The solution has been known for some time [26, 27, 28]—there is an appropriate time evolution Hamiltonian which is not the energy. This is the topic of section 2 wherein the classical dynamics is transcribed to the Wigner-Weyl quantization formalism. Section 3 adds ensemble statistics to the Wigner-Weyl time propagator, thereby incorporating Brownian statistics semi-classically. An exact expression for the propagator of the density matrix in the Wigner-Weyl picture is given. At long times it forgets its history and becomes thermal. In section 4 we consider the transition probability of the oscillator from the harmonic oscillator ground state. The Wigner-Weyl formalism is used throughout. An exact expression for this probability was worked out in [25] when there is strictly no friction in which case the oscillator cannot thermalise. In the present case, with non-zero friction, the system _can_ thermalise, but the multiple integrals involved in many cases, though often Gaussian in form, are somewhat lengthy so I resort to computation. The plot of one example is shown in Figure 1. Section 5 discusses the ‘canonical phase operator’, $\hat{\phi}$, which is, roughly speaking, the Weyl quantization of $\arctan(p/q)$ where $p$ and $q$ are the canonical coordinates in appropriate units. Also briefly discussed is what might be termed the ‘physical’ phase operator, $\hat{\overline{\phi}}$, which is, again roughly speaking, the Weyl quantization of $\arctan(m\dot{q}/q)$. Should the oscillator initially be in the ground state, Figure 2 shows, by computation, how the expectation of the canonical phase operator decays to zero as time advances. Section 5 also considers the spectra of both phase operators, again by computation, Figure 3. To my knowledge, explicit expressions for the spectral representations of these operators haven’t been given. Finally, Section 5 also considers the variance of both angle operators in the thermal limit. Section 6 briefly considers in the thermal limit the expectation \[\overline{{\rm Tr}\big{(}\hat{\rho}(t){\rm exp}(-B\hat{E}_{\rm osc})\big{)}}\] where $\hat{E}_{\rm osc}$ is the operator for the physical energy of the oscillator and $\hat{\rho}(t)$ is an arbitrary state. In the long time limit the result is, perhaps unsurprisingly, classically thermal. Section 7 gives a brief discussion. ## 2 State evolution in the Wigner-Weyl picture There are many possible formulations of quantum mechanics in phase space [3]. Generally, they can be related [9, 10] to that of Wigner and Weyl [4, 5, 6]. As in [25], where details are given, we shall adopt a formal efficient notation [11]. Denoting the Weyl transform of an operator $\hat{A}$ by $A(p,q)$, or sometimes by $(\hat{A})(p,q)$, it is given by \[\hat{A}\longleftrightarrow{\rm Tr}(\hat{A}\,\hat{\Delta}(p,q))\equiv A(p,q) \equiv(\hat{A})(p,q)\] (1) where \[\mathrm{}\hat{\Delta}(p,q) = \int_{-\infty}^{\infty}\frac{{\rm d}p^{\prime}{\rm d}q^{\prime}}{ h}\,{\rm e}^{-\frac{\rm i}{\hbar}(p^{\prime}q-q^{\prime}p)}\,\hat{D}(p^{\prime },q^{\prime})\] \[= \int_{-\infty}^{\infty}{\rm d}x\,\,{\rm e}^{\frac{\rm i}{\hbar}p \,x}\,\,\|q+\frac{x}{2}\rangle\langle q-\frac{x}{2}\|\] and $\hat{D}$ is the Weyl operator ([12]), \[\hat{D}(p,q)={\rm e}^{\frac{\rm i}{\hbar}(p\,\hat{q}-q\,\hat{p})}\,.\] (3) In this formalism, it is important to realize that $\hat{p}$ and $\hat{q}$ are canonical operators such that $[\hat{p},\hat{q}]=-{\rm i}\hbar$. Formally $\hat{\triangle}$ has the the properties [11] that its trace is unity, that \[\int_{-\infty}^{\infty}\frac{{\rm d}p\,{\rm d}q}{h}\hat{\Delta}(p,q)=1\,,\] (4) and that \[{\rm Tr}\left(\hat{\Delta}(p,q)\hat{\Delta}(p^{\prime},q^{\prime})\right)=h \delta(p-p^{\prime})\delta(q-q^{\prime})\,.\] (5) Then equation (1) can be inverted to give \[\hat{A}=\int_{-\infty}^{\infty}\frac{{\rm d}p\,{\rm d}q}{h}A(p,q)\hat{\Delta}( p,q)\quad\mbox{and}\quad{\rm Tr}\hat{A}=\int_{-\infty}^{\infty}\frac{{\rm d}p \,{\rm d}q}{h}A(p,q)\,.\] (6) From these properties one can also show [11] that, for two operators $\hat{A}$ and $\hat{B}$, \[{\rm Tr}(\hat{A}\hat{B})=\int_{-\infty}^{\infty}\frac{{\rm d}p\,{\rm d}q}{h}A( p,q)B(p,q)\,,\] (7) and that the Weyl transform of the product $\hat{A}\hat{B}$ is \[\hat{A}\hat{B}\longleftrightarrow{\rm Tr}\left(\hat{A}\hat{B}\hat{\Delta}(p,q) \right)=A(p,q)\,{\rm exp}\left[\frac{{\rm i}\hbar}{2}\left(\frac{\partial^{}} {\partial q}\frac{\partial}{\partial p}\,-\,\frac{\partial^{}}{\partial p} \frac{\partial}{\partial q}\right)\right]B(p,q)\,,\] (8) where the starred operators act to the left on $A(p,q)$. The Weyl transform makes the sensible fundamental associations $\hat{1}{\longleftrightarrow}1$, $F(\hat{p}){\longleftrightarrow}F(p)$ and $F(\hat{q}){\longleftrightarrow}F(q)$, but functions that mix $\hat{p}$ and $\hat{q}$ are more complicated. For instance $\hat{p}\,\hat{q}{\longleftrightarrow}p\,q-{\rm i}\hbar/2$ and $\hat{q}\hat{p}{\longleftrightarrow}p\,q+{\rm i}\hbar/2$ so that the Weyl operator corresponding to $p\,q$ is $(\hat{p}\,\hat{q}+\hat{q}\,\hat{p})/2$. The classical equation of motion for a damped harmonic oscillator forced by $F(t)$ is \[m\ddot{q}+m\beta\dot{q}+m\omega^{2}q=F(t).\] (9) The time generator, but not the energy, of this system is [26, 27, 28] the Hamiltonian \[H_{t}(p,q)=\frac{p^{2}}{2m}\,{\rm e}^{-\beta t}+{\rm e}^{\beta t}\left(\frac{1 }{2}m\omega^{2}q^{2}-qF(t)\right),\] (10) To verify that this is the correct Hamiltonian we note that \[\dot{q}=\frac{\partial H_{t}}{\partial p}=\frac{p}{m}{\rm e}^{-\beta t}\quad \mbox{and}\quad\dot{p}=-\frac{\partial H_{t}}{\partial q}={\rm e}^{\beta t}(F( t)-m\omega^{2}q),\] from which it follows that \[m\ddot{q}={\rm e}^{-\beta t}(\dot{p}-\beta p),\] so leading to equation (9). The Lagrangian corresponding to $H_{t}(p,q)$ is \[L(q,\dot{q})=p\dot{q}-H_{t}(p,q)\] so that the canonical variables $(p,q)$ in phase space are \[p=\frac{\partial L}{\partial\dot{q}}=m\dot{q}\,{\rm e}^{\beta t}\equiv P{\rm e }^{\beta t}\,\,{\rm and}\,\,q,\] (11) where we use the symbol $P$ to denote $m\dot{q}$, the physical momentum. For the Wigner-Weyl association the operator governing the quantum time dependence of this system is \[\hat{H}_{t}=\frac{\hat{p}^{2}}{2m}{\rm e}^{-\beta t}+{\rm e}^{\beta t}\left( \frac{1}{2}m\omega^{2}\hat{q}^{2}-\hat{q}F(t)\right).\] (12) The the Wigner function is defined [11] as \[\rho_{w}(p,q;t)\equiv\frac{1}{h}{\rm Tr}\big{(}\hat{\rho}(t)\hat{\Delta}(p,q) \big{)}\longleftrightarrow\frac{1}{h}\hat{\rho}(t)\,,\] (13) so that by (7), \[{\rm Tr}\big{(}\hat{\rho}(t)\hat{A}\big{)}=\int_{-\infty}^{\infty}{\rm d}p\,{ \rm d}q\,\rho_{w}(p,q;t)A(p,q)\,.\] (14) The evolution of the Wigner function under the action of a time-dependent Hamiltonian is is well-known [9, 11, 13]. Wave functions evolve according to \[\|\psi_{t}\rangle=\hat{U}_{t}\|\psi\rangle,\] (15) where the unitary time evolution operator ${\hat{U}}_{t}$ is governed by the equation \[{\rm i}\hbar\frac{\partial}{\partial t}\hat{U}_{t}=\hat{H}_{t}\hat{U}_{t}\,,\] (16) where, in this case, the Hamiltonian is $\hat{H}_{t}$, equation (12). Then the time-dependent density matrix $\hat{\rho}(t)$ is given by \[\hat{\rho}(t)={\hat{U}}_{t}\,\hat{\rho}(0)\,{\hat{U}}^{\dagger}_{t}\,,\] (17) and its Weyl transform is \[\rho_{w}(p,q;t)=\int{\rm d}p^{\prime}{\rm d}q^{\prime}P_{w}(p,q,t\|p^{\prime},q ^{\prime},0)\rho_{w}(p^{\prime},q^{\prime};0)\] (18) where $P_{w}(\cdot\|\cdot)$ is the Wigner propagator defined by \[P_{w}(p,q,t\|p^{\prime},q^{\prime},0)=\frac{1}{h}{\rm Tr}(\hat{U}_{t}^{\dagger} \hat{\Delta}(p,q)\hat{U}_{t}\hat{\Delta}(p^{\prime},q^{\prime})).\] (19) In particular, it is easy to show from the properties above that \[\int{\rm d}p^{\prime}{\rm d}q^{\prime}P_{w}(p,q,t\|p^{\prime},q^{\prime},0)= \int{\rm d}p\,{\rm d}qP_{w}(p,q,t\|p^{\prime},q^{\prime},0)=1\] (20) and \[P_{w}(p,q,0\|p^{\prime},q^{\prime},0)=\delta(p-p^{\prime})\delta(q-q^{\prime})\,.\] (21) From definition (19) and equation (16) we can differentiate the propagator with respect to time to get \[{\rm i}\hbar\frac{\partial}{\partial t}P_{w}(p^{\prime},q^{\prime},t\|p,q,0)= \frac{1}{h}\left(\,[\hat{H}_{t},{\hat{U}}_{t}\hat{\Delta}(p,q){\hat{U}}_{t}^{ \dagger}]\,\right)(p^{\prime},q^{\prime})\] (22) The Weyl transform of ${\hat{H}}_{t}$, equation (10), is quadratic in $p$ and $q$ and so has no derivatives with respect to $p$ and/or $q$ of order higher than second. Applying equation (8) to equation (22) is straightforward. Collecting terms gives \[\frac{\partial}{\partial t}P_{w}(p,q,t\|p^{\prime},q^{\prime},0)=\] (23) \[= \frac{\partial}{\partial q}H_{t}(p,q)\frac{\partial}{\partial p}P _{w}(p,q,t\|p^{\prime},q^{\prime},0)-\frac{\partial}{\partial p}H_{t}(p,q)\frac {\partial}{\partial q}P_{w}(p,q,t\|p^{\prime},q^{\prime},0)\,.\] This describes classical motion, under the action of $H_{t}(p,q)$, of the canonical variables $p$ and $q$, equation (11). The solution that satisfies initial condition (21) is \[P_{w}(p,q,t\|p^{\prime},q^{\prime},0)=\delta(p-p(t\|p^{\prime},q^{\prime},0)) \delta(q-q(t\|p^{\prime},q^{\prime},0))\,,\] (24) where $\big{(}p(t\|p^{\prime},q^{\prime},0),q(t\|p^{\prime},q^{\prime},0)\big{)}$ is the classical phase space solution for canonical momentum and position under the action of Hamiltonian $H_{t}(p,q)$ such that $\big{(}p(0\|p^{\prime},q^{\prime},0),q(0\|p^{\prime},q^{\prime},0)\big{)}=(p^{ \prime},q^{\prime})$. This solution also obeys equation (20), by direct integration in the first instance and, in the second, by recognizing that the Jacobian $\partial(p,q)/\partial(p^{\prime},q^{\prime})$ is unity when $(p,q)$ and $(p^{\prime},q^{\prime})$ are related by the classical motion implied by equation (24). For the cases we are considering, the equations for classical motion are \[p(t)=m\dot{q}(t)\,{\rm e}^{\beta t}\quad\mbox{and}\quad m\ddot{q}(t)+m\beta \dot{q}(t)+m\omega^{2}q(t)=F(t).\] (25) ## 3 Forcing by stationary white noise, with friction added The white noise force is a stationary Gaussian process [14] with the particular ensemble averages, \[\overline{F(t)}=0\quad\mbox{and}\quad\overline{F(t_{1})F(t_{2})}=\mu\,\delta(t _{1}-t_{2})\,.\] (26) With friction added this describes the classical theory of Brownian motion [14]. The friction term is $m\beta\dot{q}$ in (25) and represents phenologically the average effect of a heat bath. In our semi-classical theory the effects of Brownian motion are expressed by taking the ensemble average of the propagator (24). Denoting this average by an overline, from equation (24) we have \[\overline{P_{w}{(p,q,t\|p_{0},q_{0},0)}}=\overline{\delta(p-p(t\|p_{0},q_{0},0)) \delta(q-q(t\|p_{0},q_{0},0))}\equiv W(p,q,t\|p_{0},q_{0},0)\] where $W(\cdot\|\cdot)$ is the conditional probability density for $(p,q)$ at time $t$ given the initial conditions $(p_{0},q_{0})$ at $t=0$, and one must remember that $p$ and $q$ are the canonical variables (11). The connection with the physical variables is straightforward, for from (11) we can write \[W(p,q,t\|p_{0},q_{0},0) = \overline{\delta(p-m\dot{q}(t)\,{\rm e}^{\beta t})\delta(q-q(t)}\] \[= {\rm e}^{-\beta t}\,\overline{\delta(p\,{\rm e}^{-\beta t}-m\dot{ q}(t))\delta(q-q(t))},\] with initial conditions $q(0)=q_{0}$ and $m\dot{q}(0)=p_{0}$. Classical Brownian motion is random, stationary and Markovian, and can be characterized by the conditional probability density $W(P,q,t\|P_{0},q_{0},0)$ where $P=m\dot{q}$ is the physical momentum. Reference [8] gives the following efficient expression that is generally approximate, but is _exact_ when the energy is at most quadratic in $P$ and $q$: (28) \[\times\,\,{\rm exp} \left[-\frac{\mu}{2}\int_{0}^{t}{\rm d}s\Big{(}\partial_{P_{s}}\big{(}aq(t)+bP (t)\big{)}\Big{)}^{2}\right]\] where $\mu$ characterizes the random force $F(t)$, equation (26). In (28) $(P(t),q(t))$ is shorthand for solutions $\big{(}P(t\|P_{0},q_{0},0),q(t\|P_{0},q_{0},0)\big{)}$ (where $P(t)=m\dot{q}$), $\dot{q}$ follows from the solution to (25), and $\partial_{P_{s}}$ is the partial derivative with respect to the physical momentum $P_{s}$ at the intermediate time $0\leq s\leq t$. Thus $\mu$ occurs explicitly in equation (28) and $\beta$ occurs implicitly through the solution $(P(t),q(t))$. If the classical Brownian particle were to thermalize after long times [8] then¹ the relation $\mu/2=m\beta\Theta$ must obtain, where $\Theta=kT$, with Boltzmann’s constant $k$ and temperature $T$. In terms of canonical variables $p$ and $q$ the required propagator is [FOOTNOTE:1][ENDFOOTNOTE] \[W(p,q,t\|p_{0},q_{0},0)={\rm e}^{-\beta t}W({\rm e}^{-\beta t}p,q,t\|p_{0},q_{0} ,0)\] (29) In our picture we require a solution, where $\beta$ is nonzero, for the classical unforced harmonic oscillator, $P(t)=m\dot{q}(t)=P_{t}=P(t\|P_{s},q_{s},s)$ and $q(t)=q_{t}=q(t\|P_{s},q_{s},s)$. For the underdamped case ($\omega\geq\beta/2$) these are, for $0\leq s\leq t$, \[P_{t}={\rm e}^{-\frac{\beta(t-s)}{2}}\big{[}P_{s}\big{(}\cos\Omega(t-s)-\frac{ \beta}{2\Omega}\sin\Omega(t-s)\big{)}-q_{s}\frac{m\omega^{2}}{\Omega}\sin \Omega(t-s)\big{]}\] (30) and \[q_{t}={\rm e}^{-\frac{\beta(t-s)}{2}}\big{[}q_{s}\big{(}\cos\Omega(t-s)+\frac{ \beta}{2\Omega}\sin\Omega(t-s)\big{)}+P_{s}\frac{1}{m\Omega}\sin\Omega(t-s) \big{]},\] (31) where \[\Omega^{2}=\omega^{2}-\frac{\beta^{2}}{4}\,.\] (32) At this point it is convenient to transform from canonical phase space coordinates $(p,q)$ to the dimensionless coordinates $(x,y)$, such that (33) Thus we can rewrite equations (30) and (31) as \[X_{t}=\frac{P_{t}}{\hbar\alpha}={\rm e}^{-\frac{\beta(t-s)}{2}}\big{[}X_{s} \big{(}\cos\Omega(t-s)-\frac{\beta}{2\Omega}\sin\Omega(t-s)\big{)}-y_{s}\frac{ \omega}{\Omega}\sin\Omega(t-s)\big{]},\] (34) and \[y_{t}={\rm e}^{-\frac{\beta(t-s)}{2}}\big{[}y_{s}\big{(}\cos\Omega(t-s)+\frac{ \beta}{2\Omega}\sin\Omega(t-s)\big{)}+X_{s}\frac{\omega}{\Omega}\sin\Omega(t-s )\big{]},\] (35) Using this information to evaluate (28) and replacing $\simeq$ by $=$, for this result is exact [8], gives \[W(x,y,t\|x_{0},y_{0},0)={\rm e}^{-\beta t}\int\frac{{\rm d}a{\rm d }b}{(2\pi)^{2}\hbar}\,{\rm exp}\big{[}{\rm i}a\epsilon(y-y_{t})\big{]}\] (36) \[\times\,{\rm exp}\big{[}{\rm i}\frac{b}{\epsilon}(x{\rm e }^{-\beta t}-X_{t})\big{]}\,{\rm exp}\left[-\frac{N}{2}\int_{0}^{\Omega t}{\rm d }\theta\,{\rm e}^{-\frac{\beta}{\Omega}\theta}\Big{(}a\sin\theta+b(\cos\theta- \frac{\beta}{2\Omega}\sin\theta)\Big{)}^{2}\right],\] where $(X_{t},y_{t})$ are given by (34) and (35) with initial time $s=0$, $(x_{0},y_{0})=(\frac{p_{0}}{\hbar\,\alpha},\alpha q_{0})$, $N=\mu/(m\Omega^{2}\hbar)$, and $\epsilon=\sqrt{\Omega/\omega}$. This expression for $W(\|)$ is normalised with respect to integration over ${\rm d}p{\rm d}q=\hbar\,{\rm d}x{\rm d}y$. The integral in (36) has a two-dimensional Gaussian form and thus can be evaluated to give another Gaussian form in $x$ and $y$. In particular, consider the long-time limit, for which $\beta t$ is not small. Ignoring terms damped by the factor ${\rm exp}(-\beta t)$, \[\frac{N}{2}\int_{0}^{\Omega t}{\rm d}\theta\,{\rm e}^{-\frac{\beta}{\Omega} \theta}\Big{(}a\sin\theta+b(\cos\theta-\frac{\beta}{2\Omega}\sin\theta)\Big{)} ^{2}\approx N\frac{\Omega^{3}}{4\omega^{2}\beta}(a^{2}+b^{2}\frac{\omega^{2}}{ \Omega^{2}})\,.\] (37) Using this, and ignoring the terms $y_{t}$ and $X_{t}$ in (36) in this long time limit, gives the product of Gaussian integrals separately with respect to $a$ and $b$. When converted to canonical variables $p$ and $q$ via (33) and with the choice \[\mu=2m\beta\Theta\quad{\rm where}\quad\Theta=kT\] (38) the result is that, for long times, \[W(x,y,t\|x_{0},y_{0},0)\approx\frac{1}{2\pi\hbar}\frac{\hbar\omega}{\Theta}{\rm e }^{-\beta t}{\rm e}^{-\frac{\hbar\omega}{2\Theta}\big{(}x^{2}{\rm e}^{-2\beta t }+y^{2}\big{)}}\quad\quad({\rm thermal\,\,limit}),\] (39) where $x$ and $y$ are given by (33) in terms of canonical coordinates $(p,q)$. This expression is normalized with respect to integration over $\hbar\,{\rm d}x{\rm d}y$. Writing it in terms the physical variables $P=m\dot{q}(t)=\hbar\,\alpha x\,{\rm e}^{-\beta t}$ and $q$ shows it to be the Maxwell-Boltzmann distribution for the oscillator. That expression (39) is Maxwell Boltzmann underlines that this theory is semi-classical. It is a single particle with a c-number term to describe the forces acting. A more fully quantum theory would involve interaction with a heat bath [1, 2]. Notwithstanding its semi-classicality, in the following I give examples of how it can be used to model quantum effects. ## 4 Transition probabilities The probability for transition between states $\|\psi_{1}\rangle$ and $\|\psi_{2}\rangle$ is \[\|\langle\psi_{2}\|\hat{U}_{t}\|\psi_{1}\rangle\|^{2} = {\rm Tr}\left(\|\psi_{2}\rangle\langle\psi_{2}\|\hat{U}_{t}\|\psi_{1 }\rangle\langle\psi_{1}\|\hat{U}_{t}^{\dagger}\right)\] (40) \[= \int\frac{{\rm d}p\,{\rm d}q}{h}\int{\rm d}p^{\prime}{\rm d}q^{ \prime}\,\big{(}\|\psi_{2}\rangle\langle\psi_{2}\|\big{)}(p,q)P_{w}(p,q,t\|p^{ \prime},q^{\prime},0)\big{(}\|\psi_{1}\rangle\langle\psi_{1}\|\big{)}(p^{\prime} ,q^{\prime})\,,\] where $P_{w}$ is the Wigner propagator, (19). In particular, for the random driving force (26), the ensemble averaged propagator is \[W(p,q,t\|p^{\prime},q^{\prime},0)=\overline{P_{w}(p,q,t\|p^{\prime},q^{\prime},0 )}\,.\] The corresponding ensemble averaged transition probability is \[\overline{\|\langle\psi_{2}\|\hat{U}_{t}\|\psi_{1}\rangle\|^{2}}=\int\frac{{\rm d} p\,{\rm d}q}{h}\int{\rm d}p^{\prime}{\rm d}q^{\prime}\,\big{(}\|\psi_{2}\rangle \langle\psi_{2}\|\big{)}(p,q)W(p,q,t\|p^{\prime},q^{\prime},0)\big{(}\|\psi_{1} \rangle\langle\psi_{1}\|\big{)}(p^{\prime},q^{\prime})\] (41) where, for an oscillator, $W$ is given by (36). Translated to variables ($x,y$) this is \[\overline{\|\langle\psi_{2}\|\hat{U}_{t}\|\psi_{1}\rangle\|^{2}}=\int\frac{{\rm d} x\,{\rm d}y}{2\pi}\int\hbar\,{\rm d}x^{\prime}{\rm d}y^{\prime}\,\big{(}\|\psi_ {2}\rangle\langle\psi_{2}\|\big{)}(x,y)W(x,y,t\|x^{\prime},y^{\prime},0)\big{(}\| \psi_{1}\rangle\langle\psi_{1}\|\big{)}(x^{\prime},y^{\prime})\] (42) As in [25] we might suppose the oscillator were initially in the ground state, so that $\|h_{0}\rangle$, where \[\langle\xi\|h_{0}\rangle=(\alpha/\sqrt{\pi})^{1/2}{\rm exp}\left(-\alpha^{2}\xi ^{2}/2\right).\] (43) and ask for the probability $\overline{\|\langle h_{0}\|\hat{U}_{t}\|h_{0}\rangle\|^{2}}$ that it stays there. Now \[\left(\|h_{0}\rangle\langle h_{0}\|\right)(x,y)=2\,{\rm exp}\left(-(x^{2}+y^{2}) \right)=2\,{\rm exp}(-R^{2})\,\] (44) so that from this and (36) it is clear that the evaluation of this probability requires a number of Gaussian integrals only. This is easy in the limit of long times. Using (39) and (44) in (42) gives, for large $\beta t$, \[\overline{\|\langle h_{0}\|\hat{U}_{t}\|h_{0}\rangle\|^{2}}\approx{\rm e}^{-\beta t }\,\,\frac{\hbar\omega}{\Theta}\sqrt{\frac{1}{(1+\frac{\hbar\omega}{2\Theta})( 1+\frac{\hbar\omega}{2\Theta}{\rm e}^{-2\beta t})}}.\] (45) This equation applies for long times it will not, of course, be expected to equal unity at $t=0$. By contrast, when $\beta$ strictly vanishes thermalization does not occur. For that case we found [25] that for all times, (46) where $N_{o}=\mu_{o}/(m\omega^{2}\hbar)$ and $\mu_{o}$ is a parameter characteristic of the white noise forcing strength and is not necessarily equal to $2m\beta kT$. More generally, when the oscillator is initially in the ground state it is clear from (36), (44) and (42) that an exact evaluation of $\overline{\|\langle h_{0}\|\hat{U}_{t}\|h_{0}\rangle\|^{2}}$ requires the evaluation of several Gaussian integrals. As an example, the result for the parameter values $D=\frac{2\theta}{\hbar\omega}=5$ and $B=\frac{\beta}{\omega}=0.05$ is shown by the middle curve of Figure 1. ## 5 Phase ### Definition In this subsection I consider briefly the generalization of the model in [25] to include the effect of friction $\beta$ on the time evolution of the Weyl quantized phase of the oscillator. In particular, if the creation operator, \[\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(\alpha\hat{q}-{\rm i}\frac{\hat{p}}{ \alpha\hbar}\right)\] has Weyl transform (where $\alpha^{2}\equiv m\omega/\hbar.$) (47) then the canonical phase operator $\hat{\phi}$ can be defined [15] as the Weyl quantization of $\phi$. Properties of $\hat{\phi}$ have been considered previously [7, 15, 16, 17]. It is a bone-fide bounded self-adjoint operator on Hilbert Space. As befits an angle, its spectrum must be limited to a range of $2\pi$ which I shall take as $[-\pi,\pi)$. Generally in the Wigner-Weyl picture, the time-dependent average of an operator $\hat{A}$ with respect to the state $\hat{\rho}$ is given by (14) with (18), where, for an oscillator, the evolution is classical, equation (24). In particular the Weyl quantizated phase operator is $\hat{\phi}{\longleftrightarrow}\phi(p,q)$, where $\phi(p,q)$ is the harmonic oscillator phase in the plane $(p,q)$. Then, where ${\rm d}p\,{\rm d}q/h=R{\rm d}R{\rm d}\phi/2\pi$, \[\phi={\rm Tr}(\hat{\phi}\,\hat{\Delta}(R,\phi)){\longleftrightarrow}\hat{\phi} =\int_{0}^{\infty}{\rm d}RR\int_{-\pi}^{\pi}\frac{{\rm d}\phi}{2\pi}\,\phi\, \hat{\Delta}(R,\phi)\,,\] (48) where $\phi$ is given in (47) and we may consider $\hat{\Delta}$ as a function of plane polar coordinates $R$ and $\phi$. Details are given in [25]. Alternatives to $\hat{\phi}$ have been suggested to represent the phase of an harmonic oscillator, or in some sense photons, for instance by a non-projective positive operator valued measure, or POVM [19, 20, 21, 22, 23, 24]. Using a POVM to represent a quantum system may be thought of as allowing for an element of imperfection in the measurement. ### Angle operators Translating (14) and (18) to the variables $(x,y)$, combining them and taking the ensemble average gives, for an operator $\hat{O}$ \[\overline{{\rm Tr}(\hat{\rho}(t)\hat{O})}=\int\frac{{\rm d}x\,{\rm d}y}{2\pi} \int\hbar\,{\rm d}x^{\prime}{\rm d}y^{\prime}\,O(x,y)\,W(x,y,t\|x^{\prime},y^{ \prime},0)\,\rho_{w}(x^{\prime},y^{\prime};0),\] (49) where $\rho_{w}(x^{\prime},y^{\prime};0)$ is an arbitrary initial state. At long times, according to equation (39) memory of the initial state is lost and \[\overline{{\rm Tr}(\hat{\rho}(t)\hat{O})}\longrightarrow\frac{\hbar\omega}{ \Theta}{\rm e}^{-\beta t}\int\frac{{\rm d}x\,{\rm d}y}{2\pi}O(x,y){\rm e}^{- \frac{\hbar\omega}{2\Theta}\big{(}x^{2}{\rm e}^{-2\beta t}+y^{2}\big{)}}.\] (50) Writing the Weyl transformation of the operator $\hat{O}$ in polar coordinates as $M(R,\phi)$, say, then, \[\overline{{\rm Tr}(\hat{\rho}(t)\hat{O})}\longrightarrow\int_{-\pi}^{\pi}\frac {{\rm d}\phi}{2\pi}\int_{0}^{\infty}{\rm d}u\,\,M\Big{(}{\rm e}^{\beta t/2} \sqrt{u\frac{2\Theta}{\hbar\omega}},\phi\Big{)}{\rm e}^{-u\left({\rm e}^{- \beta t}\cos^{2}\phi+{\rm e}^{\beta t}sin^{2}\phi\right)}\,.\] (51) Now consider those operators, $\hat{\Phi}$, whose Weyl quantizations are functions $\Phi(\phi)$ of phase angle only, namely $\hat{\Phi}{\longleftrightarrow}\Phi(\phi)$. For these operators, from (51), at long times \[\overline{{\rm Tr}(\hat{\rho}(t)\hat{\Phi})}\longrightarrow\int_{-\pi}^{\pi} \frac{{\rm d}\phi}{2\pi}\frac{\Phi(\phi)}{({\rm e}^{-\beta t}\cos^{2}\phi+{\rm e }^{\beta t}\sin^{2}\phi)}\,.\] (52) Angle $\phi$ is defined with respect to the canonical pair $\hat{p}\leftrightarrow p$ and $\hat{q}\leftrightarrow q$ via the variables $(x,y)$. But the physical variables are $(X,y)$, equation (33). In particular we may define the physical angle as $\overline{\phi}$ such that \[\tan\overline{\phi}={\rm e}^{\beta t}\tan{\phi.}\] (53) Thus $\overline{\phi}$ is a function of $\phi$, with parametric dependence on $\beta t$. Differentiating both sides of (53) with respect to $\overline{\phi}$ and rearranging terms gives \[\frac{{\rm d}\phi}{{\rm d}\overline{\phi}}=\big{(}{\rm e}^{-\beta t}\cos^{2} \phi+{\rm e}^{\beta t}\sin^{2}\phi\big{)}=\frac{1}{\big{(}{\rm e}^{-\beta t} \sin^{2}\overline{\phi}+{\rm e}^{\beta t}\cos^{2}\overline{\phi}\big{)}}\,\,,\] (54) so that we can also make the association $\hat{\Phi}\longleftrightarrow\Phi(\overline{\phi})$, and write \[\overline{{\rm Tr}(\hat{\rho}(t)\hat{\Phi})}\longrightarrow\int_{-\pi}^{\pi} \frac{{\rm d}\overline{\phi}}{2\pi}\,\,\Phi(\overline{\phi})\,,\] (55) corresponding to a random distribution in physical angle $\overline{\phi}$. On the other hand from the form of (52) at long times the distribution of $\phi$ becomes strongly concentrated near $\phi=0$. From (52), (53), and (54) it is clear that at long times, whatever the initial state $\hat{\rho}(0)$ may be, the expectations of $\hat{\phi}$ and of $\hat{\overline{\phi}}$ vanish. Figure 2 shows for three illustrative cases, computed using the full propagator (36) in (49), the approach of the expectation of $\hat{\phi}$ to zero as functions of time when the initial state is the ground state $\|h_{0}\rangle\langle h_{0}\|$, equation(44). They are: curve A {$D=\frac{2\theta}{\hbar\omega}=1000$ and $B=\frac{\beta}{\omega}=0.02$}; curve B {$D=10$ and $B=0.1$}; curve C {$D=5$ and $B=0.05$}. Curve A especially corresponds to high temperature and low damping such that their product is 20. This is nearly identical to the case $\beta=0$ discussed in reference [25].² [FOOTNOTE:2][ENDFOOTNOTE] More generally, in the Weyl correspondence, for any angle operator $\hat{\Phi}\longleftrightarrow\Phi(\phi)$, equations (1) and (2) give $\hat{\Phi}$. For any such function of angle only it can be shown [7, 16, 29] that its matrix elements with respect to harmonic oscillator eigenstates {$\|h_{n}\rangle,n=0,1,2\ldots$} are \[\langle h_{m}\|\hat{\Phi}\|h_{n}\rangle={\rm i}^{m-n}\,g_{m,n}\int_{-\pi}^{\pi} \frac{{\rm d}\phi}{2\pi}\,\Phi(\phi)\,{\rm e}^{{\rm i}(n-m)\phi}\,.\] (56) Here $g_{m,n}$ is the real symmetric matrix \[g_{m,n}=2^{-\frac{\|m-n\|}{2}}\,\frac{\Gamma\left(\frac{n_{\ell}}{2}+s_{\ell} \right)}{\Gamma\left(\frac{n_{g}}{2}+s_{\ell}\right)}\sqrt{\frac{n_{g}!}{n_{ \ell}!}}\] (57) with \[s_{\ell}=\left\{\begin{array}[]{r@{\quad n_\ell}l}1/2\quad n_{\ell}&\,\,\, \mbox{even}\\ 1\quad n_{\ell}&\,\,\,\mbox{odd}\end{array}\right.\] (58) and $n_{\ell}\,(n_{g})$ is the lessor (greater) of the pair $(m,n)$. We may also ask for matrix elements of the operator corresponding to $\overline{\phi}$ which, by equation (53), is a function of $\phi$ only, with $\beta t$ as a parameter. Then \[\langle h_{m}\|\hat{\overline{\phi}}\|h_{n}\rangle={\rm i}^{m-n}\,g_{m,n}\int_{- \pi}^{\pi}\frac{{\rm d}\phi}{2\pi}\,\overline{\phi}(\phi)\,{\rm e}^{{\rm i}(n- m)\phi}\,.\] (59) When this is integrated by parts, use made of the fact that $\overline{\phi}(\pm\pi)=\pm\pi$, and recourse made to equation (54) one finds upon rearranging that, when $n=m$, $\langle h_{m}\|\hat{\overline{\phi}}\|h_{n}\rangle$ vanishes and when $n\neq m$, \[\langle h_{m}\|\hat{\overline{\phi}}\|h_{n}\rangle={\rm i}^{n-m-1}\frac{g_{m,n}} {{(n-m)}}\Big{\{}1-\int_{-\pi}^{\pi}\frac{{\rm d}\phi}{2\pi}\frac{{\rm e}^{{ \rm i}(n-m)\phi}}{\big{(}{\rm e}^{-\beta t}\cos^{2}\phi+{\rm e}^{\beta t}\sin^ {2}\phi\big{)}}\Big{\}}\,.\] (60) The imaginary part of the second term in curly brackets vanishes by symmetry. What remains can be re-expressed as an integral over the range $(0,\pi/2)$ and evaluated by using tables, eg [18]. The result is \[\langle h_{m}\|\hat{\overline{\phi}}\|h_{n}\rangle=(1-\delta_{m,n})\,{\rm i}^{n- m-1}\frac{\overline{g}_{m,n}}{{(n-m)}}\] (61) where \[\overline{g}_{m,n}=g_{m,n}\Big{\{}1-\sigma_{\|n-m\|}\,\big{(}\tanh(\beta t/2) \big{)}^{\|n-m\|/2}\Big{\}}\,,\] (62) with \[\sigma_{n}=\left\{\begin{array}[]{r@{\quad n}l}1\quad n&\,\,\,\mbox{even}\\ 0\quad n&\,\,\,\mbox{odd}\,.\end{array}\right.\] (63) One might be forgiven for calling calling $\hat{\phi}$ the _canonical_ phase operator because it is the Weyl quantization of angle in the phase plane defined (effectively) by the canonical coordinates $(p,q)$. It is a bounded self-adjoint operator discussed at length in [7] and [16], but properties of the operator $\hat{\overline{\phi}}$, which is based on the physical parameters $(P,q)$, where $P=m\dot{q}=p\,{\rm e}^{-\beta t}$, are open questions. We can, however, get some suggestions by computation. In particular, Figure 3 plots the computed eigenvalues of $\hat{\overline{\phi}}$ for several values of $\beta t$ when its matrix, equation (61), is truncated to size $150\times 150$. Note that the canonical operator corresponds to the case $\beta t=0$. The eigenvalues for that case appear to be spread evenly _pari passu_, as befits the canonical phase, from $-\pi$ to $\pi$, but that as $\beta t$ increases the spread of values polarises equally between the values $\pm\pi/2$ although, as $\beta t$ increases, the values $\pm\pi$ are limit points. ### Angle variance The variance of the canonical quantised angle is \[\langle h_{m}\|\hat{\phi}^{2}\|h_{m}\rangle=\sum_{n=0}^{\infty}\|\langle h_{m}\| \hat{\phi}\|h_{n}\rangle\|^{2}=\sum_{n=1}^{m}\frac{1}{n^{2}}\,(g_{m,m-n})^{2}+ \sum_{n=1}^{\infty}\frac{1}{n^{2}}\,(g_{m+n,m})^{2}\,,\] (64) In the correspondence limit, for which $n$ and $m$ are large compared to their difference, $g_{m,n}$ approaches unity. So for large $m$ \[\langle h_{m}\|\hat{\phi}^{2}\|h_{m}\rangle\longrightarrow 2\sum_{n=0}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{3}\qquad({\rm as}\,\,m\rightarrow\infty),\] consistent with a random distribution of angle. The analysis for $\hat{\overline{\phi}}^{2}$ follows directly. From (61) one has \[\langle h_{m}\|\hat{\overline{\phi}}^{2}\|h_{m}\rangle=\sum_{n=0}^{\infty}\| \langle h_{m}\|\hat{\overline{\phi}}\|h_{n}\rangle\|^{2}=\sum_{n=1}^{m}\frac{1}{n ^{2}}\,(\overline{g}_{m,m-n})^{2}+\sum_{n=1}^{\infty}\frac{1}{n^{2}}\,( \overline{g}_{m+n,m})^{2}\,.\] (65) This expectation is time-dependent, equation (62). When $\beta t$ vanishes it becomes identical to (64), but as $\beta t\rightarrow\infty$ we have the situation that $\overline{g}_{m,n}$ vanishes when $\|m-n\|$ is even but equals $g_{m,n}$ otherwise. In that case the result is that as $m$ increases, \[\langle h_{m}\|\hat{\overline{\phi}}^{2}\|h_{m}\rangle\longrightarrow\ 2\sum_{n= 0}^{\infty}\frac{1}{(2n+1)^{2}}=\frac{\pi^{2}}{4}\qquad({\rm as}\,\,m\,{\rm and }\,\,\beta t\rightarrow\infty).\] This makes sense for a distribution of angle between the values between the limiting values $\pm\pi/2$—see Figure 3—with equal probability. We may ask for the expectation of the square of the canonical phase operator, $\hat{\phi}^{2}$, in the limit of thermalisation, equation (51), when damping is weak, $\beta\to 0$. In particular, (66) Now by definition, equation (1), where ([16]) \[\langle h_{m}\|\,\hat{\Delta}(R,\phi)\,\|h_{n}\rangle=2(-1)^{n}\,{\rm i}^{\|m-n\|} \,2^{\frac{\|m-n\|}{2}}\sqrt{\frac{n_{\ell}!}{n_{g}!}}\,\,{\rm e}^{{\rm i}(n-m) \phi}R^{\|m-n\|}\,{\rm e}^{-R^{2}}L^{\|m-n\|}_{n_{\ell}}(2R^{2})\,.\] (67) Only the diagonal terms $n=m$ survive to give \[\overline{{\rm Tr}\big{(}\hat{\rho}(t)\hat{\phi}^{2}\big{)}}\longrightarrow 4 \int_{0}^{\infty}{\rm d}x\,x\,{\rm e}^{-x^{2}}\sum_{m=0}^{\infty}(-)^{m}\,{\rm e }^{-\frac{2\Theta}{\hbar\omega}x^{2}}L_{m}\left(-\frac{4\Theta}{\hbar\omega}x^ {2}\right)\langle h_{m}\|\hat{\phi}^{2}\|h_{m}\rangle\,.\] The integral is standard ([18]). The result is, where $D=\frac{2\Theta}{\hbar\omega}$, \[\overline{{\rm Tr}\big{(}\hat{\rho}(t)\hat{\phi}^{2}\big{)}}\longrightarrow \frac{2}{D+1}\sum_{m=0}^{\infty}\left(\frac{D-1}{D+1}\right)^{m}\langle h_{m}\| \hat{\phi}^{2}\|h_{m}\rangle\quad({\rm thermal\,\,limit\,\,but}\,\,\beta \to 0)\,.\] (68) Consider the high temperature limit of this result, for which $(D-1/(D+1)$ approaches unity, The sum then becomes dominated the increasing number of terms for which $\langle h_{m}\|\hat{\phi}^{2}\|h_{m}\rangle\rightarrow\pi^{2}/3$. So that, as $D$ increases we have \[\lim_{D\rightarrow\infty}\overline{{\rm Tr}\big{(}\hat{\rho}(t)\hat{\phi}^{2} \big{)}}=\frac{\pi^{2}}{3}\,.\] This is characteristic of a random distribution of phase. It is consistent with the analysis of [25] which had the white noise driving force $\mu$ but no friction $\beta$. However, for thermalization to occur in this analysis, equation (39), we require $\mu=2m\beta\Theta$, equation (38), so to retain $\mu$ at all we must keep the product $\beta\Theta$ finite. ## 6 Oscillator energy Consider the operator function \[\hat{O}={\rm exp}(-B\hat{E}_{\rm osc})\,,\] where $\hat{E}_{\rm osc}$ is the oscillator’s physical energy operator (noting equation (11)) \[\hat{E}_{\rm osc}=\frac{\hat{P}^{2}}{2m}+\frac{m\omega^{2}}{2}\hat{q}^{2}={\rm e }^{-2\beta t}\frac{\hat{p}^{2}}{2m}+\frac{m\omega^{2}}{2}\hat{q}^{2}\] (69) Then, for purposes of operator algebra and the Weyl transform we can scale $m$ and $\omega$ such that $\overline{m}\equiv m\,{\rm e}^{2\beta t}$ and $\overline{\omega}\equiv\omega\,{\rm e}^{-\beta t}$ so that \[\hat{E}_{\rm osc}=\frac{\hat{p}^{2}}{2\,\overline{m}}+\frac{\overline{m}\, \overline{\omega}^{2}}{2}\,\hat{q}^{2}\,.\] Correspondingly we can define \[(\overline{x},\overline{y})\equiv(\frac{p}{\hbar\overline{\alpha}},\overline{ \alpha}q)\] where $\overline{\alpha}^{2}\equiv\overline{m}\,\overline{\omega}/\hbar=\alpha^{2}\,{ \rm e}^{\beta t}$. Now for the basic harmonic oscillator Hamiltonian $\hat{H}=\frac{\hat{p}^{2}}{2m}+\frac{m\omega^{2}}{2}\hat{q}^{2}$ the Weyl correspondence for ${\rm exp}(-B\hat{H})$ is ([25]) \[{\rm exp}(-B\hat{H}){\longleftrightarrow}\frac{1}{\cosh\left(\frac{\hbar\omega B }{2}\right)}{\exp}\left[-R^{2}\tanh\left(\frac{\hbar\omega B}{2}\right)\right]\,.\] (70) Thus, defining $\overline{R}^{2}\equiv\overline{x}^{2}+\overline{y}^{2}=R^{2}({\rm e}^{-\beta t }\cos^{2}\phi+{\rm e}^{\beta t}\sin^{2}\phi)$, we can write an expression for the Weyl transform of ${\rm exp}(-B\hat{E}_{\rm osc})$ by replacing in (70) $R$ by $\overline{R}$ and $\omega$ by $\overline{\omega}$. The integrals in (51) are straightforward, recognizing $\frac{{\rm d}\overline{\phi}}{{\rm d}\phi}$, equation (54), gives \[\overline{{\rm Tr}\big{(}\hat{\rho}(t){\rm exp}(-B\hat{E}_{\rm osc})\big{)}} \longrightarrow\frac{1}{\cosh(\frac{\hbar\omega}{2}B{\rm e}^{-\beta t})+\frac{ 2\Theta}{\hbar\omega}{\rm e}^{\beta t}\tanh(\frac{\hbar\omega}{2}B{\rm e}^{- \beta t})}\,.\] Finally, as $\beta t\longrightarrow\infty$ this becomes \[[\overline{{\rm Tr}\big{(}\hat{\rho}(t){\rm exp}(-B\hat{E}_{\rm osc})\big{)}} \longrightarrow\frac{1}{1+B\Theta}=\frac{1}{1+BkT}\,.\] This is the the classical result for a thermalized harmonic oscillator. ## 7 Discussion By adding the friction force, this paper generalizes the discussion in [25] of an harmonic oscillator acted upon by solely by an external white noise force. In transcribing the classical Brownian dynamics to a semi-classical quantum mechanics it has proved efficient to use the Wigner/Weyl formalism. The result is semi-classical of course because the external forces are represented by c-numbers, with the twist that the white noise force is stochastic. Though phenomenological the theory can be used to model quantum effects of an oscillator interacting with a heat bath. Inclusion of friction in the model means that although the Hamiltonian still generates time translation it is not the physical energy. At long times, with friction acting, the oscillator forgets its initial state and thermalizes, (39) with (38). Section 4 considers the time-dependence of the probability that the oscillator remains in its ground state. The effect of friction is to drive this probability to zero exponentially. Figure 1 shows this. Section 5 considers implications for the phase operator $\hat{\phi}\longleftrightarrow\phi$, defined by equations (47) and (48), and more generally for operators $\hat{\Phi}\longleftrightarrow\Phi(\phi)$. At long times ($\beta t$ large) the distribution of $\phi$ peaks strongly towards zero, equation (52). But $\hat{\phi}$ is defined from the canonical variables $(\hat{p},\hat{q})\longleftrightarrow(p,q)$. Instead one might define a physical angle operator $\hat{\overline{\phi}}$ in terms of $(m\dot{q},q)\longleftrightarrow({\rm e}^{-\beta t}\hat{p},\hat{q})$ whose distribution at long times becomes random. Figure 2 shows the approach to zero in time of the expectation of $\hat{\phi}$ when the oscillator starts in its ground state. Angle operators $\hat{\phi}$ and $\hat{\overline{\phi}}$ can be represented by the set of their matrix elements between the standard harmonic oscillator states $h_{m}$, where $m=0,1,2,\cdots$. Figure 3 shows the spread of their approximate eigenvalues when these matrices are truncated to sizes $150\times 150$. A curious feature of $\hat{\overline{\phi}}$ is that its spectrum depends on time. Consistent with the results of [25], the variance ${\rm Tr}\big{(}\hat{\rho}(t)\hat{\phi}^{2}\big{)}$ approaches $\pi^{2}/3$ (as for a random distribution of phase) in the limits $\beta\to 0$ and $kT\rightarrow\infty$ such that $\mu\equiv 2m\beta kT$ remains finite. Section 6 considers for long times the expectation of the generating function $\hat{O}\equiv{\rm exp}(-B\hat{E}_{\rm osc})$ where $B$ is a c-number parameter and $\hat{E}_{\rm osc}$ is the oscillator’s physical energy, equation (69). As $\beta t\rightarrow\infty$ this becomes identical with the classical result. ## References * [1] Honda D, Nakazato H and Yoshida M 2010 _J. Math. Phys._51 072107 * [2] Guerrero P, Lopez J L and Montejo-Gamez J 2014 _J. Phys. A:Math. Theor._47 035303 * [3] Cohen L 1966 _J. Math. Phys._7 (5) 781 * [4] Weyl H 1927 _Z physik_46 1 * [5] Weyl H 1930 _The Theory of Groups and Quantum Mechanics_ (New York: Dover) * [6] Wigner E P 1932 _Phys. Rev._40 749 * [7] Dubin D A, Hennings M A and Smith T B 2000 _Mathematical Aspects of Weyl Quantization and Phase_ (Singapore: World Scientific) * [8] Smith T B 1979 _Physica_100A 153 * [9] Smith T B 2006 _J. Phys. A:Math. Gen._39 1469 * [10] Lee Hai-Woong 1995 _Phys. Rep._259 147 * [11] de Groot S R and Suttorp L G 1972 _Foundations of Electrodynamics_ (Amsterdam: North-Holland) * [12] Klauder J R and Skagerstam B 1985 _Coherent States: Applications in Physics and Mathematical Physics_ (Singapore: World Scientific) * [13] Smith T B 1978 _J. Phys. A:Math. Gen._11 (11) 2179 * [14] Wax N (editor) 1954 _Noise and stochastic processes_ (New York: Dover Publications, inc) * [15] Smith T B, Dubin D A and Hennings M A 1992 _J. Mod. Optics_39 1603 * [16] Dubin D A, Hennings M A and Smith T B 1994 _Publ. Res. Inst. Math Sci. Kyoto_30 479 * [17] Lynch R 1995 _Phys. 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Modern Physics_9 2597 ## 8 Figures <figure><img src="content_image/1709.03391/x1.png"><figcaption>Figure 1: The probability that the oscillator stays in the ground state,calculated by a series of numerical Gaussian integrals. The lower curve usesthe long time (βt≫0) approximation (39) for the propagator. The middle curveis exact with parameter values D=2θℏω=5 and B=βω=0.05. For the top curve, βstrictly vanishes, equation (46), with No≡μmω2ℏ=0.25. This result was derivedin [25].</figcaption></figure> <figure><img src="content_image/1709.03391/x2.png"><figcaption>Figure 2: The expectation of the Weyl quantized phase, equation (48), whenthe oscillator is initially in its ground state. For curve “A”, D=2θℏω=1000and B=βω=0.02. For curve “B”, D=10 and B=0.05. For curve “C” D=5 and B=0.05.Curve “A” very nearly reproduces Figure 1 of reference [25] for which β isstrictly zero and No≡μ/(mω2ℏ)=20.</figcaption></figure> <figure><img src="content_image/1709.03391/x3.png"><figcaption>Figure 3: Numerical computation of spectrum of ^¯¯¯ϕ, equation (60), asapproximated by a 150×150 matrix, equation (61), for βt=(0,2,5). The case βt=0reproduces the spectrum for the canonical Weyl quantized phase operator ^ϕ.</figcaption></figure>
1202.4411	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 26731, "num_imgs": 3, "llama3_tokens_count": 8486 }	[ "content_image/1202.4411/x1.png", "content_image/1202.4411/x2.png", "content_image/1202.4411/x3.png" ]	# Localization and spreading of diseases in complex networks A. V. Goltsev Department of Physics $\&$ I3N, University of Aveiro, 3810-193 Aveiro, Portugal Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia S. N. Dorogovtsev Department of Physics $\&$ I3N, University of Aveiro, 3810-193 Aveiro, Portugal Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia J. G. Oliveira Department of Physics $\&$ I3N, University of Aveiro, 3810-193 Aveiro, Portugal Departamento de Engenharia Física, Faculdade de Engenharia, Universidade do Porto, rua Dr. Roberto Frias, 4200-465 Porto, Portugal J. F. F. Mendes Department of Physics $\&$ I3N, University of Aveiro, 3810-193 Aveiro, Portugal ###### Abstract Using the SIS model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks. pacs: 05.10.-a, 05.40.-a, 05.50.+q, 87.18.Sn Survey of infectious diseases reveals that before an outbreak, often, if not typically, a disease is localized within a small group of individuals. Changes in environmental conditions or increase in the frequency of external contacts result in an epidemic outbreak. In the present paper we propose an approach that enables us to describe quantitatively this important localization-delocalization phenomenon. Our approach is based on the SIS model [1; 2] of spreading of diseases in weighted and unweighted networks, where the weights of edges encode frequency of contacts between vertices. It is widely accepted that in uncorrelated networks the epidemic threshold $\lambda_{c}$ of the infection rate $\lambda$ is $\lambda_{MF}=\langle q\rangle/\langle q^{2}\rangle$, where $\langle q\rangle$ and $\langle q^{2}\rangle$ are the first and second moments of the degree distribution [2]. So in networks with a finite $\langle q^{2}\rangle$ the threshold should be non-zero, while it is zero if $\langle q^{2}\rangle$ diverges. One should stress however that all these well-known results were obtained only within a mean-field theory, actually within an annealed network approximation in which a random network is substituted for its fully connected weighted counterpart [2]. Contrastingly, one can show exactly for an arbitrary graph that $\lambda_{c}$ is actually determined by the largest eigenvalue $\Lambda_{1}$ of the adjacency matrix $A_{ij}$ of the graph, and $\lambda_{c}=1/\Lambda_{1}<\lambda_{MF}$[3; 4; 5; 7; 8; 6; 9; 10; 11]. For uncorrelated networks, in particular, scale-free networks with the degree exponent $\gamma>2.5$, it was found that $\Lambda_{1}$ is determined by the maximum degree $q_{max}$, $\Lambda_{1}\propto\sqrt{q_{max}}$[3; 4; 5]. Then, if in the infinite size limit, $q_{max}$ tends to infinity, as, e.g., in the Erdős-Rényi graphs, this leads to the amazing conclusion that the epidemic threshold is absent even in (infinite) networks with a finite $\langle q^{2}\rangle$ in contrast to the mean-field result. The conclusion that the epidemic threshold may be absent even in the networks with rapidly decaying degree distributions was confirmed in numerical simulations performed in Ref. [6]. In the present paper we develop a spectral approach to the SIS model on complex networks. We show that the contradiction between the mean-field approximation and the exact result can be resolved if we take into account localization of diseases. It turns out that, in contrast to the mean field theory, in which a finite fraction of vertices are infected at $\lambda>\lambda_{c}$, there are actually two scenarios of the spreading of diseases. If $\Lambda_{1}$ corresponds to a localized eigenstate, then, at $\lambda$ right above $\lambda_{c}=1/\Lambda_{1}$, disease is mainly localized on a finite number of vertices, i.e., the fraction of infected vertices is negligibly small in large networks. With further increase of $\lambda$, the disease gradually infects more and more vertices until it will infect a finite fraction of vertices. In the second scenario, $\Lambda_{1}$ corresponds to a delocalized state. Then already at $\lambda\Lambda_{1}{-}1{\ll}1$, the disease infects a finite fraction of vertices. Analysing network models and real-world networks, we show that hubs, edges with large weights, and other dense subgraphs can be centers of localization. We consider the standard SIS model of disease spreading in a complex network of size $N$ having adjacency matrix with arbitrary entries $A_{ij}\geq 0$. Infected vertices become susceptible with unit rate, and each susceptible vertex becomes infected by its infective neighbor with the infection rate $\lambda$. Neglecting correlations between infected and susceptible vertices, the probability $\rho_{i}(t)$ that vertex $i$ is infected at time $t$ is described by the evolution equation \[\frac{d\rho_{i}(t)}{dt}=-\rho_{i}(t)+\lambda[1-\rho_{i}(t)]\sum_{j=1}^{N}A_{ij }\rho_{j}(t).\] (1) In the steady state, at $t\rightarrow\infty$, the probability $\rho_{i}\equiv\rho_{i}(\infty)$ is determined by a non-linear equation, \[\rho_{i}=\frac{\lambda\sum_{j}A_{ij}\rho_{j}}{1+\lambda\sum_{j}A_{ij}\rho_{j}},\] (2) which has a non-zero solution $\rho_{i}>0$ if $\lambda$ is larger than the so-called epidemic threshold $\lambda_{c}$. In this case, the prevalence $\rho\equiv\sum_{i=1}^{N}\rho_{i}/N$ is non-zero. _Spectral approach._—To solve the SIS model, we use the spectral properties of the adjacency matrix $\widehat{A}$. The eigenvalues $\Lambda$ and the corresponding eigenvectors $f$ with components $f_{i}$ are solutions of the equation $\Lambda\mbox{\boldmath$f$}=\widehat{A}\mbox{\boldmath$f$}$. Since the matrix $\widehat{A}$ is real and symmetric, its $N$ eigenvectors $\mbox{\boldmath$f$}(\Lambda)$ ($\Lambda_{max}\equiv\Lambda_{1}\geq\Lambda_{2}\geq\dots\Lambda_{N}$) form a complete orthonormal basis. According to the Perron-Frobenius theorem, the largest eigenvalue $\Lambda_{1}$ and the corresponding principal eigenvector $\mbox{\boldmath$f$}(\Lambda_{1})$ of a real nonnegative symmetric matrix are nonnegative [12]. The probabilities $\rho_{i}$ can be written as a linear superposition, \[\rho_{i}=\sum_{\Lambda}c(\Lambda)f_{i}(\Lambda).\] (3) The coefficients $c(\Lambda)$ are the projections of the vector $\rho$ on $\mbox{\boldmath$f$}(\Lambda)$. Substituting Eq. (3) into Eq. (2), we obtain \[c(\Lambda)=\lambda\sum_{\Lambda^{\prime}}\Lambda^{\prime}c(\Lambda^{\prime}) \sum_{i=1}^{N}\frac{f_{i}(\Lambda)f_{i}(\Lambda^{\prime})}{1+\lambda\sum_{ \widetilde{\Lambda}}\widetilde{\Lambda}c(\widetilde{\Lambda})f_{i}(\widetilde{ \Lambda})}.\] (4) In order to find the epidemic threshold $\lambda_{c}$ and $\rho(\lambda)$ near $\lambda_{c}$, it is enough to take into account only the principal eigenvector $\mbox{\boldmath$f$}(\Lambda_{1})$ in Eqs. (3) and (4), i.e., $\rho_{i}\approx c(\Lambda_{1})f_{i}(\Lambda_{1})$. Solving Eq. (4) with respect to $c(\Lambda_{1})$ gives $\lambda_{c}{=}1/\Lambda_{1}$. At $\lambda{\geq}\lambda_{c}$ in the first order in $\tau\equiv\lambda\Lambda_{1}{-}1{\ll}1$, we find $\rho\approx\alpha_{1}\tau$, where the coefficient $\alpha_{1}$ is \[\alpha_{1}=\sum_{i=1}^{N}f_{i}(\Lambda_{1})/[N\sum_{i=1}^{N}f_{i}^{3}(\Lambda_ {1})].\] (5) This expression is exact if there is a gap between $\Lambda_{1}$ and $\Lambda_{2}$ (see also Ref. [13]). Thus, at $\tau\ll 1$, $\rho$ is determined by the principal eigenvector. The contribution of other eigenvectors are of the order of $\tau^{2}$. Considering the two largest eigenvalues in Eq. (4), $\Lambda_{1}$ and $\Lambda_{2}$, and their eigenvectors, we obtain $\rho(\lambda)\approx\alpha_{1}\tau+\alpha_{2}\tau^{2}$ and so on. The usual point of view is that $\alpha_{1}$ is of the order of $O(1)$, and so a finite fraction of vertices is infected right above $\lambda_{c}$. To learn if another behavior is possible, we study whether $\Lambda_{1}$ corresponds to a localized or delocalized state. We use the inverse participation ratio \[IPR(\Lambda)\equiv\sum_{i=1}^{N}f_{i}^{4}(\Lambda).\] (6) If, in the limit $N\rightarrow\infty$, $IPR(\Lambda)$ is of the order of $O(1)$, then the eigenvector $f$$(\Lambda)$ is localized. If $IPR(\Lambda){}\to{}0$ then this state is delocalized. For a localized $f$$(\Lambda)$ the components $f_{i}(\Lambda)$ are of the order of $O(1)$ only at few vertices. For a delocalized $f$$(\Lambda)$ we usually have $f_{i}(\Lambda)\sim O(1/\sqrt{N})\ll 1$. From Eq. (5) it follows that if the principal eigenvector $f$$(\Lambda_{1})$ is localized, then $\alpha_{1}\sim O(1/N)$ and so $\rho\approx\alpha_{1}\tau\sim O(1/N)$. In this case, above $\lambda_{c}$ the disease is localized on a finite number $N\rho$ of vertices. If $f$$(\Lambda_{1})$ is delocalized, then $\rho$ is of the order of $O(1)$, and the disease infects a finite fraction of vertices right above $\lambda_{c}$. These two contrasting scenarios are shown in Fig. 1 for the SIS model on the karate-club network [14] and the weighted collaboration networks of scientists posting preprints on the astrophysics archive at arXiv.org, 1995–1999, and the condensed matter archive at January 1, 1995 – March 31, 2005 [15]. The astro-ph and karate-club nets have delocalized principal eigenstates while the cond-mat-2005 net has a localized principal eigenstate. Numerical solution of Eq. (2) gives $\alpha_{1}{=}1.8{\times}10^{-3}$ for the astro-ph net and smaller $\alpha_{1}=1.5{\times}10^{-4}$ for the cond-mat-2005 net. <figure><img src="content_image/1202.4411/x1.png"><figcaption>Figure 1: Prevalence ρ versus the infection rate λ in real networks. (a)astro-phys (upper line) and cond-mat-2005 (lower line) weighted networks [fromEq. (2)]. The eigenstate Λ1 is localized in the cond-mat-2005 network anddelocalized in the astro-phys and karate-club networks. (b) Karate-clubnetwork. The lower curve accounts for only the eigenstate Λ1 in Eq. (4).Accounting for eigenstates Λ1 and Λ2, we find the higher curve and so on. Themost upper curve is the exact ρ.</figcaption></figure> One can find $\Lambda_{1}$ and $IPR(\Lambda_{1})$ for any unweighted and weighted graph: \[\Lambda_{1}=\lim_{n\rightarrow\infty}\Lambda_{1}(n)\equiv\lim_{n \rightarrow\infty}(\mbox{\boldmath$g$}^{(n)}\widehat{A}\mbox{\boldmath$g$}^{(n )})/\|\mbox{\boldmath$g$}^{(n)}\|^{2},\] (7) \[IPR(\Lambda_{1})=\lim_{n\rightarrow\infty}\sum_{i=1}^{N}(g^{(n)} _{i})^{4}/\|\mbox{\boldmath$g$}^{(n)}\|^{4},\] (8) where $\mbox{\boldmath$g$}^{(n+1)}{=}\widehat{A}\mbox{\boldmath$g$}^{(n)}$ and $\mbox{\boldmath$g$}^{(0)}$ is a positive vector. $\Lambda_{1}(n)$ is a lower bound of $\Lambda_{1}$. In unweighted networks, i.e., $A_{ij}=0,1$, for $\mbox{\boldmath$g$}^{(0)}{=}1$, the first iteration $n=1$ gives \[\Lambda_{1}(1)=\frac{1}{\langle q^{2}\rangle N}\sum_{i,j}q_{i}A_{ij}q_{j}= \Lambda_{MF}+\frac{\langle q\rangle\sigma^{2}r}{\langle q^{2}\rangle},\] (9) where $\Lambda_{MF}{\equiv}\langle q^{2}\rangle/\langle q\rangle$, $r$ is the Pearson coefficient, and _$\sigma^{2}=\langle q^{3}\rangle/\langle q\rangle-\langle q^{2}\rangle^{2}/ \langle q\rangle^{2}$_[16; 17]. Eq. (9) shows that assortative degree-degree correlations ($r>0$) increase $\Lambda_{1}$ while disassortative correlations ($r<0$) decrease $\Lambda_{1}$. The first iteration also gives the mean-field result $IPR=\langle q^{4}\rangle/[N\langle q^{2}\rangle^{2}]\sim O(1/N)$. A few iterations already give good approximations for $\Lambda_{1}$ and $IPR$ if the principal eigenstate is delocalized but more iterations are needed if this eigenstate is localized. _Bethe lattice._—To find possible centers of localization of $\Lambda_{1}$, we use Bethe lattices as simple but representative examples of networks. The adjacency matrix of an unweighted regular Bethe lattice in Fig. 2(a) with vertices of degree $k$ has the largest eigenvalue $\Lambda_{1}{=}k$ with a delocalized eigenvector $f_{i}(\Lambda_{1}){=}N^{-1/2}$. Let us introduce a hub of degree $q{>}k$ connected to the neighbors by edges with a weight $w\geq 1$ [see Fig. 2(b)]. The other edges have weight 1. We look for such a solution $f$ of the equation $\Lambda\mbox{\boldmath$f$}=\widehat{A}\mbox{\boldmath$f$}$ that has a maximum component $f_{0}(\Lambda_{1})$ at the hub and exponentially decreases with increasing distance $n$ from the hub, $f_{i}(\Lambda_{1})=f_{n}(\Lambda_{1})\propto 1/a^{n}$. We find \[\Lambda_{1}=qw^{2}/\sqrt{qw^{2}-B},\] (10) \[IPR(\Lambda_{1})=f_{0}^{4}(\Lambda_{1})[1+qw^{4}/(a^{4}-B)],\] (11) \[f_{0}(\Lambda_{1})=[(qw^{2}/2-B)/(qw^{2}-B)]^{1/2},\] (12) \[f_{n}(\Lambda_{1})=wf_{0}(\Lambda_{1})/a^{n}.\] (13) Here $B\equiv k-1$ is the branching coefficient of the graph, $a\equiv(qw^{2}-B)^{1/2}$. Due to the exponential decay, $IPR$ is finite, so this eigenstate is localized. In the limit $qw^{2}\gg B$, we have $IPR\rightarrow(1+1/q)/4$. This solution gives the maximum eigenvalue if $\Lambda_{1}>k$. This condition can be written in the form $q>q_{loc}\equiv(B^{2}+B)/w^{2}$. The second eigenstate with $\Lambda_{2}=k$ and $f_{i}(\Lambda_{2}){\approx}N^{-1/2}$ is delocalized. <figure><img src="content_image/1202.4411/x2.png"><figcaption>Figure 2: (a) Regular Bethe lattice with degree k=3. (b) Bethe lattice withone hub of degree q>k. This hub is connected to neighbors by edges having thesame weight w≥1 (red lines). (c) Bethe lattice with two vertices of degrees q1and q2 connected by an edge with a weight w≥1 (red line).</figcaption></figure> Now we consider a Bethe lattice with two hubs of degrees $q_{1}$ and $q_{2}$ connected by an edge with weight $w\geq 1$ [see Fig. 2(c)]. Other edges have weight 1. As above, we look for an eigenvector $f$ that exponentially decays from these hubs. We find that there are two localized eigenstates with eigenvalues $\Lambda_{1}$ and $\Lambda_{2}$ above $\Lambda_{3}=k$, \[\Lambda_{1(2)}=a_{\pm}+B/a_{\pm},\] \[a_{\pm}^{2}=\frac{1}{2}(Q_{1}{+}Q_{2}{+}w^{2}){\pm}\frac{1}{2}[( Q_{1}{+}Q_{2}{+}w^{2})^{2}{-}4Q_{1}Q_{2}]^{1/2},\] \[\Psi_{1}^{2}(a_{\pm}^{2}{+}Q_{1})+\Psi_{2}^{2}(a_{\pm}^{2}{+}Q_{2 })=a_{\pm}^{2}-B,\] \[\!\!IPR(\Lambda_{1(2)}){=}[\Psi_{1}^{4}(a_{\pm}^{4}{+}Q_{1}){+} \Psi_{2}^{4}(a_{\pm}^{4}{+}Q_{2})]/(a_{\pm}^{4}{-}B).\] (14) The signs $\pm$ correspond to $\Lambda_{1}$ and $\Lambda_{2}$, respectively, and $Q_{1(2)}\equiv q_{1(2)}-B-1$. The components $f_{i}$ decrease exponentially as $\Psi_{1(2)}/a_{\pm}^{n}$ with increasing distance $n$ from the hubs 1 and 2. $\Psi_{1}$ and $\Psi_{2}$ are the components of $f$ at the hubs 1 and 2. Their ratio is $\Psi_{2}/\Psi_{1}=(a_{\pm}^{2}{-}Q_{1})/(wa_{\pm})$. The criterion for localization is $\Lambda_{1},\Lambda_{2}>k$. If $q_{1}=q_{2}$ and $w\gg 1$, then $\Psi_{1}=\Psi_{2}\to 1/\sqrt{2}$ and $IPR(\Lambda_{1})$ reaches the maximum value 0.5 that means localization on two hubs. In general, $\Lambda_{1}$ can be localized in a larger cluster. _Scale-free networks._—To study the appearance and properties of localized eigenstates in uncorrelated complex networks, we use the static model [18] that generates unweighted scale-free networks with degree distribution $P(q)\propto Cq^{-\gamma}$ at $q\gg 1$. Using software OCTAVE, for each realization of a random network of size $N$ with mean degree $\langle q\rangle$ and $\gamma=4$, we calculated eigenvalues, eigenvectors, and $IPR(\Lambda)$ of the adjacency matrix. In networks of size $N=10^{5}$, we found that several (typically, from one to three for different realizations) eigenstates appear above the upper delocalized eigenstate. These states are localized at hubs and their properties are described well by Eqs. (10)–(13) with $w=1$ if the branching coefficient $B$ in these equations is replaced by the averaged branching coefficient $B=\langle q^{2}\rangle/\langle q\rangle{-}1$. We observed that in these scale-free graphs, the upper delocalized eigenstate $\Lambda_{d}$ is slightly above the mean-field value $\Lambda_{MF}=\langle q^{2}\rangle/\langle q\rangle$. The maximum degree $q_{max}$ fluctuates from realization to realization. Localization of the principal eigenstate at a vertex with degree $q_{max}$ occurs if \[\Lambda_{1}=q_{max}/\sqrt{q_{max}-B}\geq\Lambda_{d}.\] (15) The equality here gives the threshold degree $q_{loc}$. In realizations with $q_{max}<q_{loc}$, the principal eigenvector is delocalized and $\Lambda_{1}=\Lambda_{d}$. For $N=10^{5}$, $\langle q\rangle=10$, and $\gamma=4$, our numerical calculations give $\langle q^{2}\rangle/\langle q\rangle\approx 14.1$ and $\Lambda_{d}\approx 15$. According to Eq. (15), a localized state appears above $\Lambda_{d}$ if $q_{max}$ is larger than $q_{loc}\approx 214$. Since the average value of $q_{max}$ depends on $N$, at small $N$ the probability to generate a graph with $q_{max}>q_{loc}$ is small [19]. Only large graphs can have a localized principal eigenstate. The criterion (15) is not satisfied at $\gamma{\leq}5/2$ because $\Lambda_{d}$ becomes larger than the eigenvalue $\Lambda\approx\sqrt{q_{max}}$ of a state localized at the vertex with $q_{max}$. Indeed, assuming $\Lambda_{d}\approx\Lambda_{MF}$, we find $\Lambda_{d}\propto q_{max}^{3{-}\gamma}>\sqrt{q_{max}}$ at $q_{max}{\gg}1$ when $\gamma{\leq}5/2$. Hence, the largest eigenstate is delocalized and $\Lambda_{1}=\Lambda_{d}\approx\Lambda_{MF}$ in agreement with Refs. [4; 6]. Thus, in the case of uncorrelated random graphs of sufficiently large size, the principal eigenvector is localized if $\gamma>5/2$, which includes the Erdős–Rényi graphs, and delocalized if $2<\gamma\leq 5/2$. Fig. 3 represents the results of our numerical solution of Eq. (4) for the SIS model on one typical realization of the scale-free network. The principal eigenvector is localized at the hub with $q_{max}=323$. Equations (10)–(13) and (5) give $\Lambda_{1}=18.35$, $IPR=0.23$, and $\alpha_{1}\simeq 1.4{\times}10^{-3}$. These values agree well with the measured values $\Lambda_{1}=18.47$, $IPR=0.21$, and $\alpha_{1}\simeq 1.7{\times}10^{-3}$. The eigenvector with $\Lambda_{2}$ is localized at the second largest hub with $q=254$. The third eigenvector with $\Lambda_{3}\approx 15.3$ is delocalized. The first two eigenstates allow to describe $\rho(\lambda)$ close to $\lambda_{c}{=}1/\Lambda_{1}$. Accounting for the delocalized eigenstate $\Lambda_{3}$ gives better results in a broader range of $\lambda$ (see Fig. 3). <figure><img src="content_image/1202.4411/x3.png"><figcaption>Figure 3: (a) Prevalence ρ versus λ in a scale-free network of 105 verticesgenerated by the static model with γ=4, ⟨q⟩=10. The lowest curve accounts foronly the principal eigenstate in Eq. (4), the next one accounts Λ1 and Λ2, andso on. (b) Zoom of the prevalence at λ near λc=1/Λ1.</figcaption></figure> Network \| N \| γ \| qmax \| qloc \| ⟨q2⟩/⟨q⟩ \| mixing \| Λ1 \| Λ1(1) \| IPR(Λ1) \| Λ1 \| IPR(Λ1) ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| \| \| \| \| \| \| \| \| weighted cond-mat 2005 Newman2001 \| 40421 \| 3.0 \| 278 \| 2604 \| 27.35 \| A \| 51.29 \| 35.205 \| 0.0081 \| 47.63 \| 0.3415 hep-th Newman2001 \| 8361 \| − \| 50 \| 521 \| 8.687 \| A \| 23 \| 10.632 \| 0.0417 \| 40.52 \| 0.3531 astro-ph Newman2001 \| 16706 \| − \| 360 \| 5415 \| 44.92 \| A \| 73.89 \| 56.287 \| 0.005 \| 33.7575 \| 0.0525 power grid ws1998 \| 4941 \| exponential \| 19 \| 53 \| 3.87 \| − \| 7.483 \| 3.9 \| 0.041 \| \| fp5 Almendral2007 \| 27985 \| 2.2 \| 2942 \| 38610 \| 211.0 \| − \| 197.03 \| 176.3 \| 0.0035 \| \| CAIDA (router-internet) CAIDA \| 192244 \| 2.7 \| 1071 \| 11947 \| 37.89 \| − \| 109.5 \| 42.9 \| 0.010 \| \| karate club Zachary1977 \| 34 \| − \| 17 \| 37 \| 7.77 \| D \| 6.72 \| 6.01 \| 0.073 \| \| Table 1: Characteristics of real-world networks. N is size, γ is the degree distribution exponent, qmax is the maximum degree, qloc is the localization threshold found from Eq. (15), Λ1 is the largest eigenvalue and Λ1(1) is its lower bound, Eq. (9), respectively. D and A stand for assortative and disassortative mixing. Two last columns represent weighted networks. _Real networks._—The largest eigenvalue $\Lambda_{1}$, $IPR(\Lambda_{1})$, and other parameters of a few weighted and unweighted real-world networks are given in Table 1. Note first that in all of these unweighted real networks the inverse participation ratio $IPR(\Lambda_{1})$ is small that evidences a delocalized $\Lambda_{1}$. We suggest that localization does not occur because the localization threshold $q_{loc}$ from the criterion Eq. (15) exceeds $q_{max}$. Second, in unweighted networks, $\Lambda_{1}$ differs strongly from the mean-field value $\Lambda_{MF}=\langle q^{2}\rangle/\langle q\rangle$. $\Lambda_{1}$ is larger than $\Lambda_{MF}$ in networks with assortative mixing (cond-mat 2005, hep-th, and astro-ph networks) while $\Lambda_{1}$ is smaller than $\Lambda_{MF}$ in disassortative networks (karate club network). Qualitatively, this agrees with Eq. (9). A similar observation was made in Refs. [9; 11]. Table 1 shows that in contrast to the unweighted hep-th and cond-mat-2005 networks, their weighted versions have a localized principal eigenvector with a large $IPR$. Localization occurs at vertices linked by edges with a large weight. In the cond-mat-2005 network, localization occurs at two vertices of degrees 37 and 28 connected by an edge with weight 34.3 that is much larger than the average weight $\overline{w}=0.51$. In this case, Eq. (14) gives $\Lambda_{1}\approx 34.5$ and $IPR\approx 0.49$. In the hep-th network, the strong edge has weight 34 larger than $\overline{w}=0.97$ and connects two vertices of degrees 34 and 33. Using Eq. (14), we find $\Lambda_{1}\approx 35$ and $IPR\approx 0.47$ in agreement with the data in Table 1. The components of the principal eigenvectors in these networks decay exponentially with distance from the strong edges in agreement with Eq. (14). In the astro-ph weighted network none of the edges satisfies the localization criterion. Two scenarios of behavior of the prevalence $\rho(\lambda)$ in weighted networks with localized and delocalized $\Lambda_{1}$ are shown in Fig. 1(a). Although above we considered only localization centers with one or two vertices, note that a disease may also be localized in larger finite clusters. It was concluded in Refs. [11; 9] that in unweighted networks a disease first survives inside the higher $k$-cores. By definition, $k$-cores are subgraphs containing a finite fraction of a network, and so these two works actually discussed the delocalized state of disease. The principal difference of the present work from Refs. [11; 9] is that we consider situations in which a disease takes in a finite number of vertices and not a finite fraction both in unweighted and weighted networks. In conclusion, based on a spectral approach to the SIS model, we showed that if the principal eigenvector of the adjacency matrix of a network is localized, then at the infection rate $\lambda$ right above the threshold $1/\Lambda_{1}$, the disease is mainly localized on a finite number of vertices. Importantly, a strict epidemic threshold in this case is actually absent, and a real epidemic affecting a finite fraction of vertices occurs after a smooth crossover, at higher values of $\lambda$. On the other hand, if the principal eigenvector is delocalized, the epidemic occurs in the whole region above $\lambda_{c}=1/\Lambda_{1}$. We suggest that further investigations of real-world networks will give many new examples of disease localization-delocalization phenomena. ###### Acknowledgements. This work was partially supported by the FCT projects PTDC: FIS/71551/2006, FIS/108476/2008, SAU-NEU/103904/2008, MAT/114515/2009, and PEst-C/CTM/LA0025/2011. ## References * (1) R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002); S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079 (2002); M. E. J. Newman, SIAM Review 45, 167 (2003); S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. Phys. 80, 1275 (2008). * (2) R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001); Phys. Rev. E 63 066117 (2001); M. Boguñá, R. Pastor-Satorras, and A. Vespignani, Phys. Rev. Lett. 90, 028701 (2003). * (3) Y. Wang, D. Chakrabarti, C. Wang, and C. Faloutsos, 22nd International Symposium on Reliable Distribute Systems (SRDS03) (IEEE, 2003), p. 25. * (4) F. Chung, L. Lu, and V. Vu, PNAS 100, 6313 (2003). * (5) S. N. Dorogovtsev, A. V. Goltsev, J. F. F. Mendes, A. N. Samukhin, Phys. Rev. E 68, 046109 (2003). * (6) C. Castellano and R. Pastor-Satorras, Phys. Rev. Lett. 105, 218701 (2010). * (7) S. Gómez, A. Arenas, J. Borge-Holthoefer, S. Meloni, and Y. Moreno, Europhys. Lett. 89, 38009 (2010). * (8) B. A. Prakash, D. Chakrabarti, M. Faloutsos, N. Valler, and C. Faloutsos, arXiv:1004.0060. * (9) C. Castellano and R. Pastor-Satorras, Sci. Rep. 2, 371 (2012). * (10) P. Moretti, S. Liu, A. Baronchelli, and R. Pastor-Satorras, Eur. Phys. J. B 85, 88 (2012). * (11) M. Kitsak, L. K. Gallos, S. Havlin, F. Liljeros, L. Muchnik, H. E. Stanley, and H. A. Makse, Nature Phys. 6, 888 (2010). * (12) H. Minc, _Nonnegative matrices_ (A Wiley-Interscience Publication, New York, 1988). * (13) P. Van Mieghem, EPL 97, 48004 (2012). * (14) W. W. Zachary, J. Anthropol. Res. 33, 452 (1977). * (15) M. E. J. Newman, PNAS 98, 404 (2001). * (16) M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002). * (17) S. N. Dorogovtsev, A. L. Ferreira, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. E 81, 031135 (2010). * (18) K.-I. Goh, B. Kahng, and D. Kim, Phys. Rev. Lett. 87, 278701 (2001). * (19) D.-H. Kim and A. E. Motter, Phys. Rev. Lett. 98, 248701 (2007). * (20) D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998). * (21) J. A. Almendral, J. G. Oliveira, L. López, M. A. F. Sanjuán, and J. F. F. Mendes, New J. Phys. 9, 183 (2007). * (22) CAIDA’s router-level topology measurements, http://www. caida.org/tools/measurements/skitter/router_topology/.
0906.5174	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 19207, "num_imgs": 0, "llama3_tokens_count": 5249 }	[]	# Partial Kekule Ordering of Adatoms on Graphene V. V. Cheianov${}^{1}$, V. I. Fal’ko${}^{1}$, O. Syljuåsen${}^{2},$ and B. L. Altshuler${}^{3}$ ###### Abstract Electronic and transport properties of Graphene, a one-atom thick crystalline material, are sensitive to the presence of atoms adsorbed on its surface. An ensemble of randomly positioned adatoms, each serving as a scattering center, leads to the Bolzmann-Drude diffusion of charge determining the resistivity of the material. An important question, however, is whether the distribution of adatoms is always genuinely random. In this Article we demonstrate that a dilute adatoms on graphene may have a tendency towards a spatially correlated state with a hidden Kekulé mosaic order. This effect emerges from the interaction between the adatoms mediated by the Friedel oscillations of the electron density in graphene. The onset of the ordered state, as the system is cooled below the critical temperature, is accompanied by the opening of a gap in the electronic spectrum of the material, dramatically changing its transport properties. ¹ [FOOTNOTE:1][ENDFOOTNOTE] ² [FOOTNOTE:2][ENDFOOTNOTE] ³ [FOOTNOTE:3][ENDFOOTNOTE] When can an apparently random system be considered ordered? Or can an apparently random ensemble of impurities in a system be correlated enough to force the reconstruction of the electronic band structure in a material? In this Article we predict that a dilute ensemble of adatoms sprinkled randomly over a graphene monolayer [1, 2] can establish long-range correlations between their positions, despite the fact that they may be many graphene unit-cell lengths apart. This correlation is strong enough that at a transition temperature it will induce an energy gap in the electronic spectrum despite the fact that, in the “ordered” state, the distribution of adatoms does not show any crystalline structure. It rather resembles the ferromagnetically ordered state of spins of magnetic ions in dilute magnetic semiconductors [3]. The physical mechanism behind this phenomenon is the electron-mediated interaction between the adsorbents, which prompts their partial ordering into a configurations associated with a hexagonal superlattice with a unit cell three times bigger than that of graphene. Since the density, $\rho$ of adsorbents is low, they occupy a small randomly chosen fraction of the equivalent positions on the superlattice. This ordering folds the Brillouin zone and thus opens a spectral gap for low-energy electrons. This phenomenon suggests a novel route towards engineering the band structure and controlling transport in graphene-based devices. Graphene [4, 2] is a two dimensional crystal of carbon atoms, which form a honeycomb lattice with two distinct sublattices (A and B). The first Brillouin zone (BZ) has a hexagonal form (the blue area in Fig. 1A), and the conduction band touches the valence band in six BZ corners [4] which form two non-equivalent triads of BZ corners, $\mathbf{K}$ and $\mathbf{K}^{\prime}$ connected by the reciprocal lattice vectors, $\mathbf{G}$ and $\mathbf{G}^{\prime}$. Low-energy electronic excitations in the momentum space are located in the vicinities of the points $\mathbf{K}$ and $\mathbf{K}^{\prime}$, i.e. belong to one of the two valleys with the linear ’Dirac-type’ spectrum, $\varepsilon(p)=\pm vp$ where $\mathbf{p}$ is the momentum counted from one of the $\mathrm{K}$-points and $v\approx 10^{8}$cm/sec. Both the gapless spectrum and the valley degeneracy follow directly from the symmetries of the honeycomb lattice. In pristine graphene, the honeycomb lattice is stable against spontaneous structural changes. Recently, several types of adatoms were used to dope graphene in attempts to tailor properties of graphene-based devices [5, 6, 7, 8]. Below, we consider theoretically a particular example, Fig. 2A, of adatoms, such as alkali atoms [9, 10], Ca or Al whose stable positions are above the centers of the hexagons. A single adatom of this type preserves rotational and reflection symmetries but breaks the translational symmetry of the lattice. Therefore, it can scatter electrons between valleys. The intervalley scattering generates the Friedel oscillations (FO) of the electron density, which amplitude rather slowly decays when the distance from the adatom [11]. The pattern and period of such FO are determined by the wave vectors $\delta\mathbf{K}=\pm(\mathbf{K}^{\prime}-\mathbf{K)}$ transferred upon scattering between the valleys. Due to a peculiar relation of $\mathbf{K}$, $\mathbf{K}^{\prime}$ and the reciprocal lattice vectors in graphene, $\mathbf{K}=\frac{1}{3}(\mathbf{G}+\mathbf{G}^{\prime})$ and $\mathbf{K}^{\prime}=-\frac{1}{3}(\mathbf{G}+\mathbf{G}^{\prime})$, the FO formed around an adatom have a structure of a charge density wave with the hexagonal lattice pattern and the unit cell extended over three unit cells of graphene. In graphene with zero carrier density ($\rho_{e}=0$), the amplitude of these superlattice oscillations decays as inverse cube of the distance between the adatoms. The oscillations of the electron density induced by one adatom affect other add atoms, thus leading to the pairwise interaction between adsorbents, which is sensitive to their position in the superlattice. In Fig. 2 (A and B), we compare the potential landscape for a probe adatom created by several others due to their FO, to stress that its amplitude is substantially enhanced by ordering. Note that the long-range RKKY-type [12] interaction in a low-density ’gas’ of adatoms as well as the ordering it promotes has little to do with those in dense aggregates of adsorbents. Indeed, the interaction at atomic distances is mediated by local lattice deformations - phonons. Such an interaction decays exponentially as a function of distance between the adatoms, as opposed to the power law $1/R^{3}$ decay of the electron-mediated coupling. Figure 2 illustrates an example of hidden structural ordering of adatoms sprinkled over graphene. Without mosaic coloring (or a superlattice mesh) it would be difficult to distinguish the ordered configuration of adatoms (Fig. 2D) from a disordered state (Fig. 2C). With the help of colors, one can identify a triple-size unit cell of the superlattice, with three non-equivalent adatom positions (red, blue, and green) between six carbons, which resembles a hexagonal Kekulé-type [13] lattice [14, 10]. The FO-mediated interaction $V_{ij}$ between two adatoms $i$ and $j$ depends on whether they occupy equivalent (same color) positions on the superlattice, or not. This consideration maps the problem of hidden Kekulé mosaic ordering of adsorbents onto the three-state Potts model [15] with a random in strength ’ferromagnetic’ coupling of species. To estimate critical temperature of the hidden ordering, one has to evaluate the function $V_{ij}(\mathbf{r}_{i}-\mathbf{r}_{j})$, where $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ are adatoms locations. We use the technique developed for the studies dealing with disorder in graphene [16, 17, 11] and describe electrons by four-component Bloch function $\psi$ (taking into account its sublattice composition and valley degeneracy) and the 4x4 Hamiltonian, \[H=v(\mathbf{p}\cdot\mathbf{\sigma})\Pi_{z}+\lambda hva\sum\limits_{i}\mathbf{( \mathbf{u}_{i}\cdot\Pi})\delta(\mathbf{r}-\mathbf{r}_{i}).\] (1) Here, Pauli matrices $\sigma_{x,y,z}$ act on sublattice indices and $\Pi_{x,y,z}$ on the valley indices of $\psi$[16, 17, 11]. The first term in Eq. (1) is a familiar Hamiltonian for Dirac-type electrons in pristine graphene. The second accounts for the intervalley scattering of electrons by adatoms, with $\lambda$ being the dimensionless coupling constant (realistically, $\lambda\lesssim 1$ since $hv/a$ is of the order of the bandwidth in graphene). Unit two-component vector $\mathbf{\mathbf{u}}_{i}=(\cos\frac{2\pi m_{i}}{3},\sin\frac{2\pi m_{i}}{3})$ specifies which of the three non-equivalent positions the $i$-th adatom occupies on the superlattice, with $m_{i}=-1$, $0$ and $-1$ (red, blue, and green hexagons). Using thermodynamic perturbation theory and the standard RKKY approach [12] we express the interaction between adsorbents mediated by electrons as \[V_{ij} =\] \[G(\mathbf{r},\tau) = -\frac{1}{4\pi}\frac{v\tau+i\Pi_{z}\mathbf{\sigma}\cdot\mathbf{r} }{(v^{2}\tau^{2}+r^{2})^{3/2}},\] (2) where $\tau$ is imaginary time and $G(\mathbf{r},\tau)$ is the zero-temperature Green function of Dirac-like electrons. Strictly speaking the equation 2 is valid at $T=0.$ However one can use it as long as $T<hv\rho^{1/2}.$ The, trace ($\mathrm{Tr}$) is taken over the sublattice and valley indices. The electron spin degeneracy is accounted for by the overall factor of 2. The resulting electron-mediated ’ferromagnetic’ interaction between adatoms at a distance $\|\mathbf{r}_{i}-\mathbf{r}_{j}\|\gg a$, \[V_{ij} = -J\frac{\mathbf{u}_{i}\cdot\mathbf{u}_{j}}{\|\mathbf{r}_{i}- \mathbf{r}_{j}\|^{3}\rho^{3/2}},\;\;\mathbf{u}_{i}\cdot\mathbf{u}_{j}=\cos\frac {2\pi(m_{i}-m_{j})}{3},\] (3) \[J = \frac{\lambda^{2}}{2}(a^{2}\rho)^{3/2}\frac{hv}{a},\] has a long-range tail, $V\propto\|\mathbf{r}_{i}-\mathbf{r}_{j}\|^{-3}$. The typical interaction energy scale, $J$ is the interaction strength at a mean distance between the nearest neighbors, $\sim\rho^{-1/2}$ (recall that $\rho\ll a^{-2}$). To evaluate the critical temperature, $T_{c}$, we modeled the ordering transition numerically. We used the cluster Monte Carlo algorithm [18] to compute statistical moments \[M_{2n}=\frac{\sum_{\mathbf{u}_{1},...,\mathbf{u}_{N}}\left(\sum_{i=1}^{N}\frac {\mathbf{u}_{i}}{N}\right)^{2n}e^{-\frac{1}{2T}\sum V_{ij}}}{\sum_{\mathbf{u}_ {1},...,\mathbf{u}_{N}}e^{-\frac{1}{2T}\sum V_{ij}}},\] for 10 realizations of quenched Poissonic distributions of $N=2\times 10^{4}$ adatoms. The ordering transition can be detected by a sudden rise of the order parameter $M\equiv\sqrt{M_{2}}$, from $M(T>T_{c})=0$ to $M(T<T_{c})=1$ accompanied by decrease of $\eta=M_{4}/M_{2}^{2}$, from $\eta(T\gg T_{c})=2$ (set by the central limit theorem for a large number of uncorrelated clusters) to $\eta(T\ll T_{c})=1$. Results of the numerical analysis are presented in Fig. 3. The transition temperature turned out to be \[T_{c}\approx 8J\sim 4\lambda^{2}(a^{2}\rho)^{3/2}\frac{hv}{a}.\] (4) For example, for $\lambda\sim 1$, just 1% coverage of graphene by adatoms should generate $T_{c}$ in the room temperature range. Since the mobility of adatoms on graphene strongly depends on temperature, the higher the adsorbent density $\rho$, the higher $T_{c}$ is, and the quicker the self-organization should establish upon cooling. Note that the aggregation of adsorbents, such as discussed in Refs. [19, 20], would be a much slower process in a dilute system. At $T<T_{c}$ the proposed partial ordering suppresses adatoms diffusion leading to a further slowdown of aggregation. The value of $T_{c}$ may also depend on the concentration $\rho_{e}$ of electrons (or holes) in graphene. Finite carrier density leads to the additional modulation of the FO, with the period twice as small as the electron Fermi wavelength $\sim\rho_{e}^{-1/2}$[11]. For $\rho_{e}\gtrsim\rho$ these modulations would make the sign of the interaction between adatoms random and, thus eliminate the ordering. Therefore it seems to be possible to control hidden ordering of adsorbents electrically, by filling or depleting the flake with carriers - the method already in use to fine-tune the ferromagnetic transition temperature in thin films of dilute magnetic semiconductors [21]. The self-organization of an apparently random ensemble of adatoms into a Kekulé-type ordered state drastically changes electronic spectrum in graphene. Adatoms that preferentially occupy one of the three equivalent positions in the supercells over a length scale $L\gg\sqrt{1/\rho}$ can Bragg scatter electrons between the two valleys coherently. This implies the Brillouin zone folding in Fig. 1 (B and C): all of the points $\mathbf{K}$ and $\mathbf{K}^{\prime}$ of the original BZ are projected onto the $\Gamma$-point of a smaller BZ corresponding to the superlattice with a triple unit cell. Simultaneously, a gap, $\Delta,$ opens in the electronic spectrum \[\varepsilon(p)=\pm\sqrt{(vp)^{2}+\Delta^{2}},\;\Delta=\lambda a^{2}\rho\frac{ hv}{a}.\] (5) To derive Eq. (5), one can substitute the second term of the Hamiltonian in Eq. (1) by its average, for example, for all adatoms positioned on yellow hexagons, and diagonalize the resulting matrix, $\bar{H}=v\mathbf{\sigma}\cdot\mathbf{p}+\lambda va\rho\Pi_{x}$. One can think of several ways to experimentally detect the hidden Kekulé mosaic order. One is to use the angle-resolved photoemission spectroscopy (ARPES). The latter technique is not only a natural method to reveal the formation of the spectral gap. It can also provide a direct evidence of the BZ folding. Indeed, ARPES measures simultaneously the energy and all three momentum components of the photo-emitted electrons. While at low energies only the vicinity of the BZ corners $\mathbf{K}$ and $\mathbf{K}^{\prime}$ can be seen in pristine graphene [22, 23], the Bragg scattering by the self-organized adsorbents generates an ARPES signal also at the $\Gamma$-point of the BZ in Fig. 1A. Another signature of the Kekulé ordering would be a bright appearance of the D peak in the phonon Raman scattering [24]: the excitation of a BZ phonon forbidden by momentum conservation in pristine graphene. Finally, the gap in the electronic spectrum would dramatically affect charge transport in graphene. This may offer numerous opportunities for graphene-based electronics. ## References * [1] Novoselov, K. S. _et al._ Electric field effect in atomically thin carbon films. _Science_306, 666 - 669 (2004) * [2] Novoselov, K. 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B_77, 195403 (2008) * [24] Das, A. _et al._ Monitoring dopants by Raman scattering in an electrochemically top-gated graphene transistor. _Nature Nanotechnology_3, 210-215 (2008) ## Acknowledgements The authors would like to thank George Pickett for valuable comments. The work was supported by the Lancaster-EPSRC Portfolio Partnership, ESF CRP SpiCo, and US DOE contract No. DE-AC02-06CH11357. Numerical computations were carried out using resources provided by the Notur project of the Norwegian Research Council. ## Figure legends Figure 1: (A) Brillouin zone of graphene in the reciprocal space is shown as a blue hexagon. The two valleys $K$ and $K^{\prime}$ are situated in the six corners of the hexagon, which are identified via reciprocal lattice translations generated by vectors $\mathbf{G}$ and $\mathbf{G}^{\prime}.$ (B) Brillouin zone folding due to the ordering transition. The folding leads to the identification of the valley points $K$ and $K^{\prime}$ with the $\Gamma$ point in the center of the Brillouin zone. (C) The energy surface in the folded Brillouin zone. Due to the interaction between the valleys a gap opens in the spectrum. Figure 2: Kekulé mosaic ordering of adatoms of _the same chemical element_ on graphene lattice. Panels (A) and (B) show the potential landscape that an extra atom would see in the presence of four atoms already shown. Coloring of the atoms is introduced to reveal their position within the Kekulé superlattice, as shown in panels (C) and (D). From a comparison of (A) and (B) one can see that adatoms placed on unicolor tiles enhance the potential landscape forcing other atoms to occupy tiles of the same color. Figure 3: The Kekulé mosaic order parameter $M$ as a function of temperature. The phase transition to the ordered state is characterized by a rise of $M$ accompanied by a sharp drop of the measure of finite-size fluctuations $M_{4}/M_{2}^{2}.$ The data were obtained using the cluster Monte Carlo algorithm. The error bars indicate the standard deviation of the thermodynamic quantities in an ensemble of 10 random realizations of Poisson distributions of $N=2\times 10^{4}$ atoms in the plane.
1601.06076	{ "language": "en", "source": "Arxiv", "date_download": "2024-12-03T00:00:00" }	{ "doc_length": 31300, "num_imgs": 10, "llama3_tokens_count": 7755 }	[ "content_image/1601.06076/An_exemplary_network_consisting_of_nodes_and_edges.jpg", "content_image/1601.06076/Edges_fullcells_and_nodes.png", "content_image/1601.06076/Infinite_one.png", "content_image/1601.06076/The_distribution_according_to_the_total_number_of_cars.jpg", "content_image/1601.06076/Representation_of_an_exemplary_simulation_scenario_in_CarPed.jpg", "content_image/1601.06076/traverse_edge.png", "content_image/1601.06076/solve.png", "content_image/1601.06076/traverse_node.png", "content_image/1601.06076/setFull.png", "content_image/1601.06076/DistributeLastDensities.png" ]	# Liquid Humans - Pedestrian Simulator based on the LWR-model Quirin Aumann Chair of Computational Modeling and Simulation, Technische Universität München Carlos M. Osorio Chair of Computational Modeling and Simulation, Technische Universität München Celeste Lai Chair of Computational Modeling and Simulation, Technische Universität München Dense human flow has been a concern for the safety of public events for a long time. Macroscopic pedestrian models, which are mainly based on fluid dynamics, are often used to simulate huge crowds due to their low computational costs (COLOMBO and ROSINI, 2005). Similar approaches are used in the field of traffic simulations (LIGHTHILL and WHITHAM, 1955). A combined macroscopic simulation of vehicles and pedestrians is extremely helpful for all-encompassing traffic control. Therefore, we developed a hybrid model that contains networks for vehicular traffic and human flow. This comprehensive model supports concurrent multi-modal simulations of traffic and pedestrians (BIEDERMANN, KIELAR, AUMANN, LAI and OSORIO, 2015). ## 1Introduction ### Hartmann-Sivers Model Every year, about 2000 people die due to the movement of large human crowds (HUGHES, 2003). The growing eventisation of social and public life in the 21st century (HITZLER, 2010) increases the probability and frequency of organized events attended by thousands or millions of people. In the past, fatal accidents have occurred repeatedly at such large events. Examples include the catastrophe in the Heysel stadium in 1985 (LEWIS, 1989) with 39 fatalities or the Loveparade disaster in 2010 with 21 deaths (HITZLER et al., 2011). Professional crowd control increases the safety of public events and lowers the probability of fatal accidents in the context of large human flows. Pedestrian dynamics simulations support successful crowd management by helping organizers of public events to foresee and prevent dangerous situations. Macroscopic pedestrian dynamic models like the Hartmann & Sivers mode (HARTMANN and VON SIVERS, 2013) are most suitable for simulating large numbers of people (BIEDERMANN, KIELAR and HANDEL, 2014). The same macroscopic model can be applied to simulate the motion of cars. By combining human and vehicular flow into one model, we can achieve an all-embracing simulation model, which covers crowd and traffic control at the same time (BIEDERMANN, KIELAR, AUMANN, LAI and OSORIO, 2015). Due to the macroscopic nature of these simulation models, we can simulate very large and dense scenarios faster than real time. ### Current Hybrid Modeling Two different kinds of hybrid modeling exist in the field of traffic and crowd control. The first type combines traffic or pedestrian dynamcis models at different scales in order to reduce the overall computational effort. This means that simulations in the hazardous areas of the scenario (places with high densities or small bottlenecks) are calculated with high detail but computationally costly models. At the same time, less hazardous areas of the scenario can be simulated by less detailed but more cost-effective models. This provides us precious results in the interesting parts of the scenario with a reduced total computational cost. TOLBA et al., 2005 as well as MCCREA and MOUTARI, 2010 developed such hybrid models in the field of vehicular flow. Similar models were developed for pedestrian dynamics. A wide range of hybrid modeling exists; from the coupling of very specific pedestrian simulation models (e.g. NGUYEN et al., 2012) to generic coupling frameworks (BIEDERMANN, KIELAR, HANDEL and BORRMANN, 2014). An overview of the current state of the art can be found by IJAZ et al., 2015. The second type of hybrid modeling does not combine different scales, but different kinds of simulations. GALEA et al., 2008 describe the coupling of pedestrian dynamics and fire simulations and GÖTTLICH et al., 2011 combine an evacuation simulation with the spread of hazardous gases. Our approach is the connection of vehicular traffic with human flow. Some studies have already been carried out in this field. PRETTO et al., 2011 coupled agent-based representations of pedestrians and cars. A macroscopic approach was developed by BORSCHE et al., 2014. They used the classic Lighthill-Whitham-Richards model (LIGHTHILL and WHITHAM, 1955; RICHARDS, 1956) for the vehicular flow and the model from HUGHES, 2002 for the macroscopic simulation of pedestrians. Their work is important progress in the field of hybrid macroscopic modeling. However, they considered vehicles and humans as two interacting, but separate flows. In reality, cars and pedestrians are not completely separate: drivers exit their cars after finishing driving and become pedestrians and pedestrians may enter their vehicles to become drivers. Our approach is a hybrid model, which is capable of converting humans and vehicles into each other to create a realistic simulation model. ## 2The Hybrid Simulation Model CarPed ### The Hybrid approach It is important to use a single network that combines pedestrian and traffic simulations. Therefore, we introduce an interface which links human and vehicular flow. The movement behavior of cars and pedestrians is based on the macroscopic Hartmann and Sivers model (HARTMANN and VON SIVERS, 2013) and is calculated on a discrete network. Roads for cars and walking paths for pedestrians are represented by the edges of this network. Its nodes represent the links between the different edges and can be considered crossings. Specifically, the nodes have different flow rates. There are special nodes, which represent parking lots, and serve as a connection between roads and walkways. Visitors of the event enter and leave their cars there. To put it another way, the simulated subjects get transformed from vehicles into pedestrians and vice versa. Therefore, the parking lots serve as a connection of the traffic and pedestrian networks. In Figure 1, an exemplary network is shown. The darker nodes and edges represent streets and their respective intersections, whereas the pedestrians’ walkways and intersections are represented in a lighter gray. The transformation of cars and pedestrians is carried out at special parking lot nodes. Green numbered entry nodes represent sources, which are the starting points of the visitors (e.g. their hometown). The red marked exit node is the final destination of the visitors (e.g. a large public event), where they are deleted from the network. <figure><img src="content_image/1601.06076/An_exemplary_network_consisting_of_nodes_and_edges.jpg"><figcaption>Figure 1: An Exemplary Network Consisting of Nodes and Edges</figcaption></figure> We implemented the Hartmann and Sivers model for pedestrian simulation and extended it to traffic simulation by using different model parameters. The extension was done by applying different maximum capacities to the edges and nodes, using different free flow velocities and other model constants for the simulated subjects. We used data obtained from WEIDMANN, 1993 for the pedestrians and data from WACHS, 2000 for the vehicle simulation. Additionally, we conducted a large field study to receive sufficient data for the transformation process (see Section 3). ### The Hartmann-and-Sivers Model The Hartmann and Sivers model adopts a structured continuum model on a macroscopic level. The approach basically relies on fundamental diagrams - the relation between fluxes and local densities - as well as the explicit consideration of individual velocities; thus showing better agreement with microscopic pedestrian models (HARTMANN and VON SIVERS, 2013). The free flow velocity $v_{ff}$ is an individual property of each pedestrian / vehicle, i.e. a macroscopic, time independent state variable attached to each individual. The free flow velocity describes the highest velocity an individual can reach under optimal circumstances. This property is distributed normally among pedestrians or cars. Thus for modelling human or vehicular flow, the following equation holds \[\frac{\partial{\rho(v_{ff};x,t)}}{\partial{t}}+\frac{\partial}{\partial{x}}(v \:(v_{ff};\rho(x,t))\rho(v_{ff};x,t))=f(v_{ff};x,t)\] (1) The total density of subjects is given by the sum of densities for all free flow velocities. \[\rho(x,t)=\int_{0}^{v_{ff}^{max}}\rho(v;x,t)\mathrm{d}v\] (2) The fundamental relation between velocity and density for pedestrians can be adopted according to: \[v(v_{ff};\rho)=v_{ff}\left[1-exp\left(-\gamma\left(\frac{1}{\rho}-\frac{1}{ \rho_{max}}\right)\right)\right]\] (3) and for cars: \[v(v_{ff};\rho)=v_{ff}\frac{\rho_{max}^{n}-\rho^{n}}{\rho_{max}^{n}+K\rho^{n}}\] (4) The maximum density $\rho_{max}$ and the factor $\gamma$ are unique parameters for pedestrians, as well as $n$ and $K$ for cars. The differential equations for pedestrians and cars were discretized by an upwind finite volume scheme (HARTMANN and VON SIVERS, 2013). ### Discretization The used parameters are $\rho_{max}=5.4$\mathrm{p}\mathrm{e}\mathrm{o}\mathrm{p}\mathrm{l}\mathrm{e} \mathrm{/}\mathrm{m}^{2}$$, $\gamma=1.913$, $v_{ff}=1.34$\mathrm{m}\mathrm{/}\mathrm{s}$$ in case of pedestrians, and $\rho_{max}=0.12$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{/}\mathrm{m}$$, $K=6.83$, $n=1.81$, $v_{ff}=15$ m/s for cars in the city. The following finite difference is used: \[\rho_{i}^{j+1}=\rho_{i}^{j}-\Delta t\left[d^{+}F_{\Delta}^{+}+d^{-}F_{\Delta}^ {-}\right]\] (5) In this case the proper HARTMANN and VON SIVERS, 2013 method is used: \[F_{\Delta}^{+}=\frac{\rho_{i}v(v_{ff};(1-\alpha)\rho_{i}+\alpha\rho_{i+1})- \rho_{i-1}v(v_{ff};(1-\alpha)\rho_{i-1}+\alpha\rho_{i})}{\Delta x}\] (6) \[F_{\Delta}^{-}=\frac{\rho_{i+1}v(v_{ff};(1-\alpha)\rho_{i+1}+\alpha\rho_{i})- \rho_{i}v(v_{ff};(1-\alpha)\rho_{i}+\alpha\rho_{i-1})}{\Delta x}\] (7) Being $\alpha$ a discretization parameter between 0 and 1, $d^{+}=1$ and $d^{-}=0$ always for forward moving cars and pedestrians. When the velocity is constant, this scheme reduces to first order upwind method. #### 2.3.1Simulation of structured model At first, the initial densities in the starting nodes separate into different velocity classes. If a very big event in which the number of pedestrians is very high (central limit theorem) is assumed, a normal distribution to predict how many people has each free flow velocity class can be used, using $N(1.34,0.26)$ as specified in HARTMANN and VON SIVERS, 2013. \[\rho_{j,0}=\rho_{0}\left[N.cumulativeProbability(v_{j})-N. cumulativeProbability(v_{j-1})\right]\] (8) In this case, some ’lines’ of people that walk are assumed, each of them with a different free flow velocity. A person never changes his or her own free flow velocity. Assuming that, the velocity can be further decomposed as: \[v(v_{ff};\rho(x,t))=v_{ff}\hat{v}(\rho(x,t))\] (9) The $\hat{v}$ part is dependant on the total density, and the $v_{ff}$ part is the well known normal-distributed free flow velocity for each class. ### Mass conservation The upwind schemes are in principle mass conservative within an infinite domain. That means as well as the values do not ’escape’ through the boundaries the mass is conserved. However, the edges in this case are indeed finite so a scheme that takes people close to the boundaries before they leave and move them into other edges is necessary. The spatial discretization of an edge has $N$ cells, so the array goes from $0$ to $N-1$. The cells $0$ and $N-1$ are boundary conditions and do not belong to the solver domain, and their value is zero. In order to assure the mass conservation, the cell 1 (second cell) is considered the start cell so the people from the beginning node is placed there. The cell $N-2$ (penultimate cell) is considered the end cell of an edge and those people are sent to the end node when they reach the cell. The penultimate cell $N-2$ is then emptied there after. When some people reach the penultimate cell of an incoming edge, those people are sent to the node. Then, when the node is computed, the remaining people are placed in the outcoming edges according to the weights. The way to split the people, and assuming not-infinite nodes that let the people accumulate in the incoming edges are aspects that should be taken into account. Integrating the densities in the edges, and then summing the people in the nodes can be useful if the user wants to check if people are preserved through the computation. \[people=PPC\sum_{s}w_{s}\Delta x\sum_{i=0}^{N_{s}-1}\rho_{s,i}+\sum_{w}w_{w} \Delta x\sum_{i=0}^{N_{w}-1}\rho_{w,i}+\sum_{n}\rho_{n}\] (10) Where $PPC$ means persons per car, $s$ streets, $w$ walkways and $n$ nodes. In the nodes we calculate with the number of pedestrians and in the edges with the local density which is given in persons per square metre. Initializing people and cars in the start nodes is the first step in the computation. ### Fixed distributor In this distributor, the same density is placed for all the outcoming edges so the number of people that an edge takes is proportional to the width. This density can be found using the following equation. \[\sum\rho_{in}w_{in}dx=\sum\rho_{out}w_{out}dx\] (11) \[\rho_{out}=\frac{\sum\rho_{in}w_{in}dx}{\sum w_{out}dx}\] (12) In the start nodes the entering people and cars are initialized using the information from the GUI. Sometimes the $\rho_{out}$ is greater than the maximum density allowed in the edges. This is a situation that happens quite often i.e. in a sudden change of width in two consecutive edges. If the first edge is very wide, a lot of people will want to go into the following edge (in this case a very narrow one). The first solution is assuming infinite nodes, so in that case the people will wait until the second cell of the outcoming edge allows them. ### Finite nodes In situations like the one explained before, infinite nodes shall not be used because such behavior does not happen in reality. The people should go inside the node and then leave immediately, and if they find the end node full, they should stay in the incoming edge instead of getting inside that full node. Unfortunately, the first order equation and the structured model leads the fast people go first and always the density is a decreasing function through the edge. The model equation by itself is far away from concentrating the people close to the edge exit when the node is full. Also, the parameter $\alpha=1$ (remember Hartmann and Sivers numerical method) is more ’human-wise’ because the people can ’look forward’ and use that density to calculate their velocity. This model and numerical method does not look beyond one cell density in the edge. Implementing a numerical method or another model with longer sight for the velocity, could make the people move in a more human-wise way. In order to allow the people to accumulate before a full node (see Figure 2) in the incoming edge, some new routines must be implemented. Section 7 explains these algorithms in detail. <figure><img src="content_image/1601.06076/Edges_fullcells_and_nodes.png"><figcaption>Figure 2: An Example Showing a Full Cell</figcaption></figure> Figure 3 shows the behavior of pedestrians in a narrowing example, in both infinite and finite nodes: <figure><img src="content_image/1601.06076/Infinite_one.png"><figcaption>Figure 3: Narrowing example ratio 30:1. 500 timesteps. α=1, dx=0.01, dt=0.002,L1=1, L2=1, w1=30, w2=1.</figcaption></figure> ### Routing Behavior of Pedestrian and Cars In their original paper, Hartmann and Sivers used fixed probability values to distribute the pedestrians at each intersection. These values were obtained by comparing the capacities of the adjacent edges and did not take the current density of those edges into account. It is possible that the majority of a flow gets directed into an edge which is already relatively full, and the required time for the flow to pass the system could increase in an unrealistic way. Therefore, we extend this approach and introduce a routing algorithm which determines the possible edges the pedestrians or cars will be allocated at. To find the fastest path through the system, we use the classical Dijkstra’s algorithm (DIJKSTRA, 1959). This algorithm weighs all edges of the graph to compute the shortest way through the system. We used the Dijkstra’s algorithm to calculate the route with the shortest travel time. Therefore, we used the length and the current velocity $v(v_{ff};\rho)$ as weighting factors: \[Weight=\frac{length}{v(v_{ff};\rho)}\] (13) With increasing density the velocity $v(v_{ff};\rho)$ decreases according to Equation 3. If the density of an edge reaches the maximum density $\rho_{max}$, the edge is considered as closed. Closed edges are ignored for the calculation of the routing algorithm. ### Transformation Between The Two Simulation Models Cars get transformed into pedestrians in the parking lot nodes. Typically, for ordinary cars, one to five people get out of each vehicle, with a prescribed random distribution. The distribution for the transformation was gathered from field data (BIEDERMANN, KIELAR, AUMANN, LAI and OSORIO, 2015). The transformed pedestrians and cars receive their individual parameters, like their free flow velocity or their maximum density, according to WEIDMANN, 1993 and WACHS, 2000 respectively. ## 3Visitor Distribution of Incoming Cars <figure><img src="content_image/1601.06076/The_distribution_according_to_the_total_number_of_cars.jpg"><figcaption>Figure 4: The Distribution According to The Total Number of Cars Counted andthe Distribution According to The Total Number of Passengers</figcaption></figure> To model a realistic transformation of pedestrians and cars, it is necessary to know the distribution of passengers per car. This means, that we have to know how many visitors are normally transported by one car. Therefore, we conducted an extensive field study to obtain the lacking data. We studied a large music festival in Munich over three consecutive days and counted the amount of visitors per car. We surveyed a total number of 1960 cars (for details see Table 1), which carried an average of 2.21 persons. Over 70% of the vehicles were occupied by one or two passengers. A negligible amount of cars transported more than five people, and the maximum observed was eight passengers per car. The observed distributions from all three days can be seen in Figure 4. This data was used as a configuration for the transformation process of the hybrid model. Date \| 1 \| 2 \| 3 \| 4 \| 5 \| 5+ \| Total ---\|---\|---\|---\|---\|---\|---\|--- 29.05.2015 \| 144 \| 190 \| 53 \| 33 \| 11 \| 4 \| 435 30.05.2015 \| 98 \| 263 \| 62 \| 54 \| 15 \| 3 \| 495 31.05.2015 \| 210 \| 526 \| 158 \| 98 \| 36 \| 2 \| 1030 Total \| 452 \| 979 \| 273 \| 185 \| 62 \| 9 \| 1960 Table 1: Number of Visitors Per Car on The Music Festival “Rockavaria” ## 4The CarPed-Toolbox ### Structure and Usability The software toolbox CarPed was developed as a proof of concept. Great attention was payed towards usability. Therefore, we implemented an intuitive and easy to use graphical user interface (GUI). The GUI is divided into three major parts, namely the network input, the model and solving options and the result window. The network, consisting of nodes and edges, can be easily entered through the network input. All network depending properties, like the length or the capacity of an edge, can be determined immediately after the input procedure. By using the model and solving options window, the user can determine which computational method should be used and how the data should be processed. Finally, the computed results can be accessed and visualized in the viewer. As all computed data is stored, the density distribution for each time-step can be looked at by the user. Different coloring of the edges shows the current densities. Alternatively, the user can click on an edge or node to get more detailed information about this object like the amount of pedestrians or cars on it. An exemplary screenshot of the GUI can be seen in Figure 3. <figure><img src="content_image/1601.06076/Representation_of_an_exemplary_simulation_scenario_in_CarPed.jpg"><figcaption>Figure 5: Representation of an Exemplary Simulation Scenario in CarPed</figcaption></figure> ### Workflow The network can be entered by clicking on the designated area in the GUI. Special nodes, like entry and exit nodes or parking lots, can be defined by the user. Additionally, the nodes can be linked to incoming or outgoing edges. If the input to the system is finished, a checking routine searches for errors and discrepancies. For each algorithm, the user can individually modify the settings. After everything is set up, the solver can begin its calculations by just pressing a button. At the beginning of the simulation, the network model reads the information of nodes and edges from the database previously created. It constructs a network from the infrastructure and passes the entire network to the numerical solver. Together with the received input data, the numerical solver starts generating vehicles and pedestrians in the associated entry nodes. It determines the initial density for each edge that is connected to the entry node and computes the density distribution. For each time-step, the results are passed from the numerical solver and stored in the network model. If a subject enters a node, it gets removed from this incoming edge and is put into an outgoing edge according to the route calculated by the Dijkstra algorithm. Additionally, the subject gets transformed if the visited node is a parking lot. The computations are repeated until all pedestrians and cars have reached an exit node and are dismissed from the system. During the result analysis, the GUI displays two visions: a parameter set and an animation of density distributions. All statistical information is provided by the toolbox and can be easily accessed. ## 5Conclusion and Outlook This report describes the dedicated hybrid and macroscopic simulation approach CarPed. This approach uses the macroscopic Hartmann and Sivers model on a network of nodes and edges. It is able to simulate human and vehicular flow in one hybrid model. The transformation between pedestrians and cars is carried out at special nodes, which represent parking lots. The tool is thought to play a fundamental rule in the planning phase of massive public events. Organizers can use it to identify pathways and roads, which have a higher risk of congestions. Therefore, the organizers can develop strategies to avoid dangerous situations. In future studies, the current approach will be extended to a hybrid model for all-encompassing traffic control. This extended hybrid model will include more traffic subjects, like trains or ships to receive a global and universal traffic simulator. Additionally, the CarPed simulator needs to be validated by field data. Therefore, we observed a public music festival over two consecutive years (BIEDERMANN, DIETRICH, HANDEL, KIELAR and SEITZ, 2015). The first results are promising, but need further investigations. ## 6Acknowledgements We would like to thank our supervisors Daniel H. Biedermann and Peter K. Kielar for their nice suggestions, advices, and their unconditional help at any time. Also Isabella von Sivers for her support and long conversations according to this topic. Additionally, we would like to thank Andrea Mayer, Andreas Riedl and other student assistants for their help to collect sufficient data. ## 7Program Code ### Subalgorithm for Traverse(edge) The traverse algorithm in the edge first check if the end node is full, to see if there are still people in the penultimate cell. If that’s the case, only solve the edge. If not throw the people in the penultimate cell into the node and solve the edge. The penultimate available cell should be empty before the solver, otherwise mass is not conserved. That’s why the solver (see figure 6) has inside a Distribute Last Densities function that finds and empties the available penultimate cell. After solving, check again for full node to throw people. Finally, apply traverse to the end node. <figure><img src="content_image/1601.06076/traverse_edge.png"><figcaption>Figure 6: Algorithm for Traverse(edge)</figcaption></figure> ### Subalgorithm Solve(Edge) <figure><img src="content_image/1601.06076/solve.png"><figcaption>Figure 7: Algorithm for Solve(Edge)</figcaption></figure> Check if there are full cells in edge. If thats the case, distribute last densities before and after computing the timestep. ### Sub algorithm for Traverse(Node) This traverse checks if Fixed or Dijkstra distributor is chosen. ### Sub algorithm for Distributor Fixed After all the incoming edges of the node threw people into it, we have a total amount to distribute into the outcoming ones. If that density to distribute cannot fit into the outcoming edges, the node is set full and it throws only the possible amount of people into the outgoing edge, given by a friction which is nothing else that a quantity between 0 and 1, the percentage of people that fits. If density to distribute fits all, set the node not full and throw all the people into the outcoming edges. <figure><img src="content_image/1601.06076/traverse_node.png"><figcaption>Figure 8: Algorithm for Traverse(Node)</figcaption></figure> ### Subalgorithm Node.SetFull(bool) If the input is true, check first if incoming edges have full cells. If that is the case do nothing, because there the Distribute Last Densities function in the edge already took the job of reorganizing the edge cells. If they have zero full cells, set one full cell for all of them. If false, set all full cells zero. <figure><img src="content_image/1601.06076/setFull.png"><figcaption>Figure 9: Algorithm for Node.setFull(bool)</figcaption></figure> ### Subalgorithm Edge.distributeLastDensities() The routine checks at first if both ultimate and penultimate denisties can be summed, using the amount. If this amount is more than the maximum density, define a friction and move only the possible people. After that, increase by one the number of full cells, and then call the same function again recursively. 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