aa r X i v : . [ h e p - l a t ] S e p Localization with overlap fermions
R´eka ´A. Vig
University of Debrecen, H-4032 Debrecen,Bem t´er 18/A, Hungary
Tam´as G. Kov´acs
E¨otv¨os Lor´and University, H-1117 Budapest,P´azm´any P´eter s´et´any 1/A, HungaryandInstitute for Nuclear Research,H-4026 Debrecen, Bem t´er 18/c, Hungary (Dated: September 28, 2020)We study the finite temperature localization transition in the spectrum of the overlap Dirac op-erator. Simulating the quenched approximation of QCD, we calculate the mobility edge, separatinglocalized and delocalized modes in the spectrum. We do this at several temperatures just abovethe deconfining transition and by extrapolation we determine the temperature where the mobilityedge vanishes and localized modes completely disappear from the spectrum. We find that this tem-perature, where even the lowest Dirac eigenmodes become delocalized, coincides with the criticaltemperature of the deconfining transition. This result, together with our previously obtained sim-ilar findings for staggered fermions shows that quark localization at the deconfining temperatureis independent of the fermion discretization, suggesting that deconfinement and localization of thelowest Dirac eigenmodes are closely related phenomena.
I. INTRODUCTION
Strongly interacting matter is known to undergo acrossover at high temperature. In the low temperatureregime quarks are bound together to form hadrons due tocolor confinement. During the crossover the boundariesof the hadrons become blurred and matter goes into thestate of quark-gluon plasma (QGP). At the same timethe spontaneously broken chiral symmetry becomes ap-proximately restored. Besides deconfinement and chiralrestoration, there is a third phenomenon that happens inthe crossover region. Above the crossover temperaturethe lowest lying eigenmodes of the Dirac operator be-come spatially localized [1]-[4]. This is in sharp contrastto the temperature regime below the crossover, where allthe quark eigenmodes are extended [5].In the high temperature phase the spectrum of theDirac operator can be separated into two regions. At thelow end of the spectrum there are only localized eigen-modes and their eigenvalues can be described by Poissonstatistics. In the upper part of the spectrum the eigen-modes are extended and the corresponding eigenvaluesobey Wigner-Dyson statistics [5]. At fixed temperaturethis transition in the spectrum between the localized andextended eigenmodes was shown to be a genuine secondorder transition, and its correlation length critical expo-nent was found to be compatible with that of the An-derson model in the same symmetry class [6] . Build-ing on this analogy with Anderson transitions, we call Note, however, that in contrast to the Anderson transitions incondensed matter systems, in QCD this is not a genuine physical the critical point separating the localized and extendedmodes in the spectrum, the mobility edge, λ c [7]. Whilein the Anderson model the mobility edge is controlled bythe amount of disorder in the system, in QCD an analo-gous role is played by the physical temperature. As thetemperature is lowered towards the crossover, the mo-bility edge moves down in the spectrum, the part of thespectrum corresponding to localized eigenmodes occupiesa narrower and narrower band in the spectrum aroundzero. Eventually, at a well defined temperature that wedenote by T locc , the mobility edge vanishes, implying thateven the lowest Dirac eigenmodes become delocalized.In QCD with physical quark masses the critical tem-perature of the localization transition, T locc is in thecrossover region [4]. This raises the question whetherthis is just a coincidence or there is some deeper phys-ical connection between the localization transition andthe chiral and/or the deconfinement transition. A pos-sible way to test this is to move in the parameter spaceof QCD to a regime where there is a genuine finite tem-perature phase transition and check whether its criticaltemperature coincides with T locc . The simplest way to dothat is to consider the limit of infinitely heavy quarks, i.e.the quenched approximation to QCD, which is known tohave a first order deconfining phase transition at a tem-perature of around 300 MeV.The possibility of linking the QCD transition to anAnderson-type localization transition in the Dirac spec-trum was first raised more than ten years ago by Garcia- phase transition, as λ , the location in the Dirac spectrum is nota tuneable physical control parameter. Garcia and Osborn. They studied the spectral statisticsof the Dirac operator in an instanton liquid model [3]and in quenched as well as full lattice QCD [2] and foundevidence that around the chiral transition the spectralstatistics of the Dirac spectrum changes from Wigner-Dyson towards Poisson. This indicates that the chiraltransition is accompanied by a localization transition forthe lowest eigenmodes of the Dirac operator, however,at that time no no attempt was made at a determina-tion of T locc , the critical temperature of the localizationtransition, with a precision comparable to how T c , thecritical temperature of the quenched deconfining phasetransition is available in the literature.In a previous paper we explored this possibility bystudying the spectrum of the staggered quark Dirac oper-ator in quenched gauge field backgrounds, generated justabove the finite temperature phase transition [8]. Forstaggered fermions we calculated T locc , the critical tem-perature of the localization transition and found that itcoincided with that of the deconfining transition. Ourresults, obtained on lattices with three different tempo-ral extensions, L t = 4 , SU (3) gauge group are in the same random matrix sym-metry class, the chiral unitary class, as fermions in thecontinuum. Secondly, unlike the staggered action thatis ultralocal, the overlap action couples quark degrees offreedom to arbitrarily large distances, albeit with cou-plings falling exponentially with the distance. Since inthe theory of Anderson-type models localization is gen-erally known to strongly depend on the range of the cou-plings (hopping terms in the Hamiltonian) [11], it is in-teresting to check whether the non-locality of the overlapDirac operator has any influence on the localization tran-sition in QCD. In fact, to our knowledge, this is the first β L s N c N evs L t = 6. study where the mobility edge is explicitly determined inQCD with chiral quarks .In the present work we used a subset of the gauge con-figurations that were previously generated for our earlierstaggered study. Since overlap spectra are significantlymore expensive to calculate than staggered spectra, herewe limited our study to one value of the temporal lat-tice size, L t = 6. We computed the mobility edge forgauge ensembles generated with six different values ofthe gauge coupling, β , all corresponding to temperaturesslightly above the deconfining transition. By extrapola-tion we determined the gauge coupling β locc where themobility edge vanished and all localized eigenmodes dis-appeared from the Dirac spectrum. Confirming our pre-vious staggered result, we found β locc to be compatiblewith the critical gauge coupling of the deconfining phasetransition for L t = 6.The plan of the paper is as follows. In Sec. II we de-scribe the lattice ensembles used for the calculation andshow how we computed the mobility edge from the Diracspectra. In Sec. III we discuss the determination of thecritical coupling of the localization transition. In Sec.IV we draw our conclusions and finally in the Appendixwe describe the technical details of the unfolding of thespectrum. II. CALCULATION OF THE MOBILITY EDGE
The Dirac operator that we used for this study wasthe overlap with Wilson kernel parameter M = − .
3. Assmearing of the gauge field is known to improve someproperties of the overlap and also makes the calculationsfaster [9], two steps of hex smearing [12] were applied to Indirect evidence for localization of overlap quarks has alreadybeen obtained by studying the distribution of the lowest twoeigenvalues in Ref. [13], but the transition to the delocalizedregime in the spectrum was not explicitly seen in that work. the gauge field before inserting it into the overlap. Thegauge field configurations we used here were quenchedWilson action lattices with temporal extension L t = 6.In Table I we collected the parameters of the simulations.On each gauge configuration we computed a numberof lowest eigenvalues of D † D , where D is the overlapDirac operator. In what follows we always work with theeigenvalues of D † D that are the magnitude squared of thecorresponding eigenvalues of the Dirac operator D . Sinceour analysis is based on the unfolded spectrum, which isinvariant with respect to monotonic reparametrizationsof the spectrum, it makes no difference that we performthe analysis in terms of the eigenvalues of D † D . As ex-plained in the Appendix, we take extra care to make eventhe assignment of eigenvalue pairs to spectral windowsto be reparametrization invariant. To make the nota-tion simpler and avoid having to write the absolute valuesquared everywhere, we denote by λ the eigenvalues of D † D , in terms of which we perform the entire analysis.The number of eigenvalues to be computed per con-figuration was chosen to include all the eigenvalues to apoint well above the mobility edge, λ c . Having exactchiral symmetry, the overlap possesses exact zero eigen-values in gauge field backgrounds with non-zero topolog-ical charge. Since these eigenvalues are all exactly at thelower edge of the spectrum, they do not contain any infor-mation relevant to the present study, we simply removedthem from the spectra before further analysis.Localized and delocalized eigenmodes are character-ized by different statistics of the corresponding eigen-values. To track the transition throughout the spec-trum and locate the mobility edge, we used the simplestspectral statistics, the unfolded level spacing distribu-tion (ULSD), calculated locally, within narrow spectralwindows of the spectrum. Unfolding, a transformationwell known in the theory of random matrices, is a mono-tonic mapping of the spectrum that sets the local spectraldensity to unity everywhere throughout the spectrum.In particular, by construction, the unfolded eigenvaluesare dimensionless and their average level spacing is unity.More details on how the unfolding was done are presentedin the Appendix.Unfolding is useful since both for localized and delo-calized eigenmodes, universally valid analytic results areknown for the ULSD of the corresponding eigenvalues[5]. Spectra corresponding to localized eigenmodes obeyPoisson statistics and the ULSD follow the exponentialdistribution, p ( s ) = exp( − s ) , (1)where s is the level spacing between the nearest neighborunfolded eigenvalues.In the case of extended modes the ULSD is also knownanalytically, however, it is much more complicated than This criterion could be checked only a posteriori, after determin-ing λ c FIG. 1. The probability density functions of the level spac-ings in the cases when the eigenvalues obey the Poisson statis-tics (exponential distribution, dashed line) and Wigner-Dysonstatistics (continuous line, Wigner surmise). in the localized case and also depends on the randommatrix symmetry class of the given model. A very goodapproximation to the ULSD in this case is provided bythe so called Wigner surmise that for the unitary sym-metry class, to which the overlap operator belongs, readsas p ( s ) = 32 π s exp( − π s ) . (2)Notice that both the exponential and the Wigner surmisedistribution are universal in the sense that they are freeof any adjustable parameters. In particular, the origi-nally dimensionful parameter, the local spectral densityhas been removed from the spectrum by the unfolding.For further reference we plotted the two distributions inFig. 1.Our aim here is to scan the spectrum starting fromzero and follow how the local ULSD changes from the ex-ponential distribution of Eq. (1) to the Wigner surmise ofEq. (2). To this end we divide the spectrum into narrowspectral windows, compute the ULSD separately in eachspectral window and follow how it changes throughoutthe spectrum. In Fig. (2) we show how the unfolded levelspacing distribution evolve as the spectrum is scannedstarting from the lowest eigenvalues (top panel) crossingthe critical, transition region (middle panel) and finallymoving up to the Wigner-Dyson regime (bottom panel).However, monitoring the continuous change of a func-tion (here the probability density of the unfolded levelspacings) is complicated. To make this task easier, wechoose a single parameter of this distribution and moni-tor how that changes throughout the spectrum. A simplechoice for this parameter is the integral I s = Z s p ( s ) ds (3) FIG. 2. The unfolded level spacing distribution in three differ-ent spectral windows for the β = 5 . L s = 40 ensemble. Thespectrum is scanned from the lowest eigenvalues (top panel)with Poisson statistics, through the critical region (middlepanel), up to the regime with Wigner-Dyson statistics (bot-tom panel). We also show the expected limiting distributions,the exponential (dashed line) and the Wigner surmise (con-tinuous line). of the probability density up to the lowest crossing point s ≈ .
508 of the two limiting distributions, the expo-nential and the Wigner surmise. This choice of s hasthe advantage that it maximizes the difference of the in-tegral between the two limiting cases and thereby facili-tates their clear separation.An example of how I s changes through the spectrumis shown in Fig. 3. As expected and can also be seenin the figure, in a finite volume I s changes smoothlyfrom the value I Ps ≈ .
398 corresponding to the expo-nential distribution to I Ws ≈ .
117 corresponding to theWigner surmise. However, based on the finite size scal-ing study of Ref. [6], in the thermodynamic limit we ex-pect the transition to become singular, as in a secondorder phase transition. The mobility edge, λ c that we FIG. 3. The integrated probability density (defined in Eq. 3)as a function of eigenvalues λ of D † D . The figure shows thedata for β = 5 .
95 with a spatial volume of 40 . The two shorthorizontal lines in the top left and the bottom right corner ofthe figure indicate the limiting values of I s for the Poisson(localized) and the Wigner-Dyson (delocalized) statistics. eventually want to locate is this sharply defined singu-lar transition point appearing only in the infinite volumelimit. In a finite volume the definition of the “criticalpoint” is somewhat arbitrary, however, a good choice isthe point in the spectrum where I s is equal to the value I crits = 0 . λ c , for which I s ( λ c ) = I crits , themobility edge.The quantity λ c , defined in this way, can still have avolume dependence, but it is a good approximation tothe mobility edge in the thermodynamic limit. To keepthe finite size corrections under control, we calculated λ c on lattices of spatial linear size L s = 24 , ,
40. While theresults on the smallest volume differed significantly fromthose on the other volumes, the results from the largertwo volumes agreed within the statistical uncertainties.Therefore, for the rest of the analysis we always used thedata from the largest volume, L s = 40.Since the function I s ( λ ) has an inflection point at λ c ,around this point it can be well approximated with astraight line. We could thus easily determine λ c by solv-ing the equation I s ( λ c ) = I crits by approximating thefunction I s ( λ ) with a linear fit to the data in the givenrange (see Fig. 3). III. THE CRITICAL TEMPERATURE OF THELOCALIZATION TRANSITION
So far we have shown how to calculate the mobilityedge, λ c , at a given temperature. Our final goal is todetermine the temperature where the mobility edge van- FIG. 4. The mobility edge as the function of the gauge cou-pling, approximated with a power function. ishes and localized modes completely disappear from theDirac spectrum. Since we keep the temporal size of thelattice fixed, the temperature can be controlled by thegauge coupling, β . We computed λ c for lattice ensem-bles generated at several different values of the gaugecoupling, above, but close to the deconfining phase tran-sition. The results are shown in Fig. 4. The range ofcouplings we used were limited by two factors. Firstly,even though the deconfining transition is of first order,the correlation length increases substantially towards thetransition which puts a lower limit to the couplings forwhich finite size corrections can be kept under control.Secondly, we would like to extrapolate the function λ c ( β )to find where it vanishes, and for the extrapolation onlypoints close enough to the zero of this function are use-ful. Since we expected the zero of the function λ c ( β ) tobe close to the deconfining transition, β c , we limited oursimulations to couplings not too far from this point.Finally, for the extrapolation we used the ansatz λ c ( β ) = p ( β − β locc ) p , (4)and its parameters p , p and β locc were fitted to the data.The ansatz turned out to describe the data remarkablywell and using all six data points the resulting χ perdegree of freedom was χ = 0 .
67. The fit along withthe data is shown in Fig. 4. The resulting location ofthe localization transition is β locc = 5 . β c = 5 . IV. CONCLUSIONS
We examined the localization transition of the quarksusing the quenched approximation. We computed thelowest lying eigenvalues of the overlap Dirac operatorabove the critical coupling of the deconfining transition.By calculating the mobility edge, λ c , for different gaugecouplings we determined the function λ c ( β ) and extrap-olated it to locate β locc , where the mobility edge vanishesand all the eigenmodes become delocalized. We com-pared our result with the critical coupling of the decon-fining phase transition and found that the two criticalcouplings are compatible; the localization transition anddeconfinement occur at the same temperature. This isin agreement with our previous similar results with stag-gered fermions and indicates that localization and decon-finement are strongly related phenomena.The present work was motivated by the fact that inQCD with physical dynamical quarks the localizationtransition occurs in the crossover region. On the onehand, our results clearly indicate that the localizationtransition is strongly related to deconfinement, which –at least on a qualitative level – probably carries over fromthe quenched model to real physical QCD. On the otherhand, the quenched model cannot properly account forthe other important transition, the chiral transition thatalso occurs in the QCD crossover. To see how local-ization is related to chiral restoration, it would be in-teresting to consider the other limiting case, the chirallimit. For massless light quarks, the chiral transition isexpected to become a genuine phase transition [15] andit could be tested whether its critical temperature agreeswith the critical temperature of the localization transi-tion. Although simulations in the chiral limit are tech-nically immensely challenging, such a study could alsoprovide additional insight into the physics of the restora-tion of chiral symmetry, how that happens in the mass-less (chiral) limit. Several questions related to this arecurrently under active study [16]. ACKNOWLEDGMENTS
T.G.K. was partially supported by the Hungarian Na-tional Research, Development and Innovation Office -NKFIH Grant No. KKP126769. R.A.V. was partiallysupported by the New National Excellence Program ofthe Hungarian Ministry for Innovation and TechnologyGrant No. ´UNKP-19-3-I-DE-490. T.G.K. thanks MatteoGiordano, S´andor Katz and D´aniel N´ogr´adi for helpfuldiscussions.
Appendix: Unfolding
Unfolding is a monotonic mapping of the eigenvaluesthat - by definition - renders the spectral density unitythroughout the unfolded spectrum. This transformationis useful since it removes the scale, specific to the givenspectrum and reveals universal spectral fluctuations. Inprinciple unfolding can be done in several different ways,all equivalent for a dense enough spectrum. Here we didthe unfolding by taking all the eigenvalues from all theconfigurations of the given ensemble and putting theminto ascending order according to their magnitudes. Toeach eigenvalue we assigned its rank divided by the num-ber of configurations, N c , we used this mapping to definethe unfolded spectrum. In this way the level spacingbetween successive unfolded eigenvalues is exactly 1 /N c ,which means that there are N c eigenvalues in an intervalof unit length anywhere in the unfolded spectrum. Thisimplies that the average spectral density per configura-tion is unity throughout the unfolded spectrum.In the present work we used the unfolded level spac-ing distribution (ULSD) calculated from the spectrumunfolded in the above described way. In particular, wefollowed how the ULSD changed throughout the spec-trum, starting from the Poisson statistics and going overto Wigner-Dyson statistics. This required the calculationof the local ULSD at different locations in the spectrum.In order to do this, we divided the spectrum into small spectral windows and calculated the ULSD in each win-dow separately.In principle, this method is straightforward, if the spec-trum is infinitely dense. However, for finite density, thereis an ambiguity in how we decide whether a pair of neigh-boring eigenvalues belongs to the given spectral windowor not. We could demand that both members of the pairbe within the spectral window in question. However, thiswould artificially limit the largest possible level spacings,especially for eigenvalues close to the edge of a spectralwindow. To avoid this uncontrolled truncation of the tailof the ULSD we chose the criterion that a pair of near-est neighbor eigenvalues was considered to belong to thegiven spectral window if the midpoint of the pair wasin the window. To ensure that our procedure, includingthe assignment of pairs to spectral windows, is invari-ant with respect to monotonic reparametrizations of thespectrum, we applied the midpoint rule in the unfoldedspectrum. This is easily done by mapping the endpointsof the spectral window into the unfolded spectrum. No-tice, however, that we can and do still plot the resultsin terms of the original (not the unfolded) spectrum, asseen in Fig. 3. [1] A. M. Halasz and J. J. M. Verbaarschot, Phys. Rev.Lett. , 3920 (1995) doi:10.1103/PhysRevLett.74.3920[hep-lat/9501025].[2] A. M. Garcia-Garcia and J. C. Osborn, Phys. Rev.D , 034503 (2007) doi:10.1103/PhysRevD.75.034503[hep-lat/0611019].[3] A. M. Garcia-Garcia and J. C. Osborn, Nucl. Phys.A , 141 (2006) doi:10.1016/j.nuclphysa.2006.02.011[hep-lat/0512025].[4] M. Giordano, T. G. Kovacs and F. Pittler, Int.J. Mod. Phys. A , no. 25, 1445005 (2014)doi:10.1142/S0217751X14450055 [arXiv:1409.5210 [hep-lat]].[5] J. J. M. Verbaarschot and T. Wettig,Ann. Rev. Nucl. Part. Sci. , 343 (2000)doi:10.1146/annurev.nucl.50.1.343 [hep-ph/0003017].[6] M. Giordano, T. G. Kovacs and F. Pittler,Phys. Rev. Lett. , no. 10, 102002 (2014)doi:10.1103/PhysRevLett.112.102002 [arXiv:1312.1179[hep-lat]].[7] T. G. Kovacs and F. Pittler, Phys. Rev. D ,114515 (2012) doi:10.1103/PhysRevD.86.114515[arXiv:1208.3475 [hep-lat]].[8] T. G. Kovacs and R. A. Vig, Phys. Rev. D ,no. 1, 014502 (2018) doi:10.1103/PhysRevD.97.014502[arXiv:1706.03562 [hep-lat]]. [9] T. G. Kovacs, Phys. Rev. D , 094501 (2003)doi:10.1103/PhysRevD.67.094501 [hep-lat/0209125].[10] R. Narayanan and H. Neuberger, Phys. Rev. Lett. ,no. 20, 3251 (1993) doi:10.1103/PhysRevLett.71.3251[hep-lat/9308011]; R. Narayanan and H. Neuberger,Nucl. Phys. B , 305 (1995) doi:10.1016/0550-3213(95)00111-5 [hep-th/9411108].[11] F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355 (2008) doi:10.1103/RevModPhys.80.1355[arXiv:0707.4378 [cond-mat.mes-hall]].[12] S. Capitani, S. Durr and C. Hoelbling, JHEP , 028 (2006) doi:10.1088/1126-6708/2006/11/028[hep-lat/0607006].[13] T. G. Kovacs, Phys. Rev. Lett. , 031601 (2010)doi:10.1103/PhysRevLett.104.031601 [arXiv:0906.5373[hep-lat]].[14] A. Francis, O. Kaczmarek, M. Laine, T. Neuhausand H. Ohno, Phys. Rev. D , no. 9, 096002 (2015)doi:10.1103/PhysRevD.91.096002 [arXiv:1503.05652[hep-lat]].[15] R. D. Pisarski and F. Wilczek, Phys. Rev. D , 338(1984). doi:10.1103/PhysRevD.29.338[16] H. T. Ding et al. , Phys. Rev. Lett. , no.6, 062002 (2019) doi:10.1103/PhysRevLett.123.062002[arXiv:1903.04801 [hep-lat]]; K. Suzuki et al.et al.