Ideal topological gas in the high temperature phase of SU(3) gauge theory
aa r X i v : . [ h e p - l a t ] J a n Ideal topological gas in the high temperature phase of SU(3) gauge theory
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: January 6, 2021)We show that the nature of the topological fluctuations in SU (3) gauge theory changes drasticallyat the finite temperature phase transition. Starting from temperatures right above the phase transi-tion topological fluctuations come in well separated lumps of unit charge that form a non-interactingideal gas. Our analysis is based on a novel method to count not only the net topological charge, butalso separately the number of positively and negatively charged lumps in lattice configurations usingthe spectrum of the overlap Dirac operator. This enables us to determine the joint distribution ofthe number of positively and negatively charged topological objects, and we find this distributionto be consistent with that of an ideal gas of unit charged topological objects. The presence of topologically nontrivial gauge fieldconfigurations is a peculiar feature of QCD that has im-portant phenomenological consequences. Most recentlythis was highlighted by calculations to estimate the axionmass [1]-[3], an essential ingredient of which was the de-termination of the temperature-dependence of the topo-logical susceptibility up to temperatures well above theQCD crossover temperature to the quark-gluon plasma.At very high temperatures, fluctuations of the topo-logical charge are strongly suppressed and occur in theform of localized lumps of action, carrying topologicalcharge ±
1. These objects are probably close in theirshape and other properties to solutions of the classicalEuclidean gauge field equations, i.e. instantons, or rathertheir finite temperature counterparts, calorons [4]-[6].Moreover, since at high temperatures, fluctuations of thetopological charge are strongly suppressed, calorons (andanticalorons) are expected to form a dilute gas and theirsize is limited by the inverse temperature.These properties of the caloron gas motivate theso called dilute instanton gas approximation (DIGA)which –in principle– makes it possible to calculate thetemperature-dependence of the topological susceptibilityperturbatively. However, the topological susceptibilitydetermined in lattice simulations differs by an order ofmagnitude from the DIGA predictions even at temper-atures as high as 5 − T c [1]-[3]. It is thus clear thatat least one of the two assumptions that the DIGA is Since these objects are not exact solutions of the field equations,they are not exactly calorons or anticalorons, it would be moreappropriate to call them topological objects . Nevertheless, forsimplicity we will mostly use the word instanton or caloron forthem. based on is not satisfied. These two assumptions, bothof which are expected to be valid at high enough tempera-tures are: A1:
The probability of one instanton occurringin a given volume can be calculated perturbatively in thesemiclassical approximation.
A2:
The instantons gas isso dilute that interactions among (anti)instantons can beneglected, the gas of topological objects is an ideal gas.A1 has been recently reconsidered, but despite thecorrection of the previously grossly underestimated un-certainty of the semiclassical one-instanton calculation,there is still at least a 3 σ discrepancy between the latticeand the DIGA result for the topological susceptibility [7].Thus the still outstanding question is whether this dis-crepancy is due to interactions among instantons or thesemiclassical approximation being still so poor even atthe highest temperatures considered.In the present paper we focus on A2 and study inter-actions among topological objects in the quenched ap-proximation of QCD, just above the finite temperaturephase transition. In a free, non-interacting topologicalgas the number distribution of topological objects canbe characterized with a single parameter, the topologi-cal susceptibility χ = h Q i V , where Q = n i − n a is thetopological charge (the number of instantons minus thenumber of antiinstantons), V is the space-time volumeand h . i denotes the expectation with respect to the pathintegral. In an ideal topological gas all higher cumulantsof the distribution can be exactly calculated in terms of χ . Any deviations from these ideal-gas cumulants are aresult of interactions among topological objects.In the recent literature several quenched lattice calcu-lations of the lowest non-trivial cumulant B = h Q i − h Q i h Q i (1)appeared [8, 9]. The most precise calculation reports thateven though above 1 . T c the value of B is consistentwith 1, its ideal-gas value, just above the phase transi-tion, at 1 . T c it is still 1.27(7) [9].For a more complete assessment of the situation, moreinformation would be desirable about the distribution ofthe number of topological objects, beyond the first non-trivial cumulant of Q . However, higher cumulants ofthe distribution are notoriously hard to calculate, andeven the full topological charge distribution can in prin-ciple miss subtle correlations among instantons and an-tiinstantons. Full information about that is containedonly in the joint distribution of the number of instan-tons and antiinstantons. The problem is that while thereare well-established methods to compute the topologi-cal charge Q = n i − n a in lattice simulations, there isno easy way to determine n i and n a separately in lat-tice configurations . In the present paper we introducea novel method to compute n i and n a separately, anddetermine their joint distribution. Our method is basedon the low-lying spectrum of the overlap Dirac opera-tor. In particular, our main observation is that mixinginstanton-antiinstanton zero modes constitute a distinctpart of the Dirac spectrum close to zero, and can be reli-ably separated from the rest of the spectrum. Countingthe number of these close to zero modes together withthe exact zero modes of the overlap operator provides away to determine not only the topological charge Q , butalso the total number of topological objects n i + n a .For the present study we used quenched lattice config-urations generated at T = 1 . T c on lattices of temporalextension N t = 8 and aspect ratio 3 and 4. For both spa-tial volumes we determined n i and n a on 5k lattice config-urations. In the smaller volume, we found that the num-ber distribution of topological objects significantly devi-ated from the expectation based on a free non-interactinggas. In contrast, the larger volume showed no such de-viation. Although in finite temperature lattice QCD anaspect ratio of 3 is usually considered safe in terms offinite (spatial) volume corrections, we show here that theunexpectedly large finite volume corrections are due tothe proximity of the phase transition. We conclude thatin quenched QCD already slightly above T c the numberdistribution of topological objects is consistent with thatof a gas of free topological objects. We note that in the literature usually the value of b = − B / One possibility would be to analyze the structure of the topolog-ical charge density and locate individual lumps in it. However,this would be rather cumbersome and would be plagued by un-certainties.
Let us first motivate the main tool used in our study,the separation of the bulk of the spectrum and thetopology-related close to zero modes. It is known that inthe presence of an isolated instanton (or antiinstanton)the Euclidean Dirac operator has an exact zero modewith chirality +1 ( −
1) [11]. In the field of a well sepa-rated instanton and antiinstanton the two would be zeroeigenvalues split slightly and produce two complex conju-gate eigenvalues. The splitting is controled by the spatialdistance of the topological objects (relative to their size),as well as their orientation in group space. Generally thefarther away the two objects are, the smaller the split-ting is, and in the limit of infinite separation the splittingtends to zero [10]. In this way a dilute gas of topologicalobjects is expected to produce not only | Q | = | n i − n a | exact zero modes, corresponding to the net topologicalcharge, but also n i + n a − | Q | small Dirac eigenvalues,from the mixing of opposite chirality instanton and an-tiinstanton would be zero modes. Motivated by the in-stanton liquid model, we call the region in the spectrumcontaining these modes the Zero Mode Zone (ZMZ).In what follows, we will demonstrate that in contrastto the low temperature case, at high temperature theZMZ can be reliably separated from the rest of the Diracspectrum, provided a chirally symmetric Dirac operator,such as the overlap [12] is used. Already in the earlydays of the overlap it was noticed that above T c , besidesthe expected exact zero modes, the overlap Dirac spec-trum also contains an unexpectedly large number of verysmall close to zero eigenvalues [13]. This enhancementof the low end of the Dirac spectrum resulted in a spikein the spectral density, well separated from the bulk ofthe spectrum. This came as a surprise, since above T c the restoration of chiral symmetry would imply a vanish-ing spectral density at zero virtuality, due to the Banks-Casher relation. The spike in the spectral density wasconjectured to contain mixing would be zero modes of in-stantons and antiinstantons. Subsequent work confirmedthat this enhancement of the spectral density is neithera discretization, nor a quenched artifact [14]. More re-cently the appearance of this spike in the Dirac spectrumwas speculated to signal a genuinely new state of stronglyinteracting matter, intermediate between the low tem-perature hadronic and the high temperature quark-gluonplasma state [15].In the present work we analyze the statistical proper-ties of the eigenvalues in this spike of the spectral densityin a high statistics quenched SU (3) lattice study. Weshow that the statistics of these eigenvalues is to a highprecision consistent with the assumption that they areproduced by mixing instanton and antiinstanton wouldbe zero modes. To this end we use quenched gaugeensembles generated with the Wilson gauge action at β = 6 .
09 and temporal lattice extension N t = 8. Thiscorresponds to a temperature of T = 1 . T c , just abovethe finite temperature transition that in the quenched SU (3) case is a first order phase transition. For thedetailed statistical analysis we used two ensembles ofgauge configurations with spatial extension L = 24 and32, both containing 5000 configurations. In addition, tocheck finite volume effects in the spectral density, we alsohad an ensemble of 600 configurations on a larger spa-tial volume L = 40 . The negative mass parameter ofthe overlap Wilson kernel was set to M = − .
3, andtwo steps of hex smearing [16] were performed on thegauge links before inserting them into the Wilson kernel.The statistical analysis we report here was performed onthe overlap Dirac eigenvalues of smallest magnitude with | λ | /T c < . n i andthat of antiinstantons n a are expected to follow indepen-dent and identical Poisson distributions with mean V χ/ V is the four-volumeof the system and χ will turn out to be the topologicalsusceptibility. The joint distribution P ( n i , n a ) = e − V χ ( V χ/ n i + n a n i ! n a ! (2)of n i and n a can be used to compute all the relevantphysical quantities of this free topological gas in termsof the single parameter χ . In particular the topologicalsusceptibility is1 V h Q i = ∞ X n i =0 ∞ X n a =0 P ( n i , n a )( n i − n a ) == n i − n a − n i n a = χ, (3)where expectations like n i are understood to be with re-spect to the respective Poisson distribution. The topo-logical charge distribution for Q ≥ P ( Q ) = ∞ X n =0 e − V χ ( V χ/ Q +2 n ( Q + n )! n ! = e − V χ I Q ( V χ ) , (4)where I Q are the Bessel functions of imaginary argument.Due to time-reversal symmetry the distribution is sym-metric, P ( Q ) = P ( − Q ).Another interesting quantity to consider is the distri-bution of the total number of topological objects n = n i + n a , P ( n ) = n X n i =0 e − V χ ( V χ/ n n i !( n − n i )! = e − V χ ( V χ ) n n ! . (5)Let us now confront the lattice data with these ex-pectations. In Fig. 1 we show the spectral density ofthe overlap Dirac operator on the previously mentionedlattice ensembles. Although in the spontaneously bro-ken phase of the pure gauge theory that we considerhere, the three Polyakov loop sectors are equivalent, herewe restrict the analysis to configurations in the physical ρ / T c λ /T c L=24L=32L=40
FIG. 1. The spectral density of the overlap Dirac operator onquenched SU (3) gauge ensembles just above the phase tran-sition, at T = 1 . T c . The shaded region indicates λ ZMZ /T c ,the boundary of the Zero Mode Zone. Eigenmodes below thispoint in the spectrum are related to mixing topological wouldbe zero modes. P ( | Q | ) |Q|32 x8 simulationnon-interacting FIG. 2. The distribution of the topological charge in our lat-tice simulations and the distribution expected in a free topo-logical gas with the same susceptibility. As the distributionis expected to be symmetric, positive and negative charges ofthe same magnitude are counted together. The inset showsthe tail of the distribution on logarithmic scale. Re P > . The enhancement of thespectral density near zero is clearly seen in Fig. 1. Wenote that the exact zero eigenvalues are not shown here, We are planning to return to this question in a forthcoming pub-lication. P ( n ) n=n i +n a x8 simulationnon-interacting gas FIG. 3. The distribution of the number of topological ob-jects computed from the number of eigenvalues in the zeromode zone. The inset shows the tail of the distribution onlogarithmic scale. they would appear as a delta function exactly at zero.Counting the number of zero eigenvalues allows us tocompute the topological susceptibility χ , as well as thedistribution of the topological charge. In Fig. 2 we com-pare the distribution obtained in the lattice simulationwith the one expected in a free topological gas with sus-ceptibility χ . This is essentially a one-parameter fit ofthe function in Eq. (4), the fit parameter being V χ andthe chisquared per degree of freedom of the fit turns outto be 0.85.Encouraged by the good agreement between the lat-tice data and the free topological gas, we assume thatthe exact zero modes and the small Dirac eigenvalues,up to a point λ ZMZ in the spectrum, are the eigenvaluesassociated to the topological objects. In this way, bycounting them we count the number of topological ob-jects n present in the gauge field. To make this pictureconsistent, we have to choose λ ZMZ such that h n i = V χ ,as predicted by Eq. (5) for an ideal topological gas. Re-quiring this, results in λ ZMZ a = 0 . | λ | < λ ZMZ (includingthe exact zero modes) with n , the number of topologicalobjects. Counting the eigenvalues in the ZMZ configura-tion by configuration, we obtain the distribution of n andin Fig. 3 we compare it with the one expected in a gas ofnoninteracting topological objects, given by Eq. (5). Weemphasize that at this point no fitting is involved, sincethe only parameter of this distribution, V χ , had alreadybeen determined independently from the charge distribu-tion. We do not find a significant deviation from the freetopological gas distribution, as the chisquared per degreeof freedom of the deviation is 0.62. P r obab ili t y den s i t y |P|L=24L=32L=40 FIG. 4. The probability distribution of the Polyakov loop forthree different spatial volumes with linear size L = 24 ,
32 and40.
The finite temperature SU (3) transition is a first or-der phase transition, so the correlation length does notdiverge, however, large finite volume corrections cannotbe excluded. To assess their importance, we repeatedthe analysis in a smaller volume with linear extension L = 24. In that case we found significant deviations fromthe expected free topological gas behavior. The resultingchisquared per degree of freedom was 1.99 and 6.29 inthe case of the charge distribution and the distributionof the total number of topological objects, respectively.To understand finite volume corrections in the vicin-ity of a phase transition, it is instructive to look at thevolume dependence of the distribution of the order pa-rameter. In Fig. 4 we show the probability distributionof the magnitude of the Polyakov loop, the order pa-rameter of the quenched finite temperature transition.Besides the widening of the distribution, expected forsmaller volumes, the data for L = 24 ,
32 lattices alsoshow an unusual enhancement of smaller values of thePolyakov loop. The reason for this is that in the hightemperature phase the Z (3) center symmetry is sponta-neously broken and the system randomly chooses one ofthe three Z (3) sectors. However, in a finite volume tun-neling among the sectors is still possible, the tunnelingprobability is enhanced in smaller volumes, and configu-rations in the process of tunneling have small magnitudesof the Polyakov loop.As also seen in Fig. 4, in larger volumes these tunnelingstates get suppressed, however, in smaller volumes theycan still give significant contributions to physical quan-tities, resulting in large finite-size effects. To see, howthese tunneling states can affect the topological charge,we looked at the correlation between topology and thePolyakov loop. The simplest quantity to study is thetopological susceptibility. We computed its dependenceon the Polyakov loop by restricting the averaging of Q to configurations with Polyakov loop magnitudes in in-tervals of length 0.01. The results for the L = 24 and 32 χ / T c |P| L=24L=32 FIG. 5. The dependence of the topological susceptibility onthe Polyakov loop for two different spatial volumes with linearsize L = 24 ,
32. The two horizontal stripes indicate the sus-ceptibility (with its uncertainty) computed for the full ensem-bles without restricting the Polyakov loop. The lower value,indicated with the darker band corresponds to the larger vol-ume. ensembles, shown in Fig. 5 reveal a strong dependenceof the susceptibility on the Polyakov loop. The previ-ously seen enhanced contribution of the tunneling region(small Polyakov loop), where the susceptibility is larger,will thus result in significantly larger topological suscep-tibilities in smaller volumes. To have a feeling about therelative importance of the enhanced region, we note thatthe probability that | P | < . L = 24 ,
32 and 40respectively. For the two ensembles shown in Fig. 5 wealso indicated the overall susceptibilities and their uncer- tainties with the horizontal stripes. Since on the L = 24lattices even the susceptibility suffers large finite-size ef-fects, it is not surprising that the same occurs for thedistribution of the topological charge and the number oftopological objects.In the present paper we used a novel way to computethe joint distribution of the number of topological ob-jects in lattice simulations. We showed that right abovethe critical temperature of pure SU (3) gauge theory thedistribution is consistent with the one expected in anideal gas of non-interacting charges of unit magnitude.It is remarkable that while below the phase transitiontopological fluctuations form a dense medium withouteasily identifiable individual lumps [17], right above thephase transition an ideal gas of well separated topologi-cal lumps emerges. Our result also implies that the mostlikely explanation of the large discrepancy between thelattice and DIGA based calculation of the topologicalsusceptibility is that the topological lumps we found arenot close enough in shape to ideal calorons to warrant asemiclassical treatment.We expect that –at least on a qualitative level– thispicture of the topological fluctuations that we found inthe quenched case carries over to QCD with dynamicalquarks. However, the fermion determinant might intro-duce some interaction even among well separated topo-logical lumps, but to study that one would need to usea chiral Dirac operator also for the simulation of the seaquarks. Acknowledgments
TGK was partially supported by theHungarian National Research, Development and Innova-tion Office - NKFIH grant KKP126769. TGK thanksMatteo Giordano, S´andor Katz and D´aniel N´ogr´adi forhelpful discussions. [1] S. Borsanyi, Z. Fodor, J. Guenther, K. H. Kampert,S. D. Katz, T. Kawanai, T. G. Kovacs, S. W. Mages,A. Pasztor and F. Pittler, et al.
Nature , no.7627,69-71 (2016) doi:10.1038/nature20115.[2] C. Bonati, M. D’Elia, M. Mariti, G. Martinelli, M. Mesiti,F. Negro, F. Sanfilippo and G. Villadoro, JHEP , 155(2016) doi:10.1007/JHEP03(2016)155.[3] P. Petreczky, H. P. Schadler and S. Sharma, Phys. Lett. B , 498-505 (2016) doi:10.1016/j.physletb.2016.09.063.[4] T. C. Kraan and P. van Baal, Nucl. Phys. B , 627-659(1998) doi:10.1016/S0550-3213(98)00590-2.[5] C. Gattringer, M. Gockeler, P. E. L. Rakow, S. Schae-fer and A. Schaefer, Nucl. Phys. B , 205-240 (2001)doi:10.1016/S0550-3213(01)00509-0.[6] C. Gattringer and S. Schaefer, Nucl. Phys. B , 30-60(2003) doi:10.1016/S0550-3213(03)00083-X.[7] A. Boccaletti and D. Nogradi, JHEP , 045 (2020)doi:10.1007/JHEP03(2020)045.[8] S. Borsanyi, M. Dierigl, Z. Fodor, S. D. Katz,S. W. Mages, D. Nogradi, J. Redondo, A. Ringwaldand K. K. Szabo, Phys. Lett. B , 175-181 (2016)doi:10.1016/j.physletb.2015.11.020. [9] C. Bonati, M. D’Elia, H. Panagopoulos and E. Vi-cari, Phys. Rev. Lett. , no.25, 252003 (2013)doi:10.1103/PhysRevLett.110.252003.[10] T. Sch¨afer and E. V. Shuryak, Rev. Mod. Phys. , 323-426 (1998) doi:10.1103/RevModPhys.70.323.[11] M. F. Atiyah, I. M. Singer, Annals Math. (1971) 139.[12] R. Narayanan and H. Neuberger, Phys. Rev. Lett. ,no. 20, 3251 (1993) doi:10.1103/PhysRevLett.71.3251.[13] R. G. Edwards, U. M. Heller, J. E. Kiskis andR. Narayanan, Phys. Rev. D , 074504 (2000)doi:10.1103/PhysRevD.61.074504.[14] A. Alexandru and I. Horv´ath, Phys. Rev. D , no.4,045038 (2015) doi:10.1103/PhysRevD.92.045038.[15] A. Alexandru and I. Horv´ath, Phys. Rev. D , no.9,094507 (2019) doi:10.1103/PhysRevD.100.094507.[16] S. Capitani, S. Durr and C. Hoelbling, JHEP , 028(2006) doi:10.1088/1126-6708/2006/11/028.[17] I. Horvath, N. Isgur, J. McCune andH. B. Thacker, Phys. Rev. D65