J-PARC-TH-0246,UTHEP-758

Finite-size scaling around the critical point in the heavy quark region of QCD

Atsushi Kiyohara,1 Masakiyo Kitazawa,1, 2 Shinji Ejiri,3 and Kazuyuki Kanaya4

1Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan2J-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies,

KEK, 203-1, Shirakata, Tokai, Ibaraki 319-1106, Japan3Department of Physics, Niigata University, Niigata 950-2181, Japan

4Tomonaga Center for the History of the Universe,University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

(Dated: August 3, 2021)

Finite-size scaling is investigated in detail around the critical point in the heavy-quark region ofnonzero temperature QCD. Numerical simulations are performed with large spatial volumes up tothe aspect ratio Ns/Nt = 12 at a fixed lattice spacing with Nt = 4. We show that the Bindercumulant and the distribution function of the Polyakov loop follow the finite-size scaling in theZ(2) universality class for large spatial volumes with Ns/Nt ≥ 9, while, for Ns/Nt ≤ 8, the Bindercumulant becomes inconsistent with the Z(2) scaling. To realize the large-volume simulations in theheavy-quark region, we adopt the hopping tails expansion for the quark determinant: We generategauge configurations using the leading order action including the Polyakov loop term for Nt = 4,and incorporate the next-to-leading order effects in the measurements by the multipoint reweightingmethod. We find that the use of the leading-order configurations is crucially effective in suppressingthe overlapping problem in the reweighting and thus reducing the statistical errors.

I. INTRODUCTION

One of the interesting features of the medium describedby quantum chromodynamics (QCD) is the existence ofphase transitions of various orders. While the finite-temperature QCD transition is an analytic crossover atzero quark chemical potential µq [1, 2], this phase transi-tion is expected to become of first order in dense mediumwith large µq [3]. The end point of the first-order tran-sition is called the critical point (CP) at which the tran-sition is of second order. The singularity in thermody-namic observables associated with the second order CPare believed to be useful in detecting the CP in heavy-ion collision experiments [4, 5]. Accordingly, researchescalled the beam-energy scan are actively performed inexperimental facilities all over the world to search for thecritical fluctuations around the CP [5–7].

The order of the finite temperature QCD transitionchanges also with variation of quark masses [8]. For the2 + 1-flavor QCD, it is known that the crossover at thephysical quark masses becomes of first order both in thelight- and heavy-quark limits; the phase diagram repre-senting this feature is known as the Columbia plot [9, 10].Revealing the nature of the phase transitions with thevariation of quark masses is an important subject of QCDat nonzero temperature since it provides us with variousinsights into the transition at the physical quark masses.

Pinning down the boundaries of the first-order tran-sitions in 2 + 1-flavor QCD in the light [11–23] andheavy [24–31] quark regions is a longstanding subject inlattice QCD simulations. It, however, has been foundthat the location of the boundaries are strongly depen-dent on the lattice cutoff of the simulations [22, 23, 31],and their quantitative determination in the continuumlimit has not been established yet.

One of the difficulties in these analyses is that observ-

ables near the CP are strongly dependent on the spatialvolume of the system. The spatial volume dependenceis in part described by the finite-size scaling (FSS) [32].However, the FSS is applicable only for describing thesingular part of thermodynamic quantities that domi-nates over the non-singular part only in the vicinity of theCP for sufficiently large spatial volumes. When the spa-tial volume is not large enough the FSS of observables isviolated due to the contributions of the non-singular partand this makes their analysis based on the FSS problem-atic. In fact, although the CP of QCD is believed to be-long to the three-dimensional Z(2) universality class [8],a clear FSS in this universality class has not been ob-served in latest numerical studies around the CP in thelight [23] and heavy [31] quark regions. These resultssuggest the necessity to perform numerical analyses withyet larger spatial volumes.

In the present study, we focus on the CP in the heavyquark region and study the behavior of observables bynumerical simulations with large spatial volumes corre-sponding to the aspect ratios up to Ns/Nt = 12. Tocarry out analysis on the large spatial volumes with highprecision, we fix the temporal lattice extent to be Nt = 4in this study.

We also employ the hopping parameter expansion(HPE) to deal with the quark determinant. In thisstudy, we generate gauge configurations using the lead-ing order (LO) action of the HPE including the Polyakovloop term for Nt = 4, and then incorporate the next-to-leading order (NLO) effects by a multipoint reweightingmethod [33]. In our previous study, we generated theconfigurations in quenched QCD (zero-th order of theHPE) and reweighted them to incorporate LO and NLOeffects [27, 29, 30]. We show that the use of the LO actionto generate configurations is quite effective in suppress-ing the overlapping problem of the reweighting method.

arX

iv:2

108.

0011

8v1

[he

p-la

t] 3

1 Ju

l 202

1

2

FIG. 1. Phase diagrams of (a) QCD in heavy-quark regionand (b) three-dimensional Ising model.

This is essential to carry out simulations with large sys-tem volumes as studied in the present paper. We alsoverify the convergence of the HPE by comparing the LOand NLO results.

We perform the Binder cumulant analysis of thePolyakov loop around the CP and find that the numericalresults for Ns/Nt ≥ 9 follow well the FSS in the Z(2) uni-versality class. On the other hand, inclusion of the dataat Ns/Nt ≤ 8 in the analysis gives rise to a statistically-significant deviation from the scaling behavior. We fur-ther investigate the scaling behavior of the distributionfunction of the Polyakov loop. We find that the distri-bution function follows the FSS in the Z(2) universalityclass for Ns/Nt ≥ 8. From the deviation pattern of thedistribution function for LT = 6 from the Z(2) FSS, wediscuss that the violation of the scaling behavior in theBinder cumulant is caused by the deviation in the tailsof the distribution for small LT .

This paper is organized as follows. In the next sec-tion we give a brief review on the FSS. We then explainthe setup of our lattice simulation and analyses using theHPE in Sec. III. In Sec. IV, we determine the transitionline and perform the Binder cumulant analysis to deter-mine the location of the CP and the critical exponent. InSec. V, we investigate the FSS of the distribution func-tion of the Polyakov loop. The last section, Sec. VI, isdevoted to a summary. In Appendix A, we give definitionof cumulants. In Appendix B, the HPE of the quark de-terminant is calculated up to the NLO. In Appendix C,the convergence of the HPE is examined by comparingthe Binder cumulants at the LO and the NLO.

II. FINITE-SIZE SCALING

Let us first give a brief review on the FSS and its ap-plication to the CP in the heavy-quark region of QCD.

The heavy-quark limit of QCD corresponds to theSU(3) Yang-Mills theory (quenched QCD). This the-ory has a first-order deconfinement phase transition atnonzero temperature T . When the quark mass mq is fi-nite (throughout this section we assume that the quarkmasses are degenerate), with increasing 1/mq, this first-order transition becomes weaker and eventually termi-

nates at the CP, as schematically shown in the phasediagram on the (T, 1/mq) plane in Fig. 1 (a). This CP,as well as that in the light quark region, is believed tobelong to the Z(2) universality class, i.e. the universalityclass of the three-dimensional Ising model [8].

Near the CP of the three-dimensional Ising model, therelevant scaling parameters are the reduced temperaturet and external magnetic field h; extensive variables con-jugate to these parameters are the energy and the mag-netization, respectively. As shown in Fig. 1 (b), the CPis located at (t, h) = (0, 0) and the first-order transitionexists on the t axis for t < 0. The singular part of ther-modynamic quantities near the CP is described by thescaling function of t and h. According to the universal-ity, the singular part of thermodynamic quantities nearthe CP of heavy-quark QCD is described by the samescaling function, where the scaling parameters t and hare encoded into the (T, 1/mq) plane as schematicallyshown in Fig. 1 (a); the t axis is parallel to the first-order line at the CP while the direction of the h axis isnot constrained from the universality.

According to the FSS argument [32] the singular partof the dimension-less free energy F (t, h, L−1) around theCP obtained at a finite volume V = L3 has a scaling

F (t, h, L−1) = F (tbyt , hbyh , L−1b), (1)

for arbitrary scale factor b. The values of the exponentsyt and yh are specific for each universality class. In theZ(2) universality class these parameters are numericallyobtained as [32]

yt = 0.110, yh = −1.237. (2)

By setting b = L one has

F (t, h, L−1) = F (tLyt , hLyh , 1) ≡ F (tLyt , hLyh). (3)

Derivatives of F (t, h, L−1) with respect to t and h de-fine the cumulants of the corresponding extensive vari-ables. For example, cumulants of the magnetization Mare given by

〈M(t, h, L−1)n〉c = ∂nhF (t, h, L−1), (4)

with ∂h = ∂/∂h. From Eq. (3), L dependence of thecumulants 〈Mn〉c near the CP are written as

〈M(t, h, L−1)n〉c = ∂nhF (t, h, L−1)

= Lnyh∂nh F (tLyt , hLyh). (5)

In Ref. [34], it is suggested that the so-called (fourth-order) Binder cumulant

B4(t, h, L−1) =〈M(t, h, L−1)4〉c

(〈M(t, h, L−1)2〉c)2+ 3 (6)

plays a useful role to determine the location of the CPfrom numerical results obtained at finite L. When the

3

distribution of M obeys the Gauss distribution or the dis-tribution composed of two delta functions with an equalweight, we have

B4 =

3 Gauss distribution,

1 two delta functions,(7)

respectively. Since the distribution of M approachesthese functions in the L → ∞ limit on the crossoverand first-order lines at h = 0, respectively, B4 should ap-proach Eq. (7) on these lines in the L→∞ limit. More-over, from Eq. (4), B4 at h = 0 behaves as a function oft and L as

B4(t, 0, L−1) =∂4hF (t, 0, L−1)

(∂2hF (t, 0, L−1))2

+ 3

=∂4hF (tLyt , 0)

(∂2hF (tLyt , 0))2

+ 3

= b4 + ctL1/ν +O(t2), (8)

for small t, where b4 = ∂4hF (0, 0)/(∂2

hF (0, 0))2 + 3, ν =1/yt, and c is a constant. Eq. (8) shows that B4(t, 0, L−1)obtained for various L at h = 0 has a crossing at t = 0.The parameter b4 is given only from F (t, h) and thus arespecific for each universality class. For the Z(2) univer-sality class, the value is known to be [32]

b4 = 1.604. (9)

Equation (4) means that F (t, h, L−1) is the cumulantgenerating function of M up to an additive constant.Then, as shown in Eqs. (A2) and (A4) in Appendix A,this function is related to the probability distributionfunction pM (M ; t, h, L−1) of M as

eF (t,h′−h,L−1) = cF

∫dM eh

′MpM (M ; t, h, L−1), (10)

where cF is a constant determined from the normaliza-tion condition

∫dM pM (M) = 1. Here, let us define

another probability distribution pM (M ; t) as

eF (t,h′) = cF

∫dMeh

′mpM (M ; t). (11)

From Eq. (3), one finds

pM (M ; t, 0, L−1) = Lyh pM (ML−yh ; tLyt). (12)

When we consider magnetization per unit volume m =M/V , the probability distribution of m is given by

pm(m; t, 0, L−1) = Lyh−3pM (mL3−yh ; tLyt). (13)

At the CP, (t, h) = (0, 0), one finds from Eq. (13) that

pm(m; 0, 0, L−1) = Lyh−3pM (mL3−yh ; 0). (14)

Equation (13) also suggests that, when pM (M ; t) has a

local extremum at M = M(t), pm(m; t, 0, L−1) has cor-responding local extremum at

m = Lyh−3M(tLyt). (15)

This implies that the t and L dependences of the maxi-mum of pm(m; t, 0, L−1) are described by a single func-

tion M(t).

III. SETUP

A. Lattice action and basic observables

In this study we investigate the four-dimensional sys-tem described by the lattice action of QCD

S = Sg + Sq, (16)

with Sg and Sq being the gauge and quark actions. ForSg we employ the plaquette action

Sg = −6Nsite β P , (17)

with the gauge coupling parameter β = 6/g2 and thespace-time lattice volumeNsite = N3

s×Nt. The plaquette

operator P is given by

P =1

6NcNsite

∑x, µ<ν

Re trC

[Ux,µUx+µ,νU

†x+ν,µU

†x,ν

],

(18)

where Ux,µ is the link variable in the µ direction at sitex, x+ µ is the next site in the µ direction from x, Nc = 3,and trC is the trace over color indices.

For Sq, we adopt the Wilson quark action

Sq =

Nf∑f=1

∑x, y

ψ(f)x Mxy(κf )ψ(f)

y , (19)

with the Wilson quark kernel

Mxy(κf ) = δxy − κfBxy, (20)

Bxy =

4∑µ=1

[(1− γµ)Ux,µ δy,x+µ + (1 + γµ)U†y,µ δy,x−µ

],

(21)

where x, y represent lattice sites. The color and Dirac-spinor indices are suppressed for simplicity. κf is thehopping parameter for the fth flavor. The bare quarkmass mf is related to κf as

κf =1

2amf + 8, (22)

with the lattice spacing a. The matrix Bxy has nonzerovalues only when lattice sites x and y are located in an

4

adjacent sites. Therefore, this term represents the “hop-ping” of a quark between adjacent sites. The heavy quarklimit mf →∞ corresponds to κf → 0.

In the following, we consider degenerated Nf flavorswith a common hopping parameter κ = κf correspond-ing to a common quark mass mq — generalization tonon-degenerate cases is straightforward. In this case, theexpectation value of a gauge operator O(U) is calculatedas

〈O(U)〉 =1

Z

∫DUDψDψ O(U) e−Sg−Sq

=1

Z

∫DU O(U)[detM(κ)]Nf e−Sg

=1

Z

∫DU O(U) e−Sg+Nf ln detM(κ), (23)

with the partition function Z =∫DU e−Sg+Nf ln detM(κ).

In the heavy quark limit κ = 0 (mq = ∞), the de-confinement phase transition at nonzero temperature ischaracterized by the spontaneous symmetry breaking ofthe global Z(3) center symmetry of the SU(3) gaugesymmetry. The most conventional choice for the orderparameter of this phase transition is the Polyakov loop

Ω =1

NcN3s

∑~x

trC

[U~x,4U~x+4,4U~x+2·4,4 · · ·U~x+(Nt−1)·4,4

],

(24)

where the summation∑~x is over the spatial lattice sites

on one time slice. In the heavy quark limit, 〈Ω〉 = 0 be-

low the critical temperature Tc, while 〈Ω〉 takes a nonzerovalue at T > Tc. For finite mq, the Z(3) symmetry is

explicitly broken by the quark term, and thus 〈Ω〉 be-comes non vanishing for all T . Even in this case, whenmq is sufficiently large, 〈Ω〉 jumps discontinuously at thefirst-order transition and thus can be used to detect thefirst-order transition line and its CP [27, 29, 30]. In thepresent study, we focus on the real part of the Polyakovloop

ΩR = Re Ω, (25)

and study its probability distribution function [27, 29]

p(ΩR) = 〈δ(ΩR − ΩR)〉. (26)

B. Hopping parameter expansion

To calculate Eq. (23) around the heavy-quark limit,in the present study we adopt the hopping parameterexpansion (HPE) for ln detM(κ):

ln

[detM(κ)

detM(0)

]= −

∞∑n=1

1

nTr [Bn]κn, (27)

where the matrix Bxy is defined by Eq. (21) and Tr isthe trace over all indices. In Eq. (27), the contribution

FIG. 2. Six-step Wilson loops; rectangle (a), chair-type (b)and crown-type (c).

FIG. 3. Bent Polyakov loops, Ω1 (left) and Ω2 (middle) forNt = 4.

at κ = 0 is subtracted for convenience. Since Bxy takesnonzero values only for adjacent lattice sites x and y, Bn

is graphically represented by trajectories of n links [35].Because of the trace in Eq. (27), non-vanishing contribu-tions are given by closed trajectories. The lowest-ordercontributions of Eq. (27) start from n = 4, and all contri-butions for odd n vanishes when Nt is even. By writing

− ln

[detM(κ)

detM(0)

]=SLO + SNLO +O(κ8), (28)

one finds for Nt = 4 that the LO contribution is given bythe plaquette and the Polyakov loop as

SLO = −2Nc

(48NsiteP + 32N3

s ΩR

)κ4. (29)

The NLO term SNLO consists of the six-step Wilson loopsand bent Polyakov loops as

SNLO =− 2Nc

(384 Wrec + 768 Wchair + 256 Wcrown

+ 192 Re Ω1 + 96 Re Ω2

)Nsiteκ

6. (30)

Here, Wrec, Wchair, and Wcrown represent the six-stepWilson loops of the rectangular, chair, and crown types,respectively, as illustrated in Fig. 2. Ωn are the bentPolyakov loops illustrated in Fig. 3, which run one stepin a spatial direction, n steps in the temporal directionand return to the original line. All the Wilson loopsand Polyakov-loop-type loops are normalized such thatWrec = Wchair = Wcrown = 1 and Ωn = 1 in the weakcoupling limit, Ux,µ = 1. Explicit definitions of these op-erators as well as the derivation of Eqs. (29) and (30) aregiven in Appendix B.

5

(a) Xµ(x) (b) Y0(x)

FIG. 4. Staple Xµ(x) and the operator Y0(x) in Eq. (36)corresponding to the link variable Uµ(x) shown by the blueline.

C. Numerical implementation with HPE

In this study, we generate the gauge configurationswith respect to the action at the LO in the HPE, i.e.

Sg+LO = Sg +NfSLO

= −2NcNsite

(β + 48Nfκ

4)P − 64NcNfN

3s κ

4ΩR

= −2NcNsiteβ∗P − λN3

s ΩR, (31)

with

β∗ = β + 48Nfκ4, λ = 64NcNfκ

4. (32)

We then perform the measurements at the NLO by incor-porating the effect of SNLO by the multipoint reweightingmethod [29, 33, 36]. In this subsection we discuss the nu-merical implementation of these analyses.

In the Monte Carlo simulations of pure gauge theory,thanks to the locality of the action Sg, it is possible toadopt the pseudo-heat-bath (PHB) and over-relaxation(OR) algorithms for updating gauge configurations. Fo-cusing on a link variable Uµ(x), the dependence of Sg onUµ(x) is given by

∆Sg(Uµ(x)) = − β

NcRe trC[Uµ(x)Xµ(x)], (33)

where the staple

Xµ(x) =∑ν 6=µ

∑s=±1

Ux+µ,sνU†x+sν,µU

†x,sν , (34)

with Ux,−µ = U†x−µ,µ for µ > 0, is graphically shown

in Fig. 4 (a). In the PHB and OR algorithms, the linkvariable Uµ(x) is updated according to the probabilitydetermined by Eq. (33). The fact that Eq. (33) is rep-resented only by local variables near Uµ(x) enables thisprocedure efficient especially on the memory-distributedparallel computing.

When the LO term, Eq. (29), is included into the ac-tion, the contribution of a temporal link U0(x) to Sg+LO

is modified as

∆Sg+LO(Uµ(x))

= − β∗

NcRe trC

[U0(x)

(X0(x) +

λ

β∗Y0(x)

)](35)

with

Y0(x) = U0(x+ 0)U0(x+ 2 · 0)U0(x+ 3 · 0), (36)

which is schematically shown in Fig. 4 (b). For Nt > 4,Y0(x) is given by the product of Nt−1 link variables alongthe temporal direction. The contribution of a spatial linkto Sg+LO is unchanged from Eq. (33).

These results on ∆Sg+LO(Uµ(x)) suggest that theMonte Carlo updates of Uµ(x) can be performed by thePHB and OR efficiently even for Sg+LO, provided thatthe temporal direction is not separated into different par-allel nodes and Y0(x) can be calculated efficiently. Sat-isfying this condition is not difficult to attain for large-volume simulations. By taking this advantage, in thisstudy we perform update of gauge fields at the LO bycombining PHB and OR1. Compared with the pure-gauge simulation, the increase of the numerical cost todeal with Sg+LO in this method is small since the addi-tional multiplications of SU(3) matrices required for anupdate are only Nt − 2 times for the temporal links andthe cost to update the spatial links is unchanged.

In the measurement of observables, we include all thecontribution of SNLO using the multipoint reweightingmethod2. The expectation value of a gauge observableO(U) at the NLO at the parameter set (β, κ) is given

from the LO simulations at (β, κ) as

〈O(U)〉NLOβ,κ

=

∫DU O(U) e−Sg+LO(β,κ)−SNLO(β,κ)∫DU e−Sg+LO(β,κ)−SNLO(β,κ)

=

∫DU O(U) e−∆Sg+LO−SNLO(β,κ)e−Sg+LO(β,κ)∫DU e−∆Sg+LO−SNLO(β,κ)e−Sg+LO(β,κ)

=〈O(U) e−∆Sg+LO−SNLO(β,κ)〉LO

β,κ

〈e−∆Sg+LO−SNLO(β,κ)〉LOβ,κ

, (37)

with

∆Sg+LO =Sg+LO(β, κ)− Sg+LO(β, κ)

=− 2NcNsite

(β − β +Nf(κ

4 − κ4))P

+ 64NcNfN3s (κ4 − κ4)ΩR, (38)

and 〈·〉LOβ,κ is the expectation value taken with the action

Sg+LO(β, κ). We generate gauge configurations for the

LO action for several values of (β, κ), and evaluate the ex-pectation values at the NLO by the multipoint reweight-ing method.

1 After finishing our numerical analyses, we knew that a similaridea is suggested in Ref. [37]. We thank F. Karsch for notifyingthis literature.

2 In Ref. [30], the effect of SNLO is included in part effectively bythe effective NLO method. In this study, we deal with it exactly.

6

D. Simulation parameters

TABLE I. Simulation parameters: lattice size N3s × Nt, β∗,

and λ. The value of κ corresponding to each λ is also listedfor the case Nf = 2.

lattice size β∗ λ κNf=2

483 × 4 5.6869 0.004 0.0568

5.6861 0.005 0.0601

5.6849 0.006 0.0629

403 × 4, 363 × 4 5.6885 0.003 0.0529

5.6869 0.004 0.0568

5.6861 0.005 0.0601

5.6849 0.006 0.0629

5.6837 0.007 0.0653

323 × 4 5.6885 0.003 0.0529

5.6865 0.004 0.0568

5.6861 0.005 0.0601

5.6845 0.006 0.0629

5.6837 0.007 0.0653

243 × 4 5.6870 0.0038 0.0561

5.6820 0.0077 0.0669

5.6780 0.0115 0.0740

In this study, we perform Monte Carlo simulations withfixed temporal lattice size Nt = 4, while the spatial ex-tent Ns is changed from 24 to 48. This allows us toperform simulations with large aspect ratio Ns/Nt = LTup to 12, where L is the lattice size along the spatialdirection in physical units. For each Ns, the gauge con-figurations are generated for 3 to 5 sets of (β∗, λ) shownin Table. I, which are chosen so that β∗ is close to thetransition line at the LO.

The gauge configurations are updated with the LO ac-tion Sg+LO using the PHB and OR algorithms as dis-cussed in Sec. III C. Gauge configurations are updatedby five OR steps after each PHB step. We measure ob-servables every two sets of the PHB+OR updates, i.e.totally ten OR steps and two PHB steps. For all param-eters we have performed 6 × 105 measurements in thisway. In the following, we set Nf = 2 to show the numer-ical results unless otherwise stated.

In Monte Carlo simulations near a first-order transi-tion, because the transition between the coexisting twophases becomes rare when the spatial volume of the sys-tem is large, observables averaged over the two phasestend to have quite long autocorrelations. In Fig. 5 weshow the Monte Carlo time history of ΩR for Ns/Nt =LT = 10 and 12 at the smallest λ, at which the auto-correlation is the longest. The horizontal axis representsthe Monte Carlo time in the unit of measurements. Thefigure shows that flip-flops between the two phases oc-cur several times in 10, 000 measurements. The autocor-relation lengths estimated from the correlation functionare about 1, 900 and 800 measurements for these time

0 2 4 6 8Monte Carlo history ×10−4

0.000.020.040.060.080.100.12

ΩR

LT= 10, λ= 0.003

0 2 4 6 8Monte Carlo history ×10−4

0.000.020.040.060.080.100.12

ΩR

LT= 12, λ= 0.004

FIG. 5. Examples of the Monte Carlo history of ΩR. Hor-izontal axis represents the Monte Carlo time in the unit ofmeasurements which are made every two sets of PHB+ORupdates.

histories, respectively. Throughout this study, we esti-mate the statistical errors of observables by the jackknifemethod unless otherwise stated, adopting the binsize of10, 000 measurements which is sufficiently larger than theestimated autocorrelation lengths. We checked that thestatistical errors thus estimated are roughly stable withina variation of the binsize from 5, 000 to 30, 000 measure-ments.

E. Overlapping problem

Our main objective to use the LO action for configu-ration generation is to avoid the overlapping problem inthe reweighting. In Fig. 6 we show the contour plot forthe probability distribution function of P and ΩR,

p(P,ΩR) =⟨δ(P − P )δ(ΩR − ΩR)

⟩, (39)

obtained at λ = 0.003 (red) and 0.007 (blue) on 403 × 4lattices. The solid lines are the contours for the dis-tribution measured on the LO configurations generatedwith Sg+LO. Each contour curve is drawn such thatthe probability inside the contour is 0.9, 0.7, · · · , and0.1. To smoothen the plot, we approximate the deltafunction in Eq. (39) by the Gauss function, δ(x) '

7

0.546 0.547 0.548 0.549 0.550 0.551P

0.00

0.02

0.04

0.06

0.08

0.10

0.12Ω

R

0.9

0.7

0.5

0.3

0.10.9

0.7

0.7

0.5

0.5

0.3

0.3

0.1

LT= 10

λ= 0.007

NLO

λ= 0.003

NLO

FIG. 6. Probability distribution p(P,ΩR) of the gauge con-figurations on the 403 × 4 lattice for the simulation pointsat λ = 0.003 and 0.007. Solid lines represent the contours ofp(P,ΩR) obtained with the LO action. The label on each con-tour shows the probability inside the contour. Dotted linesrepresents the contour lines of p(P,ΩR) with the NLO actionat the same parameters obtained by reweighting the LO data.

exp[−(x/∆)2]/(∆√π), with the width along ΩR and P

directions ∆ΩR= 0.002 and ∆P = 0.0001, respectively.

In Fig. 6, we also show by the dotted lines the contoursof p(P,ΩR) at the NLO, which is obtained by reweight-ing the LO data at the same (β∗, λ). The meaning of thecontours is the same as the solid lines. We find that thedeviation of the NLO distribution from the original oneat the LO is not significant, suggesting that the effectsof the NLO contribution are not large. The large overlapof the LO and NLO distributions ensures that, for ob-servables constructed from P and ΩR, the NLO resultsobtained by reweighting the LO data at the same β∗ andλ are statistically reliable. In the analysis of the CP, theoverlapping of the distributions is even more improvedafter adjusting the parameters to the transition line.

On the other hand, from this figure, we find that theoverlapping of the distributions at λ = 0.003 and 0.007 isquite poor — the regions with probability larger than 0.5are not overlapping at all with each other. This meansthat, if we were to calculate observables at λ = 0.007 byreweighting data obtained at λ = 0.003, or vice versa,the statistical quality of the results would be quite low.In Refs. [27, 29, 30], the CP in heavy quark region wasinvestigated on lattices with Ns/Nt = 4–6, by reweight-ing from pure gauge configurations, i.e. those obtained atλ = 0. Because the overlapping problem becomes quicklysevere as the system volume becomes large, the samestrategy is not applicable to the present study in whichmuch larger system volumes up to Ns/Nt = LT = 12are simulated. Figure 6 shows that, to incorporate theNLO effects by reweighting, the use of the LO action forconfiguration generation is sufficiently effective in sup-pressing the overlapping problem. The smallness of the

NLO effects in Fig. 6 further suggests that the effectsof dynamical quarks are dominated by the LO term forthese values of λ.

IV. BINDER CUMULANT ANALYSIS

A. Transition line

We first determine the location of the transition linethat corresponds to h = 0 in terms of the Ising parame-ters; see Fig. 1. In the coupling parameter space (β∗, λ),we denote the transition line as β∗tr(λ). In this study,we determine β∗tr at each λ adopting the following threeconventional choices:

• Maximum of 〈Ω2R〉c

• Zero point of 〈Ω3R〉c

• Minimum of BΩ4 = 〈Ω4

R〉c/〈Ω2R〉2c + 3

In Fig. 7, we show the LT dependence of β∗tr deter-mined by these definitions for several values of λ. Thefigure shows that the maximum of 〈Ω2

R〉c has a visibleLT dependence. On the other hand, the zero point of〈Ω3

R〉c and the minimum of BΩ4 do not have statistically-

significant LT dependence for LT ≥ 8. This result showsthat the zero point of 〈Ω3

R〉c and minimum of BΩ4 are suf-

ficiently close to the β∗tr in the L→∞ limit in this rangeof LT . In the following, we employ the minimum of BΩ

4

for the definition of the transition line β∗tr for each Nt.In Fig. 8, we show the transition line on the (β∗, λ) and(β, λ) planes obtained at LT = 10.

In Fig. 9, we show the distribution function of ΩR,Eq. (26), on the transition line for several values ofλ, where the delta function in Eq. (26) is smeared bythe Gaussian approximation as before with the width∆ΩR

= 0.002. The shaded bands represent the statisticalerrors. At λ = 0.003, we see a clear two-peak structurein p(ΩR) and find that the peaks become sharper as LTbecomes larger. This behavior suggests the first-orderphase transition at λ = 0.003. At λ = 0.007, on theother hand, while two peaks are observed for LT ≤ 9,the two peaks cease to exist as LT becomes large. Thissuggests the crossover transition in the L → ∞ limit atthis λ.

B. Binder cumulant

Next, let us determine the position of the CP on the(β, κ) plane. As discussed in Sec. II, it is convenient toemploy the Binder cumulant B4 of ΩR

BΩ4 =

〈Ω4R〉c

〈Ω2R〉2c

+ 3. (40)

This quantity approaches the known values given inEq. (7) in the L→∞ limit depending on the order of the

8

0.000 0.001 0.002(LT )−3

5.68845

5.68850

5.68855β∗ tr

λ= 0.003

BΩ4 min.⟨Ω3

R

⟩c zero⟨

Ω2R

⟩c max.

0.000 0.001 0.002(LT )−3

5.68580

5.68585

5.68590

β∗ tr

λ= 0.005

BΩ4 min.⟨Ω3

R

⟩c zero⟨

Ω2R

⟩c max.

0.000 0.001 0.002(LT )−3

5.68320

5.68325

5.68330

5.68335

β∗ tr

λ= 0.007

BΩ4 min.⟨Ω3

R

⟩c zero⟨

Ω2R

⟩c max.

FIG. 7. Transition line β∗tr(λ) as function of the aspect ratio LT = Ns/Nt, determined through three definitions: (1) minimum

of BΩ4 , (2) zero of 〈Ω3

R〉c, and (3) maximum of 〈Ω2R〉c. The results for λ = 0.003, 0.005, 0.007 are shown from left to right.

0.003 0.004 0.005 0.006 0.007λ

5.682

5.684

5.686

5.688

βtr,β∗ tr

λc

NLO, Nf = 2

β ∗tr

βtr

FIG. 8. Transition line β∗tr on the (β, λ) and (β∗, λ) plane

obtained at LT = 10. The points with the error bar arethe results obtained on each measurement. The shaded ar-eas show the result obtained by the multipoint reweightingmethod with the error band representing the statistical error.The dotted vertical line shows the value of λc determined bythe analysis in Sec. IV B.

transition. Furthermore, provided that ΩR correspondsto m = M/V of the Ising model, BΩ

4 should obey Eq. (8)near the CP.

In the upper panel of Fig. 10, we show BΩ4 along the

transition line as a function of λ for five values of LT .λ is varied continuously by the multipoint reweightingmethod. The lower panel is an enlargement of the upperpanel around the crossing point. The figure shows thatBΩ

4 has a crossing at λ = λc ' 0.005 and is an increasing(decreasing) function of LT for λ > λc (λ < λc). Theexistence of the CP at λ ' 0.005 is suggested from thisresult.

To determine λc and the critical exponent ν quanti-tatively, we fit the numerical results of BΩ

4 by a fittingfunction motivated by Eq. (8):

BΩ4 (λ, LT ) = b4 + c(λ− λc)(LT )1/ν , (41)

where b4, λc, ν, c are the fit parameters. In this study,

TABLE II. Results of the four parameter fits of BΩ4 with

Eq. (41). Three largest volumes with LT = Ns/Nt =12, 10, 9 are used for the fits. The left column shows thevalues of λ used for the fit.

fit λ (×104) b4 λc ν χ2/dof

49, 52 1.631(24) 0.00503(14) 0.614(49) 0.45

48, 53 1.630(24) 0.00503(14) 0.614(48) 0.46

45, 55 1.629(24) 0.00502(14) 0.622(48) 0.46

45, 50 1.630(24) 0.00503(15) 0.634(47) 0.48

50, 55 1.631(24) 0.00504(14) 0.610(51) 0.38

45, 50, 55 1.620(23) 0.00494(14) 0.626(41) 7.9

we can vary λ continuously by the multipoint reweight-ing method. However, because data at different λ on thesame volume are correlated, it is not meaningful to usetoo many λ values. Using the data at three largest vol-umes, LT = 12, 10, 9, two or three λ values (6 or 9 datapoints, respectively) should be sufficient for the four pa-rameter fit of Eq. (41). We thus repeat the fit for severalchoices of λ values, taking the covariance between dataat different λ into account in the calculation of χ2.

In Table II, we summarize the results of the fit usingthe data at three largest volumes, LT = 12, 10, 9, andat λ values listed in the left column of the table. Thestatistical error in the table is estimated by the stan-dard chi-square analysis. The table shows that the valueof χ2/dof are smaller than unity in the fits with two λvalues, but χ2/dof is unacceptably large with three λvalues, while all the results for the fitting parameters arewell consistent within errors. We choose the fit resultfor λ = (0.0048, 0.0053) depicted by bold characters inTable II as the central value and include the uncertaintyin the fits with two λ values as the systematic error.

We repeat the analysis also with other sets of systemvolumes. The results of the fits with four and five largestvolumes, together with the fit with three largest volumes,are summarized in Table III. For the fit with four andfive largest volumes, LT = 12–8 and 12–6, we now have8 and 10 data points for the four parameter fit with twoλ values. For these fits, we choose λ = (0.0048, 0.0053)

9

0.00 0.05 0.10 0.15ΩR

0

5

10

15

20p(Ω

R)

λ= 0.0030

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

0.00 0.05 0.10 0.15ΩR

0

5

10

15

p(Ω

R)

λ= 0.0050

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

0.00 0.05 0.10 0.15ΩR

0

5

10

15

p(Ω

R)

λ= 0.0070

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

FIG. 9. Distribution of ΩR at λ = 0.003, 0.005 and 0.007.

0.003 0.004 0.005 0.006 0.007λ

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

BΩ 4

NLO, Nf = 2

LT= 6

LT= 8

LT= 9

LT= 10

LT= 12

0.0048 0.0050 0.0052λ

1.58

1.60

1.62

1.64

1.66

1.68

BΩ 4

NLO, Nf = 2

LT= 12, 10, 9

LT= 12− 8

LT= 12− 6

LT= 6

LT= 8

LT= 9

LT= 10

LT= 12

FIG. 10. Binder cumulant BΩ4 as a function of λ obtained

at five LT = Ns/Nt. The statistical errors are shown by theshaded area. The lower panel is an enlargement of the upperpanel around the crossing point, where the dotted rectanglein the upper panel represents the region of the lower panel.The points in the lower panel with error bars show the resultsof the four parameter fit with Eq. (41). See text for details.

TABLE III. Results of the four parameter fit of BΩ4 using

the data points at three and four largest and all volumes.The first parentheses are for statistical errors estimated bythe jackknife method, and the second parentheses are for sys-tematic errors due to the choice of λ values for the fit. SeeSec. IV B for details.

LT = Ns/Nt b4 λc ν χ2/dof

12, 10, 9 1.630(24)(2) 0.00503(14)(2) 0.614(48)(3) 0.46

12, 10, 9, 8 1.643(15)(2) 0.00510(10)(2) 0.614(29)(3) 0.37

12, 10, 9, 8, 6 1.645(11)(2) 0.00511(8)(2) 0.593(18)(3) 0.67

as the central value again.The results of b4 and λc obtained by these fits are

shown in the lower panel of Fig. 10. The thick sym-bols show the central value, and thin symbols with lightcolors show the results obtained by the variation of λin Table II. The results with the three largest volumesare shown by black triangles, while those with four andfive largest volumes are shown by blue squares and greenpentagons, respectively. In the figure, b4 expected fromthe Z(2) universality, Eq. (9), is shown by the dashedhorizontal line.

From Table III and Fig. 10, we find that, when weadopt the fit with the three largest volumes, LT ≥ 9, thefit result of b4 is consistent with the Z(2) value withinabout 1σ. On the other hand, when we include smallervolumes, LT ≥ 8 or LT ≥ 6, b4 from the fits show sta-tistically significant deviation from the Z(2) value. InTable III, we also summarize the results for the criticalexponent ν. From the Z(2) universality class, we expectν = 1/yt ≈ 0.630. We find that the result of the fitwith LT ≥ 9 is consistent with the Z(2) value within theerror, while the result of the fit with LT ≥ 6 has a sig-nificant deviation from the Z(2) value, though the valuesof χ2/dof are all smaller than unity.

We thus conclude that the FSS in the Z(2) universal-ity class is confirmed when the system volume is largeenough, LT ≥ 9 — lattices with LT ≤ 8 are not largeenough to apply a FSS analysis for BΩ

4 . The value of λc

thus determined is also shown in Fig. 8.In Appendix C, we perform the analysis of BΩ

4 at the

10

TABLE IV. Location of the critical point (βc, κc) for variousNf . For λc, the first parentheses are for statistical errors andthe second parentheses are for systematic errors from the fitas discussed in Sec. IV B. The errors for βc and κc include thesystematic errors.

Nf βc κc λc

1 5.68446(22) 0.0714(5) 0.00498(14)(2)

2 5.68453(22) 0.0602(4) 0.00503(14)(2)

3 5.68456(21) 0.0544(4) 0.00505(14)(2)

LO of the HPE and compare the results with those atthe NLO discussed in this Section. We find that the LOresult for λc is about 2.6% larger than the NLO value.This small difference suggests that the truncation errorof the HPE is well under control at the NLO around λc.

C. Mixing with energy-like observable

So far, we have performed the analyses of BΩ4 assuming

that ΩR corresponds to the magnetization m = M/V ofthe Ising model. Although our numerical results thus farare in good agreement with this assumption, a possiblemixing with the energy-like observable [14, 22] in ΩR isnot excluded in general. In this case, the behavior of BΩ

4

near the CP is modified from Eq. (8) as [22]

BΩ4 (λ, LT ) =

(b4 + c(λ− λc)(LT )1/ν

)(1 + d(LT )yt−yh

).

(42)

To investigate the effect of this possible mixing, we tryfits of BΩ

4 based on Eq. (42). We use the values of BΩ4 at

three λ for the fits to increase the number of data points.We find that the six parameter fits with Eq. (42) withthe fitting parameters b4, λc, ν, c, d, yt − yh are quiteunstable, suggesting that χ2 has many local minima. Themodel space of Eq. (42) would be too large against thedata. As a next trial, we perform five parameter fits withEq. (42) by fixing yt− yh = −0.894. In this case, we findthat χ2 still has many local minima, and χ2/dof becomeslarger compared with the four parameter fit. It is alsofound that the value of d is consistent with zero withinthe error for all trials with the variation of λ values. Thissuggests that the mixing of the energy-like observable inΩR is negligible around the CP in the heavy-quark region.

D. Nf dependence

In Table IV, we summarize our final results for thelocation of the CP, (βc, κc). In the table, we also show theresults forNf = 1 and 3. We note that theNf dependenceof the HPE is trivial at the LO in the sense that Nf

enters the action Eq. (31) at this order only through thecombination λ = 64NcNfκ

4 after the replacement β →β∗. Therefore, λc does not depend on Nf . At the LO, this

allows us to obtain the value of κc for various Nf from thevalue of κc at Nf = 2 [29]. Because such a simple scalingis no longer applicable at the NLO, we made individualnumerical analyses at Nf = 1 and 3. From Table IV, wefind that the results of λc are almost insensitive to Nf .This means that the NLO effects on λc are small.

V. DISTRIBUTION FUNCTION OF ΩR

In this section, we study the scaling behavior of the dis-tribution function p(ΩR) to further investigate the con-sistency with the Z(2) universality class around the CP.

A. Scaling of distribution function

Let us first focus on the LT dependence of p(ΩR) atthe CP. In the following, instead of p(ΩR) itself, we studythe effective potential defined from p(ΩR):

V (ΩR;λ, LT ) = − ln p(ΩR)λ,LT , (43)

as this quantity is more convenient in comparing the re-sults at different LT [27, 29]. From Eq. (14), the LTdependence of V (ΩR, λ, LT ) at the CP will be described

by a single function V (x) as

V (ΩR;λc, LT ) = V((

ΩR − 〈ΩR〉)(LT )3−yh

), (44)

up to an additive constant, where 〈ΩR〉 is subtracted fromΩR to adjust the center of the distribution.

To see if the scaling behavior of Eq. (44) is satisfied,we show in Fig. 11 the effective potential V (ΩR;λc, LT )at the CP obtained at five values of LT , as a func-tion of

(ΩR − 〈ΩR〉

)(LT )3−yh , where we set 3 − yh =

0.504. For the figure, we adjust the arbitrary con-stant term of V (ΩR;λc, LT ) such that V (Ω(1);λc, LT ) +V (Ω(2);λc, LT ) = 0, where Ω(1) and Ω(2) (> Ω(1)) are thevalues of ΩR at the two local minima of V (ΩR;λc, LT ).No further adjustments are made in the figure. The errorbands do not include the uncertainty of the additive con-stant. The lower panel is an enlargement of the region in-dicated by the dotted rectangle in the upper panel. FromFig. 11, we find that the numerical results for LT = 8–12 agree almost completely within the errors with thescaling relation Eq. (44). This result nicely supports theFSS in the Z(2) universality class at the CP. From theupper panel of Fig. 11, we note that the effective poten-tial for LT = 6 shows a clear deviation from the resultsfor larger volumes at ΩR Ω(1) and ΩR Ω(2), whileit agrees well with them in the range Ω(1) . ΩR . Ω(2).This suggests that the deviation from the Z(2) FSS bylattices with small LT , discussed in Sec. IV B, is due tothat in the tails of the distribution p(ΩR) for small LT .

11

0.2 0.1 0.0 0.1 0.2(ΩR −

⟨ΩR

⟩)/(LT )yt − 3

0

1

2

3

4

V(Ω

R)

λ= λc

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

0.15 0.10 0.05 0.00 0.05 0.10 0.15(ΩR −

⟨ΩR

⟩)/(LT )yt − 3

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

V(Ω

R)

λ= λc

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

FIG. 11. Effective potential V (ΩR) = − ln p(ΩR). Bottompanel is an enlargement of the region enclosed by a dottedrectangle in the top panel.

B. Gap between the two minima

Using Eq. (13), the argument of Sec. V A on the effec-tive potential can be extended away from the CP alongthe transition line. In this subsection, we study the gapbetween the two local minima of V (ΩR;λc, LT ),

∆Ω = Ω(2) − Ω(1). (45)

According to Eq. (15), this quantity should behavearound the CP as

∆Ω(λ, LT ) = (LT )yh−3∆Ω((λ− λc)(LT )1/ν

)(46)

provided that p(ΩR) obeys the FSS.In Fig. 12, we show the λ dependence of Ω(1) and Ω(2).

As seen from Fig. 9, a clear two peak structure of p(ΩR)disappears when λ exceeds some value depending on LT .

0.003 0.004 0.005 0.006 0.007 0.008λ

0.00

0.02

0.04

0.06

0.08

0.10

peak

pos

ition

s Ω

(1) ,

Ω(2

)

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

FIG. 12. Positions of peaks of the distribution functionp(ΩR) measured on the transition line.

0.10 0.05 0.00 0.05(λ− λc)(LT )1/ν

0.15

0.20

0.25

0.30

∆Ω

(LT

c)3−y t

LT= 12

LT= 10

LT= 9

LT= 8

LT= 6

FIG. 13. Scaling of the gap ∆Ω around Tc.

Even before the disappearance of the two peaks, identi-fication of local maxima of p(ΩR) becomes unstable bystatistical fluctuations. In Fig. 12, we thus truncate theplots for Ω(1) and Ω(2) at finite λ. The shaded areasin the figure represent statistical errors estimated by thejackknife method, for which we repeat the analysis ofΩ(1,2) for p(ΩR) obtained in each jackknife sample withthe smearing width of ∆ΩR

= 0.002. 3

From Fig. 12 we extract ∆Ω as a function of λ. In

3 We see that the errors in Fig. 12 become occasionally large. Wefind that this is due to statistical oscillations in the shape ofp(ΩR) around the peak: Though the oscillations are within thestatistical errors, the peak position in each jackknife bin canjump discontinuously when oscillation appears just at the peakposition as we vary λ. This makes the resulting jackknife errorlarge there. From this observation, we think that these largeerrors are overestimated.

12

0.1 0.2 0.3 0.4(LT )3− yt

0.02

0.04

0.06

0.08

0.10∆

Ωλ= 0.0025

λ= 0.0030

λ= 0.0035

λ= 0.0040

λ= 0.0045

λ= 0.0050

λ= 0.0055

λ= 0.0060

λ= 0.0065

FIG. 14. Gap ∆Ω around Tc as a function of (LT )3−yh . TheZ(2) value 3− yh = 0.519 is assumed.

Fig. 13, we show ∆Ω for five different volumes. To seethe FSS, the vertical and horizontal axes are adjustedaccording to Eq. (46), where the Z(2) values 3 − yh =0.519 and ν = 0.630, and λc = 0.00503 determined in theprevious section are used. The figure shows that, for awide range of λ−λc and LT , the results of ∆Ω obtainedon different volumes are on top of each other within theerrors. This supports the FSS of p(ΩR) around the peakpositions over a wide range of LT and λ.

It is interesting to note that the scaling behavior of ∆Ωis observed even at LT = 6, although the FSS of BΩ

4 isviolated already at LT = 8. As discussed in the previoussubsection, we may understand this when the violation ofthe FSS for BΩ

4 is due to the violation in the tails of thedistribution function p(ΩR). As seen in Fig. 11, V (ΩR) atvarious volumes agrees well for Ω(1) . Ω . Ω(2) even forsmall values of LT . As the higher-order cumulants aresensitive to the whole structure of the distribution, BΩ

4

will be more sensitive to the violation of FSS at the tailsof the distribution. On the other hand, ∆Ω is insensitiveto them by definition. We also note that the statisticalerror of ∆Ω is naturally small because it is defined bythe peaks of the distribution. Therefore, ∆Ω is useful inseeing the FSS around the CP.

From Eq. (46), ∆Ω should behave linearly as a functionof (LT )3−yh at the CP λ = λc. In Fig. 14, we show ∆Ωon the transition line at various values of λ, as a functionof (LT )3−yh . In the same figure, the dashed lines showlinear functions ∆Ω = k(LT )3−yh for various values ofk. Figure 14 suggests that the linear behavior is realizedat λ ' 0.005, which is consistent with our estimationλc = 0.00503(14)(2) from the analysis of BΩ

4 .

C. Discussions

Let us comment on the relation of the present resultswith those given in Refs. [27, 29, 30]. In these studies, theCP is defined as the point at which the two peak structureof p(ΩR) disappears. On finite lattice with finite LT ,this leads to λ which is larger than our value of λc inthe L → ∞ limit. In fact, in Refs. [27], the location ofthe CP is estimated as κc = 0.0658(3)(+4

−11) for Nf = 2and Nt = 4, which is about 10% larger than that givenin Table IV. Values of λc (κc) which are smaller thanRef. [30] for each Nt are also reported by a recent study ofthe CP in the heavy quark region on fine lattices (Nt = 6–10) using the Binder cumulant method [31]. Though thedifference may be removed in the L→∞ limit, a carefulextrapolation will be required. Because the FSS is clearlyidentified in this study, we think that the extrapolationto the large volume limit is stably performed with thepresent analysis.

We also note that the latent heat at the deconfinementtransition in the SU(3) Yang-Mills theory (κ = 0) hasbeen measured in Ref. [38] recently. It was found thatthe latent heat becomes larger with increasing the spatialvolume. This may be attributed to a remnant of the FSSaround the Z(2) CP like ∆Ω studied in the present study.

VI. CONCLUSIONS

In this paper, we studied the distribution function ofthe Polyakov loop and its cumulants around the CP inthe heavy quark region of QCD. Large volume simula-tions up to the aspect ratio Ns/Nt = LT = 12 have beencarried out to see the finite-size scaling, while the latticespacing is fixed to Nt = 4. We have performed the mea-surement of observables using the hopping parameter ex-pansion for the quark determinant; the measurement hasbeen performed at the next-to-leading order of the hop-ping parameter expansion by the multipoint reweightingmethod evaluated on the gauge configurations generatedfor the action at the leading order. We found that thisanalysis is quite effective in reducing statistical errorsby avoiding the overlapping problem of the reweightingmethod, while the numerical cost hardly changes fromthe pure Yang-Mills simulations. The convergence of thehopping parameter expansion at the next-to-leading or-der at the critical point is also verified by the comparisonwith the leading order result.

Using the data on p(ΩR) thus obtained, we have per-formed the Binder cumulant analysis for determining thelocation of the critical point and evaluating the criticalexponent. We found that the critical exponent ν andthe value of the Polyakov-loop Binder cumulant BΩ

4 atthe critical point is consistent with the Z(2) universalityclass when LT ≥ 9 data are used for the analysis. Onthe other hand, statistically-significant deviation fromthe Z(2) scaling is observed when the data at LT = 8is included, which suggests that this spatial volume is

13

not large enough to apply the finite-size scaling.The scaling behavior near the critical point is further

studied using the structure of the distribution function ofthe real part of the Polyakov loop, p(ΩR). We found thatthe structure of p(ΩR) for various LT obeys the finite-sizescaling well especially near the peaks of p(ΩR). We havealso proposed the use of the gap ∆Ω between the peaks ofp(ΩR) for the finite-size scaling analysis. We showed thatthe λ and LT dependence of ∆Ω is in good agreementwith the Z(2) scaling over a wide range of λ and LT . Onthe other hand, the deviation of p(ΩR) around the tailsof the distribution is observed on small lattices, whichwould give rise to the violation of the finite-size scalingof BΩ

4 in small volumes.

AcknowledgmentsThe authors thank Frithjof Karsch, Makoto Kikuchi,

Yoshifumi Nakamura, and the members of WHOT-QCD Collaboration for useful discussions. Thiswork was supported by in part JSPS KAKENHI(Grant Nos. JP17K05442, JP18K03607, JP19H05598,JP19K03819, JP19H05146, JP21K03550), the UchidaEnergy Science Promotion Foundation, the HPCI Sys-tem Research project (Project ID: hp170208, hp190036,hp200089, hp210039), and Joint Usage/Research Cen-ter for Interdisciplinary Large-scale Information Infras-tructures in Japan (JHPCN) (Project ID: jh190003,jh190035, jh190063, jh200049). This research used com-putational resources of OCTPUS of the large-scale com-putation program at the Cybermedia Center, Osaka Uni-versity, and ITO of the JHPCN Start-Up Projects at theResearch Institute for Information Technology, KyushuUniversity.

Appendix A: Cumulants

Let us consider a probability distribution function p(x)of a stochastic variable x. Since p(x) represents probabil-ity, it satisfies the normalization condition

∫dx p(x) = 1.

The mth-order moment 〈xm〉 of p(x) is defined by

〈xm〉 =

∫dxxmp(x). (A1)

Using the moment generating function

G(θ) =

∫dx exθp(x) = 〈exθ〉, (A2)

the moments are also given by

〈xm〉 = ∂mθ G(θ)|θ=0, (A3)

with ∂θ = ∂/∂θ.The cumulants are defined from the cumulant gener-

ating function

K(θ) = lnG(θ) = ln〈exθ〉 (A4)

FIG. 15. “Appendix” structure of trajectories.

as

〈xm〉c = ∂mθ K(θ)|θ=0. (A5)

From Eqs. (A3) and (A5), one easily finds that the cu-mulants are related to the moments as

〈x〉c = 〈x〉, (A6)

〈x2〉c = 〈x2〉 − 〈x〉2 = 〈δx2〉, (A7)

〈x3〉c = 〈δx3〉, (A8)

〈x4〉c = 〈δx4〉 − 3〈δx2〉2, (A9)

and etc. with δx = x − 〈x〉. The cumulants are usefulin representing properties of p(x) than the moments forvarious purposes. In particular, in statistical mechan-ics the cumulants of an extensive variable are extensivevariables; see, for example, Ref. [5].

Appendix B: Hopping parameter expansion

In this appendix we derive Eqs. (29) and (30).Throughout this Appendix we assume general value forNt.

As in Eq. (27), the HPE of ln[detM(κ)] is givenby Tr[Bn]. Since the matrix B has nonzero contribu-tions only between neighboring lattice sites, Tr[Bn] aregraphically represented by the closed trajectories with nlinks [35]. However, trajectories including “appendices”shown in Fig. 15 do not contribute to the HPE becausefor such a path the product of the matrix in Dirac spacevanishes at the tip of the appendix as (1− γµ)(1 + γµ) =0 [35]. With this exception, all possible closed trajecto-ries composed of n links contribute to the HPE at theorder κn. In the following, we calculate their contribu-tions by classifying the trajectories by the shape.

Let us start from the plaquette, i.e. 1 × 1 rectangle,which gives a lowest-order contribution to Eq. (27) at

the order κ4. The plaquette operator P in Eq. (18) is

defined in such a way that 〈P 〉 = 1 in the weak couplinglimit with Uµ(x) = 1. To satisfy this condition Eq. (18)has a coefficient 1/(NcMplaq) = 1/18, where Mplaq is thenumber of different plaquettes per lattice site; Mplaq =

4C2 = 6, which is the number of combinations of axes(µ, ν) at which the plaquettes are located.

The contribution from all plaquettes to Eq. (27) is cal-culated to be

−2NcMplaqDplaqNsiteκ4P . (B1)

14

Here, Dplaq is the coefficient from the trace in the Diracspace

Dplaq = trD[(1− γµ)(1− γν)(1 + γµ)(1 + γµ)] = −8,(B2)

where trD means the trace over the Dirac indices. Thefactor 2 in Eq. (B1) comes from two directions for eachtrajectory, which have to be distinguished in the HPE.

The factor 1/n in Eq. (27) is canceled by the numberof four starting points of a trajectory; this cancellationoccurs for all trajectories.

Next, let us consider the Wilson loops of length 6 with-out windings along the temporal direction. At this or-der there are three types of trajectories; 1× 2 rectangle,chair, crown, which are shown in Fig. 2. We define theoperators, Wrect, Wchair, Wcrown, corresponding to thesetrajectories as

Wrect =1

NcMrectNsite

∑x

∑µ6=ν

Re trC

[Ux,µUx+µ,µUx+2·µ,νU

†x+µ+ν,µU

†x+ν,νU

†x,ν

], (B3)

Wchair =1

NcMchairNsite

∑x

∑µ,ν<ρ,ν 6=µ 6=ρ

∑s,t=±1

Re trC

[Ux,sνUx+sν,µU

†x+µ,sνUx+µ,tρU

†x+tρ,µU

†x,tρ

], (B4)

Wcrown =1

NcMcrownNsite

∑x

∑µ<ν<ρ

∑s,t=±1

Re trC

[Ux,µUx+µ,sνUx+µ+sν,tρU

†x+sν+tρ,µU

†x+tρ,sνU

†x,tρ

], (B5)

with Ux,−µ = U†x−µ,µ and U†x,−µ = Ux−µ,µ for µ > 0.

Eqs. (B3)–(B5) are defined so that 〈Wrect〉 = 〈Wchair〉 =

〈Wcrown〉 = 1 in the weak coupling limit again; to satisfythis condition, the operators are divided by Mrect = 12,Mchair = 48, and Mcrown = 16, respectively, correspond-ing to the number of trajectories per lattice site.

The contribution of these trajectories to the HPE ofln detM(κ) is given by

−2Nc

∑s

MsDsNsiteκ6Ws, (B6)

with s =rect, chair, crown. Ds is the coefficient from thetrace in the Dirac space. For the 1× 2 rectangle we have

Drect = trD

[(1− γµ)(1− γµ)(1− γν)

× (1 + γµ)(1 + γµ)(1 + γν)]

= −32, (B7)

and similar manipulations lead to Dchair = Dcrown =−16.

Next we consider the Polyakov-loop type operatorshaving a winding along the temporal direction. Thelowest-order contribution among them is the Polyakovloop, i.e. the straight lines of length Nt. To calculateits contribution, one needs to pay a special attention tothe fact that there is only one independent Polyakov loopfor each spatial coordinate on one time slice, not for eachlattice site. Therefore, their contributions to the HPE isgiven by

2NcDpolN3s κ

NtΩR, (B8)

where the factor −1 is to be applied because of the anti-periodic boundary condition of the quark determinant.The real part of Ω has to be taken after multiplying thefactor 2 because two directions of a trajectory are inde-pendently taken into account. The factor from the Diractrace for the Polyakov loop is calculated to be

Dpol = trD

[(1− γ4)Nt

]= 2Nt+1. (B9)

Finally, we consider the contribution of the bentPolyakov loops shown in Fig. 3, whose explicit definitionsare given by

Ω1 =1

NcMbentNsite

∑x

3∑i=1

∑s=±1

trC

[Ux,siUx+si,4U

†x+4,si

Ux+4,4Ux+2·4,4 · · ·Ux+(Nt−1)·4,4

], (B10)

Ω2 =1

NcMbentNsite

∑x

3∑i=1

∑s=±1

trC

[Ux,siUx+si,4Ux+si+4,4U

†x+2·4,siUx+2·4,4 · · ·Ux+(Nt−1)·4,4

], (B11)

and so on. The factor Mbent = 6 is needed to make 〈Ωn〉 = 1 in the weak coupling limit. From the defini-

15

0.0045 0.0050 0.0055λ

1.45

1.50

1.55

1.60

1.65

1.70

1.75B

4

Nf = 2

LO

NLO

LT= 6

LT= 8

LT= 9

LT= 10

LT= 12

FIG. 16. Binder cumulant B4 as a function of λ calculatedat the LO. The results at the NLO are also plotted by thethin dashed lines. The square with errors shows the fit resultwith three largest volumes. The NLO result on the same fitresult is also shown by the thin circle.

tion of Ωn we have Ωn = ΩNt−n. Also, when Nt is even

ΩNt/2 counts each trajectory twice, and thus its contri-bution to the HPE should be divided by 2. Bearing thesefacts in mind, the contribution from Ωn to the HPE ofln detM(κ) is given by

2NcMbentDbentNsiteκNt+2

×(Nt/2−1∑

n=1

Re Ωn +1

2Re ΩNt/2

)(B12)

where the last term Re ΩNt/2 should be omitted for oddNt. The contribution of the Dirac trace is calculated tobe Dbent = 2Nt+1.

Accumulating these results gives Eqs. (29) and (30).

Appendix C: Comparison of LO and NLO results

In this appendix, to see the convergence of the HPEat the NLO, we repeat the analyses in Sec. IV B at theLO and compare its results with those at the NLO. InFig. 16, we show the Binder cumulant BΩ

4 obtained atthe LO along the transition line. In this figure, the NLOresults of Fig. 10 are also shown by the thin dashed lines.We find that, though the difference between the LO andthe NLO results grows as λ becomes larger, the deviationis within a few percent level around the crossing point λc.

In Fig. 16, we also show the result of the four parameterfit with Eq. (41) using data at the LO on LT = 12, 10and 9 lattices by the black square. The same result at theNLO is shown by gray circle for comparison. We find thatthe LO result of the fit is consistent with the NLO resultwithin statistical errors: The values of b4 = 1.630(24)and ν = 0.620(47) at the LO are hardly changed fromthe corresponding NLO values given in Table III. Thoughthe central value of λc = 0.00516(15) at the LO is about2.6% larger than the NLO value, it is consistent with theNLO result within errors, suggesting that the truncationerror of the HPE at the NLO is well suppressed in thesequantities.

The success of the BΩ4 fit together with the consistency

of the fit results with the Z(2) values suggests that theZ(2) scaling is realized also with the LO action when thesystem volume is sufficiently large. This is reasonablesince the scaling properties near the CP are insensitiveto detailed structures of the theory.

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