On the universality class of the deconfinement transition in lattice gauge theories at finite temperature

Volume 148B, number 6

PHYSICS LETTERS

6 December 1984

ON T H E UNIVERSALITY CLASS OF THE DECONFINEMENT TRANSITION IN L A T r I C E GAUGE THEORIES AT FINITE TEMPERATURE Stephan WANSLEBEN lnstitut fiir Theoretische Physik, Universiti~t zu K$ln, Ziilpicher Str. 77, 5000 Cologne 41, West Germany Received 3 August 1984

We study the (2 + 1)-dimensional Z(2) lattice gauge theory at a finite temperature by Monte Carlo simulation with system size up to three million variables. Our data indicate that the critical exponent fl of the Polyakov line correlation function is greater than 1/8, in contradiction to a recent conjecture based on renormalization group arguments.

1. Introduction. The canonical partition function of a D-dimensional gauge theory in lattice formulation is equivalent to the partition function of a spin model on a (D + 1)-dimensional lattice. The lattice direction (thermal or 0-direction) added to the space directions has finite extension N t = 1/T with periodic boundary conditions [1]. Consider a gauge group G. Let U., ~denote the element of G which is associated to the link between lattice point n and n + e~ and U~ ~, = U. ~- U , + ,

~. U ÷

. U+

exp (S)

=f[dU]exp[~Re(at~i

trU,,,oi

where the temperature T and the coupling constant g is incorporated in a s and a t through

T = (atlas),

2/g 2= (atas) 1/2.

L(x)

= t r I - I Ux+i,o,0 i=1

via

F=-Tlimln[(L(x)L+(x+rej))]

1/2.

(4)

(1)

the product around an elementary plaquette. Then, the partition function is given by z = f[dU]

The deconfinement transition is indicated by the behaviour of the free energy of an isolated heavy quark, which can be evaluated from the expectation value of the Polyakov line operator

(3)

Infinite F means that an isolated quark needs infinite energy to exist, whereas the existence of isolated quarks is possible when F is finite. Thus, L is one of the interesting observables in lattice gauge theories at finite temperatures. Moreover, ( L ) can be regarded as an order parameter which signals the spontaneous breaking of the (global) "center symmetry" [1,2]. The transformation corresponding to this symmetry is performed by multiplication of all variables on thermal links between two adjacent temperature slices with an element of the center of G. Obviously, the action S is invariant under this transformation while L is not. The critical behaviour at the deconfinement transition point has been discussed in several papers during the last years. Svetitsky and Yaffe [1] made a conjecture concerning the critical

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exponent /3 of {L). They argue that when integration over all variables except L is carried out an effective theory with short-ranged interactions will be obtained. Thus, applying renormalization group arguments they conclude that the remaining model should belong to the universality class of a spin model with short-ranged interactions on a lattice with a dimension reduced by one and the center of G as the global symmetry group. Particularly the critical exponent /3 should be equal to that of the spontaneous magnetization in these models. This conjecture is obviously true in the limit N t = 1. Furthermore, its correctness for arbitrary but finite N t is supported by renormalization group calculations for the (2 + 1)-dimensional U(1) lattice gauge model in the limit a t >> a s by Svetitsky and Yaffe [1], by calculations in the strong coupling limit within the hamiltonian formulation [3], and by mean-field calculations [4]. Monte Carlo simulations of (3 + 1)-dimensional Z(2) [5], SU(2) [6,7] and SU(3) [8] lattice gauge models at finite temperatures show that the conjecture correctly predicts the order of the deconfinement transition for these models. The critical exponent/3 for the case of second order transitions [SU(2) and Z(2)], however, has not yet been calculated because of insufficient statistics of the Monte Carlo data. For the SU(4) gauge model there has been published Monte Carlo data which indicate that the deconfinement transition is of first order in contradiction to the conjecture [9]. In this letter we report on Monte Carlo simulations of the simplest lattice gauge model which undergoes a deconfinement transition at finite temperatures, the (2 + 1)-dimensional Z(2) gauge model. We found that /3 is 0.20 + 0.04 which indicates that /3 is greater than 0.125 conjectured by Svetitsky and Yaffe.

2. (2 + 1)-dimensional Z(2) lattice gauge theory at finite temperature. For G = Z(2) eq. (1) is simplified to

Z = ~ exp(a t {o }

+ as

~

Y'~ spacel, plaq.

462

oooo

therm, plaq.

oooo)

with o = + 1.

(5)

6 December 1984

The models with values of a s at infinity and zero, respectively, are exactly solvable. At vanishing a s the model is dual to a two-dimensional Ising model with coupling constant K defined by the equation t a n h ( K ) = [ t a n h ( a t ) ] N,

(6)

The dual hamiltonian represents the effective theory for Polyakov line operators. Thus, according to the conjecture of Svetitsky and Yaffe the critical exponent /3 has the value 1/8. It is reasonable to assume that the critical properties for vanishing a smoothly continue into the region of small but nonzero a s. This would generalize the results of Svetitsky and Yaffe for the (2 + 1)dimensional U(1) lattice gauge model [1] to the Z(2) model. When a s is infinite all spacelike plaquettes are forced to be not frustrated. Thus, the variables on spacelike links can be gauged to unity. Since the model decouples into N t independent two-dimensional Ising models the singularities of the free energy are described by the critical exponents of the two-dimensional Ising model:

= m N',

and thus 13 = Nt/8.

(7)

This shows that due to the explicit dependence of on N t the critical exponent fl is not that of the two-dimensional Ising model although the transition is of the Ising type. One cannot expect the properties of the model evaluated while spacelike plaquettes are frozen to continue smoothly into the region of finite a s [2]. Thus, from the above considerations it is not clear whether or not the critical exponent fl for all finite values of a s and a t is equal to 1/8.

3. Monte Carlo simulation. We simulated the model defined above with periodic boundary conditions in all directions at constant N t and variable coupling constant g with a s = a t = a. It can easily be shown by application of GKS inequalities [10,11] that eq. (7) gives an upper

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PHYSICS LETTERS

bound for the spontaneous "magnetization" ( L ) . Thus, when a < oo ( L ) decreases exponentially with increasing N t. The spacelike size of the lattice, N S × N~, must be sufficiently large to reach the scaling region of ( L ) without perturbation by finite size effects when approaching to the transition point. Since in a finite system ( L ) is always zero a calculation of the spontaneous " m a g netized" ( L ) within the ordered phase is possible only when the system is in a metastable state with fixed sign of ( L ) . The time of simulation, therefore, must be smaller than the time between succeeding changes of sign. Thus, the available simulation time increases with increasing lattice size but decreases when the distance to the transition point decreases. On the other hand, the time must be large enough to ensure equilibrium. Thus, system sizes have to be very large. Considering these points we chose N t = 4 and the spacelike extension as large as can be simulated in reasonable computer time. In order to save computer time we used a multispin coding algorithm on the vector computer C D C Cyber 205 which reaches a speed of 11 million updates per second [12]. So we could simulate a lattice, N~ x N S × Nt, with N~ = 512, i.e. three million variables, in runs with 30000 Monte Carlo steps per spin. In addition, we simulated a system with Ns = 256 with 20000 M o n t e Carlo steps per spin in order to have a check for finite size effects. We calculated L every 10th Monte Carlo step per variable (iteration) and

6 December 1984

stored all these values. For the final calculation of the averages we ignore the first n 0 iterations, n o was estimated by comparing all values with the average L of the last 1000 iterations. The first time when a value lies within a range around L defined by the fluctuation of the last 1000 values the n u m b e r of the corresponding iteration is denoted as no; typically n o = 5000 for the points nearest to the transition point. Fig. 1 shows the results for both values of N s. In order to find the critical exponent fl we performed a least squares fit of (L)=bt(A-Ac)•[I+b2(A-Ac) A = tanh ( a ) , to the data points including a term for corrections to scaling. The full line represents the result which was obtained, assuming A = 1 and using the data of the large system, with A c = 0.6228 and B = 0.20 + 0.04. This result is confirmed by a plot (fig. 2) of ( L ) N f l versus N s ( A - A c ) on logarithmic scales for both values of N~ (the critical exponent u is assumed to be unity). According to the finite size scaling theory [13] our two different systems should follow the same scaling function. Moreover, this function should follow a power law behaviour with exponent /3 in the asymptotic region as it does in fig. 2. In order to check whether fl = 0.125 lies within the range of statisti200

1.0
l

I

I

I

e

~

1

I

I

I

I

g÷

I

Iit1:

1

I Ill

/ ,.,' ,'-I,++" ,÷

• Ns =512 + Ns =256

I

÷/slope

0.2

V"

0.2 0

I

,¢+

~'1 04

IIIIIII

• Ns =512 + Ns = 256 A c = 0.6228

20

0.6

I Illlll

.N °.2

e~

0.8

a+ -..],

" 062 0624

0.2 I

I

I

I

0.628 Q632 0.636 0.64 A

Fig. 1. ( L ) v e r s u s A = tanh(a). The data points are based on about 50 hours CPU time on a CDC Cyber 205.

I

0

I

IIIIIII

I

0.1

I

I IIIIII

I

1

Ns.(A-Ac)

10

Fig. 2. (L)Nfl versus Ns(A - Ac) on logarithmic scales for /~ ~ 0.20, A = tanh(a). 463

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PHYSICS LETTERS

~125r

. . . . . . . . . . . . . . . . . . . . . . . . . .

tn" ~ Z

L l/ 12.5 ~_ I~

.25

0125

° Ns=512 + Ns=256 ®A
/ . / ../ X'," + °s°

,~°÷ Slope 0.125

f ÷../(÷ ""

j~~ ®* + O0 ~ 0.1

1

10 Ns" (A-Ac)

Fig. 3. ~ L ) N f versus Ns(A - Ac) on logarithmic scales for fl = 0.125, according to the dashed line in fig. 1 [A = tanh(a)].

cal errors we made the same plot for the values obtained by a least squares fit to the data at fixed fl = 0.125 (dashed line in fig. 1). This plot shows that fl = 0.125 is not compatible with our data (fig. 3). Note that the points corresponding to coupling constants less than the critical one do not lie on a smooth c o m m o n curve in contradiction to finite size scaling theory [13]. Finally, we should mention that the value for fl increases when the exponent A is assumed to be less than unity which might be possible (for example, fl = 0.25 when A is assumed to be 0.5).

4. Summary. We have performed Monte Carlo simulations which indicate that the conjecture of Svetitsky and Yaffe [1] concerning the critical exponent fl of the Polyakov line correlation function is not correct for the (2 + 1)-dimensional Z(2) lattice gauge theory in general, although it

464

6 December 1984

could be verified in the limit of vanishing spacelike coupling constant. Thus, we think that while the critical exponent a could have the value of the two-dimensional Ising model, the critical exponent fl depends on N t, i.e. the inverse temperature. In order to determine a and to investigate the dependence of fl and A c on N t calculations by means of Monte Carlo renormalization group methods are in preparation. Moreover, we hope that similar calculations for the (2 + 1)-dimensional SU(2) lattice gauge theory at finite temperatures will be carried out because for this model the same critical exponents as those of the model studied here are conjectured by Svetitsky and Yaffe [1]. We thank J. Zittartz and D. Stauffer for helpful discussions and P, von Brentano for support of this work by providing the DATEX P connection to the CDC Cyber 205 at Bochum University, West Germany.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

B. Svetitsky and L.G. Yaffe, Nucl. Phys. B210 (1982) 423. A. Weinkauf and J. Zittartz, Z. Phys. B45 (1982) 223. L. Susskind, Phys. Rev. D20 (1979) 2610. T. Banks and A. Ukawa, Nucl. Phys. B225 (1983) 145. R.V. Gavai and F. Karsch, Phys. Lett. 125B (1983) 406. L.D. McLerran and B. Svetitsky, Phys. Lett. 98B (1981) 195. J. Kuti, J. Polonyi and K. Szlachanyi, Phys. Lett. 98B (1981) 199. T. (~elik, J. Engels and H. Satz, Phys. Lett. 125B (1983) 411. A. Goksch and M. Okawa, Phys. Rev. Lett. 52 (1984) 1751. R.B. Grifliths, J. Math. Phys. 8 (1967) 478,484. D.G. Kelly and S. Sherman, J. Math. Phys. 9 (1968) 466. S. Wansleben, J.G. Zabolitzky and C. Kalle, J. Stat. Phys. 37 (1984), to be published. D.P. Landau, Phys. Rev. B13 (1976) 2997.