Non-perturbative thermodynamics of SU (N) gauge theories

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Non-perturbative thermodynamics of SU (N) gauge theories

J. Engels, J. Fingberg, F. Karsch, D.

Miller and M. Weber Fakultiit J~r Physik, Universitdt Bielefeld, W-4800 Bielefeld 1, FRG Received 20 September 1990

The pressure near the deconfinement transition as determined up to now in lattice gauge theories shows unphysical behaviour:

it can become negative and may in SU (3) even have a gap at the transition. This has been attributed to the use of only perturba- tively known derivatives of coupling constants. We propose a method to evaluate the pressure, which works without these derivatives, and is valid on large lattices. In SU(2) we study the finite-volume effects and show that for lattices with spatial extent N, ~> 15 these effects are negligible. In SU ( 3 ) we then obtain a positive and continuous pressure. The influence of non-perturbative corrections to the fl-function on the energy density are investigated and found to be important, in particular for the latent heat.

1. Introduction

M o n t e Carlo s i m u l a t i o n s o f lattice gauge theories have p r o v e n to be a powerful tool to analyze the non- p e r t u r b a t i v e aspects o f these theories. This a p p r o a c h also o p e n e d for the first t i m e the p o s s i b i l i t y to study the f i n i t e - t e m p e r a t u r e Q C D phase t r a n s i t i o n f r o m first principles a n d to o b t a i n q u a n t i t a t i v e results for the t r a n s i t i o n t e m p e r a t u r e . The analysis o f o t h e r im- p o r t a n t t h e r m o d y n a m i c q u a n t i t i e s like energy density, e n t r o p y d e n s i t y o r pressure h a v e also b e e n stud- ied on the lattice. T h e o p e r a t o r s used to extract these quantities have, however, an essential drawback: they involve in a d d i t i o n to certain m a t r i x e l e m e n t s o f the field strength t e n s o r also d e r i v a t i v e s o f the b a r e couplings with respect to t e m p e r a t u r e a n d / o r v o l u m e [ 1 ]. The relevant d e r i v a t i v e s o f the space-like (g~) a n d time-like (g~) couplings in the s t a n d a r d W i l s o n a c t i o n have been calculated p e r t u r b a t i v e l y [ 2 ]. O n e finds to O(g 2) **l

(0g;2

u g / ~ ' ¢ = L = c ' + O ( g 2); (1)

~ The derivative with respect to the temperature Tcan be writ- ten in terms of the lattice anisotropy ~= a/a,, where a and a~

are the lattice spacings in the space-and time-like directions, respectively. Details about this and our notation can be found in ref. [ l ].

( 0 g 7 2

- - - ~ J ¢ = 1 = C'~ + O ( g 2 ) . ( 1 c o n t ' d ) In m o s t n u m e r i c a l calculations o f t h e r m o d y n a m i c a l q u a n t i t i e s p e r f o r m e d in the past the l e a d i n g - o r d e r w e a k - c o u p l i n g expressions for these d e r i v a t i v e s (c~,, c'~) have been used. Results for t h e r m o d y n a m i - cal q u a n t i t i e s o b t a i n e d in this way are thus not entirely n o n - p e r t u r b a t i v e .

N o n - p e r t u r b a t i v e results for the relevant derivatives o f the couplings have b e e n calculated at some selected p o i n t s [ 3 ]. These calculations indicate that at least for the SU ( 3 ) gauge t h e o r y at i n t e r m e d i a t e couplings ( g 2 ~ l ) d e v i a t i o n s from the p e r t u r b a - tively calculated values can be large. This does not c o m e as a surprise since it is k n o w n t h a t the derivatives o f g~ a n d g~ are related to the Q C D fl-function t h r o u g h

a~_~g a = g 3 ( ~ + Og~-2~

.

(

²

)

As there are large d e v i a t i o n s from the p e r t u r b a t i v e fl-function o f the S U ( 3 ) gauge t h e o r y for g~> 1, it is to be expected that this is also true for the d e r i v a t i v e s ofg~ a n d g~.

A n o t h e r i n d i c a t i o n o f the i n a d e q u a c y o f the per- t u r b a t i v e relations for the d e r i v a t i v e s Og;c~) / O~ at in- t e r m e d i a t e couplings comes f r o m recent high-statistics calculations o f t h e r m o d y n a m i c a l quantities for

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the SU ( 3 ) gauge theory on t h e r m a l lattices o f size N 3 × 4 [4,5]. Close to the critical coupling for the first-order d e c o n f i n e m e n t transition, which on lattices o f this size occurs close to 6 / g 2~ - 5.69, one ob- serves that the pressure, P, b e c o m e s negative. More- over, P is d i s c o n t i n u o u s at the critical point. This a p p r o a c h also leads to i n c o m p a t i b l e results for the latent heat o f the transition, if it is extracted either from the i n t e r a c t i o n measure d = ( ~ _ - 3 P ) / T 4 [ 1 ], which reflects the d e v i a t i o n s o f the t h e r m o d y n a m i c a l system from a massless ideal gas, or the enthalpy density, e + P [4,5]. M o n t e Carlo s i m u l a t i o n s for these quantities have thus reached an accuracy where in- consistencies resulting from the usage o f operators for t h e r m o d y n a m i c a l observables, which are not entirely n o n - p e r t u r b a t i v e , b e c o m e visible.

2. Thermodynamics

One way o f arriving at c o m p l e t e l y n o n - p e r t u r b a - rive expressions for t h e r m o d y n a m i c a l observables is o u t l i n e d in ref. [ 3 ]: one first calculates the couplings given in eq. ( 1 ) n o n - p e r t u r b a t i v e l y at zero t e m p e r a - ture. These results can then be inserted in the stan- d a r d expressions for the t h e r m o d y n a m i c a l observables.

An alternative a p p r o a c h , which we w a n t to discuss here, is based on a calculation o f the free-energy den- s i t y , f and the i n t e r a c t i o n measure A. The free energy is related to the p a r t i t i o n function via

f = - T V -1 In Z . ( 3 )

On the lattice the l o g a r i t h m o f the p a r t i t i o n function m a y be calculated f r o m the e x p e c t a t i o n value o f the s t a n d a r d W i l s o n action, S, since the d e r i v a t i v e with respect to the bare coupling B = 2 N / g 2 is [ 1 ]

- 0 In Z / O f f = ( S ) = 3 N 3 N~( P~ + P~) , ( 4 ) where P~(~) is the space ( t i m e ) p l a q u e t t e with P = N -1 ( T r ( 1 - UUU*U t ) ) .

The physical free-energy density is then o b t a i n e d up to an integration constant, i.e. the value at Bo, from

= - 3 N , 4 d B ' [ 2 P o - ( P , + P d ] . ( 5 ) Bo

Here we have n o r m a l i z e d f r e m o v i n g the v a c u u m contribution at a p p r o x i m a t e l y T = 0 by subtracting the plaquette value Po on a s y m m e t r i c lattice ( N ~ = N o ) .

The i n t e r a c t i o n measure A can be o b t a i n e d as [ 1 ]

dg 4

A = ~ - 3 P -2

T 4 - a ~ - a 6 N N ~ [ 2 P o - ( P ~ + P O ] . ( 6 ) This relation involves the Q C D B-function, adg/da.

In the past it has been a p p r o x i m a t e d in the leading o r d e r by the p e r t u r b a t i v e weak-coupling expression, eq. ( 2 ) . However, as discussed a b o v e we should take into account d e v i a t i o n s from a s y m p t o t i c scaling in the ff-function in o r d e r to achieve a truly n o n - p e r t u r - b a t i v e calculation o f the t h e r m o d y n a m i c a l quantities. Starting with a calculation o f f a n d A has the a d v a n t a g e that we need to know only the n o n - p e r t u r - b a t i v e form o f t h e / ? - f u n c t i o n and the average plaquette Po+ P,, i.e. the expectation value o f the action on a lattice with N~<N~ and on the c o r r e s p o n d i n g s y m m e t r i c lattice with N~ = N~.

Other t h e r m o d y n a m i c a l quantities can then be ob- t a i n e d from s t a n d a r d t h e r m o d y n a m i c a l relations, if we assume in a d d i t i o n h o m o g e n e i t y o f the system ~2 This general p r o p e r t y can usually be expected to hold in a given phase o f a very large system o f a single par- ticle type when only isotropic interactions are acting.

An i m p o r t a n t consequence o f the h o m o g e n e i t y o f the system is then

0 In Z In Z

0-I7 T-- V ' ( 7 )

which in fact relates the pressure, P, and the free-energy density, f through the identity

P = - f . ( 8 )

G i v e n eq. ( 8 ) a n d the t h e r m o d y n a m i c relation

f = ¢ - T s , ( 9 )

we can then d e t e r m i n e the e n t r o p y density, s, using eqs. ( 5 ) a n d ( 6 ) ,3. In general we can expect eq. ( 8 ) to h o l d in the t h e r m o d y n a m i c limit. In a finite vol-

~2 An extensive quantity is said to be homogeneous of order one when an increase of the size of the system by a factor 2 leads to an increase of the quantity by the same factor.

~3 We note that the quantity (e+P)/T calculated on finite lat- tices is usually taken to be the entropy density, though this is true only for homogeneous systems.

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ume, however, this relation is v i o l a t e d due to surface effects. F o r a d e r i v a t i o n o f n o n - p e r t u r b a t i v e ther- m o d y n a m i c s o f S U ( N ) gauge theories on finite lattices we therefore have to look s o m e w h a t closer at the p r o p e r t i e s o f the t h e r m o d y n a m i c a l functions:

( 1 ) The relation between P a n d f g i v e n in eq. ( 8 ) has to be investigated for finite volumes.

( 2 ) We need a n o n - p e r t u r b a t i v e fl-function.

We shall discuss these p o i n t s in the following sections.

3. The free-energy density and pressure

As p o i n t e d out a b o v e we w a n t to d e t e r m i n e the pressure f r o m the free-energy d e n s i t y by s i m p l y using eq. ( 8 ) . A first test o f eq. ( 8 ) consists in c o m p a r i n g the w e a k - c o u p l i n g e x p a n s i o n s o f P / T 4 a n d - - f i t 4.

T h e e x p a n s i o n s are o b t a i n e d from those o f the plaquettes, which were calculated up to o r d e r g4 in ref.

[ 6 ]. O f course, the i n t e g r a t i o n c o n s t a n t to be used in eq. ( 5 ) is the value at g2 = 0, i.e. we can only test the a p p r o a c h to this point. The i n t e g r a n d c o n t a i n s first a g2-term, which m u s t d i s a p p e a r to leave a finite integral. Indeed, with the values f r o m ref. [ 6 ] , we f i n d for N~= 16, N ~ = 4 a c a n c e l l a t i o n o f the g2-factor with an accuracy o f 10 - 6 . T h e next t e r m f r o m the integral a n d the c o r r e s p o n d i n g one f r o m the pressure weak- coupling e x p a n s i o n (see, e.g., ref. [7 ] ) c o i n c i d e within 5%.

We have s t u d i e d the p r o b l e m o f finite-size corrections to eq. ( 8 ) at i n t e r m e d i a t e fl-values in the case o f SU ( 2 ) . T h e r e we have a large set o f d a t a on lattices with N ~ = 8 , 12, 18, 26 a n d N ~ = 4 f r o m a finite- size analysis [ 8 ] a n d an i n v e s t i g a t i o n o f the interac- t i o n m e a s u r e A [ 7 ]. In a d d i t i o n we have calculated s o m e new p o i n t s on the 83)<4 lattice below f l = 2 . 2 4 . In fig. 1 we show the pressure d a t a d i v i d e d by T 4 resulting from the usual formula, ref. [ 1 ], involving the p e r t u r b a t i v e coupling derivatives. I n s i d e the fluctuation o f the d a t a p o i n t s (the error b a r s have been o m i t t e d for clarity, their size is o f the o r d e r o f the fluctuation for the d a t a f r o m each lattice size) the P / T 4 values show no significant finite-size effects. In- d e e d this is e x p e c t e d f r o m finite-size scaling t h e o r y [ 8 ] . The c o r r e s p o n d i n g d a t a for A = ( E - 3 P ) / T 4 are gathered in fig. 2. T h e y show a c o n s i d e r a b l e finite- size effect in the n e i g h b o u r h o o d o f the critical cou-

0 . 3

0 . 2

0 . 1

i

2 . 2 0

. . . . I . . . . I . . . . I

p / T 4 o x

8 u (2)

X 0

O 2 6 n

0 IB

X 't2 0 X,,,

Ix

A

~ o ~

**~ ~ ~*X~×~ x ~ ~**

~.~...a...~.~...,, ... ~ ... ~ ...

A

4 / g e

I I I I , , , i I , i i i I i i

2 . 2 5 2 . 3 0 2 . 3 5

Fig. 1. Pressure divided by T 4 for the SU(2) gauge theory plotted against B on N~=8, 12, 18, 26 and N,=4 lattices calculated with the conventional formula involving the perturbative coupling derivatives.

~.0

0 . 5

I I I I I I

SU (2)

AX X O 2 6

~ o 18

~ × ~

A a 8

A o

~ 0 v ~1

o ......................................................... ....

I I I I I I

2 . 2 2 . 3 2 . 4 2 . 5 2 . 6 4 / g 2

Fig. 2. The interaction measure A = ( ~ - 3 P ) / T 4 for SU(2) gauge theory plotted against fl, eq. (6), with adg-2/da calculated from the weak-coupling scaling relation. The lines are fits through the small-l/points, used for the integral in eq. (5) for No=8 (long dashes), 12 (short dashes) and 18 (solid line).

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piing, tic = 2.3, up to about fl= 2.35. Since A is propor- tional to the integrand in eq. (5) we expect a similar finite-size effect for f / T 4.

We have e v a l u a t e d f f r o m eq. (5) in two steps: in the regions where our data populations were dense enough, we have used the histogram technique [9]

to find the integral contributions from the N 3 × 4 lattices, the contributions o f the symmetric lattice were calculated from spline fits to high-statistics 164 lattice plaquette values in the same fl-ranges; in the small-fl region we have supplemented the A data by fits through the remaining few points (the fits are also shown in fig. 2) and then integrated these fits to obtain the lower integral parts. The errors from the histogram method are difficult to determine. Most probably they are much smaller than the fluctuation o f data points, since information from m a n y overlap- ping histograms (in the case o f the 83X 4 lattice we had 38, for 183X 4 still 20) was used simultaneously.

Errors from the fits to A and their integrals could shift the free-energy density curves slightly in the y-direc- tion but would not change their shapes.

The resulting curves for - f i T 4 on N~=8, 12, 18 lattices (for N~= 26 too few points were available) are compared in fig. 3 to the direct data for P / T 4 on the 183X4 lattice from fig. 1. We observe a strong volume dependence o f the free-energy density indi- cating that F = V/is not an extensive quantity on small lattices. On the other hand, the N o = 18 curve is in perfect agreement with the direct data above and already at the transition point, i.e. for SU (2) we see in this region essentially no violation of the perturbative relations for the coupling derivatives. Below the phase transition the direct data should, however, de- viate more and more from the true non-perturbative result.

We conclude that for SU (2) on lattices with N~> 15 (and N~= 4) the finite-size dependence o f the free- energy density becomes negligible and therefore that eq. (8) is applicable. For SU (3) gauge theory we expect similar finite-size effects; in contrast to SU (2), however, a noticeable difference to the perturbative calculation is anticipated.

To calculate the pressure via eq. (8) we have taken the SU (3) data on 163 × 4 and 16 4 lattices from ref.

[4]. In fig. 4a we show N 4 ( 2 P o - P , - P ~ ) as obtained from the A-data o f ref. [4], and the known factors in front o f this quantity in eq. (6). The solid

'i . . . .

-f/T 4 anc/

I . . . . ^{P / T 4} I . . . . I ⁰

S U

(a) /

/ ^o

- - 18 0 /

O.t

. . . . 12

// ,~

8 / / / , , , /

0 . . .

,4/g 2

l t i i t i i i i I r t i l i i i

2.20 2.25 2.30 2.35

Fig. 3. Finite-size dependence of the free-energy density for the SU(2) gauge theory plotted against fl, Compared a r e - f i T 4 from eq. (5) for N,,=8 (long dashes), 12 (short dashes) and 18 (solid line) with P/T 4 (circles) from the direct formula with perturbative coupling derivatives for the 183× 4 lattice (same as in fig. 1).

curve in the plot is an interpolation o f these data points, with a gap assumed at f l = 5.6925. F r o m this interpolation we evaluated the integral in eq. (5). The resulting non-perturbative P I T 4 is shown in fig. 4b together with the Monte Carlo data ofref. [ 4 ], which were calculated with perturbative coupling derivatives. We note that by construction this pressure is always continuous across the deconfinement transition. Evidently, our approach solved the unsatisfac- tory situation of a negative, discontinuous pressure.

4. Non-perturbative//-function and energy density Already the early Monte Carlo renormalization group ( M C R G ) studies [ 10,11 ] of the Q C D fl-function have shown that there are considerable deviations from the weak-coupling scaling relation. In particular in the case of SU (3) large deviations have been observed for fl> 5.7, which, however, seem to disap- pear rapidly above fl> 6.1. The numerical results for the discrete fl-function, Afl(fl), obtained from a start-

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1 . 5

l . O

0 . 5

0 . 0

- 0 . 5 5 . 6

, I , i

N4 (2p o- (Pa+Pr))

p/T4

[]

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a SU (3)

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

5 . 8 6 , o

6/g 2

Fig. 4. (a) The difference between plaquette expectation values on asymmetric and symmetric lattices times N 4 from Monte Carlo S U ( 3 ) data [4] on 1 6 a x 4 and 16 4 lattices, respectively. The solid line is an interpolation through these data, used for the integral in eq. (5). (b) The non-perturbative result for the pressure in SU ( 3 ) gauge theory as a function offl along with the Monte Carlo data of ref. [4], based on perturbative coupling derivatives.

d a r d M C R G analysis, have been fitted with Pad6 ap- p r o x i m a n t s , in o r d e r to extract a n o n - p e r t u r b a t i v e r - function [ 12,13 ]. The a p p r o x i m a n t s are chosen such that they are consistent with the p e r t u r b a t i v e f o r m at large fl a n d r e p r o d u c e the o b s e r v e d scaling v i o l a t i o n s d o w n to f l ~ 5.7. To be specific we use the following p a r a m e t r i z a t i o n given in ref. [ 13 ]:

dg (1-alg2)2+a~g 4

a ~ a a = b o g 3 [l_(al+bl/2bo)gZ]2+a3g4, ( 1 0 ) where bo= 11N/48x 2 a n d bt = 34N2/768/~ 4 are the first two coefficients in the p e r t u r b a t i v e Q C D B-function for SU(N) gauge theories. In the case o f SU(3) we use the set o f coefficients { a ~ = 0 . 8 5 3 5 7 2 , a 2 = 0.0000093, a3 = 0 . 0 1 5 7 9 9 3 } , given in ref. [13].

We have used this n o n - p e r t u r b a t i v e r - f u n c t i o n a n d o u r i n t e r p o l a t i o n o f N 4 ( 2 P o - P , - P , ) , shown in fig.

4a, to extract the i n t e r a c t i o n m e a s u r e A from eq. ( 6 ) . This is p l o t t e d in fig. 5a as a solid curve. F o r c o m p a r - ison we show the u n c h a n g e d d a t a o f ref. [ 4 ] , which are b a s e d on the a s y m p t o t i c weak-coupling scaling relation. N e a r the transition point, a r o u n d r = 5.7, we observe a d r o p in A by about a factor two, which arises f r o m the n o n - p e r t u r b a t i v e r - f u n c t i o n . By construc- t i o n the two results a p p r o a c h each o t h e r at higher r - values. The non-perturbative energy density was then found by adding three times the pressure from eq. ( 8 ) ( s o l i d curve in fig. 4 b ) . In fig. 5b we c o m p a r e the result ( s o l i d c u r v e ) again with the u n c h a n g e d d a t a f r o m ref. [4 ]. A s i m i l a r d r o p as in A occurs; in particular the slight p e a k in ~ / T 4 n e a r r = 5.83 has dis-

6

u ( e - 3 P ) / T 4

1

4 [] (a)

• [ ] ~

0 - - -

p I ~ I r

[] m [] o

e/T 4

^m

°

:

3 4 ~

^{( b )}

s u (3)

2

1 ~ 16ax4

0 I . . .

5,6

^{5 , 8} ^{6 . 0}

6/g 2

Fig. 5 T h e i n t e r a c t i o n m e a s u r e A = ( e - 3 P ) / T 4, ( a ) , a n d the energy density divided by T 4, ( b ), plotted against fl for SU ( 3 ) gauge theory. The data are from ref. [4] and were calculated using perturbative coupling derivatives. The solid lines show the corre- sponding results when the pressure is computed from eq. (8) and the non-perturbative r-function, eq. (10), is taken into account.

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a p p e a r e d - the n o n - p e r t u r b a t i v e energy d e n s i t y is a very m o n o t o n o u s l y rising function a n d the gap A e /

T 4 is r e d u c e d considerably.

5. Summary and discussion

In s u m m a r i z i n g , we have seen that, u n d e r the as- s u m p t i o n o f the h o m o g e n e i t y o f the system, it is pos- sible to calculate the pressure n o n - p e r t u r b a t i v e l y on the lattice. By construction it is also c o n t i n u o u s at the first-order t r a n s i t i o n p o i n t s and it turns out to be positive everywhere. The i n f o r m a t i o n n e e d e d on an N 3 × N~ lattice is j u s t the e x p e c t a t i o n value o f the average p l a q u e t t e (Po+P~) and a c o r r e s p o n d i n g value Po on a s y m m e t r i c N4~ lattice, i.e. essentially the actions.

F r o m the same lattice data, the average plaquettes, one then d e t e r m i n e s the i n t e r a c t i o n m e a s u r e e - 3 P a n d after that by the simple a d d i t i o n o f 3P the energy density. In this second step, however, one needs the //-function. The n o n - p e r t u r b a t i v e fl-function m a y change the shape o f the energy density a n d the size o f the latent heat density in SU ( 3 ) considerably. In fact it m a y resolve the p r o b l e m o f the N~-dependence o f A ~ / T 4 f o u n d in ref. [ 14]. Because o f the c o n t i n u i t y o f the n o n - p e r t u r b a t i v e pressure there is no further a m b i g u i t y in the d e t e r m i n a t i o n o f the latent heat density, as is the case in the usual gap calculations from the enthalpy density, e + P , or ~ - 3 P involving the d e r i v a t i v e s c~, c'~ and the a s y m p t o t i c fl-function.

This a m b i g u i t y a n d its origin, the i n c o m p l e t e knowledge o f the d e r i v a t i v e s o f the couplings with respect to the a n i s o t r o p y ~, was a l r e a d y n o t i c e d in one o f the first d e t e r m i n a t i o n s o f the latent heat d e n s i t y in SU ( 3 ) , ref. [ 15 ]. The cure in b o t h refs. [ 14 ] a n d [ 15 ] was to i m p o s e the c o n d i t i o n o f c o n t i n u i t y o f the pressure at the t r a n s i t i o n point. The two calculations still differ in so far as in ref. [ 14] the q u a n t i t y e + P , a n d in ref. [ 15 ] the q u a n t i t y E - 3P, was used to extract the gap in the energy density. The e n t h a l p y density e + P d e p e n d s on the difference o f the partial derivatives given in eq. ( 1 ) , while the i n t e r a c t i o n measure ~ - 3P d e p e n d s on the sum o f these derivatives via eq. ( 2 ) . I f one insists on using p e r t u r b a t i v e coupling derivatives, the second q u a n t i t y is to be pre- ferred, because, owing to eq. ( 2 ) , and the weak-coupling fl-function the sum o f the d e r i v a t i v e s is known up to o r d e r g2, the difference only to o r d e r gO.

In full Q C D theory, the same difficulties in the de- t e r m i n a t i o n o f the pressure appear, see, for example, ref. [ 16 ]. T h e r e too our p r o c e d u r e to o b t a i n a physical pressure is applicable a n d certainly s u p e r i o r to the c o n v e n t i o n a l a p p r o a c h as long as M o n t e Carlo s i m u l a t i o n s for t h e r m o d y n a m i c quantities are re- stricted to rather small t e m p o r a l lattices (N~ < 10).

Acknowledgement

We are i n d e b t e d to H L R Z Jiilich, the B o c h u m U n i v e r s i t y c o m p u t e r centre a n d the J. von N e u m a n n c o m p u t e r centre, P r i n c e t o n for p r o v i d i n g the neces- sary c o m p u t e r time. We t h a n k the Deutsche F o r - schungsgemeinschaft for partial s u p p o r t u n d e r grant En 164/2.

References

[ 1 ] J. Engels, F. Karsch, I. Montvay and H. Satz, Nucl. Phys. B 205 [FS5] (1982) 545.

[2] F. Karsch, Nucl. Phys. B 205 [FS5] (1982) 285.

[3] G. Burgers, F. Karsch, A. Nakamura and I.O. Stamatescu, Nucl. Phys. B 304 (1988) 587.

[4] Y. Deng, Nucl. Phys. B (Proc. Suppl. ) 9 (1989) 334;

F.R. Brown, N.H. Christ, Y. Deng, M. Gao and T.J. Woch, Phys. Rev. Lett. 61 (1988) 2058.

[5] M. Fukugita, A. Ukawa and M. Okawa, Nucl. Phys. B 337 (1990) 181.

[6] U. Heller and F. Karsch, Nucl. Phys. B 251 [FSI3] (1985) 254.

[ 7 ] J. Engels, J. Fingberg, K. Redlich, H. Satz and M. Weber, Z.

Phys. C42 (1989) 341.

[8] J. Engels, J. Fingberg and M. Weber, Nucl. Phys. B 332 (1990) 737.

[9] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 63 (1989) 1195.

[ 10 ] A. Hasenfratz, P. Hasenfratz, U. Heller and F. Karsch, Phys.

Lett. 140B (1984) 76; B 143 (1984) 193.

[ 11 ] R. Gupta, G. Guralnik, A. Patel, T. Warnock and C. Zemach, Phys. Rev. Lett. 53 (1984) 1721; Phys. Lett. B 161 (1985) 352.

[12] D. Petcher, Nucl. Phys. B 275 [FS17] (1986) 24l.

[ 13 ] J. Hoek, Nucl. Phys. B 339 (1990) 732.

[14] N.H. Christ and H.Q. Ding, Phys. Rev. Lett. 60 (1988) 1367.

[ 15 ] B. Svetitsky and F. Fucito, Phys. Lett. 131 B ( 1983 ) 165.

[ 16 ] J.B. Kogut and D.K. Sinclair, Pressures, energy densities and chiral condensates in SU(3) lattice gauge theory with fermions, Illinois preprint ILL-(TH)-90-7 dynamical

(1990).