Monte Carlo renormalisation group studies of SU(3) lattice gauge theory: CERN-DESY-Edinburgh Collaboration

Volltext

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Nuclear Physics B257 [FS14] (1985) 155-172

© North-Holland Publishing Company

M O N T E C A R L O R E N O R M A L I S A T I O N G R O U P S T U D I E S O F S U ( 3 ) L A T I ' I C E G A U G E T H E O R Y *

C E R N - D E S Y - E d i n b u r g h C o l l a b o r a t i o n

K.C. BOWLER l, A. HASENFRATZ 2.*, P. HASENFRATZ 3"4, U. HELLER 4, F. KARSCH 4'~, R.D. KENWAY l, H. MEYER-ORTMANNS 6, I. MONTVAY 6, G.S. PAWLEY I

and D.J. WALLACE 1 Received 2 January 1985

Results are reported of Monte Carlo renormalisation group studies of the approach to asymptotic scaling in SU(3) lattice gauge theory. By comparing measurements on 84 and 16 4 lattices, estimates are obtained for the shift, Aft, in the fundamental plaquette coupling,/3, corresponding to a change of length scale by a factor of 2. The definitions of block link variables contain a free parameter whose value can be optimised to minimise the transient flow to a renormalised trajectory.

Our results, at/3 = 6.0, 6.3 and 6.6, are consistent with those obtained previously with the improved ratio method, which is also briefly discussed. In both methods simulation is performed only with the standard Wilson action. An important feature of the results is the appearance of a pronounced dip in A/3 which implies that in the presently accessible range of /3 the asymptotic value is approached from below, and its onset is delayed.

1. Introduction

I n t h e l a r g e c u t - o f f l i m i t o f r e n o r m a l i s a b l e t h e o r i e s it is p o s s i b l e t o t u n e t h e c u t - o f f a n d t h e c o u p l i n g ( s ) t o g e t h e r i n s u c h a w a y t h a t t h e p h y s i c a l c o n t e n t o f t h e t h e o r y r e m a i n s u n c h a n g e d . T h e f u n c t i o n a l r e l a t i o n b e t w e e n t h e c o u p l i n g ( s ) a n d t h e c u t - o f f is g i v e n b y t h e / 3 - f u n c t i o n ( s ) o f t h e t h e o r y . P e r t u r b a t i o n t h e o r y s u g g e s t s t h a t i n a n S U ( N ) g a u g e t h e o r y t h e l e a d i n g c u t - o f f d e p e n d e n t c o r r e c t i o n s a r e e x p o n e n t i a l l y s m a l l i n t h e i n v e r s e o f t h e b a r e c o u p l i n g c o n s t a n t g2. O n l y t h e t w o l e a d i n g t e r m s o f t h e / 3 - f u n c t i o n

f l ( g ) = - b o g 3 - b i g 5 + . • • (1)

• A preliminary version of this work appeared as CERN preprint TH.3952 and was submitted to the XXII International Conference on High Energy Physics, Leipzig 1984.

• * On leave of absence from the Central Research Institute for Physics, Budapest.

Physics Dept., University of Edinburgh, Edinburgh EH9 3JZ, Scotland.

2 Physics Dept., University of Michigan, Ann Arbor, MI 48109, USA.

3 Institut fiir Theor. Physik, Universit~it Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland.

4 CERN, 1211 Geneva 23, Switzerland.

5 Present address: Dept. of Physics, University of Illinois at Urbana-Champaign, 11 l0 W. Green Street, Urbana, IL 61801, USA.

6 II. Inst. f. Theor. Physik, Univ. Hamburg, Notkestrasse 85, D-2 Hamburg 52, Germany.

155

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156 K.C. Bowler et al. / Lattice gauge theory

are universal: the higher-order corrections are power-like and not necessarily small in the region where the cut-off-dependent corrections are already negligible. There might also be sizeable contributions to the /3-function from non-perturbative phenomena.

In the lattice formulation of Yang-Mills theories, where numerical studies are performed at moderate correlation lengths (i.e. at intermediate coupling constant values), the quantitative knowledge of the/3-function is of fundamental importance for the correct interpretation of the results. One should confirm also that the /3-function approaches the asymptotic form of eq. (1) without passing through a phase transition, assuring a continuum limit with the expected properties of asymptotic freedom and confinement.

In this p a p e r first results obtained by an extended collaboration for a Monte Carlo renormalisation group ( M C R G ) study of SU(3) lattice gauge theory are reported. In this study block loop expectation values o n 164 and 8 4 lattices are matched in order to determine the shift A/3 = A/3(/3) in the fundamental plaquette coupling /3 ( = 6 / g 2) of the standard Wilson action, corresponding to a change of scale by a factor of 2. The function

zafl(fl)

is directly related to the integral o f the inverse of the /3-function and contains the same information:

~ dx - -',/~2 In 2. (2)

The anticipated renormalisation group (RG) flow [1] in a reduced (3-dimensional) coupling constant space is indicated schematically in fig. 1. The axes refer to a parametrisation of the action in terms of a bare coupling, g2, and dimensionless couplings, ci, characterising the relative strengths o f fundamental, adjoint, 6-1ink etc. couplings. The continuum fixed point (FP) lies in the g2= 0 hyperplane and is stable to all perturbations in this hyperplane. The renormalised trajectory (RT) is the one-dimensional unstable manifold emerging from the fixed point. The transient

C 3

F p RT

_ C 2

gl g2

Fig. 1. Schematic diagram of the anticipated RG flow in a three-dimensional coupling constant space.

FP denotes the fixed point in the critical hypersurface, whilst RT denotes the one-dimensional unstable manifold.

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K.C. Bowler et al. / Lattice gauge theory 157 flOW f r o m two points on the

g2

axis (which we c o n v e n t i o n a l l y c h o o s e to c o r r e s p o n d to the s t a n d a r d Wilson action for the pure g a u g e theory) into the r e n o r m a l i s e d trajectory is indicated. The positions o f the fixed point a n d r e n o r m a l i s e d trajectory are not universal - they d e p e n d on the choice o f R G t r a n s f o r m a t i o n - but flow transverse to the R T is contractive so that all lattice actions describe the same long-distance physics.

F o r fixed g~, p r o v i d e d that there exists a universal fl-function so that all physical quantities scale in the same way, we are g u a r a n t e e d to be able to find a value o f g~

such that m a t c h i n g is achieved after a suitably large n u m b e r o f blockings, due to the contractive nature o f the R G flow. The resulting Aft is universal. H o w e v e r , there is a finite-size limitation arising f r o m the fact that o u r starting configurations are on 164 a n d 84 lattices, so that we can carry out at most three blockings o f the smaller lattice. It is c o n s e q u e n t l y i m p o r t a n t to take a d v a n t a g e o f o u r f r e e d o m in the choice o f the R G t r a n s f o r m a t i o n to extend the definition o f the block link variables to include a free p a r a m e t e r which can be used to maximise the rate o f c o n v e r g e n c e o f the trajectories t h r o u g h g2 a n d g2 [2]*. If, by varying this parameter, we could get perfect m a t c h i n g , after one blocking o f the larger lattice, o f all physical quantities which can be fitted on to the finite lattice, i.e. the points g2 a n d g2 2 lie on the same R G trajectory as s h o w n in fig. 2, then we w o u l d have d e t e r m i n e d ,aft e x a c t l y from the first blocking. All s u b s e q u e n t blockings w o u l d yield the same Aft, as the two lattices " f o l l o w each o t h e r " along the same trajectory. In practice o f course perfect m a t c h i n g is never achieved but optimisation speeds up c o n v e r g e n c e o f the sequence o f estimates {Aft (")} by m a k i n g the trajectory t h r o u g h g~ initially flow close to the g2 axis (and hence to g2). Furthermore, k n o w l e d g e o f the sequences {Aft(")} for a range o f b l o c k t r a n s f o r m a t i o n s a r o u n d the o p t i m u m can be used to obtain m o n o t o n i - cally increasing (decreasing) lower (upper) b o u n d s on Aft = A/3 ~ ) , which are impor- tant for estimating the systematic error i n d u c e d by this finite-size effect.

FP

2 2 g

g g

1 2

Fig. 2. Schematic diagram of the effect of optimising the RG transformation as discussed in the text.

* In [2] a similar idea to that of Hasenfratz et al. has been put forward and applied to the 3d Ising model recently by Swendsen. The basic idea of a MCRG analysis is suggested by Ma and by Swendsen in the second reference by him.

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158 K.C. Bowler et al. / Lattice gauge theory

O u r m a i n results a r e t h e f o l l o w i n g m a t c h i n g v a l u e s at t h r e e / 3 - v a l u e s : Aft (/3 = 6.0) = f {0.35 + 0.02,

L 0 . 3 4 + 0 . 0 2 , A/3 (/3 = 6.3) = 0.43 + 0 . 0 3 , A/3 (/3 = 6.6) = 0.56 ± 0 . 0 6 ,

s c h e m e 1, s c h e m e 2 , s c h e m e 1 ,

s c h e m e 1, (3)

w h e r e the e r r o r s i n c l u d e o u r e s t i m a t e o f b o t h s t a t i s t i c a l a n d s y s t e m a t i c u n c e r t a i n t i e s . R e s u l t s a t / 3 -- 6.9, 7.2, a n d o f o t h e r c a l c u l a t i o n s will b e p r e s e n t e d e l s e w h e r e [3].

T h e r e s u l t s q u o t e d a b o v e are c o n s i s t e n t w i t h t h o s e o b t a i n e d e a r l i e r b y a different M C R G m e t h o d , the i m p r o v e d r a t i o m e t h o d (fig. 3) [4]. T h e s t a r t i n g p o i n t o f this l a t t e r p r o c e d u r e is t h e o b s e r v a t i o n [5] t h a t t h o s e r a t i o s o f W i l s o n l o o p e x p e c t a t i o n v a l u e s f r o m w h i c h t h e s e l f - m a s s a n d c o r n e r c o n t r i b u t i o n s c a n c e l satisfy the h o m o g e n e o u s r e n o r m a l i s a t i o n g r o u p ( R G ) e q u a t i o n , at l e a s t if the l o o p s i n v o l v e d a r e large c o m p a r e d to t h e lattice s p a c i n g . S i n c e the h o m o g e n e o u s R G e q u a t i o n is

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 I l l l l

5.7 6,0

w I I i

• optimised blocking: scheme1 o o p t i m i s e d blocking: scheme 2 [] l - l o o p i m p r o v e d r o t i o m e t h o d

I I i i I' , ,

6.3 6.6

Fig. 3. The shift zlfl as a function of/3 obtained from l-loop improved ratios (~) [4] and from optimised blocking scheme 1 (O) and scheme 2 (C)) in this work. In the case of the ratio results the thin error bars refer to the statistical error while the thick error bars refer to the average fluctuation of the large number

of different ratios included in the analysis. The dashed line is the asymptotic prediction.

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K.C. Bowler et al. / Lattice gauge theory 159 linear in these ratios, arbitrary linear combinations of the basic ratios satisfy it also.

The method can be optimised by taking particular combinations of the basic ratios to cancel lattice artifacts order by order in perturbation theory. An advantage of this method is its simplicity. Any high statistics m e a s u r e m e n t of the potential or of the string tension can easily be extended to p e r f o r m this analysis, and it is expected to work even at large correlation lengths. Its disadvantage is that it is difficult to see how to do the optimisation non-perturbatively and the method requires very good statistics. The statistical error of the block loop matching results at /3 = 6.0 quoted in eq. (3) is about 4 times smaller than the corresponding error quoted in ref. [4].

2. The block transformations

The block variable VAB associated with the block link A - B in fig. 4a is chosen with the probability

p r o b ( VAB ) ~ exp ~P Tr ( V*AaX + h.c.). (4)

( 0 ) x * x " x

2 3

I 4

:i

5 6

i B

1o

8 9

(b)

X * X " X

B A B

A B

Fig. 4. (a) Construction of the block link in scheme 1, as described in the text. (b) The additional classes of paths used in the definition of the block link in scheme 2.

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160 K.C. Bowler et al. / Lattice gauge theory

In blocking scheme 1, X is taken to be the sum of the matrix products along 7 different paths connecting the sites A and B:

UsU6, U~U2U3U4, UTUsUgUto

and the corresponding paths in the orthogonal planes [6]. To test for possible systematic effects arising from this choice of blocking scheme, a second scheme, 2, was tried at/3 = 6.0, in which the set of paths connecting A and B was extended to include a further 36 paths of the classes indicated in fig. 4b. A/3 should of course be i n d e p e n d e n t of the blocking scheme. The p a r a m e t e r P is used for optimisation as described in sect. 1.

3. Configurations and statistics

The 164 SU(3) configurations at /3 =6.0, 6.3 and 6.6 were created on the DAPs at Edinburgh. After every 112 p s e u d o - h e a t b a t h [7] sweeps the configuration was stored for later blocking and other measurements [3]. The first 1500 sweeps were discarded. The limited m e m o r y of the Edinburgh DAPs forced us to store the link variables as 16 bit integers after multiplication by a scale factor N = 32 000, while the matrix multiplications in the updating were done in 3 byte real arithmetic. The corresponding rounding errors introduce additional randomness into the system resulting in a slight systematic error: the configuration looks s o m e w h a t " h o t t e r "

than the nominal/3-value would require. H o w e v e r this effect is very small. Theoretical considerations and test runs with an artificially decreased scale factor, N, suggest that the corresponding error in the plaquette expectation value is O ( t 0 -8) - well below our statistical accuracy. Details are given in table 1.

The 84 configurations at /3 = 5.4, 5.6, 5.7, 5.8, 5.9, 6.0 and 6.1 were created at C E R N and D E S Y starting from the last, well-equilibrated configurations of earlier studies. These configurations were separated by 10 p s e u d o - h e a t b a t h sweeps.

Using scheme 1 the blocking was done on the C E R N IBM machines at /3 = 6.0 at four values of the free parameter, P: 20, 30, 35 and 40. These values were picked after a few trial runs. Similarly at/3 = 6.6, the values 22, 25, 30 and 40 were chosen but a preliminary analysis clearly indicated that all the matching results behaved linearly in 1/P, as observed at /3 = 6.0. Thus the complete analysis at /3 = 6.6 and at 6.3 was done at just two P values: 25 and 40.

TABLE 1

Effect of various scale factors, N, used for integer storage as described in the text, on the average plaquette for a single 164 configuration at/3 = 6.0

N 32 000 284 248 124 62 31 15 11 9

([]) 1.781 1.781 1.780 1.779 1.781 1.773 1.768 1.755 1.744

AE 0 0.000 0.001 0.002 0.000 0.008 0.013 0,026 0.037

A E is the shift in the average plaquette relative to the value obtained with the maximal scale factor of 32 000.

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K.C. Bowler et al. / Lattice gauge theory

The results quoted in eq. (3)

161

for scheme 1 are based on the following statistics:

164: /3 =6.0 /3 =6.3 /3 =6.6

84: /3 = 5.4 /3=5.6 /3=5.7

/3 =5.8,

/3 = 5 . 9 , /3 = 6 . 0 , /3 = 6 . 1 ,

50 configurations 59 configurations 99 configurations 32 configurations 96 configurations 64 configurations 96 configurations, 288 configurations, 96 configurations, 96 configurations.

The result at 13 = 6.0 for the second blocking scheme is based on an analysis of 36 164 configurations using three P-values: 26, 21 and 17. Linear interpolation was used throughout to get intermediate fl-values.

The statistical errors were estimated by measuring time correlations and also by the usual binning.

4. Results

If for some block transformation the renormalised trajectory runs along the line of the standard Wilson action in the multi-parameter coupling constant space*, then after the first blocking step the effective action is again a standard action at some coupling

/3'=/3- A/3.

In this case the block loop expectation values are equal to the corresponding Wilson loop expectation values o f the standard action at coupling /3' on a lattice of half the size. The parameter P is fixed by requiring that one gets as close as possible to this situation. To say it in another way: an optimal value of P at each /3 is determined by requiring the best possible consistent matching for many observables after the first blocking step.

Fig. 5 illustrates the matching values of 12 different loops (1 × 1, 1 × 2 , . . . , 4 x 4, 8 , ~ ) after the first blocking step at /3 =6.0, using scheme 1 while fig. 6 shows the effect of subsequent blockings on the matching of the 1 x 1 Wilson loop.

Decreasing P-dependence at large length scales is manifested in two ways in these figures. Firstly, at a given level of blocking, matching for the larger loops is in general less P-dependent than for smaller loops. Secondly, matching becomes less P-dependent as the level of blocking increases. The results suggest

p o p t +1o

= 3 5 _ 5 . ( 5 )

It should be emphasised that in principle any value of P is appropriate; the different block transformations should give the same final prediction for A/3.

* Even in principle this is possible only up to the exponentially small corrections discussed in the introduction. We are indebted to J. Kripfganz for a discussion of this point.

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162 K.C. Bowler et al. / Lattice gauge theory

~ = 6 . 0

0.6

0.5

0.4

matching a f t e r f i r s t btocking

Ixl

/ 2xl

3 x l / 4xl

2x2

3 × 2 4 x 2

2'

. 3x3 4x3

l x l

2xl

3 x l

÷ 4 x l + 2x2

4 3x2 4 x 2 3x3

I I I I I

1140 1130' 1125 1/20

1/35 l I P

Fig. 5. The matching predictions obtained in scheme 1 from 12 different block loops after the first blocking step at/3 = 6.0. P is the free parameter in the block transformation. For P ~ 30 the predictions obtained from a given loop are linear in I/P. The mean deviation of the matching predictions has a

broad minimum in the region P = 3 5 ~ °.

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K.C. Bowler et aL / Lattice gauge theory 163

0.6

0.5

0.4

SU(3): ~=6.0 1x1 block loop motching

+ Blocking step 1

I +

1 + ~ Blocking step 2

1 +

3

Blocking step 3

I I I I I

1/40 I/30 I/20

I/35 I / P

Fig. 6. The matching predictions obtained for the 1 x I block loop in scheme 1 at subsequent blocking levels as a function of l / P for/3 =6.0.

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164 K.C. Bowler et al. / Lattice gauge theory

A similar analysis a t / 3 = 6.3 yields, after the first blocking step,

popt _ 27+35 (6)

I - -

whilst after the s e c o n d b l o c k i n g step

p o p t _ 32+_8 (7)

2 - -

A t / 3 = 6.6 the c o r r e s p o n d i n g results are

plopt - - - - 23.5 +-4 , (8)

popt _ 28+_8 (9)

2 - -

Figs. 7-10 s h o w the raw data on which these estimates are based. T h e y illustrate that as /3 increases, so also does the P - d e p e n d e n c e o f a/3 (for particular Wilson loops at a given b l o c k i n g level); hence it is increasingly i m p o r t a n t to optimise the b l o c k i n g p r e s c r i p t i o n at the h i g h e r / 3 - v a l u e s .

The predictions for A/3 (/3 = 6.0) o b t a i n e d f r o m m a t c h i n g f o u r different block loops (plaquette, 61( = ~ ), 6 2 ( = ~ ' 2 ) a n d 63( = ~ f ' ) ) are given for P = 3 0 , 35 a n d 40 at s u b s e q u e n t b l o c k i n g levels in table 2. As the statistical errors o f the block loops 6, a n d 63 after the third b l o c k i n g step are very large no m a t c h i n g value is q u o t e d there. The P - d e p e n d e n c e is linear in 1 / P (as suggested by p e r t u r b a t i o n theory) f o r P ~ 30, which makes it possible to follow the predictions for large P even w i t h o u t actually m e a s u r i n g them. F o r the 1 × 1 block l o o p m a t c h i n g s p o p t 50.

This is the value where the predicted A/3 is the same after the first a n d s e c o n d b l o c k i n g step: 3/3 = 0.35 + 0.01 a n d is consistent with the e x t r a p o l a t e d third b l o c k i n g result. A simple averaging o f the third b l o c k i n g step predictions at P = 30, 35 a n d 40 gives za/3 = 0.359 + 0.013. It is also e n c o u r a g i n g that using the alternative b l o c k i n g scheme yields results for A/3 (/3 = 6.0) which are c o m p l e t e l y consistent with those o f s c h e m e 1 (see eq. (3)).

TABLE 2

The matching predictions A/3 (fl = 6.0) are summarised for 4 different block loops at different blocking levels and different values of P in scheme 1

P Blocking ~ ~ ~ .~7

step

30 l 0.461 (3) 0.471 (3) 0.414 (3) 0.392 (3)

2 0.382 (6) 0.375 (7) 0.372 (5) 0.368 (5)

3 0.359 (13) 0.383 (25)

35 1 0.420 (2) 0.444 (2) 0.386 (3) 0.370 (3)

2 0.371 (6) 0.369 (5) 0.365 (5) 0.362 (5)

3 0.368 (13) 0.348 (21)

40 1 0.388 (2) 0.424 (2) 0.363 (3) 0.352 (3)

2 0.361 (5) 0.360 (6) 0.367 (5) 0.355 (5)

3 0.351 (13) 0.342 (20)

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K.C. Bowler et al. / Lattice gauge theory 165

0.6

0.5

0.4

~=6.3 matching a f t e r f i r s t blocking

3xl

l x l

2xl / 2x2

3x3

4x4

Notation:

7--f

i n (5,n) I

i

I 1

I I I

1/40 1/25 1/20 1 / P

Fig. 7. The m a t c h i n g predictions obtained from various block loops after the first blocking step at fl = 6.3.

Error bars have been omitted for clarity but are broadly comparable with those s h o w n in fig. 5.

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166 K.C. Bowler et al. / Lattice gauge theory

(1= 6.3

0.6

0.5

0.4

matching a f t e r second btocking

Ixi

4x3

(5, 1)

2)

I I I

1/40 1/25 1/20

1/P

Fig. 8. The m a t c h i n g predictions obtained from various block loops after the second blocking step at /~ = 6.3.

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H

O

[ >

o .o o t l ~

| !

II CY~

~JJ × 3

c~ V~ ~ ~

Lt~

0 "

c~

. ~ .~. L,,J V X W

s~

s~

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168 K.C. Bowler et al. / Lattice gauge theory

0.6

0.5

0.4

(3 = 6.6 matching

4x2'

3x2

4x1//

3 x l / ( 5 , 2 ) /

(5,~)

after second btocking

3×3

I 1 I

1/40 1/25 1/20

Fig. 10. As fig. 8 but for/3 =6.6.

l x l 2xl

5,3) 4x3

,4)

4x4

lIP

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K.C. Bowler et al. / Lattice gauge theory TABLE 3

/tfl (n = co) is given as obtained by assuming that the subleading eigenvalue of the linearised RG transformation is ¼

30 0.358 (8) 0.346 (9) 0.358 (7) 0.360 (7) 35 0.356 (8) 0.346 (7) 0.358 (7) 0.359 (7) 40 0.353 (7) 0.340 (8) 0.355 (7) 0.356 (7)

169

T h e f o l l o w i n g s i m p l e c o n s i d e r a t i o n h e l p s to give us c o n f i d e n c e t h a t the s y s t e m a t i c e r r o r s a r e r e a l l y u n d e r c o n t r o l . It is e x p e c t e d t h a t in the c o n t i n u u m limit the s u b l e a d i n g e i g e n o p e r a t o r is o f d i m e n s i o n 6 with an e i g e n v a l u e o f 1 ( u p to n e g l i g i b l e l o g a r i t h m i c c o r r e c t i o n s ) . This i m p l i e s t h e b e h a v i o u r

Aft ( " ) = Aft('=°°) + a ( P ) ( 1 ) " , (10) w h e r e n is the n u m b e r o f b l o c k i n g steps a n d Aft ("-~) is t h e P - i n d e p e n d e n t result a f t e r n = oo b l o c k i n g steps. Eq. (10) gives

A f l ( , = ~ ) = ½[4Aft(, =2) _ Aft(,= 1)]. ( 1 1)

I f this p r o c e d u r e is c o n s i s t e n t the p r e d i c t e d Aft ('=~) s h o u l d b e i n d e p e n d e n t o f P a n d the b l o c k l o o p c o n s i d e r e d . T h e n u m b e r s are s u m m a r i s e d in t a b l e 3, w h e r e the e r r o r s q u o t e d are statistical. O n the basis o f this t a b l e a n d o f the p r e v i o u s c o n s i d e r a t i o n s we feel t h a t the e r r o r e s t i m a t e s in eq. (3) are r a t h e r c o n s e r v a t i v e . T h e m a t c h i n g results for the s a m e f o u r b l o c k l o o p s at fl = 6.3 a n d at fl = 6.6 are s u m m a r i s e d in t a b l e s 4 a n d 5 r e s p e c t i v e l y . T h e e s t i m a t e o f Aft (" =~¢) in eq. (1 l ) d o e s n o t w o r k q u i t e so well at t h e s e l a r g e r v a l u e s o f ft. (The p r e v i o u s a r g u m e n t u s i n g the d i m e n s i o n o f the s u b l e a d i n g e i g e n o p e r a t o r is v a l i d o n l y i f finite-size effects are n e g l i g i b l e in c o r r e l a t i o n f u n c t i o n s , w h i c h is c e r t a i n l y n o t the case w h e n fl is as large

TABLE 4

The matching predictions za/3 (/3 = 6.3) together with estimates for A/3 ~"~°°) as discussed in the text

P Blocking step [~ [ ~ ~_~ .~f

25 l 0.592 (6) 0.600 (6) 0.534 (5) 0.503 (5)

2 0.487 (9) 0.475 (10) 0.474 (9) 0.467 (10)

3 0.44 (2) 0.45 (_+32) 0.44 (+-5) 0.45 (+-32)

~(4A13("-2)- Afl(n=l) ) 0.451 (14) 0.433 (15) 0.454 (13) 0.456 (14)

l(4Afl(n=3)--z~fl(n=2))

0.43 (3) 0.44 (-4)+3 0.43 (+_43 ) 0.44 (-4)+3

40 1 0.356 (3) 0.439 (5) 0.347 (3) 0.348 (4)

2 0.421 (9) 0.426(11) 0.419(10) 0.419(12)

3 0.428 (+_,4) 0.43 (2) 0.43 (2) 0.43 (+_2)

~(4Afl(":2)-- Aft ~":t)) 0.442 (13) 0.422 (17) 0.443 (14) 0.443 (17) 31-(4Aft ("=3) --A/3 (n=2)) 0.43 (2) 0.43 (3) 0.43 (3) 0.44 (+_43)

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170 K.C. Bowler et al. / Lattice gauge theory TABLE 5

As table 4 but for/3 = 6.6

P Blocking step ~] ~ ~ ' ~ .~7

25 l 0.592 (5) 0.625 (7) 0.544 (5) 0.520 (5)

2 0.551 (10) 0.554 (21) 0.547 (16) 0.545 (19)

3 0.55 (5) 0.56 (6) 0.56 (6) 0.56 (6)

~(4A/3("-2)- Aft ("-~1) 0.538 (15) 0.53 (3) 0.548 (23) 0.553 (28) 1(4A/3(,,-3)_ dfl(, =2)) 0.55 (7) 0.56 (9) 0.56 (8) 0.57 (9)

40 l 0.285 (3) 0.401 (5) 0.289 (3) 0.303 (4)

2 0.453 (17) 0.474 (26) 0.457 (20) 0.459 (22)

3 0.52 (5) 0.53 (6) 0.53 (6) 0.53 (7)

1(4//3 (~=2)- Aft ("=D) 0.508 (23) 0.499 (36) 0.512 (28) 0.511 (30)

½(4dfl(n =3) _ Afl(.=z)) 0.57 (7) 0.55 (9) 0.55 (9) 0.55 (9)

as 6.6.) A more reliable estimate of A/3 (~-~) based u p o n A/3 (n=2) a n d A/3 (n=3) is also given. An alternative a p p r o a c h is to note that even without any extrapolation A/3 (n) is decreasing as n increases for P = 25, whereas Aft (n) is increasing as n increases for P = 40, in agreement with the expectation that poor lies somewhere between 25 and 40.

There is one trivial type of systematic error which we did not check, however:

the error coming from the linear interpolation between /3-values on the 84 lattice.

This error is very easy to avoid completely (by blocking 84 configurations at the estimated value o f / 3 ' ) and we intend to do so in the future. The effect is expected to be small.

5. Discussion

Fig. 3 shows a p r o n o u n c e d dip in A/3(fl) around /3 = 6 . 0 which implies that in the accessible range of/3 the asymptotic value is a p p r o a c h e d from below and that its onset is delayed. For /3 = 6.6 this deviation is rather small, which is supported by recent M C R G m e a s u r e m e n t s by G u p t a and Patel [8] using a special "x/3" block transformation [9] at /3 = 6.5 and 7.0. However, whilst for us optimisation of the block t r a n s f o r m a t i o n a p p e a r s to be an essential ingredient, this is not the case in ref. [8]; this aspect requires clarification e.g. by perturbative analysis. Recent precise string tension [ 10] and critical temperature [ 1 l] (in [ 1 l] deviations from asymptotic scaling for the deconfinement temperature have also been observed by M o n t v a y and Pietarinen) measurements show the same qualitative b e h a v i o u r for A/3(/3). A similar structure seems to be emerging in SU(2) according to string tension [12] and preliminary M C R G results [13]. (However, it appears [14] that the SU(3) mass gap has a qualitatively different behaviour and is consistent with asymptotic scaling in the range 5.5 </3 < 5.9.) It is an interesting theoretical p r o b l e m to understand the

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K.C. Bowler et al. / Lattice gauge theory 171 o r i g i n o f this " u n n a t u r a l " b e h a v i o u r , w h i c h is u n l i k e t h a t f o u n d in the d = 2 s t a n d a r d n o n - l i n e a r t r - m o d e l , w h e r e the a s y m p t o t i c v a l u e is a p p r o a c h e d s m o o t h l y f r o m a b o v e [2, 15]. It is e v e n m o r e i m p o r t a n t to d e t e r m i n e /3min a b o v e w h i c h the different m e t h o d s give q u a n t i t a t i v e l y the s a m e A / 3 ( / 3 ) . F o r /3 >/3min--A/3(/3min) a u n i q u e /3-function c a n be d e f i n e d a n d the t h e o r y reflects the c o n t i n u u m p r o p e r t i e s .

C o n c e r n i n g the first q u e s t i o n it is a n a t u r a l a s s u m p t i o n t h a t the d i p in A/3 is r e l a t e d to t h e critical p o i n t at the e n d o f the f i r s t - o r d e r t r a n s i t i o n line in the f u n d a m e n t a l - a d j o i n t c o u p l i n g c o n s t a n t p l a n e . T h e flow a w a y f r o m the s p u r i o u s critical p o i n t is e x p e c t e d to slow d o w n the flow f r o m fir = ~ until t h e flow has p a s s e d t h e n e i g h b o u r h o o d o f t h e s p u r i o u s critical p o i n t , w h e n the two flows r e i n f o r c e a n d s p e e d u p t h e flow t o w a r d s the fixed p o i n t a t / 3 f - - f l a = 0. T h e s l o w i n g d o w n i m p l i e s t h a t A / 3 f a p p r o a c h e s its a s y m p t o t i c v a l u e f r o m b e l o w . A r e l a t e d e x p l a n a t i o n was s u g g e s t e d b y M a k e e n k o a n d P o l i k a r p o v r e c e n t l y [16]. A c c o r d i n g to t h e s e i d e a s s m o o t h e r b e h a v i o u r a n d e a r l i e r o n s e t o f a s y m p t o t i c s c a l i n g is e x p e c t e d a l o n g the lines /3f//3a = - c ( c > 0 ) . It is k n o w n t h a t the p e a k in the specific h e a t is s t r o n g l y r e d u c e d in this r e g i o n [17].

T h e e x p l a n a t i o n o f the d i p in A/3 in t e r m s o f h i g h e r - o r d e r p e r t u r b a t i v e t e r m s o f t h e / 3 - f u n c t i o n is v e r y i m p r o b a b l e [18].

T h e a n s w e r to the s e c o n d q u e s t i o n r e q u i r e s p r e c i s i o n d a t a . It is e x c i t i n g a n d r e a s s u r i n g t h a t the k i n d o f p r e c i s i o n q u o t e d h e r e a n d in r e l a t e d w o r k s m i g h t p i n d o w n flmin a n d p r e d i c t t h e / 3 - f u n c t i o n with a r e a s o n a b l e a c c u r a c y .

P.H. is i n d e b t e d to A. P e t e r m a n n for v a l u a b l e d i s c u s s i o n s . W e are g r a t e f u l to E d i n b u r g h R e g i o n a l C o m p u t i n g C e n t r e for c o n t i n u e d D A P s u p p o r t . T h e r e s e a r c h was s u p p o r t e d in p a r t b y S E R C grants N G 1 1 8 4 9 a n d NG14703.

References

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[12] F. Gutbrod and I. Montvay, Phys. Lett. 136B (1984) 411

[13] P.B. Mackenzie, in Gauge theory on a lattice: 1984, Proc. Argonne National Laboratory Workshop (ANL, 1984)

[14] P. de Forcrand, G. Schierholz, H. Schneider and M. Teper, Phys. Lett. 152B (1985) 107 [15] S.H. Shenker and J. Tobochnik, Phys. Rev. B22 (1980) 4462;

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