Efficient calculation of critical parameters in SU (2) gauge theory

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North-Holland

Efficient calculation of critical parameters in SU (2) gauge theory

J. Engels

a

j. Fingberg b and V.K. Mitrjushkin c,1 a Fakultiitfur Physik, Universitdt Bielefeld, W-4800 Bielefeld, FRG b HLRZ, Forschungszentrum Jiilich, Pf 1913, W-5170 Jiilich, FRG c Fachbereich Physik, Humboldt-Universitat, 0-1040 Berlin, FRG Received 28 September 1992

We show how the critical point and the ratio ~,/u of critical exponents of the finite temperature deconfinement transition of SU (2) gauge theory may be determined simply from the expectation value of the square of the Polyakov loop. In a similar way we estimate the ratio ( a - 1 )/~. The method is based on a consistent application of finite size scaling theory to results obtained with the density of states technique. It may also be used in other lattice theories at second order transitions.

F i n i t e size scaling ( F S S ) techniques have now be- c o m e a well-established tool for the i n v e s t i g a t i o n o f critical p r o p e r t i e s in S U ( 2 ) [ 1 - 3 ] a n d S U ( 3 ) [ 4 - 6 ] lattice gauge theories at finite t e m p e r a t u r e . M a n y ingenious m e t h o d s have been d e v i s e d using FSS theory to extract the infinite v o l u m e critical p o i n t a n d ratios o f critical e x p o n e n t s from v a r i o u s variables.

M o s t o f these m e t h o d s were i n v e n t e d in statistical physics a n d a p p l i e d to high p r e c i s i o n M o n t e Carlo d a t a o f Ising [7,8] a n d other c o m p a r a t i v e l y s i m p l e models. Especially B i n d e r ' s f o u r t h - o r d e r c u m u l a n t [9] o f the m a g n e t i z a t i o n or the energy has b e c o m e a f a v o u r i t e o b s e r v a b l e for the d e t e r m i n a t i o n o f the critical point. Besides that the p e a k p o s i t i o n s o f ther- m o d y n a m i c d e r i v a t i v e s p r o v i d e finite lattice transition points, which m a y be e x t r a p o l a t e d by FSS for- m u l a s to the a s y m p t o t i c critical point.

Both the c u m u l a n t and the susceptibility, which are the most used observables for this p u r p o s e are quan- tities, which i n v o l v e differences o f powers o f directly m e a s u r e d o b s e r v a b l e s a n d require thus higher statistics to reach the s a m e accuracy. M o r e o v e r , i n s t e a d o f the true susceptibility a p s e u d o s u s c e p t i b i l i t y is c o m - m o n l y used, where the e x p e c t a t i o n value o f the mag-

Work supported by the Deutsche Forschungsgemeinschaft.

Permanent address: Joint Institute for Nuclear Research, 101 000 Dubna, Russian Federation.

n e t i z a t i o n is replaced by the one o f the m o d u l u s o f the m a g n e t i z a t i o n .

In this letter we w a n t to show that the FSS behav- i o u r o f the e x p e c t a t i o n value o f the square o f the m a g n e t i z a t i o n , though it is not p e a k e d at the transition, allows to d e t e r m i n e the asymptotic critical point a n d the ratio o f the critical e x p o n e n t s 7 a n d u.

We a p p l y our idea to S U ( 2 ) gauge theory on N 3 X N ~ , N ~ = 4 lattices using the s t a n d a r d W i l s o n a c t i o n

S ( U ) = ~ 5 ~ ( 1 - ½ T r U p ) , 4 ( 1 )

where Up is the p r o d u c t o f link o p e r a t o r s a r o u n d a plaquette. The n u m b e r o f lattice p o i n t s in the space ( t i m e ) d i r e c t i o n N~(~) a n d the lattice spacing a fix the v o l u m e a n d t e m p e r a t u r e as

V = ( N o a ) 3, T = I / N ~ a . ( 2 )

O n an infinite v o l u m e lattice the o r d e r p a r a m e t e r o f m a g n e t i z a t i o n for the d e c o n f i n e m e n t t r a n s i t i o n is the e x p e c t a t i o n value o f the P o l y a k o v loop

N r

L ( x ) = ½ T r I-[ U~,x;4, ( 3 )

z = l

o r else, that o f its lattice average

L = - ~ o ~ L ( x ) , 1 ( 4 )

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Volume 298, number 1,2 PHYSICS LETTERS B 7 January 1993 where Ux; 4 are the SU (2) link matrices o f four-posi-

tion x in time direction.

Since, due to system flips between the two ordered states in finite lattices the expectation value ( L ) is always zero, the true susceptibility

z = N ~ ( ( L 2 ) - ( L ) 2),

(5) reduces there to

z v = N 3 ( L 2) . (6)

The quantity Z~ is monotonically rising as a function

offl=4/g 2

or the temperature T. At first sight there is no hint of the transition point. However, below the critical point

Z~=Z, (7)

and we expect therefore Z~ to have the FSS behaviour o f the susceptibility [ 1 ] for T < Tc

Z - N ~ / ~ o _v_{- - ~ ,} _a ~ X \ - . . " ~,-Tv~/~ x~NY') G ~

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Here

T - T c ~ 4/g 2 4 2

- / g c , ~

x = - - or x = (9)

Tcoo , 4 / g c , ~ ² ^'

is the reduced temperature and

Qz

is a scaling function with possible additional dependencies on irrelevant scaling fields xi and exponents y i < 0 leading to correction-to-scaling terms.

Taking into account only the largest irrelevant exponent y~ = - t o and expanding the scaling function

Qz

a r o u n d x = 0 one arrives at

xv=NJV[Co+(Cl +c2Nj~)xN~/V+c3N~°~].

(10) I f we take now the logarithm of the last equation at x = 0 we find

lnzv=ln(N3~(L 2) )

= l n c o + ~ l n N ~ + C a N j ~ +

....

(11) Co

i.e., apart from probably small correction-to-scaling terms ( t o ~ 1 [3] ), we have a linear dependence on In No with slope

y/v,

whereas for x ~ 0 the N~-behaviour is drastically changed due to the presence o f

xN1/~-terms

( 1 / u ~ 1.59 [ 8 ] ). We shall take advan- tage o f this fact to determine the critical point as that fl-value where a linear fit o f l n Z~ as a function o f l n No has the least minimal X2 a n d / o r highest goodness o f

fit. In addition we obtain then the value

ofy/v

from the slope at the critical point.

The SU (2) Monte Carlo data, which we want to use here for a demonstration o f the above m e t h o d were c o m p u t e d on

N3XNT

lattices with N~=8, 12, 18, 26 and N , = 4 and have already been reported on in refs. [ 1,3 ], apart from one new point. Though these data were taken at m a n y (except on the largest lattice) fl-values in the neighbourhood o f the critical point this is not sufficient for a systematic search for the asymptotic critical point. The necessary interpolation may, however, be performed with the density o f states method ( D S M ) [ 10 ]. Following the same lines as in ref. [3], we have reevaluated the data in the very close vicinity (2.2980~<fl~<2.3005) o f the transition. In fig. 1 we show Xv as a function offl for the different lattices. The error corridors were calculated with the jackknife method. The results from the N o = 26 lattice have the largest errors. There only four overlapping histograms were available from refs.

[ 1,3 ]. To improve this situation and to test the sta- bility o f the DSM interpolation we calculated one new point at fl= 2.2988 on the 263 X 4 lattice with a 9 times higher statistics as compared to the point at f l = 2.30.

We observed no change in the DSM interpolation of

150

lO0

50

I I I

2 6

N a < L 2 > ... iiii~ii ...

~ 8

I I E

2 . 2 9 8 2 , 2 9 9 2 . 3 0 0

Fig. 1. The expectation value of the square o f the lattice averaged Polyakov loop N ~ as a function o f f l = 4 / g 2 for N~= 8, 12, 18, 26 and N~= 4. The solid lines were calculated from the DSM, the dotted lines indicate the errors.

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Z~, however the error corridor decreased by 50%. The data points in fig. 1 are the directly measured quan- tities and indicate the fl-values o f the corresponding histograms falling into our fl-interval.

In fig. 2 we present the result o f a linear Z 2 fit o f In Z~ as a function o f In No. At each fl-value we have determined the minimal

z2/Nf,

with N f = 2 the number o f degrees o f freedom. We see that there is a unique fl-value, where the four data points for N~= 8, 12, 18, 26 lie on a perfect straight line. We consider this value,

flm~0=2.2988(1), (12)

to be a very good estimate o f the infinite volume critical point. Indeed it is in excellent agreement with the best determination from the cumulant [ 3 ]

f l c , ~ = 2 . 2 9 8 6 ( 6 ) , (13)

and, as is evident from fig. 2, ~min is fixed by the data with extreme precision. The error in eq. ( 12 ) was es- timated from the change in

flmin

which was induced due to the inclusion o f our high statistics point on the largest lattice. The slope o f each linear fit is compared to

7/u

from the three-dimensional Ising model.

That model and SU (2) gauge theory are supposed to

I I / I _

n/Nf

2 . 2 9 8 2 . 2 9 9 2 . 3 0 0

ha

Fig. 2. The minimal ~(2 per degree of freedom if at each fl a linear fit of the logarithm

ofz~=N~(L 2)

^{a s}a function ofln No is performed; Q is the goodness of fit. The ratio ),/u is the slope of the fit; the dotted line the 3D Ising model value.

be in the same universality class [ 11 ] and therefore to have coinciding critical exponents.

The slope of the linear fit which we find at flmi, is

y / U = 1.931 ( 1 5 ) , (14)

i.e. less than 2% different from the value

y / ~ t = 1 . 9 7 0 ( 1 1 ) , (15)

o f the three-dimensional Ising model [ 8 ]. The error in eq. ( 14 ) comes from the error in flmi, and the error o f the slope from the linear fit. In fig. 3 we show the fits at flmi~ with these two slopes, respectively. We ob- serve only for the highest In N~ ( N o = 2 6 ) a slight difference.

The critical exponent a o f the specific heat is dif- ficult to determine directly. F r o m the hyperscaling relation

a = 2 - d u ,

(16)

one estimates a ~ 0.1 I. The non-singular parts in the specific heat and the energy density are therefore dominating. This leads to a modified scaling ansatz for the energy density,

~.=~.regular-.b N(aa-1)/V Q,(xW 1/u) ,

(17) where we have already neglected the irrelevant scal-

I I I I

zl'

s L

4

3 ~ 7 / u = 1. g 3 1

7/~) = 1 . 9 7 0 I S I N S

2 I I I I

2 . o 2 . 5 a . o L n N a

Fig. 3. The best linear fit at the minimum ofz 2, i.e. at flmi. (solid line) and a fit with the slope fixed to the 3D Ising model value (dashed line ).

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Volume 298, number 1,2 PHYSICS LETTERS B 7 January 1993 ing fields. The regular part o f the energy density is

assumed to be, up to exponentially d a m p e d contri- butions, independent o f No. The respective singular part o f the pressure is proportional to N g a, d = 3 and therefore also m u c h less size dependent than the singular part of the energy. Linear combinations o f the energy and the pressure P are then generally expected to behave like

Ca q - N ( a - 1 ) / v ¢ 2 , ¢1,2 = c o n s t . , (18) at x = 0 . Since apart from only g2 but not N , dependent factors [ 12 ]

P o - P , ~ ( e + P ) / T 4 , (19)

and

P , ~ + P , : ~ ( e - 3 P ) / T 4 , (20)

where Po and P, are the expectation values o f the space and time plaquettes, one m a y use the last three rela- tions to determine ( o e - 1 ) / v in a similar m a n n e r as we have done it before to obtain 7/u. There are, however, some modifications. First, due to the regular term represented by the constant Cl we cannot just fit the logarithm to a straight line in In No. Instead we shall use as variable N~ '~-a)/", where we prefix the exponent and then look for the best linear approxi- mation at each fl-value. As a consequence we have a three parameter fit for Ca, cz and the exponent ( a - 1 ) / v . The measured values for P~-P~ and Po+P~ were again interpolated with the density o f states m e t h o d in the close vicinity o f the deconfinement transition. Quite similarly as for the energy of the three-dimensional Ising model [8] we observed noticeable systematic errors in the interpolation results for P,+P,. This was not so p r o n o u n c e d in the case o f the plaquette difference Po-P~. We have therefore only evaluated the latter quantity. In con- trast to the DSM interpolation for ( L 2) on the largest lattice, which was unaffected by the inclusion o f the additional histogram, we found a considerable change in the D S M interpolation leading to a steeper slope o f the plaquette difference. The resulting critical exponent ratios which we shall calculate below will therefore have to be taken with some care.

A plot o f the DSM results for P ~ - P~ versus fl for the different lattice sizes is shown in fig. 4. To take into account the systematic errors due to the D S M interpolation and the uncertainties coming from pos-

.0006

.0005

P o-Pr

.0004 ~ 12

.0003 ~ 18

26 .00O2 i I I

2.298 2.299 2.300

Fig. 4. The difference of the space and time plaquette expectation values as obtained ~om the DSM for the same lattices as in fig.

I.

sibly too low statistics we proceed as follows. Since the interpolations for N o = 18 and 26 are probably the least reliable, we make three different fits: one where we discard the N ~ = 2 6 data, one where the N o = 18 data are omitted - in these cases we have three parameters and three points and therefore exact solutions - and a fit with all data. The corresponding solutions for the exponent ( a - 1 ) / u are compared in fig. 5 to the Ising value [ 8 ] obtained from the hyperscaling relation

( a - 1)/u= 1 / v - d , . ~ - 1.41 . (21) Averaging the three solutions at flmin we find

( a - 1 ) / u = - 1 . 3 6 ( 1 4 ) . (22) In addition to the ratio o f critical exponents we obtain information on the size o f the regular part at the transition point. This is of importance for theoretical models o f the deconfinement transition. The constant Cl from the fit is 1.93(19) × 10 -4, or expressed with the energy density and pressure

(E'k-P)/Tregular

4 = 0 . 3 8 ( 4 ) . (23) Here, the main contribution will be that o f the energy density, because the size o f the total pressure is only about 7% o f the size o f the regular sum in eq. (23).

157

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- 0 . 5

- i . O

- 1 . 5

- 2 . 0

I I I

... .~ .': n - ~ u -. L.."~. ,<:.Q ...

- - - _

I I I

2 . 2 9 8 2 . 2 9 9 2. 300

Fig. 5. The ratio ( a - l ) / v obtained from linear fits to P~-P, with the ansatz eq. (24), using only the N, = 8, 12, 18 data (solid line ), only the N~= 8, 12, 26 data (dashed line) and from all lattices (dot-dashed line). The points indicate the 3D Ising model value from the hyperscaling relation.

We are indebted to the computer center of the Uni- versity o f Cologne, where we calculated our addi- tional high statistics histogram on the largest lattice.

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