Three-dimensional deconfinement transitions and conformal symmetry

Physics Letters B 276 ( 1992) 472-478 North-Holland

P H YSIC S L ETT ER $ B

Three-dimensional deconfinement transitions and conformal symmetry J. Christensen

1 G. Thorleifsson

Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-21O0 Copenhagen, Denmark

P.H. Damgaard MID1T, Physics Laboratory II1, Technical University of Denmark, DK-2800 Lyngby, Denmark

and J.F. W h e a t e r Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK Received 14 November 1991

Conformal field theory and high-statistics Monte Carlo simulations are used to determine the conformal dimension of the Polyakov line operator at the finite temperature SU(3) gauge theory transition in three dimensions. Results are well in accord with the hypothesis placing this transition in the universality class of the 3-state Potts model in two dimensions.

One o f the unexpected applications o f two-dimensional conformal field theory is in the study o f threedimensional deconfinement phase transitions in lattice gauge theories [ 1 ]. From a numerical point of view, one particularly useful result o f conformal field theory in two dimensions ~1 is the explicit expression for any two-point function (¢)i(z)f)i(z')) o f operators O;(z) ( o f conformal dimensions xi) on certain f i n i t e or semi-infinite geometries [3,4]. These expressions o f course turn out to be consistent with ordinary finite size scaling theory, but go much beyond this general formalism in that they provide the actual precise analytical expression for the quoted Green function on any finite size at the coupling of the infinite volume fixed point. The only assumption is that of conformal symmetry at this critical point. The purpose o f this letter is to further illustrate the

usefulness o f these conformal field theory results in the realm o f lattice gauge theories. Our main objective shall be to compute numerically the conformal dimension x o f the Polyakov loop Tr W ( z ) (where the trace is taken in the fundamental representation) o f ( 2 + 1 )-dimensional SU ( 3 ) lattice gauge theory at the deconfinement phase transition point. This transition has earlier been observed to be o f second order [ 5,6 ], and the universality hypothesis [ 7 ] then associates the transition with the critical fixed point of the 3-state Ports model in two dimensions. This theory in turn has completely known critical behavior, with thermal and magnetic indices Yt and Yh, respectively, given by (see e.g. ref. [8 ] for some reviews) 3(l-u) Yt=

2--U

'

Yh=

(3-u)(5-u) 4(2-u)

'

(1)

where for a q-state Potts model (with q ~<4), 1 Present address: National Environmental Research Institute, Riso, DK-4000 Roskilde, Denmark. a' See e.g. ref. [2] for a comprehensive collection of relevant papers. 472

2 0~
1

I

I--

(~x/q) ~< 1 .

(2)

From this follow all leading critical exponents. O f

0370-2693/92/$ 05.00 © i 992 Elsevier Science Publishers B.V. All rights reserved.

Volume 276, number 4

PHYSICS LETTERSB

particular relevance for the present work (with q = 3) is the conformal dimension of the spin operator x = ~ , the susceptibility exponent 7= ~ , and the correlation length exponent v = -~. Since the 3-state Potts model can be completely analyzed using the tools of conformal field theory, it follows - if the universality hypothesis is valid - that also the three-dimensional SU (3) deconfinement phase transition can be analyzed within this framework. As mentioned above, this holds in particular for the finite size scaling expressions at the two-dimensional fixed point. To carry such a program through, the first step consists in determining the infinite volume phase transition point of the theory under study. Also here finite size scaling theory is a useful tool, since one is free to choose a ratio of variables which directly forms a scaling function with no size-dependent prefactors. We shall restrict ourselves to what is probably the simplest of these, Binder's fourth-order cumulant ("the renormalized charge" ) [ 9 ]. To define it properly we first need to introduce our notation. As lattice action we choose the standard summation over all plaquettes Uee SU (3) in the form of

S=ti Z (1-~ R e T r Up)

(3)

P

and in order to extract magnetic susceptibilities we also consider the addition of a "magnetic field" term Sh=½h ~ [Tr W ( z ) + T r

W(z)*]

z

= h ~. Re Tr W(z),

(4)

z

where Tr W(z)=]]-It Uo(z, t) is the Polyakov line referred to earlier, and the product is taken over all timelike links in the periodic time direction of period N~. The physical temperature is then just T=N~-~ in units of the lattice spacing. Since a test of the universality hypothesis is equally non-trivial at any value of N , we choose for simplicity to work with N~= 2. This places the deconfinement transition in the region around t i - 8.1 [6]. We are now ready to define the magnetic susceptibilities

x(,> = 0% 0h4 h=o and

02f~ X= 0h2 h=O'

(5)

20 February 1992

where the free energy is f s = N ~ "2 In Z ( Z being the gauge theory partition function), and N~ is the spatial volume. Finally we can then introduce the renormalized coupling

•(4) (ti, Me ) gr(ti, No) = N 2 z ( t i ' No)2 =Qg( AtiN1/") ,

(6)

where Ati denotes the (small) distance from the critical point, and where we have explicitly indicated the form of the scaling function Qg. Note that at Aft= 0 this gives just a size-independent constant, thus offering an efficient method for locating the infnite volume critical point ~2. One searches for the coupling constant value at which gr(tic, No) =gr(tic, N~ ) for all spatial volumes No and N~,. We have therefore first performed a series of Monte Carlo simulations on a spatially toroidal geometry of size No × No. From an earlier investigation [ 6 ] we were already able to give some rough estimates of the finite volume "critical coupling" tic(No) for various lattice sizes. These gave #3 tic (20) - 8.04, tic (30) --8.11, tic(40) - 8.12 and tic(60) ---8.14. These numbers should of course not be taken as more than giving a rough estimate of the region in which to start the more precise search based on the fourth-order cumulant defined above. Using lattices ranging in sizes from No= 20 to No= 40, we have made a number of relatively high-statistics runs (50000 to 400000 sweeps) for various r-values in the above neighborhood. The results are best summarized graphically, and are shown in fig. 1. The numbers have been obtained by averaging over measurements after excluding the first 25000 thermalization sweeps. The errors indicated are estimated by blocking the data into four sub-blocks, and comparing the average of these subblocks to the total average, as suggested in ref. [ 10 ] #4 Since the scaling function Qg(y) is an analytic function even around the critical point, we can Taylor expand it, and then keep only the leading linear term close to tic. The (hopefully) common crossing point between the lines obtained in this manner for ~2 There are of course corrections to this result when the volumes considered are very small, i.e., when the correlation length is far from the critical regime. ~3 Our precisedefinition offl~(No)based on magnetizationmeasurements on a finite volume is describedin ref. [6 ]. ,M For footnote 4 see next page. 473

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PHYSICS LETTERS B

i,illllllll

-0.5

J

,

I i

i

i

I

i

~ i

o

lv=2o

o

N=30

o

N = 40

\ \

20 February 1992

takes at the infinite v o l u m e critical point is universal [ 11 ]. This n u m b e r follows as a simple by-product o f the above analysis. In this highly indirect m a n n e r we thus, on the basis o f the universality hypothesis, predict a value which we find to be gr(fl¢, N ~ ) = 1.856 + 0.018 for the 3-state Potts m o d e l in two dimensions. We are not aware o f any i n d e p e n d e n t det e r m i n a t i o n o f this number. It appears to be computable using the technique o f ref. [12], and the i n f o r m a t i o n about the four-point function o f the 3state Potts m o d e l p r o v i d e d in ref. [ 13 ]. Having established the infinite v o l u m e critical point, we are now ready to p e r f o r m the finite size scaling analysis based on conformal field theory. The idea, first considered by C a r d y [3,4], is that although conformal invariance is intimately linked with an infinite volume ( a n d infinite correlation length), one can perform a conformal m a p p i n g from the full ( c o m p l e x ) two-dimensional plane to certain finite or semi-infinite geometries. F r o m a numerical p o i n t o f view the most useful geometry is probably that o f an infinitely long c y l i n d e r - the infinitely long strip with periodic b o u n d a r y conditions on the finite width. In this case the two-point function takes the simple form -

\ \ \

-1.5

\ 't \ \ \ .N.

I

-2

i 8.13

[

[

i 8.14

i

i

I

I 8.15

I

I

I 8.16

I

I

[ 8.17

I

I

I

IN~ 8.18

I

I 8.19

P Fig. 1. Enlarged view of the crossing point of the fourth-order cumulant gr, which yields the infinite volume critical point. different N , then determines tic. The corresponding straight-line fits are also indicated in fig. 1. The points o f intersection turn out to be f l = 8 . 1 7 7 + 0 . 0 0 3 , 8.173 + 0.016 and 8.179 + 0.006. Taking the average o f these three numbers, which are already very close to each other, yields our best estimate for the infinite volume critical coupling: flc=8.175 +0.002. As we shall see shortly, this level o f accuracy is m o r e than sufficient for our present purposes. It should be noted that the actual value gr(fl, N,~)

*~ We have also tried a more sophisticated method in estimating the effects of the thermalization. Assuming that the average (as a function of thermalization sweeps) approaches the correct value exponentially, and thus fitting gr to the function g,(N~h) =gr®+A exp(-Ntb/No), where Nth is the number of sweeps excluded and No is an estimate of the non-thermalized initial sweeps•We were not able to fit all the runs to this equation as in some cases the average, even after large amount of sweeps, showed some irregularities. But in those cases where the fit was meaningful the estimate of the non-thermalized sweeps was between 5000 and 20000, thus motivating our choice of using 25000. 474

[3] (~(U, V)~(U', V') > = (21r/L) 2x X (2 cosh{ (2~r/L) [ u - u ' ]} - 2 cos{ ( 2 n / L ) [ v - v '

]})-x

(7)

for an o p e r a t o r ~ (z) o f conformal d i m e n s i o n x. Here the complex coordinate z has been split up into the two c o m p o n e n t s u and v along and orthogonal to the strip, respectively. The fields ~ ( z ) are here normalized such that

<¢~(z)O(O) ) = 1/Izl 2x

(8)

in the infinite plane. The operator we wish to focus on is the P o l y a k o v line defined earlier. Since for S U ( 3 ) this is a complex number, a n d since we wish to m i n i m i z e difficulties with different complex phases at different domains on the lattice during our M o n t e Carlo simulations, we choose more precisely q~(z) = ] Re T r W ( z ) . This object has the same scaling d i m e n s i o n as the complex-valued Polyakov line, a n d is more convenient for our purposes. We shall restrict our attention to this particular geometry, a n d these particular b o u n d a r y conditions. A nice review

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o f other possibilities m a y be found in a p a p e r by C a r d y [ 8 ]. O f course, in numerical simulations we want both spatial directions to be o f finite extent, b u t the idea is now to choose the longitudinal direction far larger than the width. T h e n the correlation length, which from eq. ( 7 ) above is seen to equal ~= (2nx)-~L

/"

20 February 1992

0.5

0.4

(9)

should also be on the o r d e r o f L, a n d the deviations from choosing a truly infinite cylinder are expected to be negligible. It follows from this simple argument that the relevant n u m b e r to keep small is L/Loo, where Loo is the length o f the cylinder. This is all based on a c o n t i n u u m formalism, which ignores short-distance deviations due to the finite lattice spacing. O n a lattice we are also faced with a lower limit on L, on the order o f 3 - 4 lattice units [ l, 14 ] if we wish to a v o i d such lattice-distance artifacts. In our M o n t e Carlo simulations we m u s t find a c o m p r o m i s e between these two requirements. We choose p e r i o d i c b o u n d ary conditions also in the longitudinal direction. In the first part o f o u r numerical study o f eq. ( 7 ) we have therefore c o n c e n t r a t e d on quantifying the above considerations. Precisely how small do we need to keep the ratio f = L/Loo in o r d e r to effectively simulate an infinitely long cylinder? One way o f determ i n i n g this is to establish at which ratio f the correlation function in the longitudinal direction at length ½Lo~ has decreased to zero within errors. The ratio f at which this occurs is clearly in no way universal, a n d needs to be established on a case-by-case basis. In fig. 2 we show results for a series o f runs on lattices o f sizes 6 0 × L , with L taking values between 4 a n d 12. Since we at this p o i n t do not directly need the error bars, we have not included t h e m in the figure. There is a clear t r e n d in the way the longitudinal correlation function decreases as a function o f distance. Only for lattices with L = 4 do we observe a correlation at half-distance c o m p a t i b l e with zero. We have repeated the same exercise at other values o f the Loo, with the same conclusion as concerns the value o f the ratio f. F o r the present theory it needs to be smaller than or on the o r d e r o f 0.07. This result is consistent with the m o r e naive estimate above ,5. #5 During the runs needed to establish these facts we also collected data which we in any case will use for an ordinary finite size scaling analysis [ 15] (where an infinite longitudinal direction need not be assumed). See later.

0.3

0.2

0.1

0

5

10

15

20

25

30 zlu

Fig. 2. The two-point function Fas a function of the longitudinal distance Au for a lattice of length L~ = 60. Because of periodic boundary conditions we can probe only half the lattice. Note that only the lattice of width L = 4 gives a correlation at half-distance which is compatible with zero, within errors. As our next step we have then m a d e extended M o n t e Carlo runs on lattices c o m p a t i b l e with the b o u n d s given above. F o r the actual test o f eq. ( 7 ) above we have used lattices o f sizes 150 × L, L = 4, 6, 8, 10, 12 a n d 2 0 0 × L , L = 4 , 6, 8. The n u m b e r o f sweeps used has for all these runs been 150000 or more. To m i n i m i z e the possible error introduced by considering only the infinite v o l u m e transition couplingpc quoted above, we have also m a d e several runs at nearby fl-values. As expected, a n d in accord with previous investigations [ 1 ], this h a d no measurable influence on the results. In total we are here going to report on runs done at fl= 8.170, 8.175 and 8.180. We have also m a d e a n u m b e r o f M o n t e Carlo simulations on lattices not c o m p a t i b l e with the above restrictions on the lattice sizes, since these runs in any case could be used for an o r d i n a r y finite size scaling 475

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PHYSICS LETTERS B

analysis. These results will be reported at the end o f our paper. We measure the correlation function in both u- and v-directions, but due to the periodicity o f the lattice we only get the correlation function in h a l f o f each direction. We have then p e r f o r m e d a two-point fit o f the d a t a to eq. ( 7 ) , fitting both the conformal d i m e n sion x a n d the overall normalization. We fit simultaneously the Au and Av dependence. To a v o i d finite lattice spacing artifacts, we introduce an a p p r o p r i a t e short distance cutoff, here chosen as ( A u ) 2 + (Av) 2 >19. In fig. 3 we have plotted the results o f these fits, as a function o f f , for all the runs we have perf o r m e d (in some cases the points plotted are the weighted average for m a n y different lattices, having the same ratio f ) . I f we limit ourselves to ratios such thatf~<0.07, we find the values quoted in table 1. F r o m fig. 3 we see a tendency o f a slight drift in the

L

±

l

~=0.1~

0.1

20 February 1992

Table 1 The conformal dimension x of the Polyakov line, obtained by first using bare data from lattices with f~<0.1, and then linearly extrapolating the data to f=0, as shown in fig. 3. The errors indicated are those of the 95% confidence level. Taking the weighted average of these last numbers gives x= 0.134 _+0.005, to be compared with the x= 0.133... value of the 2D 3-state Potts model. fl

x, f~< 0.07

x, extrapolated

8.170 8.175 8.180

0.109+0.006 0.118_+0.003 0.106 _+0.007

+o.oo6 0.131_o.oo9 +0.003 0.125_o.olo +0.007 0.141_o.oo6

numbers we obtain for various f-values, consistent with our earlier remarks on the b e h a v i o r o f the correlation function. It is clear from the graphs that our present results are well described by a small linear decrease in the measured value o f the exponent x for all three r - v a l u e s in question. Just for completeness we have therefore also m a d e a linear extrapolation down to f = 0 on the basis o f these numbers, as is shown in fig. 3. Extrapolating in this m a n n e r to f = 0 (corresponding to L ~ o = o o ) , leads to the estimates for x shown in the third column o f table 1. Taking the average o f these numbers yields X=0.134+0.005 ,

fl = &175

0.1

0

-~ =8.17o

0.1 ~

0

0

~

1

0.2

0.3

0.4 L/L

Fig. 3. Our measure values of the conformal dimension x as a function of the ratio fdescribed in the main text. Also shown is a linear fit to these measurements, from which the extrapolated values ofx which are listed in table 1 were extracted. 476

(10)

which is in r e m a r k a b l y good agreement with the conformal d i m e n s i o n o f the spin operator in the 3-state Potts m o d e l in two dimensions ( x = 2 = 0 . 1 3 3 . . . ) . Even i f we only make the comparison with the raw numbers o f table 1, the agreement is still excellent. Having so m a n y sets o f data at our disposal we have also been able to perform a fit o f the explicit L-dependence o f eq. (7). This obviously tests this equation in a new a n d again highly non-trivial manner. Keeping L ~ , Au a n d Av fixed, we thus again p e r f o r m a t w o - p a r a m e t e r fit to the data, this time using the different values o f L available. The results, averaged over different values o f Au and Av, and using the same cutoff as before, are shown in table 2. The total average o f these results is x = 0.123 + 0 . 0 2 2 , again consistent with the value o f the 3-state Potts m o d e l in two dimensions. As an i m p o r t a n t a posteriori check on the insensitivity to the very precise value o f tic, we note that for none o f the fits do we see any systematic deviations for the three different ]/-values we have considered.

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PHYSICS LETTERS B

ruble 2 Numerical results for the conformal dimension x, obtained by fitting the L-dependence in eq. (7) using a fixed L~. The results are averaged over different Au and Av values, and a short-distance cutoff r>~ 3 is used. The errors indicate one standard deviation. A weighted average of these numbers gives x = 0.123 _+0.022. fl

L~

x

8.170 8.175 8.180 8.175 8.175

60 60 60 150 200

0.123_+0.031 0.118+0.008 0.146+0.026 0.184+0.060 0.219_+0.080

During the preparation and excecution of the above program to determine the conformal dimension of the Polyakov line operator for this theory, we have collected a number of high-statistics results on many different lattice sizes. As mentioned earlier, many of these results fall outside the domain of validity ofeq. (7), but they can of course be used to investigate the usual finite size scaling formalism. These runs have (mostly) been performed on rectangular lattices with varying values for both L and L~. Although finite size scaling can be tested by means of such lattices (as opposed to symmetric lattices, which are more commonly used), it is important to realize that within any given finite size scaling analysis the ratio f=L/Loo must be keptfixed [ 15 ]. With the amount of data we have available, this is fortunately no serious restriction. We have thus been able to consider two different ratios f = ~o and f = ~, with L taking the values 4, 6, 8, 10 in the former case and 6, 8, 10, 12 in the latter. This we have done for all three B-values quoted earlier. An illustrative example of the measured susceptibility versus L is shown in a log-log plot in fig. 4, here for fl= 8.17 5 and f = ~. The other cases are very similar. From finite size scaling theory we would expect the following scaling behavior to leading order

z

20 February 1992

loo

I

I

I

I

I

I

I

I

I

t

~

I

I

f = 0.1

=8.175

I l

4

6

8

10

Fig. 4. Ordinary finite size scaling test of the magnitude of the susceptibility Z at the infinite volume critical point, here shown for f = ~ . A least-z 2 fit yields in this case an exponent ratio y / v = 1.675 _+0.010. See table 2 for the other measured values. Table 3 Estimate of the ratio ? / v obtained by applying finite size scaling to the susceptibility Z. The errors indicate 95% confidence limits. A weighted average of these numbers gives y / v = 1.690+0.011. fl

Y,r=l/lO

gr=l/6

8.170 8.175 8.180

1.664_+0.031 1.730__ 0.031 1.607 _+0.039

1.756_+0.017 1.675 +_0.010 1.675 + 0.026

[151: z( L, Aft) = L ~/~Qx( Afl L 1/~) .

( 11 )

Note that in this case we are sitting right on the infinite volume peak, with Aft=0. In order to estimate the ratio 7/v we have therefore performed a straight line fit to the logarithm of the susceptibility (this fit is also shown in fig. 4). The results for all measurements are summarized in table 3. Taking the average of those numbers gives

y/v= 1.690+0.011,

(12)

which should be compared with the expected 3-state Potts model value of~/v = 1~ = 1.733...). By means of the hyperscaling relation (in 2D) 7= ( 2 - r / ) v ,

orx=l-7/2v,

(13)

this also gives us a consistency check on the value of x quoted earlier. 477

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In the c o n t i n u u m one can o f course derive an e x a c t expression for the susceptibility Z on the geometry o f an infinitely long cylinder, by integration o f eq. ( 7 ) [ 8 ]. This expression is not useful on a lattice realization such as ours, since the largest contribution evidently comes from the short-distance region in which lattice artifacts are p r e d o m i n a n t . Thus even for the lattice sizes for which a test o f eq. ( 7 ) is possible, it is not meaningful to c o m p a r e with the integrated formula. O f course, we could numerically evaluate the expected finite-lattice susceptibility by s u m m i n g eq. ( 7 ) over the lattice, but this can hardly be said to test eq. ( 7 ) in any new way. To conclude, we feel that we have presented rather convincing evidence that the recent conformal field theory results concerning two-dimensional critical p h e n o m e n a can with great advantage be applied to the study o f three-dimensional deconfinement transitions, as was suggested earlier in ref. [ 1 ]. O u r numerical results cannot o f course pin down with complete certainty the relevant fixed point o f the threed i m e n s i o n a l SU ( 3 ) deconfinement phase transition we have studied, but all results are well in accord with 2D 3-state Potts m o d e l exponents. O u r statistics was accumulated during a p p r o x i m a t e l y 300 C P U hours on an A m d a h l 1100 vector processor. It will clearly be difficult to i m p r o v e substantially on the accuracy we have presented here. This work was s u p p o r t e d partly by EEC Science Twinning G r a n t no. SC1-000337, a n d D a n i s h Sci-

478

20 February 1992

ence Research Council ( S N F ) G r a n t no. 11-8668.

References [ 1] J. Christensen and P.H. Damgaard, Phys. Rev. Lett. 65 (1990) 2495; Nucl. Phys. B 354 (1991) 339. [2] C. Itzykson, H. Saleur and J.B. Zuber, eds., Conformal invariance and applications to statistical mechanics (World Scientific, Singapore, 1988). [3] J.L. Cardy, J. Phys. A 17 (1984) L385. [ 4 ] J.L. Cardy, J. Phys. A 17 (1984) L961; A 19 (1986) L1093; H.W. Blrte, J.L. Cardy and M.P. Nightingale, Phys. Rev. Lett. 56 (1986) 742; I. Affieck, Phys. Rev. Lett. 56 (1986) 746. [ 5 ] J.F. Wheater and M. Gross, Z. Phys. C 28 ( 1985 ) 471. [6] J. Christensen, G. Thorleifsson, P.H. Damgaard and J.F. Wheater, CERN preprint CERN-TH-6106-91. [ 7 ] B. Svetitsky and L.G. Yaffe, Nuci. Phys. B 210 [FS6 ] ( 1982) 423. [8] F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235; J.L. Cardy, in: Phase transitions and critical phenomena, Vol. 11, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1987). [91 K. Binder, Z. Phys. B 43 ( 1981 ) 119. [101 J. Engels, J. Fingberg and M. Weber, Nucl. Phys. B 332 (1990) 737. [111 V. Privman and M.E. Fisher, Phys. Rev. B 30 (1984) 322. [121 T.W. Burkhardt and B. Derrida, Phys. Rev. B 32 (1985) 7273. [131 V1.S. Dotsenko and V.A. Fateev, Phys. Lett. B 154 (1985) 291. [14] P. Suranyi, Nucl. Phys. B 300[FS22] (1988) 289. [15] M.N. Barber, in: Phase transitions and critical phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz, (Academic Press, New York, 1983).