Volume 254, number 1,2 PHYSICS LETTERS B 17 January 1991

Existence of a massless phase in a nonabelian spin model in two dimensions

A. Pat rasc io iu , J.-L. R i c h a r d Physics Department and Center for the Study of Complex Systems, University of Arizona, Tucson, AZ 85721, USA

and

E. Seiler Max-Planck-lnstitutJ~r Physik undAstrophysik, Werner-Heisenberg-lnstitutJ~r Physik, P.O. Box40 12 12, W-8OOO Munich, FRG

Received 14 September 1990; revised manuscript received 29 October 1990

Numerical evidence supportive of the existence of an intermediate phase characterized by algebraic decay of correlations in the dodecahedron spin model in two dimensions is presented. In view of the nonabelian character of the model the result is unex- pected, yet it is strongly suggested by a rigorous inequality proven recently.

It is a standard belief among particle- and con- densed-matter physicists that in two dimensions (2D) there is a dramatic difference between the phase structure of the classical ferromagnets enjoying O (2), respectively O (3) symmetry. Indeed, as the temper- ature is lowered, the first one is supposed to undergo the famous Kosterlitz-Thouless transition to a phase characterized by algebraic decay of correlations, while the latter to exhibit forever exponential decay. The difference is explained as stemming from two sources: (i) 0 ( 3 ) is asymptotically free, while 0 ( 2 ) has a vanishing Callan-Symanzik fl-function [1 ] and (ii) O ( 3 ) has topological excitations of finite energy (in- stantons), while 0 ( 2 ) only of infinite energy (vor- tices) [2,3]. Since the two reasons stated above are logically independent, one may wonder which one, if any, could be the true cause of this supposed differ- ence? The rigorous facts established so far are: (i) there is such a phase transition in 0 ( 2 ) [4] and (ii) instanton computations are plagued by uncontrolla- ble infrared divergences [ 5]. Moreover, while it has

Permanent address: Centre de Physique Throrique, CNRS- Luminy, Case 907, F-13288 Marseille Cedex 9, France.

been possible to prove rigorously for O (2) that or- dinary perturbation theory does provide the correct asymptotic expansion as the temperature T = 1/fl goes to 0 [6 ], no such proof exists for O (3) and there are good reasons to believe that perturbation theory gives a false answer [7,8].

In some previous work Patrascioiu [ 9 ] and Patras- cioiu, Seiler and Stamatescu [ 10 ], gave heuristic rea- sons for the existence of a phase characterized by al- gebraic decay in all O ( N ) N>~2 models in 2D. The arguments were based on rough estimates of the en- ergy-entropy balance for the formation of defects. Support for this conjecture came from a rigorous in- equality derived by Richard [ 11 ]; it shows that in a model interpolating between O (2) and O ( 3 ) and en- joying asymptotic freedom, such a phase does exist. The limitation of the result is that the model is not O (3) invariant.

In an earlier paper [ 12 ], we derived a new rigorous inequality relating the dodecahedron and the Z(10) spin models. We recall that the dodecahedron ~ is the largest regular polyhedron (20 vertices) embed- ded in $2. It is invariant under the largest nonabelian subgroup of O (3), the icosahedron group Y with 60

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland ) 173

Volume 254, number 1,2 PHYSICS LETTERS B 17 January 1991

elements. We orient ~ such that its vertices are given by the following unit vectors:

S = (sin a cos -~nn, sin a sin ~nn, cos a ) ( 1 )

and

S = (sin a cos(~nn+~n) ,

sin a s in(Znn+]n) , - c o s a ) , (2)

with a=O~, 02 and n=0 , 1, 2, 3, 4 (sin 01=0.607, sin 02 = 0.982). For the case of nearest neighbor in- teraction we gave a rigorous proof [ 12 ] of the follow- ing inequality:

>/sin z0~ z~ ~o) ( COS ( 00 -- On ) ) fl sin201 • ( 3 )

If we denote by tim(Z(10) ) respectively tim(H) the inverse temperature below which Z(10) , respec- tively ~ lose long range order (1.r.o.) and by tic(Z(10) ) the one below which Z(10) exhibits ex- ponential decay, then inequality (3) implies that provided tim ( ~ ) > tic (Z (10) )/sin20~, the ~ model must possess an intermediate phase with algebraic decay of its two-point function (just as Z (10) does). Since ~ is a non-abelian model, as stated above, the existence of such a phase is contrary to common expectations.

We therefore set up to determine numerically the value of tim ( ~ ) . As we shall report shortly, as a by- product, we also searched directly for the existence of such an intermediate massless phase. The studies were carried out on square lattices of linear size L as large as 336. The Monte Carlo algorithm employed was of the new percolation type [ 13,14]. Specifi- cally, considering a given orientation of ~, we ran- domly select one of its 12 faces and rotate it into the Sz=cos 01 plane. Next we perform a random Z(5) rotation around the z-axis. Finally we re-express the spin in eqs. (1) and (2) as

S = (sin a cos ~nn, a sin a sin ~nn, cos a ) (4)

and

S = (sin a cos(~nn+½n) ,

asin a sin(~nn+½n) , - c o s a ) , (5)

where a = + 1 and n=O, 1, 2. In terms of the Ising

variables a, the ensuing action is ferromagnetic, hence amenable to the Fortuin-Kasteleyn transformation. We therefore update those variables in the standard fashion introduced by Swendsen and Wang [ 15 ]; this procedure reduces critical slowing down, allowing to investigate larger systems, at larger correlation lengths. We would like to emphasize though that our investigation reveals that critical slowing down is still present; hence, the errors quoted reflect only (suffi- ciently) short term fluctuations and ignore possible slow drifts of the results.

Our measurements pertained to the following three categories:

( 1 ) Thermodynamic values of the susceptibility X and correlation length ~. We used the following definitions:

Z= S(x, y) (6)

and

L

where

Zp= ~ ~ S(x , y ) exp(2~ix/L . (7) x , y = 1

The measurements were taken at 1.4-..< fl-..< 1.7 on lat- tices of size L such that L/~> 7. We measured Z and Zp both on the cluster and on the lattice (see ref. [ 13 ] for a discussion of these alternative methods) and found no significant differences in the value of the observables or their errors. The data are given in ta- ble 1. The increase in both Z and Z~ is faster than that predicted by asymptotic freedom for O (3). Attempts to fit the data with a power or exponential (Koster- litz-Thouless) singularity in ti give somewhat un- stable results that depend on the data points in- cluded. In our opinion this is due to the fact that our data points are still rather far from the critical point, which, depending upon the fit used, should occur be- tweenti=2.1 and 2.3.

(2) Binder's scaling variable. Following Binder [ 16 ] one can define the following variable:

U=~- ([(ESx)2]2) ((ZSx)2) 2 (8)

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Volume 254, number 1,2 PHYSICS LETTERS B 17 January 1991

Table 1 Susceptibility X and correlation length ~from fl= 1.4 to 1.7. L denotes the linear size of the (square) lattice used. To compare with the predictions of asymptotic freedom, we also give c¢ = ~fl exp ( - 2nil) × 104 and c z = ~f14 exp ( - 4nil) X 106, which should have been inde- pendent of ft.

fl L ~ X c¢ c x

1.4 54 7.09(3) 82.7(7) 15.01 1.45 72 9.17(9) 126.8(1.8) 14.69 1.5 94 11.67(5) 193.0(1.6) 14.14 1.55 134 15.38(8) 313.8(2.3) 14.05 1.6 172 21.08(13) 540.3(4.9) 14.52 1.65 232 29.53(11) 969.6(4.5) 15.32 1.7 336 41.98(13) 1796.1(6.3) 16.39

7+27 6.84 6.36 6.29 6.56 7.11 7.91

It can easily be verif ied that as L ~ : (a ) Ugoes to 0 if Green 's functions decay exponentially, (b) Ugoes to z i f there is 1.r.o., and (c) no a pr ior i s ta tement can be made on the behavior o f U(L) i f there is al- gebraic decay. We measured U for 1.7~<fl~<2.9 and L = 24, 48, 96, 144. The results shown in fig. 1 indi-

0.05

5 /3 - U L

O.OL,

0.03

0.02

0.01

13=1

= .

2.0

• - - = ~ 13:2.1

~ , ~ - ~ ~ - - - - - - - ~ 13= 2.2

" -~ P=2.3

- " ~/[3=2.75 t +- ~ , l ~ p = 2 . 9

24 48 72 96 120 144 L

Fig. 1 .5 /3- U (eq. (8)) as L at various values offl between 1.8 and 2.9.

cate that this model has a phase transit ion at tic g 2.15. The data are quite s imilar to those previously ob- ta ined for O (2) [ 17 ].

(3) Size of the majority spin cluster. For a given configurat ion we de termine the direct ion o f the total spin of the lattice. We assign 1 (0) to each site i f the angle of its spin with respect to the total is smaller ( larger) than 20.9 ° (ha l f the min imal angle between two vertices of ~ ) . Then we construct all clusters of l ' s and identify the size of the largest one relative to L 2. Only in the 1.r.o. phase should this size converge to a nonzero value. We measured this size (using the cluster algori thm ment ioned above) on a variety o f lattices with L values between 10 and 200. Depend- ing on the lattice size we took between 20000 and 70000 clusters, such that the statistical error in the da ta points was about 1% or less. The data are shown in fig. 2; error bars are not displayed because they would be smaller than the symbols. The figure indi- cates that for fl~> 2.9 there is convergence to a non- zero value, while at f l= 2.7 the L dependence persists and strongly suggests convergence to 0. This suggests tim (c j ) _~ 2.8. (We used the same method to est imate t im(Z (10 ) ) =3 .9 , in agreement with the inequal i ty tim ( Z ( 1 0 ) ) ~<4.6 proven in ref. [ 12]; also t i c (Z(4) ) = 0.88, in agreement with the exact value. Of course we cannot rule out that eventually, for very large L, the da ta would go to zero even at f l=2 .9 , but that would be irrelevant. The crucial point is that they do go to zero at f l= 2.7 which is very strongly suggested by the data. )

In conclusion we believe that the numerics, as they stand, give consistent indicat ions o f the existence of an in termedia te massless phase in the ~ model be- tween f l=2 .15 and f l=2.8 . Our result disagrees with

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Volume 254, number 1,2 PHYSICS LETTERS B 17 January 1991

( I C I ) / L 2

I

10-1

I I I ~11 I I I I l I III~

0 0 0 ~ x x + •

+

I I ~ I I I I I I I

Largest majority cluster

D o 13=3.1 x 13=2.9 • 13=2.8

• + 13=2.7

10_ 2 , , , , ~ , , I , , , , , ~ , , I , , , , , ~ , , I l

10 100 L

Fig. 2. Size of the largest majority spin cluster divided by the volume L 2 for/~= 2.7, 2.8, 2.9, and 3. h

a phase d i ag ram proposed [18 ] on the basis o f the M i g d a l - K a d a n o f f a p p r o x i m a t i o n of the r eno rma l i - za t ion group. Whi le the presen t inequa l i ty (3) can- no t strictly be employed to give a r igorous p roo f of

this fact, the large value of t i m ( ~ ) suggests that a mi ld i m p r o v e m e n t of the inequal i ty would accompl ish this goal. As discussed in an earl ier paper [ 12 ], the in ter- media te phase mus t enjoy Y inva r i ance by analogy with the Z ( N ) , N>~ 5 models , we expect its long-dis- tance behav io r to exhibi t full O (3) symmet ry .

J . - L Richa rd would like to t hank the Un ive r s i t y of Ar izona a n d especially P. Car ru thers for the k i n d hospi ta l i ty ex tended to h im. A Pat rasc io iu is grateful for the hospi ta l i ty of the Max-P lanck- Ins t i tu te . The num er i ca l work was pe r fo rmed part ly at the J. v o n

N e u m a n n S u p e r c o m p u t e r Cen te r a n d par t ly on the CRAY XMP-48 o f the IPP in Garching .

R e f e r e n c e s

[ 11 E. Brezin and J. Zinn-Justin, Phys. Rev. B 14 ( 1976 ) 3110.

[2]J.M. Kosterlitz and D.J. Thouless, J. Phys. (Paris) 32 (1975) 581.

[ 3] A.M. Polyakov, Phys. Lett. B 59 (1975) 79. [4]J. Fr6hlich and T. Spencer, Commun. Math. Phys. 81

(1981) 527. [5] A. Patrascioiu and A. Rouet, Lett. Nuovo Cimento 35

(1982) 107; Nucl. Phys. B 214 (1983) 481. [6] J. Bricmont, J.-R. Fontaine, J.L. Lebowitz, E.H. Lieb and

T. Spencer, Commun. Math. Phys. 78 ( 1981 ) 545. [7] A. Patrascioiu and J.-L. Richard, Phys. Lett. B 149 (1984)

167. [8] A. Patrascioiu, Phys. Rev. Lett. 56 (1985) 1023. [9] A. Patrascioiu, Phys. Rev. Lett. 58 (1987) 228.

[10]A. Patrascioiu, E. Seiler and I.O. Stamatescu, Nuovo Cimento l ID (1989) 1165.

[ 11 ] J.-L Richard, Phys. Lett. B 184 (1987) 75. [ 12] A. Patrascioiu, J.-L. Richard and E. Seiler, Phys. Lett. B 241

(1990) 241. [ 13 ] U. Wolff, Phys. Rev. Lett. 62 (1989) 361. [14] A. Patrascioiu, Nuovo Cimento 105B (1990) 91. [ 15 ] R.H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58 ( 1987 )

86. [ 16 ] K. Binder, Z. Phys. B 43 ( 1981 ) 119. [ 17 ] W. Bernreuther and M. G6ckeler, Phys. Lett. B 214 ( 1988 )

109. [ 18 ] A. Margaritis and A. Patkos, J. Phys. A 19 ( 1986 ) 2855.

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