Volume 241, number 2 PHYSICS LETTERS B 10 May 1990

SOME INEQUALITIES FOR DISCRETE SPIN MODELS

A. PATRASCIOIU, J.-L. R I C H A R D l Physics Department and Center for the Study of Complex Systems, University of Arizona. Tucson, AZ 85721, USA

and

E. SEILER Max-Planck-lnstitut J~r Physik und Astrophysik, Werner-Heisenberg-lnstitut J~r Physik, P.O. Box 40 12 12. D-8000 ,14unich 40, FRG

Received 19 January 1990; revised manuscript received 26 February 1990

We consider classical ferromagnets with nearest neighbor interaction in D dimensions and single site measure restricted to either Z(N) or dodecahedron symmetry. By suitable parametrization of the spin, we re-express the measure in a form amenable to the application of Ginibre's inequalities. The ensuing inequalities relate the Z(N) models to the lsing model and the dodeca- hedron to the Z(10) model. The latter result suggests that contrary to common expectations, in 2I) this nonabelian model pos- sesses a massless intermediate phase.

It has been rigorously proven that the 2D O ( 2 ) ferromagnet, as well as the Z ( N ) ferromagnets for N suffi- ciently large possess a phase in which correlation functions decay algebraically, rather than exponentially, as is demonstrably the case at high temperature [ 1 ]. It is generally accepted that the corresponding nonabelian models, such as O ( N ) N > 2 or the dodecahedron, do not exhibit algebraic decay for any temperature. In the absence o f rigorous proof, these beliefs stem from two sources: (i) the existence of asymptotic freedom [2 ], and (ii) the existence of topological excitations of finite energy (instantons) for N = 3, as opposed to the infinite energy ones (vortices) for N = 2 [ 3,4]. However, the justification of perturbation theory as the correct procedure for pro- ducing the asymptotic expansion as T ~ 0 has not been achieved for the nonabelian cases N > 2 [5 ], while in- stanton computat ions have been shown to be hopelessly plagued by infrared divergences [6] and would lead to the violation of the cluster decomposing property (if only instanton contributions were summed up).

Motivated originally by the above stated difficulties, an opposite point of view has been expressed [7,8] according to which for all N>~ 2 at sufficiently large inverse temperature r , the O (N) ferromagnets are massless and their two point functions exhibit algebraic decay in 2D. The heuristic arguments rely upon energy-entropy estimates of the type Peierls used to prove the existence of a phase with long range order (LRO) in the Ising model [9 ]. A strong indication that this may indeed be the case is contained in a rigorous inequality proven by Richard [ 10], showing that a model interpolating between O (2) and O (3), which is asymptotically free, does indeed possess a massless phase at large ft. This model consists of modifying the O (3) model by restricting the single site measure so that, say sin O>~c for c> 0 when using spherical coordinates. It has the unpleasant feature of being only O (2) invariant.

In this paper we derive a new rigorous result which, while not proving outright, gives credence to the existence of a massless phase in the O (N) N>~ 3 models in 2D. We consider the dodecahedron ( •J ) spin model, which is invariant under the largest discrete nonabelian subgroup of O (3), the icosahedron group Y with 60 elements.

On leave from Centre de Physique Throrique, Luminy, Case 907, F-13288 Marscille Cedex 9, France.

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We orient the dodecahedron so that its vertices are given by the following set of unit vectors (on the sphere $2):

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

and

S=(sinacos(2zcn+~zt),sina • 2 l s m ( : E n + : O , - c o s a ) , (2)

with a=Ot, 02 and n=0 .... ,4. The angles 0t and 02 are given by sin 0t =0.607, sin 02=0.982. We notice then that all the vertices of the dodecahedron are in fact given by

S= (sin a cos ~, sin a sin ¢, cos a cos (Sq~)) , (3)

where now ~ takes its values in Z(10) (~= ~oZrn, n = 0, ..., 9 ) and a remains unchanged. The partition function being given by

Z= ~ c x p ( f l ~ S , ~ ) , (4) (Si} <ij>

then reads

Z = ~ e xp ( f l ~ [cosa, cosajcos(SODcos(SOj)+sina, s inajcos(¢,-¢:)]) . (5) {a,~} ( ij>

We see that at fixed { a }, this model is a particular case of the following, more general, Z ( 1 O) model defined by the measure

d/z(H,.o ( 0 ) = exp (<~> [ll,jcos(5~)cos(5¢?/)+J,,cos(¢,-¢j)])dq). (6)

Since H e and Jo are nonnegative, the usual Ginibre inequalities [ 1 l ] apply and guarantee monotonicity in H and J separately. A simple algebra leads to

(S(o I ) ~ ( I ) .4..~'~2)~'(2) -~ - . . . . )~>~sin20, ( c o s ( 0 o - - ~ ) \z(l°)/psm2o, , (7)

where ( ) ~ ( t ° ) denotes the expectation value for the standard Z(10) model (Iio-0, Jo=fl) at inverse temper- aturefl, while ( )~ denotes the expectation value for the ~ model under investigation.

We will denote by tim (Z (10) ) and tim ( ,@ ) the inverse temperatures below which Z (10) and :~, respectively, lose LRO and by tic(Z(10)) the inverse temperature below which Z(10) exhibits exponential decay. Since numerical simulations [ 12,13 ] yield/3~ (Z ( 10 ) ) = 1.1, while tim (Z (10) ) = 3.9, the rigorous inequality ( 7 ) would imply that forfl in the interval

&(Z(10) )/sin20t >~ fl>~flm( ~ ) , (8)

the expectation value ( S~ ~) S~, 1) + S~2)S (2)) ff would exhibit algebraic decay. The proviso is that tim ( ~ ) = 2.8 should be larger than tic(Z( 10))/sin20t. Unfortunately the numerical data [ 14] yield flm(~) =2.8 [while tic (Z ( l 0) )/sin20~ = 3.0 ]. Consequently inequality (7) does not imply the existence of a massless phase in the cj model. However, the same numerical data [ 14] suggest that the model exhibits exponential decay of its two point function for fl<fl~(9)-~ 2.15. Therefore, an intermediate massless phase does exist in this nonabelian model in 2D and probably if the inequality ( 7 ) could be improved, this would follow rigorously from a compar- ison with Z(10).

One may wonder if the intermediate phase we are describing corresponds to the breaking of the group Y to one of its subgroups - this effect can be shown rigorously to occur in Z (4) with a suitable action [ 15 ]. With the nearest neighbor action we have considered, at sufficiently small temperatures, the ~ model possesses 20 pure

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Gibbs states. They are invariant under Z(3) and under suitable reflections. Since one would expect the sym- metry to bc enhanced in the intermediate phase, a candidate subgroup [containing Z (3 ) ] is T (tetrahedron group). However, this subgroup does not contain the reflections, hence, the intermediate phase must have full Y symmetry.. We would expect that, in complete analogy with the Z(N) N > 5 case [16], in the intermediate massless phase at large distances the Y symmetry is enhanced to O (3). The only rigorous statement that can be made, though, is that since any polynomial of degree 2 or 4 which is invariant under Y is also invariant under O (3), the two and four point function are O (3) invariant in the intermediate phase.

The next topic wc address is the Z(2N) models. Patrascioiu [7] and Patrascioiu and Seiler [ 17], using en- ergy-entropy estimates, argued that for D > 1 these models (with nearest neighbor interaction) should exhibit a phase transition at tip (Z (2N)) obeying

~p(Z(2N) ) >~ 2,8c(Z(2) ) / [1 - c o s ( n / N ) ] . (9)

Here fie (Z (2) ) is the critical temperature of the I sing model and the (nonrigorous) inequality (9) is obtained by arguing that the first phase transition encountered in increasing the temperature away from zero is when the smallest allowed (nonzero) value of the spin gradient becomes entropically favored; moreover, one would ex- pect more entropy for defects in Z (2N), hence (9). In this paper we will prove that whether or not the transition described above occurs, the one associated with the loss of LRO does not obey cq. (9) and in fact obeys the opposite inequality. This follows simply from Griffiths' inequality [ 18] if we orient the x-axis so that

S x = ~ c o s ( n n / N ) ,

with n = - ½ ( N - 1 ), - ½ ( N - 3), ..., ~ ( N - 1 ) and a = _+ 1. Indeed in terms of the a variables the action

[a, aj c o s ( n n J N ) c o s ( n n J N ) + s i n ( n n J N ) s i n ( n n J N ) ] (10) <ij>

is that of an lsing ferromagnet of minimal inverse temperature,8 cos 2 [ ~r ( N - 1 ) / 2 N ] , so that one has

( S , ( O )S:,( n ) ) {{2N) >. • , .~ s tn-(~/2N) z 2) <&(0)&(n) >~)s~°~(,/2,,,). (11)

If we denote by,sm(Z(2N) ) the inverse temperature below which LRO is lost, inequality ( 11 ) implies

2,8¢(Z(2)) , s m ( Z ( 2 N ) ) ~ 1 - - c o s ( g / N ) " ( 1 2 )

We conclude by pointing out that the same observation can be used in a variety of discrete spin models (such as the discrete gaussian or equivalently the Villain Coulomb gas) to relate them to the lsing model and forms the basis of several new types of Monte Carlo algorithms [ 19,20], which employ the Fortuin-Kasteleyn repre- sentation of the Ising model as a percolation model. Our own data for the .c2 and Z ( I 0 ) models [ 14] were obtained using this type of algorithm, which reduce dramatically critical slowing down.

J.-L. Richard would like to thank the University of Arizona and especially P. Carruthers for the kind hospi- tality extended to him. A. Patrascioiu is grateful for hospitality of the Max-Planck-Institut fiir Physik (Munich).

References

[ 1 ] J. Fr6hlich and T. Spencer, Commun. Math. Phys. 81 ( 1981 ) 527. [2] E. Br6zin andJ. Zinn-Justin, Phys. Rcv. B 14 (1976) 3110. [3] J.M. Kosterlitz and D.J. Thouless, J. Phys. (Paris) 32 (1975) 581. 14] A.M. Polyakov, Phys. Left. B 59 (1975) 79. [ 51J. Bricmont, J.R. Fontaine, J.L. Lebowitz, E.H. Lieb and T. Spencer, Commun. Math. Phys. 87 ( 1981 ) 545. [ 61 A. Patrascioiu and A. Rouet, Nuovo Cimento 35 (1982) 117; Nucl. Phys. B 214 ( 1982 ) 481.

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[ 7 ] A. Patrascioiu, Phys. Rev. Lctt. 58 (1987) 2285. [ 8 ] A. Patrascioiu, E. Seiler and I.O. Stamatescu, Nuovo Cimento 11D ( 1989 ) 1165. [9] R. Peierls, Proc. Camb. Phil. Soc. 32 (1936) 477.

[ 10 ] J.-L. Richard, Phys. Left. B 134 ( 1987 ) 75. [11 ] J. Ginibre, Commun. Math. Phys. 16 (1970) 310. [12] E. Seiler, 1.O. Stamatescu, A. Patrascioiu and V. Linke, Nucl. Phys. B 305 (1988) 623. [ 13 ] R. Gupta et al., Phys. Rev. Lett. 61 ( 1988 ) 1996. [ 14] A. Patrascioiu, E. Seiler and J.-L. Richard, Existence of a massless phase in a 21) nonabelian fcrromagnet, University of Arizona

preprint ( 1989 ). [ 15] C.E. Pfister, Commun. Math. Phys. 86 (1982) 391. [ 16 ] J. Fr6hlich and T. Spencer, Commun. Math. Phys. 81 ( 1981 ) 527. [ 17] A. Patrascioiu and E. Seiler, Phys. Rev. Lctt. 60 (1988) 875. [ 181R. Griffiths, J. Math. Phys. 8 (1967) 478. [ 191U. Wolff, Phys. Rev. Lett. 62 (1989) 361. [20] A. Patrascioiu, Employing the lsing representation to implement nonlocal Monte Carlo updating in O(N) model, University of

Arizona preprint (1989).

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