Critical time-dependent fluctuations in Heisenberg spin systems

Volume25A, number 2 PHYSICS L E T T E R S 31 July 1967

C R I T I C A L T I M E - D E P E N D E N T F L U C T U A T I O N S IN H E I S E N B E R G S P I N S Y S T E M S

P . RI~.SIBOIS and M. D E L E E N E R * Universitb Libre de Bruxelles, Belgique

Received 31 May 1967

In the Heisenberg spin system near the critical point, the small wave-number (q) component of the auto- correlation function does not satisfy any diffusion equation but exhibits an oscillatory decay with a char- acteristic time Tq~q-~.

Since the bas ic work of Van Hove [1], cons i - derab le at tention has been given to the spin auto- cor re la t ion function in Heisenberg f e r romagne t s

= (1)

where S~(t) denotes the Heisenberg r e p r e s e n t a - t ion of spin component a(t~ = z,+, -) at la t t ice point a, and the b racke t indica tes an average over the canonical equi l ibr ium dis t r ibut ion.

In pa r t i cu l a r , much work has been devoted to the ana lys i s of eq. (1) nea r the c r i t i ca l point , where the spin f luctuat ions a re known to be ve ry large. Yet the exis t ing theor ies , e i ther pheno- menologica l [1] o r mic roscop ic [2], seem unable to account for the observed neut ron sca t te r ing c r o s s - s e c t i o n s which a r e d i rec t ly connected to eq. (1) (see ref . 2 for an excel lent review).

Recent ly , the authors [3,4] have der ived k ine- tic equations desc r ib ing the behavior of eq. (1) in the h igh - t empera tu re l imi t (flzJ = 0, where J is the exchange in tegra l and z is the number of neighbors) . It is the purpose of the p r e sen t note to d i scuss the extension of this calculat ion valid nea r the c r i t i ca l point.

Let us f i r s t br ie f ly s u m m a r i z e the r e su l t s obtained fo r the inf ini te t empe ra tu r e region, in the Weiss l imi t where z --* oo. We have shown that the d i rec t au tocor re la t ion function r(t) (~ 4 I~aaZ(t)) obeys a kinetic equation

a t r ( t ) = - f:Co(T/r)r(t-r)~r (2)

* Chargd de Recherches au Fonds National de la Re- cherche Scientifique de Belgique.

where the kerne l , Go(T/F ) i s a non l inear funct io- nal of F i tse l f ; on the other hand, the F o u r i e r t r ans fo rm Fq(t) of the au tocor re la t ion function obeys an equation s i m i l a r to eq. (2) with a ke rne l [Go('r/1-~-Gq(T,/I~], this second equation being l i - nea r once r(t) has been de te rmined by eq. (2). In pa r t i cu l a r , we have studied the sma l l wave- number l imi t of F q ( t ) ; b e c a u s e of the separa t ion of the t ime sca les de te rmin ing the evolution of F(T) and of Fq(r), we then obtained a markoff ian diffusion equation [4, eqs. (IH-21) and (III-25)].

The genera l iza t ion of this t r ea tmen t nea r the c r i t i ca l point r equ i r e s two impor tan t modif ica- t ion s: 1. One has to take into account the equilibrium long range co r re l a t ions which a re p re sen t in the sys tem at t ime t = 0 and which influence the dy- namica l evolution of the sys tem; in the Weiss l imi t , these co r re l a t ions a re descr ibed by the well-known O r n s t e i n - Z e r n i k e law [1]; 2. One should also keep in mind that dynamical cor re l a t ions a r e a lso of long range (both in t ime and in space); subsequent ly , it may be shown that the method which expresses the ke rne l Gq(7) in t e r m s of the d i rec t au tocor re la t ion function F(t) is no longer adequate, one should instead desc r ibe the ke rne l in t e r m s of the Four i e r t rans . form Fq(t) of the autocorre la t ion function (now defined in such a way that Fq(0) = 1).

These two steps can be r igorous ly formula ted for z --. oo and one a r r i v e s again at an equation for Fq( t ) s i m i l a r to eq. (2), with a new ke rne l [~O(r/Fq') - Gq(T/Fq')]. Near the c r i t i ca l point , this ke rne l leads to qualitatively different r e - sui ts for the behavior of rq(t) in the sma l l wave- number l imi t . Indeed, for T >~ Tc(T c = zJ/2k) and q ~ 0, we a r r i v e d at the following equation:

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Volume25A. number 2 PHYSICS L E T T E R S 31July 1967

[t[a. 5(kTc)2q 2 ~tFq (t) =- ~o~ ~ x

× ~ (q-ql)2-q~ d3ql q-~l rql(T)rq-ql(z)]Fq(t-r)dT(3)

It is of course imposs ib le to solve eq. (3) ex- act ly. P r e l i m i n a r y inves t iga t ions indicate how- ever that: I. The dominant contr ibut ions to the bracketed ke rne l of eq. (3) come f r o m smal l va lues of q l ~ O(q). This imp l i e s that the c h a r a c t e r i s t i c decay t ime of this ke rne l i s the same as the one which d e t e r m i n e s the evolution of rq(t) i tsel f . The si tuation is thus f o r m a l l y much m o r e s i m i - l a r to that desc r ibed at infinite t e m p e r a t u r e by eq. (2) fo r the d i r e c t a . f . , than to the usual d i f - fusion p r o c e s s . We have shown in ref . 4 that eq. (2) shows an o sc i l l a t o ry approach to equi l i - b r ium. S imi la r ly , we expect thus that eq. (3) wil l neve r lead to a diffusion reg ime . Moreove r we may infer f r o m this o sc i l l a to ry behavior that the t ime F o u r i e r t r a n s f o r m of r q(t) wil l exhibit a peak c e n t e r e d at a non -ze ro f requency (see ref . 5 for a s i m i l a r example) ; this might explain the pseudo sp in-wave behavior s o m e t i m e s advo- cated above the c r i t i c a l t e m p e r a t u r e [ 2].

H. Although it is v e r y diff icult to obtain a r igo- rous evaluation of the c h a r a c t e r i s t i c t i m e sca le 7q governing the decay of Fq( t ) , approximate ca lcula t ions lead to rq ~ q-~, in cont ras t to rq ~ q -4 as obtained by c l a s s i c a l t heo r i e s and in much be t te r qual i ta t ive a g r e e m e n t with exper i - ment. Let us point out that a s i m i l a r t ime sca le has been obtained by Bennett and Mart in [6] and by Kawasaki [7].

References 1. L.VanHove, Phys.Rev. 95 (1954} 1374. 2. W. Marshall. in Proc. Conf. on Critical phenomena.

Washington. 1965 (National Bureau of Standards Misc. Pub. 273 (1966) p. 135.

3. P.R~sibois and M.De Leener. Phys. Rev. 152 (1966) 305.

4. M,De Leener and P.Rdsibois, Phys.Rev. 152 (1966} 318.

5. B.J. Berne, J. P. Boon and S.A. Rice. J. Chem. Phys. 45 (1966) 1086.

6. H.Bennett and P.Martin. Phys.Rev. 138 (1965)607. 7. K.Kawazaki, Kyushu University, Fukuota, Japan.

preprint.

ON THE BEHAVIOR OF THERMODYNAMIC FUNCTIONS NEAR THE CRITICAL POINT OF A FERROMAGNET

M. H. COOPERSMITH Physics Department. Case Institute of Technology and Western Reserve University

Cleveland, Ohio, USA

Received 26 May 1967

The Rushbrooke and Griffiths inequalities for critical exponents of a ferromagnet are demonstrated using a simple graphical approach. As in the original derivations, possitivity of the susceptibility and the spe- cific heat at constant magnetization are assumed.

Recently, a great deal of effort has been made to derive relations between the exponents in the so-called scaling for various thermodynamic quantities near a critical point. Two of the most well-known of these relations are due to Rush- brooke [1] and Griffiths [2], which, in the nora-

66

tion of Essam and Fisher [3], are

~' + 2fl+T' >12

and

a' + fi(8+l) >I 2.

(1)

(2)