Theory of disordered spin systems - UCI Math

22 downloads 0 Views 1MB Size Report
THEORY OF DISORDERED. SPIN SYSTEMS. L.A. Pastur and A.L. Figotin. Disordered, i.e., containing random parameters, lattice spin systems are considered.
THEORY

OF

L.A.

DISORDERED Pastur

SPIN

and A . L .

SYSTEMS

Figotin

D i s o r d e r e d , i . e . , containing r a n d o m p a r a m e t e r s , lattice spin s y s t e m s a r e c o n s i d e r e d . It is shown that the f r e e e n e r g y in the m a c r o s c o p i c l i m i t b e c o m e s nonrandom if the p r o b a b i l i t y distribution of the r a n d o m p a r a m e t e r s s a t i s f i e s conditions of spatial homogeneity on the a v e r a g e and vanishing of s t a t i s t i c a l c o r r e l a t i o n s at distant points. The p o s s i b l e orientations of the spins in these s y s t e m s a r e d i s c u s s e d in t e r m s of r a n d o m fields. An a s y m p t o t i c a l l y exactly solvable model of such a s y s t e m is proposed; it d e m o n s t r a t e s different types of orientation, including one c o r r e s p o n d i n g to the spin g l a s s state in which t h e r e is no m a c r o s c o p i c magnetization but the magnetic m o m e n t of individual regions of the c r y s t a l is nonzero. 1.

Introduction

Dilute solutions of atoms of transition metals with large magnetic moment (Fe, Co, Mn) in paramagnets (Cu, Au) have for long attracted the interest of experimentalists and theoreticiams (see [1-4] and the literature quoted there). These systems have a number of rather unusual properties, among which we mention the very sharp peak in the graph • ) of the magnetic susceptibility in zero field and the linear dependence in the limit T --~ 0 of the specific heat on T with a slope independent of the impurity concentration. It was recognized comparatively long ago [2] that these and many other properties of these solutions are due to the indirect Ruderman-Kittel-Kasuya-Yosidainteraction of the spins of the impurity atoms brought about by the exchange of matrix electrons; it has the form

](lx-y])s=s~,

](r)=(kFr)-~cos2kFr,

(1.1)

where s x a r e the spins of the i m p u r i t i e s and k F is the F e r m i m o m e n t u m . The rapid oscillations and weak d e c r e a s e of J ( r ) (i. 1), and also the r a n d o m distribution of the i m p u r i t i e s lead at sufficiently low t e m p e r a . t u r e s to a " f r e e z i n g " of the spins in r a n d o m d i r e c t i o n s and to an i n c r e a s e in • The resulting magnetic s t r u c t u r e is called a spin g l a s s . It is a " c o n g l o m e r a t e " of blocks of only slightly disoriented "frozen" spins whose total orientation however v a r i e s f r o m block to block, so that the m a c r o s c o p i c m o m e n t is z e r o . In [3], it was suggested that this f r e e z i n g of the spins should be r e g a r d e d as a phase transition. It is not h o w e v e r c l e a r how one can solve the s t a t i s t i c a l p r o b l e m c o r r e s p o n d i n g to the interaction (1.1), i . e , a Hamiltonian of the f o r m l

i I - y y'

(1.21

x~

x

where c x is the "population n u m b e r " of site x, equal to 1 with p r o b a b i l i t y c and 0 with p r o b a b i l i t y 1 - e, where c is the concentration of the i m p u r i t y a t o m s . The difficulty of solving this p r o b l e m is also i n c r e a s e d by the fact that b e c a u s e the c o n s i d e r e d s y s t e m is d i s o r d e r e d * it is n e c e s s a r y to a v e r a g e o v e r the positions of the i m p u r i t y a t o m s , i . e . , o v e r the values of {Cx}, the f r e e e n e r g y and not the partition function (the e a s e of "quenched" i m p u r i t i e s , i . e . , i m p u r i t i e s that a r e not in e q u i l i b r i u m with the m a t r i x ) . In [3], it was t h e r e f o r e suggested that (1.2) should be replaced by the e x p r e s s i o n -~- 2/=~s~sy -- 2

hsx,

(1.3)

* T o avoid misunderstanding, note that when h e r e and below we speak of d i s o r d e r e d (respectively, ordered) s y s t e m s we mean s y s t e m s that have (or do not have) in t h e m r a n d o m p a r a m e t e r s (the positions of i m p u r i t i e s , etc), and the t e r m i n o l o g y should not be confused with the e x p r e s s i o n d i s o r d e r e d (respectively, ordered) s t a t e frequently used in statistical physics to designate the state of m a t t e r above (below) the phase transition point. Physicotechnical Institute of Low T e m p e r a t u r e s , A c a d e m y of Sciences of the Ukrainian SSR. T r a n s lated f r o m T e o r e t i c h e s k a y a i M a t e m a t i c h e s k a y a Fizika, Vol.35, No.2, pp. 193-210, May, 1978. Original a r t i c l e submitted April 4, 1977.

0040-5779/78/3502-0403507.50

9 1978 Plenum Publishing C o r p o r a ~ o n

403

in which Jxy a r e independent random v a r i a b l e s with s y m m e t r i c Gaussian dis~ribu~i(:a a~:: ~ : ~ ) ~ the opinion of Edwards and Anderson [3] this would p r e s e r v e the basic feature of (1.2) -. r.he strongly oscillating nature of J ( r ) in (1.1).

in

Using the s e l f - c o n s i s t e n t field approximation, Edwards and Anderson [3] found in this model a point of inflection in the t e m p e r a t u r e dependence of • and they interpreted this as evidence for the existence of a phase transition to the spin glass state. L a t e r , in [4] an attempt was :r~de to give the calculations in [3] an asymptotically exact meaning, and for this it was a s s u m e d that

I.~=~N-'Y.+N-"3"~,

{1.4)

where J0 is a positive constant, and J ~ a r e random variables as in (1.3). It is natural to expect such a model in the limit N -e :~ to give an exact a n s w e r agreeing with the one obtained ha the average field a p p r o x i marion since such a situation obtains in the case of an o r d e r e d f e r r o m a g n e t [5] when J~ = Q. According to [4], this really is so - the result obtained in [4] a g r e e s with the one found in [3]. However, the calculations made in [4J include some assumptions that a r e difficult to v e r i f y and a r e apparently not completely c o r r e c t , which has the consequence that the entropy obtained in [4] is negative at low t e m p e r a t u r e s , as ~he authors themselves note. But this, as will be shown below on the basis of v e r y simple general considerations, is impossible in the c a s e of the Ising model considered in [4]. Note also that a spherical model with the i n t e r action (1.4) was considered in [6], which also contains a discussion of [4]. The p r e s e n t p a p e r consists of two p a r t s . In Sec.2, we study the general p r o p e r t i e s of d i s o r d e r e d spin s y s t e m s , both c l a s s i c a l and quantum. We show that if the random interaction J~y has the p r o p e r t i e s of spatial homogeneity and the statistical c o r r e l a t i o n s between its values at p a i r s of distant points tend to z e r o , then the free e n e r g y of such a s y s t e m in the m a c r o s c o p i c limit is nonrandom, i . e . , it is in p r a c t i c e a certain quantity. Under the specified conditions, this justifies the usually employed p r o c e d u r e for calculating the mean value with r e s p e c t to the r a n d o m p a r a m e t e r s of the free e n e r g y and the identification of it with the observable free e n e r g y . In the s a m e section, we prove that the entropy in the Ising model and in quantum s y s t e m s is positive at all t e m p e r a t u r e s . In Sec.3 (in its f i r s t part} we give an heuristic descri:ption of the possible states of d i s o r d e r e d spin s y s t e m s . The main p a r t of this section is devoted to considering models (classical and quantum) of such s y s t e m s that differ f r o m (1.4). Namely, we a s s u m e that the interaction i n these models has the "separable" f o r m nt

nz

where fh and a k a r e positive constants, and a, (~) a r e r a n d o m and, in general, dependent v a r i a b t e s , satisfying for each k the conditions of spatial homogeneity and vanishing of c o r r e l a t i o n s as ix - y! -~ ~ . The p r e s e n c e of the f a c t o r N -~ suggests that, as in (1.4), this model in the limit N --> ~ must give r e s u l t s that coincide with the ones obtained in the s e l f - c o n s i s t e n t field approximation. It can be shown that this is r e a l l y so, and, in contrast to (1.4), this can be proved r i g o r o u s l y . The a r g u m e n t s employed here a r e a generalization of the method developed in [7] for investigating model Hamiltonians that admit an asymptotically exact solution in the m a c r o s c o p i c limit. Our results, in p a r t i c u l a r the point of inflection in the graph of • at a c e r t a i n t e m p e r a t u r e Tc (see (3.29)) a r e valid in the general case of statistically dependent and a r b i t r a r i l y distributed a(~~ and to a considerable extent do not depend on the f o r m of this distribution. However, it s e e m s to us that the following f o r m of the probability density a~~) is of p a r t i c u l a r interest:

p (a) = ( t - c ) ~ (~) +cq(~),

(!. ~)

where 0 < c < l , q(0:)~0, ~q(u)dcz=l. In a c c o r d a n c e with (1.2), this f o r m of p ( a ) c o r r e s p o n d s to the c a s e when an impurity a t o m is p r e s e n t o r absent at e v e r y point of the lattice with probability e o r t - c. F r o m the mathematical point of view, this assumption means that the r a n d o m variables aN ) in (1.5) a r e replaced by ?~)c,, where the two spatially homogeneous sequences ?~) and cx are a s s u m e d to be independent of one another, which, of c o u r s e , does not rule out the possibility of a statistical dependence between the t e r m s of each sequence (it is only n e c e s s a r y that the statistical c o r r e l a t i o n s between these t e r m s tend to zero as Ix - yl ~ .o, the rate at which this happens being unimportant for what follows). Thus, in our model ~(~) simulates the oscillations of the exchange integrals (1. I), and cx determine the configuration of the impUritY a t o m s in the sample. By means of these quantities, we introduce into the theory a dependence on the impurity concentration, which was not considered in [3, 4]. It can then be shown that the critical t e m p e r a t u r e

404

at which the graph of • has a point of inflection is proportional to the concentration (formula (3.18b)), and in the Ising model and in the quantum models the l o w - t e m p e r a t u r e specific heat has the observed [2] l i n e a r dependence with slope that is independent of the concentration (Eq. (3.33)). 2.

General

Properties

of Disordered

Spin

Systems

1. F o r simplicity, we shall c o n s i d e r the c l a s s i c a l isotropic Heisenberg model [51, although all the results obtained below also r e m a i n true for general c l a s s i c a l and quantum spin models. Thus, in d-dimensional (d = 1, 2, 3) space we c o n s i d e r a simple lattice with unit cell formed by the basis v e c t o r s a~ . . . . . ad, and we take V to be a parallelepiped with sides of length N~a~. . . . , N~a~, a~=laj[. F o r e v e r y point x of the lattice and e v e r y pair of points (x, y) we specify r a n d o m variables hx(w) and Jxy(w), which are an external field and the exchange integrals (here, w is a point of the space ~2 of the possible realizations of these r a n d o m variables). We shall a s s u m e that h x and Jxy have the p r o p e r t i e s of spatial homogeneity and vanishing of the c o r r e l a t i o n s at distant points. The first of them e x p r e s s e s translational invariance on the average, which holds in a d i s o r d e r e d s y s t e m of m a c r o s c o p i c dimensions, and can be formulated as the condition that all mean values* a r e independent of a: into a product of the c o r r e s p o n d i n g mean values. It is readily seen that the r a n d o m v a r i a b l e s that o c c u r in (1.2), i . e . , h = const and Jxy of the f o r m J ( I x - yl)cxCy, where J ( I x l ) is a nonrandom function and cx are the population numbers, satisfy these c o n ditions if the r a n d o m v a r i a b l e s cx satisfy them. The simplest but important example of such cx is provided by statistically independent and identically distributed r a n d o m v a r i a b l e s , which is a sensible choice for cx at low impurity c o n c e n t r a t i o n s . The p r o p e r t i e s we have a s s u m e d for h x and Jxy can be conveniently f o r m a l i z e d by introducing, as is usually done in ergodic theory [8], a shift o p e r a t o r Ta, which acts on the space of realizations ~ is such a way that h~ (T~0)) =h~+=(r (2.1)

J=. ~(To~o) =I=§ ~+o(0)).

(2.2)

Then spatial homogeneity is equivalent [8] to the o p e r a t o r T a p r e s e r v i n g the probabilities of all events, and the ergodic t h e o r e m is true: for any function F(w) on ~ there exists the limit limN -I ~"I,F,,(T,,G)) -------P((0)

(2.3)

aEV

(N = N~N2. .. Nd) , and the p r o p e r t y of c o r r e l a t i o n weakening means that this limit is a determinate quantity, equal to the limit of the mean values of F(w), i . e . , _~= (F).

(2.4)

Note that the v e r y fact of the existence of the limit (2.3) is also true in the case when the summation is p e r f o r m e d o v e r only a c e r t a i n sublattice since invariance under shifts over the sublattice follows f r o m invariance under all shifts. We shall make use of this r e m a r k in what follows. 2.

Having h x and Jxy, we can write down the Hamiltonian of the c l a s s i c a l Heisenberg model: (2.5) x ~ y , x,yOV

* The angular b r a c k e t s contained within them.

x~lv

( . . . ) here and below denote averaging o v e r the realization of the random variables

405

where s x is the spin at the point x, which we shall a s s u m e is a D-dimensional unit v e c t o r (in ~.he c a s e D = I, s x takes the values ~-1, which c o r r e s p o n d s to the Ising model). The f r e e e n e r g y f~ c o r r e s p o n d i n g to (2,5) is defined in the usual m a n n e r as

fN=-(~IN)-'lnZ~, Z~= fe "a~' H ds~,-

(2.6}

~v

where fl-~ is the t e m p e r a t u r e , and ds x is the surface e l e m e n t of the (D - : ) - d i m e n s i o a a : s p h e r e of uni~ a r e a (for D = 1, we have 2 -~ 2 THEOREM.

"'')

in (2.6).

Suppose

2 (ilo:,[) < ~,

(]hol)< ~.

(2,7)

x

Then in the limit V - o ~ * the f r e e e n e r g y fN in ( 2 . 6 ) t e n d s with p r o b a b i l i t y i to a nonrandom l i m i t f, and 1-----lira ~O,~=i/kT, i . e . , In Zn is a convex function of ft. But then -].~-=(~A')-t In Z,~ is a convex function of fi-~, i . e . , the t e m p e r a t u r e , and this means in a c c o r d a n c e with (2.14) that aS.~./~T>~O. (2.15) T h e r e f o r e , there exists the limit S~-(~

In the quantum c a s e and in the Ising model, when T'+0

Z.~=Spexp(--~H~), S!~)=N-~kln • T h e r e f o r e S(~ N

where ~ is the multiplicity of the g r o u n d - s t a t e d e g e n e r a c y ( i . e . , ~ >-- 1). and then by virtue of (2.15)

s,~0

(2.16)

at all t e m p e r a t u r e s . By what we have proved above, fN iS a concave function of the t e m p e r a t u r e (O~tN/OTZrr~ In - ~ - ( ~N) -~ In Sp e-~n" .o to h(]a~ ~) 1) and a r e t h e r e f o r e finite. the l i m i t f r e e e n e r g y is given b y the following m i n i m a x p r i n c i p l e :

T h e r e f o r e , as in [7], we find that

n~

f = rain max ~ ' - 6 - ~ hF~ --

akA~--~-~

Thus, the d i f f e r e n c e between the c l a s s i c a l and q u a n t u m c a s e s is d e t e r m i n e d s o l e l y by the f o r m of the function go(r) in (3.7). * We r e q u i r e s o m e p r o p e r t i e s of this function that a r e c o m m o n to all c a s e s : (p (r) i>0;

(3. lla)

(p'(r) >0, r>0, q/(0) =0;

(3. l l b )

q/(r) ~. (3.17) A nontrivial solution F ~ 0 of this equation, and hence a c e r t a i n p h a s e t r a n s i t i o n a p p e a r at the t e m p e r a I m r e

To=](p" (0) ,

(3. lSa)

which with a l l o w a n c e f o r (1.6) is a l i n e a r function of the i m p u r i t y c o n c e n t r a t i o n :

To=I~" (0) #0. (3.21) Indeed, denoting JFI a x I by ~0, we obtain

I~-t~ol~