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~
