On classical spin-glass models

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On classical spin-glass models

D. Grensing, R. Kühn

To cite this version:

D. Grensing, R. Kühn. On classical spin-glass models. Journal de Physique, 1987, 48 (5), pp.713-721.

�10.1051/jphys:01987004805071300�. �jpa-00210490�

On classical spin-glass ^models

D. Grensing ^{and R. Kühn}

Institut für Theoretische Physik ^und Stemwarte, Universität Kiel, D-2300 Kiel 1, ^F.R.G.

(Requ ^{le 1er} septembre 1986, accepté ^{le 5} janvier 1987)

Résumé.

Nous présentons

^méthode simple ^et générale pour résoudre des modèles de champ ^{moyen de}

verres de spin, ^dans lesquels le caractère aléatoire des liens peut s’exprimer

^{termes de} variables aléatoires de sites. Nous observons que l’hamiltonien de

modèles est

forme quadratique dans les aimantations de sous-réseaux et que

pouvons évaluer l’énergie ^libre

^termes des valeurs propres et vecteurs propres,

sans utiliser la méthode de réplique. Alors que, dans le ^cas de modèles séparables, le nombre de paramètres

d’ordre nécessaires à la description ^du système ^est indépendant de la distribution de probabilité des variables de site, ^{dans le cas} de modèles

séparables, ^il augmente lorsqu’on s’approche de distributions continues.

Abstract.

A simple general ^{method is} presented ^for solving mean-field spin-glass models where the bond- randomness is expressible ^{in terms of}

underlying site-randomness. The method is based

the observation that the Hamiltonian of these models is

quadratic form of sublattice magnetizations and that the free energy

can

be evaluated in terms of eigenvalues ^and eigenvectors ^without using replicas. ^Both separable ^and

separable ^models

be solved. While for separable models the number of order-parameters necessary ^to describe

system ^is independent ^{of the} probability distribution for the site-variables this proves ^{not to} be the

case

for the non-separable ^models, where this number increases,

continuous distributions

approached.

Classification

Physics ^Abstracts

75.40

1. Introduction.

If one considers mean-field models of spin-glasses,

one can divide the available models into two classes in the following way : the first class consists of the

true random-bond models, ^{where the} couplings

between interacting spins ^{are taken to be} indepen-

dent random variables [1, 2]. ^{The solution of these}

models can be obtained by ^the n-replica ^trick [1, 3],

and has required the invention of sophisticated

schemes of (hierarchical) replica-symmetry breaking [3, 4]. ^{In models of the second class, the}

bond-randomness is expressed ^{in terms of some} underlying hidden site-randomness and is thus of a more superficial ^{nature. This entails} that bond-ran- domness in these models is in general ^{not uncor-} related, ^{even if the} underlying site-randomness is. It has been pointed ^out [5, 6], however, ^{that this} feature retains an important physical aspect ^{of true}

spin-glasses, viz. that they ^{are random with} respect

to the positions ^of magnetic impurities.

The available models in the

random-site class » share the important feature that the random part ^of the interaction is a bilinear function of the underlying

site-randomness, ^{hence can} quite generally ^{be ex-} pressed ^as [6]

Jij = N-’(gi, Jgj) , ⁽¹⁾

where the t’s ^are stochastic vectors in RP, ^{J a real} symmetric p x p matrix and N denotes the number of spins ^{in the} system. Allowing for shifts of the t’s by ^some non-random vector to, equation (1) ^includes

for instance the models of Mattis [7], Luttinger [8]

and van Hemmen [9]. Various methods have been invented to solve these so-called separable models, exploiting e.g. Gaussian linearization techniques [5, 10-12], ^the theory ^of large deviations [9, 13] ^{or the}

fact that, because of the bilinear nature of the ansatz

equation (1), separable ^models possess

natural >>

order parameters [6, 14], ^{in terms} of which their solution can be obtained.

While separable ^{models are} capable ^of reproduc- ing ^certain thermodynamic properties ^{of the} spin- glass phase, ^{such as the} plateau ^{in the} dc-suscep- tibility [13], they invariably ^{lack a} major ^{feature of} spin-glasses, namely ^{the existence of a} large ^number

of metastable low-temperature phases. ^{In a recent}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004805071300

714

paper, however, Benamira et al. [6] showed how this

deficiency may be ^overcome in the framework of

separable models, ^but only ^{at the cost of} introducing

an infinite number (p --+ oo ) of random variables per lattice site.

In this paper ^{we take a} different approach ^{to the}

solution of mean-field spin-glass models in the random-site class. Utilizing ^discrete probability ^dis- tributions, ^we analyse ^{and solve} spin-glass ^{models of}

a novel type, where the random part ^{of the interac-} tion is given by

As before the t’s ^{are taken to} be stochastic vectors in RP with some common distribution, ^but f (t ; t’) ^is

now an arbitrary ^real symmetric function and no

longer ^{needs to be a} bilinear form.

Our method of solution is elementary ^and requires

neither replicas ^{nor the} theory ^of large deviations. It is based on the observation that _every random-site class Hamiltonian

with Ji¡ ^{given by} equation (2), ^{can be} expressed ^{as a} quadratic ^{form of} magnetizations of certain sublat- tices which are of macroscopic ^{size, if the} probability

distributions of the t’s ^are suitably chosen. The

principal ^{result of our} approach is that every ^non-

zero eigenvalue ^{of the} quadratic ^{form must be}

associated with an order parameter ^{of the} system.

Moreover, ^{it will be shown} that, ^for non-separable

models in the random-site class, i.e. those where the function f ⁱⁿ equation (2) ^{is of} higher ^{than bilinear}

order in the t’s, ^the thermodynamic properties depend ^{in a sensitive} manner on the probability

distribution for the random variables. In particular,

the number of order parameters necessary ^to de- scribe the system ^{and thus the} possible ^{number of}

metastable low-temperature phases increases, ^as

continuous probability distributions are approached.

Besides being ^{in marked} contrast to the situation in

separable ^models, this feature _may help ^to incorpo-

rate into spin-glass models the effects of atomic short range order (ASRO), which is known to affect the

magnetic phase diagrams ^of typical spin-glasses in a quite ^intimate way [15].

We have organized ^our paper ^as follows. In section 2 we present ^a solution of the general

random-site model described by equations (2), (3)

for the case of discrete random variables with a finite number of values. Using ^{van Hemmen’s model} [9] ^as

an example, ^we demonstrate in section 3 that our

approach ^also provides elementary ^{solutions to exist-} ing separable models. Section 4, finally, ^{serves to}

illustrate some of the novel features that _{may be}

expected ^from non-separable models in the random- site class.

2. General solution of random-site models.

In this section we present ^the solution of random-site class models described by equation (3) ^{in the case of}

discrete random variables. To be specific, ^{we assume}

the components ^{of the} p-dimensional vectors ti ⁱⁿ equations (1), (2) ^{to be} independent ^{random vari-}

ables drawn from the set An

{at, ^a2,

an} ^with

uniform probability. ^The generalization ^{to non-uni-}

form distributions is trivial (see below). ^{The ele-}

ments of An ^{must, of course, be} disposed ^{of in a} way which depends ^{on the desired} probability distribution for the Jij’s.

Our method of solution is based on the obser- vation that every quenched configuration ^{of the}

random vectors ti ^{leads to a} partition ^of the lattice

12N ^{into nP} disjoint sublattices and that the Hamil- tonian depends ^{on a} given configuration fgi) ^only

through ^this partitioning, ^{so that the sum over all} spin configurations ^{can be} performed ^{in an essen-} tially ^trivial way. Indeed, given p ^and An, ^the C’s ^can only be drawn from a finite set A of nP different vectors. Introducing ^a single ^{index y to enumerate}

the nP vectors ay ⁱⁿ A, ^we find that the sublattices

are disjoint ^and together ^make up the whole lattice.

If we introduce nP corresponding ^block spins ^or

sublattice magnetizations

the Hamiltonian takes a particularly compact ^form in terms of the MY

Here we have defined

and

and have omitted irrelevant terms of order unity,

which do not contribute to the free _energy density ⁱⁿ

the thermodynamic ^limit.

Since V is a symmetric ^{matrix, we can reduce it to}

a diagonal ^form by ^an orthogonal transformation

so that

with

The primed summation in equation (10) ^indicates

that non-contributing ^terms with A "

0 have been omitted. To evaluate the partition ^function

we linearize the exponent ⁱⁿ equation (12) by ^the

well-known Gaussian transformation

- r- r- -- I .. -

and obtain

The sum-over-states is readily performed ^to yield

’

J ....I

where

denotes the fraction of lattice sites which belong ^to

the sublattice f2l., ^{According to the strong law of}

large ^numbers [16] ^the quantities w y converge with

probability ^{one to their mean value n-p in the} thermodynamic limit. In the case of non-uniform

probability distributions the Cd ’Y ^would simply ^con-

verge ^{to some} pY according ^{to their} probability.

The integrals ⁱⁿ equation (17) ^{are evaluated} by ^the

method of steepest descents, ^{and the} thermodynamic

limit of the quenched free energy density ^{is found to}

be

The maximizing parameters t IA ^{are among the solu-}

tions of the following ^set of transcendental equations

and again ^{we note that} only ^those t,’s ^{are involved}

which correspond ^{to non-zero} eigenvalues A ,.

It is easy ^to show that the t 1L ^{as solutions of}

equations (20) ^are linear combinations of sublattice

magnetization densities defined by

Since sublattice magnetizations ^{are real} quantities, ^it

follows from equations (20) ^that those t,-, ^which correspond ^to negative eigenvalues, ^hence having

s2 ^{= - 1, must be} purely imaginary. It is therefore convenient to rewrite the fixed point equations (20)

in terms of the real parameters

which leads to

In terms of the solution {Y IL} ^{of the above equations}

which corresponds ^{to the} maximum in equation (19)

the free energy per spin ^is simply given by

Whenever equations (23) allow several solutions,

one still has, ^{of course to decide which of the} phases (solutions) ^{minimizes the free} energy, hence is

absolutely stable, and which of them must be

regarded ^as metastable (corresponding ^{to local mini-}

ma of the free energy) ^{or even unstable.}

The most important consequence of the above results is that the number of order-parameters

necessary ^to describe the system ^is equal ^{to the}

number of non-zero eigenvalues of the matrix V.

Since the dimension of V is nP, ^{where n} denotes the

number of elements of An, ^{one would} expect ^the

716

rank of V to be, ^{to a} large extent, ^{at our} disposal ^and

that it would ultimately ^{increase with n. There is in}

this respect, however, ^a fundamental difference between separable ^and non-separable ^{models in the}

random-site class. While for non-separable ^models

the rank of V does indeed increase with _n, this proves ^{not to} be the case for separable models. This is easily ^seen by noting ^{that for} separable ^{models the}

matrix V is of the form (ignoring, ^{for the moment,}

the non-random interaction Jo)

with

V is thus the sum of p rank-1 dyadic matrices with elements ay,, b ’Y’ JL ^so that its rank is at the most p,

irrespective ^{of the dimension of V} and hence inde-

pendent ^{of the} probability distribution for the

Jij’s. ^{This is, of course, related to} the fact that

separable ^{models can be solved in terms of} their p

natural order parameters [6, 14].

It should perhaps ^{be noted that it} is easy and sometimes advantageous ^to eliminate the diagonaliz- ing ^matrix Q ^{from the final results} equations (23), (24) ^{so as to formulate} the free energy and the associated set of fixed point equations ^{in terms of the} sublattice-magnetizations _my ^{and the} « interaction matrix

V. The number of fixed point equations, however, ^will then increase from rank (V) ^to

dim (V).

3. The separable ^{model of van Hemmen.}

The general theory developed ^{in the} preceding

section is applicable ^to any mean-field model of the random-site class and to arbitrary ^discrete prob- ability distributions of the hidden random variables.

The results summarized ’ in equations (24)-(25) ^re- quire ^the calculations of eigenvalues ^and eigenvec-

tors of the quadratic ^form equation (7). ^In general

these calculations have to be done numerically.

Results for a non-separable ^{model are described}

below in section 4. This section demonstrates the

workability ^{of the} above method in the case of the

separable spin-glass ^{model of van Hemmen, where}

the non-zero eigenvalues ^{and their} corresponding eigenvectors ^can be calculated analytically. ^{In van}

Hemmen’s model the random exchange coupling ^is

chosen to be

where §i i and q i denote ^the components ^{of the two-} dimensional vector ti. They ^are independent, ^identi- cally distributed random variables, ^{their values} being

at, a2, ..., an with equal probability. ^{We assume the}

elements of this set An ^{to have zero mean and finite} variance, _say ^{one :}

Due to the first assumption ⁱⁿ equation (28) ^the

matrix V in equation (8) ^{can be} decomposed ^into

two commuting ^matrices according ^to

where

denotes the ratio Jo/J of interaction

strengths ^and the elements of W and I are given by

Here, ^{the indices} characterizing ^{the rows and}

columns of V are given by y = n (i -1 ) + j ^and y’ = n (i’ -1 ) + j’ ^where i, j, i’ and j’ ^{each take}

values 1,

^n.

Exploiting ^the decomposition of V and the as-

sumptions ^{made in} equation (28), ^we get, ^{after a} little algebra, ^the following ^three non-vanishing eigenvalues ^{of V:}

The corresponding eigenvectors ^{are denoted} by xo, x:t, ^their components ^{are found to be}

where C stands for a normalization constant, its numerical value being ^C

(2 n2)- ^{ln. The} remaining eigenvalues ^{of V are zero and are} dummy quantities

as far as the thermodynamics ^{of the} spin-glass ^are

concerned.

Equipped ^{with the} knowledge ^{of the non-zero} eigenvalues ^{and the} corresponding eigenvectors ^of

the matrix V, ^{we are able to} write down the free energy density ^{and the} associated set of fixed point equations for the three order parameters ^{related to} the non-zero eigenvalues. ^{If we} substitute the above

eigenvalues ^{and the} eigenvectors x+ , xo, ^{X into the} saddle-point equations (23), bearing ^{in mind that the}

components xl, xo, x- ^{must be} identified with elements of the diagonalizing ^matrix Q (cf. Eq. (9)),

we obtain

where

If we introduce the averaging ^notation

and, ^for comparison purposes, define the following quantities

we can express the saddle point equations ^{in the}

more compact ^form

where F, ^{as a} function of the transformed order parameters, ^is given by

Now, using equation (39), ^{it can be seen that the}

r.h.s. of equation (38c) ^{is zero} for q

^{0 and,} differentiating ^with respect ^to q, ^{that its} slope ^{as a}

function of 4 ^is always negative, ^so that 4

^{0 is the} only possible ^solution regardless of the values of q

and m.

Thus, ^{the set} (38) of transcendental equations

reduces to

These equations ^are equivalent ^to equations (6.15)- (6.16) presented by ^{van Hemmen et al.} [13] ^{in their} chapter ^on general probability distributions.

We close this section by merely quoting results for the phase boundaries : the paramagnetic (PM) ^solu-

tion m = q

0 of equation (40) is unstable (i) ^with

respect ^{to the} ferromagnetic (FM) phase m # 0 ^at

,8 c Jo ^and (ii) ^{to the} spin-glass (SG) phase q :A ^{0 at} Kc- 1 = 1. (iii) The line where the pure FM solution becomes unstable against the admixture of a

SG phase ^is given by ^the parametric representation :

and thus proves ^to be independent ^{of the actual}

choice of the set An. It is marked by d ⁱⁿ figure ^1.

The line, where the pure SG phase ^{becomes unstable} against the admixture of a FM phase, depends ^{on the}

actual choice of the set An ^{in the} following way :

This phase boundary ^is illustrated in figure ^{1 for the} following ^{choice of the set A :}

Here, c (n ) ^{denotes a} normalization factor ensuring

variance 1. The labels _a, b, ^{c in} figure ¹ correspond

to n

2, 4, ⁶ respectively. ^{The solid} parts ^{of these}

lines represent equilibrium phase transitions,

whereas their dashed continuations indicate where the pure SG phase ^{would cease to exist as a}

metastable state.

Fig. ^1.

^Phase diagram ^of

Hemmen’s model with

q ^E A,,

given ⁱⁿ equation (44) ^{for n}

² (a ), n

⁴ (b) and n = 6 (c ).

Finally, ^{there are the} equilibrium phase ^bound-

aries separating ^{the FM from the SG} phase ^{or mixed} phase (MP). They ^{start as} FM-SG boundaries at the

triple point K

^=1, become FM-MP boundaries,

where they ^{intersect the lines marked a, b and c,} respectively, ^{and end on the a-axis at a critical value}

a c which is easily ^{found to be}

718

Note that _ac (n

2 )

^ac (no oo )

2/3 and that these equilibrium boundaries are rather insensitive

to the number of elements in A. Since the mixed

phase is bounded from above by ^{the critical line} (43), ^{which in the} low-temperature region ^is simply ^a straight ^line K-1= n-1 a, ^{it dies out for} large ^values

of n as indicated in figure ^1.

4. A non-separable ^model.

In this section we investigate ^a non-separable model, where in close analogy ^{with van} Hemmen’s model

[9] (VH) ^{the random part of the} interaction is chosen to be

Recall that in the VH-model _every discrete prob- ability distribution of the 6’s ^and q’s gives ^{rise to} only ⁴ phases, ^viz.

with MP dying ^{out in the limit} of continuous

probability distributions. The overall characteristics of the phase diagram ^were (cf. Fig. 1) ^{rather insensi-}

tive to the choice of the probability distribution P (J), ^{which is} ultimately ^{due to the} dyadic ^structure

of the interaction matrix V

« I + W defined in

equation (29).

The thermodynamics ^{of the} non-separable ^model

described by equation (46), ^{however, does} depend

in a sensitive manner on the probability distribution for the 6’s and q’s. ^In particular, the number of

order-parameters necessary ^to describe the system

increases with n, the number of values the random

variables § ^and r. ^are allowed to take on. Choosing

the set An, ^{from which} the random variables can be taken, ^{to be of the form}

we find the following ^{results : the} spectrum ^{of the} matrix W, with elements now given by

is symmetric ^with respect ^{to zero. There are}

n 2/4 positive eigenvalues A a

0 (IL =1, ... , ⁿ 2/4 ),

each having ^a negative partner Å JL

^{A, while}

the remaining n 2/2 eigenvalues ^{of W are zero. These}

observations follow from the symmetry ^of the argu- ment of the sinh-function in equation (48) ^{and the} symmetric ^{choice of the set} An ^{alone. Another,}

consequence of these symmetries ^{is that the fixed} point equations only allow trivial solutions y-,

⁰

for the order-parameters associated with the negative eigenvalues k , ^{of W, just as} in the VH-model. We are, therefore, ^{left with} n2/4 ^relevant spin-glass

order-parameters Y JL and the additional ferromagne-

tic order-parameter yo associated with the eigenvalue Tao

« n 2 of the matrix a 1. These n 2/4 ⁺ 1 par- ameters govern ^the thermodynamics ^{of the non-} separable ^model proposed ⁱⁿ equation (46).

It is readily ^{seen that the case n}

^{2 leads to} exactly ^{the same} phase diagram ^{as in the linear VH-}

model with n

2, ^the only modification being ^that

temperature ^and interaction-ratio a have to be rescaled by the factor sinh (2)/2. ^{So n}

^{4 is the}

first non-trivial case, where a more complicated

behaviour at low temperatures may be expected,

since there are now n2/4

⁴ spin-glass order-par-

ameters instead of 1 in the corresponding ^VH-

model.

We have diagonalized the matrix V of equation (7)

and solved the set of fixed point equations (23) numerically. ^{Results are} summarized in figure ^2.

Fig. ^2.

^Phase diagram ^{of the} non-separable ^model

described by equation (44), with §, q E A n ^of equation (47) for n = 4.

The analytical results available in this case are the

following :

(i) ^The phase transition from the paramagnetic phase (PM) ^{into an ordered state is ruled} by ^the largest eigenvalue ^{of V}

^{« I + W} according ^to

which follows immediately ^from equations (23). ^For

small a the largest eigenvalue ^{of V is} A 1= 36.452,

i.e. the maximum eigenvalue ^{of W, and a} phase transition ^{from the PM} phase ^{into a} spin-glass phase (SG) ^{with zero} magnetization occurs on the line

Kc- ⁼ 36.452/16. If the

ferromagnetic eigenvalue

ago ^{an 2 exceeds the maximum}

spin-glass eigen-

value

A ₁ of W, the stable ordered state will be the

ferromagnetic ^{one and the critical} temperature ^is

given by Kc- 1 =

At T

0 the function tanh (x/T) ^{can be} replaced by ^{the function} sign (x). This facilitates an analytic

discussion of the transcendental fixed point equations, leading ^{to the} following ^T

0-results :

(ii) ^{There are} three different mixed phases ^with

and where we have adopted ^the abridged ^notation s (k)

^sinh (J2/5 k). ^These limiting ^{values for}

are marked by ^{arrows in} figure ^2.

(iii) ^{For small}

^{the mixed} phase MP2/8 proves ^to be the state of lowest (free) energy. At ^a critical a 1 given by

the state of lowest free energy changes ^from MP2/8 ^to MP4/8. ^As

^varies through

it is the pure ferromagnetic phase (FM) ^which

becomes absolutely ^stable.

(iv) For a values larger ^than a 2 the mixed phases only ^{exist as} metastable states. Up ^{to a third critical}

value a 3 given by

MP4/8 proves ^to be the mixed phase ^with lowest free energy, exchanging ^relative stability ^with MP5/8 ^for

a ::-- a 3 -

The numerical results illustrated in figure ^{2 con-}

firm that these (meta)stable ^{states found at T}

⁰

survive in a small range of finite temperatures.

In the vicinity ^of the 2nd-order phase ^transition

from the PM state to an ordered state, ^the phase diagram ⁱⁿ figure ^{2 differs} only quantitatively ^from

the VH-diagram ^of figure ^{1. There} is . also, ^{in both}

cases, ^a line, ^marked (a), where the pure ^SG phase

becomes unstable to the admixture of a FM phase.

The solid part ^of (a) represents ^{a line of 2nd-order}

equilibrium transitions, whereas its dashed continu- ation indicates where the pure SG phase ^{ceases to}

exist as a metastable state. There is a point ^on (a),

marked by ^a circle in figure 2, locating ^{a critical}

ac, below which the SG phase ^enters continuously

T

0-magnetizations ^{2/8, 4/8 and} 5/8, ^{to be denoted} by MP2/8, MP4/8 ^and MP5/8 respectively. They ^{exist as}

stable or metastable states for

where

into a mixed phase ^{when the} temperature ^is lowered, while above this critical _ac we find a discontinuous SG - FM transition. Below the line (a) ^{we find the}

three mixed phases already encountered at T

0

(cf. Eqs. (49)-(52)). ^{There are the} absolutely ^stable phases MP2/8 ^and MP4/8 ^{which have T}

0-magneti-

zations 2/8 and 4/8 respectively. ^These phases ^are separated by ^an equilibrium ^lst-order phase ^bound-

ary starting at T = 0, a = a 2 ^and ending ^{at a critical} end-point ^{in the mixed} phase regime. Moreover, there is the mixed phase MP5/8 which, however, exists as a metastable state only. ^The phases MP4/8 ^and MP5/8 ^are separated by ^{a lst-order} phase boundary starting at T = 0, a = a 3 ^and ending ^{at a}

critical end-point, ^{which now} is in the metastable mixed phase regime. Finally, ^{with each of the critical} end-points (including ^the triple point) ^{there is as-}

sociated a pair ^of spinodal ^lines indicating, ^where phases ^{cease to exist as} metastable states.

It is interesting ^{to note} that, ^for suitable values of a, the magnetization ^{can exhibit} reentrance behavi-

our as illustrated in figure ^{3. This} always happens, ^if by lowering ^the temperature one crosses one of the 1st order phase boundaries separating ^the MP2/8 ^and MP4i8 ^or MP4/8 ^and MP5j8, respectively, ^{and the}

solution of the fixed point equations ^{becomes meta-}

stable. It should be emphasized, however, ^{that there}

are only ^{two narrow} windows of a-values, ^where such behaviour can occur, ^{as can} be seen from

figure ^2.

Summarizing, ^we find that the resulting phase diagram, ^as compared with that of figure 1, ^has gained structure, already ^{in the} simple ^{case n}

^4.

The complexity ^{of the} phase diagram may ^{be ex-}

pected ^to increase with n due to the increasing

number of order-parameters (as ^discussed above),

thus demonstrating ^a strong ^{influence of the} prob- ability distribution on the thermodynamics ^{of the}

system.

So far we were dealing ^with symmetric ^distribu-

tions of the random variables § and q and, hence, ^of

the random coupling Jij. ^{Surely, for a stochastic}

720

distribution of the impurity ^atoms it is reasonable to assume a symmetric probability distribution P (J) ^of

the coupling ^{constants which follows from}

relating P (J ) ^with J (r ), ^the strength of the RKKY

coupling ^{at distance r via} 9 (r), ^the impurity ^corre-

lation function. The fact, however, ^{that even so-} called archetypical spin glass systems ^{such as} AuFe have a tendency ^{towards nearest} neighbour clustering (ASRO), generally ^renders P (J) ^non- symmetric according ^to equation (53). ^{This should}

serve as a motivation to include the possibility ^of non-symmetric probability distributions P (J). ^One

rather direct _{way of} doing ^{this in the} present ^context is to allow the elements ai ^E An ^{to be} non-symmetri- cally distributed about zero. Destroying ^the sym- metry ^{of the set} An ^{has several} consequences. First,

the matrices a 1 and W will no longer ^commute.

Second, ^the order-parameters Y IL corresponding ^to negative eigenvalues ^{of V will no} longer ^{be irrelevant}

for the thermodynamics ^{of the} system, ^{since the set} of transcendental equations ^{now allows non-zero}

solutions. Third, ^all eigenvalues ^{of V will} generally

be non-zero. To illustrate, this, figure ^{3 shows the} spectrum ^{of V} corresponding ^{to the} non-separable

Ansatz equation (46) ^{for the set}

Here ri (i

^{± 1, ±} 2) ^are positive ^{random numbers} and 4 ^{is a} parameter measuring ^the asymmetry ^of the probability distribution constructed from the set

A4. ^For non-zero >, all eigenvalues ^{of V are non-}

zero and 16 order-parameters ^are necessary ^to describe the system (instead ^{of 5} for >

0). ^This

Fig. ^3.

Magnetization

function of temperature ^for various values of

Fig. ^4.

Eigenvalues ^{of V} plotted against ^the parameter

>. The eigenvalues

^scaled by ^0.1 (dotted lines), 1 (solid lines), ¹⁰ (short-dashed lines), ¹⁰⁰ (long-dashed lines)

1 000 (dot-dashed lines). The ratio of interaction strengths

is a

JOIJ

^0.5.

behaviour is to be contrasted with that of the

corresponding separable ^case, where the rank of V does not increase and only ^{one extra order} parameter q appears (cf. Eq. (38)), ^if nonsymmetric probability

distributions are chosen. A non-separable ^Ansatz

for the random coupling ^thus promises ^{both, a}

sensitive variation of the phase diagram ^{with the}

choice of the set An ^{and a rich structure}

^for large ^n.

5. Discussion.

We have presented ^a simple ^{method for} solving

mean-field spin-glass ^{models in the} random-site class. The available methods to solve random-site class models [5-9, 11-14] ^are invariably restricted to the separable ^{models, since} they utilize, ^{in one} way

or another, ^{the fact that the random} part ^{of the} interaction is a bilinear function of ^{the site random-}

ness, whereas our method is free of such a restriction, requiring nothing ^but bilinearity ^{in the} spins. ^Conse- quently ^our approach also allows to solve models which must be regarded ^as non-separable ^{in the}

conventional sense. This is ultimately facilitated by

the different nature of our order-parameters, ^which

are sublattice magnetizations ^{or suitable linear com-}

binations thereof. As it stands, ^{our method is} applicable only ^{to discrete} probability distributions.

Continuous distributions can be handled, ^{but this} requires ^further ingredients ^{and a somewhat dif-}

ferent approach [17]. Nevertheless the class of

exactly ^soluble models has been considerably ^exten-

ded.

Interpreting the Hamiltonian as a quadratic ^form

of sub-lattice magnetizations ^and evaluating ^{the free}

energy ^{in terms} of eigenvalues ^and eigenvectors, ^we

are able to classify random-site models by ^{the rank}

of their quadratic ^form, thereby ruling out, ^for instance, the existence of many valley structures in certain separable ^{models, without} actually solving

them : neither the mean-field version of the Mattis model [7] ^{nor the models of} Luttinger [8] ^{and van}

Hemmen [9] ^{can, for} example ^{exhibit a} large ^number

of metastable phases ^{at low} temperatures [12], ^since they ^are rank - 1, - ^{2 and - 3 models} respectively,

which severely ^{limits the number of} possible phases

in these models.

It should be emphasized ^{that the} non-separable models, ^{where the random} _part ^{of the} interaction is of higher ^{than bilinear} order in the randomness, exhibit a strong ^{influence of the} probability ^distri-

bution for the site variables on the thermodynamics

of the system. ^In particular, ^{the number of order-}

parameters and, thus, ^the possible ^{number of} phases

increases as continuous probability distributions are

approached. Non-separable ^{models are, in this re-}

spect, very different from the separable ^{models in}

the random-site class. Sections 3 and 4 contain illustrative examples. Finally, ^{the strong} influence of

probability distributions in non-separable ^models

may help ⁱⁿ adapting spin-glass ^{models to} exper- imental facts, ^{such as} reentrance phenomena ^{or the} correspondence ^{known to exist} between atomic

ordering ^{and the} magnetic phase diagram.

Acknowledgments.

We would like to thank Prof. H. Koppe ^for having

introduced us to the key ^idea developed in section 2 and for helpful ^and stimulating discussions.

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