Spin glasses: recent advances in mean-field theory1

B. W. SOUTHERN Department of Physics, University of Matzitoba, Winnipeg, Man., Canada R3T 2N2

Received September 29, 1986

A survey of recent advances in the mean-field theory of Ising spin glasses is presented. The physical picture of the spin-glass phase predicted by this theory is described, and its relationship to real three-dimensional systems is discussed.

Cet article prCsente un vue d'ensemble des progres rCcents I'approximation du champs moyen. L'irnage physique corresl avec les vrais matCriaux tridimensionels.

Can. J . Phys. 65, 1245 (1987)

1. Introduction Spin glasses have posed a challenge to both theorists and

experimentalists for the past 25 years. The history of the development of this subject can roughly be divided into three basic periods of time. The period before 1975, which could be called the "prereplica" age, was concerned mainly with experimental investigations of the magnetization, magnetic susceptibility, specific heat, and Mijssbauer effect in dilute alloys of noble metals dissolved randomly in nonmagnetic hosts. The results that were obtained from these various studies suggested that a cooperative magnetic phase transition occurred in these dilute systems at a well-defined temperature Tf, called the freezing temperature, whose value depended on the concentration of magnetic atoms. However, below this freezing temperature, there was no evidence of any long-range spatial magnetic order among the spins. A sharp cusp was observed in the low-frequency ac susceptibility at Tf, but no anomalous behaviour was observed in the specific heat at this same temperature. Thus, if Tf was signalling a phase transition in these systems, then it was quite different from the familiar type of transition that occurs in ferromagnets at the Curie temperature, where the spins become aligned over macroscopic distances and a spontaneous magnetization appears. Early theoretical attempts to explain the transition using the simplest mean-field approaches failed. There was a great deal of contro- versy as to whether the experimental results should be inter- preted in terms of a genuine thermodynamic phase transition or simply in terms of a nonequilibrium freezing phenomenon in which the relaxation times to attain true equilibrium greatly exceeded typical measuring times, as is the case in real glasses.

A major advance in the theoretical approach to spin glasses occurred in 1975 with the publication of a paper by Edwards and Anderson (1) in which they proposed an extremely simple model to describe the spin-glass state. The model includes two basic ingredients: "randomness" and "frustration." An important characteristic of all spin glasses is that the exchange interactions between pairs of spins are not all of the same sign but are in conflict with one another owing to a frozen-in, or quenched, disorder. At low temperatures, there is no configura- tion of the magnetic spins that simultaneously minimizes the

- exchange energy of each pair, and the system is said to be frustrated (2). The actual ground-state energy lies much higher than the value it would have if all bonds were satisfied, and more importantly, the system has a large number of quite different spin configurations that are nearly degenerate in energy and

h his paper is dedicated to Allan H. Morrish on the occasion of his 65th birthday.

dans la description thkorique des verres de spin de type Ising dans ~ondante de la phase verre de spin est dCcrite, ainsi que son rapport

[Traduit par la revue]

close to the ground-state value. Even for the simplest models, however, the calculation of the configurationally averaged free energy in a quenched random system is a very difficult task. Edwards and Anderson (1) (hereafter referred to as EA) introduced a novel method, called the "replica method," to perform this average and solved their model within a mean-field approach. This paper marked the beginning of the replica age, and the period from 1975 to 1985 witnessed a renewed interest by both theorists and experimentalists in the spin-glass problem.

During this second period, experimental studies on the measurement of the ac and dc magnetic susceptibilities below Tf multiplied, and it was confirmed that they depended quite strongly on how the measurements were performed. The same was true for the remanent magnetization, which can be measured either by cooling the system from above Tf to well below Tf in the presence of a field and then removing the field (field cooled) or by cooling in zero field (zero-field cooled) and then applying a field. The results depended on the sample's history of magnetic field and temperature changes; extremely long relaxation times were observed in both types of experi- ment, indicating that major rearrangements of the spins were required to reach true equilibrium. Higher order terms in the expansion of the magnetization as a function of the magnetic field were also studied experimentally, and these nonlinear susceptibilities seemed to be strongly divergent at Tf. At the same time, numerical investigations of the EA model revealed that it shared many of these same properties and hence contained the main physical ingredients to describe real systems.

The theoretical advances in the mean-field theory of spin glasses took another leap forward in 1979 with a paper by Parisi (3). He presented an elaborate self-consistent solution of the Ising mean-field version of the EA model using the replica method. The solution had several remarkable properties, but its physical interpretation remained rather obscure until about 1983. Since then, a coherent theoretical description of the spin-glass phase has emerged, but the replica approach used by Parisi still suffers because it cannot. be placed on a firm mathematical foundation. At the present time, we are entering a postreplica period where the understanding that was gained from the replica approach is being utilized to derive the mean-field solutions in a more physical way.

It remains to be seen if the mean-field solution of the EA model and its predictions are in accord with the experimental results on real systems. The mean-field theory has not resolved the question as to whether or not a genuine phase transition occurs in real systems. The model predicts a transition provided the space dimension is large enough. However, it is not yet known whether the mean-field solution is correct for three

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1246 CAN. J . PHYS. VOL. 65. 1987

dimensions (d). Recent numerical studies of the short-range EA model in d = 3 suggest that a phase transition occurs in Ising systems, but its nature is not completely understood.

The present survey is not intended to be a complete review of this subject. Detailed accounts of both the theoretical and experimental developments can be found in the reviews by Rammal and Souletie (4), Van Hemmen and Morgenstern (5), Fischer (6,7), and Binder and Young (8). We only describe the basic characteristics of spin glasses and summarize the current status of the theory. A comparison of the predictions with real experiments is given in the following article by Williams.

2. Basic properties The classical spin-glass systems are dilute alloys of noble

metals dissolved in nonmagnetic hosts such as AuFe and CuMn, where the exchange mechanism involves the conduc- tion electrons and is both long ranged and oscillatory. However, the term spin glass is now also used to describe concentrated nonmetallic systems such as (EuxSrl-,)S, where the inter- actions are due to superexchdnge and are short ranged. Both systems share the essential feature that is required for spin- glass behaviour even though the exchange mechanisms and concentration of magnetic atoms are quite different. This feature is conflicting signs of the interactions due to a quenched randomness. The most striking characteristics shared by these different systems are a relatively sharp cusp in the low- frequency ac susceptibility at a freezing temperature Tf and the onset of strong irreversibility and time-dependent effects in the magnetization below this temperature. There is no apparent anomaly in the specific heat at Tf, and no evidence of any long-range spatial order of the spins below Tf. The nonlinear susceptibility also appears to be strongly divergent at Tf.

The main question that still remains to be answered for spin glasses is whether this behaviour, which occurs in a wide class of magnetic materials, corresponds to a new type of phase transition as predicted by mean-field theory or is simply a dynamical freezing, where relaxation times exceed the time of observation. If it is a phase transition, then what is the order parameter and is there a breaking of some symmetry at the transition? What is the lower critical dimension below which fluctuations destroy the transition? Is the transition in the presence of a field different from the transition in zero field? While many of these questions remain unanswered, a great deal of progress towards resolving them has been achieved during the past five years in the mean-field approach to spin glasses.

3. Mean-field theory The model introduced by Edwards and Anderson (1) is

described by the following Hamiltonian:

[ l a ] H = - ~ J ~ , S ~ . S ~ i C j

where the Sf's are classical vector spins located at the sites i of a regular lattice and the Ji,'s are exchange interactions between nearest neighbours. Each Jij is assumed to be an independent random variable with the following Gaussian distribution:

112

[I b] P(Jij) = ( ) - ~ X ~ ( - J $ Z / ~ J ' )

where z is the coordination number of each site. The equilibrium properties of the model are obtained by calculating the free energy of the system for a given configuration of the Ji,'s and then averaging over the distribution of the Jij's. This corres-

ponds to averaging the logarithm of the partition function and not the partition function itself. It is much easier to average the moments of Z than to calculate the logarithm, because averaging Z n is related to the problem of a pure system with n variables at each site or, equivalently, n replicas of the original system. The replica method is based on the identity

- - zn- 1

[2] In Z = lim- n + ~ n

where the bar indicates an average over the quenched disorder, and thus the replica method corresponds to deducing the average free energy from the averaged moments of the partition function. However, the continuation of n from integer values to noninteger values is generally not unique and hence cannot be justified on mathematical grounds. Edwards and Anderson have used this approach to evaluate Z" within a mean-field approximation where a single order parameter was introduced. The results predict a phase transition at a well-defined tempera- ture Tf, with a cusp in the susceptibility but no long-range spatial order of the spins below Tf. Instead, the spins are frozen in random directions below Tf. Edwards and Anderson chose the simplest order parameter that would reflect this order:

where indicates the thermal average of the spin at site i. However, the mean-field solution also predicts a cusp in the specific heat at Tf, which was thought to occur because the mean-field approximations generally underestimate fluctuations.

The classical EA model was generalized to quantum Heisen- berg spins by Sherrington and Southern (9) and Fischer ( lo) , but more attention was focussed on the Ising case. In an attempt to justify the mean-field solution of EA, Sherrington and Kirkpatrick (1 1) (hereafter referred to as SK) introduced an Ising version of the same model in which the Jij's were assumed to be long ranged and each spin interacted with every other spin. The Weiss molecular-field theory for ferromagnets becomes exact in this limit. It was hoped that this would also be true for the EA spin-glass model and that the infinite-range model might provide a starting point for the study of the effects of fluctuations. The SK model is described by the following Hamiltonian:

where Si = ? 1 are Ising spins and the Jij7s have the same distribution as in [I b] with z = N. In the infinite-range limit, the moments of P(J i j ) must be normalized by the number of neighbours (N) to obtain a sensible thermodynamic limit. The method of solution used by SK followed that of EA by expressing the average of Z" as'an integral over n(n - 1)/2 variables gap ( a Z. P), where a , P = 1, 2, ..., n. The integrals were then evaluated using the method of steepest descents, and SKchose the symmetric saddle point with all gap's the same and equal to the EA order pararneter. This choice is referred to as the replica symmetric solution.

The SK solution predicted a continuous phase transition at Tf = J , where the single-order pararneter qEA became nonzero. Near Tf, it behaved as qEA - (Tf - T)/Tf, and the susceptibility could be expressed in the form

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FIG. I . The line of stability T f ( H ) in the H-T plane for the SK model. The replica symmetric solution is stable for T > T f ( H ) .

for all T's and exhibited a cusp at Tf. In the presence of a weak field H, the magnetization at T > Tf could be written as

H 2 T ~ + ~ T : [6] M = ( H I T ) 1 - - ( 3 T 2 ( T 2 - T : ) ' " ' )

indicating that the nonlinear susceptibility x , , = d 2 ~ / d ~ 2 diverged as Tf is approached from above. In the presence of a finite field, the transition disappears because q,, + 0 at all values of T and the cusp in the susceptibility becomes a rounded maximum.

In spite of these appealing features near TI, the solution had some unphysical properties below Tf. At low values of T, the entropy was found to be negative; and in addition, the solution with q,, + 0 had a higher free energy than the paramagnetic phase ( q E A = 0 ) . This led de Almeida and Thouless (12) to an examination of the stability of the SK solution by expanding to quadratic order the fluctuations of gap about the symmetric saddle point. The resulting stability matrix had three different eigenvalues that should be positive if the solution is stable. In the limit n + 0 , one of the eigenvalues became negative, indicating that the solution was unstable to small fluctuations. In the presence of a field H , this solution could be stabilized. The line corresponding to the limit of stability in the H-T plane is called the AT line and is shown in Fig. 1. For small values of H, this line approaches Tf as

Hence in the absence of a field, the SK solution is unstable at all values of T < Tf.

This instability of the SK solution indicated that within the replica approach, the permutation symmetry of the replicas had to be broken below Tf . Various schemes ( 13, 14) were proposed in the period from 1975 to 1979 but none led to a stable solution. In 1979, Parisi (3) introduced a sophisticated scheme for breaking the replica symmetry. The solution proposed by Parisi is not the most general solution, and it remains to be shown if it is the exact mean-field solution to the SK model. However, the solution has remedied many of the difficulties suffered by the replica symmetric solution and has an interesting physical interpretation.

FIG. 2. One step of Parisi's replica symmetry-breaking scheme for the case mk- = 3 m k . At stage k - 1, the matrix elements within each diagonal block all have the same value Q ( m k - , ) . At stage k , this symmetry of the replicas is broken by introducing three smaller diagonal blocks within which the matrix elements have a new value Q ( m k ) and outside of which the value remains unchanged from Q(mk- 1 1 .

The Parisi (15-17) solution was constructed as follows. The order parameter was represented by the n X n matrix Q a p , whereas in the SK solution, all Q a p (a + (3)'s were taken to be the same. Parisi divided the n x n matrix into ( n / m l ) x ( n / m l ) blocks of size m , x m l , where ml is an arbitrary integer. The values of Q in the blocks off the diagonal were taken to be the same, say Q ( m o ) , whereas the values in the blocks on the diagonal took another value, Q ( m l ) . This procedure was then repeated for the diagonal blocks by introducing another integer m2 and a corresponding Q ( m 2 ) , as shown schematically in Fig. 2. If one interated this construction k times, then there would be k + 1 values of Q ( m k ) and k different integers, mk.

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1248 CAN. I . PHYS. VOL. 65. 1987

FIG. 3. Qualitative form of the Parisi function Q ( x ) near Tf.

Parisi then considered the limit n+ 0 and k-+ =of this scheme, where the mk's were replaced by a continuous variable x in the interval 0 S x S 1 and Q becomes a continuous function of x. Hence, the simple order parameter of EA or SK became a continuous function in the Parisi solution. The free energy in the n + 0 limit was a functional of Q ( x ) and had to be extremized to obtain Q ( x ) . It was not possible to deduce Q ( x ) analytically except near T f , where the free energy could be expanded about the paramagnetic solution Q ( x ) = 0 .

The solution for Q ( x ) near Tf is shown in Fig. 3 and has the following general form:

where xl = 2Q(1) and xo = 2Q(0) are determined from the extremum of the free energy. In a zero applied field, the stable solution has Q(0) = 0 and Q ( 1 ) = ( T f - T ) / T F = qEA. The susceptibility can be written in the same form as [5] with qEA replaced by

Note that if Q ( x ) is independent of x, then we recover the replica symmetric solution. In the presence of a field H, Q(0) becomes Q(0) = (3/4)'13 ( ~ 1 5 ) ~ ' ~ . Above and on the AT line, which is given by [7] near T f , Q ( x ) reduces to the replica symmetric solution obtained by SK; below this line, replica symmetry breaking occurs and the order parameter becomes a function. The value of Q at x = 1 is q ~ ~ , but the susceptibility depends on the whole function Q ( x ) . For H = 0 , the fact that Q(0) = 0 leads to the surprising result that the susceptibility is constant for all values of T < T f . This is, in fact, similar to the behaviour observed experimentally in field-cooled measurements ( 4 ) .

Sompolinsky (18) has given a completely different formula- tion of replica symmetry breaking at the AT line. Sompolinsky used time-dependent order parameters, which displayed a broad spectrum of relaxation times and which diverged in the thermodynamic limit as the AT line was approached from above. This latter approach has been shown to be mathemati- cally equivalent to the Parisi solution, and the large increase in relaxation times corresponds to the instability of the replica

symmetric solution at the AT line. However, a physical inter- pretation of these order parameters was not given immediately.

Many earlier attempts (19, 20) at developing a mean-field theory for spin glasses without the use of replicas appeared after the difficulties of the SK solution became apparent. At zero temperature, these approaches reduce to a simple Weiss mean-field theory, where the average value of the spin at each site mi = (Si)T is determined self-consistently from equations of the following form:

[ lo ] mi = sign ( F Jijmj)

The solutions must be obtained for a given distribution of the Jij's and then averaged over all distributions. For the SK model, it has been shown by Bray and Moore (21) that there are an exponentially large number of solutions that are stable at T = 0 . Bray and Moore also studied the equations derived by Thouless et al. (19) at finite temperatures and found that this is also true everywhere below the AT line. These solutions correspond to equilibrium states, or valleys in phase space, which are characterized by the values of the local magnetizations my in each state a and which are separated by free-energy barriers that diverge in the thermodynamic limit.

One possible interpretation of replica symmetry breaking is that it corresponds to the appearance of a large number of stable solutions below Tf and that these states are separated in phase space by infinite energy barriers. The AT line thus corresponds to a breaking of ergodicity in that all configurations in phase space are no longer accessible and the system becomes trapped in a single valley. This behaviour is similar to that in simple Ising ferromagnets where the magnetization becomes nonzero below the Curie temperature T, . There are two equilibrium states corresponding to all spins up or all spins down, which are equivalent and related by spin-reversal symmetry. Above T,, these states have equal weight in the partition function and the total magnetization is zero. Below T, , this symmetry is broken and long-range order develops as the system settles into one of the two equivalent states. In the SK model, this breaking of ergodicity is more complicated and there are an infinite number of equilibrium states that are not related to one another by any symmetry.

A physical interpretation of the Parisi order parameter Q ( x ) was given by Parisi (22) . He argued that the phase space in spin glasses consists of many free-energy valleys, which can be labelled by an index a. Each valley, or pure state, is characterized by the local magnetizations my of that state, and its weight Pa in the partition function is determined by the value of its free energy Fa as follows:

[ l l a ] P a = exp ( - F a I T ) c exp ( - F y l T ) Y

If one then considers the overlap of two such states a and P, which could be characterized by the quantity

then the probability P J ( q ) of finding an overlap q for a given distribution of couplings can be evaluated in terms of the probabilities Pa of the pure states as follows:

Owing to the Ising symmetry, we can confine our attention to

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SOUTHERN 1249

q min

FIG. 4. Expected shape of the average probability of overlap q between any two pure states for a given distribution of couplings.

the range 0 G q G 1. Parisi demonstrated that the average of PJ(q) was, in fact, dx/dQ, where Q(x) is the Parisi order parameter. Hence, the Parisi function Q(x) has a physical interpretation in terms of the overlap of the many valleys, or pure states, with one another. Using the form of Q(x) in Fig. 3 , we have sketched the predicted shape of PJ(q) is Fig. 4. The peak at q,,, corresponds to the plateau from kl to 1, whereas the broad structure between qmi, and q,,,, is due to the structure between xo and x , . If there is no replica symmetry breaking, then Q(x) is independent of x and PJ(q) is a delta function at q = ~ E A . Hence, information on the Parisi order parameter ~ ( x ) can, in principle, be obtained from the overlap function Pj(q). Young (23) has attempted to verify these features of - Pj(q) numerically and has found that for the SK model above the AT line, there is only a single delta function, but a much broader distribution appears below this line. Although there are large finite-size effects in the calculation. the resdts seem to ., support the many-valley picture of the ~ ~ i n - ~ l a s s phase for the infinite-range model.

4. Conclusions The mean-field theory of spin glasses seems to be well

understood at present. There is a line in the H-T plane above which the replica symmetric solution of SK is stable and that corresponds to a phase in which the simple order parameter q,, is nonzero. As this line is approached from above, an instability in this solution occurs and the order parameter becomes the Parisi function Q(x). There are a large number of equilibrium states, or valleys, in phase space below the AT line, and Q(x) not only describes the order within each valley but also their overlap with one another. Relaxation times also diverge at this replica symmetry-breaking transition because there are infinite free-energy barriers between the valleys. Although the Parisi solution on the SK model was obtained within the replica method, the physical interpretation of the results seems sensi- ble. Mezard et al. (24) have recently obtained the same results

using a different mean-field approach that is based on the physical insight afforded by the replica solution.

Even if the mean-field solution is well understood, its relationship to real spin glasses in three dimensions (3d) has not yet been established. The mean-field solution corresponds to an infinite dimensional system, whereas in finite dimensions, fluctuations can completely destroy the order at all nonzero temperatures. Many of the experimental results on real 3d spin-glass systems have been interpreted as supporting the existence of an AT line and hence a replica symmetry-breaking transition. This transition would then account for the onset of strong irreversibility in the magnetization below Tr . Extensive numerical investigations by Binder and Kinzel (25) of the short-range Ising version of the EA model in both d = 2 and d = 3 have shown that it indeed displays behaviour qualitatively similar to the real systems and the results of mean-field theory. It has also been established by Morgenstem (26) that in d = 2 there is, in fact, no phase transition at any finite temperature and thus the apparent AT line observed in the numerical simulations is a purely dynamic effect. There are a large number of metastable states, or valleys, but the barriers between them are finite at nonzero values of T. The system can spend a long time within each valley, but thermal fluctuations eventually allow the system to sample all of the phase space and there is no breaking of ergodicity. Early studies of the short-range EA model using approximate real-space rescaling methods by Southem and Young (27) suggested that the lower critical dimension for Ising spin glasses was between 2 and 3 dimensions. Recent numerical studies of the EA model by Bhatt and Young (28) and Ogielski (29) also support the idea that a transition does, in fact, occur at a finite Tr in d = 3.

Although we have only discussed the mean-field theory for Ising spin glasses in this survey, whereas the best studied experimental spin-glass systems are characterized by a Heisen- berg Hamiltonian with weak anisotropy, it has recently been argued by Bray et al. (30) that all d = 3 experimental spin-glass systems should have a phase transition that is in the same universality class as the short-range Ising spin-glass model. However, even if there is an Ising-like transition in d = 3, it is not clear if it would correspond to a replica symmetry-breaking transition as predicted by mean-field theory. It has recently been suggested (3 1, 32) that for all dimensions less than d = 6, there is no transition in the presence of a magnetic field and hence no AT line. It remains to be verified if the interpretation of the experimental results in terms of an AT line are really correct or if the transition is simply a dynamical effect as in d = 2.

Acknowledgements I would like to thank B. Derrida, P. Mottishaw, and C . De

Dominicis for many enjoyable discussions about spin glasses. Financial support from the Natural Sciences and Engineering Research Council of Canada is also gratefully acknowledged.

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