Most of our knowledge of spin glass dynamics is within MFT. This chapter is therefore devoted to a fairly complete review of the dynamics of the soft-spin SK model introduced in Section 4.3, focusing especially on the description of the AT line as a dynamical instability and on the dynamical description of broken ergodicity below it. Now we should note that the SK model is not a priori expected to give a good description of experimental reality in two and three dimensions: We are now quite sure that the two-dimensional EA model does not order at all except at zero temperature, and in the three-dimensional system the observed Tf is much lower than mean field theory would predict, indicating relatively weak order. Nevertheless, effects can be observed, in both experiments and Monte Carlo simulations, which can be identified in a qualitative way as something like an AT line. Therefore we spend some time in this chapter reviewing the comparison between MFT and these measurements.

As we have noted, the dynamical theory as formulated here does not require replicas. There is, however, a similarity between the dynamical formalism and the replica approach: The bond averaging will now lead to quantities which play a role analogous to the matrix qαβ of the static theory but have two time arguments instead of the two replica indices. In both cases, one uses the method of steepest descent, which is exact for the SK model.

Specific heat, sound propagation, and transport properties have in common that they all are determined by the spin excitations, provided one subtracts contributions due to phonons and conduction electrons. For the specific heat, this is obvious since the heat capacity is a direct measure of the degrees of freedom of the system. The magnetic contribution to the resistivity and other transport properties of metallic spin glasses is due to the exchange coupling between the localized spins and the spins of the conduction electrons. This coupling leads to the scattering of conduction electrons by spin excitations and therefore to a resistivity which has some similarity with that due to electron–phonon scattering. The mechanism which leads to ultrasound attenuation and to a change of sound velocity is slightly more complicated. The sound waves modulate the exchange interactions Jij between the spins and therefore ‘feel’ the spin excitations. In the specific heat and transport properties, the spin excitations enter in the form of integrals, which makes it plausible that no dramatic effects are seen in the freezing temperature. This is in contrast to sound propagation, which is directly determined by the dynamic susceptibility, as shown below.

Specific heat

We mentioned already in the Introduction that (at least at first sight) the magnetic contribution CM to the specific heat shows no anomaly at the freezing temperature Tf (see Fig. 1.8). This was taken as evidence against a possible phase transition.

In this chapter we will put the concepts we have developed in Chapter 2 to work in constructing the mean field theory for an Ising spin glass. The term ‘mean field theory’ (henceforth frequently abbreviated ‘MFT’) can be interpreted in many ways. Here we will take it to mean the exact solution of a model in which the forces are of infinitely long range, so that each spin interacts equally strongly with every other one. For spin glasses, this is the Sherrington–Kirkpatrick (SK) model (1975), which we solve heuristically (though unfortunately incorrectly) in Section 3.1. The error is a somewhat subtle one, as is evident from the fact that we obtain exactly the same result in the more systematic calculations of Sections 3.2 (a direct summation of the leading terms in N−1 in high-temperature perturbation theory) and 3.3 (which uses the replica formalism). In both these approaches, we can also see how the theory itself reveals that it is wrong, since they lead to negative values of quantities which are necessarily positive.

We then study the correct mean field theory, with replica symmetry breaking, in Section 3.4, including the remarkable nature of the broken ergodicity it implies and an examination of its stability. Further physical insight into the problem is obtained in Section 3.5 from the mean field equations first introduced by Thouless, Anderson and Palmer (TAP) (1977).

In most of this book the important randomness we were concerned with was random exchange. In this chapter we will examine systems characterized by two other kinds of randomness: random external fields and random anisotropy. They both share some general features with random exchange, but they have different properties as well. They also differ significantly from each other.

We studied random anisotropy, especially of the Dzyaloshinskii–Moriya sort, in Chapter 6, and again, briefly, in Chapter 7. However, there we were interested in it as a small perturbation on a system whose physics was determined primarily by random exchange. Its importance lay in the fact that it changed the overall symmetry of the system, and this had qualitative consequences at long lengthscales.

A similar remark applies to random fields: Whenever we have been interested in the effects of uniform external fields (as in the AT line, for example) they might as well have been random fields; for symmetrically distributed exchange interactions, the only condition on the external field was that it be uncorrelated with the interactions. Thus we could say that we have implicitly been studying the effects of random fields in spin glasses, but, as with the random anisotropies we studied, we were mainly interested in their effects as perturbations on the spin glass state produced by the random exchange.

Both these kinds of randomness are local, so they only produce trivial behaviour by themselves.

The mean field theory discussed in the preceding chapters revealed a surprisingly rich structure and in particular a very complex ordered phase with many ‘pure’ states. However, we know from periodic systems that the mean field theory does not always give the right answer. In this and the next two chapters, we will discuss alternative approaches to the study of spin glasses with short-range interactions. The most important one will be the renormalization group which led to a deep understanding of phase transitions (and in particular critical behaviour) of nonrandom systems and which has also turned out to be extremely useful for the study of spin glasses. Together with Monte Carlo simulations and experimental data, this will give us a fairly complete picture, at least of static spin glass properties below Tf. Critical behaviour near Tf and scaling arguments for T < Tf have also been considered for the spin glass dynamics but here the situation seems to be less clear. Depending on the system, the remanent magnetization decays with quite different decay laws (see Chapter 1), which indicates that not all dynamic processes are universal. However, all spin glasses have a huge range of characteristic relaxation times, ranging from 10−13s (the Korringa relaxation in a metallic spin glass) to 10−6 – 10−8s. This long-time limit is not inherent to the system but is simply determined by the patience of an experimentalist (or the average time of a student's thesis).

As in many problems, one-dimensional models can give some insight into the physics of spin glasses because they are easier to solve than those in higher dimensionality, but this advantage can be limited because the physics of the one-dimensional problem may be essentially different. Typically, for example, the higher-dimensional problem we really want to study has a broken symmetry phase at low temperature, while one-dimensional systems (with short-range interactions) do not exhibit broken symmetry at finite temperature.

Nevertheless, short-range one-dimensional systems can have broken symmetry at zero temperature, so one can study this relatively simple kind of order and the approach to it as the temperature is lowered to zero. This zero-temperature phase transition can be compared with finite-temperature transitions in higher dimensionality. This is the motivation for Section 12.1, where we will see that the critical point at T = h = 0 in a one-dimensional Ising spin glass with nearest-neighbour interactions has interesting features in common with finite-temperature spin glass transitions, and that they can be calculated analytically, sometimes rather simply.

Further insight can be gained by studying one-dimensional systems with long-range (typically power law) interactions. Then if the exponent describing the falloff of the interaction strength with distance is small enough, finite-temperature phase transitions are possible. Though such problems are harder to solve than nearest-neighbour ones, they can still be easier than the full higher-dimensional ones.

So far we have considered only the statics of spin glasses. However, the nonergodicity discussed in Sections 3.4 and 3.5 for the SK model suggests that the dynamics of spin glasses is rather unusual, at least below the freezing temperature. One expects transitions over energy barriers between the metastable states discussed in Section 3.5, leading to a new class of very slow relaxation times, some of which become infinite, at least in MFT. In this chapter we introduce the basic models and formal techniques necessary for describing the dynamics of spin glasses, corresponding to what we did for statics in Chapter 2. We focus on Ising systems here, both for the sake of simplicity and because, as we have mentioned, the anisotropic interactions found in most systems which one would expect to be Heisenberg spin glasses make them look rather Ising-like in many of their properties.

The Glauber model

The classical Ising model has no inherent dynamics, so to make a dynamical model one has to couple the spins to an additional ‘heat bath’ which induces spin flips (Glauber, 1963; Suzuki and Kubo, 1968). For metallic spin glasses this heat bath can be identified with the conduction electrons, which produce single-spin flip processes with the impurity spins via the exchange interaction Jsd. For a single magnetic impurity this leads to the so-called Korringa relaxation with relaxation time τ0.

Spin glasses are a fascinating new topic in condensed matter physics which developed essentially after the middle of the 1970's. The aim of this book is to give an introduction to it which will both attract the newcomer to the field (say, a student with a basic knowledge of solid state physics and statistical mechanics) and give a comprehensive survey to the expert who perhaps has worked on a very specific problem. It is a field which is still open to new ideas and concepts and in which important new experiments can certainly still be done.

Our understanding of spin glasses is based on three approaches: theory, experiment, and computer simulation. We have tried to present the most important developments in all of them. One possibility is to take the theory as a guide and to check it by comparison with experimental data and simulations. This is roughly what we do in the first part of this book (Chapters 3 to 6), after introducing the basic experiments, models and concepts which define what we are talking about. (Spin glasses are disordered systems, so we have to introduce several concepts which are unknown in the ‘classical’ theory of ideal solids.)

In Chapters 3 to 6 we discuss a mean field theory, which is so far the only well-established spin glass theory. It turns out to be highly nontrivial and has been developed over more than a decade.

One of the dominant themes in the history of physics in this century has been the effort to understand condensed states of matter. This began with very simple systems — the Van der Waals description of the liquid–gas transition and the Weiss mean field theory of ferromagnetism — and has gradually developed to include more and more complex and subtle states and phenomena. Spin glasses are the current frontier in this development, the most complex kind of condensed state encountered so far in solid state physics.

In trying to understand these systems, experimentalists have used a wide spectrum of probes in ingenious ways, and theorists have invented an equally wide variety of models and new theoretical concepts. The resulting developments have had an impact, not only on other parts of physics, but also on other fields such as computer science, mathematics, and biology. It is because of this widespread influence and the interest in spin glasses that it has aroused that we are writing this book.

We expect that many people who read this book will be condensed matter physicists. However, we also have in mind as a typical reader someone from another area in physics, or perhaps a graduate student looking for a research topic, who wants to find out what all the excitement is about.

Spin glasses differ from most magnetic materials in having dynamics on many timescales. That is, for example, if one measures the susceptibility of a spin glass to fields oscillating with frequencies ω = 1012, 108, 104, 1 and 10−4 sec−1 one may get different results at all the measuring frequencies, indicating that the system has characteristic excitation and relaxation times over (at least) 16 orders of magnitude, ranging from the typical times associated with microscopic spin motions in solids to macroscopic times. For sufficiently low temperatures, this spectrum can extend at the longtime end to geological timescales. (A practical limit is set by the length of time society considers reasonable for a graduate student to complete a Ph.D.) This situation is completely different from that of conventional materials, where for practical purposes measurements at all frequencies less than about 1010 sec−1 would give indistinguishable results.

How are we to understand such properties? In this chapter we review the various kinds of experiments that can be done to probe the dynamical behaviour on timescales ranging from microscopic ones to the longest ones experimenters have the patience to study, and we try to sketch a theoretical picture that enables us to get some insight into what the experimental results mean.

We have actually met this theoretical framework before (often) in the preceding chapters of this book. It is nothing other than our old friend broken ergodicity (introduced in Section 2.4), adapted to the kinds of experimental situations we meet here.

The spin-glass properties discussed so far are observed only in a restricted concentration range of the magnetic atoms. Very dilute metallic magnetic alloys exhibit the Kondo effect (see Section 10.3) and very dilute insulating systems with short-range exchange interactions like EuxSr1−xS with x ≤ 0.13 remain superparamagnetic at all temperatures. In the opposite limit of large concentrations, one has magnetic order. However, there is an interesting concentration range in which one observes a competition between spin-glass and ferro- or antiferromagnetic long range order. This is the subject of this chapter. We will be particularly interested in whether a system which has made a paramagnetic to ferro- or antiferromagnetic transition will go into a ‘reentrant’ spin glass state at some lower temperature or whether there is a coexistence of spin-glass and conventional magnetic order.

Mean field theory

The first hint of an answer to this question comes from MFT. The SK model for Ising spins with the Gaussian bond distribution (3.29), as discussed in Section 3.3, predicts the phase diagram of Fig. 3.7. If the width J of the bond distribution is smaller than the variance J0, one has a spin glass state without spontaneous magnetization Ms. For J0 > J, the system goes with decreasing temperature first into a ferromagnetic state with finite order parameters Ms and q and at a lower temperature into a spin glass state with replica symmetry breaking. In this state, the order parameter q has to be replaced by the Parisi order parameter function q(x) (Section 3.4), whereas the spontaneous magnetization Ms remains finite.

We have all been delighted and fascinated by the variety of crystal forms that abound in nature and please the eye. Each pattern in this vast panorama appears unique. The variations seen in the growth forms presented in Fig. 1.1 give us some idea of how the dendritic growth form can change in appearance as we change from material to material or from one crystal growth process to another. Even for a single material and a single process we often find great variability as illustrated by Nakaya's eight common snow-flake patterns given in Fig 1.2. Such variability and uniqueness of form caused earlier scientists to speculate on and argue about the relative importance of heredity factors versus environment factors in the ultimate form of such crystals.

A relatively recent experiment by Mason tended to illuminate this issue and strongly favor the environmental factor control point of view. He placed a nylon fiber in the water vapor diffusion chamber illustrated in Fig. 1.3 and observed the form of the ice crystals that developed at the different temperature locations along the fiber. This great range of forms is listed in Table 1.1. After these forms were growing stably in the chamber, he lifted the fiber some distance so that those crystals growing between temperatures T1 and T2 on the fiber were shifted to another temperature range T3 - T4 characteristic of a different crystal form on the earlier result.