Historically, this book started as a series of lectures given in the Service de Physique Théorique at Saclay, one of the lectures (Chapter 5 here) delivered by Marc Mézard. This explains the strong field theory bias adopted in its approaches and the use of some techniques rarely found in standard literature. It deals with the theory of disordered magnetic systems and for a large part of it with the Random Field Ising Model (RFIM) and the Ising Spin Glass, paradigmatic systems of frozen disorder. Such systems enjoy nontrivial properties, different from and richer than those observed in their pure (nondisordered) counterpart, that dramatically affect the thermodynamic behaviour and require specific theoretical treatment.

Disorder induces frustration and a greater difficulty for the system to find optimal configurations. Consider, for example, the case of spin glasses. These systems are dilute magnetic alloys where the interactions between spins are randomly ferromagnetic or anti-ferromagnetic. They can be modelled using an Ising-like Hamiltonian where the bonds between pairs of spins can be positive or negative at random, and with equal probability. Due to the heterogeneity of the couplings, there are many triples or loops of spin sequences which are frustrated, that is for which there is no way of choosing the orientations of the spins without frustrating at least one bond (Toulouse, 1977). As a consequence, even the best possible arrangement of the spins comprises a large proportion of frustrated bonds.

In the previous chapters we have seen how the presence of disorder may be responsible for a novel behaviour at low temperature where bound states appear. In the case of the Random Field Ising Model we have been able to trace evidence of this nontrivial spin glass phase within a field theory, in both approaches from statics and dynamics. However, we did not go far enough to describe this phase and characterize its properties. In the rest of this book our aim will be precisely to address this problem, restricting ourselves to systems where it has been mostly studied in the last thirty years: spin glasses. For spin glasses, contrary to the RFIM, there is no random-site magnetic field; instead, the heterogeneity occurs in the exchange interactions between the spins that are then modelled as quenched random variables.

In this chapter we consider a first simple model of spin glass, the p = 2 spherical model, that is constrained to spins interacting by pairs. As usual, understanding is greatly helped if one is able to obtain an exact solution for a model that possesses some of the characteristic features of interest. This is the case of this spin model that, despite its coupling randomness, is exactly soluble, both for the statics and the dynamics, and does not require the replica method.

As we shall see, the p = 2 spherical model has not a true spin glass behaviour and is rather a disguised ferromagnet.

In the previous chapter we have started to look at the difficult task of inverting the Gaussian fluctuation matrix around a Parisi-like RSB equilibrium Qab. As we have seen, the core of the difficulty is manifest in the unitarity equations (8.27). In Section 8.4 we have shown that in the near infrared regime these equations become linearly coupled equations that can easily be solved to get the propagators. In general, however, the unitarity equations contain nontrivial integrals over replica overlaps (see Eq. (8.37)). To overcome this problem and solve the equations in full generality one resorts once again to the technique of the Replica Fourier Transform. This is what we shall do in this chapter. Even if the complete resolution of the unitarity equations is beyond the scope of this book, we wish at least to compute explicitly the propagators in the most dangerous replicon sector, giving a few hints on what can be done for the other sectors. With this aim in mind, we proceed in the following way. In Section 9.1 we return to the RFT technique which was used in Chapter 6 to diagonalize the Hessian (in the absence of RSB) and we generalize it for R steps of RSB. We exhibit how it works on a simple toy model and in Section 9.2 we use it to obtain the propagators in the replicon sector.

I vividly remember the academic year 1977–1978. I was a Loeb lecturer at Harvard that year. The Wilsonian revolution had been blossoming everywhere and I was teaching ‘field theory approach to critical phenomena’ in the wake of the works of my colleagues and friends Edouard Brezin and Jean Zinn Justin. I had not yet been exposed to the novel intricacies that were being uncovered in the critical behaviour of quenched random systems. But, during that year, several seminars were to deal with them and I began learning and interacting with Mike Stephen, Jo Rudnick and the late Sheng Ma. This is how it all started for me. A good quarter of a century later, the two central problematic systems of the field, the Random Field Ising Model and the Ising Spin Glass, despite several thousand papers and a huge amount of efforts dedicated to them, remain objects of controversy for what concerns how to describe their glassy phase. So why add a book on top of that? Perhaps I will tell how it all occurred.

At the origin the book was a mere set of lecture notes for a course given in this laboratory, a course that was largely repeated two years ago in the theory group of the physics department at UFRS in Porto Alegre. The Lecture Notes Series of Cambridge University Press having been discontinued, it was gracefully suggested that the notes be transformed into a book.

Let us now consider mean field models of spin glasses where, contrary to the simple case studied in the previous chapter, the quenched disorder dramatically affects the thermodynamics. For these models, as we shall see, the replica method not only represents a necessary mathematical tool to perform the computations, but also offers an appropriate description of the nontrivial low temperature physics.

When looking at the thermodynamics of the p = 2 spin model using replicas we ended up with a Lagrangian (Eq. (4.83)) expressed in terms of an overlap matrix Qab. To perform the calculation we had to make an ansatz on the structure of this replica matrix, and we chose the simplest possible one, the Replica Symmetric (RS) ansatz. In that case, this ansatz proved to be appropriate, however, this is in general not true: whenever real spin glass behaviour is encountered the RS solution is not the appropriate one, being unstable and leading to unphysical predictions (such as, for example, negative entropy). Alternative ansatzes for the Qab matrix necessarily involve a breaking of the replica symmetry, and are usually referred to as Replica Symmetry Broken (RSB) solutions.

The correct way to break the symmetry was the main focus of most theoretical research on spin glasses in the late seventies. As it turned out later on, the degree of symmetry breaking necessary to find a stable solution actually depends on the model considered.

The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results).

Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).

In the previous chapters we have analyzed in detail the SK model and its solution within the replica method and via mean field-like TAP equations. The physical scenario unveiled by the RSB solution is novel and intriguing, depicting a low temperature phase where ergodicity is broken in a multiplicity of pure states with a nontrivial structure. Yet, so far we have been dealing with a mean field model and one may wonder whether all these surprising results are just an artefact of the long range interaction. This question is indeed very much debated and different points of view exist with conflicting conclusions. In this book hereafter we shall embrace what seems a most natural approach for the finite dimensional model, developing a field theory that has as mean field limit the SK solution described in Chapter 6, and building up a perturbation expansion around it. This is justified if we assume that the physics of the SK model remains qualitatively relevant for a finite dimensional system. Vice versa, we may say that, if we are able to build a well defined field theory and control its perturbation expansion, this is a strong indication of the physical relevance of its content.

Even if our program is conceptually standard routine in field theory, from a practical point of view it is far from simple, given the complex nature of the order parameter. As we shall see, even the analysis of Gaussian fluctuations becomes cumbersome.

In the previous chapter we have explicitly computed components of the (bare) propagators in the replicon sector and found that we had a spectrum of masses bounded below by zero modes. In this chapter we show that these massless modes are indeed Goldstone modes related to the spontaneous breaking of a continuous symmetry of the Lagrangian, and thus are to remain massless under loop corrections. We have already analyzed a case of spontaneous symmetry breaking in the different context of the TAP free energy in Chapter 7. Here, we focus on the invariance properties of the spin glass Lagrangian and exhibit the continuous symmetry spontaneously broken below the AT line, which entails corresponding Ward–Takahashi Identities. We start in Section 10.1 by describing the problem in the simpler setting of a system with a two-component order parameter. In Section 10.2 we generalize to the spin glass case, by defining the symmetry which undergoes spontaneous breaking. In Section 10.4 we give a derivation of the Ward–Takahashi Identities, expressing the fact that the massless transverse modes are indeed Goldstone modes. Details of what follows can be found in De Dominicis, Kondor and Temesvari (1998).

The Legendre Transform and invariance properties

To explain the general problem we would like to address, it is convenient to consider first the case of a simpler field theory (at least for what concerns the notation).

The droplet picture

Throughout this book we have developed a spin glass field theory for the fluctuation field around the mean field RSB Parisi solution. The justification for such a theory, as we have stated at the beginning of our discussion, is intimately related to the validity in finite dimension of the nontrivial multi-ergodic physical scenario depicted by the mean field solution. Whether this is the case or not is still not clear. We have commented in the chapters previous to this conclusion what are the main features of the spin glass field theory, the consistency it exhibits, some reasonable extrapolations to dimensions lower than six and some possible predictions for measurable observables. Despite the great effort and the rich results obtained so far, the validity of the theory in three dimensions is still, however, a debated point, as much as the assumption of a nontrivial spin glass state at low temperature. It therefore seems important to us to briefly mention a few alternative points of view where the low temperature phase has different features from the ones we have extensively described.

The droplet model

The richest and most interesting alternative picture for the EA model was developed along the years by various authors (Bray and Moore, 1984, 1986; McMillan, 1984; Fisher and Huse, 1986, 1987, 1988; Newman and Stein, 1992, 1996, 1998) and is generally referred to as the ‘droplet model’, from the Fisher and Huse paper of 1986. The physical scenario described by this model is in striking contrast to the mean field one.

The replica trick leaves an unsatisfactory taste, on account of its unphysical character. We will now introduce an alternative, more physical approach, based on dynamics (De Dominicis, 1978). Recently, the dynamical approach has proved extremely powerful in addressing the complex behaviour of glassy systems at low temperature (see Bouchaud et al., 1998, for a review of recent results). For mean field spin glass systems the dynamics has been fully solved, revealing interesting new patterns of off-equilibrium behaviour where time translational invariance is broken and fluctuation–dissipation relations are violated. Interestingly, the theoretical framework developed for mean field systems seems to be an adequate starting point to describe a great variety of finite dimensional systems, from real spin glasses to glass forming liquids, driven granular matter, critical and coarsening systems, etc.

In this chapter, we describe the dynamical approach from a general perspective, building the formalism step by step, introducing the dynamical observables and their properties. Our starting point is the Langevin equation, that describes the relaxational dynamics of a system in contact with a thermal bath (Section 3.1). Then, for a Ginzburg–Landau–Wilson Hamiltonian, we illustrate how to perform a perturbative analysis via a tree-like expansion of the dynamical equation (Section 3.1.2), or through the more sophisticated Martin–Siggia–Rose generating functional (Section 3.2). Finally, the whole approach is generalized for the Random Field Ising Model where complications arise due to the presence of the quenched disorder.