Abstract

We introduce the basic spin glass models, namely the Edwards–Anderson model on a finite-dimensional lattice with short-range interaction and the Sherrington–Kirkpatrick model on the complete graph. The quenched equilibrium state which is used to describe the thermodynamical properties of a general disordered system is defined, together with the concept of real replicas. The notion of mean-field for a spin glass model is discussed. Finally, the original computations for the Sherrington–Kirkpatrick model based on the replica method are presented – namely the replica symmetric solution and the Parisi replica symmetry breaking scheme.

The spin glass problem

Spin glass models have been considered in different scientific contexts, including experimental condensed matter physics, theoretical physics, mathematical statistical physics and, more recently, probability. They have also been used to solve problems in fields as diverse as theoretical computer science (combinatorial optimization, traveling salesman, Boolean satisfiability, number partitioning, random assignment, error correcting codes, etc.), biology (Hopfield model), population genetics (hierarchical coalescence), and the economy (modelization of financial markets). Thus spin glasses represent a true example of a multi-disciplinary topic.

The study of spin glasses began after experiments on magnetic alloys, for instance metals like Fe, Mn and Cr weakly diluted in metals such as Au, Ag and Cu. It was observed that their thermodynamical behavior was not compatible with the theory of ferromagnetism and showed peculiar dynamical out-of-equilibrium properties such as aging and rejuvenation effects (for a recent account of spin glass dynamics and connection to experimental data see Cugliandolo and Kurchan (2008)).

Abstract

In this chapter we present some results on numerical simulation in three-dimensional spin glasses. In fact, there are problems for which the analytical approach is out of reach. In such cases numerical simulations are a source of hints and suggestions. Their robustness is based on the fact that the asymptotic properties of the model for large volumes are identified. After describing the standard algorithm used (parallel tempering) we address the following problems: overlap equivalence among site and link overlaps, ultrametricity or hierarchical organization of the equilibrium states, decay of correlations, pure state identification, energy interfaces, and stiffness exponents related to the lower critical dimension.

Introduction

Numerical simulations have played a crucial role in the development of spin glass theory, especially in those cases in which exact results are unavailable due to formidable mathematical difficulties. Some of the first systematic works were by Bray and Moore (1984), Ogielski and Morgenstern (1985), and Bhatt and Young (1988). For a recent review, see Marinari et al. (1997) and the references therein. It is also interesting to check Newman and Stein (1996) and Marinari et al. (2000) for examples of the difficulties of reconciling theory with numerical simulations in finite-dimensional spin glasses.

The general outcome of these (and other) numerical studies is that many features of themean-field theory are seen in the ever larger finite-volume systems accessible to numerical simulations. However, no definite conclusions can be reached by means of the sole use of computers.

Abstract

In this chapter we deal with the large-volume limit, often called the thermodynamic limit. While the problem has received a lot of attention in deterministic systems from the early 1960s both in statistical mechanics (Ruelle (1999)) and Euclidean quantum field theory (Guerra (1972)), in random systems a major breakthrough has been the introduction of the quadratic interpolation method by Guerra and Toninelli (2002). We review the results for finite-dimensional systems with Gaussian or,more generally, centered interactions.We then extend the analysis to quantum models and to non-centered interactions satisfying a thermodynamic stability condition. Special attention is devoted to the correction to the leading term, i.e. the surface pressure, which is investigated for various boundary conditions and for a wide class of models. A complete result is obtained on the Nishimori line where we can make use of the full correlation inequalities set introduced in the previous chapter. Finally the mean-field case is analyzed for the relevant models that appear in the literature.

Introduction

The infinite-volume limit in spin glasses has been tackled for some time (Vuillermot (1977); Ledrappier (1977); Pastur and Figotin (1978); Khanin and Sinai (1979); van Enter and van Hemmen (1983); Zegarlinski (1991)). The difficulties with respect to the deterministic case arose due to the randomness of the interaction which requires both the study of the averaged quantities as well as the random ones. For quantities like pressure (free energy) and ground state energy per particle, the fluctuation between samples vanishes for large volumes (self-averaging), therefore it suffices to study the limit of the average value.

Abstract

In this chapter we analyze some properties of the overlap probability distribution by looking at its factorization rules. It has been now understood that the disputed features of the spin glass phase are, apart from the triviality issue of the single overlap distribution, related to the structure of the joint overlap distribution. For example, a centered Gaussian family distribution is completely identified by its covariance, thanks to the Wick theorem factorization law. In the spin glass phase, the factorization structure first appeared within the Parisi replica symmetry breaking solution of the Sherrington–Kirkpatrick model. In this chapter we show that some of those properties can indeed be recovered by using a stability argument which amounts to proving how the equilibrium state is left unchanged by small perturbations. In turn, this is equivalent to controlling the size of the fluctuations for suitable thermodynamic quantities. After introducing the general method we review the stochastic stability, the control of thermal fluctuations, and the graph-theoretical description of the emerging identities. We then analyze the fluctuations due to the disorder and establish the self-averaging of the random internal energy. The extension to interactions with non-zero averages and the specific analysis on the Nishimori line follows. The chapter ends with newidentities derived from the control of the fluctuations for free energy differences involving flips of the interactions.

The stability method and the structural identities

The solution of the Sherrington–Kirkpatrick model obtained via the replica approach showed that the order parameter of the theory, namely the overlap distribution, is sufficient to fully describe the quenched state.

Spin glasses are statistical mechanics systems with random interactions. The alternating sign of those interactions generates a complex physical behavior whose mathematical structure is still largely uncovered. The approach we follow in this book is that of mathematical physics, aiming at the rigorous derivation of their properties with the help of physical insight.

The book starts with the theoretical physics origins of the spin glass problem. The main models are introduced and a description of the replica approach is illustrated for the Sherrington–Kirkpatrick model.

Chapters 2 and 3 contain the starting points of the mathematical rigorous approach leading to the control of the thermodynamic limit for spin glass systems. Correlation inequalities are introduced and proved in various settings, including the Nishimori line. They are then used to prove the existence of the large-volume limit in both short-range and mean-field models.

Chapter 4 deals with exact results which belong to the mean-field case. The methods and techniques illustrated span from the Ruelle probability cascades to theAizenman–Sims–Starr variational principle. In this framework theGuerra upper bound theorem for the pressure is presented and the Talagrand theorem is reported.

Chapter 5 deals with the structural identities characterizing the spin glass phase. These are obtained by an extension of the stochastic stability method, i.e. an invariance property of the system under small perturbations, together with the self-averaging property.

Chapter 6 features some problems which are still out of analytical reach and are investigated with numerical methods: the equivalence among different overlap structures, the hierarchical organization of the states, the decay of correlations, and the energy interface cost.

A number of experiments and observations have been undertaken to test for SOC in the ‘real world’. Ultimately, these observations motivate the research based on analytical and numerical tools, although the latter provide the clearest evidence for SOC whereas experimental evidence is comparatively ambiguous. What evidence suffices to call a system self-organised critical? One might be inclined to say scale invariance without tuning, but as discussed in Sec. 9.4, the class of such systems might be too large and comprise phenomena that traditionally are regarded as distinct from criticality, such as, for example, diffusion.

In most cases, systems suspected to be self-organised critical display a form of scaling and a form of avalanching, suggesting a separation of time scales. Because of the early link to 1/f noise (Sec. 1.3.2), some early publications regard this as sufficient evidence for SOC. At the other end of the spectrum are systems that closely resemble those that are studied numerically and whose scaling behaviour is not too far from that observed in numerical studies. Yet it remains debatable whether any numerical model is a faithful representation of any experiment or at least incorporates the relevant interactions.

At first sight, solid experimental evidence for scaling or even universality is sparse among the many publications that suggest links to SOC. This result is even more sobering as evidence for SOC is heavily biased – there are very few publications (e.g. Jaeger, Liu, and Nagel, 1989; Kirchner and Weil, 1998) on failed attempts to identify SOC where it was suspected.