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)).