Abstract:
Mean-field theory is usually the first attempt to assess the behavior of statistical systems with a large number degrees of freedom. Mean-field approximation suppressing spatial fluctuations was formalized by the limit to an infinite-dimensional lattice and is for most lattice models analytically solvable. Not for spin glasses with randomly distributed spin exchange. The main objective of the lecture is to discuss the reasons why spin glass models evade solubility even in the simplest mean-field limit.

We first discuss the standard way of derivation of the mean-field solution via the replica trick (replica-symmetry breaking solution). In the effort to understand the meaning of the order parameters introduced in the replica trick we present an alternative derivation of the mean-field solution using thermodynamic principles. We show that successive replications of the original phase space are needed to reach thermodynamic homogeneity of the mean-field free energy. Within the thermodynamic approach we propose a plausible physical interpretation of the order parameters of the spin-glass phase. Finally, we compare two forms of the replica-breaking schemes: discrete and continuous distributions of the hierarchies of replicas of the original phase space. We show that the latter can be understood as a limit of the former, but the solution with the continuous distribution of replica hierarchies exists independently of the discrete scheme and their stability.