Stability of solutions of the Sherrington-Kirkpatrick model with respect to replications of the phase space

TL;DR

This paper investigates the stability of solutions in the Sherrington-Kirkpatrick model using real replicas, revealing a natural emergence of an ultrametric hierarchy of order parameters consistent with the Parisi RSB scheme, governed by thermodynamic homogeneity.

Contribution

It introduces a thermodynamically motivated approach to derive the ultrametric hierarchy of order parameters in the SK model, connecting stability conditions to the Parisi RSB scheme.

Findings

Hierarchy of order parameters emerges from thermodynamic homogeneity.

The number of hierarchical levels is determined by generalized de Almeida Thouless conditions.

The hierarchical structure provides a physical interpretation of the order parameters.

Abstract

We use real replicas within the Thouless, Anderson and Palmer construction to investigate stability of solutions with respect to uniform scalings in the phase space of the Sherrington-Kirkpatrick model. We show that the demand of homogeneity of thermodynamic potentials leads in a natural way to a thermodynamically dependent ultrametric hierarchy of order parameters. The derived hierarchical mean-field equations appear equivalent to the discrete Parisi RSB scheme. The number of hierarchical levels in the construction is fixed by the global thermodynamic homogeneity expressed as generalized de Almeida Thouless conditions. A physical interpretation of a hierarchical structure of the order parameters is gained.