Replica trick with real replicas: A way to build in thermodynamic homogeneity

TL;DR

This paper introduces a method using real replicas to analyze and ensure thermodynamic homogeneity in spin glass models, revealing the necessity of hierarchical solutions for stability at low temperatures.

Contribution

It proposes a novel approach with real replicas to study thermodynamic homogeneity, connecting it to the hierarchical Parisi solution in spin glasses.

Findings

Averaged free energy is not thermodynamically homogeneous at low temperatures.

Minimizing inhomogeneity naturally leads to the Parisi hierarchical solution.

Conditions for global thermodynamic homogeneity are established for SK and p-spin models.

Abstract

We use real replicas to investigate stability of thermodynamic homogeneity of the free energy of the Sherrington-Kirkpatrick (SK) model of spin glasses. Within the replica trick with the replica symmetric ansatz we show that the averaged free energy at low temperatures is not thermodynamically homogeneous. The demand of minimization of the inhomogeneity of thermodynamic potentials leads in a natural way to the hierarchical solution of the Parisi type. Conditions for the global thermodynamic homogeneity are derived and evaluated for the SK and $p$-spin infinite range models.