Replica-symmetric solutions of a dilute Ising ferromagnet in a random field

TL;DR

This paper applies the replica method to analyze the stability of replica-symmetric solutions in a dilute Ising ferromagnet subjected to random external fields, revealing conditions under which the paramagnetic state remains stable.

Contribution

It introduces a variational free energy expression for the model and analyzes the stability of solutions, extending understanding beyond spin glass models.

Findings

Paramagnetic solution is fully stable in large random fields.

Analysis performed at zero temperature with specific restrictions on random fields.

Provides a detailed stability analysis contrasting with spin glass behavior.

Abstract

We use the replica method in order to obtain an expression for the variational free energy of an Ising ferromagnet on a Viana-Bray lattice in the presence of random external fields. Introducing a global order parameter, in the replica-symmetric context, the problem is reduced to the analysis of the solutions of a nonlinear integral equation. At zero temperature, and under some restrictions on the form of the random fields, we are able to perform a detailed analysis of stability of the replica-symmetric solutions. In contrast to the behaviour of the Sherrington-Kirkpatrick model for a spin glass in a uniform field, the paramagnetic solution is fully stable in a sufficiently large random field.