Maximal Mean Field Solutions in the Random Field Ising Model: the Pattern of the Symmetry Breaking

TL;DR

This paper investigates the mean field equations of the 3D Random Field Ising Model, identifying critical points for magnetization and multiple solutions, and analyzing the nature of correlations within solutions.

Contribution

It provides a detailed analysis of the phase diagram and symmetry breaking patterns in the mean field solutions of the 3D RFIM, highlighting conditions for multiple solutions.

Findings

Two critical values of β identified for magnetization and solution multiplicity

Within each solution, correlation lengths remain finite

The phase diagram exhibits distinct symmetry breaking regimes

Abstract

In this note we study the mean field equations for the $3d$ Random Field Ising Model. We discuss the phase diagram of the model, and we address the problem of finding if such equations admit more than one solution. We find two different critical values of $\beta$: one where the magnetization takes a non-zero expectation value, and one where we start to have more than one solution to the mean field equation. We find that, inside a given solution, there are no divergent correlation lengths.