Analytical solution to Heisenberg spin glass models on sparse random graphs and their de Almeida-Thouless line

TL;DR

This paper introduces a numerical method to solve Heisenberg spin glass models on sparse random graphs, enabling analysis of their phase transitions and critical behavior, which was previously limited to fully connected or Ising models.

Contribution

The authors develop a discretized Belief Propagation approach for continuous Heisenberg spins on sparse graphs, including algorithms for zero-temperature and high-connectivity regimes, and locate the de Almeida-Thouless line.

Findings

Locates the de Almeida-Thouless line for Heisenberg spin glasses.

Provides a consistent method for analyzing zero-temperature critical fields.

Extends solution techniques to models with continuous spins on sparse graphs.

Abstract

Results regarding spin glass models are, to this day, mainly confined to models with Ising spins. Spin glass models with continuous spins exhibit interesting new physical behaviors related to the additional degrees of freedom, but have been primarily studied on fully connected topologies. Only recently some advancements have been made in the study of continuous models on sparse graphs. In this work we partially fill this void by introducing a method to solve numerically the Belief Propagation equations for systems of Heisenberg spins on sparse random graphs via a discretization of the sphere. We introduce techniques to study the finite-temperature, finite-connectivity case as well as novel algorithms to deal with the zero-temperature and large connectivity limits. As an application, we locate the de Almeida-Thouless line for this class of models and the critical field at zero temperature, showing the full consistency of the methods presented. Beyond the specific results reported for Heisenberg models, the approaches presented in this paper have a much broader scope of application and pave the way to the solution of strongly disordered models with continuous variables.