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

Results regarding spin glass models are, to this day, mainly confined to models with discrete (usually 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 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.