Finitely connected vector spin systems with random matrix interactions

TL;DR

This paper develops a theoretical framework using finite connectivity replica theory to analyze finitely connected vector spin systems with random orthogonal interactions, providing explicit phase diagrams and moments for XY and Heisenberg spins.

Contribution

It introduces a general theory for vector spins with finite connectivity and random orthogonal interactions, extending previous models to continuous spins and arbitrary dimensions.

Findings

Derived phase diagrams for XY and Heisenberg spins.

Calculated moments of the order parameter explicitly.

Numerical simulations support theoretical predictions.

Abstract

We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Since the spins are continuous and the connectivity c remains finite in the thermodynamic limit, the replica-symmetric order parameter is a functional. The general theory is developed for arbitrary values of the dimension d of the spins, and arbitrary choices of the ensemble of random orthogonal matrices. We calculate phase diagrams and the values of moments of the order parameter explicitly for d=2 (finitely connected XY spins with random chiral interactions) and for d=3 (finitely connected classical Heisenberg spins with random chiral interactions). Numerical simulations are shown to support our predictions quite satisfactorily.