The replica symmetric solution for Orthogonally Constrained Heisenberg Model on Bethe lattice

TL;DR

This paper analyzes the thermodynamic behavior of a Heisenberg spin model on Bethe lattices with an unconventional interaction, using a novel replica method to determine phase transition bounds.

Contribution

It introduces a new approach based on the replica method to compute free energy and analyze phase stability in a complex spin system with orthogonal constraints.

Findings

Derived the paramagnetic free energy using the new replica approach

Identified an instability line indicating phase transition boundaries

Provided insights into the nature of the observed instability

Abstract

In this paper, we study the thermodynamic properties of a system of $D$-components classical Heisenberg spins lying on the vertices of a random regular graph, with an unconventional first neighbor non-random interaction $J(\mathbf{S}_i\cdot \mathbf{S}_k)^2$. We can consider this model as a continuum version of anti-ferromagnetic $D$-states Potts model. We compute the paramagnetic free energy, using a new approach, presented in this paper for the first time, based on the replica method. Through the linear stability analysis, we obtain an instability line on the temperature-connectivity plane that provides a bound to the appearance of a phase transition. We also argue about the character of the instability observed.