Equilibrium analysis of a symmetric bipartite mean-field Ising model

This paper investigates a symmetric bipartite mean-field Ising model in which two interacting subsystems are coupled through averaged inter-group interactions. The study focuses on the evaluation of the thermodynamic limit of the pressure and the characterization of equilibrium states obtained from the corresponding mean-field equations. Using large-deviation techniques, we provide an explicit computation of the pressure and establish the conditions under which phase transitions occur. The analysis further identifies the structure and stability of the critical points of the mean-field potential and determines how they depend on temperature and coupling parameters. The results confirm that the system undergoes a continuous symmetry-breaking transition at a critical temperature set by the largest eigenvalue of the reduced interaction matrix. When the smaller eigenvalue is also positive, a second, aligned ordered state appears; we show that it is a saddle point at first, becomes locally stable (metastable) below a characteristic temperature, but never carries the global maximum of the pressure at zero field. A sufficiently strong common field makes it the equilibrium state through a first-order transition. This work contributes to the theoretical understanding of phase behaviour in symmetric bipartite mean-field systems and their equilibrium properties.