Phase transitions in the three-state Hard-Core model with competing interactions on the Cayley tree

In this paper, we investigate the thermodynamic properties of a three-state Hard-Core (HC) model with competing nearest-neighbor and next-nearest-neighbor (NNN) interactions on a Cayley tree of order two. By mapping the problem to specific graph constraints (‘Wand’ and ‘Hinge’), we analyze the conditions for the existence of unique and multiple Gibbs measures. A key contribution of this study is the derivation of the phase diagram in the (θ,λ) plane, where θ represents the NNN interaction strength and λ is the activity parameter. Our results establish that the critical activity is a decreasing function of the interaction parameter, explicitly given by λcr=3θ+62θ+2. This relationship demonstrates that stronger attractive NNN interactions (θ>1) significantly lower the threshold for symmetry breaking, facilitating the transition from a disordered gas-like phase to an ordered phase characterized by three distinct Gibbs measures.