Symmetric and antisymmetric Ising models on closed Cayley trees: Exact solutions and phase transitions

In this paper, we study the Ising model on a closed Cayley tree with a branching ratio of three. An exact solution of the model is found in the framework of which it is shown that the critical temperatures of the first-order phase transitions of the model are determined by a system of recursive equations. Unlike the open Cayley tree, the closed structure introduces additional interactions that significantly impact the system’s phase transition behavior. We consider two distinct symmetry cases: the symmetric closed Cayley tree, where the upper and lower subtrees are structurally identical, and the antisymmetric closed Cayley tree, where interactions in the upper and lower subtrees are opposite in nature. We emphasize that the antisymmetric model introduces fundamentally different recursion structures compared to the symmetric case when the closed nature of the tree is retained, leading to distinct phase behaviors. Using recursion relations, we derive exact expressions for the limiting Gibbs measures and analyze their behavior to determine critical conditions for phase transitions. Our findings reveal that the closed Cayley tree supports phase transitions in both ferromagnetic and antiferromagnetic cases, in contrast to the open tree, where phase transitions typically occur only in the ferromagnetic regime. Additionally, we classify the different types of solutions for the recursion equations, providing a characterization of the system’s equilibrium states.