Gibbs measures and free energies of the Ising-Vannimenus Model on the Cayley tree

TL;DR

This paper rigorously analyzes the Ising-Vannimenus model on a Cayley tree, establishing conditions for Gibbs measures, phase transitions, and calculating free energies using a measure-theoretical approach.

Contribution

It introduces a rigorous measure-theoretical framework for the Ising-Vannimenus model on Cayley trees, previously only studied numerically.

Findings

Conditions for existence of Gibbs measures

Existence of phase transitions established

Explicit calculations of free energies

Abstract

In this paper, we consider Ising-Vannimenus model on a Cayley tree for order two with competing nearest-neighbor, prolonged next-nearest neighbor interactions. We stress that the mentioned model was investigated only numerically, without rigorous (mathematical) proofs. One of the main point of this paper is to propose a measure-theoretical approach the considered model. We find certain conditions for the existence of Gibbs measures corresponding to the model. Then we establish the existence of the phase transition. Moreover, the free energies of the found Gibbs measures are calculated.