Phase transition and Gibbs Measures of Vannimenus model on semi-infinite Cayley tree of order three

TL;DR

This paper investigates phase transitions and Gibbs measures of the Ising-Vannimenus model with competing interactions on a Cayley tree of order three, identifying conditions for phase transitions and describing invariant Gibbs measures.

Contribution

It introduces a new approach to describe translation-invariant Gibbs measures for the model and determines when phase transitions occur on a Cayley tree of order three.

Findings

Identification of conditions for phase transition.

Description of translation-invariant Gibbs measures.

Some measures are proven to be extreme Gibbs distributions.

Abstract

Ising model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree has long been studied but there are still many problems untouched. This paper tackles new Gibbs measures of Ising-Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures for the model. We show that some of the measures are extreme Gibbs distributions. In this paper we take up with trying to determine when phase transition does occur.