On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

TL;DR

This paper analyzes Gibbs measures for an Ising model with competing binary interactions on a Cayley tree, classifies the associated von Neumann algebras, and explores their structural properties and examples.

Contribution

It provides a complete description of periodic Gibbs measures and classifies the von Neumann algebras generated by these measures, including their types and relationships.

Findings

All periodic Gibbs measures are characterized.

Von Neumann algebras associated with these measures are classified by type.

Factors related to minimal and maximal Gibbs states are isomorphic.

Abstract

In the present paper the Ising model with competing binary ($J$) and binary ($J_1$) interactions with spin values $\pm 1$, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model considered is studied. We completely describe the set of all periodic Gibbs measures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type III$_\lambda$ then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too. \\[10mm] {\bf Keywords:} Cayley tree, Ising model, competing interactions, Gibbs measure, GNS-construction, Hamiltonian, von Neumann algebra.