Gibbs measures of the Ising model with mixed spin-1 and spin-1/2 on a Cayley tree

TL;DR

This paper studies the Gibbs measures of a mixed spin-1 and spin-1/2 Ising model on a Cayley tree, revealing phase transitions and the existence of multiple translation-invariant Gibbs measures, including in anti-ferromagnetic regimes.

Contribution

It constructs splitting Gibbs measures for the mixed-spin Ising model on a Cayley tree and demonstrates the existence of phase transitions with multiple Gibbs measures, unlike the classical model.

Findings

Existence of three translation-invariant Gibbs measures in certain regimes

Presence of a disordered Gibbs measure similar to the classical Ising model

Identification of non-extremity and extremity of Gibbs measures via tree-indexed Markov chains

Abstract

In the present paper, the Ising model with mixed spin-(1,1/2) is considered on the second order Cayley tree. A construction of splitting Gibbs measures corresponding the model is given which allows to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model has three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, while the classical Ising model does not possesses such Gibbs measures in the anti-ferromagnetic regime. It turns out that the considered model, like the Ising model, exhibits a disordered Gibbs measure. Therefore, non-extremity and extremity of such disordered Gibbs measures is investigated by means of tree-indexed Markov chains.