Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree

TL;DR

This paper proves the existence of phase transitions in quantum Markov chains for Ising models on Cayley trees, highlighting unique phenomena in the disordered phase and extending the construction of forward QMCs.

Contribution

It introduces a general construction of forward quantum Markov chains and demonstrates phase transitions specific to Cayley tree Ising models with boundary condition dependence.

Findings

Existence of multiple non-quasi-equivalent QMCs indicating phase transition

Identification of non-overlapping supports for different QMCs

Discovery of algebraic properties in the disordered phase

Abstract

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting.