On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three

TL;DR

This paper constructs quantum Markov chains on a Cayley tree and demonstrates a phase transition for the XY-model on a Cayley tree of order three within the QMC framework, indicating multiple quasi-equivalent states.

Contribution

It provides a novel construction of quantum Markov chains on Cayley trees and proves the existence of phase transitions for the XY-model in this setting.

Findings

Existence of two quasi-equivalent QMC states for the XY-model

Construction method for quantum Markov chains on Cayley trees

Identification of phase transition in the XY-model on a Cayley tree of order three

Abstract

In the present paper we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two now quasi equivalent QMC for the given family of interaction operators $\{K_{<x,y>}\}$.