# Phase Transitions for quantum Ising model with competing XY -interactions on a Cayley tree

## TL;DR

This paper proves the existence of phase transitions in a quantum Ising model with XY interactions on a Cayley tree using a $C^*$-algebraic quantum Markov chain approach, highlighting unique properties of these models.

## Contribution

It establishes phase transition conditions for the quantum Ising model with XY interactions on Cayley trees, a case lacking one-dimensional analogues, using a novel algebraic framework.

## Key findings

- Phase transition exists when XY interactions are present.
- No phase transition occurs if Ising interactions vanish.
- QMC states exhibit clustering and are associated with factor von Neumann algebras.

## Abstract

The main aim of the present paper is to establish the existence of a phase transition for the quantum Ising model with competing XY interactions within the quantum Markov chain (QMC) scheme. In this scheme, we employ the $C^*$-algebraic approach to the phase transition problem. Note that these kinde of models do not have one-dimensional analogues, i.e. the considered model persists only on trees. It turns out that if the Ising part interactions vanish then the model with only competing XY -interactions on the Cayley tree of order two does not have a phase transition. By phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model have clustering property which implies that the von Neumann algebras corresponding to the states are factors.