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

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 $C^{\ast }$-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.

30 pages. arXiv admin note: substantial text overlap with arXiv:1605.04546; text overlap with arXiv:cond-mat/0401295 by other authors

References in corpus (5)

• The Quantum Transverse Field Ising Model on an Infinite Tree from Matrix Product States
• Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree
• Quantum Markov States on Cayley trees
• AKLT Models with Quantum Spin Glass Ground States
• XY ring exchange model with frustrated Ising coupling on the triangular lattice

Cited by in corpus (3)

• Open Quantum Random Walks and Quantum Markov chains on Trees I: Phase transitions
• Refinement of quantum Markov states on trees
• Quantum Markov Chains on the Comb graphs: Ising model