A quantum Markov chain approach to phase transitions for quantum Ising model with competing XY -interactions on a Cayley tree

The main aim of the present paper by means of the quantum Markov chain (QMC) approach is to establish the existence of a phase transition for the quantum Ising model with competing XY interaction. In this scheme, the C*-algebraic approach is employed to the phase transition problem. Note that these kinds of models do not have one-dimensional analogs, 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 QMCs that are not quasi-equivalent and their supports do not overlap. Moreover, it is also shown that the QMC associated with the model has a clustering property, which implies that the von Neumann algebras corresponding to the states are factors.