Quantum Ising model with generalized competing XY -interactions on a Cayley tree

In this paper, the phase transition phenomena for the Ising model (with nearest-neighbor interaction J0) but with quantum generalized competing XY-interactions (J1 and J2 coupling constants) are treated by means of a quantum Markov chain (QMC) approach. We point out that the case J1 = J2 has been carried out in Ref. 32. Note that if J2 = 0, then it turns out that phase transition exists, for any value of J1, while the Ising coupling constant should satisfy 2J0β >ln 3. This means that the Ising interaction is dominated in the considered situation, i.e. the X-competing interactions' role is negligible. This kind of phenomena was not detected in this mentioned paper. Phase transition means the existence of at least two distinct QMCs which are not quasi-equivalent and their supports do not overlap. To prove the quasi-equivalence, it is first established that the QMCs satisfy clustering property.