Quantum Ising model with generalized competingXY-interactions on a Cayley tree

Abstract

In this paper, the phase transition phenomena for the Ising model (with nearest-neighbor interaction [Formula: see text]) but with quantum generalized competing [Formula: see text]-interactions ([Formula: see text] and [Formula: see text] coupling constants) are treated by means of a quantum Markov chain (QMC) approach. We point out that the case [Formula: see text] has been carried out in Ref. 32. Note that if [Formula: see text], then it turns out that phase transition exists, for any value of [Formula: see text], while the Ising coupling constant should satisfy [Formula: see text]. 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.