Abstract
In this paper, we consider the Ising-XY model with competing interactions on the Cayley tree of order two. This model can be seen as a non-commutative (i.e. J-XY-interactions on next-neighbor vertices) perturbation of the classical Ising model on the Cayley tree. For the considered model we establish the existence of three translation-invariant quantum Markov chains. We notice that if the XY-interactions vanish, i.e. J = 0, then one gets the Ising model. If the classical Ising model vanishes in the considered model, then we obtain XY-model for which it turns out there exists only one translation invariant QMC.