Mixed quantum Ising–XY model on a Cayley tree of order two

This paper deals with a quantum Markov chain (QMC) associated with mixed quantum Ising–XY model on a Cayley tree of order two. The considered mixed model has the nearest-neighbor Ising interaction \(J_I\) at odd levels of the tree, and the nearest-neighbor XY-interaction \(I_{XY}\) at even levels of the tree. It is known that for the nearest-neighbor Ising model on the Cayley tree, there occurs a phase transition. However, for the quantum XY model on the same tree there is unique quantum Markov chain. Therefore, it is natural to investigate the mixture of these models on the Cayley tree of order two. It turns out that for this mixed model there is only unique periodic QMC, which is a translation-invariant one. Moreover, one can construct non-translation-invariant QMC for the model.