Quantum Markov Chains for an Asymmetric Mixed Ising–XY Model on a Cayley Tree

Abstract

We study a mixed quantum Ising–XY model on the semi-infinite rooted Cayley tree of order two. For every vertex u, the edge \(\langle u,(u,1)\rangle \) carries an XY interaction and the edge \(\langle u,(u,2)\rangle \) carries an Ising interaction. Using the compatibility criterion for tree-indexed quantum Markov chains and consistently working with the normalized trace, we derive the translation-invariant boundary equation and compute explicitly the associated local transfer operator, namely the one-step partial-trace map which propagates successor boundary data to the parent vertex. We prove that the boundary equation has a unique positive translation-invariant solution for all \(J_I,J_{XY}\in \mathbb {R}\) and \(\beta >0\). Hence the model admits a unique translation-invariant quantum Markov chain generated by a positive translation-invariant boundary condition. We also show that the reduced boundary-law dynamics, i.e. the induced finite-dimensional recursion for the boundary-law parameters, has no admissible periodic points of period greater than one and compute the local two-site entanglement on the natural three-site cluster of the tree.