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

• URL: http://arxiv.org/abs/2607.14343v1
• Date: Wed, 15 Jul 2026 20:15:08 GMT
• Title: Quantum Markov Chains for an Asymmetric Mixed Ising-XY Model on a Cayley Tree
• Abstract summary: We study a mixed quantum Ising-$XY$ model on the semi-infinite rooted Cayley tree of order two.<n>We derive the translation-invariant boundary equation and compute the associated local transfer operator.<n>We show that the reduced boundary-law dynamics has no admissible periodic points of period greater than one.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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 $β>0$. 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.

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