In the present paper we consider a Quantum Markov Chain (QMC) corresponding to the XY -model with competing Ising interactions on the Cayley tree of order two. Earlier, using finite volumes states one has been constructed QMC as a weak limit of those states which depends on the boundary conditions. It was proved that the limit state does exist and not depend on the boundary conditions, i.e. it is unique. In the present paper, we establish that the unique QMC has the clustering property, i.e. it is mixing with respect to translations of the tree. This means that the von Neumann algebra generated by this state is a factor.

中文翻译：

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二阶凯莱树上具有竞争交互作用的XY模型相关的量子马尔可夫链的聚类特性

在本文中，我们考虑对应于 XY 模型的量子马尔可夫链 (QMC)，在二阶凯莱树上具有竞争性的 Ising 相互作用。早先，使用有限体积状态已经构建了 QMC 作为依赖边界条件的那些状态的弱极限。证明了极限状态确实存在并且不依赖于边界条件，即它是唯一的。在本文中，我们确定唯一的 QMC 具有聚类特性，即它在树的翻译方面是混合的。这意味着这个状态产生的冯诺依曼代数是一个因子。