On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three

@article{Accardi2010OnQM,
  title={On Quantum Markov Chains on Cayley Tree II: Phase Transitions for the Associated Chain with XY-Model on the Cayley Tree of Order Three},
  author={Luigi Accardi and Farrukh Mukhamedov and Mansoor Saburov},
  journal={Annales Henri Poincar{\'e}},
  year={2010},
  volume={12},
  pages={1109-1144}
}
In the present paper, we study forward quantum Markov chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on a Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators $${\{K_{\langle x,y\rangle}\}}$$. 

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