On quantum Markov chains on Cayley tree II: phase transitions for the associated chain with XY -Model on the Cayley tree of order tree - Accardi, Luigi; Mukhamedov, Farrukh; Saburov, Mansoor (Birkhaeuser Verlag AG, 2011)

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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_{x,y}}.