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

Luigi Accardi; Farrukh Mukhamedov; Mansoor Saburov

doi:10.1007/s00023-011-0107-2

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

Luigi Accardi
, Farrukh Mukhamedov
, Mansoor Saburov

Research output: Contribution to journal › Article › peer-review

34 Citations (Scopus)

Abstract

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〉}.

Original language	English
Pages (from-to)	1109-1144
Number of pages	36
Journal	Annales Henri Poincare
Volume	12
Issue number	6
DOIs	https://doi.org/10.1007/s00023-011-0107-2
Publication status	Published - Sept 2011
Externally published	Yes

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Nuclear and High Energy Physics
Mathematical Physics

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10.1007/s00023-011-0107-2

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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",

abstract = "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〉\}.",

author = "Luigi Accardi and Farrukh Mukhamedov and Mansoor Saburov",

note = "Funding Information: The present study has been done within the grant FRGS0308-91 of Malaysian Ministry of Higher Education. The authors also acknowledge the MOSTI grant 01-01-08-SF0079. This work was done while the second named author (F.M.) was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, as a Junior Associate. He would like to thank the Centre for hospitality and financial support.",

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AU - Saburov, Mansoor

N1 - Funding Information: The present study has been done within the grant FRGS0308-91 of Malaysian Ministry of Higher Education. The authors also acknowledge the MOSTI grant 01-01-08-SF0079. This work was done while the second named author (F.M.) was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, as a Junior Associate. He would like to thank the Centre for hospitality and financial support.

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N2 - 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〉}.

AB - 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〉}.

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

Abstract

ASJC Scopus subject areas

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