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On Quantum Markov Chains on Cayley Tree III: Ising Model

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Abstract

In this paper, we consider the classical Ising model on the Cayley tree of order \(k\) (\(k\ge 2\)), and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found critical temperature coincides with the classical critical temperature.

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Notes

  1. We remark that backward quantum Markov chains on lattices and trees have been investigated in [2, 8].

  2. Note that similar kind of constructions of QMC on integer lattice were known in the literature (see for example [1–5].

  3. For the definition of quantum Markov chain we refer [6].

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Acknowledgments

The present study have been done within the Grant ERGS13-024-0057 of Malaysian Ministry of Higher Education. The authors (F.M. and M.S) would like to thanks to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for offering a Junior Associate Scheme fellowship. The authors are grateful to an anonymous referee whose valuable comments and remarks improved the presentation of this paper.

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Authors and Affiliations

  1. Centro Interdisciplinare Vito Volterra, II Università di Roma “Tor Vergata”, Via Columbia 2, 00133 , Roma, Italy

    Luigi Accardi

  2. Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P. O. Box 141, 25710 , Kuantan, Pahang, Malaysia

    Farrukh Mukhamedov & Mansoor Saburov

Authors
  1. Luigi Accardi
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  2. Farrukh Mukhamedov
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  3. Mansoor Saburov
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Correspondence to Farrukh Mukhamedov.

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Accardi, L., Mukhamedov, F. & Saburov, M. On Quantum Markov Chains on Cayley Tree III: Ising Model. J Stat Phys 157, 303–329 (2014). https://doi.org/10.1007/s10955-014-1083-y

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  • DOI: https://doi.org/10.1007/s10955-014-1083-y

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