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Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree

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Abstract

The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting.

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Notes

  1. Recall that a representation \(\pi _1\) of a \(C^*\)-algebra \({\mathfrak {A}}\) is normal w.r.t. another representation \(\pi _2\), if there is a normal \(*\)- epimorphism \(\rho :\pi _2({\mathfrak {A}})''\rightarrow \pi _1({\mathfrak {A}})''\) such that \(\rho \circ \pi _2=\pi _1\). Two representations \(\pi _1\) and \(\pi _2\) are called quasi-equivalent if \(\pi _1\) is normal w.r.t. \(\pi _2\), and conversely, \(\pi _2\) is normal w.r.t. \(\pi _1\). This means that there is an isomorphism \(\alpha :\pi _1({\mathfrak {A}})''\rightarrow \pi _2({\mathfrak {A}})''\) such that \(\alpha \circ \pi _1=\pi _2\). Two states \(\varphi \) and \(\psi \) of \({\mathfrak {A}}\) are said be quasi-equivalent if the GNS representations \(\pi _\varphi \) and \(\pi _\psi \) are quasi-equivalent [22].

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Acknowledgments

The authors are grateful to professors L. Accardi and F. Fidaleo for their fruitful discussions and useful suggestions on the definition of the phase transition. The authors also thank referees whose valuable comments and remarks improved the presentation of this paper.

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Authors and Affiliations

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

    Farrukh Mukhamedov

  2. Department of Mathematics, Nabeul Preparatory Engineering Institute, Campus Universitairy, Carthage University, Mrezgua, 8000, Nabeul, Tunisia

    Abdessatar Barhoumi

  3. Carthage University, Tunis, Tunisia

    Abdessatar Barhoumi

  4. Department of Mathematics, Marsa Preparatory Institute for Scientific and Technical Studies, Carthage University, Tunis, Tunisia

    Abdessatar Souissi

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  1. Farrukh Mukhamedov
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Correspondence to Farrukh Mukhamedov.

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Mukhamedov, F., Barhoumi, A. & Souissi, A. Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree. J Stat Phys 163, 544–567 (2016). https://doi.org/10.1007/s10955-016-1495-y

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