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Open Systems & Information Dynamics

Uniqueness of Quantum Markov Chain Associated with XY-Ising Model on Cayley Tree of Order Two

Farrukh Maksutovich Mukhamedov, El Gheteb Soueidi

Published Jun 1, 2017• 19 citations
Publisher

Abstract

In this paper, we consider backward and forward Quantum Markov Chains (QMC) associated with XY -Ising model on the Cayley tree of order two. We construct finite volume states with boundary conditions, and define QMC as a weak limit of those states which depend on the boundary conditions. We prove that the limit state is a unique QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Moreover, we observe the relation between backward and forward QMC.

DOI
10.1142/s123016121750010x

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In this paper, we consider backward and forward Quantum Markov Chains (QMC) associated with XY -Ising model on the Cayley tree of order two. We construct finite volume states with boundary conditions, and define QMC as a weak limit of those states which depend on the boundary conditions. We prove that the limit state is a unique QMC associated with such a model, this means the QMC does not depend on the boundary conditions. Moreover, we observe the relation between backward and forward QMC.
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The main aim of the present paper is to provide a new construction of quantum Markov chain (QMC) on arbitrary order Cayley tree. 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. Note that this construction reminds statistical mechanics models with competing interactions on trees. If one considers one dimensional tree, then the provided construction reduces to well-known one, which was studied by the first author. Our construction will allow to investigate phase transition problem in a quantum setting.
In this work, we construct a quantum Markov chain (QMC) associated by the classical Ising model with competing interactions on the Cayley tree of order two. In the construction QMC is defined as a weak limit of finite volume states on quasi-local algebras with boundary conditions. We point out that phase transitions in a quantum setting play an important role to understand quantum spin systems We have defined a notion of phase transition in QMC scheme. Namely, such a notion is based on the quasi-equivalence of QMC. Therefore, such a phase transition is purely non-commutative. In this work we establish 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 usual critical temperature.