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

Accardi, Luigi; Mukhamedov, Farrukh; Saburov, Mansoor

doi:10.1007/s00023-011-0107-2

Cited by 34 publications

(26 citation statements)

References 42 publications

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“…The present paper is a continuation of our previous works [6,7]. In [6] we have introduced forward type of quantum Markov chains (QMC) defined on the Cayley tree was studied 1 .…”

Section: Introductionmentioning

confidence: 85%

On Quantum Markov Chains on Cayley Tree III: Ising Model

2014

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In this paper, we consider the classical Ising model on the Cayley tree of order k (k ≥ 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 usual critical temperature.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

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“…The present paper is a continuation of our previous works [6,7]. In [6] we have introduced forward type of quantum Markov chains (QMC) defined on the Cayley tree was studied 1 .…”

Section: Introductionmentioning

confidence: 85%

On Quantum Markov Chains on Cayley Tree III: Ising Model

2014

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“…Moreover, these states should be not quasi-equivalent and their supports do not overlap. Note that in our earlier papers (see [12,13]) we have proved only non quasi-equivalence of the states. In this paper, we additionally prove that the corresponding states do not have overlapping supports.…”

Section: Introductionmentioning

confidence: 82%

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

2016

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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.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

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“…In this section, we are going to demonstrate that localized QMS cannot be reduced to the one dimensional case. More precisely, we explicitly consider the potential (10).…”

Section: Why Localized Quantum Markov States On Trees Are Not Reducible To the One Dimensional Casementioning

confidence: 99%

Quantum Markov states on Cayley trees

Mukhamedov

Souissi

2019

Journal of Mathematical Analysis and Applications

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It is known that any locally faithful quantum Markov state (QMS) on one dimensional setting can be considered as a Gibbs state associated with Hamiltonian with commuting nearest-neighbor interactions. In our previous results, we have investigated quantum Markov states (QMS) associated with Ising type models with competing interactions, which are expected to be QMS, but up to now, there is no any characterization of QMS over trees. We notice that these QMS do not have one-dimensional analogues, hence results of related to one dimensional QMS are not applicable. Therefore, the main aim of the present paper is to describe of QMS over Cayley trees. Namely, we prove that any QMS (associated with localized conditional expectations) can be realized as integral of product states w.t.r. a Gibbs measure. Moreover, it is established that any locally faithful QMS associated with localized conditional expectations can be considered as a Gibbs state corresponding to Hamiltonians (on the Cayley tree) with commuting competing interactions.Mathematics Subject Classification: 46L53, 60J99, 46L60, 60G50, 82B10, 81Q10, 94A17.

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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

Cited by 34 publications

References 42 publications

On Quantum Markov Chains on Cayley Tree III: Ising Model

On Quantum Markov Chains on Cayley Tree III: Ising Model

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

Quantum Markov states on Cayley trees

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