Quantum Markov Fields on Graphs

Accardi, Luigi; Ohno, Hiromichi; Mukhamedov, Farrukh

doi:10.1142/s0219025710004000

Cited by 34 publications

(35 citation statements)

References 18 publications

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“…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].…”

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confidence: 88%

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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confidence: 88%

On Quantum Markov Chains on Cayley Tree III: Ising Model

2014

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“…The present paper is a continuation of the study of quantum Markov chains begun in [16]. Nevertheless, in the present paper, we propose another construction (differing from the one in [16]) of quantum Markov chains which corresponds to a "forward" quantum Markov chain.…”

Section: Introductionmentioning

confidence: 89%

“…It should be noted that quantum Markov chains over Cayley trees were first studied in [14]; similarly, the thermodynamic limit of a VBS-model on a Cayley tree was considered [15]. The notion of Markov property of states defined on a quasilocal algebra over hierarchical trees was introduced in [16]. We have generalized the construction proposed in [9] to the case of trees.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2

2011

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Abstract-We propose the construction of a quantum Markov chain that corresponds to a "forward" quantum Markov chain. In the given construction, the quantum Markov chain is defined as the limit of finite-dimensional states depending on the boundary conditions. A similar construction is widely used in the definition of Gibbs states in classical statistical mechanics. Using this construction, we study the quantum Markov chain associated with an XY -model on a Cayley tree. For this model, within the framework of the given construction, we prove the uniqueness of the quantum Markov chain i.e., we show that the state is independent of the boundary conditions.

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“…where as before A x,(x,1),...,(x,k) is given by (8). Then the functionals {ϕ (n) w 0 ,h } satisfy the compatibility condition (14). Moreover, there is a unique forward quantum Markov chain…”

Section: Construction Of Quantum Markov Chains On Cayley Treementioning

confidence: 95%

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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Quantum Markov Fields on Graphs

Cited by 34 publications

References 18 publications

On Quantum Markov Chains on Cayley Tree III: Ising Model

On Quantum Markov Chains on Cayley Tree III: Ising Model

Uniqueness of quantum Markov chains associated with an XY-model on a cayley tree of order 2

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

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