Ising model on Cayley trees: a new class of Gibbs measures and their comparison with known ones

Рахматуллаев, Музаффар Мухаммаджанович; Rozikov, U. A.

doi:10.1088/1742-5468/aa85c2

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…Let A (x,(x,1),(x,2)) := K <x,(x,1)> K <x,(x,2)> L >(x,1),(x,2)< (20) The operator A (x,(x,1),(x,2)) is localized on the algebra of observable associated with the ternary (x, (x, 1), (x, 2)).…”

Section: Why Localized Quantum Markov States On Trees Are Not Reducib...mentioning

confidence: 99%

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Refinement of quantum Markov states on trees

Mukhamedov¹,

Souissi²

2021

J. Stat. Mech.

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In the present paper, we propose a refinement for the notion of quantum Markov states (QMS) on trees. A structure theorem for QMS on general trees is proved. We notice that any restriction of QMS in the sense of reference Accardi and Fidaleo (2003 J. Funct. Anal. 200 324–347) is not necessarily to be a QMS. It turns out that localized QMS has the mentioned property which is called sub-Markov states, this allows us to characterize translation invariant QMS on regular trees.

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Section: Why Localized Quantum Markov States On Trees Are Not Reducib...mentioning

confidence: 99%

“…On the other hand, in [14][15][16][17] a particular class of QMC associated to the Ising types models on the Cayley trees have been explored (see [18][19][20][21] for recent development on models over such trees). It turned out that the above considered QMCs fall to a special class called quantum Markov states (QMS) (see [22,23]).…”

Section: Introductionmentioning

confidence: 99%

Refinement of quantum Markov states on trees

Mukhamedov¹,

Souissi²

2021

J. Stat. Mech.

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“…On the other hand, in [37,38,45,46] a particular class of quantum Markov chains (QMC) associated to the Ising types models on the Cayley trees have been explored (see [17,29,53,55] for recent development on models over such trees). It turned out that the above considered QMCs fall to a special class called quantum Markov states (QMS) (see [42,44]).…”

Section: Introductionmentioning

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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“…In [27], [28] authors constructed new set real values Gibbs measures for the Ising model on the Cayley tree. In [29] for the Ising model on the Cayley tree of order k, a new sets of non translation-invariant ((k 0 )−translation-invariant) Gibbs measures is constructed.…”

Section: Introductionmentioning

confidence: 99%