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Mat. Zametki, 2011, Volume 90, Issue 2, Pages 168–182 (Mi mz8862)  

This article is cited in 5 scientific papers (total in 5 papers)

Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order $2$

L. Accardia, F. M. Mukhamedovb, M. Kh. Saburovb

a Università degli Studi di Roma — Tor Vergata
b International Islamic University Malaysia

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.

Keywords: quantum Markov chain, Cayley tree, $XY$-model, Gibbs state, phase transition, quasiconditional expectation, graph, dynamical system, quasilocal algebra

DOI: https://doi.org/10.4213/mzm8862

Full text: PDF file (552 kB)
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English version:
Mathematical Notes, 2011, 90:2, 162–174

Bibliographic databases:

UDC: 517.98+531
Received: 01.09.2010
Revised: 17.02.2011

Citation: L. Accardi, F. M. Mukhamedov, M. Kh. Saburov, “Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order $2$”, Mat. Zametki, 90:2 (2011), 168–182; Math. Notes, 90:2 (2011), 162–174

Citation in format AMSBIB
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\paper Uniqueness of Quantum Markov Chains Associated with an $XY$-Model on a Cayley Tree of Order~$2$
\jour Mat. Zametki
\yr 2011
\vol 90
\issue 2
\pages 168--182
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\crossref{https://doi.org/10.4213/mzm8862}
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\jour Math. Notes
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Mukhamedov F., Barhoumi A., Souissi A., “Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree”, J. Stat. Phys., 163:3 (2016), 544–567  crossref  mathscinet  zmath  isi  elib  scopus
    2. Mukhamedov F., Barhoumi A., Souissi A., “On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree”, Math. Phys. Anal. Geom., 19:4 (2016), 21  crossref  mathscinet  isi  elib  scopus
    3. Accardi L., Mukhamedov F., Souissi A., “On Construction of Quantum Markov Chains on Cayley trees”, Algebra, Analysis and Quantum Probability, Journal of Physics Conference Series, 697, eds. Ayupov S., Chilin V., Ganikhodjaev N., Mukhamedov F., Rakhimov I., IOP Publishing Ltd, 2016, 012018  crossref  isi  scopus
    4. Mukhamedov F., Souissi A., “Quantum Markov States on Cayley Trees”, J. Math. Anal. Appl., 473:1 (2019), 313–333  crossref  mathscinet  isi  scopus
    5. Mukhamedov F., El Gheteb S., “Clustering Property of Quantum Markov Chain Associated to Xy-Model With Competing Ising Interactions on the Cayley Tree of Order Two”, Math. Phys. Anal. Geom., 22:1 (2019), 10  crossref  mathscinet  zmath  isi  scopus
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