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

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

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English version:
Mathematical Notes, 2011, 90:2, 162–174

Bibliographic databases:

UDC: 517.98+531
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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• http://mi.mathnet.ru/eng/mz8862
• https://doi.org/10.4213/mzm8862
• http://mi.mathnet.ru/eng/mz/v90/i2/p168

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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
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
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
4. Mukhamedov F., Souissi A., “Quantum Markov States on Cayley Trees”, J. Math. Anal. Appl., 473:1 (2019), 313–333
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
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