[Transitions de phases pour un modèle XY sur un arbre de Cayley dʼordre trois dans un schéma de chaines de Markov quantiques]
Dans cette Note on étudie des chaines de Markov directes (QMC) définies sur un arbre de Cayley. En utilisant la structure en arbre des graphes on donne une construction de chaines de Markov quantiques sur un arbre de Cayley. Au moyen de telles constructions on démontre lʼexistence dʼune transition de phases pour un modèle XY sur un arbre de Cayley dʼordre trois dans un schéma QMC. La transition de phases correspond ici à lʼexistence de deux QMC distinctes pour une famille dʼopérateurs dʼinteractions.
In the present Note we study forward Quantum Markov Chains (QMC) defined on a Cayley tree. Using the tree structure of graphs, we give a construction of quantum Markov chains on the Cayley tree. By means of such constructions we prove the existence of a phase transition for the XY-model on a Cayley tree of order three in QMC scheme. By the phase transition we mean the existence of two distinct QMC for the given family of interaction operators .
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@article{CRMATH_2011__349_7-8_425_0,
author = {Mukhamedov, Farrukh and Saburov, Mansoor},
title = {Phase transitions for {\protect\emph{XY}-model} on the {Cayley} tree of order three in quantum {Markov} chain scheme},
journal = {Comptes Rendus. Math\'ematique},
pages = {425--428},
publisher = {Elsevier},
volume = {349},
number = {7-8},
year = {2011},
doi = {10.1016/j.crma.2011.02.010},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2011.02.010/}
}
TY - JOUR AU - Mukhamedov, Farrukh AU - Saburov, Mansoor TI - Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme JO - Comptes Rendus. Mathématique PY - 2011 SP - 425 EP - 428 VL - 349 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2011.02.010/ DO - 10.1016/j.crma.2011.02.010 LA - en ID - CRMATH_2011__349_7-8_425_0 ER -
%0 Journal Article %A Mukhamedov, Farrukh %A Saburov, Mansoor %T Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme %J Comptes Rendus. Mathématique %D 2011 %P 425-428 %V 349 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2011.02.010/ %R 10.1016/j.crma.2011.02.010 %G en %F CRMATH_2011__349_7-8_425_0
Mukhamedov, Farrukh; Saburov, Mansoor. Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 425-428. doi : 10.1016/j.crma.2011.02.010. http://www.numdam.org/articles/10.1016/j.crma.2011.02.010/
[1] On the noncommutative Markov property, Funct. Anal. Appl., Volume 9 (1975), pp. 1-8
[2] Quantum Markov fields, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 6 (2003), pp. 123-138
[3] Markovian cocycles, Proc. Roy. Irish Acad. Sect. A, Volume 83 (1983), pp. 251-263
[4] On Quantum Markov Chains on Cayley tree I: uniqueness of the associated chain with XY-model on the Cayley tree of order two, Infin. Dimens. Anal. Quantum Probab. Relat. Top., in press | arXiv
[5] Quantum Markov fields on graphs, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 13 (2010), pp. 165-189
[6] Valence bond ground states in isotropic quantum antiferromagnets, Comm. Math. Phys., Volume 115 (1988), pp. 477-528
[7] Operator Algebras and Quantum Statistical Mechanics. 1, Texts and Monographs in Physics, Springer-Verlag, New York, 1987
[8] Ground states of VBS models on Cayley trees, J. Stat. Phys., Volume 66 (1992), pp. 939-973
[9] Phase transitions and reflection positivity. I. General theory and long range lattice models, Comm. Math. Phys., Volume 62 (1978), pp. 1-34
[10] V. Liebscher, Markovianity of quantum random fields in the case, in: W. Freudenberg (Ed.), Proceedings of the Conference “Quantum Probability and Infinite-Dimensional Analysis”, Burg, Germany, 15–20 March 2001, QP–PQ Series, vol. 15, World Scientific, 2003, pp. 151–159.
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