Comptes Rendus
no. 7-8
Functional Analysis/Mathematical Physics
Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme
[Transitions de phases pour un modèle XY sur un arbre de Cayley dʼordre trois dans un schéma de chaines de Markov quantiques]

Présenté par : Jean-Michel Bony

Farrukh Mukhamedov1 ; Mansoor Saburov1
1 Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, PO Box 141, 25710 Kuantan, Pahang, Malaysia
Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 425-428.

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 {K〈x,y〉} 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 {K〈x,y〉}.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.02.010

Farrukh Mukhamedov 1 ; Mansoor Saburov 1

1 Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, PO Box 141, 25710 Kuantan, Pahang, Malaysia 
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Farrukh Mukhamedov; Mansoor Saburov. Phase transitions for XY-model on the Cayley tree of order three in quantum Markov chain scheme. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 425-428. doi : 10.1016/j.crma.2011.02.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.010/

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