Abstract
In the present paper we consider a Quantum Markov Chain (QMC) corresponding to the XY -model with competing Ising interactions on the Cayley tree of order two. Earlier, using finite volumes states one has been constructed QMC as a weak limit of those states which depends on the boundary conditions. It was proved that the limit state does exist and not depend on the boundary conditions, i.e. it is unique. In the present paper, we establish that the unique QMC has the clustering property, i.e. it is mixing with respect to translations of the tree. This means that the von Neumann algebra generated by this state is a factor.
Similar content being viewed by others
References
Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a Theorem of Takesaki. J. Funct. Anal. 45, 245–273 (1982)
Accardi, L, Mukhamedov, F., Saburov, M.: Uniqueness of quantum Markov chains associated with an X Y-model on the Cayley tree of order 2. Math. Notes 90, 8–20 (2011)
Accardi, L., Mukhamedov, F., Saburov, M.: On Quantum Markov Chains on Cayley tree I: uniqueness of the associated chain with X Y-model on the Cayley tree of order two. Inf. Dim. Analysis, Quantum Probab. Related Topics 14, 443–463 (2011)
Accardi, L., Mukhamedov, F., Souissi, A.: On construction of quantum Markov chains on Cayley trees. J. Phys.: Conf. Ser. 697, 012018 (2016)
Accardi, L., Ohno, H., Mukhamedov, F.: Quantum Markov fields on graphs. Inf. Dim. Analysis, Quantum Probab. Related Topics 13, 165–189 (2010)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, New York (1987)
Chakrabarti, B.K., Dutta, A., Sen, P.: Quantum Ising Phases and Transitions in Transverse Ising Models. Springer, Berlin (1996)
Duneau, M., Iagolnitzer, D., Souillard, B: String cluster properties for classical systems with finite range interactions. Commun. Math. Phys. 35, 307–320 (1974)
Georgi, H.-O.: Gibbs Measures and Phase Transitions De Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter, Berlin (1988)
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, 544–567 (2016)
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), 21 (2016)
Mukhamedov, F., El Gheteb, S.: Uniqueness of Quantum Markov Chain associated with XY-Ising model on the Cayley tree of order two. Open Sys. & Infor. Dyn. 24(2), 175010 (2017)
Mukhamedov, F.M., Rozikov, U.A.: On Gibbs measures of models with competing ternary and binary interactions on a Cayley tree and corresponding von Neumann algebras II. J. Stat. Phys. 119, 427–446 (2005)
Park, Y.M., Yoo, H.J.: Uniqueness and clustering properties of gibbs states for classical and quantum unbounded spin systems. J. Stat. Phys. 80, 223–271 (1995)
Spitzer, F.: Markov random fields on an infinite tree. Ann. Prob. 3, 387–398 (1975)
Acknowledgments
The authors are grateful to anonymous referees whose useful suggestions and comments improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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, 10 (2019). https://doi.org/10.1007/s11040-019-9308-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-019-9308-6