Clustering Property of Quantum Markov Chain Associated to XY-model with Competing Ising Interactions on the Cayley Tree of Order Two

  • Farrukh MukhamedovEmail author
  • Soueidy El Gheteb


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.


Quantum Markov chain Cayley tree XY model Competing interaction Clustering property Uniqueness 

Mathematics Subject Classification (2010)

46L53 60J99 46L60 60G50 82B10 81Q10 



The authors are grateful to anonymous referees whose useful suggestions and comments improve the presentation of the paper.


  1. 1.
    Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a Theorem of Takesaki. J. Funct. Anal. 45, 245–273 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Accardi, L., Mukhamedov, F., Souissi, A.: On construction of quantum Markov chains on Cayley trees. J. Phys.: Conf. Ser. 697, 012018 (2016)Google Scholar
  5. 5.
    Accardi, L., Ohno, H., Mukhamedov, F.: Quantum Markov fields on graphs. Inf. Dim. Analysis, Quantum Probab. Related Topics 13, 165–189 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, New York (1987)CrossRefGoogle Scholar
  7. 7.
    Chakrabarti, B.K., Dutta, A., Sen, P.: Quantum Ising Phases and Transitions in Transverse Ising Models. Springer, Berlin (1996)zbMATHGoogle Scholar
  8. 8.
    Duneau, M., Iagolnitzer, D., Souillard, B: String cluster properties for classical systems with finite range interactions. Commun. Math. Phys. 35, 307–320 (1974)ADSCrossRefGoogle Scholar
  9. 9.
    Georgi, H.-O.: Gibbs Measures and Phase Transitions De Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter, Berlin (1988)CrossRefGoogle Scholar
  10. 10.
    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)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHGoogle Scholar
  13. 13.
    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)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    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)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Spitzer, F.: Markov random fields on an infinite tree. Ann. Prob. 3, 387–398 (1975)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, College of ScienceUnited Arab Emirates UniversityAl AinUAE
  2. 2.Department of MathematicsCarthage UniversityCarthageTunisia

Personalised recommendations