Skip to main content
SpringerLink
Log in

Factors Generated by XY-Model with Competing Ising Interactions on the Cayley Tree

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

In the present paper, we consider a quantum Markov chain corresponding to the XY-model with competing Ising interactions on the Cayley tree of order two. Earlier, it was proved that this state does exist and is unique. Moreover, it has clustering property. This means that the von Neumann algebra generated by this state is a factor. In the present paper, we establish that the factor generated by this state may have type \(\mathop {\mathrm{III_{\lambda }}}\)\(\lambda \in (0,1)\) which is unusual for states associated with models with nontrivial interactions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Accardi, L.: On the noncommutative Markov property. Funct. Anal. Appl. 9, 1–8 (1975)

    Article  MathSciNet  Google Scholar 

  2. Accardi, L., Cecchini, C.: Conditional expectations in von Neumann algebras and a theorem of Takesaki. J. Funct. Anal. 45, 245–273 (1982)

    Article  MathSciNet  Google Scholar 

  3. Accardi, L., Fidaleo, F.: Quantum Markov fields. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6, 123–138 (2003)

    Article  MathSciNet  Google Scholar 

  4. Accardi, L., Fidaleo, F.: Non-homogeneous quantum Markov states and quantum Markov fields. J. Funct. Anal. 200, 324–347 (2003)

    Article  MathSciNet  Google Scholar 

  5. Accardi, L., Fidaleo, F.: On the structure of quantum Markov fields. In: W. Freudenberg (ed.) Proceedings Burg Conference 15-20 March 2001, 15, pp.1–20. World Scientific, QP-PQ Series (2003)

  6. Accardi, L., Ohno, H., Mukhamedov, F.: Quantum Markov fields on graphs. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13, 165–189 (2010)

    Article  MathSciNet  Google Scholar 

  7. Blekher, P.M.: Some applications factors, a commentary to. In: J. von Neuman(Ed.) Selected Works on Functional Analysis. II [in Russian], pp. 353-359. Nauka, Moscow (1987)

  8. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics I. Springer, New York (1987)

    Book  Google Scholar 

  9. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II. Springer, New York (1987)

    Book  Google Scholar 

  10. Fannes, M., Nachtergaele, B., Werner, R.F.: Finitely correlated states on quantum spin chains. Commun. Math. Phys. 144, 443–490 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  11. Fidaleo, F., Mukhamedov, F.: Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras. Probab. Math. Stat. 24, 401–418 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Ganikhodzhaev, N.N., Mukhamedov, F.M.: On some properties of a class of diagonalizable states of von Neumann algebras. Math. Notes 76, 329–338 (2004)

    Article  MathSciNet  Google Scholar 

  13. Georgi, H.-O.: Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter, Berlin (1988)

    Book  Google Scholar 

  14. Mukhamedov, F.M.: On factor associated with the disordered phase of \(\lambda \)-model on a Cayley tree. Rep. Math. Phys. 53, 1–18 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  15. Mukhamedov, F.M.: Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice. Theor. Math. Phys. 123, 489–493 (2000)

    Article  Google Scholar 

  16. 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)

    Article  MathSciNet  Google Scholar 

  17. Mukhamedov, F., El Gheteb, S.: Uniqueness of quantum Markov chain associated with XY-Ising model on the Cayley tree of order two. Open Syst. Inf. Dyn. 24(2), 175010 (2017)

    Article  MathSciNet  Google Scholar 

  18. Mukhamedov, F., El Gheteb, S.: Clustering property for quantum Markov chain associated to XY-model with competing interaction Ising model on Cayley tree of order two. Math. Phys. Anal. Geom. 22, 10 (2019)

    Article  Google Scholar 

  19. 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)

    Article  ADS  MathSciNet  Google Scholar 

  20. Mukhamedov, F., Souissi, A.: Quantum Markov states on Cayley trees. J. Math. Anal. Appl. 473, 313–333 (2019)

    Article  MathSciNet  Google Scholar 

  21. Ohno, H.: Factors generated by \(C^*\)-finitely correlated states. Int. J. Math. 18, 27–41 (2007)

    Article  MathSciNet  Google Scholar 

  22. Powers, R.: Representation of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. Math. 81, 138–171 (1967)

    Article  MathSciNet  Google Scholar 

  23. Stratila, S.: Modular Theory in Operator Algebras. Abacus Press, Bucuresti (1981)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to an anonymous referee whose valuable comments and remarks improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O.Box, 15551 , Al Ain, Abu Dhabi, UAE

    Farrukh Mukhamedov

  2. Department of Mathematics, Nouakchott University, Nouakchott, Mauritania

    Soueidy El Gheteb

Authors
  1. Farrukh Mukhamedov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Soueidy El Gheteb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farrukh Mukhamedov.

Additional information

Communicated by Claude-Alain Pillet.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mukhamedov, F., Gheteb, S.E. Factors Generated by XY-Model with Competing Ising Interactions on the Cayley Tree. Ann. Henri Poincaré 21, 241–253 (2020). https://doi.org/10.1007/s00023-019-00853-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-019-00853-9

Mathematics Subject Classification

Search

Navigation