Skip to main content
SpringerLink
Log in

On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

It is known that the disordered phase of the classical Ising model on the Caley tree is extreme in some region of the temperature. If one considers the Ising model with competing interactions on the same tree, then about the extremity of the disordered phase there is no any information. In the present paper, we first aiming to analyze the correspondence between Gibbs measures and QMC’s on trees. Namely, we establish that states associated with translation invariant Gibbs measures of the model can be seen as diagonal quantum Markov chains on some quasi local algebra. Then as an application of the established correspondence, we study some algebraic property of the disordered phase of the Ising model with competing interactions on the Cayley tree of order two. More exactly, we prove that a state corresponding to the disordered phase is not quasi-equivalent to other states associated with translation invariant Gibbs measures. This result shows how the translation invariant states relate to each other, which is even a new phenomena in the classical setting. To establish the main result we basically employ methods of quantum Markov chains.

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  MATH  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  MATH  Google Scholar 

  3. Accardi, L., Fidaleo, F.: Quantum Markov fields. Infin. Dim. Analysis, Quantum Probab. Related Topics 6, 123–138 (2003)

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

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

    Google Scholar 

  6. Accardi, L., Fidaleo, F., Mukhamedov, F.: Markov states and chains on the CAR algebra. Infin. Dim. Analysis, Quantum Probab. Related Topics 10, 165–183 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Accardi, L., Frigerio, A.: Markovian cocycles. Proc. Royal Irish Acad. 83A, 251–263 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Accardi, L., Mukhamedov, F., Saburov, M.: On Quantum Markov Chains on Cayley tree I: uniqueness of the associated chain with XY-model on the Cayley tree of order two. Infin. Dim. Analysis, Quantum Probab. Related Topics 14, 443–463 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Accardi, L., Mukhamedov, F., Saburov, M.: Uniqueness of quantum Markov chains associated with an XY-model on the Cayley tree of order 2. Math. Notes 90, 8–20 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Accardi, L., Mukhamedov, F., Saburov, M.: On quantum Markov chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three. Ann. Henri Poincare 12, 1109–1144 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Accardi, L., Mukhamedov, F., Saburov, M.: On quantum Markov chains on Cayley tree III: Ising model. J. Stat. Phys. 157, 303–329 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Accardi, L., Ohno, H., Mukhamedov, F.: Quantum Markov fields on graphs. Infin. Dim. Analysis, Quantum Probab. Related Topics 13, 165–189 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bleher, P.M.: Extremity of the disordered phase in the Ising model on the Bethe lattice. Commun. Math. Phys. 128, 411–419 (1990)

    Article  ADS  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Dobrushin, R.L.: Description of Gibbsian Random fields by means of conditional probabilities. Probab. Theory Appl. 13, 201–229 (1968)

    MATH  Google Scholar 

  16. Fannes, M., Nachtergaele, B., Werner, R.F.: Ground states of VBS models on Cayley trees. J. Stat. Phys. 66, 939–973 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Ganikhodzhaev, N.N., Rozikov, U.A.: On Ising model with four competing interactions on cayley tree. Math. Phys. Anal. Geom. 12, 141–156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

  20. Gandolfo, D., Haydarov, F.H., Rozikov, U.A., Ruiz, J.: New phase transitions of the Ising model on Cayley trees. J. Stat. Phys. 153, 400–411 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Ioffe, D.: On the Extremality of the Disordered State for the Ising Model on the Bethe Lattice. Lett. Math. Phys. 37, 137–143 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Matsui, T.: A characterization of pure finitely correlated states. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, 647–661 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mossel, E.: Reconstruction on trees: Beating the second eigenvalue. Ann. Appl. Probab. 11, 285–300 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mossel, E.: Peres Y. Information flow on trees. In: Graphs, Morphisms and Statistical Physics, pp. 155–170. AMS (2004)

  25. 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  MathSciNet  MATH  Google Scholar 

  26. Mukhamedov, F.: On factor associated with the unordered phase of λ-model on a Cayley tree. Rep. Math. Phys. 53, 1–18 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Mukhamedov, F., Rozikov, U.: On Gibbs measures of models with competing ternary and binary interactions on a Cayley tree and corresponding von Neumann algebras. J. Stat. Phys. 114, 825–848 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Mukhamedov, F., Rozikov, U.: 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  MATH  Google Scholar 

  30. Ohya, M., Petz, D.: Quantum Entropy and its Use. Springer, Berlin-Heidelberg-New York (1993)

    Book  MATH  Google Scholar 

  31. Ostilli, M.: Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists. Physica A 391, 3417–3423 (2012)

  32. Ostilli, M., Mukhamedov, F., Mendes, J.F.F.: Phase diagram of an Ising model with competitive interactions on a Husimi tree and its disordered counterpart. Physica A 387, 2777–2792 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  33. Preston, C.: Gibbs States on Countable Sets. Cambridge University Press, London (1974)

    Book  MATH  Google Scholar 

  34. Rozikov, U.A.: Gibbs Measures on Cayley Trees. World Scientific, Singappore (2013)

    Book  MATH  Google Scholar 

  35. Spitzer, F.: Markov random fields on an infinite tree. Ann. Prob. 3, 387–398 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box, 141, 25710, Kuantan, Pahang, Malaysia

    Farrukh Mukhamedov

  2. College of Science, The United Arab Emirates University, P.O. Box, 15551, Al Ain, Abu Dhabi, UAE

    Farrukh Mukhamedov

  3. Department of Mathematics, Nabeul Preparatory Engineering Institute, Carthage University, Campus Universitairy, Mrezgua, 8000, Nabeul, Tunisia

    Abdessatar Barhoumi

  4. Department of Mathematics, Marsa Preparatory Institute for Scientific and Technical Studies, Carthage University, Nabeul, Tunisia

    Abdessatar Souissi

Authors
  1. Farrukh Mukhamedov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdessatar Barhoumi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Abdessatar Souissi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farrukh Mukhamedov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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, 21 (2016). https://doi.org/10.1007/s11040-016-9225-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-016-9225-x

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation