Journal of Statistical Physics

, Volume 119, Issue 1–2, pp 427–446 | Cite as

On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding Von Neumann Algebras II

  • Farruh Mukhamedov
  • Utkir Rozikov


In the present paper the Ising model with competing binary (J) and binary (J1) interactions with spin values ±1, on a Cayley tree of order 2 is considered. The structure of Gibbs measures for the model is studied. We completely describe the set of all periodic Gibbs easures for the model with respect to any normal subgroup of finite index of a group representation of the Cayley tree. Types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to the translation invariant Gibbs measures, are determined. It is proved that the factors associated with minimal and maximal Gibbs states are isomorphic, and if they are of type IIIλ then the factor associated with the unordered phase of the model can be considered as a subfactors of these factors respectively. Some concrete examples of factors are given too.


Cayley tree Ising model competing interactions Gibbs measure GNS-construction Hamiltonian von Neumann algebra 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bleher, P.M., Ganikhodjaev, NN. 1990On pure phases of the Ising model on the Bethe lattice, TheorProbab. Appl.35216227CrossRefGoogle Scholar
  2. 2.
    Bleher, PM., Ruiz, J., Zagrebnov, VA. 1995On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice, JStat. Phys.79473482Google Scholar
  3. 3.
    Bleher, PM., Ruiz, J., Zagrebnov, VA. 1998On the phase diagram of the random field Ising model on the Bethe lattice, JStat. Phys.933378CrossRefGoogle Scholar
  4. 4.
    Bratteli, O., Robinson, D. 1979Operator algebras and Quantum Statistical Mechanics ISpringer-VerlagBerlin/New-York.Google Scholar
  5. 5.
    Bratteli, O., Robinson, D. 1981Operator algebras and Quantum Statistical Mechanics IISpringer-VerlagBerlin/New-York.Google Scholar
  6. 6.
    Ganikhodjaev, NN., Pah, CH., Wahiddin, M.R.B. 2003Exact solution of an Ising model with competing interactions on a Cayley tree, JPhys. A: Math. Gen.3642834289CrossRefGoogle Scholar
  7. 7.
    GanikhodjaevN.N. Rozikov, UA. 1997A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree, TheorMath. Phys.111480486Google Scholar
  8. 8.
    Lyons, R. 2000Phase transitions on nonamenable graphs, JMath. Phys.4110991126CrossRefGoogle Scholar
  9. 9.
    Mariz, M., Tsalis, C., Albuquerque, AL. 1985Phase diagram of the Ising model on a Cayley tree in the presence of competing interactions and magnetic field, JStat. Phys.40577592CrossRefGoogle Scholar
  10. 10.
    Mukhamedov, FM. 2000Von Neumann algebras corresponding translation-invariant Gibbs states of Ising model on the Bethe lattice, TheorMath. Phys.123489493Google Scholar
  11. 11.
    Mukhamedov, F.M., Rozikov, UA. 2004On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebrasJ. Stat. Phys.114825848CrossRefGoogle Scholar
  12. 12.
    Ramagge, J., Robertson, G. 1997Farctors from trees, ProcAm. Math. Soc.12520512055CrossRefGoogle Scholar
  13. 13.
    Rozikov, UA. 1997Partition structures of the group representation of the Cayley tree into cosets by finite-index normal subgroups and their applications to the description of periodic Gibbs distributions, TheorMath. Phys.112929933Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Dipartimento di Matematica, Department of MathematicsII Universita di Roma (“Tor Vergata”), National University of UzbekistanTashkent, RomeUzbekistan, Italy
  2. 2.Institute of MathematicsTashkentUzbekistan

Personalised recommendations