## Abstract

In the present paper the Ising model with competing binary (J) and binary (J _{1}) 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 III _{λ} 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.

Original language | English |
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Pages (from-to) | 427-446 |

Number of pages | 20 |

Journal | Journal of Statistical Physics |

Volume | 119 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Apr 2005 |

Externally published | Yes |

## Keywords

- Cayley tree
- Competing interactions
- GNS-construction
- Gibbs measure
- Hamiltonian
- Ising model
- Von Neumann algebra

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics