## Abstract

In the present paper a model with competing ternary (J)_{2} and binary (J)_{1} interactions with spin values ± 1, on a Cayley tree is considered. One studies the structure of Gibbs measures for the model considered. It is known, that under some conditions on parameters J_{1}, J_{2} (resp. in the opposite case) there are three (resp. a unique) translation-invariant Gibbs measures. We prove, that two of them (minimal and maximal) are extreme in the set of all Gibbs measures and also construct two periodic (with period 2) and uncountable number of distinct non-translation- invariant Gibbs measures. One shows that they are extreme. Besides, types of von Neumann algebras, generated by GNS-representation associated with diagonal states corresponding to extreme periodic Gibbs measures, are determined. Namely, it is shown that an algebra associated with the (unordered phase is a factor of type III_{λ}, where λ = exp {-2βJ_{2}}, β > 0 is the inverse temperature. We find conditions, which ensure that von Neumann algebras, associated with the periodic Gibbs measures, are factors of type III_{δ}, otherwise they have type III_{1}.

Original language | English |
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Pages (from-to) | 825-848 |

Number of pages | 24 |

Journal | Journal of Statistical Physics |

Volume | 114 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Feb 2004 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics