On Gibbs measures of models with competing ternary and binary interactions and corresponding von Neumann algebras

Farruh Mukhamedov, Utkir Rozikov

Research output: Contribution to journalArticlepeer-review

39 Citations (Scopus)

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 J1, J2 (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 IIIλ, where λ = exp {-2βJ2}, 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 IIIδ, otherwise they have type III1.

Original languageEnglish
Pages (from-to)825-848
Number of pages24
JournalJournal of Statistical Physics
Volume114
Issue number3-4
DOIs
Publication statusPublished - Feb 2004
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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