On Gibbs Measures of Models with Competing Ternary and Binary Interactions and Corresponding von Neumann Algebras View Full Text


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Article Info

DATE

2004-02

AUTHORS

Farruh Mukhamedov, Utkir Rozikov

ABSTRACT

In the present paper a model with competing ternary (J2) and binary (J1) 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. More... »

PAGES

825-848

References to SciGraph publications

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  • 2001-02. Von Neumann algebra corresponding to one phase of the inhomogeneous Potts model on a Cayley tree in THEORETICAL AND MATHEMATICAL PHYSICS
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  • 1995-04. On the purity of the limiting gibbs state for the Ising model on the Bethe lattice in JOURNAL OF STATISTICAL PHYSICS
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  • 1981-06. Modulated phase of an ising system with competing interactions on a Cayley tree in ZEITSCHRIFT FÜR PHYSIK B CONDENSED MATTER
  • 1985-08. Phase diagram of the Ising Model on a Cayley tree in the presence of competing interactions and magnetic field in JOURNAL OF STATISTICAL PHYSICS
  • 2002-03. Exact Solution of the Ising Model on the Cayley Tree with Competing Ternary and Binary Interactions in THEORETICAL AND MATHEMATICAL PHYSICS
  • 1998-07. Non-Bernoullian Quantum K-Systems in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1998-10. On the Phase Diagram of the Random Field Ising Model on the Bethe Lattice in JOURNAL OF STATISTICAL PHYSICS
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    http://scigraph.springernature.com/pub.10.1023/b:joss.0000012509.10642.83

    DOI

    http://dx.doi.org/10.1023/b:joss.0000012509.10642.83

    DIMENSIONS

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