On an Algebraic Property of the Disordered Phase of the Ising Model with Competing Interactions on a Cayley Tree

Farrukh Mukhamedov, Abdessatar Barhoumi, Abdessatar Souissi

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)

Abstract

It is known that the disordered phase of the classical Ising model on the Caley tree is extreme in some region of the temperature. If one considers the Ising model with competing interactions on the same tree, then about the extremity of the disordered phase there is no any information. In the present paper, we first aiming to analyze the correspondence between Gibbs measures and QMC’s on trees. Namely, we establish that states associated with translation invariant Gibbs measures of the model can be seen as diagonal quantum Markov chains on some quasi local algebra. Then as an application of the established correspondence, we study some algebraic property of the disordered phase of the Ising model with competing interactions on the Cayley tree of order two. More exactly, we prove that a state corresponding to the disordered phase is not quasi-equivalent to other states associated with translation invariant Gibbs measures. This result shows how the translation invariant states relate to each other, which is even a new phenomena in the classical setting. To establish the main result we basically employ methods of quantum Markov chains.

Original languageEnglish
Article number21
JournalMathematical Physics Analysis and Geometry
Volume19
Issue number4
DOIs
Publication statusPublished - Dec 1 2016

Keywords

  • Cayley tree
  • Competing interaction
  • Disordered phase
  • Ising type model
  • Quantum Markov chain
  • Quasi-equivalence

ASJC Scopus subject areas

  • Mathematical Physics
  • Geometry and Topology

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