Phase Transitions for Quantum Markov Chains Associated with Ising Type Models on a Cayley Tree

Farrukh Mukhamedov, Abdessatar Barhoumi, Abdessatar Souissi

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered model persists only on trees. In this paper, we provide a more general construction of forward QMC. In that construction, a QMC is defined as a weak limit of finite volume states with boundary conditions, i.e. QMC depends on the boundary conditions. Our main result states the existence of a phase transition for the Ising model with competing interactions on a Cayley tree of order two. By the phase transition we mean the existence of two distinct QMC which are not quasi-equivalent and their supports do not overlap. We also study some algebraic property of the disordered phase of the model, which is a new phenomena even in a classical setting.

Original languageEnglish
Pages (from-to)544-567
Number of pages24
JournalJournal of Statistical Physics
Issue number3
Publication statusPublished - May 1 2016
Externally publishedYes


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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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