Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree

Abstract

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.

24 pages. arXiv admin note: text overlap with arXiv:1011.2256

References in corpus (2)

• Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras
• Phase diagram of an Ising model with competitive interactions on a Husimi tree and its disordered counterpart

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