{
  "id": "1605.04546",
  "version": "v1",
  "published": "2016-05-15T13:30:07.000Z",
  "updated": "2016-05-15T13:30:07.000Z",
  "title": "Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley tree",
  "authors": [
    "Farrukh Mukhamedov",
    "Abdessatar Barhoumi",
    "Abdessatar Souissi"
  ],
  "comment": "24 pages. arXiv admin note: text overlap with arXiv:1011.2256",
  "journal": "Jour. Stat. Phys. 163 (2016), 544--567",
  "doi": "10.1007/s10955-016-1495-y",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.FA",
    "math.MP",
    "math.OA",
    "quant-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-15T13:30:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46L53",
      "60J99",
      "46L60",
      "60G50",
      "82B10",
      "81Q10",
      "94A17"
    ],
    "keywords": [
      "quantum markov chain",
      "ising type models",
      "phase transition",
      "cayley tree",
      "boundary conditions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}