{
  "id": "cond-mat/9909168",
  "version": "v2",
  "published": "1999-09-13T16:02:09.000Z",
  "updated": "1999-09-23T16:18:58.000Z",
  "title": "Combinatorial and topological approach to the 3D Ising model",
  "authors": [
    "Tullio Regge",
    "Riccardo Zecchina"
  ],
  "comment": "33 pages, 5 figures",
  "journal": "J.Phys.A33:741-761,2000",
  "doi": "10.1088/0305-4470/33/4/308",
  "categories": [
    "cond-mat.stat-mech",
    "hep-th",
    "math.CO"
  ],
  "abstract": "We extend the planar Pfaffian formalism for the evaluation of the Ising partition function to lattices of high topological genus g. The 3D Ising model on a cubic lattice, where g is proportional to the number of sites, is discussed in detail. The expansion of the partition function is given in terms of 2^{2 g} Pfaffians classified by the oriented homology cycles of the lattice, i.e. by its spin-structures. Correct counting is guaranteed by a signature term which depends on the topological intersection of the oriented cycles through a simple bilinear formula. The role of a gauge symmetry arising in the above expansion is discussed. The same formalism can be applied to the counting problem of perfect matchings over general lattices and provides a determinant expansion of the permanent of 0-1 matrices.",
  "revisions": [
    {
      "version": "v2",
      "updated": "1999-09-23T16:18:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "3d ising model",
      "topological approach",
      "combinatorial",
      "simple bilinear formula",
      "planar pfaffian formalism"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2000,
      "month": "Feb",
      "volume": 33,
      "number": 4,
      "pages": 741
    },
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 507124,
      "adsabs": "2000JPhA...33..741R"
    }
  }
}