{
  "id": "1602.07076",
  "version": "v1",
  "published": "2016-02-23T08:09:27.000Z",
  "updated": "2016-02-23T08:09:27.000Z",
  "title": "Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs",
  "authors": [
    "Vladimir V. Bazhanov",
    "Andrew P. Kels",
    "Sergey M Sergeev"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper we give an overview of exactly solved edge-interaction models, where the spins are placed on sites of a planar lattice and interact through edges connecting the sites. We only consider the case of a single spin degree of freedom at each site of the lattice. The Yang-Baxter equation for such models takes a particular simple form called the star-triangle relation. Interestigly all known solutions of this relation can be obtained as particular cases of a single \"master solution\", which is expressed through the elliptic gamma function and have continuous spins taking values of the circle. We show that in the low-temperature (or quasi-classical) limit these lattice models reproduce classical discrete integrable systems on planar graphs previously obtained and classified by Adler, Bobenko and Suris through the consistency-around-a-cube approach. We also discuss inversion relations, the physicical meaning of Baxter's rapidity-independent parameter in the star-triangle relations and the invariance of the action of the classical systems under the star-triangle (or cube-flip) transformation of the lattice, which is a direct consequence of Baxter's Z-invariance in the associated lattice models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-23T08:09:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "star-triangle relation",
      "quasi-classical expansion",
      "classical discrete integrable systems",
      "reproduce classical discrete integrable",
      "quad-graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1425880
    }
  }
}