{
  "id": "cond-mat/0012035",
  "version": "v3",
  "published": "2000-12-04T09:27:28.000Z",
  "updated": "2001-01-03T08:38:37.000Z",
  "title": "The Importance of being Odd",
  "authors": [
    "Yu. Stroganov"
  ],
  "comment": "9 pages, based on the talk given at NATO Advanced Research Workshop \"Dynamical Symmetries in Integrable Two-dimensional Quantum Field Theories and Lattice Models\", 25-30 September 2000, Kyiv, Ukraine. New references are added plus some minor corrections",
  "journal": "J.Phys. A34 (2001) L179-L186",
  "doi": "10.1088/0305-4470/34/13/104",
  "categories": [
    "cond-mat.stat-mech",
    "hep-th",
    "nlin.SI"
  ],
  "abstract": "In this letter I consider mainly a finite XXZ spin chain with periodic boundary conditions and \\bf{odd} \\rm number of sites. This system is described by the Hamiltonian $H_{xxz}=-\\sum_{j=1}^{N}\\{\\sigma_j^{x}\\sigma_{j+1}^{x} +\\sigma_j^{y}\\sigma_{j+1}^{y} +\\Delta \\sigma_j^z\\sigma_{j+1}^z\\}$. As it turned out, its ground state energy is exactly proportional to the number of sites $E=-3N/2$ for a special value of the asymmetry parameter $\\Delta=-1/2$. The trigonometric polynomial $q(u)$, zeroes of which being the parameters of the ground state Bethe eigenvector is explicitly constructed. This polynomial of degree $n=(N-1)/2$ satisfy the Baxter T-Q equation. Using the second independent solution of this equation corresponding to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function $<\\sigma_j^z\\sigma_{j+1}^z> =-1/2+3/2N^2$. In its turn this correlation function is related to an average number of spin strings for the ground state of the system under consideration: $<N_{string}> = {3/8}(N-1/N)$. I would like to stress once more that all these simple formulas are \\bf wrong \\rm in the case of even number of sites. Exactly this case is usually considered.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2001-01-03T08:38:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground state energy",
      "finite xxz spin chain",
      "ground state bethe eigenvector",
      "correlation function",
      "asymmetry parameter"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 538487
    }
  }
}