{
  "id": "1604.00948",
  "version": "v1",
  "published": "2016-04-04T17:09:46.000Z",
  "updated": "2016-04-04T17:09:46.000Z",
  "title": "Finite-size corrections to scaling of the magnetization distribution in the $2d$ $XY$-model at zero temperature",
  "authors": [
    "G. Palma",
    "F. Niedermayer",
    "Z. Rácz",
    "A. Riveros",
    "D. Zambrano"
  ],
  "comment": "9 pages, 7 figures, to be submitted to Phys. Rev. E",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "The zero-temperature, classical $XY$-model on an $L \\times L$ square-lattice is studied by exploring the distribution $\\Phi_L(y)$ of its centered and normalized magnetization $y$ in the large $L$ limit. An integral representation of the cumulant generating function, known from earlier works, is used for the numerical evaluation of $\\Phi_L(y)$, and the limit distribution $\\Phi_{L \\rightarrow \\infty} (y) = \\Phi_0(y)$ is obtained with high precision. The two leading finite-size corrections $\\Phi_L (y) -\\Phi_0 (y) \\approx a_1(L)\\, \\Phi_1(y) + a_2(L)\\,\\Phi_2(y)$ are also extracted both from numerics and from analytic calculations. We find that the amplitude $a_1(L)$ scales as $\\ln(L/L_0) /L^2$ and the shape correction function $\\Phi_1 (y)$ can be expressed through the low-order derivatives of the limit distribution, $\\Phi_1 (y) = [\\,y\\, \\Phi_0 (y) + \\Phi'_0 (y)\\,]'$. The second finite-size correction has an amplitude $a_2(L)\\propto 1/L^2$ and one finds that $a_2\\,\\Phi_2(y) \\ll a_1 \\,\\Phi_1(y)$ already for small system size ($L> 10$). We illustrate the feasibility of observing the calculated finite-size corrections by performing simulations of the $XY$-model at low temperatures, including $T = 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-04T17:09:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zero temperature",
      "magnetization distribution",
      "limit distribution",
      "shape correction function",
      "small system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160400948P",
      "inspire": 1442361
    }
  }
}