{
  "id": "0903.1988",
  "version": "v2",
  "published": "2009-03-11T14:49:10.000Z",
  "updated": "2009-05-28T07:51:46.000Z",
  "title": "Statistics of the two-point transmission at Anderson localization transitions",
  "authors": [
    "Cecile Monthus",
    "Thomas Garel"
  ],
  "comment": "v2=final version with two new appendices with respect to v1; 12 pages, 10 figures",
  "journal": "Phys. Rev. B 79, 205120 (2009)",
  "doi": "10.1103/PhysRevB.79.205120",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L \\sim 1/L^{\\kappa}$ behaves as $L^{\\Phi(\\kappa)}$, where the multifractal spectrum $\\Phi(\\kappa)$ terminates at $\\kappa=0$ as a consequence of the physical bound $T_L \\leq 1$; (ii) the exponents $X(q)$ that govern the moments $\\overline{T_L^q} \\sim 1/L^{X(q)}$ become frozen above some threshold: $X(q \\geq q_{sat}) = - \\Phi(\\kappa=0)$, i.e. all moments of order $q \\geq q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($\\kappa=0$). In the present paper, we test numerically these predictions for the ensemble of $L \\times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/| i-j |)^a$. This model is known to present an Anderson transition at $a=1$ between localized ($a>1$) and extended ($a<1$) states, with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra $\\Phi_b(\\kappa)$ for various $b$ are in agreement with the relation $\\Phi(\\kappa \\geq 0) = 2 [ f(\\alpha= d+ \\frac{\\kappa}{2}) -d ]$ in terms of the singularity spectrum $f(\\alpha)$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation $\\kappa_{typ}(b)= 2 (\\alpha_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-05-28T07:51:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "71.30.+h",
      "71.23.An",
      "72.15.Rn",
      "05.45.Df"
    ],
    "keywords": [
      "anderson localization transitions",
      "two-point transmission",
      "statistics",
      "multifractal spectrum",
      "power-law random banded matrices"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review B",
      "year": 2009,
      "month": "May",
      "volume": 79,
      "number": 20,
      "pages": 205120
    },
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009PhRvB..79t5120M"
    }
  }
}