{
  "id": "1609.01121",
  "version": "v1",
  "published": "2016-09-05T12:10:34.000Z",
  "updated": "2016-09-05T12:10:34.000Z",
  "title": "Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models : application to Levy matrices",
  "authors": [
    "Cecile Monthus"
  ],
  "comment": "15 pages",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "The delocalized non-ergodic phase existing in some random $N \\times N$ matrix models is analyzed via the Wigner-Weisskopf approximation for the dynamics from an initial site $j_0$. The main output of this approach is the inverse $\\Gamma_{j_0}(N)$ of the characteristic time to leave the state $j_0$ that provides some broadening $\\Gamma_{j_0}(N) $ for the weights of the eigenvectors. In this framework, the localized phase corresponds to the region where the broadening $\\Gamma_{j_0}(N) $ is smaller in scaling than the level spacing $\\Delta_{j_0}(N) $, while the delocalized non-ergodic phase corresponds to the region where the broadening $\\Gamma_{j_0}(N) $ decays with $N$ but is bigger in scaling than the level spacing $\\Delta_{j_0}(N) $. Then the number $\\frac{\\Gamma_{j_0}(N)}{\\Delta_{j_0}(N)} $ of resonances grows only sub-extensively in $N$. We describe how the multifractal spectrum of eigenstates can be then explicitly computed in two models. For the Generalized-Rosenzweig-Potter (GRP) Matrix model, the present approach allows to recover the multifractal spectrum of Ref [V.E. Kravtsov, I.M. Khaymovich, E. Cuevas and M. Amini, New. J. Phys. 17, 122002 (2015)]. For the L\\'evy matrix model, where matrix elements are drawn with some heavy-tailed distribution $P(H_{ij}) \\propto N^{-1} \\vert H_{ij} \\vert^{-1-\\mu}$, we derive the multifractal spectrum in the delocalized non-ergodic phase existing in some energy region for $1<\\mu<2$ (while the region $0<\\mu<1$ containing only localized eigenvectors is characterized by the multifractality studied in our previous work arxiv:1606.03241).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-05T12:10:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrix models",
      "levy matrix",
      "multifractal spectrum",
      "eigenstates",
      "delocalized non-ergodic phase existing"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}