{
  "id": "1510.08322",
  "version": "v1",
  "published": "2015-10-28T14:38:38.000Z",
  "updated": "2015-10-28T14:38:38.000Z",
  "title": "Level repulsion exponent $β$ for Many-Body Localization Transitions and for Anderson Localization Transitions via Dyson Brownian Motion",
  "authors": [
    "Cecile Monthus"
  ],
  "comment": "20 pages",
  "categories": [
    "cond-mat.dis-nn"
  ],
  "abstract": "The generalization of the Dyson Brownian Motion approach of random matrices to Anderson Localization (AL) models [Chalker, Lerner and Smith PRL 77, 554 (1996)] and to Many-Body Localization (MBL) Hamiltonians [Serbyn and Moore arxiv:1508.07293] is revisited to extract the level repulsion exponent $\\beta$, where $\\beta=1$ in the delocalized phase governed by the Wigner-Dyson statistics, $\\beta=0$ in the localized phase governed by the Poisson statistics, and $0<\\beta_c<1$ at the critical point. The idea is that the Gaussian disorder variables $h_i$ are promoted to Gaussian stationary processes $h_i(t)$ in order to sample the disorder stationary distribution with some time correlation $\\tau$. The statistics of energy levels can be then studied via Langevin and Fokker-Planck equations. For the MBL quantum spin Hamiltonian with random fields $h_i$, we obtain $\\beta =2q^{EA}_{off}(N)/q^{EA}_{diag}(N) $ in terms of the diagonal $(n=m)$ and off-diagonal $(n \\ne m)$ matrix elements of the Edwards-Anderson matrix $q^{EA}_{nm}(N) \\equiv \\frac{1}{N} \\sum_{i=1}^N \\vert < \\phi_n \\vert \\sigma_i^z \\vert \\phi_m> \\vert^2 $. For the Anderson Localization tight-binding Hamiltonian with random on-site energies $h_i$, we find $\\beta =2 Y_{off}(N)/(Y_{diag}(N)-Y_{off}(N)) $ in terms of the diagonal $(n=m)$ and off-diagonal $(n \\ne m)$ matrix elements of the Density Correlation matrix $Y_{nm}(N) \\equiv \\sum_{i=1}^N \\vert < \\phi_n \\vert i> \\vert^2 \\vert <i \\vert \\phi_m> \\vert^2 $, the diagonal elements being the Inverse Participation Ratios $Y_{nn}(N) \\equiv \\sum_{i=1}^N \\vert < \\phi_n \\vert i> \\vert^4 $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-28T14:38:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "level repulsion exponent",
      "many-body localization transitions",
      "anderson localization transitions",
      "dyson brownian motion approach",
      "matrix elements"
    ],
    "publication": {
      "doi": "10.1088/1742-5468/2016/03/033113",
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2016,
      "month": "Mar",
      "volume": 3,
      "pages": 33113
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016JSMTE..03.3113M"
    }
  }
}