{
  "id": "2004.00064",
  "version": "v1",
  "published": "2020-03-31T19:20:18.000Z",
  "updated": "2020-03-31T19:20:18.000Z",
  "title": "Beyond universal behavior in the one-dimensional chain with random nearest neighbor hopping",
  "authors": [
    "Akshay Krishna",
    "R. N. Bhatt"
  ],
  "comment": "14 pages, 7 figures",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the one-dimensional nearest neighbor tight binding model of electrons with independently distributed random hopping and no on-site potential (i.e. off-diagonal disorder with particle-hole symmetry, leading to sub-lattice symmetry, for each realization). For non-singular distributions of the hopping, it is known that the model exhibits a universal, singular behavior of the density of states $\\rho(E) \\sim 1/|E \\ln^3|E||$ and of the localization length $\\xi(E) \\sim |\\ln|E||$, near the band center $E = 0$. (This singular behavior is also applicable to random XY and Heisenberg spin chains; it was first obtained by Dyson for a specific random harmonic oscillator chain). Simultaneously, the state at $E = 0$ shows a universal, sub-exponential decay at large distances $\\sim \\exp [ -\\sqrt{r/r_0} ]$. In this study, we consider singular, but normalizable, distributions of hopping, whose behavior at small $t$ is of the form $\\sim 1/ [t \\ln^{\\lambda+1}(1/t) ]$, characterized by a single, continuously tunable parameter $\\lambda > 0$. We find, using a combination of analytic and numerical methods, that while the universal result applies for $\\lambda > 2$, it no longer holds in the interval $0 < \\lambda < 2$. In particular, we find that the form of the density of states singularity is enhanced (relative to the Dyson result) in a continuous manner depending on the non-universal parameter $\\lambda$; simultaneously, the localization length shows a less divergent form at low energies, and ceases to diverge below $\\lambda = 1$. For $\\lambda < 2$, the fall-off of the $E = 0$ state at large distances also deviates from the universal result, and is of the form $\\sim \\exp [-(r/r_0)^{1/\\lambda}]$, which decays faster than an exponential for $\\lambda < 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-31T19:20:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random nearest neighbor hopping",
      "one-dimensional chain",
      "universal behavior",
      "neighbor tight binding model",
      "nearest neighbor tight binding"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}