Sample-size dependence of the ground-state energy in a one-dimensional localization problem

C. Monthus 1, G. Oshanin 2, A. Comtet 2, 3, S. F. Burlatsky 4

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 54 (1996) 231

We study the sample-size dependence of the ground-state energy in a one-dimensional localization problem, based on a supersymmetric quantum mechanical Hamiltonian with random Gaussian potential. We determine, in the form of bounds, the precise form of this dependence and show that the disorder-average ground-state energy decreases with an increase of the size $R$ of the sample as a stretched-exponential function, $\\exp( - R^{z})$, where the characteristic exponent $z$ depends merely on the nature of correlations in the random potential. In the particular case where the potential is distributed as a Gaussian white noise we prove that $z = 1/3$. We also predict the value of $z$ in the general case of Gaussian random potentials with correlations.

  • 1. Service de Physique Théorique (SPhT),
    CNRS : URA2306 – CEA : DSM/SPHT
  • 2. Division de Physique Théorique, IPN,
    Université Paris XI - Paris Sud
  • 3. LPTPE,
    Université Paris VI - Pierre et Marie Curie
  • 4. Department of Chemistry, BG-10,
    University of Washington