Information-Length Scaling in a Generalized One-Dimensional Lloyd's Model.

J Méndez-Bermúdez,R Aguilar-Sánchez

doi:10.3390/e20040300

J Méndez-Bermúdez, R Aguilar-Sánchez

Open Access

https://doi.org/10.3390/e20040300

Copy DOI

Abstract

We perform a detailed numerical study of the localization properties of the eigenfunctions of one-dimensional (1D) tight-binding wires with on-site disorder characterized by long-tailed distributions: For large , with ; where are the on-site random energies. Our model serves as a generalization of 1D Lloyd’s model, which corresponds to . In particular, we demonstrate that the information length of the eigenfunctions follows the scaling law , with and . Here, is the eigenfunction localization length (that we extract from the scaling of Landauer’s conductance) and L is the wire length. We also report that for the properties of the 1D Anderson model are effectively reproduced.

Highlights

There is a class of disordered systems characterized by random variables {e} whose density distribution function exhibits a slow decaying tail: P(e) ∼ | e |1+ α (1)for large |e|, with 0 < α < 2
Hamiltonian with Cauchy-distributed on-site potentials (which corresponds to the particular value α = 1 in Equation (1))
The recent experimental realizations of the so-called Lévy glasses [31] as well as Lévy waveguides [32,33] has refreshed the interest in the study of systems characterized by Lévy-type disorder; see some examples in Refs. [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]

Summary

Introduction

There is a class of disordered systems characterized by random variables {e} whose density distribution function exhibits a slow decaying tail: P(e) ∼. It is important to point out that one-dimensional (1D) tight-binding wires with power-law distributed random on-site potentials, characterized by power-laws different from α = 1 (which corresponds to the 1D Lloyd’s model), have been scarcely studied; for prominent exceptions see. In this paper, we perform a detailed numerical study of the localization properties of the eigenfunctions of disordered wires defined as a generalization of the 1D Lloyd’s model as follows. For weak disorder (σ2 1), the only relevant parameter for describing the statistical properties of the transmission of the finite 1D Anderson model is the ratio L/ξ [53], a fact known as single parameter scaling. [1/Γ(α)]2−α exp(−1/2e)/e1+α (where Γ is the Euler gamma function), that we used in [27]

Extraction of the Localization Length ξ

Calculation of the Information Length β

Discussion