Transport properties of 1D disordered models: a novel approach

TL;DR

This paper introduces a novel method for analyzing transport in 1D disordered systems by transforming the Schrödinger equation into a classical oscillator map, providing more accurate statistical descriptions of transmission.

Contribution

A new exact transformation technique reduces the 1D disordered Schrödinger equation to a classical Hamiltonian map, enabling improved analysis of transmission statistics.

Findings

Derived more accurate expressions for transmission mean and variance in the ballistic regime.

Showed that the distribution of f3f3 T_L approaches Gaussian as L increases.

Identified deviations from Gaussian law due to correlation effects at finite L.

Abstract

A new method is developed for the study of transport properties of 1D models with random potentials. It is based on an exact transformation that reduces discrete Schr\"odinger equation in the tight-binding model to a two-dimensional Hamiltonian map. This map describes the behavior of a classical linear oscillator under random parametric delta-kicks. We are interested in the statistical properties of the transmission coefficient $T_L$ of a disordered sample of length $L$. In the ballistic regime we derive expressions for the mean value of the transmission coefficient $T_L$, its second moment and variance, that are more accurate than the existing ones. In the localized regime we analyze the global characteristics of $\ln T_L$, and demonstrate that its distribution function approaches the Gaussian form if $L\to \infty$. For any finite $L$ there are deviations from the Gaussian law that originate from the subtle correlation effects between different trajectories of the Hamiltonian map.