Transport in the 3-dimensional Anderson model: an analysis of the dynamics on scales below the localization length

TL;DR

This paper analyzes single-particle transport below the localization length in the 3D Anderson model, examining how disorder and energy influence transport properties like mean free path, velocities, and diffusion constants, using various theoretical and numerical methods.

Contribution

It applies both Boltzmann and TCL projection techniques to study energy dependence of transport in the 3D Anderson model under different disorder regimes, providing new insights.

Findings

Transport quantities depend on disorder and energy.

Weak disorder shows known energy dependencies via Boltzmann equation.

Strong disorder reveals less pronounced energy dependencies.

Abstract

Single-particle transport in disordered potentials is investigated on scales below the localization length. The dynamics on those scales is concretely analyzed for the 3-dimensional Anderson model with Gaussian on-site disorder. This analysis particularly includes the dependence of characteristic transport quantities on the amount of disorder and the energy interval, e.g., the mean free path which separates ballistic and diffusive transport regimes. For these regimes mean velocities, respectively diffusion constants are quantitatively given. By the use of the Boltzmann equation in the limit of weak disorder we reveal the known energy-dependencies of transport quantities. By an application of the time-convolutionless (TCL) projection operator technique in the limit of strong disorder we find evidence for much less pronounced energy dependencies. All our results are partially confirmed by the numerically exact solution of the time-dependent Schroedinger equation or by approximative numerical integrators. A comparison with other findings in the literature is additionally provided.