Length scale dependent diffusion in the Anderson model at high temperatures

Abstract

We investigate a single particle on a 3-dimensional, cubic lattice with a random on-site potential (3D Anderson model). We concretely address the question whether or not the dynamics of the particle is in full accord with the diffusion equation. Our approach is based on the time-convolutionless (TCL) projection operator technique and allows for a detailed investigation of this question at high temperatures. It turns out that diffusive dynamics is to be expected for a rather short range of wavelengths, even if the amount of disorder is tuned to maximize this range. Our results are partially counterchecked by the numerical solution of the full time-dependent Schrödinger equation.

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

“…In such a system diffusive dynamics is expected to emerge only, if the coupling constant λ is much decreased. For λ ≪ 1, in complete analogy to the above numerical simulation, the time evolution of pure initial states has comprehensively been shown to be in full accord with all predictions of the TCL-based theory [24,21]. However, since it still remains to resolve the energy dependencies of the dynamics, we consider the quantities…”

“…But in general already the direct evaluation of the fourth order term turns out be extremely difficult, both analytically and numerically. However, by the use of the techniques in [21,23,24] the fourth order term can be approximated by…”