Wave-packet dynamics at the mobility edge in two- and three-dimensional systems

TL;DR

This paper investigates wave-packet evolution at the mobility edge in disordered systems, confirming scaling theory predictions through numerical simulations in 2D and 3D, revealing power-law behaviors in probability densities.

Contribution

It provides numerical evidence supporting scaling theory predictions for wave-packet dynamics at the mobility edge in disordered systems in two and three dimensions.

Findings

Wave packet moments scale as t^{k/d} in d dimensions.

Return probability scales as t^{-D_2/d}.

Probability density exhibits power-law scaling at long times and short distances.

Abstract

We study the time evolution of wave packets at the mobility edge of disordered non-interacting electrons in two and three spatial dimensions. The results of numerical calculations are found to agree with the predictions of scaling theory. In particular, we find that the $k$-th moment of the probability density $<r^k >(t)$ scales like $t^{k/d}$ in $d$ dimensions. The return probability $P(r=0,t)$ scales like $t^{-D_2/d}$, with the generalized dimension of the participation ratio $D_2$. For long times and short distances the probability density of the wave packet shows power law scaling $P(r,t)\propto t^{-D_2/d}r^{D_2-d}$. The numerical calculations were performed on network models defined by a unitary time evolution operator providing an efficient model for the study of the wave packet dynamics.