A renormalization approach for the 2D Anderson model at the band edge: Scaling of the localization volume

TL;DR

This paper investigates how the localization volume of electronic wave functions in the 2D Anderson model scales near the band edges, revealing a universal inverse proportionality to the disorder variance and confirming a scaling law through simulations.

Contribution

It introduces a renormalization approach to demonstrate the inverse relationship between localization volume and disorder variance at the band edges, establishing a scaling law near these energies.

Findings

Localization volume $V$ is inversely proportional to disorder variance $ ext{var}$ at band edges.

Scaling law $V= ext{var}^{-1} g((4 - |E|)/ ext{var})$ is confirmed by numerical simulations.

Scaling function $g(x)$ describes the behavior of $V$ near the band edges.

Abstract

We study the localization volumes $V$ (participation ratio) of electronic wave functions in the 2d-Anderson model with diagonal disorder. Using a renormalization procedure, we show that at the band edges, i.e. for energies $E\approx \pm 4$, $V$ is inversely proportional to the variance $\var$ of the site potentials. Using scaling arguments, we show that in the neighborhood of $E=\pm 4$, $V$ scales as $V=\var^{-1}g((4-\ve E\ve)/\var)$ with the scaling function $g(x)$. Numerical simulations confirm this scaling ansatz.