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

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 $\text{var}$ of the site potentials. Using scaling arguments, we show that in the neighborhood of $E=\pm 4$, $V$ scales as $V={\text{var}}^{-1}g((4-\text{ve}E\text{ve})\mathrm{/}\text{var})$ with the scaling function $g(x)$. Numerical simulations confirm this scaling ansatz.