Renormalization approach for the two-dimensional 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 $⟨{ϵ}^2⟩$ of the site potentials. Using scaling arguments, we show that in the neighborhood of $E=\pm 4$, $V$ scales as $V={⟨{ϵ}^2⟩}^{-1}g((4-\mid E\mid )∕⟨{ϵ}^2⟩)$ with the scaling function $g\left(x\right)$. Numerical simulations confirm this scaling ansatz.

Figure 1

Sketch of the Anderson lattice according to Eqs. (1, 8) as explained in the text. The circles represent the different lattice sites, the straight lines between them indicate the usual nearest-neighbor coupling, whereas the couplings of Eq. (8) are shown by the oval lines. The black circles stand for the site potentials that form the block potential [see Eq. (9)].

Figure 2

Renormalization scheme for $\nu =4$: the diagonal terms $f_{i,j}\equiv {ϵ}_{i,j}{\psi }_{i,j}$ that form the block potential are painted black and show a chess-board pattern. The sites $(n,m)$, $(n+4,m)$, $(n-4,m)$, $(n,m+4)$, and $(n,m-4)$ that couple via a Laplace operator of distance $\nu$ are symbolized by larger circles.

Figure 3

The effective localization length $V^{1∕2}$ in the 2D Anderson model at the band edge is plotted versus the variance $⟨{ϵ}^2⟩$ of the site potentials in a double-logarithmic plot. $V$ was calculated numerically for lattices of size $500\times 500$ with Dirichlet boundary conditions and averaged over 40 systems. The line of slope $-1∕2$ is a guide to the eye and shows the theoretical behavior. Finite size effects occur for small values of $⟨{ϵ}^2⟩$ (large values of $V^{1∕2}$).

Figure 4

As a test of Eq. (18), ${\left(V⟨{ϵ}^2⟩\right)}^{1∕2}$ is plotted versus the argument $x\equiv (4-E)∕⟨{ϵ}^2⟩$ for different disorder widths $w=0.6$ (circles), $w=0.8$ (squares), $w=1.0$ (diamonds), $w=1.2$ (triangles up), and $w=1.5$ (triangles down) with $⟨{ϵ}^2⟩=w^2∕12$ and for different values for $4-E$ between $0.001$ and $0.06$. The average was again taken over 40 systems of size $500\times 500$.