Anderson localization at large disorder

The localization of one-electron states in the large (but finite) disorder limit is investigated. The inverse participation number shows a nonmonotonic behavior as a function of energy owing to the anomalous behavior of few-site localization. The two-site approximation is solved analytically and is shown to capture the essential features found in numerical simulations on one-, two-, and three-dimensional systems. Further improvement has been obtained by solving a three-site model.

Figure 1

(a) The IPR of the states as a function of energy for $W=32$. The red dots correspond to the states; the black curve is the average. (b) The same as (a), but zoomed to the band edge. Here and thereafter energy and disorder are measured in units of hopping.

Figure 2

The domain of (a) $p({\varepsilon }_1,{\varepsilon }_2)$ and (b) $p(E_1,I)$ for $W=8$.

Figure 3

For $p(E,I)$, (a) is the analytical result of the two-site model, (b) is the numerical result for system size $L=512$, and (c) is the difference between (a) and (b). (d) The red points are the numerically obtained $E$ and $I$ values at system size $L=512$; the black curve is the domain of the two eigenstates in the framework of the two-site model.

Figure 4

The domain of definition for variables (a) ${\varepsilon }_1,{\varepsilon }_2,{\varepsilon }_3$, (b) $E_1,u,v$, (c) $E_2,u,v$, and (d) $E_3,u,v$ in a three-site model.

Figure 5

Left side: IPR as a function of energy (a) in $d=1$ with $L=512$ for $W=32$, (c) in $d=2$ with $L=20$ for $W=50$, and (e) in $d=3$ with $L=8$ for $W=100$. The red dots correspond to single states; the black curve is the average. Right side: $\langle I\rangle$ as the function of energy (b) in $d=1$, (d) in $d=2$, and (f) in $d=3$. The dots correspond to the numerically obtained curve for a big system; the black curve corresponds to the one-site model, the blue curve corresponds to the two-site model, and the green curve corresponds to the three-site model. The insets are the same, but zoomed to the left band edge.