On the coexistence of localized and delocalized states in the Anderson model with long hops

Abstract

We study the states with different energies $E$ arising due to fluctuations of disorder potential in the systems with long-range hopping. We demonstrate that, in contrast to the conventional systems with short-range hops, the optimal fluctuations of disorder, responsible for creation of the states in the gap, do not become shallow and long-range when $E$ approaches the band edge ($E\to 0$), but remain deep and short-range ones. The corresponding electronic wave functions also remain short-range localized for all $E<0$ up to the very band edge. The most intriguing question that we address in this paper is the structure of the wave functions slightly above $E=0$. To get a comprehensive answer to this question it was necessary to perform the corresponding study for a finite system of size $L$. We demonstrate that upon crossing the $E=0$ level the wave functions $\Psi_E$ undergo transformation from localized to quasilocalized type. The quasilocalized $\Psi_{E>0}(r)$ consists of two parts: (a) a short range core which is basically the same as $\Psi_{E=0}$, (b) a delocalized tail that spans to the boundaries of the system. The amplitude of the tail is small for small $E$, but the tail decreases with $r$ too slowly, so that its contribution to the norm of the wave function dominates for large enough systems with $L\gg L_c(E)$; such systems therefore behave as delocalized ones. On the other hand, small systems with $L\ll L_c(E)$ are dominated by localized core and are effectively localized. Moreover, if one is interested in the Inverse Participation Ratio, then the latter is dominated by the localized core of the wave function even for large systems with $L\gg L_c(E)$.