Coexistence of localized and extended states in the Anderson model with long-range hopping

Abstract

We study states arising from fluctuations in the disorder potential in systems with long-range hopping. Here, contrary to systems with short-range hopping, the optimal fluctuations of disorder responsible for the formation of the states in the gap, are not rendered shallow and long-range when E approaches the band edge (E→0). Instead, they remain deep and short-range. The corresponding electronic wave functions also remain short-range-localized for all E<0. This behavior has striking implications for the structure of the wave functions slightly above E=0. By a study of finite systems, we demonstrate that the wave functions ΨE transform from a localized to a quasi-localized type upon crossing the E=0 level, forming resonances embedded in the E>0 continuum. The quasi-localized ΨE>0 consists of a short-range core that is essentially the same as ΨE=0 and a delocalized tail extending to the boundaries of the system. The amplitude of the tail is small, but it decreases with r slowly. Its contribution to the norm of the wave function dominates for sufficiently large system sizes, L≫Lc(E); such states behave as delocalized ones. In contrast, in small systems, L≪Lc(E), quasi-localized states are overwhelmingly dominated by the localized cores and are effectively localized.