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

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 Psi(E) 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 Psi(E>0) consists of a short-range core that is essentially the same as Psi(E=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 >> L-c(E); such states behave as delocalized ones. In contrast, in small systems, L << L-c(E), quasi-localized states are overwhelmingly dominated by the localized cores and are effectively localized.