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 (\(E\to 0\)). Instead, they remain deep and short-range. The corresponding electronic wave functions also remain short-range-localized for all \(E0\) 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, \(L\gg L_c(E)\); such states behave as delocalized ones. In contrast, in small systems, \(L\ll L_c(E)\), quasi-localized states are overwhelmingly dominated by the localized cores and are effectively localized.