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

V. Temkin; A. S. Ioselevich

doi:10.1016/j.aop.2024.169620

paper

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

arXiv:2309.06345 · doi:10.1016/j.aop.2024.169620

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 ( $E \to 0$ ). 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 $Ψ_{E}$ 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 $Ψ_{E > 0}$ consists of a short-range core that is essentially the same as $Ψ_{E = 0}$ 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.

arXiv comment18+1 pages, 9+1 figures

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

Abstract

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