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 longrange hopping. We demonstrate that, in contrast to the conventional systems with shortrange hops, the optimal fluctuations of disorder, responsible for creation of the states in the gap, do not become shallow and longrange when $E$ approaches the band edge ($E\to 0$), but remain deep and shortrange ones. The corresponding electronic wave functions also remain shortrange 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 $L\gg L_c(E)$.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.06345
 arXiv:
 arXiv:2309.06345
 Bibcode:
 2023arXiv230906345T
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks