A fractional Anderson model

TL;DR

This paper investigates how fractional order and disorder influence localization and spectral properties in a one-dimensional Anderson model, revealing a transition from extended to localized states and the formation of spectral gaps.

Contribution

It introduces a fractional Anderson model, analyzing the effects of fractional order on spectral gaps, localization, and state extension, which is a novel approach in disordered quantum systems.

Findings

Decreased fractional order causes eigenvalues to detach and form a spectral gap.

Disorder width affects the spectrum's width, with lower s reducing it.

Participation ratio decreases with s, indicating localization, while MSD shows a hump indicating extended states.

Abstract

We examine the interplay between disorder and fractionality in a one-dimensional tight-binding Anderson model. In the absence of disorder, we observe that the two lowest energy eigenvalues detach themselves from the bottom of the band, as fractionality $s$ is decreased, becoming completely degenerate at $s=0$, with a common energy equal to a half bandwidth, $V$. The remaining $N-2$ states become completely degenerate forming a flat band with energy equal to a bandwidth, $2V$. Thus, a gap is formed between the ground state and the band. In the presence of disorder and for a fixed disorder width, a decrease in $s$ reduces the width of the point spectrum while for a fixed $s$, an increase in disorder increases the width of the spectrum. For all disorder widths, the average participation ratio decreases with $s$ showing a tendency towards localization. However, the average mean square displacement (MSD) shows a hump at low $s$ values, signaling the presence of a population of extended states, in agreement with what is found in long-range hopping models.