$f(\alpha)$ Multifractal spectrum at strong and weak disorder

TL;DR

This study numerically analyzes the multifractal spectrum $f( ext{alpha})$ in disordered 1D and 2D systems, revealing a transition from parabolic to non-parabolic forms as disorder strength increases.

Contribution

It provides the first detailed numerical investigation of how the $f( ext{alpha})$ spectrum varies with disorder strength in low-dimensional disordered systems.

Findings

$f( ext{alpha})$ is parabolic in weak disorder regimes.

$f( ext{alpha})$ deviates from parabolicity under strong disorder.

Corrections to parabolicity vanish at finite coupling strength.

Abstract

The system size dependence of the multifractal spectrum $f(\alpha)$ and its singularity strength $\alpha$ is investigated numerically. We focus on one-dimensional (1D) and 2D disordered systems with long-range random hopping amplitudes in both the strong and the weak disorder regime. At the macroscopic limit, it is shown that $f(\alpha)$ is parabolic in the weak disorder regime. In the case of strong disorder, on the other hand, $f(\alpha)$ strongly deviates from parabolicity. Within our numerical uncertainties it has been found that all corrections to the parabolic form vanish at some finite value of the coupling strength.