Multifractal analysis with the probability density function at the three-dimensional Anderson transition

TL;DR

This paper investigates the probability density function of critical wavefunction amplitudes in the 3D Anderson model, establishing a formal link to multifractal spectra and demonstrating the PDF's effectiveness as an alternative analysis method.

Contribution

It introduces a formal expression connecting the PDF to the multifractal spectrum, accounting for finite-size effects, and highlights the PDF's non-Gaussian features and symmetry properties at criticality.

Findings

PDF exhibits non-Gaussian behavior

Existence of symmetry relation in the PDF

PDF-based analysis can replace traditional methods

Abstract

The probability density function (PDF) for critical wavefunction amplitudes is studied in the three-dimensional Anderson model. We present a formal expression between the PDF and the multifractal spectrum f(alpha) in which the role of finite-size corrections is properly analyzed. We show the non-gaussian nature and the existence of a symmetry relation in the PDF. From the PDF, we extract information about f(alpha) at criticality such as the presence of negative fractal dimensions and we comment on the possible existence of termination points. A PDF-based multifractal analysis is hence shown to be a valid alternative to the standard approach based on the scaling of general inverse participation ratios.