Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging

TL;DR

This paper confirms the existence of a symmetry relation in the multifractal spectrum at the 3D Anderson metal-insulator transition, using large-scale numerical analysis of typical eigenstate properties.

Contribution

It demonstrates the validity of the symmetry relation in the multifractal spectrum at the 3D Anderson transition with high numerical accuracy and large system sizes.

Findings

Symmetry relation in f(alpha) confirmed at the MIT

Large system sizes improve the accuracy of symmetry detection

Typical averaging via system-size scaling is most effective

Abstract

The multifractality of the critical eigenstate at the metal to insulator transition (MIT) in the three-dimensional Anderson model of localization is characterized by its associated singularity spectrum f(alpha). Recent works in 1D and 2D critical systems have suggested an exact symmetry relation in f(alpha). Here we show the validity of the symmetry at the Anderson MIT with high numerical accuracy and for very large system sizes. We discuss the necessary statistical analysis that supports this conclusion. We have obtained the f(alpha) from the box- and system-size scaling of the typical average of the generalized inverse participation ratios. We show that the best symmetry in f(alpha) for typical averaging is achieved by system-size scaling, following a strategy that emphasizes using larger system sizes even if this necessitates fewer disorder realizations.