Multifractal analysis of the metal-insulator transition in the three-dimensional Anderson model. I. Symmetry relation under typical averaging

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 $f\left(\alpha \right)$. Recent works in one-dimensional and two-dimensional critical systems have suggested an exact-symmetry relation in $f\left(\alpha \right)$. 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 $f\left(\alpha \right)$ from the box-size and system-size scalings of the typical average of the generalized inverse participation ratios. We show that the best symmetry in $f\left(\alpha \right)$ 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.