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

We study the multifractal analysis (MFA) of electronic wave functions at the localization-delocalization transition in the three-dimensional Anderson model for very large system sizes up to 240(3). The singularity spectrum f(alpha) is numerically obtained using the ensemble average of the scaling law for the generalized inverse participation ratios P-q, employing box-size and system-size scaling. The validity of a recently reported symmetry law [Mirlin , Phys. Rev. Lett. 97, 046803 (2006)] for the multifractal spectrum is carefully analyzed at the metal-insulator transition. The results are compared to those obtained using different approaches, in particular the typical average of the scaling law. System-size scaling with ensemble average appears as the most adequate method to carry out the numerical MFA