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

TL;DR

This paper investigates the multifractal properties of electronic wavefunctions at the metal-insulator transition in the 3D Anderson model, focusing on symmetry relations and ensemble averaging for large system sizes.

Contribution

It provides a detailed numerical analysis of the multifractal spectrum using ensemble averaging and compares different methods, confirming the symmetry law at the transition.

Findings

Ensemble averaging yields reliable multifractal spectra.

The symmetry law for the multifractal spectrum holds at the transition.

System-size scaling with ensemble average is most effective for MFA.

Abstract

We study the multifractal analysis (MFA) of electronic wavefunctions at the localisation-delocalisation transition in the 3D Anderson model for very large system sizes up to $240^3$. The singularity spectrum $f(\alpha)$ is numerically obtained using the \textsl{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 [Phys. Rev. Lett. 97, 046803 (2006)] for the multifractal spectrum is carefully analysed at the metal-insulator transition (MIT). 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. Some conjectures about the true shape of $f(\alpha)$ in the thermodynamic limit are also made.