Optimisation of multifractal analysis using box-size scaling

TL;DR

This paper optimizes box-size scaling methods for multifractal analysis of critical eigenstates at the metal-insulator transition in the 3-D Anderson model, focusing on numerical reliability and efficiency.

Contribution

It introduces an optimized, reliable, and computationally efficient box-scaling technique for multifractal analysis in the 3-D Anderson model.

Findings

Equal division into cubic boxes of size l is most reliable.

The method is less computationally expensive.

It improves the accuracy of multifractal property estimation.

Abstract

We study various box-size scaling techniques to obtain the multifractal properties, in terms of the singularity spectrum f(alpha), of the critical eigenstates at the metal-insulator transition within the 3-D Anderson model of localisation. The typical and ensemble averaged scaling laws of the generalised inverse participation ratios are considered. In pursuit of a numerical optimisation of the box-scaling technique we discuss different box-partitioning schemes including cubic and non-cubic boxes, use of periodic boundary conditions to enlarge the system and single and multiple origins for the partitioning grid are also implemented. We show that the numerically most reliable method is to divide a system of linear size L equally into cubic boxes of size l for which L/l is an integer. This method is the least numerically expensive while having a good reliability.