Multifractal finite-size-scaling and universality at the Anderson transition

TL;DR

This paper introduces a new multifractal finite-size scaling method for analyzing the Anderson transition, providing highly precise estimates of critical parameters and confirming theoretical predictions about multifractal spectra.

Contribution

The paper presents a novel MFSS procedure that accurately estimates critical parameters and multifractal exponents at the Anderson transition, including correlation effects.

Findings

Critical disorder W_c=16.530 with tight confidence interval

Critical exponent nu=1.590 with high precision

Confirmed symmetry and non-parabolicity of multifractal spectrum

Abstract

We describe a new multifractal finite size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L^3=120^3 and involving nearly 10^6 independent wavefunctions have yielded unprecedented precision for the critical disorder W_c=16.530 (16.524,16.536) and the critical exponent nu=1.590 (1.579,1.602). We find that the multifractal exponents Delta_q exhibit a previously predicted symmetry relation and we confirm the non-parabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.