Multifractal finite-size scaling and universality at the Anderson transition

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 wave functions have yielded unprecedented precision for the critical disorder W₂=16. 530 (16. 524, 16. 536) and the critical exponent =1. 590 (1. 579, 1. 602). We find that the multifractal exponents ₐ exhibit a previously predicted symmetry relation and we confirm the nonparabolic 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.