Critical exponents of the three-dimensional Anderson transition from multifractal analysis

We use high-precision, large system-size wave function data to analyse the scaling properties of the multifractal spectra around the disorder-induced three-dimensional Anderson transtion in order to extract the critical exponent $\nu$ of the localisation length. We study the scaling law around the critical point of the generalized inverse participation ratios $P_q=\langle |\Psi_i|^2\rangle$ and the singularity exponent $\alpha_0$, defined as the position of the maximum of the multifractal spectra, as functions of the degree of disorder $W$, the system size $L$ and the box-size $\ell$ used to coarse-grained the wave function amplitudes. The values of $\alpha_0$ are calculated using a new method entirely based on the statistics of the wave function intensities [Phys.~Rev.~Lett.~102, 106406 (2009)]. Using finite size scaling analysis we find agreement with the values of $\nu$ obtained from transfer matrix calculations.