Wave functions in the Anderson model and in the quantum percolation model: a comparison

We compare numerically the localization behavior of electronic eigenfunctions in the Anderson model and on self-similar percolation clusters at criticality. We find that the distributions of the local wave function amplitudes \psi\ at fixed distances from the localization center are very similar for both models; The amplitude distributions are well approximated by log-normal fits, which seem to become exact at large distances. From the distributions, we can calculate analytically the behavior of the averages at sufficiently large distances. We observe two different localization regimes. In the first regime, at intermediate distances from the localization center, we find stretched exponential localization ('sublocalization'), ln[\psi\] similar to -r(d psi), with effective localization exponents d(psi) < 1. In the second regime, for very large r, the averages strongly depend on the number of configurations N, and superlocalization (d(psi) > 1) is observed, converging to simple exponential behavior asymptotically as expected. The crossover from the intermediate to the asymptotic regime depends logarithmically on the number of configurations.