Distribution of fractal dimensions at the Anderson transition

Abstract

We investigated numerically the distribution of participation numbers in the 3d Anderson tight-binding model at the localization-delocalization threshold. These numbers in {\em one} disordered system experience strong level-to-level fluctuations in a wide energy range. The fluctuations grow substantially with increasing size of the system. We argue that the fluctuations of the correlation dimension, $D_2$ of the wave functions are the main reason for this. The distribution of these correlation dimensions at the transition is calculated. In the thermodynamic limit ($L\to \infty$) it does not depend on the system size $L$. An interesting feature of this limiting distribution is that it vanishes exactly at $D_{\rm 2max}=1.83$, the highest possible value of the correlation dimension at the Anderson threshold in this model.