Distribution of fractal dimensions at the Anderson transition

TL;DR

This paper numerically studies the distribution of participation numbers at the Anderson transition in a 3D disordered system, revealing size-dependent fluctuations and a limiting distribution of correlation dimensions.

Contribution

It provides the first detailed calculation of the distribution of correlation dimensions at the Anderson transition, showing size independence in the thermodynamic limit.

Findings

Fluctuations of correlation dimension grow with system size.

Distribution of correlation dimensions converges in the thermodynamic limit.

Distribution vanishes at the maximum correlation dimension of 1.83.

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.