Multifractal characteristics of electronic wave functions in disordered systems

Abstract

We investigate the multifractal properties of electronic wave functions in the Anderson model of localisation. Different sample sizes are used to analyse the dependence of the multifractal properties on the system size. It is shown that at the mobility edge in 3D systems the singularity spectrum does not depend on the system size, rather it takes a universal form. However, away from the mobility edge in the conducting regime we find a characteristic dependence of the multifractal properties on the system size, which leads to the suggestion that in the thermodynamic limit the extended wave functions show no multifractal behaviour at all. Rather we expect the singularity spectrum to collapse to a single point, which indicates that the states will appear homogeneously extended on large enough length scales. We carried out the same analysis in the insulating regime but the results are blurred by fluctuations due to different disorder configurations, so no final conclusion can be reached for this case. However, the numerical data suggest that in the limit of infinite system size the singularity spectrum collapses to two points as one would expect for a reasonably localised state.