Multifractals in the Anderson Transition

Abstract

In Chap. 9, we showed the partial similarity between the Anderson transition and the thermal or percolation transition. Critical properties near/at the thermal phase transition or the percolation transition are related to the fractality at the critical point. This is because the local order parameter of such a transition distributes in a fractal manner. It is natural to suppose that the fractality is relevant in the case of the Anderson transition as well. However, it is not easy to find an appropriate order parameter for the Anderson transition itself. What kinds of distribution are fractal at the Anderson transition? The answer is the squared amplitudes of the wavefunction at the transition point and the energy distribution of the spectral measure. These distributions are actually multifractal rather than conventional fractals. Although the multifractalities of the critical wavefunction and the spectral measure should generally be independent, they are closely related in the case of the Anderson transition. This is because the one-parameter scaling hypothesis holds for the Anderson transition.