Anderson Transition

Abstract

The concept of multifractals described in Chap. 4 dramatically improves our understanding of complex distributions of quantities in physical systems. The importance of multifractal analysis is ensured by the fact that an entire spectrum of exponents τ (q) or f (α) describes not only the profile of the distribution but also dynamical properties of the system. This is well demonstrated by the case of the Anderson transition. The Anderson transition is a disorder-induced metal—insulator transition in a non-interacting electron gas at zero temperature. The insulating phase is a consequence of the localization of electron wavefunctions, which is called Anderson localization. Anderson localization is caused by quantum interference of an electron wave scattered by disordered potentials. At the Anderson transition point, the squared amplitude of the electron wavefunction distributes in a multifractal manner. Critical properties of the Anderson transition are deeply related to the multifractality of critical wavefunctions. It is therefore important to study the multifractal nature of the Anderson transition. In this and the next chapter, we show that distributions of a critical wavefunction at the Anderson transition point and the energy spectrum are multifractal. Some exponents characterizing their multifractality are related to dynamical properties of electrons. This chapter aims to explain what the Anderson transition is, and thus serves as an introduction to the next chapter.