A problem that has always been central to solid state physics is the insulator–metal transition. The question to answer here is why some materials conduct and others do not. Metals are characterized by a non-zero dc conductivity σ(0) at zero temperature, whereas σ(0) = 0 in an insulator. There are currently three standard models that describe a transition between these two extremes. Anderson (A1958) was first to point out that scattering from a static but random potential can disrupt metallic conduction and lead to an abrupt localization of the electronic eigenstates. Mott (M1949), on the other hand, proposed that an insulating state can obtain even in a material such as NiO which possesses a partially filled valence band. The insulating state arises from strong electron correlations which induce a gap at the Fermi energy. The closing of the gap, signaling the onset of a metallic state, results typically in the intermediate-coupling regime in which the kinetic energy effects can destroy the ordering tendencies of the potential energy. Finally, a structural transition in which the lattice periodicity doubles can also thwart metallic transport. While all of these mechanisms are of considerable interest in their own right, our focus in this chapter will be the disorder-driven insulator–metal or Anderson transition.

We start by reviewing the essential physics and some of the key controversies surrounding the disorder-induced localization transition. Two controversies we address are the role of perturbation theory and whether or not the conductivity is continuous in the vicinity of the Anderson transition.