Quantum interference can completely suppress the diffusion of a particle in random potential, a phenomenon known as Anderson localization. For a given energy and disorder strength the quantum states are either all localized or all delocalized. This implies the existence of Anderson transitions between insulating and metallic phases in disordered electronic systems. Anderson localization is a very rich phenomenon. The properties of Anderson transitions are sensitive to the symmetry and dimensionality of the disordered system. Moreover, subtle topological effects may significantly alter the critical behavior.
One of the most recent arenas where novel peculiar localization phenomena have been studied is physics of topological insulators. Topological insulators are bulk insulators with delocalized (topologically protected) states on their surface. As discussed above, the critical behavior of a system depends on the underlying topology. This is particularly relevant for topological insulators. The famous example of a topological insulator is a two-dimensional (2D) system on one of quantum Hall plateaus in the integer quantum Hall effect. Such a system is characterized by an integer (Chern number) n = …, -2, -1, 0, 1, 2,… which counts the edge states (here the sign determines the direction of chiral edge modes). The integer quantum Hall edge is thus a topologically protected one-dimensional (1D) conductor realizing the group Z.
Another (Z2) class of topological insulators can be realized in systems with strong spin-orbit interaction and without magnetic field (symplectic symmetry class) – and was discovered in 2D HgTe/HgCdTe structures. A 3D Z2 topological insulator has been found and investigated for the first time in Bi1âˆ’xSbx crystals. Both in 2D and 3D, Z2 topological insulators are band insulators with the following properties: (i) time reversal invariance is preserved (unlike ordinary quantum Hall systems); (ii) there exists a topological invariant, which is similar to the Chern number in quantum Hall effect; (iii) this invariant belongs to the group Z2 and reflects the presence or absence of delocalized edge modes (Kramers pairs). Topological insulators exist in all ten symmetry classes in different dimensions. Very generally, the classification of topological insulators in d dimensions can be constructed by studying the Anderson localization problem in a (d - 1)-dimensional disordered system. Indeed, absence of localization of surface states due to the topological protection implies the topological character of the insulator.