The quantum Hall state constitutes a paradigm for a topological state of matter, the Hall conductance of which is insensitive to continuous changes in the parameters and depends only on the number of edge states, which are unidirectional because of the breaking of the time reversal symmetry due to the magnetic field. Recently, an analogous effect was predicted in a time reversal symmetric situation: it was shown that a class of insulators, such as graphene with spin orbit coupling and an "inverted" semiconductor HgTe/CdTe quantum well, possess the topological property that they have a single pair of counter-propagating or helical edge state, exhibiting the phenomenon of the quantum spin Hall effect. This "topological insulator" is distinguished from an ordinary band insulator by a Z₂topological invariant, analogous to the Chern number classification of the quantum Hall effect. The prediction of non-zero conductance in a band-insulating region of an "inverted" HgTe/CdTe quantum well has been verified experimentally, although the origin of the observed deviation from an exact quantization is not yet fully understood.
In view of its importance in 2D, it is natural to ask how disorder affects the stability of the helical edge states in the topological insulator. As expected, we find that the physics of topological insulator is unaffected by the presence of weak disorder but is destroyed for large disorder. More surprisingly, however, our results show that disorder can create a topological insulator for parameters where the system was metallic in the absence of disorder, and the bulk electrons are believed to be localized by disorders. We call this phase “topological Anderson insulator”. In this talk I shall introduce relevant properties of this novel quantum state of matter, and recent progresses on this topic.