UC Berkeley

This dissertation introduces some basic characteristics of a class of materials known as topological insulators. These materials were introduced theoretically as crystalline band insulators, in which electrons do not interact with each other and the atomic cores form a perfectly ordered, fixed background potential for the electrons. It is shown that, in the two-dimensional case, this definition can be relaxed to the case of disordered, noninteracting insulators. It is further shown numerically that the introduction of disorder to these two- dimensional insulators eliminates the direct transition between the topological and ordinary insulating phases, consistent with the presence of an intervening metallic phase.

In the three-dimensional case, one formulation of the distinction between the topological and ordinary phases involves a quantized response function (the magnetoelectric polarizability). It is shown here that this characterization holds on quite general grounds, and therefore allows an extension of the topological class to disordered and interacting systems. Finally, a relatively rigorous derivation of the (fixed-ion, linear, orbital) magnetoelectric response of crystalline band insulators is presented.