Orbital magnetoelectric effects and topological insulators

David Vanderbilt

Rutgers

In this talk I will discuss two kinds of topological insulators. The first is one in which the Chern number, defined in terms of the integral of the Berry curvature over the Brillouin zone, is non-zero.  Such a system is known as a quantum anomalous Hall (QAH) insulator because it would exhibit a quantum Hall effect in the absence of a magnetic field.  While no examples of a QAH insulator are yet known in nature, theoretical models are readily constructed, and there is no known reason why they should not exist.  Second, there has been a great deal of interest recently in another kind of topological insulator, the "Z2 insulator", also called "quantum spin Hall" in 2D or "strong topological insulator" in 3D.  The 2D version can be conceptualized by imagining that a spin-up system of electrons having Chern number +1 coexists with a spin-down system having Chern number -1 in such a way that the system as a whole has total Chern number zero and obeys time-reversal (T) symmetry. Even when the spin-orbit interaction is turned on, the system carries a topological "even-odd" (Z2) label that distinguishes it from a normal insulator.  Some of the recent excitement about this subject is due to the discovery of experimental realizations in the BiSb alloys and in Bi2Te3 and related systems.  I will discuss such systems from the point of view of a theory of the orbital contribution to the linear magnetoelectric effect (or equivalently, to the surface Hall conductivity) in insulators having broken T symmetry in general, and then specialize to the case that T symmetry is present. From this point of view, it emerges that the surface of a 3D Z2 topological insulator, if it is gapped by a T-breaking perturbation, will exhibit a half-integer quantum Hall effect.