Magnetoelectric polarizability and axion electrodynamics in crystalline insulators

Spin-orbit coupling can lead in two- and three-dimensional solids to time-reversal-invariant insulating phases that are ``topological'' in the same sense as the integer quantum Hall effect and similarly have protected edge or surface states. The three-dimensional topological insulator is known to have unusual magnetoelectric properties referred to as ``axion electrodynamics'': it supports an electromagnetic coupling $\Delta{\cal L}_{EM} = (\theta e^2 / 2 \pi h) {\bf E} \cdot {\bf B}$ with $\theta=\pi$, giving a half-integer surface Hall conductivity $\sigma_{xy}=(n + 1/2) e^2 / h$. An approach to $\theta$ in any three-dimensional crystal is developed based on the Berry-phase theory of polarization: $\theta e^2/ 2 \pi h$ is the bulk orbital magnetoelectric polarizability (the polarization induced by an applied magnetic field). We compute the orbital magnetoelectric polarizability for a simple model and show that it predicts the fractional part of surface $\sigma_{xy}$, computed using a slab geometry. Although $\theta$ is not quantized once time-reversal and inversion symmetries are broken, it remains a bulk quantity for the same reasons as ordinary polarization.