Magnetoelectric Polarizability and Axion Electrodynamics in Crystalline Insulators

Abstract

The orbital motion of electrons in a three-dimensional solid can generate a pseudoscalar magnetoelectric coupling $\ensuremath{\theta}$, a fact we derive for the single-particle case using a recent theory of polarization in weakly inhomogeneous materials. This polarizability $\ensuremath{\theta}$ is the same parameter that appears in the ``axion electrodynamics'' Lagrangian $\ensuremath{\Delta}{\mathcal{L}}_{EM}=(\ensuremath{\theta}{e}^{2}/2\ensuremath{\pi}h)\mathbf{E}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{B}$, which is known to describe the unusual magnetoelectric properties of the three-dimensional topological insulator ($\ensuremath{\theta}=\ensuremath{\pi}$). We compute $\ensuremath{\theta}$ for a simple model that accesses the topological insulator and discuss its connection to the surface Hall conductivity. The orbital magnetoelectric polarizability can be generalized to the many-particle wave function and defines the 3D topological insulator, like the integer quantum Hall effect, in terms of a topological ground-state response function.