Magnetoelectric Polarizability and Axion Electrodynamics in Crystalline Insulators

The orbital motion of electrons in a three-dimensional solid can generate a pseudoscalar magnetoelectric coupling $\theta$, a fact we derive for the single-particle case using a recent theory of polarization in weakly inhomogeneous materials. This polarizability $\theta$ is the same parameter that appears in the “axion electrodynamics” Lagrangian $\Delta {\mathcal{L}}_{EM}=(\theta e^2/2\pi h)\mathbf{E}·\mathbf{B}$, which is known to describe the unusual magnetoelectric properties of the three-dimensional topological insulator ($\theta =\pi$). We compute $\theta$ 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.

Figure 1

The magnetoelectric polarizability $\theta$ (in units of $e^2/2\pi h$). The filled squares are computed by the Chern-Simons form, Eq. (4). The open squares are $dP/dB$ from Eq. (9). The points are obtained by layer-resolved ${\sigma }_H$ calculations using Eq. (12). The curve is obtained from Eq. (10).

Figure 2

The layer-resolved Hall conductivity (in units of $e^2/h$) at $\beta =\pi$ in a slab of 20 layers, with $m=t/2$ and ${\lambda }_{SO}=t/4$, terminated in ($\overline{1}11$) planes.