Chern-Simons orbital magnetoelectric coupling in generic insulators

The isotropic Chern-Simons coupling $\theta$ is a component of the orbital contribution to the magnetoelectric coupling.\footnote{A. Malashevich {\it et al.}, New J. Phys. {\bf 12}, 053032 (2010); A.~M. Essin {\it et al.}, Phys. Rev. B {\bf 81}, 205104 (2010). } In a generic insulator it can have any value, while it must be exactly $\pi$ in a strong Z$_2$ topological insulator. The results of our first-principles density-functional calculations for the ordinary magnetoelectrics Cr$_2$O$_3$, BiFeO$_3$ and GdAlO$_3$ confirm that the Chern-Simons contribution is quite small in these materials.\footnote{S. Coh {\it et al.}, arXiv:1010.6071.} We discuss various strategies for finding insulators for which $\theta$ is large but not equal to $\pi$. For example, we show that if the spatial inversion and time-reversal symmetries of the Z$_2$ topological insulator Bi$_2$Se$_3$ are broken by hand, large induced changes appear in the Chern-Simons magnetoelectric coupling. We also perform an analysis based on space-group representation theory to determine the simplest possible magnetic structures which allow for a non-zero and possibly large value of $\theta$.