Calculation of the axion magnetoelectric coupling

Recently it was shown [X.-L.\ Qi {\it et al.}, PRB {\bf 78}, 195424 (2008); A.M.\ Essin {\it et al.}, PRL {\bf 102}, 146805 (2009)] that there exists a purely isotropic (``axionic'') component $\theta$ to the magnetoelectric coupling (MEC). Furthermore, this $\theta$ arises only from the electron orbital motion, and in strong Z$_2$ topological insulators it is unusually large and equals exactly half a quantum ($\theta=\pi$). Experimental observation of this large MEC would require some peculiar breaking of the time-reversal ($T$) symmetry at the surfaces, but $\theta$ might be observed in normal insulators that have $T$ already broken in the bulk. Since there are by now several examples of strong Z$_2$ topological insulators having $\theta=\pi$, we believe there is no strong reason why $\theta$ should necessarily be small in a normal insulator with broken $T$. For this reason, we have used density-functional theory to calculate $\theta$ in various materials. We first consider Cr$_2$O$_3$, a widely studied magnetoelectric material, but we find $\theta$ to be very small there. We attribute this to a weak spin-orbit effect in Cr (and to the fact that even a strong spin-orbit effect by itself does not guarantee a large $\theta$). To calculate $\theta$ we express it in terms of well localized Wannier functions to ensure smoothness of the gauge and also to allow for decomposition of contributions to $\theta$ coming from various electronic bands. The calculation of $\theta$ for BiFeO$_3$ and other materials is currently ongoing.