Magnetic circular dichroism and the orbital magnetization of ferromagnets

The spontaneous magnetization of ferromagnets has both spin and orbital contributions, ${\bf M}={\bf M}_{\rm spin}+{\bf M}_{\rm orb}$, which can be separated out via gyromagnetic measurements. Recently\footnote{D. Ceresoli, T. Tonhauser, D. Vanderbilt, and R. Resta, {\it Phys. Rev. B} {\bf 74}, 024408 (2006).} it was found that, when expressed as a bulk property of the Bloch electrons, the orbital magnetization itself consists of two terms, ${\bf M}_{\rm orb}= \widetilde{\bf M}_{\rm LC}+\widetilde{\bf M}_{\rm IC}$, which can be loosely interpreted as the localized and itinerant contributions, respectively. Interestingly, $\widetilde{\bf M}_{\rm LC}$ and $\widetilde{\bf M}_{\rm IC}$ are separately gauge-invariant, which raises the possibility that they may be independently measurable. We show that indeed they are related to the magnetic circular dichroism (MCD) spectrum by a subtle sum rule. MCD, the difference in absorption between left- and right-circularly-polarized light, is given by $\sigma_{{\rm A},\alpha\beta}^{(2)}(\omega)$, the absorptive part of the antisymmetric conductivity. We derive the following sum rule for the interband contribution: $\int_0^\infty \vec\sigma_{\rm A}^{(2)}(\omega)d\omega= (2\pi e c/\hbar)\big( \widetilde{\bf M}_{\rm LC}-\widetilde{\bf M}_{\rm IC}\big)$, where $\vec\sigma_{\rm A}^{(2)}(\omega)$ is a pseudo-vector. Hence, by combining the results of gyromagnetic and magneto-optical experiments, $\widetilde{\bf M}_{\rm LC}$ and $\widetilde{\bf M}_{\rm IC}$ can in principle be measured independently.