Dichroic f-sum rule and the orbital magnetization of crystals

We consider the magnetic circular dichroism spectrum of a crystal with broken time-reversal symmetry in the electric-dipole approximation. Using the Kubo-Greenwood formula for the absorptive part of the antisymmetric optical conductivity, its frequency integral is recast as a ground-state property. We show that in insulators this quantity is proportional to the circulation of the occupied Wannier orbitals around their centers (more precisely, to the gauge-invariant part thereof). This differs from the net circulation, or ground-state orbital magnetization, which has two additional contributions: (i) the remaining Wannier self-rotation, and (ii) the "itinerant" circulation arising from the center-of-mass motion of the Wannier orbitals, both on the surface and in the interior of the sample. Contributions (i) and (ii) are not separately meaningful, since their individual values depend on the particular choice of Wannier functions. Their sum is, however, gauge invariant, and can be inferred from a combination of two experiments: a measurement of the magneto-optical spectrum over a sufficiently wide range to evaluate the sum rule, and a gyromagnetic determination of the total orbital magnetization.