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.