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.

Figure 1

Schematic model of a molecular crystal with one molecule per cell and four WFs per molecule. The range of the orbitals is indicated by the overlapping circles, and their center-of-mass velocities ${\overline{\mathbf{v}}}_n$ are denoted by arrows. The two panels show in bold two possible choices of “Wannier basis.”

Figure 2

A finite sample cut from the bulk crystal of Fig. 1. With the choice of Wannier basis of Fig. 1a, interior and surface WFs lie inside and outside the dashed line, respectively; with that of Fig. 1b, they are denoted by solid and open circles, respectively. Right panel: open and solid arrows show the extra “itinerant currents” [${\mathbf{M}}_{\mathrm{IC}}^{\left(\mathrm{surf}\right)}$ and ${\mathbf{M}}_{\mathrm{IC}}^{\left(\mathrm{int}\right)}$, respectively] associated with the latter choice.