Band theory of spatial dispersion in magnetoelectrics

Working in the crystal-momentum representation, we calculate the optical conductivity of noncentrosymmetric insulating crystals at first order in the wave vector of light. The time-even part of this tensor describes natural optical activity and the time-odd part describes nonreciprocal effects such as gyrotropic birefringence. The time-odd part can be uniquely decomposed into magnetoelectriclike and purely quadrupolar contributions. The magnetoelectriclike component reduces in the static limit to the traceless part of the frozen-ion static magnetoelectric polarizability while at finite frequencies it acquires some quadrupolar character in order to remain translationally invariant. The expression for the orbital contribution to the conductivity at transparent frequencies is validated by comparing numerical tight-binding calculations for finite and periodic samples.