  We study the optical properties of topological flat and dispersive bands. Due
to their topological nature, there exists an anomalous Hall response which
gives rise to a transverse current without applied magnetic field. The
dynamical Hall conductivity of systems with flat bands exhibits a sign change
when the excitation energy is on resonance with the band gap, similar to the
magnetotransport Hall conductivity profile. The sign change of the Hall
conductivity is located at the frequency corresponding to the singularity of
the joint density of states, i.e., the van Hove singularity (VHS). For
perfectly flat bands, this VHS energy matches the band gap. On the other hand,
in the case of dispersive bands, the VHS energy is located above the band gap.
As a result, the two features of the Hall conductivity, i.e., the resonant
feature at the band gap and the sign change at the VHS energy, become
separated. This anomalous Hall response rotates the polarization of an electric
field and can be detected in the reflected and transmitted waves, as Kerr and
Faraday rotations, respectively, thus allowing a simple optical
characterization of topological flat bands.
