  The optical Hall response is theoretically studied for spin-orbit coupled
metals with ferroic orders of cluster-type magnetic multipoles. We find that
different magnetic multipoles give rise to distinct spectra in the optical Hall
conductivity. In the cases of monopole and quadrupole orders, the optical Hall
response appears predominantly in high- and low-energy regions, which
correspond to the energy scales of electron correlation and kinetic energy,
respectively, while the response is dispersed and rather weak in the case of
toroidal order. By decomposing the spectra into different interband
contributions, we reveal selection rules stemming from the interplay between
the antisymmetric spin-orbit coupling and the underlying multipoles. Our
results suggest that the optical Hall measurement is useful to detect and
distinguish the cluster-type magnetic multipole orders.
