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The optical Hall response is theoretically studied for spin-orbit coupledmetals with ferroic orders of cluster-type magnetic multipoles. We find thatdifferent magnetic multipoles give rise to distinct spectra in the optical Hallconductivity. In the cases of monopole and quadrupole orders, the optical Hallresponse appears predominantly in high- and low-energy regions, whichcorrespond to the energy scales of electron correlation and kinetic energy,respectively, while the response is dispersed and rather weak in the case oftoroidal order. By decomposing the spectra into different interbandcontributions, we reveal selection rules stemming from the interplay betweenthe antisymmetric spin-orbit coupling and the underlying multipoles. Ourresults suggest that the optical Hall measurement is useful to detect anddistinguish the cluster-type magnetic multipole orders.

Optical Hall response in spin-orbit coupled metals: Comparative study of magnetic cluster monopole, quadrupole, and toroidal orders