Non-Loudon-Fleury Raman scattering in spin-orbit coupled Mott insulators

We revisit the theory of magnetic Raman scattering in Mott insulators with strong spin-orbit coupling, with a major focus on Kitaev materials. We show that, in order to obtain the correct leading contributions to the Raman vertex operator $\mathcal{R}$, one must take into account the precise, photon-assisted microscopic hopping processes of the electrons and that, in systems with multiple hopping paths, $\mathcal{R}$ contains terms beyond those appearing in the traditional Loudon-Fleury theory. It is further shown that Kitaev materials with non-collinear magnetic orders are generally expected to show both one- and two-magnon responses. Most saliently, a numerical implementation of the revised formalism to the case of the three-dimensional hyperhoneycomb Kitaev material $\beta$-Li$_2$IrO$_3$ reveals that the non-Loudon-Fleury scattering terms actually dominate the Raman intensity. In addition, they induce a qualitative modification of the polarization dependence, including, e.g., the emergence of a sharp one-magnon peak at low energies which is not expected in the traditional Loudon-Fleury theory. This peak has been seen in recent experiments in the expected polarization channel. This peak is shown to arise from microscopic photon-assisted tunneling processes that are of similar type with the ones leading to the symmetric off-diagonal interaction $\Gamma$ (known to be present in many Kitaev materials), but take the form of a bond-directional magnetic dipole term in the Raman vertex. These results are expected to apply across all Kitaev materials and mark a drastic change of paradigm for the understanding of Raman scattering in materials with strong spin-orbit coupling and multiple exchange paths.