  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 Kitaev materials with bond-anisotropic interactions are generally expected
to show both one- and two-magnon responses. It is further shown that, in order
to obtain the correct leading contributions to the Raman vertex operator
$\mc{R}$, one must take into account the precise, photon-assisted microscopic
hopping processes of the electrons and that, in systems with multiple hopping
paths, $\mc{R}$ contains terms beyond those appearing in the traditional
Loudon-Fleury theory. 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 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.
