The Loudon-Fleury form of the Raman operator has been used to compute magnetic Raman responses inside magnetic insulators for decades. The formalism provided by Loudon and Fleury  only considers the light-induced direct hopping between two magnetic ions. However, this is an oversimplified scenario for spin-orbit coupled Mott insulators. For example, the microscopic origin of the superexchange interaction inside Kitaev materials shows multiple indirect superexchange paths involving ligand ions can also produce a significant anisotropic interaction . Therefore, to correctly construct the Raman operator for spin-orbit coupled Mott insulators requires considering all the non-negligible direct and indirect superexchange paths.
In this talk, I will present our work on constructing the Raman operator for the spin-orbit coupled Mott insulators which involve multiple superexchange paths , and I will also show how our revised theory can be applied to the three-dimensional hyperhoneycomb Kitaev material β−Li2IrO3 , where we show 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.
 P. A. Fleury and R. Loudon, Phys. Rev. 166, 514 (1968).
 G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
 Yang Yang, Mengqun Li, Ioannis Rousochatzakis, and Natalia B. Perkins Phys. Rev. B 104, 144412 (2021).
 Yang Yang, Yiping Wang, Ioannis Rousochatzakis, Alejandro Ruiz, James G. Analytis, Kenneth S. Burch, and Natalia B. Perkins Phys. Rev. B 105, L241101(2022).