Footprints of the Kitaev spin liquid in the Fano lineshapes of the Raman active optical phonons

We develop a theoretical description of the Raman spectroscopy in the spin-phonon coupled Kitaev system and show that it can provide intriguing observable signatures of fractionalized excitations characteristic of the underlying spin liquid phase. In particular, we obtain the explicit form of the phonon modes and construct the coupling Hamiltonians based on $D_{3d}$ symmetry. We then systematically compute the Raman intensity and show that the spin-phonon coupling renormalizes phonon propagators and generates the salient Fano linshape. We find that the temperature evolution of the Fano lineshape displays two crossovers, and the low temperature crossover shows pronounced magnetic field dependence. We thus identify the observable effect of the Majorana fermions and the $Z_2$ gauge fluxes encoded in the Fano lineshape. Our results explain several phonon Raman scattering experiments in the candidate material $alpha$-RuCl$_3$.