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We develop a theoretical description of the Raman spectroscopy in thespin-phonon coupled Kitaev system and show that it can provide intriguingobservable signatures of fractionalized excitations characteristic of theunderlying spin liquid phase. In particular, we obtain the explicit form of thephonon modes and construct the coupling Hamiltonians based on $D_{3d}$symmetry. We then systematically compute the Raman intensity and show that thespin-phonon coupling renormalizes phonon propagators and generates the salientFano linshape. We find that the temperature evolution of the Fano lineshapedisplays two crossovers, and the low temperature crossover shows pronouncedmagnetic field dependence. We thus identify the observable effect of theMajorana fermions and the $Z_2$ gauge fluxes encoded in the Fano lineshape. Ourresults explain several phonon Raman scattering experiments in the candidatematerial $\alpha$-RuCl$_3$.

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