  In this study, we propose that the signatures of spin fractionalization in
quantum magnets can be identified through a detailed analysis of the
temperature dependence of the asymmetric Fano lineshape of optical phonons
overlapping with a continuum of spin excitations. We focus on the
hyperhoneycomb magnet $\beta$-Li$_2$IrO$_3$, a promising candidate for being in
proximity to a three-dimensional Kitaev quantum spin liquid. The Raman response
in $\beta$-Li$_2$IrO$_3$ notably displays a distinctive asymmetric Fano
lineshape in the 24 meV Raman-active optical phonon. This asymmetry arises from
the interaction between the discrete phonon mode and the spin excitation
continuum, which could be fractionalized if the material is indeed near a
quantum spin-liquid phase. Our theoretical model considers the coupling of this
optical phonon to Majorana fermions in the Kitaev model on the hyperhoneycomb
lattice. Our findings reveal that the temperature-dependent Fano lineshape is
consistent with the fractionalization of spins into Majorana fermions and
$\mathbb{Z}{_2}$ fluxes.
