  The search for fractionalization in quantum spin liquids largely relies on
their decoupling with the environment. However, the spin-lattice interaction is
inevitable in a real setting. While the Majorana fermion evades a strong decay
due to the gradient form of spin-lattice coupling, the study of the phonon
dynamics may serve as an indirect probe of fractionalization of spin degrees of
freedom. Here we propose that the signatures of fractionalization can be seen
in the sound attenuation and the Hall viscosity. Despite the fact that both
quantities can be related to the imaginary part of the phonon self-energy,
their origins are quite different, and the time-reversal symmetry breaking is
required for the Hall viscosity. First, we compute the sound attenuation due to
a phonon scattering off of a pair of Majorana fermions and show that it is
linear in temperature ($\sim T$). We argue that it has a particular angular
dependence providing the information about the spin-lattice coupling and the
low-energy Majorana fermion spectrum. The observable effects in the absence of
time-reversal symmetry are then analyzed. We obtain the phonon Hall viscosity
term from the microscopic Hamiltonian with time-reversal symmetry breaking
term. Importantly, the Hall viscosity term mixes the longitudinal and
transverse phonon modes and renormalize the spectrum in a unique way, which may
be probed in spectroscopy measurement.
