  Here we present a study of the phonon dynamics in the honeycomb Kitaev spin
model at finite temperatures. We show that the fractionalized spin excitations
of the Kitaev spin liquid, the itinerant Majorana fermions and static $Z_2$
fluxes, have distinct effects on the phonon dynamics, which makes the phonon
dynamics a promising tool for exploring the Kitaev spin liquid candidate
materials. In particular, we will focus on the signature of the fractionalized
excitations in the thermodynamic behaviour of the sound attenuation and the
phonon Hall viscosity: The former describes the phonon decay into the
fractionalized excitations, and the later is the leading order time reversal
symmetry breaking effect on the acoustic phonon. We find that the angular
dependence of the attenuation coefficient and its magnitude are modified by the
thermal excitation of the $Z_2$ fluxes. The strength of this effect strongly
depends on the relative magnitude of the sound velocity and the Fermi velocity
characterizing the low-energy Majorana fermions. We also show that the Hall
viscosity is strongly suppressed by the increase of the density of the $Z_2$
fluxes at finite temperatures. All these observations reflect the effects of
the emergent disorder on the Majorana fermions introduced by the $Z_2$ fluxes.
Our analysis is based on the complementary analytical calculations in the
low-temperature zero-flux sector, and numerical calculations in the
inhomogeneous flux sectors at intermediate and high temperatures with
stratified Monte Carlo (strMC) method.
