  Magnons and phonons are two fundamental neutral excitations of magnetically
ordered materials which can significantly dominate the low-energy thermal
properties. In this work we study the interplay of magnons and phonons in
honeycomb and Kagome lattices. When the mirror reflection with respect to the
magnetic ordering direction is broken, the symmetry-allowed in-plane
Dzyaloshinskii-Moriya (DM) interaction will couple the magnons to the phonons
and the magnon-polaron states are formed. Besides, both lattice structures also
allow for an out-of-plane DM interaction rendering the uncoupled magnons to be
topological. Our aim is to study the interplay of such topological magnons with
phonons. We show that the hybridization between magnons and phonons can
significantly redistribute the Berry curvature among the bands. Especially, we
found that the topological magnon band becomes trivial while the hybridized
states at lower energy acquire Berry curvature strongly peaked near the avoided
crossings. As such the thermal Hall conductivity of topological magnons shows
significant changes due to coupling to the phonons.
