  We theoretically study magnon-phonon hybrid excitations (magnon-polarons) in
two-dimensional antiferromagnets on a honeycomb lattice. With an in-plane
Dzyaloshinskii-Moriya interaction (DMI) allowed from mirror symmetry breaking
from phonons, we find non-trivial Berry curvature around the anti-crossing
rings among magnon and both optical and acoustic phonon bands, which gives rise
to finite Chern numbers. We show that the Chern numbers of the magnon-polaron
bands can be manipulated by changing the magnetic field direction or strength.
We evaluate the thermal Hall conductivity reflecting the non-trivial Berry
curvatures of magnon-polarons and propose a valley Hall effect resulting from
spin-induced chiral phonons as a possible experimental signature. Our study
complements prior work on magnon-phonon hybridized systems without optical
phonons and suggests possible applications in spin caloritronics with
topological magnons and chiral phonons.
