  We theoretically investigate magnon-phonon hybrid excitations in
two-dimensional ferromagnets. The bulk bands of hybrid excitations, which are
referred to as magnon-polarons, are analytically shown to be topologically
nontrivial, possessing finite Chern numbers. We also show that the Chern
numbers of magnon-polaron bands and the number of band-crossing lines can be
manipulated by an external magnetic field. For experiments, we propose to use
the thermal Hall conductivity as a probe of the finite Berry curvatures of
magnon-polarons. Our results show that a simple ferromagnet on a square lattice
supports topologically nontrivial magnon-polarons, generalizing topological
excitations in conventional magnetic systems.
