  We theoretically study the intrinsic thermal Hall and spin Nernst effect in
collinear ferrimagnets on a honeycomb lattice with broken inversion symmetry.
The broken inversion symmetry allows in-plane Dzyaloshinskii-Moriya interaction
between the nearest neighbors, which does not affect the magnon spectrum in the
linear spin wave theory. However, the Dzyaloshinskii-Moriya interaction can
induce large Berry curvature in the magnetoelastic excitation spectrum through
the magnon-phonon interaction to produce thermal Hall current. Furthermore, we
find that the magnetoelastic excitations transport spin, which is inherited
from the magnon bands. Therefore, the thermal Hall current is accompanied by
spin Nernst current. Because the magnon-phonon interaction does not conserve
the spin, we also study the spin density induced by thermal gradient in the
presence of magnon-phonon interaction. We find that the intrinsic part of the
spin density shows no asymmetric spin accumulation near the boundary of the
system having a stripe geometry. However, because of the magnon-phonon
interaction, we find nonzero total spin density in the system having armchair
edges. The extrinsic part of the spin density, on the other hand, shows
asymmetric spin accumulation near the boundary for both armchair and zigzag
edges because of the magnon-phonon interaction. In addition, we find nonzero
total spin density in the system having zigzag edge.
