  We obtain exact results for the orbital angular momentum (OAM) of magnons at
the high symmetry points of ferromagnetic (FM) and antiferromagnetic (AF)
honeycomb lattices in the presence of Dzyallonshinskii-Moriya (DM)
interactions. For the FM honeycomb lattice in the absence of DM interactions,
the values of the OAM at the corners of the Brillouin zone (BZ) (${\bf
k}_1^*=(0,2\sqrt{3}/9)2\pi/a $, ${\bf k}_2^*=(1/3,\sqrt{3}/9)2\pi/a ,\ldots $)
are alternately $\pm 3\hbar /16$ for both magnon bands. The presence of DM
interactions dramatically changes those values by breaking the degeneracy of
the two magnon bands. The OAM values are alternately $3\hbar /8$ and 0 for the
lower magnon band and $-3\hbar /8$ and 0 for the upper magnon band. For the AF
honeycomb lattice, the values of the OAM at the corners of the BZ are $\mp
(3\hbar /16)\kappa $ on one of the degenerate magnon bands and $\pm (3\hbar /8)
(1+\kappa /2)$ on the other, where $\kappa $ measures the anisotropy and the
result is independent of the DM interaction.
