Exact Results for the Orbital Angular Momentum of Magnons on Honeycomb Lattices

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) (${\mathbf{k}}_1^{\ast }=(0,2\sqrt{3}\mathrm{/}9)2\pi \mathrm{/}a$, ${\mathbf{k}}_2^{\ast }=(1\mathrm{/}3,\sqrt{3}\mathrm{/}9)2\pi \mathrm{/}a,\dots$) are alternately $\pm 3\mathrm{\hslash }\mathrm{/}16$ 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 $3\mathrm{\hslash }\mathrm{/}8$ and 0 for the lower magnon band and $-3\mathrm{\hslash }\mathrm{/}8$ and 0 for the upper magnon band. For the AF honeycomb lattice, the values of the OAM at the corners of the BZ are $\mp (3\mathrm{\hslash }\mathrm{/}16)\kappa$ on one of the degenerate magnon bands and $\pm (3\mathrm{\hslash }\mathrm{/}8)(1+\kappa \mathrm{/}2)$ on the other, where $\kappa$ measures the anisotropy and the result is independent of the DM interaction.