Magnon Orbital Angular Momentum of Ferromagnetic Honeycomb and Zig-Zag Lattices

By expanding the gauge $\lambda_n(k)$ for magnon band $n$ in harmonics of momentum ${\bf k} =(k,\phi )$, we demonstrate that the only observable component of the magnon orbital angular momentum $O_n({\bf k})$ is its angular average over all angles $\phi$, denoted by $F_n(k)$. For both the FM honeycomb and zig-zag lattices, we show that $F_n(k)$ is nonzero in the presence of a Dzyalloshinzkii-Moriya (DM) interaction. The FM zig-zag lattice model with exchange interactions $0<J_1< J_2$ provides a new system where the effects of orbital angular momentum are observable. For the zig-zag model with equal exchange interactions $J_{1x}$ and $J_{1y}$ along the $x$ and $y$ axis, the magnon bands are degenerate along the boundaries of the Brillouin zone with $k_x-k_y =\pm \pi/a$ and the Chern numbers $C_n$ are not well defined. However, a revised model with $J_{1y}\ne J_{1x}$ lifts those degeneracy and produces well-defined Chern numbers of $C_n=\pm 1$ for the two magnon bands. When $J_{1y}=J_{1x}$, the thermal conductivity $\kappa^{xy}(T)$ of the FM zig-zag lattice is largest for $J_2/J_1>6$ but is still about four times smaller than that of the FM honeycomb lattice at high temperatures. Due to the removal of band degeneracies, $\kappa^{xy}(T)$ is slightly enhanced when $J_{1y}\ne J_{1x}$.