  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 $J_{1y}\ne
J_{1x}$.
