Abstract
Unlike the Berry phase, the orbital angular momentum (OAM) of magnons with two-dimensional wave vector $\mathbf{k}$ in band $n$ is not gauge invariant for arbitrary phase ${\ensuremath{\lambda}}_{n}(\mathbf{k})$ and so is not physically observable. However, by integrating the OAM over the orientation $\ensuremath{\phi}$ of wave vector $\mathbf{k}$, we construct a gauge-invariant function ${F}_{n}(k)$. Like ${F}_{n}(k)$, the average OAM for magnon band $n$ in a circle of radius $k$ is also gauge invariant and can be directly observed. We demonstrate these results for a ferromagnet on a honeycomb lattice with Dzyalloshinskii-Moriya interactions between next-nearest neighbor spins. With wave vectors $\mathbf{k}$ restricted to the first Brillouin zone, the angular averaged OAM ${F}_{n}(k)$ then has opposite signs for lower and upper bands $n=1$ and 2 for all $k$.
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