Gauge-invariant measure of the magnon orbital angular momentum

Unlike the Berry phase, orbital angular momentum (OAM) of magnons with two-dimensional wavevector k in band n is not gauge invariant for arbitrary phase lambda_n(k). However, by integrating OAM over orientation $\phi $ k, we construct a gauge-invariant function F_n(k). Like F_n(k), average magnon circle radius also invariant. We demonstrate these results ferromagnet on honeycomb lattice Dzyalloshinskii-Moriya interactions between next-nearest neighbor spins. With wavevectors restricted to first Brillouin zone, averaged F_n(k) then has opposite signs lower and upper bands n=1 2 all k.