Abstract
By expanding the gauge ๐๐โก(๐ค) for magnon band ๐ in harmonics of momentum ๐ค=(๐,๐), we demonstrate that the only observable component of the magnon orbital angular momentum ๐๐โก(๐ค) is its angular average over all angles ๐, denoted by ๐น๐โก(๐). Although ๐น๐โก(๐) vanishes for antiferromagnetic honeycomb and zigzag (0<๐ฝ1<๐ฝ2) lattices, it is nonzero for the ferromagnetic (FM) versions of those lattices in the presence of Dzyaloshinskii-Moriya interactions. For a FM zigzag model with equal exchange interactions ๐ฝ1โข๐ฅ and ๐ฝ1โข๐ฆ along the ๐ฅ and ๐ฆ axes, the magnon bands are degenerate along the boundaries of the Brillouin zone with ๐๐ฅโ๐๐ฆ=ยฑ๐/๐ and the Chern numbers ๐ถ๐ are not well defined. However, a revised model with ๐ฝ1โข๐ฆโ ๐ฝ1โข๐ฅ lifts those degeneracies and produces well-defined Chern numbers of ๐ถ๐=ยฑ1 for the two magnon bands. When ๐ฝ1โข๐ฆ=๐ฝ1โข๐ฅ, the thermal conductivity ๐ ๐ฅโข๐ฆโก(๐) of the FM zigzag lattice is largest for ๐ฝ2/๐ฝ1>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, ๐ ๐ฅโข๐ฆโก(๐) is slightly enhanced when ๐ฝ1โข๐ฆโ ๐ฝ1โข๐ฅ.
