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Topological magnon insulators in insulating Kagom\'e ferromagnets have beenextensively studied in a series of papers. It has been established thatDzyaloshinskii-Moriya interaction (DMI) is the key ingredient to observe anontrivial topological magnon with edge modes. However, insulatingantiferromagnets on the Kagom\'e lattice are frustrated systems considered as aplayground for studying quantum spin liquid physics. In these systems the DMIcan induce a coplanar but noncollinear magnetic orders with a $\mathbf{q}=0$propagating wavevector. We show that topological magnon bands are absent inthis coplanar spin texture in sharp contrast to collinear ferromagnets withDMI. Hence, geometrically frustrated Kagom\'e antiferromagnets can be deemedtopologically trivial. The presence of an out-of-plane magnetic field in thesefrustrated magnets induces noncoplanar spin textures exhibiting a nonzero spinscalar chirality. We show that the field-induced spin chirality providestopological magnon bands in Kagom\'e antiferromagnets without the need of DMIand survives in the chiral spin liquid phase of frustrated magnets. Possibleexperimentally material includes iron jarosite KFe$_3$(OH)$_{6}$(SO$_{4}$)$_2$.

Topological Magnon Insulator in NonCoplanar Kagom\'e Antiferromagnets: Supplemental Material of arXiv:1608.04561