Topology and surface states of spin waves

While the topology of electron band structures is well understood, a systematic framework is still missing from the topology of spin waves. By establishing a map from a spin wave (or magnon) system to an electron system, we classify the topology of magnon bands according to the unbroken symmetries of the magnetic orders. We demonstrate it in various magnetic orders, including colinear, coplanar and chiral ones, and discuss its application to certain magnetic materials.