At sufficiently low temperatures, magnetic materials often enter correlated phases hosting collective, coherent magnetic excitations such as magnons or triplons. Drawing on the enormous progress on topological materials of the past few years, recent research has led to new insights into the geometry and topology of these magnetic excitations. Berry phases associated with magnetic dynamics can lead to observable consequences in heat and spin transport, whereas analogs of topological insulators and semimetals can arise within magnon band structures from natural magnetic couplings. Magnetic excitations offer a platform to explore the interplay of magnetic symmetries and topology, drive topological transitions using magnetic fields, examine the effects of interactions on topological bands, and generate topologically protected spin currents at interfaces. In this review, we survey progress on all these topics, highlighting aspects of topological matter that are unique to magnon systems and the avenues yet to be fully investigated.

中文翻译：