Magnon band structure and topology from a symmetry perspective

The group theory of magnetic materials has a long history dating back to the early days of quantum mechanics and culminating in the relatively recently completed representation theory for the 1651 magnetic space groups. But we know from old work by Bill Brinkman and Roger Elliott that the magnetic space groups do not complete the set of symmetry groups relevant to magnetism. Instead we must consider the larger set of spin space groups. In this talk I discuss magnon band structures, how symmetries constrain and enrich their features, how the magnetic interactions feed into the relevant symmetries and I give examples of natural models with magnetically enhanced degeneracies, gapped topology and magnonic alternmagnetism.