  Topological magnons have received substantial interest for their potential in
both fundamental research and device applications due to their exotic uncharged
yet topologically protected boundary modes. However, their understanding has
been impeded by the lack of fundamental symmetry descriptions of magnetic
materials, of which the spin Hamiltonians are essentially determined by the
isotropic Heisenberg interaction. The corresponding magnon band structures
allows for more symmetry operations with separated spin and spatial operations,
forming spin space groups (SSGs), than the conventional magnetic space groups.
Here we developed spin space group (SSG) theory to describe collinear magnetic
configurations, identifying all the 1421 collinear SSGs and categorizing them
into four types, constructing band representations for these SSGs, and
providing a full tabulation of SSGs with exotic nodal topology. Our
representation theory perfectly explains the band degeneracies of previous
experiments and identifies new magnons beyond magnetic space groups with
topological charges, including duodecuple point, octuple nodal line and
charge-4 octuple point. With an efficient algorithm that diagnoses topological
magnons in collinear magnets, our work offers new pathways to exploring exotic
phenomena of magnonic systems, with the potential to advance the
next-generation spintronic devices.
