Ao Zhang

Magnonic systems provide a fertile playground for bosonic topology¹, for example, Dirac^2,3,4,5,6 and Weyl^7,8 magnons, leading to a variety of exotic phenomena such as charge-free topologically protected boundary modes^6,7, the magnon thermal Hall effect⁹ and the magnon spin Nernst effect¹⁰. However, their understanding has been hindered by the absence of fundamental symmetry descriptions of magnetic geometries and spin Hamiltonians primarily governed by isotropic Heisenberg interactions. The ensuing magnon dispersions enable gapless magnon band nodes that go beyond the scenario of representation theory of the magnetic space groups^11,12, thus referred to as unconventional magnons. Here we developed spin space group^{13,14,15,16,17} theory to elucidate collinear magnetic configurations, classifying the 1,421 collinear spin space groups into 4 types, constructing their band representations and providing a comprehensive tabulation of unconventional magnons, such as duodecuple points, octuple nodal lines and charge-4 octuple points. On the basis of the MAGNDATA database¹⁸, we identified 498 collinear magnets with unconventional magnons, among which more than 200 magnon band structures were obtained by using first-principles calculations and linear spin wave theory. In addition, we evaluated the influence of the spin–orbit-coupling-induced exchange interaction in these magnets and found that more than 80 per cent are predominantly governed by the Heisenberg interactions, indicating that the spin space group serves as an ideal framework for describing magnon band nodes in most 3d, 4d and half-filled 4f collinear magnets.