Spurious Symmetry Enhancement and Interaction-Induced Topology in Magnons

Linear spin wave theory (LSWT) is the standard technique to compute the spectra of magnetic excitations in quantum materials. In this paper, we show that LSWT, even under ordinary circumstances, may fail to implement the symmetries of the underlying ordered magnetic Hamiltonian leading to spurious degeneracies. In common with pseudo-Goldstone modes in cases of quantum order-by-disorder these degeneracies tend to be lifted by magnon-magnon interactions. We show how, instead, the correct symmetries may be restored at the level of LSWT. In the process we give examples, supported by nonperturbative matrix product based time evolution calculations, where symmetries dictate that there should be a topological magnon gap but where LSWT fails to open up this gap. We also comment on possible spin split magnons in MnF$_2$ and similar rutiles by analogy to recently proposed altermagnets.