Magnon spectrum of altermagnets beyond linear spin wave theory: Magnon-magnon interactions via time-dependent matrix product states vs. atomistic spin dynamics

The energy-momentum dispersion of magnons, as collective low-energy excitations of magnetic material, is computed from an effective quantum spin Hamiltonian but simplified via linearized Holstein-Primakoff transformations to describe noninteracting magnons. The dispersion produced by such linear spin wave theory (LSWT) is then plotted as ``sharp bands'' of infinitely long-lived quasiparticles. However, magnons are prone to many-body interactions with other quasiparticles -- such as electrons, phonons or other magnons -- which can lead to shifting (i.e., band renormalization) and broadening of ``sharp bands'' as the signature of finite quasiparticle lifetime. The magnon-magnon interactions can be particularly important in antiferromagnets (AFs), and, therefore, possibly in newly classified altermagnets sharing many features of collinear AFs. Here, we employ nonperturbative quantum many-body calculations, via time-dependent matrix product states (TDMPS), to obtain magnon spectral function for RuO$_2$ altermagnet whose effective quantum spin Hamiltonian is put onto 4-leg cylinder. Its upper band is shifted away from upper ``sharp band'' of LSWT, as well as broadened, which is explained as the consequence of {\em hybridization} of the latter with three-magnon continuum. This implies that two-magnon Raman scattering spectra {\em cannot} be computed from LSWT bands, which offers a litmus test for the relevance of magnon-magnon interactions. Finally, we employ atomistic spin dynamics (ASD) simulations, based on classical Landau-Lifshitz-Gilbert (LLG) equation, to obtain magnon spectrum at finite temperature and/or at a fraction of the cost of TDMPS calculations. Despite including magnon-magnon interactions via nonlinearity of LLG equation, ASD simulations {\em cannot} match the TDMPS-computed magnon spectrum, thereby signaling {\em nonclassical} effects harbored by AFs and altermagnets.