Acoustic magnons in the long-wavelength limit: Investigating the Goldstone violation in many-body perturbation theory

Collective spin excitations in magnetic materials arise from the correlated motion of electron-hole pairs with opposite spins. The pair propagation is described by the transverse magnetic susceptibility, which we calculate within many-body perturbation theory from first principles employing the full-potential linearized augmented-plane-wave formalism. Ferromagnetic materials exhibit a spontaneously broken global rotation symmetry in spin space leading to the appearance of acoustic magnons (zero gap) in the long-wavelength limit. However, due to approximations used in the numerical scheme, the acoustic magnon dispersion exhibits a small but finite gap at Gamma. We analyze this violation of the Goldstone mode and present an approach that implements the magnetic susceptibility using a renormalized Green function instead of the Kohn-Sham one. This much more expensive approach shows substantial improvement of the Goldstone-mode condition. In addition, we discuss a possible correction scheme, which involves an adjustment of the Kohn-Sham exchange splitting, which is motivated by the spin-wave solution of the one-band Hubbard model. The new exchange splittings turn out to be closer to experiment. We present corrected magnon spectra for the elementary ferromagnets Fe, Co, and Ni.