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 $\mathrm{\Gamma }$. 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.

Figure 1

Representation of Eq. (4) for the transverse magnetic susceptibility in Feynman diagrams. Successive reinsertion of $R$ on the right-hand side and $\delta \mathrm{\Sigma }/\delta G=iW$ generates the ladder approximation.

Figure 2

DOS spectra for the bulk $3d$ transition metals iron, cobalt, and nickel. The upper and the lower panel show the majority and the minority spin channel, respectively. The Fermi level is set to zero.

Figure 3

Spin-wave spectra for Fe, Co, and Ni obtained with the LSDA (blue triangles), corrected LSDA (black circles), and COHSEX (red crosses) Green function as starting points. The spin-wave dispersion is shown along the high-symmetry line $\mathrm{P}\to \mathrm{\Gamma }\to \mathrm{N}$ and $\mathrm{L}\to \mathrm{\Gamma }\to \mathrm{X}$ for the bcc and fcc structures, respectively.

Figure 4

Spin-wave dispersions that have been calculated using two different Goldstone correction schemes: LSDA corr. as explained in the text and by scaling of $W$ as suggested in Ref. [16]. Experimental values taken from Refs. [57, 58, 59] are shown for comparison.