Electron–plasmon and electron–magnon scattering in ferromagnets from first principles by combining GW and GT self-energies

This work combines two powerful self-energy techniques: the well-known GW method and a self-energy recently developed by us that describes renormalization effects caused by the scattering of electrons with magnons and Stoner excitations. This GT self-energy, which is fully k-dependent and contains infinitely many spin-flip ladder diagrams, was shown to have a profound impact on the electronic band structure of Fe, Co, and Ni. In the present work, we refine the method by combining GT with the GW self-energy. The resulting GWT spectral functions exhibit strong lifetime effects and emergent dispersion anomalies. They are in an overall better agreement with experimental spectra than those obtained with GW or GT alone, even showing partial improvements over local-spin-density approximation dynamical mean-field theory. The performed analysis provides a basis for applying the GWT technique to a wider class of magnetic materials.


INTRODUCTION
Many-body spin excitations in interacting electron systems are complex quantum mechanical processes that are fundamental for the description of the properties of magnetic materials. In the field of spintronics, the interaction between conduction electrons and collective spin excitations is critical for explaining the temperature dependence of resistivity and magnetotransport 1 . There is experimental evidence that the scattering of propagating electrons and holes on collective and single-particle spin excitations in magnetic materials lead not only to a renormalization of the quasiparticle band dispersion but also to the appearance of characteristic band anomalies in the quasiparticle spectra measured by the angleresolved photoemission spectroscopy (ARPES) 2-6 , a claim that was very recently corroborated by a common theoretical and experimental study of photoemission in iron 7 . Furthermore, spin fluctuations are considered as one of the key mechanisms to mediate the Cooper pairing in high-T c superconductors such as cuprates and iron-based pnictides [8][9][10] .
The elementary ferromagnets Fe, Co, and Ni are prototypical model systems to study the effects of the electron-magnon interaction. Progress in photoemission spectroscopy made it possible to obtain ARPES spectra in high energy and momentum resolution, sufficient to resolve anomalies that may appear in the quasiparticle band dispersion as indicators of genuine many-body scattering processes, such as magnon-induced renormalization effects. This opens an important source of information for theorists to develop and test new theoretical approaches.
Although Kohn-Sham (KS) spin-density-functional theory (DFT) is a well-established and powerful tool to describe magnetic materials 11 , it is limited to groundstate properties in practice, unable to yield accurate excitation energies and lifetime broadenings, which would be relevant for the interpretation of photoemission spectra. Nevertheless, it does provide a reference system on which more advanced first-principles approaches are based. One of these methods is the dynamical mean-field theory (DMFT), which maps the interacting many-body system onto a minimal Anderson impurity model 12,13 containing only a small number of electrons, making it possible to include correlation effects to high orders. However, DFT+DMFT suffers from several restrictions: the self-energy is strictly local, its k dependence is therefore lost. The electron-electron interaction is described by parameters (U and J), which are often fitted to experiment or treated as free parameters, limiting the predictive power of the method, and there is a double-counting problem related to the treatment of exchange and correlation between DMFT and the DFT reference. Therefore, we resort to an alternative theoretical framework, the many-body perturbation theory (MBPT), which is not subjected to these restrictions. Practical calculations rely on an approximation to the electronic self-energy Σðr; r 0 ; ωÞ, which describes the manybody exchange and correlation processes that an electron or hole experiences while propagating through a material 14 . The widely used GW self-energy approximation 15 is one such approximation. It has been applied to Fe, Co, and Ni [16][17][18] and yielded results in better agreement with the experimental values for the bandwidths and the effective masses than KS DFT. Importantly, the GW self-energy predicts the bands to be broadened due to lifetime effects. The predicted broadening is, however, way too small compared with what is seen in photoemission spectra. Furthermore, the theoretical spectra do not exhibit any band anomalies, of which there is experimental evidence.
In a recent publication 19 , we showed that a different approximation, which we called GT self-energy, improves on these shortcomings. The GT self-energy describes the scattering of electrons with spin waves and single-particle spin-flip excitations and is thus complementary to GW, in which the scattering processes do not involve a spin-flip. Collective spin waves and single-particle excitations are described jointly by a magnetic T matrix, constructed from solutions of a spin-flip Bethe-Salpeter equation (BSE). Applying the GT self-energy to the elementary present the k-resolved GWT spectral functions and analyze emergent band dispersion anomalies.
The GWT method has been implemented in the SPEX code 18 within the framework of the full-potential linearized augmentedplane-wave method, which provides a highly accurate basis set 23 for all-electron calculations, equally suitable for itinerant s states as for localized d states, making the LAPW basis the optimal basis set for the treatment of transition elements. The GW method employs a mixed product basis 18,24,25 , derived from the LAPW basis. The GT method, on the other hand, employs a Wannier basis set for the representation of the T matrix, a quantity that depends on four points in space. This spatial representation allows an efficient truncation of the short-range screened interaction without sacrificing numerical accuracy.
The article is organized as follows. The theory and the relevant computational details are outlined in the Methods section. The results of applying the GW, GT, and GWT self-energies to the electronic structure of bcc Fe, fcc Co, and fcc Ni are presented in the Results section. We analyze the momentum-and energydependent quasiparticle spectral functions and discuss the effects of the combined action of the spin-flip and non-spin-flip scattering processes. As part of our analysis, we provide a detailed comparison between the results obtained with different theoretical approaches and the available experimental values for the magnetic moments, exchange splittings, and d bandwidths. The Discussion section concludes the paper with a summary.

RESULTS
Band structure of iron Figure 1 presents, for iron, the momentum-and energy-resolved spectral function [Eq. (8)] along with P-Γ-N, as obtained from the three self-energies GW, GT, and GWT. For comparison, the LSDA bands are included as red lines. Owing to the explicit electron-electron scattering described by the different selfenergies, all renormalized band structures exhibit lifetime broadening of the line widths, however of a rather different magnitude.
As seen in the top panels, the GW bands exhibit a well-defined quasiparticle character with the least broadening among the selfenergies. The line widths are small close to the Fermi energy and grow towards higher (positive and negative) energies, whereas there is not much difference in lifetime broadening between the two spin channels in contradiction to the experimental observations 3 .
The picture changes drastically with the GT self-energy. The lifetime broadening still vanishes at the Fermi energy as it must, but, from there, it grows very rapidly, particularly in the majority spin channel, where, in the binding energy range 1-2 eV (negative energies in the plot), the bands disappear completely. Figure 2 shows the renormalized band structure along Γ-N on a smaller energy range and with measured photoemission peak positions. Indeed, between −1 eV and −2 eV there are no experimental points, just where the theoretical spectrum shows the loss of quasiparticle character. This is in marked contrast to the GW quasiparticle bands, which remain well-defined throughout this energy range. Similar suppression of quasiparticle character can be seen, albeit to a much lesser degree, in the spin-down channel above the Fermi energy at~0.5 eV. The lifetime broadening decreases again for larger energies, and the quasiparticle bands reappear. We see a much stronger difference in lifetime broadening between the two spin channels than in GW. One can also observe a similar difference between particle and hole (empty and filled) states. Both observations can be explained by the spin selectivity of the electron-magnon scattering 19 , which predicts the strongest renormalizations to occur for occupied spin-up and unoccupied spin-down states. The spin asymmetry of the line broadening is in accordance with experimental measurements 3,7 and also with LSDA+DMFT calculations 26,27 .
Combining the two self-energies in GWT, we see that the main features of GT remain intact. In addition, we observe an important renormalization effect originating from the GW self-energy: one of the bands that cross the Fermi energy in LSDA is pushed below the Fermi energy and never crosses it. This effect is seen in both spin channels and for GW and GWT but not for GT. In fact, the consequential missing Fermi surface sheets (forming a hole pocket in the spin-down case) are confirmed in experimental ARPES measurements, where such Fermi surface sheets are not found 28,29 . Moreover, Fig. 2 corroborates this finding, as it does not show any photoemission peaks that would indicate the bands in question to cross the Fermi energy. It does show a photoemission peak position close to the N point that is in favor of the spin-down band to remain below the Fermi energy. This is also consistent with recent quasiparticle self-consistent GW calculations 30,31 .
It is interesting to note that for the respective spin-up band to show a physically correct band dispersion, it is important to take off-diagonal elements 32 of the GW self-energy into account according to Eq. (8) because of strong hybridization effects (that are different in GW and LSDA). With the usual GW approach of considering only diagonal elements, the bands show unphysical dispersion anomalies, see Fig. 3, where we zoom into the respective energy region along the Γ-N path. The correct dispersion is fully recovered when the 'full' self-energy matrix (diagonal+off-diagonal elements) is taken into account. The reason for this behavior can be understood from comparing the avoided crossings of the (red) LSDA and the (green) GW (full) bands close to N. They clearly take place at different momenta. As a consequence, the standard procedure of approximating the quasiparticle wave functions by the KS wave functions [GW (diag)] breaks down at that point, and we are forced to calculate the quasiparticle wave function explicitly by diagonalization of the quasiparticle equation [GW (full)]. We also see from Fig. 3 that, for the most part, the two GW approaches give nearly identical results. This is also true for the spin-down channel of iron as well as for cobalt and nickel. Therefore, we have employed GW (full) only for the band structure of iron.
Comparing different self-energies in Fig. 4, it might be surprising at first glance that the combined GWT self-energy seems to show less intensity and lifetime broadening than GT, whereas one might expect the lifetime effects to increase, rather than decrease, when more diagrams, i.e., more scattering processes, are included. On the one hand, the observation is not completely accurate. For binding energies larger than~3 eV we clearly see that the lifetime broadening of GW dominates, making the GW and GWT band structures look nearly the same at  larger energies, whereas the GT bands become very sharp there. On the other hand, we do see an apparent decrease in the peak broadening at lower binding energies. The bands are narrower again, and certain features become less pronounced, though without disappearing altogether, see, e.g., the spin-down band anomaly just above the Fermi energy. To understand this seeming contradiction, we have to remember that not only do the imaginary parts of the two self-energies add up but also their real parts. The imaginary part gives rise to the line broadening, whereas the self-energy's real part shifts the quasiparticle bands (more generally, the spectral function) energetically. The GW self-energy has a rather sizable real part, which, in the present case, shifts the quasiparticle bands away from the energies where the self-energy's imaginary part is large. This detuning effect can ultimately lead to weaker lifetime effects in the GWT spectral function.
Comparing the computed band structures with experimental measurements in Fig. 2, one notices that the best agreement is achieved with the GWT self-energy. We thus see the strengths of GW and GT gainfully combined in the GWT approach. It is particularly noteworthy that GWT yields a flat majority band at a binding energy of 2.34 eV, nicely matching with several photoemission peak positions at this energy. Interestingly, in LSDA +DMFT 33 this part of the spectrum is completely washed out, offering no explanation for the experimentally measured points.
The inclusion of the electron-magnon scattering does not only lead to pronounced lifetime effects, but it can also give rise to dispersion anomalies in the quasiparticle bands. One such anomaly is a waterfall (kink) structure in the dispersion of a spin-down band of Fe along Γ-H at a binding energy of~1.5 eV, caused by electron scattering with virtual Stoner excitations. It was recently predicted by us 7,19 and subsequently confirmed experimentally in high-resolution ARPES 7 . In Fig. 4, the anomaly is shown together with the respective photoemission data. The GW renormalization (top panel) moves the band towards the experimental points but does not show any anomaly. The GT renormalization alone (middle panel) affects the formation of a kink structure, in agreement with the experiment, albeit at a slightly lower momentum and energy. Furthermore, the shape of the kink appears exaggerated in GT compared with the experimental points. If the two renormalizations are combined in the GWT method (lower panel), both the shape and location of the dispersion anomaly improve significantly. The band anomaly gets flattened but does not disappear, its shape being in very good agreement with the experimental observation.
Comparing the different renormalized band structures of Fig. 1 reveals a curious and unexpected behavior: although the GW and GT quasiparticle bands (where they are well-defined) are clearly different from the LSDA band structure (red lines), we find the GWT quasiparticle bands surprisingly close to the LSDA bands again. First, we have to emphasize that this is not a general behavior. Even in the same material, a clear difference between GWT and LSDA is seen along a different k path, Γ-H (Fig. 4). However, the observation hints at a compensating effect that we will encounter again later-on and that deserves to be investigated in more detail.
We do this here by an analysis of the renormalization of the Fe Γ 25 states at (LSDA) binding energies of 2.19 eV (spin-up) and 0.42 eV (spin-down). Figure 5 presents the GW, GT, and GWT selfenergies together with the resulting spectral functions, see the caption for the exact definitions. We also plot the function ω À ϵ σ kn (dotted lines). Following the Methods section, the spectral function exhibits the main quasiparticle peak approximately where this linear function intersects with the real part (solid lines) of the respective self-energy. (It is approximate because the imaginary part of the self-energy also has a role regarding the exact peak position-as would the off-diagonal self-energy matrix elements, which are neglected in the present analysis, though.) In the following, we focus on the more interesting spin-up case. At ϵ σ kn (the node of the linear function), we see that the real parts of the GW and GT self-energies have approximately the same magnitude but opposite signs. Their sum (green solid line) is thus close to zero and its intersection with the linear function is nearly at the KS energy ϵ σ kn . One also notices that the rate of change of the two real parts is nearly the same but, again, of the opposite sign so that their sum stays in a relatively small window over a quite large energy range (from −2.5 to −0.5 eV). In this range, it exhibits some oscillatory structure, which clearly stems from GT, whereas GW is rather featureless in the displayed energy range. We can say that while GW dominates the overall behavior, it is the GT self-energy that provides the fine structure.
Similarly, it is the GT contribution that is predominantly responsible for the lifetime effects in the respective energy range. They are nearly an order of magnitude larger than in GW and cause an extreme broadening of the quasiparticle states as represented in the corresponding spectral function plot ( Fig. 5c-d). However, the imaginary part of GT falls off quickly to either side so that it can be seen as a modulation of the smooth parabolic-like behavior of the imaginary part of GW. It is the GW self-energy that prevails at large binding energies and prevents the imaginary part of GWT from becoming negative as happens in the case of the GT self-energy below −4 eV. Such a forbidden sign change can be traced back to a violation of the time ordering and, thus, to a violation of causality. We will later come back to this point.
The GWT quasiparticle peak is located between the ones from GT and GW, but its intensity is strongly suppressed. This is because, at the position of the peak, the imaginary part of GWT happens to be large, contrary to the cases of GW and GT. Here, we have an example where the lifetime broadening in GWT is not simply the sum of the broadenings from GW and GT. The GWT spectral function, similarly to the GT case, shows two side peaks at~−2 and −1 eV, which bear resemblance to satellite replicas. However, in GWT they appear rather as extended shoulders to the main peak than as two distinct peaks as in GT. Considering the strong suppression of the GWT quasiparticle peak, one might wonder where the quasiparticle weight goes to. The shrinking of the peak is accompanied by a redistribution of weight around the peak, for example, forming the two shoulders. It indeed turns out that the (numerically) integrated areas under the GW and GWT peaks, despite their different shapes, are very nearly identical. On the other hand, the integral under the GT peak is larger (as is obvious to the naked eye) and very close to one. As a consequence, the GW and GWT peaks integrate to less than one in the presented energy range, namely to~0.6, which leads to the question: where are the remaining 40%? The question is resolved when one extends the plot to a much larger energy scale: The GW (and GWT) spectral function displays a very long tail towards larger binding energies where it additionally exhibits a shallow peak (plasmon satellite). If this long tail is fully taken into account, the spectral function, in fact, correctly integrates to one. The long tail can be attributed to the large plasmon energy, present in GW and GWT but not in GT, in which the much smaller magnon energy sets the scale.
The spin-down spectra are less exotic than their spin-up counterparts. The GT quasiparticle peak is moderately renormalized to lower energy and has a larger width due to the higher value of the imaginary part of the self-energy with respect to the GW peak. The location of the GWT peak is close to the one from GW. The peak broadening is intermediate between that of GW and GT, another example that lifetime broadenings do not simply sum, even if the self-energies do.
The renormalized band structures of Fig. 2 look remarkably similar to respective results from LSDA+DMFT 26,27,33 . However, there are differences in the details: (1) we have already commented on the flat majority quasiparticle band at −2.34 eV in GWT, which nicely agrees with photoemission peak positions at this energy. LSDA+DMFT, on the other hand, does not show a distinct feature there, which led the authors of ref. 33 to suspect the peaks to be a surface effect. Similarly, in a plot for Γ − P − H (not shown), we find a flat band at P~−3.2 eV, which loses intensity quickly towards Γ but persists nearly halfway to the H point. For this k range and energy, one indeed observes peaks in ARPES, whereas the LSDA+DMFT ('LDA++') spectrum is basically featureless there 26 . (2) We observe band anomalies caused by electron-magnon scattering, e.g., at~−1.5 eV in a minority hole band and another at 0.5 eV in a minority electron band. The former is seen in the experiment. We attribute the absence of both effects [(1) and (2)] in DMFT to the spatial restriction imposed by the Anderson impurity. The impurity cannot host extended excited states such as plasmons or spin waves. So, instead of as well-defined many-body states, they rather appear as an incoherent background in the excitation spectrum. (3) LSDA and LSDA+DMFT (and also GT) predict extra Fermi surface sheets (a hole pocket in the minority channel) close to the N point, which, according to all available experimental evidence 28,29 , do not exist. GW and GWT pushes the respective bands below the Fermi energy, making the spurious sheets disappear. With all these being said, it should be noticed that GWT overestimates the binding energies of the photoemission peak positions in the minority channel. This overestimation is stronger than in LSDA+DMFT. However, one should keep in mind that the present GWT approach is free of any adjustable parameters, while the LSDA+DMFT method relies on the U parameter, which was chosen in the cited works so as to maximize the agreement of the binding energies with the experiment.
In ref. 19 , we presented color plots for the imaginary part of the GT self-energy of iron as a function of crystal momentum and frequency. It is the k dependence of the self-energy that is one of the fundamental differences between MBPT and DMFT approaches (present in MBPT but missing in DMFT). The addition of the GW self-energy into GWT gives rise to some important changes and improvements. Figure 6 shows the imaginary part of the GWT selfenergy expectation value Σ σ kn ðω À Δ v Þ ¼ hφ σ kn jΣðω À Δ v Þjφ σ kn i as a function of k and ω for six low-lying bands n of Fe and both spins.  We use the same plot arrangement as in ref. 19 (Figs. 10 and 11 therein). The plots show a striking similarity to the GT plots in a small energy range around the Fermi energy (~−2 eV < ω < 2 eV), again demonstrating that the electron-magnon scattering (GT) dominates for small binding energies with intensive peaks below (above) the Fermi energy for the spin-up (spin-down) channel.
The main observations made about the plots in ref. 19 hence apply here, too, and we do not want to repeat them.

Fig. 6
Momentum and energy dependence of the correlation self-energy in Fe. Im Σ σ kn ðω À Δ v Þ is shown for the lowest bands and both spins (cf. Fig. 1). The bands are ordered with respect to their energy at Γ from bottom to top (same order as in Fig. 10 of ref. 19 ). D. Nabok et al. However, at larger binding energies, GW takes over, and here the GWT plots clearly deviate from the GT ones. One of the most important amendments brought about by including GW is seen for larger binding energies in the spin-up channel: the imaginary part of the GT self-energy becomes negative at around ω = −4 eV. This is perhaps not so readily seen in Fig. 10 of ref. 19 because the plots show the absolute value of the imaginary part. However, the sign change reveals itself by white lines traversing the diagrams from left to right, marking the regions where the imaginary part vanishes. This change of sign is unphysical. Causality demands the imaginary part of the self-energy to be non-negative (non-positive) for ω < 0 (ω > 0). In ref. 19 , we expressed the expectation that the inclusion of GW diagrams would restore the correct sign. Indeed, as is evident from Fig. 5, no unphysical sign change is observed at larger binding energies. So, it is the GW self-energy that provides the lifetime effects necessary to rectify the picture. (Had we chosen a larger energy scale towards positive ω in Fig. 10 of ref. 19 , we would see an analogous effect in the spin-down channel.) Another noteworthy effect can be observed in the third plot from the top (spin-up and spin-down). These two plots pertain to bands that cross the Fermi surface twice. (The crossings are about halfway between P and Γ as well as close to the N point in the spin-up case, and symmetrically and close to Γ for spin-down.) In the GWT plots, one can make out step-like features that go all the way from top to bottom just where the bands cross the Fermi energy. Such features are not found in the corresponding GT plots. Closer inspection shows that the step-like feature is not a discontinuous step. It is smoothed out to some extent but remains discernible as a step nonetheless. It turns out that this step has the same mathematical origin as the well-known logarithmic divergence of the group velocity in metallic Hartree-Fock (HF) bands where they cross the Fermi energy. The smoothing of the step has a logarithmic form (with a logarithmic divergence). GWT (and GW) quasiparticle bands, on the other hand, do not exhibit a logarithmic divergence because there is no divergence right at the Fermi energy (ω = 0). A more-detailed discussion of the logarithmic divergence in GW together with its efficient numerical treatment will be presented elsewhere.

Band structure of cobalt and nickel
To complete the series of elementary ferromagnets, we have also carried out GW, GT, and GWT calculations for fcc cobalt and fcc nickel. The corresponding renormalized band structures are compared in Figs. 7 and 8. Cobalt and nickel have fewer empty d states than iron. As a consequence, the phase space of spin excitations is smaller and, hence, we observe a considerably smaller impact of electron-magnon scattering on the electronic structure than in iron. The reduced effect of electron-magnon scattering is particularly evident in nickel, where the GWTrenormalized majority and minority band structures look identical to the GW ones. They are, of course, not exactly identical, but, by eyesight, it is hard to see any difference. This might be surprising since the GT band structures clearly show broadening and energy renormalization. Here, we see an effect that we have already encountered before and that we called a detuning effect: the (comparatively) large real part of the GW self-energy shifts the quasiparticle energy into a region where the GT self-energy is small and where GW dominates. In ref. 19 , we discussed weak magnon satellite features in nickel at binding energies of~0.8 and 1.7 eV. Fig. 8 shows that the GWT calculation washes these features out so that they are not visible anymore in the color plot. However, the satellites are still present in the raw data as shoulders to the main quasiparticle peaks (not shown), but certainly too small to be detected in the experiment.
Similar to nickel, there is hardly any difference between the spin-down GW and GWT band structures of cobalt. Compared  with that, however, we observe a strong lifetime-broadening effect in the spin-up channel from just below to the Fermi energy down to binding energy of~3 eV: the GWT bands are significantly broader than in the case of GW, though not completely losing their quasiparticle character as in iron. The band energies, on the other hand, turn out to be not very different from the GW ones. Furthermore, we see an effect in Co that had gone unnoticed when we collected the results for our previous publication 19 . The GT renormalization produces a tiny spin-up hole pocket at the X point that is neither present in LSDA nor in GW. Renormalizing with the combined GWT self-energy pushes the band in question back below the Fermi energy, and the hole pocket disappears again.

Tables
One of the most important quantities to characterize a spinpolarized system is its magnetic moment. The spin magnetic moment is directly given by an integration over the spin density and is, as such, perfectly computable by spin-density functional theory. In fact, the LSDA magnetic moments of iron, cobalt, and nickel are very close to the experimental results as shown in Table 1, so close that they are hard to match by any theory. (The moments are up to 0.05 μ B lower than in a previous publication of ours 34 . The small deviations are due to different basis sets used in the two studies. The present study employs a more precise basis for the wave functions.) The table also lists the respective magnetic moments obtained with the GW, GT, and GWT methods. Two kinds of GW results are given. One includes-for consistency with the present GT and GWT calculations-the Δ x correction (see Methods) for the reference system, and the other does not. The latter value is the one that should be compared to GW values from the literature. We want to emphasize at this point that, as explained in the Methods section, the Δ x correction is applied to the mean-field reference system but later subtracted in the solution of the Dyson equation. One can say that the two GW values are based on two different starting points, and their difference shows the well-known starting point dependence of GW. We will come back to this point at the end of this section. Neither of the two GW magnetic moments improves on the LSDA values. The ones in the brackets are systematically larger, the others nearly identical to LSDA. The GT moments, on the other hand, quite strongly underestimate the experimental values, far more than LSDA overestimates. Finally, the highest-level theory, the GWT approach, represents a clear improvement over LSDA. We find magnetic moments in very good agreement with the experimental values for iron and cobalt. Nickel is the only exception with a magnetic moment nearly identical to the LSDA value.
When comparing the effect of the different self-energies on the band structures of Fe, Co, and Ni (Figs. 1, 7, and 8), we notice a common behavior: GW shifts the d bands up and contracts them, giving rise to smaller d bandwidths, whereas GT tends to increase the bandwidth. In fact, earlier GW studies [16][17][18] of the simple ferromagnets reported that the d bandwidths shrink upon renormalization, bringing them closer to experiment for some of the materials. Since we now seem to have a competing effect by the GT self-energy, it is interesting to investigate the bandwidths obtained with the different self-energies. Table 2 presents a comparison of our values with experimental and theoretical results from the literature. For consistency, we determine the d bandwidth in the same manner as in the experimental studies, namely based on the electron binding energies taken at specific critical points (see footnote 33 of ref. 35 ), N 1 for bcc Fe and L 1 for fcc Co, and Ni. The d bandwidths of Fe are given for both spins separately because of the large exchange splitting in this material, whereas the values for Co and Ni are spin-averaged. In agreement with previous studies, we find the GW d bandwidths to be consistently smaller than the LSDA values, thus improving the agreement with the experiment for Co and Ni but leading to underestimated values for Fe. The GT bandwidths, in contrast, tend to be larger than in LSDA. This opposite effect partially compensates for the d-band shrinkage caused by GW in the results obtained from the combined GWT self-energy. In this way, the iron GWT bandwidths now end up very close to the experiment, showing the best agreement among the theoretical methods. In the case of Ni, an early GW study 16 already demonstrated that the d bandwidth decreases by the selfenergy renormalization, bringing it in close proximity to the experimental value. As we have seen before, the GT self-energy is marginal in nickel. As a consequence, the bandwidths obtained from GW and GWT are nearly identical. For cobalt, although closer to the experiment than LSDA, there is still a significant overestimation in both GW and GWT. On the one hand, this might be due to different crystal structures, hcp in experiment vs. fcc in the present study. On the other hand, the corresponding experimental value is given with a large inaccuracy of measurement, and the GWT bandwidth is only just outside the error range. Furthermore, the authors of ref. 36 , whereas not giving an explicit value, maintain that the bandwidth is larger than 4 eV, in accordance with all theoretical values. Table 3 presents the exchange splittings of several valence states at special k points, for which experimental measurements are available. LSDA mostly overestimates the experimental exchange splittings, reaching a maximal difference of 0.81 eV in the case of cobalt. GW yields exchange splittings systematically smaller than the ones from LSDA. In particular, with the Δ x correction in the reference system (values without brackets), GW underestimates nearly all experimental splittings, in some cases quite strongly. We find the opposite trend for the GT exchange splittings. They systematically overestimate the corresponding experimental values. So again, the GWT method is seen to benefit from a compensating effect between GW and GT, and the respective exchange splittings turn out to be close to the   Ref. 16  The cases of cobalt and nickel deserve a second look. LSDA is known to overestimate the exchange splittings in these materials rather strongly 16,37 . In nickel, the LSDA value is larger by about a factor of two. From the values in the table, we see that the standard GW renormalization (values in brackets) corrects the values in the right direction but only by a small step. Applying GW on the Δ x -corrected reference system, however, yields exchange splittings close to the experiment. The GWT splittings are nearly identical to the GW ones because the GT self-energy has only a minor role in nickel. This is different in the case of cobalt. While the GW correction is seen to have an effect similar to that of Ni, the exchange splittings change significantly by the GT renormalization. With GW alone, deviations are −0.22, 0.35, and −0.15 eV, and after applying GT (in GWT) they are reduced to merely −0.02, 0.19, and 0.03 eV, respectively.
Finally, we want to have a closer look at the two types of GW values shown in Tables 1-3. Both types rely on one-shot GW approaches but are based on two different reference systems, the standard LSDA mean-field system, which however violates the Goldstone condition 14,38 , and the same system where the spin-up and spin-down eigenenergies are shifted with respect to each other with a universal parameter Δ x (see Methods). In ref. 34 , we showed that a self-consistent Coulomb-hole screened-exchange (COHSEX) calculation leads to a change in the exchange splitting that can effectively be mimicked by the parameter Δ x used in the present work. In this way, besides its original purpose to enforce the Goldstone condition, the parameter can also be interpreted as simulating the missing self-consistency of the one-shot approach. It is important to emphasize that Δ x is thus introduced and determined to fulfill a physical constraint, and no tuning or fitting between theory and experiment is involved.
The difference between the two GW exchange splittings in Table 3 shows an interesting correlation: for all systems, it amounts approximately to Δ x /2, which is just half the difference between the underlying two reference systems (Δ x by definition). To understand this correlation, we write the quasiparticle energy (see the last paragraph of Methods) in a simplified way as where we have suppressed Δ v and the band and k dependencies. Furthermore, we have only retained the diagonal self-energy matrix elements here (whereas the full self-energy matrix, including off-diagonal elements, has been taken into account in the one-shot GW calculations for the iron band structure). Making another standard approximation by linearizing the self-energy around ϵ σ eliminates the quasiparticle energy from the right-hand side and gives E σ ¼ ϵ σ þ Z σ ½Σ σ ðϵ σ Þ À v σ xc with the renormalization factor Z σ ¼ ½1 À dΣ σ ðωÞ=dωj ϵ σ À1 . We are interested in the difference of the exchange splitting ΔE x = E ↑ − E ↓ between the two GW calculations where the tilde denotes quantities of the GW calculation based on the Δ x -corrected reference system. We have used the approximation Z ¼Z " ¼Z # ¼ Z " ¼ Z # (found to be valid in the present cases) and the fact thatṽ " . We find the second term of Eq. (1) to be significantly smaller (by a factor of three to six) than the first term, which thus dominates. This, together with the fact that Z ≈ 0.5 for d states, explains the observed behavior This leaves us with the question of why the second term of Eq. (1) is small compared with the first, which might appear particularly surprising in light of the fact that the GW approximation contains the HF exchange potential Σ σ HF (the subscript commonly used for the HF self-energy is 'x', however, we use 'HF' here to avoid confusion with the subscripts in Δ x and the exchange splitting ΔE x ), and the (one-shot) HF exchange splittings are several times larger than the ones from LSDA, so typically Σ " Further, the 'difference of differences'Σ " HF ÀΣ # HF À Σ " HF þ Σ # HF [cf. the right side of Eq. (1)] is typically in the order of Δ x , reaching a multiple of it for some states, and is thus certainly not smaller than (1−Z)Δ x in general. It turns out that the corresponding term in GW is much smaller because of a rather effective cancellation betweeñ Σ σ HF À Σ σ HF andΣ σ c À Σ σ c (separately for σ = ↑ and σ = ↓), where the subscript 'c' refers to the correlation part of the self-energy. The latter can formally be written as Σ c = iG(W − v). Because of metallic screening, we have W ≪ v, and the dominant term − iGv cancels with the HF self-energy Σ HF = iGv. This explains the relative smallness of the second term on the right-hand of Eq. (1) with respect to the first term.

DISCUSSION
In summary, we implemented a full frequency-and k-dependent (non-Hermitian) non-local self-energy that combines the GW with the GT (T-matrix) diagram. These diagrams describe electron-plasmon and electron-magnon scattering processes, respectively. The GWT self-energy (as we call it) is doublecounting free by construction and does not rely on adjustable parameters. It has been implemented in the SPEX code, which is based on the LAPW basis set. As the first application of this method, we studied the electronic properties of the elementary ferromagnets bcc Fe, fcc Co, and fcc Ni and compared with results from LSDA, GW, GT, LSDA+DMFT, and experiment.
In a previous work 19 , we had already investigated the effect of the GT self-energy alone and found that it gives rise to strong lifetime broadening close to the Fermi energy and anomalies in the electronic band dispersions such as kinks, waterfall structures, and magnon satellite features. In the present work, we demonstrated that the inclusion of the GW self-energy further improves the description of the electronic structure in several ways: (i) the lifetime broadening, which appears somewhat exaggerated in GT, is less extreme in GWT. (ii) we find a good quantitative agreement with photoemission peak positions in iron. (iii) An important improvement concerns a band anomaly at a binding energy of 1.5 eV that we had found in the previous GT study and that was subsequently observed in ARPES spectra 7 . We revisited this band anomaly and found it to benefit from the inclusion of GW in terms of location in energy and momentum and its shape. (iv) A violation of causality (a wrong sign of spectral function) that had been found in GT is healed in the GWT self-energy. (v) GWT magnetic moments, exchange splittings, and d bandwidths are in very good agreement with the experiment, better than those obtained with LSDA or GW and GT alone. Interestingly, one notices a favorable compensating effect between GW and GT benefitting the GWT method, which is particularly obvious in the tables for the magnetic moments, exchange splittings, and d bandwidths (but also noticeable in the spectral functions). The GWT band structures are similar to results from LSDA+DMFT. However, we do find some improvements in the details: (1) GWT yields well-defined quasiparticle bands at binding energies of 2.34 eV (at Γ) and 3.2 eV (at P) in good agreement with ARPES peak positions, whereas LSDA+DMFT does not show any distinct feature at the respective energies. (2) Several bands show dispersion anomalies, e.g., the 1.5 eV anomaly in iron mentioned above. (3) In contrast to LSDA+DMFT, GWT (also GW) corrects wrong LSDA band dispersions in iron, which would falsely predict the existence of Fermi surface sheets (including a spin-down hole pocket) that are not observed in the experiment.
The GT self-energy contributes significantly only in the vicinity of the Fermi energy at the energy scale of magnon (and Stoner) excitations. In contrast, the GW self-energy becomes dominant beyond that region with a characteristic energy scale of plasmonic excitations. This explains the importance of the combined GW + GT treatment to cover the entire energy range of electronic excitations. We emphasize that the GWT lifetime broadenings (scattering rates) are not simply the sum of the broadenings from GW and GT. The combination of self-energies that have distinct characteristics in different energy regions can cause a detuning effect: for example, the large real part of the GW self-energy can shift (detune) a quasiparticle peak away from an energy region where GT would cause a strong lifetime broadening. This can lead to the somewhat counter-intuitive observation that GWT may show less (more) lifetime broadening in certain energy regions than any of (the sum of) GW and GT. Few examples of this effect have been discussed. A more-detailed discussion of the lifetime effects in these materials induced by the combined electron-plasmon and electron-magnon scattering processes is planned for a forthcoming article.
We showed the frequency and k dependence of the GWT selfenergy in the form of a color plot, which, by comparison with an analogous plot 19 for GT, demonstrated that no sign problem (violation of causality) appears in GWT. Moreover, the color plot revealed a curious step-like behavior of the GWT self-energy at a momentum where the corresponding quasiparticle band crosses the Fermi energy. This step is not really a discontinuity in the selfenergy but exhibits, in its gradient, a logarithmic divergence, which has the same mathematical origin as the famous logarithmic divergence in the group velocity of metallic HF bands. It is important to note that the corresponding quasiparticle bands themselves are unaffected by this divergence.
The presented method of combining the GW and GT diagrams are implemented in a one-shot (non-self-consistent) approach, but self-consistency is partially taken into account. To this end, a parameter Δ v is introduced, which is fixed by a self-consistency condition for the Fermi energy. Another parameter Δ x needs to be included to enforce the Goldstone condition, which is otherwise violated when using the LSDA Green function in the BSE. This latter parameter can also be interpreted as a second selfconsistency condition. Both parameters are thus fixed by exact physical constraints and are therefore no adjustable parameters.
To go beyond this type of self-consistency would require a full iterative scheme. Quasiparticle self-consistent GW 31 (complemented by the GT self-energy) may spring to one's mind. However, as we have seen, the electron-magnon scattering strongly renormalizes the electronic bands, to the extent that some may completely lose their quasiparticle character. In this case, it is questionable whether the quasiparticle self-consistent GW scheme would be applicable, and one might need to employ a fully self-consistent GWT approach (in the spirit of fully selfconsistent GW [39][40][41], which would retain all lifetime effects encoded in the imaginary part of the Green function. On the other hand, if we compare our GWT results with the selfconsistent DMFT results, we find many similarities which suggest that a fully self-consistent GWT treatment would not drastically change any conclusions made in our work.
The GWT method and its implementation within FLAPW offer new opportunities for re-investigating the electronic structure of magnetic materials with possible applications to antiferromagnets and magnetic insulators. It may also help in elucidating the mechanism of superconductivity in high-T c materials. The inclusion of spin-orbit coupling into the formalism would make the treatment of more complex non-collinear spin structures possible, which are of importance for spintronic applications. The implementation of further scattering channels, such as described by non-spin-flip electron-hole, electron-electron, and hole-hole ladder diagrams, may prove necessary to describe other complex materials, for example, low-dimensional systems. The GWT method is particularly attractive due to its relatively low computational cost. The extra time needed for the GT part is similar to a standard GW calculation.

METHODS Theory
The main focus of the present study is the GWT self-energy, comprising GW and GT diagrams (see Fig. 9), and its application to elementary ferromagnets. The theory of the GT self-energy and its double-counting-free combination with GW was laid out already in a foregoing work. For completeness, we recapitulate the main points in this section. For more details, the reader is referred to ref. 19 . Fig. 9 GWT self-energy. Feynman diagrams representing the GW (electron-plasmon scattering) and the GT self-energy (electron-magnon scattering) are included in the combined GWT self-energy.
The self-energy approximations are based on a systematic diagrammatic expansion following the Hedin equations 15 . Although a full self-consistent calculation is not feasible in practice, the Hedin equations can serve as an invaluable tool to derive and improve approximations to the self-energy. The first iteration yields the famous GW approximation 15 , commonly used in computational condensed-matter physics 14,42 , with the Green function G σ (1, 2) for spin-σ particles and the screened interaction W (1, 2), calculated within the random-phase approximation.
(We use the shorthand 1 = r 1 , t 1 .) However, when applied to the elementary ferromagnets, it becomes obvious that, although GW is an undeniable improvement over KS DFT, it lacks higher-order scattering effects, in particular close to the Fermi energy. We conjectured these scattering processes to be the ones between electrons and magnons, comprising collective spin waves but also renormalized single-particle Stoner excitations, and we argued that these are described by a summation over spin-flip ladder diagrams. We use the term 'magnon' in this general way because the solution of the BSE yields both types of spin excitations at the same time. In general, the resulting many-body excitations are of a mixed collective and single-particle nature.
The ladder diagram of lowest order appears in the second iteration of the Hedin equations, each following iteration generating successively higher orders Summing these diagrams to all orders yields the GT self-energy where the electron-hole T matrix is determined as a solution of the BSE and σ 0 ¼ Àσ. The free electron-hole propagator K σσ 0 is given by It is instructive to make a parallel between the GT and the GW selfenergies. The T matrix can be viewed as an effective interaction that acts through the exchange of magnons in a similar sense as W incorporates the exchange of plasmons. It is important to note here that the leading order of the GT self-energy is seen to be the diagram of third order in W. Iterating the Hedin equations does not generate any lower-order direct ladder diagrams. This is because: (1) these are essentially already present in other diagrams, namely in the Hartree 'tadpole' diagram and the GW self-energy. (2) Furthermore, closer inspection shows that they would contain diagrammatic double-counting errors in themselves. This demonstrates the usefulness of the Hedin equations in that the resulting self-energy diagrams are guaranteed to be free of such double-counting errors. In the present case, the doublecounting of diagrams is removed by the fact that the GT series starts with the third order in W.
Starting from the third-order GT diagram, however, can give rise to a wrong sign in the imaginary part of the self-energy, which can be interpreted as a violation of causality 19,21 . In ref. 19 , we argued that the unphysical sign change occurs in an energy region far away from the Fermi energy, outside the energy range of interest. Therefore, we accepted the wrong sign with the expectation that the inclusion of the GW diagram would restore the correct sign, as we show in Results to be indeed the case.
The present combined GWT self-energy still lacks a number of manybody scattering effects that involve particle-particle (electron-electron, hole-hole, and non-spin-flip electron-hole) scattering as well as highorder exchange processes. These can, in principle, be included with a general T-matrix formulation [20][21][22] , which would, however, go beyond the present paper.

Computational details
MBPT requires as a reference a non-interacting mean-field system. We choose the ground state KS system as that reference system. The selfconsistent KS DFT calculations are carried out with the FLEUR code, an allelectron DFT implementation within the FLAPW method. Lattice constants of 2.87 Å, 3.54 Å, and 3.53 Å are employed for Fe, Co, and Ni, respectively. We use the LSDA in the parameterization of Perdew and Zunger 43 for the exchange-correlation functional. The LSDA has been chosen for the reference system to stay consistent with previous works where the same functional was employed. This choice also facilitates the comparison with the LSDA+DMFT method. For completeness, we have performed GWT calculations for iron with PBE 44 instead of LSDA for the reference meanfield system. The resulting renormalized band structures are very similar to the ones presented here for the LSDA starting point. The LAPW basis set cutoff is 6.2 Bohr −1 , and the muffin-tin radius is 2.2 Bohr for all systems. The Brillouin zone (BZ) is sampled with a 14 × 14 × 14 k-point grid.
The quasiparticle band structures are computed perturbatively on top of the LSDA eigensolutions, an approach commonly called 'single-shot' or (specifically for the GW approximation) G 0 W 0 calculations. However, a selfconsistency condition for the Fermi energy is included, as explained further below. The BZ is sampled with a 10 × 10 × 10 k-grid and 200 empty states are used in summations over unoccupied states (involving KS states up to 15 Ha). To represent the high-energy states accurately, we employ additional basis functions given as high-energy local orbitals in the LAPW formalism [45][46][47] . Two extra sets of local orbitals are added to each angular momentum channel l from 0 to 3, and one set to l = 4, amounting to 41 additional basis functions. As in the previous publication, we employ the contour deformation technique 18,19,48 for evaluating the frequency convolution of the self-energy. Frequency meshes along the imaginary axis are chosen to consist of 34 non-uniformly distributed points in intervals from 0 to 2i Ha for GT and from 0 to 10i Ha for GW. Integrals along the real axis are computed using uniform frequency grids with increments of 1 mHa and 20 mHa for GT and GW, respectively.
In the T matrix, the screened interaction W(ω) is approximated by its static limit W(ω → 0). This is common practice in BSE calculations and a choice out of necessity: without the static approximation, the BSE would contain three coupled frequencies, which would make the equation untractable for real systems. The static approximation eliminates two of the frequencies, and the equation becomes solvable by matrix inversion. The static approximation is also motivated by the ω dependence of the screened interaction. Besides, the screened interaction W(ω) is nearly constant in the range of the low-energy magnonic excitations ω, which indicates that for all practical purposes the static approximation is a viable approximation, as is indeed corroborated by our results.
The screened Coulomb interaction W is short-range in metallic systems. This fact has motivated us to employ an on-site approximation for the interaction, i.e., only electrons that occupy the same atomic site are allowed to interact with each other, whereas the interaction of electrons on different sites is neglected. We have tested this approximation 34 and found it to be a very good one for the systems studied in this work. It should be noted that the formalism can be generalized straightforwardly in a way that avoids the on-site approximation, but the notation would become more complex, which is why we restrict ourselves to the on-site treatment in this paper.
The non-self-consistent one-shot self-energy renormalization, as it is carried out in this work, is known to (potentially) violate particle-number conservation in metals 49 . As a simple remedy, one could determine a new Fermi energy according to the renormalized bands. However, this comes with the problem that the condition of vanishing lifetime broadening at ω = 0 [ImΣ(ω = 0) = 0] is fulfilled for the old Fermi energy and not for the new one. Furthermore, we will see that band anomalies (kinks, satellites, etc.) can occur close to the Fermi energy. By the re-adjustment of the Fermi energy, these features might shift to a wrong energetic position or even end up on the wrong side of the Fermi surface. For these reasons, we employ a correction scheme originally proposed by Hedin 15 , which solves the aforementioned problems in an elegant way. One introduces an energy shift Δ v in Σ(ω) → Σ(ω − Δ v ), which is determined such that the Fermi energy remains unchanged by the self-energy renormalization. In principle, this is a non-linear problem, which, in itself, requires a selfconsistent solution, since Δ v affects the Fermi energy (both before and after renormalization), which, in turn, influences the choice of Δ v . The selfenergy, once calculated, can be reused in each step, however. So, this selfconsistent cycle can be performed with little computational overhead. One should take into account, though, that the self-energy needs to be evaluated at all k points and at all states around the Fermi energy for this calculation. The energy shift Δ v is identical in the two spin-channels. It can be interpreted 19 as a simple constant shift of the exchangecorrelation potential (v σ xc ! v σ xc þ Δ v ), used for the underlying KS meanfield system. (It might be surprising that Δ v should have any effect at all, as it seems to simply shift the energy zero. However, one should not forget that quasiparticle energies are defined absolutely and that such a change, therefore, does have an effect. This issue is further discussed in ref. 19 ). In this way, v xc is seen to incorporate, to some extent, information about the renormalized system, which can be understood as low-level The parameter Δ v ensures the conservation of the particle number, and Δ x enforces the Goldstone condition in the magnon excitation spectra (negative sign corresponds to reduced exchange splitting). See text for details.