Asymmetry in band widening and quasiparticle lifetimes in ${\mathrm{SrVO}}_3$: Competition between screened exchange and local correlations from combined $GW$ and dynamical mean-field theory $GW + \mathrm{DMFT}$

The very first dynamical implementation of the combined GW and dynamical mean-field scheme “$GW + \mathrm{DMFT}$” for a real material was achieved recently [Tomczak et al., Europhys. Lett. 100, 67001 (2012)], and applied to the ternary transition metal oxide ${\mathrm{SrVO}}_3$. Here, we review and extend that work, giving not only a detailed account of full $GW + \mathrm{DMFT}$ calculations, but also discussing and testing simplified approximate schemes. We give insights into the nature of exchange and correlation effects: dynamical renormalizations in the Fermi liquid regime of ${\mathrm{SrVO}}_3$ are essentially local, and nonlocal correlations mainly act to screen the Fock exchange term. The latter substantially widens the quasiparticle band structure, while the band narrowing induced by the former is accompanied by a spectral weight transfer to higher energies. Most interestingly, the exchange broadening is more pronounced in the unoccupied part of the spectrum than in the occupied one. In addition, shorter lifetimes for unoccupied states further contribute to making the corrections to the Kohn-Sham band structure asymmetric with respect to the chemical potential. As a result, the $GW + \mathrm{DMFT}$ electronic structure of ${\mathrm{SrVO}}_3$ resembles the conventional density functional based dynamical mean-field ($\mathrm{DFT} + \mathrm{DMFT}$) description for occupied states but is profoundly modified in the empty part. Our work leads to a reinterpretation of inverse photoemission spectroscopy (IPES) data. Indeed, we assign a prominent peak at about 2.7 eV dominantly to ${\mathrm{e}}_g$ states, rather than to an upper Hubbard band of ${\mathrm{t}}_{2g}$ character. Similar surprises can be expected for other transition metal oxides. This prediction urgently calls for more detailed investigations of conduction band states in correlated materials.

Figure 1

Density of states within LDA in comparison with experimental spectra: PES [73] and PES+BIS [74].

Figure 2

Dynamical screening in ${\mathrm{SrVO}}_3$. From top to bottom: (a) comparison of the inverse dielectric function of ${\mathrm{SrVO}}_3$ within RPA with the experimental EELS spectrum of ${\mathrm{SrTiO}}_3$ [116]. (b)/(c) real/imaginary part of the fully (partially) screened interaction $W^{\mathrm{loc},d}$ ($\mathcal{U}$) in the Wannier basis. (d) Bosonic factor (see text for definition) and density of screening modes $\mathrm{Im}\mathcal{U}\left(\omega \right)/{\omega }^2$.

Figure 3

The $GW$ spectral function in comparison to the same experiments as in Fig. 1.

Figure 4

Momentum resolved spectral function (a) within the $GW$ approximation and (b) taking into account only the nonlocal part of the $GW$ self-energy. Superimposed is the LDA band structure.

Figure 5

Comparison of the local $GW$ spectral function with the LDA density of states. Also shown is the spectral function Eq. (46) and density of states Eq. (57) of nonlocal-$GW$. See text for details.

Figure 6

The $GW$ self-energy at several high symmetry points resolved into the three ${\mathrm{t}}_{2g}$ (Wannier) orbitals as a function of frequency. Also shown is the local projection (real parts: top panel, imaginary parts: middle panel). The lower panel displays the standard deviation of the frequency dependent generalization of the quasiparticle weight $Z_{\mathbf{k}}\left(\omega \right)$ with respect to its local projection $Z^{\mathrm{loc}}\left(\omega \right)$ [see Eq. (48)]. The origin of energy corresponds to the Fermi level and the shaded area roughly delimits the Fermi liquid regime within $GW$.

Figure 7

The $GW$ self-energy correction to DFT as a function of real space distance at the Fermi level ($\omega =0$) and the energy region of the plasmon satellite ($\omega =15$ eV). The occurrence of several values per distance owes to different orbital orientations at growing numbers of neighbours. Also shown are the absolute values of the ($\omega =0$) correlation and exchange self-energies of the $GW$.

Figure 8

Local spectral function from standard LDA + DMFT with static interactions (left). LDA + DMFT with dynamical interactions (right). Spectral functions are normalized to one, such that the filling corresponding to ${\mathrm{SrVO}}_3$ is 1/6.

Figure 9

$GW + \mathrm{DMFT}$ spectral function for a large (left) and low-energy (right) windows, in comparison to (inverse) photoemission spectra of Refs. [73, 74]. Note in particular the appearance of a lower Hubbard at the correct energy in the occupied part of the spectrum, and the fact that the peak at $+3$ eV in the inverse photoemission spectrum is not derived from a ${\mathrm{t}}_{2g}$ upper Hubbard band, but rather from the $e_g$ orbitals.

Figure 10

Momentum-resolved spectral function from $GW$ and $GW + \mathrm{DMFT}$ in comparison to angle resolved photoemission experiments [78, 95]. Also shown is the position of the LDA band at the $\Gamma$ (lower band edge) and $X$ points. The arrows indicate how these states are renormalized by $GW$ (middle panel) and $GW + \mathrm{DMFT}$ (lower panel). Note that the position of the unoccupied LDA band at $X$ coincides with the corresponding quasiparticle peak within $GW + \mathrm{DMFT}$.

Figure 11

The DMFT@nonlocal-$GW$ approach. The left panel shows the spectral function resolved along the usual momentum path, while the right panel displays a low-energy zoom around the $\Gamma$-point. There, the dashed (blue) lines are guides to the eye highlighting the reduced group velocity for energies below the kink in the dispersion (see text for details). The calculation was performed at an inverse temperature $\beta =10$/eV. Owing to the large temperature, the Fermi vector $k_F$ is slightly reduced with respect to our LDA (and $GW$) results.

Figure 12

$k$-summed spectrum of the DMFT@nonlocal-$GW$ approach in comparison with the full $GW + \mathrm{DMFT}$ and LDA + $\mathcal{U}\left(\omega \right)$ + DMFT. All DMFT calculations were performed at $\beta =10$ ${\mathrm{eV}}^{-1}$.

Figure 13

Comparison of local ${\mathrm{t}}_{2g}$ self-energies on the Matsubara axis: usual LDA + DMFT, LDA + DMFT with dynamical interactions, DMFT@nonlocal-$GW$ and local part of full $GW + \mathrm{DMFT}$.