$GW$ approximations and vertex corrections on the Keldysh time-loop contour: Application for model systems at equilibrium

We study the effects of self-consistency and vertex corrections on different $GW$-based approximations for model systems of interacting electrons. For dealing with the most general case, we use the Keldysh time-loop contour formalism to evaluate the single-particle Green's functions. We provide the formal extension of Hedin's $GW$ equations for the Green's function in the Keldysh formalism. We show an application of our formalism to the plasmon model of a core electron within the plasmon-pole approximation. We study in detail the effects of the diagrammatic perturbation expansion of the core-electron/plasmon coupling on the spectral functions in the so-called S model. The S model provides an exact solution at equilibrium for comparison with the diagrammatic expansion of the interaction. We show that self-consistency is essential in $GW$-based calculations to obtain the full spectral information. The second-order exchange diagram (i.e., a vertex correction) is also crucial to obtain the good spectral description of the plasmon satellites. We corroborate these results by considering conventional equilibrium $GW$-based calculations for the pure jellium model. We find that with no second-order vertex correction, one cannot obtain the full set of plasmon side-band resonances. We also discuss in detail the formal expression of the Dyson equations obtained for the time-ordered Green's function at zero and finite temperature from the Keldysh formalism and from conventional equilibrium many-body perturbation theory.

Figure 1

Different levels of approximation for the one-particle self-energy $\Sigma ={\Sigma }^{\left(1\right)}+{\Sigma }^{\left(2\right)}$ within the plasmon model. First-order diagrams: (a) ${\Sigma }^{\left(1\right)}=G_0{W}_p$ with $G_0$ being the bare core-electron Green's function, and (b) ${\Sigma }^{\left(1\right)}=G{W}_p$ for self-consistent calculations. Second-order diagrams with vertex corrections (c) ${\Sigma }^{\left(2\right)}=G{\Gamma }_{\left(1\right)}^{GW}{W}_p$ for non-self-consistent calculations, and (d) ${\Sigma }^{\left(2\right)}=G{\Gamma }_{\left(1\right)}^{\mathrm{SC}}{W}_p$ for the full self-consistent calculations (see Appendix C2).

Figure 2

Zero-temperature equilibrium spectral functions $A\left(\omega \right)$ for the high-density limit with $r_S=2$, corresponding to medium core electron-plasmon coupling ${\gamma }_0/{\omega }_p=0.62$. Top panel: exact results and $GW$ calculations with and without self-consistency $\Sigma =G{W}_p$, $G_0{W}_p$. Bottom panel: results for different levels of approximation for the self-energy $\Sigma =G_0{W}_p$, $G{W}_p$, $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{GW})W_p$, and $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{\mathrm{SC}})W_p$ (see Fig. 1) with fewer grid points ($N_{\omega }=1579$), giving an extra broadening.

Figure 3

Zero-temperature equilibrium spectral functions $A\left(\omega \right)$ for the low-density limit with $r_S=4$, corresponding to very strong core electron-plasmon coupling ${\gamma }_0/{\omega }_p=1.20$. Top panel: exact results and $GW$ calculations for the different self-energies $\Sigma =G_0{W}_p$, $G{W}_p$. Bottom panel: results for different levels of approximation for the self-energy $\Sigma =G_0{W}_p$, $G{W}_p$, $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{GW})W_p$, and $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{\mathrm{SC}})W_p$ (see Fig. 1) with fewer grid points ($N_{\omega }=1579$), giving an extra broadening, in comparison to the top panel.

Figure 4

Finite-temperature equilibrium spectral functions $A\left(\omega \right)$ for the high-density electron gas with $r_S=2$ and a finite temperature $kT=0.2$ corresponding to ${\omega }_p/kT=3.062$. Top panel: exact results and calculations for different self-energies $\Sigma =G{W}_p$ and $G{W}_p^{\left(2\right)\mathrm{SC}}$. Bottom panel: results for different self-energies $\Sigma =G{W}_p$, $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{GW})W_p$, and $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{\mathrm{SC}})W_p$ (see Fig. 1) with fewer grid points ($N_{\omega }=1579$), giving an extra broadening.

Figure 5

Finite-temperature equilibrium spectral functions $A\left(\omega \right)$ for the low-density electron gas with $r_S=4$ and a finite temperature $kT=0.2$ (corresponding to ${\omega }_p/kT=1.083$) and with fewer grid points $N_{\omega }=1579$. Calculations for different self-energies $\Sigma =G{W}_p$, $G{W}_p^{\left(2\right)\mathrm{SC}}$, $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{GW})W_p$, and $G({\Gamma }_{\left(0\right)}+{\Gamma }_{\left(1\right)}^{\mathrm{SC}})W_p$ (see Fig. 1) are shown.

Figure 6

Spectral function $A_k\left(\omega \right)$ at $k=0$ and for (a) $r_{\mathrm{S}}=2$ and (b) $r_{\mathrm{S}}=4$ for the pure jellium model. Each set of curves shows a one-shot calculation for $G_0$ (black dash curves) and self-consistent iterations for $G$ (red solid curves). The bottom panel of plots (a) and (b) show standard $G_0{W}_0$ and $G{W}_0$, the top panels show $G_0{\tilde{W}}_0$ and $G{\tilde{W}}_0$ with a local vertex correction in ${\tilde{W}}_0$ as described in Ref. 5. $G_0{W}_{\mathrm{S}}$ and $G{W}_{\mathrm{S}}$ refer to a momentum-dependent vertex correction in $W_{\mathrm{S}}$ defined in Ref. 45 to approximate the exact $W$ of jellium. The results including the different vertex corrections are indicated with a line (for ${\tilde{W}}_0$) and a symbol (for $W_{\mathrm{S}}$). All chemical potentials are aligned at the Fermi energy ${\epsilon }_F$ of the noninteracting gas. The positive (negative) deviation of the main quasiparticle peak from the origin indicates a narrowing (broadening) of the occupied bandwidth. None of these approximations provides more than one plasmon satellite in contrast to the expected exact result.