Analysis of exact vertex function for improving on the GW$\Gamma$ scheme for first-principles calculation of electron self-energy

In order to propose a sophisticated scheme for the self-consistent calculation of the electron self-energy $\Sigma$, a detailed analysis of the analytical structure of the three-point vertex function $\Gamma$ is made with full respect for the Ward identity from the perspective of Fermi-liquid theory. Our scheme may be regarded as an improvement on the gauge-invariant self-consistent approximation to the exact theory for obtaining $\Sigma$ as a fixed point of the self-energy revision operator $\mathcal{F}$, indicating an intrinsically nonperturbative approach applicable to both Fermi and Tomonaga-Luttinger liquids in a unified manner, but it may also be considered as providing a general framework for constructing an accurate functional form for $\Gamma$ in the GW$\Gamma$ method for the first-principles calculation of $\Sigma$. Our result on the momentum distribution function in the homogeneous electron gas is compared with the one recently obtained by the reptation Monte Carlo method.

Figure 1

Self-consistent iteration loops to determine the self-energy $\Sigma \left(K\right)$ in the GW$\Gamma$ scheme: (a) the original version and (b) the improved one, named the G$\tilde{\mathrm{W}}{\Gamma }_{\mathrm{WI}}$ scheme.

Figure 2

One-electron spectral function $A(\mathbf{k},\omega )$, plotted as a function of $\omega$ at (a) $k=0$, (b) $k=k_{\mathrm{F}}$, and (c) $k=1.4k_{\mathrm{F}}$ at $T=0.001{\epsilon }_{\mathrm{F}}$ in the electron gas for $r_s=4$, 5, 6, and 8.

Figure 3

Overall structure of the one-electron spectral function $A(\mathbf{k},\omega )$ in the electron gas at (a) $r_s=4$ and (b) $r_s=8$ with $T=0.001{\epsilon }_{\mathrm{F}}$.

Figure 4

Momentum distribution function $n\left(\mathbf{k}\right)$ for the electron gas at $r_s=1$ (diamond), 2 (ellipse), 4 (plus), 5 (star), 6 (cross), and 8 (rectangle) in the GW$\Gamma$ method. For comparison, we also plot the recent results for $r_s=1$ (longer dashed), 2 (dashed), $3.99$ (dotted-dashed), 5 (dotted), and 10 (solid) in the reptation Monte Carlo method (Ref.20).

Figure 5

Self-consistent iteration loop in the GI$\Gamma$ scheme to determine the self-energy $\Sigma \left(K\right)$ with use of the known information on $\Pi \left(Q\right)$.