Description of quasiparticle and satellite properties via cumulant expansions of the retarded one-particle Green's function

A cumulant-based $GW$ approximation for the retarded one-particle Green's function is proposed, motivated by an exact relation between the improper Dyson self-energy and the cumulant generating function. Qualitative aspects of this method are explored within a simple one-electron independent phonon model, where it is seen that the method preserves the energy moment of the spectral weight while also reproducing the exact Green's function in the weak-coupling limit. For the three-dimensional electron gas, this method predicts multiple satellites at the bottom of the band, albeit with inaccurate peak spacing. However, its quasiparticle properties and correlation energies are more accurate than both previous cumulant methods and standard $G_0{W}_0$. Our results point to features that may be exploited within the framework of cumulant-based methods and suggest promising directions for future exploration and improvements of cumulant-based $GW$ approaches.

Figure 1

A comparison of three distinct $GW$ schemes. (a), (c) The spectral weight for the 3D electron gas with $r_s=4.0$. (b), (d) The absolute value of the imaginary part of the proper self-energy, solved using Dyson's equation, corresponding to the spectral weights in (a) and (c).

Figure 2

Band structure for various $GW$ schemes at $r_s=4$. Note that the $G_0{W}_0$ IRC contains multiple satellites, similar to the $G_0{W}_0$ PRC case, but with a spacing that mimics the incorrect position of the $G_0{W}_0$ plasmaron peak.

Figure 3

The $k$-dependent occupation number $n\left(k\right)$ for all three approximation schemes at $r_s=4.0$, plotted against the “exact” QMC result [40, 41].

Figure 4

Correlation energies of the 3D electron gas per particle ${\epsilon }_{\mathrm{corr}}/N$ for various $GW$ schemes and $r_s$ values compared with the “exact” QMC result. Exact values are reported in Table 1.

Figure 5

Band structure for various $GW$ schemes at $r_s=4$ and $k=3k_F$. The sharp quasiparticle resonance predicted by $G_0{W}_0$ IRC is broadened by the $b\mathrm{IRC}$ and $r\mathrm{IRC}$ schemes.