Spectral properties from an efficient analytical representation of the $GW$ self-energy within a multipole approximation

We propose an efficient analytical representation of the frequency-dependent $GW$ self-energy $\Sigma$ via a multipole approximation (MPA-$\Sigma$). The multipole-Pad\'e model for the self-energy is interpolated from a small set of numerical evaluations of $\Sigma$ in the complex frequency plane, similarly to the previously multipole representation developed for the screened Coulomb interaction (MPA-$W$) [Phys. Rev. B \textbf{104}, 115157 (2021)]. We show that, likewise MPA-$W$, an appropriate choice of frequency sampling in MPA-$\Sigma$ is critical to guarantee computational efficiency and high accuracy. The combined MPA-$W$ and MPA-$\Sigma$ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range. Crucially, MPA-$\Sigma$ enables a multipole representation for the interacting Green's function $G$ (MPA-$G$), providing a straightforward evaluation of all the spectral properties, and a more general way to define the renormalization factor $Z$. We validate the MPA-$\Sigma$ and MPA-$G$ approaches for diverse systems: bulk Si, Na and Cu, monolayer MoS$_2$, the NaCl ion-pair and the F$_2$ molecule. Moreover, we introduce toy MPA-$\Sigma$/$G$ models to examine the quasiparticle picture in different regimens of weak and strong correlation. With these models, we expose limitations in defining $Z$ from the local derivative of $\Sigma$.