Exact benchmarks for the plasmon-pole approximation: Multi-scale screening and the breakdown of the quasiparticle picture

Abstract

The $GW$ approximation is the gold standard for calculating quasiparticle band structures, yet its computational cost frequently necessitates the use of the plasmon-pole approximation (PPA). While PPA is known to be highly accurate for simple metals and weakly correlated semiconductors, its regime of validity in materials with competing energy scales remains poorly quantified. Here, we construct an exact, numerical benchmark of the PPA using the Lehmann representation of the density response on finite one-dimensional Hubbard clusters. By analytically convolving the exact non-interacting Green's function $G_0$ with the exact pole representation of the screened interaction $W$, we compute the $GW$ self-energy without numerical frequency integration. We demonstrate that in single-scale Mott insulators, the moment-conserving PPA is essentially exact. However, in multi-band semiconductors where interband transitions introduce a low-energy screening channel that competes with high-energy Mott fluctuations, the PPA systematically misjudges the quasiparticle gap by several electron-volts. Furthermore, we show that strong multi-pole screening can drive the exact quasiparticle weight $Z \to 0$, destroying the quasiparticle picture-an effect entirely missed by the PPA, which artificially stabilizes sharp quasiparticles. Finally, we propose a computationally inexpensive diagnostic based on the polydispersity of the loss function's spectral weight, which accurately predicts PPA failure \textit{ab initio}. Our results provide a rigorous framework for assessing the validity of dynamical screening approximations in strongly correlated materials.