Scrutinizing $GW$-based methods using the Hubbard dimer

TL;DR

This paper investigates the $GW$ approximation using the Hubbard dimer, addressing quasiparticle solution multiplicity, analyzing neutral excitations, and comparing correlation energy methods, revealing strengths and limitations of the $GW$ and BSE approaches.

Contribution

It demonstrates that full self-consistency resolves multiple quasiparticle solutions and compares the accuracy and stability of different correlation energy formulas within the $GW$ and BSE frameworks.

Findings

Full self-consistency solves multiple quasiparticle solutions issue.

Trace (plasmon) formula yields accurate correlation energies over a range of U.

Trace formula is sensitive to complex excitation energies, while ACFDT is more stable.

Abstract

Using the simple (symmetric) Hubbard dimer, we analyze some important features of the $GW$ approximation. We show that the problem of the existence of multiple quasiparticle solutions in the (perturbative) one-shot $GW$ method and its partially self-consistent version is solved by full self-consistency. We also analyze the neutral excitation spectrum using the Bethe-Salpeter equation (BSE) formalism within the standard $GW$ approximation and find, in particular, that i) some neutral excitation energies become complex when the electron-electron interaction $U$ increases, which can be traced back to the approximate nature of the $GW$ quasiparticle energies; ii) the BSE formalism yields accurate correlation energies over a wide range of $U$ when the trace (or plasmon) formula is employed; iii) the trace formula is sensitive to the occurrence of complex excitation energies (especially singlet), while the expression obtained from the adiabatic-connection fluctuation-dissipation theorem (ACFDT) is more stable (yet less accurate); iv) the trace formula has the correct behavior for weak (\ie, small $U$) interaction, unlike the ACFDT expression.