Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.24838

Accidental accuracy and vertex corrections in $GW$: Exact benchmarks for the extended Hubbard model

Michael O. Atambo

Published Aug 25, 2026• 0 citations
arXivHTMLPDF

Abstract

The $GW$ approximation is the standard tool for quasiparticle predictions in materials, yet its regime of validity in correlated systems remains poorly quantified, because \textit{ab initio} vertex corrections are computationally prohibitive. Using exact diagonalization of the half-filled extended Hubbard model on finite rings as a numerically exact reference, we construct the corresponding model-space $GW$ theory on the identical Hilbert space and quantify its error as a function of local ($U$) and non-local ($V$) interaction strength. We find that the required vertex correction changes character across the phase diagram: in the weak-coupling regime the effective vertex $Γ_{\rm eff} < 1$, reflecting the suppression of RPA charge fluctuations by exact short-range correlations, whereas in the Mott regime $Γ_{\rm eff}$ grows monotonically (to $\sim 3$ at $U=8t$ for $N=6$), reflecting the local dynamical self-energy structure required to open the Hubbard gap. Vertex corrections in the electron-hole (polarizability) channel are shown to \emph{worsen} the gap error, indicating that reproducing the Mott gap requires dynamical self-energy structure rather than improved screening. For $V=0$, static COHSEX is accidentally exact at a single crossover $U^* \approx 3.5\,t$; finite $V$, through non-local Fock exchange, splits this point into a double-crossover window that collapses toward weak coupling. Even at the crossover, however, the exact spectral function retains Hubbard-band structure that no static functional reproduces, so gap agreement does not imply functional accuracy. These results yield quantitative diagnostics for the reliability of $GW$ in correlated materials.

DOI
10.48550/arxiv.2608.24838

Related papers

Accidental accuracy and formal consistency in $GW$+BSE: Exact benchmarks and regime-dependent error cancellation

Michael O. Atambo

Standard many-body perturbation theory for optical excitations combines the $GW$ approximation for quasiparticle energies with the Bethe-Salpeter equation (BSE) for the electron-hole response. Formally, the BSE kernel must equal the functional derivative of the self-energy, $K=δΣ/δG$. In routine practice, however, the kernel is built from a statically screened direct interaction and a bare exchange interaction, which breaks this consistency. Using exact diagonalization of the extended Hubbard dimer, we construct a controlled benchmark that isolates the price of this inconsistency. We show that the internally inconsistent practical $GW$+BSE construction frequently yields accurate optical gaps through accidental cancellation between an underestimated $GW$ gap and an oversized bare-exchange kernel, and we map the regions of parameter space where this accidental accuracy occurs. Upon switching on the nearest-neighbor interaction and binding the exciton, the practical construction remains the accurate one throughout the weak-binding regime, with cancellation optima at parts in $10^{3}$, while the frozen-$W$ construction fails at every binding strength. Beyond a binding threshold both static constructions deteriorate, showing that deeply bound excitons require physics beyond any static kernel. Accidental accuracy is therefore a property of the weak-binding regime, and the systematic failure of the frozen-$W$ derivative kernel within the explored parameter space traces to its overscreening of the exchange channel. The accuracy of the practical kernel is an accidental cancellation of two independent errors, an underestimated quasiparticle gap and an oversized exchange term, not a hidden consistency. These results provide a controlled taxonomy of approximation failure in $GW$+BSE and a precise diagnostic for when standard workflows can be trusted.

arXiv:2609.38247•Sep 29, 2026•0 citations

The self-energy beyond GW: Local and nonlocal vertex corrections

Pina Romaniello, Steve Guyot, Lucia Reining

It is commonly accepted that the GW approximation for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approximation: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the atomic limit, in which, instead, correlation is directly responsible for the observed problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be corrected by a two-point vertex contribution to the self-energy derived from time-dependent density functional theory (TDDFT). We explain why the addition of a particle, instead, requires the use of a different approximate vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the atomic limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qualitatively. We discuss the effects of the self-screening error as well as the atomic limit using GW for the exactly solvable two-site Hubbard model.

The Journal of Chemical Physics•Oct 20, 2009•131 citations

Self-consistent G W + extended dynamical mean field theory for semiconductors and insulators

Viktor Christiansson, Francesco Petocchi, Philipp Werner

Theoretical studies of semiconductors and band insulators are usually based on variants of the $GW$ method without full self-consistency, like single-shot $G^0W^0$ or quasiparticle self-consistent $GW$. Fully self-consistent $GW$ provides a poor description of the gap size and electronic structure due to the lack of vertex corrections. While it is hard to predict at which order corrections can be neglected, local vertex corrections to all orders can be consistently included by combining $GW$ with extended dynamical mean field theory (EDMFT). Here, we show that \textit{ab initio} multitier $GW$+EDMFT calculations, which achieve full self-consistency in a suitably defined low-energy space, provide an accurate description of semiconductors and band insulators, comparable to the established methods which are typically used for this class of materials. Our results demonstrate that the range of applicability of $GW$+EDMFT extends to weakly correlated systems, and they imply that despite the weak correlations, local vertex corrections are an important ingredient in the diagrammatic treatment of semiconductors and band insulators.

arXiv:2410.19579•Physical review. B./Physical review. B•Jun 12, 2026•0 citations

Vertex corrections in localized and extended systems

Andrew J. Morris, Martin Stankovski, Kris T. Delaney +3 more

Within many-body perturbation theory, we apply vertex corrections to various closed-shell atoms and to jellium, using a local approximation for the vertex consistent with starting the many-body perturbation theory from a Kohn-Sham Green's function constructed from density-functional theory in the local-density approximation. The vertex appears in two places---in the screened Coulomb interaction $W$ and in the self-energy $\ensuremath{\Sigma}$---and we obtain a systematic discrimination of these two effects by turning the vertex in $\ensuremath{\Sigma}$ on and off. We also make comparisons to standard $GW$ results within the usual random-phase approximation, which omits the vertex from both. When a vertex is included for closed-shell atoms, both ground-state and excited-state properties demonstrate little improvement over standard $GW$. For jellium, we observe marked improvement in the quasiparticle bandwidth when the vertex is included only in $W$, whereas turning on the vertex in $\ensuremath{\Sigma}$ leads to an unphysical quasiparticle dispersion and work function. A simple analysis suggests why implementation of the vertex only in $W$ is a valid way to improve quasiparticle energy calculations, while the vertex in $\ensuremath{\Sigma}$ is unphysical, and points the way to the development of improved vertices for ab initio electronic structure calculations.

arXiv:cond-mat/0702294•Physical Review B•Oct 8, 2007•59 citations

Modified GW method in electronic systems

Zhipeng Sun, Zhenhao Fan, Hui Li +2 more

A modified $GW$ approximation to many-body systems is developed. The approximation has the same computational complexity as the traditional $GW$ approach, but uses a different truncation scheme. This scheme neglects the high-order connected correlation functions. A covariant (preserving the Ward identities due to the charge conservation) scheme for the two-body correlators is employed, which holds the relation between the charge correlator and the charge susceptibility. The method is tested on the two-dimensional one-band Hubbard model. The results are compared with exact diagonalization, the $GW$ approximation, the fluctuation-exchange (FLEX) theory, and determinantal Monte Carlo approach. The comparison for the (one-body) Green's function demonstrates that it is more precise in the strong-coupling regime (especially away from half filling) than the $GW$ and FLEX approximations, which have a similar complexity. More importantly, this method indicates a Mott-Hubbard gap as the Hubbard $U$ increases, whereas the $GW$ and FLEX methods fail. In addition, the charge correlator obtained from the covariant scheme not only holds the consistency of the static charge susceptibility, but also makes a significant improvement over the random phase approximation calculations.

arXiv:2105.14775•Physical review. B./Physical review. B•Sep 24, 2021•6 citations

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

Michael O. Atambo

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.

arXiv:2609.29635•Aug 29, 2026•0 citations
Accidental accuracy and vertex corrections in GWGWGW: Exact benchmarks for the extended Hubbard model
The GWGWGW approximation is the standard tool for quasiparticle predictions in materials, yet its regime of validity in correlated systems remains poorly quantified, because \textit{ab initio} vertex corrections are computationally prohibitive. Using exact diagonalization of the half-filled extended Hubbard model on finite rings as a numerically exact reference, we construct the corresponding model-space GWGWGW theory on the identical Hilbert space and quantify its error as a function of local (UUU) and non-local (VVV) interaction strength. We find that the required vertex correction changes character across the phase diagram: in the weak-coupling regime the effective vertex Γeff<1Γ_{\rm eff} < 1Γeff​<1, reflecting the suppression of RPA charge fluctuations by exact short-range correlations, whereas in the Mott regime ΓeffΓ_{\rm eff}Γeff​ grows monotonically (to ∼3\sim 3∼3 at U=8tU=8tU=8t for N=6N=6N=6), reflecting the local dynamical self-energy structure required to open the Hubbard gap. Vertex corrections in the electron-hole (polarizability) channel are shown to \emph{worsen} the gap error, indicating that reproducing the Mott gap requires dynamical self-energy structure rather than improved screening. For V=0V=0V=0, static COHSEX is accidentally exact at a single crossover U∗≈3.5 tU^* \approx 3.5\,tU∗≈3.5t; finite VVV, through non-local Fock exchange, splits this point into a double-crossover window that collapses toward weak coupling. Even at the crossover, however, the exact spectral function retains Hubbard-band structure that no static functional reproduces, so gap agreement does not imply functional accuracy. These results yield quantitative diagnostics for the reliability of GWGWGW in correlated materials.
Accidental accuracy and formal consistency in GWGWGW+BSE: Exact benchmarks and regime-dependent error cancellation
Standard many-body perturbation theory for optical excitations combines the GWGWGW approximation for quasiparticle energies with the Bethe-Salpeter equation (BSE) for the electron-hole response. Formally, the BSE kernel must equal the functional derivative of the self-energy, K=δΣ/δGK=δΣ/δGK=δΣ/δG. In routine practice, however, the kernel is built from a statically screened direct interaction and a bare exchange interaction, which breaks this consistency. Using exact diagonalization of the extended Hubbard dimer, we construct a controlled benchmark that isolates the price of this inconsistency. We show that the internally inconsistent practical GWGWGW+BSE construction frequently yields accurate optical gaps through accidental cancellation between an underestimated GWGWGW gap and an oversized bare-exchange kernel, and we map the regions of parameter space where this accidental accuracy occurs. Upon switching on the nearest-neighbor interaction and binding the exciton, the practical construction remains the accurate one throughout the weak-binding regime, with cancellation optima at parts in 10310^{3}103, while the frozen-WWW construction fails at every binding strength. Beyond a binding threshold both static constructions deteriorate, showing that deeply bound excitons require physics beyond any static kernel. Accidental accuracy is therefore a property of the weak-binding regime, and the systematic failure of the frozen-WWW derivative kernel within the explored parameter space traces to its overscreening of the exchange channel. The accuracy of the practical kernel is an accidental cancellation of two independent errors, an underestimated quasiparticle gap and an oversized exchange term, not a hidden consistency. These results provide a controlled taxonomy of approximation failure in GWGWGW+BSE and a precise diagnostic for when standard workflows can be trusted.
It is commonly accepted that the GW approximation for the electron self-energy is successful for the description of the band structure of weakly to moderately correlated systems, whereas it will fail for strongly correlated materials. In the present work, we discuss two important aspects of this approximation: first, the "self-screening error," which is due to an incorrect treatment of induced exchange, and second, the atomic limit, in which, instead, correlation is directly responsible for the observed problem. Using the example of the removal of a particle from a box, we show that the self-screening error stems from the use of test charge-test charge screening and that it can be corrected by a two-point vertex contribution to the self-energy derived from time-dependent density functional theory (TDDFT). We explain why the addition of a particle, instead, requires the use of a different approximate vertex. This illustrates why the general vertex function, valid both for valence and conduction states, must be a three-point function. Moreover, we show that also the bad performance of GW in the atomic limit is due to the neglect of the vertex in the self-energy; in that case, the TDDFT-derived vertex correction is not sufficient in order to remove the error even qualitatively. We discuss the effects of the self-screening error as well as the atomic limit using GW for the exactly solvable two-site Hubbard model.
Theoretical studies of semiconductors and band insulators are usually based on variants of the GWGWGW method without full self-consistency, like single-shot G0W0G^0W^0G0W0 or quasiparticle self-consistent GWGWGW. Fully self-consistent GWGWGW provides a poor description of the gap size and electronic structure due to the lack of vertex corrections. While it is hard to predict at which order corrections can be neglected, local vertex corrections to all orders can be consistently included by combining GWGWGW with extended dynamical mean field theory (EDMFT). Here, we show that \textit{ab initio} multitier GWGWGW+EDMFT calculations, which achieve full self-consistency in a suitably defined low-energy space, provide an accurate description of semiconductors and band insulators, comparable to the established methods which are typically used for this class of materials. Our results demonstrate that the range of applicability of GWGWGW+EDMFT extends to weakly correlated systems, and they imply that despite the weak correlations, local vertex corrections are an important ingredient in the diagrammatic treatment of semiconductors and band insulators.
Within many-body perturbation theory, we apply vertex corrections to various closed-shell atoms and to jellium, using a local approximation for the vertex consistent with starting the many-body perturbation theory from a Kohn-Sham Green's function constructed from density-functional theory in the local-density approximation. The vertex appears in two places---in the screened Coulomb interaction WWW and in the self-energy $\ensuremath{\Sigma}$---and we obtain a systematic discrimination of these two effects by turning the vertex in $\ensuremath{\Sigma}$ on and off. We also make comparisons to standard GWGWGW results within the usual random-phase approximation, which omits the vertex from both. When a vertex is included for closed-shell atoms, both ground-state and excited-state properties demonstrate little improvement over standard GWGWGW. For jellium, we observe marked improvement in the quasiparticle bandwidth when the vertex is included only in WWW, whereas turning on the vertex in $\ensuremath{\Sigma}$ leads to an unphysical quasiparticle dispersion and work function. A simple analysis suggests why implementation of the vertex only in WWW is a valid way to improve quasiparticle energy calculations, while the vertex in $\ensuremath{\Sigma}$ is unphysical, and points the way to the development of improved vertices for ab initio electronic structure calculations.
A modified GWGWGW approximation to many-body systems is developed. The approximation has the same computational complexity as the traditional GWGWGW approach, but uses a different truncation scheme. This scheme neglects the high-order connected correlation functions. A covariant (preserving the Ward identities due to the charge conservation) scheme for the two-body correlators is employed, which holds the relation between the charge correlator and the charge susceptibility. The method is tested on the two-dimensional one-band Hubbard model. The results are compared with exact diagonalization, the GWGWGW approximation, the fluctuation-exchange (FLEX) theory, and determinantal Monte Carlo approach. The comparison for the (one-body) Green's function demonstrates that it is more precise in the strong-coupling regime (especially away from half filling) than the GWGWGW and FLEX approximations, which have a similar complexity. More importantly, this method indicates a Mott-Hubbard gap as the Hubbard UUU increases, whereas the GWGWGW and FLEX methods fail. In addition, the charge correlator obtained from the covariant scheme not only holds the consistency of the static charge susceptibility, but also makes a significant improvement over the random phase approximation calculations.
The GWGWGW 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 G0G_0G0​ with the exact pole representation of the screened interaction WWW, we compute the GWGWGW 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→0Z \to 0Z→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.