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arXiv:2501.09121

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

Dario A. Leon, Kristian Berland, Cláudia Cardoso

Published Jan 15, 2025• 0 citations
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Abstract

We propose an efficient analytical representation of the frequency-dependent $GW$ self-energy $Σ$ via a multipole approximation (MPA-$Σ$). The multipole-Padé model for the self-energy is interpolated from a small set of numerical evaluations of $Σ$ 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-$Σ$ is critical to guarantee computational efficiency and high accuracy. The combined MPA-$W$ and MPA-$Σ$ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range. Crucially, MPA-$Σ$ 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-$Σ$ 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-$Σ$/$G$ 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 $Σ$.

DOI
10.48550/arxiv.2501.09121

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Spectral properties from an efficient analytical representation of the GWGWGW self-energy within a multipole approximation
We propose an efficient analytical representation of the frequency-dependent GWGWGW self-energy ΣΣΣ via a multipole approximation (MPA-ΣΣΣ). The multipole-Padé model for the self-energy is interpolated from a small set of numerical evaluations of ΣΣΣ in the complex frequency plane, similarly to the previously multipole representation developed for the screened Coulomb interaction (MPA-WWW) [Phys. Rev. B \textbf{104}, 115157 (2021)]. We show that, likewise MPA-WWW, an appropriate choice of frequency sampling in MPA-ΣΣΣ is critical to guarantee computational efficiency and high accuracy. The combined MPA-WWW and MPA-ΣΣΣ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range. Crucially, MPA-ΣΣΣ enables a multipole representation for the interacting Green's function GGG (MPA-GGG), providing a straightforward evaluation of all the spectral properties, and a more general way to define the renormalization factor ZZZ. We validate the MPA-ΣΣΣ and MPA-GGG approaches for diverse systems: bulk Si, Na and Cu, monolayer MoS2_22​, the NaCl ion-pair and the F2_22​ molecule. Moreover, we introduce toy MPA-ΣΣΣ/GGG models to examine the quasiparticle picture in different regimens of weak and strong correlation. With these models, we expose limitations in defining ZZZ from the local derivative of ΣΣΣ.
We propose an efficient analytical representation of the frequency-dependent GWGWGW self-energy $\mathrm{\ensuremath{\Sigma}}$ via a multipole approximation ($\text{MPA-}\mathrm{\ensuremath{\Sigma}}$). The MPA self-energy model is interpolated from a small set of numerical evaluations of $\mathrm{\ensuremath{\Sigma}}$ in the complex frequency plane, similar to the MPA interpolation developed for the screened Coulomb interaction (MPA-W\text{MPA-}WMPA-W) [D. A. Leon et al., Phys. Rev. B 104, 115157 (2021)]. Crucially, $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ enables a multipole representation for the interacting Green's function GGG (MPA-G\text{MPA-}GMPA-G) and, in turn, access to all the spectral properties, including quasiparticle energies (QP) and renormalization factors beyond the linearized QP equation. We validate the $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ and MPA-G\text{MPA-}GMPA-G approaches for a diverse set of systems: bulk Si, Na, and Cu, monolayer MoS2{\mathrm{MoS}}_{2}MoS2​, the NaCl ion pair, and the F2{\mathrm{F}}_{2}F2​ molecule. We show that, just as for MPA-W\text{MPA-}WMPA-W, an appropriate choice of frequency sampling in $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ is critical to guarantee computational efficiency and high accuracy. Moreover, the combined MPA-W\text{MPA-}WMPA-W and $\text{MPA-}\mathrm{\ensuremath{\Sigma}}$ scheme considerably reduces the cost of full-frequency self-energy calculations, especially for spectral band structures over a wide energy range.
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.
The dynamical GW self-energy approximation is a key computational tool to provide the fundamental spectrum of electronic systems. We reformulate this approximation, representing the particle and hole parts of the GW self-energy through a highly compact set of Cayley-transformed moment constraints. The Cayley transformation maps real frequencies to the unit circle, keeping the moments bounded as their order increases, ensuring numerical stability and allowing resolution to be focused on an energy range of interest. We calculate these Cayley-transformed moments via an efficient O[N4^44] scaling contour integration, and from them, construct a Hermitian upfolded Hamiltonian with a linearly scaling dimensionality with system size. A single-shot diagonalization of this effective Hamiltonian gives an explicit full-frequency G0W0 Green's function with manifestly real poles and non-negative spectral weights. This enables quasiparticle energies, satellite features, and their spectral weights to be obtained across the full G0W0 spectrum. Comparisons with exact G0W0 calculations and convergence across the GW100 test set and the larger Chlorophyll A molecule demonstrate substantially faster and more reliable convergence with moment order than an earlier monomial-moment approach. These Cayley moment representations therefore provide a stable, compact, and systematically improvable route to the complete spectral information of zero-temperature GW, without explicit frequency grids, plasmon-pole models and other common approximations, or analytic continuation.
Besides the proof of a modified Luttinger–Ward–Klein variational principle and a related self-consistency idea, there is not much new in principle in this paper. L. Hedin, Phys. Rev . 139, A796–823 (1965) Summary In this chapter a set of equations is formulated that determine the self-energy and the one-body Green's function in terms of the screened Coulomb interaction between classical charges. The equations contain a correction to the classical picture in terms of a vertex function. The physical meaning of the various contributions is discussed. The simplest approximation for the vertex yields the random phase approximation for the polarizability and the GW approximation for the self-energy. Various aspects of the GWA are analyzed, with a focus on the physics that is added beyond Hartree– Fock. A brief summary of model cases illustrates the domain of validity and the limits of the GWA. In this chapter we elaborate in more detail on the question of how to calculate the onebody Green's function from a Dyson equation with a self-energy kernel. In the previous chapter a scheme was introduced to design approximations to the self-energy. However, the question of where to stop, which pieces of physics to include and which to neglect, is not yet settled. Of course, there is no unique answer, besides the exact solution, but different strategies can be more or less advantageous in practice. In a system with a few electrons, for example, different aspects will be important than in a system with many electrons. Here we are mostly interested in solids, or more generally in extended systems. In such systems, screening plays an essential role: the interaction between two charges is strongly modified, in general reduced, by the rearrangement of all the other charges. It is therefore most convenient to reformulate the equations such that screening appears explicitly. Some steps in this direction can be found in earlier chapters, in particular in Sec. 8.3, the formulation of the Ψ [ G , W ]-functional of the screened interaction W instead of the Ф[ G , v c ]-functional of the bare v c . In Sec. 10.5 the screened interaction approximation for the self-energy is derived, with the self-energy as a product of the one-body Green's function and the screened interaction W .
In this thesis, the GW approximation (GWA, Green's function G times screened interaction W) and the Hubbard I approximation (HIA) are combined in a non-self-consistent one-shot calculation to determine the electronic structure of a one-dimensional strongly-correlated model. The scheme was chosen to incorporate both screening effects through the GWA and strong on-site correlations through the atomic HIA. The resulting self-energies of both methods are summed together and another term is subtracted to correct double-counting. The examined double-counting terms were the local self-energy of the GWA result, the GWA self-energy of the atomic model used in the HIA and lastly the impurity self-energy within the GWA. The self-energy of the atomic model in the HIA was solved analytically. It was found that all three approaches yield non-causal features, which increase with the on-site Coulomb repulsion, in the resulting spectral function. The local GWA self-energy was observed to perform best in terms of causality and computational effort. Changes in the resulting quasi-particle structure in the spectral function showed that screening effects and local correlations were included successfully.
The self-energy method for quantum impurity models expresses the correlation part of the self-energy in terms of the ratio of two Green's functions and allows for a more accurate calculation of equilibrium spectral functions than is possible directly from the one-particle Green's function [Bulla et al., J. Phys.: Condens. Matter 10, 8365 (1998)], for example, within the numerical renormalization group method. In addition, the self-energy itself is a central quantity required in the dynamical mean field theory of strongly correlated lattice models. Here, we show how to generalize the self-energy method to the time-dependent situation for the prototype model of strong correlations, the Anderson impurity model. We use the equation-of-motion method to obtain closed expressions for the local Green's function in terms of a time-dependent correlation self-energy, with the latter being given as a ratio of a one-particle time-dependent Green's function and a higher-order correlation function. We benchmark this self-energy approach to time-dependent spectral functions against the direct approach within the time-dependent numerical renormalization group method. The self-energy approach improves the accuracy of time-dependent spectral function calculations, and the closed-form expressions for the Green's function allow for a clear picture of the time evolution of spectral features at the different characteristic time scales. The self-energy approach is of potential interest also for other quantum impurity solvers for real-time evolution, including time-dependent density matrix renormalization group and continuous-time quantum Monte Carlo techniques.