GW+C: Beyond-GW Cumulant Expansion

• GW+C is a many-body technique that extends the GW approximation by incorporating cumulant expansion to restore satellite features and preserve spectral moments.
• It systematically includes multi-boson shake-up processes, accurately capturing quasiparticle weight reductions and spectral weight distribution in systems like solids, nanostructures, and molecules.
• Algorithmic optimizations such as Padé fitting and efficient FFT enable computationally efficient spectral function assembly, aligning theoretical results closely with experimental photoemission data.

The GW plus cumulant (GW+C) approach is a beyond-GW method in many-body perturbation theory for the calculation of one-particle Green’s functions, spectral functions, and satellite features in the electronic structure of solids, molecules, and low-dimensional systems. GW+C combines the non-perturbative exponential cumulant resummation with the GW approximation for the self-energy, restoring crucial satellite structures and correcting spectral features that are inadequately described within GW alone. The method has gained prominence due to its ability to preserve spectral moments, correct quasiparticle weights, systematically capture multi-boson (plasmon or shake-up) effects, and yield improved agreement with experimental photoemission data across diverse material systems.

1. Theoretical Foundation: Cumulant Expansion and GW Self-Energy

The GW+C methodology is grounded in the cumulant expansion of the one-particle Green’s function, which encapsulates the effects of dynamic screening and vertex corrections missed by the GW approximation. For a retarded Green’s function in the time domain, the cumulant ansatz is

$G_k^r(t) = G_{0,k}^r(t) e^{C_k(t)}$

with

$G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$

and $C_k(t)$ is the cumulant. The cumulant is formally related to the improper Dyson self-energy via an exact mapping,

$$C_k(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} d\omega \frac{Im\,\Sigma_k^{imp}(\omega+\varepsilon_k)}{\omega^2}\left(e^{-i\omega t} + i\omega t -1\right)$$

where $\Sigma_k^{imp}$ includes multiple scattering terms ($\Sigma_k^* + \Sigma_k^* G_0 \Sigma_k^* + \cdots$).

This cumulant formalism preserves the first energy moment of the spectral function,

$A_k(\omega) = -\frac{1}{\pi} Im G_k^r(\omega)$

ensuring that the center of gravity of the spectral weight remains unmoved (i.e., $M_1 = \varepsilon_k$), thus preventing spurious static shifts and maintaining sum rules (Mayers et al., 2016).

In practice, $Im\,\Sigma_k^{imp}$ is computed using non-self-consistent $G_0W_0$ with RPA screening, and $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$0 is obtained numerically. Fourier transformation yields the GW+C spectral function, which naturally divides into a quasiparticle peak and an infinite ladder of satellites (Gumhalter et al., 2016, Caruso et al., 2016).

2. Practical Implementation and Algorithmic Structure

The core GW+C computational workflow involves:

1. Mean-field calculation to obtain ground-state electronic structure (DFT or Hartree-Fock).
2. RPA dielectric function and $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$1 self-energy: Computation of $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$2 on a fine frequency grid, using either full-frequency integration or a plasmon-pole model (Nakamura et al., 2015, Caruso et al., 2016).
3. Cumulant construction: Numerical evaluation of the time-domain cumulant via the Landau form or integral over $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$3, requiring efficient quadrature or FFT.
4. Spectral function assembly: Fourier transform of $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$4 to frequency space, yielding $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$5.
5. Convolution with the QP propagator: When separating the cumulant into QP and satellite parts (common in molecular applications) (Loos et al., 2024, Kockläuner et al., 2024).

Several algorithmic optimizations have been developed, including Padé fitting for pole structure, symmetry-based decoupling of core and valence spaces, and efficient $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$6 scaling for large molecules via basis function screening (Kockläuner et al., 2024).

3. Applications in Solids, Nanostructures, and Molecules

GW+C has been widely deployed in the ab initio simulation of core-level spectra, valence satellites, and quasiparticle renormalization in diverse systems:

• Homogeneous Electron Gas (HEG): GW+C remedies the spurious “plasmaron” satellite in pure GW theory, generating broadened plasmonic polaron bands accurately shifted by the plasmon energy, with satellite intensity and renormalization in accord with experiment (Caruso et al., 2016).
• 2D Electron Gas (2DEG): The GW+C method removes the unphysical plasmaron pole predicted by GW, predicting smooth satellite tails and matching tunneling spectra and derivatives observed experimentally (Lischner et al., 2014).
• Doped Graphene: The separation between the main quasiparticle and the first plasmon-induced satellite is considerably reduced in GW+C (e.g., $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$7 eV vs $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$8 eV in GW), and only GW+C eliminates the unphysical extra root in Dyson’s equation (Lischner et al., 2013).
• Bulk Semiconductors (Si): GW+C recovers plasmon satellites in Si valence bands at correct energies and with asymmetric lineshapes, improving on $G_{0,k}^r(t) = -i \theta(t) e^{-i\varepsilon_k t}$9 which lacks satellite structure (Gumhalter et al., 2016).
• Organic Conductors and Transition-Metal Oxides: GW+C captures the broad incoherent satellites and transfer of spectral weight seen in ARPES of (TMTSF)$C_k(t)$0PF$C_k(t)$1 and SrVO$C_k(t)$2, giving Z-factors and satellite intensities in better agreement with experiment than GW (Nakamura et al., 2015).
• Molecules: GW+C corrects GW’s satellite positions and intensities in molecular photoemission. For example, in small 10-electron systems and in $C_k(t)$3-conjugated acenes, it yields satellite splittings and intensities within $C_k(t)$4 eV of experiment when core/valence decoupling and basis set convergence are properly addressed (Kockläuner et al., 2024, Loos et al., 2024).

4. Physical Interpretation: Multi-Boson Effects and Vertex Corrections

The GW+C approach systematically resums an infinite series of bosonic shake-up processes (plasmon, phonon, or electron-hole pair emission) through its exponential cumulant structure. In the spectral function, this yields:

• A main quasiparticle peak (with renormalized weight $C_k(t)$5)
• Multiple satellite features (Poisson-like hierarchy for pure-boson vertex cases)
• Proper distribution of spectral weight between coherent and incoherent parts
• Elimination of unphysical satellite branches (e.g., plasmaron modes present in GW)

The cumulant expansion provides a leading-order "vertex correction" in the spirit of conserving diagrams, correcting for deficiencies of the GW self-energy, and capturing essential many-boson coupling absent in a simple Dyson solution (Gumhalter et al., 2016, Lischner et al., 2013).

5. Innovations, Limitations, and Extensions

Key strengths of GW+C include:

• Rigorous conservation of spectral moments and QP renormalization factors (Mayers et al., 2016).
• Robust reproduction of multi-satellite structure and improved agreement with experiment for both satellites and main lines across solids and molecules (Loos et al., 2024, Kockläuner et al., 2024).
• Computational efficiency: Post-processing step after GW calculations, with negligible extra scaling.
• Systematic inclusion of long-range plasmon fluctuation diagrams and multi-boson corrections, crucial for featureless backgrounds and asymmetric lineshapes (Nakamura et al., 2015, Caruso et al., 2016).

Limitations and open directions:

• Satellite spacing errors: For the improper-retarded cumulant, satellite separation inherits G$C_k(t)$6W$C_k(t)$7 plasmon energy (e.g., $C_k(t)$8 in the HEG) rather than the physical plasmon pole; higher-order cumulants or self-consistency may improve this (Mayers et al., 2016).
• Missing lifetime/broadening for $C_k(t)$9: Spurious sharp features may arise due to the on-shell structure of $$C_k(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} d\omega \frac{Im\,\Sigma_k^{imp}(\omega+\varepsilon_k)}{\omega^2}\left(e^{-i\omega t} + i\omega t -1\right)$$0.
• No explicit non-linear screening response included in standard linear cumulant approaches: Recent work utilizes real-time time-dependent DFT to include non-linear density feedback in the cumulant, showing essential corrections in core-level strong-coupling limits (Tzavala et al., 2020).
• Basis set convergence and CVS-like decoupling: For molecules, high-quality description of satellites requires diffuse-augmented and core-polarized basis sets, and removal of core–valence cross-terms (Kockläuner et al., 2024).
• Quasiparticle energies: GW+C slightly reduces QP energies versus G$$C_k(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} d\omega \frac{Im\,\Sigma_k^{imp}(\omega+\varepsilon_k)}{\omega^2}\left(e^{-i\omega t} + i\omega t -1\right)$$1W$$C_k(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} d\omega \frac{Im\,\Sigma_k^{imp}(\omega+\varepsilon_k)}{\omega^2}\left(e^{-i\omega t} + i\omega t -1\right)$$2, occasionally worsening already good QP predictions (Loos et al., 2024).

6. Summary Table: Comparison of GW and GW+C Spectral Features

Feature	GW	GW+C
Quasiparticle energies	Accurate	Slightly reduced or unchanged
Satellite structure	Often one (spurious)	Multiple, correct positions and intensities
Plasmon/phonon coupling	Single emission	Multi-boson shake-up series
Vertex corrections	Absent	Included at leading order via cumulant
Physical plasmon dispersion	Can be incorrect	Largely correct, with some limitations
Computational cost	Baseline	$$C_k(t) = \frac{1}{\pi} \int_{-\infty}^{\infty} d\omega \frac{Im\,\Sigma_k^{imp}(\omega+\varepsilon_k)}{\omega^2}\left(e^{-i\omega t} + i\omega t -1\right)$$3 same as GW, efficient post-processing

GW+C hence provides a non-perturbative, physically consistent treatment of dynamic correlation effects in spectral functions, applicable to metallic, semiconducting, and molecular systems, and has become a reference standard for interpreting and predicting photoemission and XPS/ARPES spectral phenomena where satellites and many-body coupling are dominant (Mayers et al., 2016, Loos et al., 2024, Nakamura et al., 2015, Kockläuner et al., 2024). For strongly non-linear regimes or when higher accuracy is needed, extensions beyond linear-response or to higher cumulant orders are actively being developed (Tzavala et al., 2020).