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GW Approximation

Overview

The GW approximation goes beyond Hartree-Fock by including frequency-dependent (dynamical) screening of the Coulomb interaction. The name "GW" reflects that the self-energy is constructed as a convolution of the Green's function \(G\) and the dynamically screened interaction \(W\). While HF treats the Coulomb interaction as static, the GW approximation replaces \(V\) with a frequency-dependent screened interaction \(W\) that accounts for the polarization response of the electron gas.

The GW self-consistent workflow extends the Hartree-Fock calculation by incorporating three additional classes -- PolLat, WLat, and SigmaGWC -- that work together with the existing GreenInt, SigmaHartree, and SigmaFock classes.

Non-Interacting Green's Function

The calculation begins with the non-interacting Green's function \(G_0\), constructed from the non-interacting Hamiltonian \(H_0\). In imaginary time, \(G_0\) is defined as:

\[ G^{p,q}_{0\,ij\sigma}(\mathbf{R}-\mathbf{R}', \tau - \tau') = -\langle T_\tau c^p_{i\sigma}(\mathbf{R}, \tau)\, c^{\dagger\, q}_{j\sigma}(\mathbf{R}', \tau') \rangle_0 \]

where \(T_\tau\) is the imaginary-time ordering operator and \(\langle \cdots \rangle_0\) is the expectation value in the grand-canonical ensemble of \(H_0\). The Matsubara-frequency representation is obtained via Fourier transform:

\[ G^{pq}_{0\,ij\sigma}(\mathbf{k}, i\omega_n) = \frac{1}{N_\mathbf{k}} \int_0^\beta d(\tau - \tau') \sum_{\mathbf{R},\mathbf{R}'} G^{pq}_{0\,ij\sigma}(\mathbf{R}-\mathbf{R}', \tau-\tau') e^{i(\mathbf{k}\cdot(\mathbf{R}-\mathbf{R}') - \omega_n \tau)} \]

QAssemble uses the discrete Lehmann representation (DLR) for compact and accurate representation of both imaginary-time and Matsubara-frequency Green's functions. The GreenBare class constructs \(G_0\) from \(H_0\) produced by the NIHamiltonian class.

Dyson Equation

The interacting Green's function \(G\) is obtained through the Dyson equation:

\[ G^{pq}_{ij\sigma}(\mathbf{k}, i\omega_n) = G^{pq}_{0\,ij\sigma}(\mathbf{k}, i\omega_n) + \sum_{r,s} \sum_{k,l} G^{pr}_{0\,ik\sigma}(\mathbf{k}, i\omega_n)\, \Sigma^{rs}_{kl\sigma}(\mathbf{k}, i\omega_n)\, G^{sq}_{lj\sigma}(\mathbf{k}, i\omega_n) \]

where \(\Sigma\) is the irreducible electron self-energy. In the GW approximation, the total self-energy includes three contributions:

\[ \Sigma = \Sigma^H + \Sigma^F + \Sigma^{C,GW} \]

The GreenInt class solves the Dyson equation at each iteration, receiving \(G_0\) from GreenBare along with \(\Sigma^H\) from SigmaHartree, \(\Sigma^F\) from SigmaFock, and \(\Sigma^{C,GW}\) from SigmaGWC.

Irreducible Polarizability

The irreducible polarizability \(P\) is computed from the current Green's function by evaluating the two-particle correlation function (particle-hole bubble). In the Matsubara frequency domain:

\[ P^{p\sigma;q\sigma'}_{ijkl}(\mathbf{k}, i\nu_n) = \int_0^\beta d\tau \sum_{\mathbf{R}} G^{qp}_{ki\sigma'}(-\mathbf{R}, -\tau) \times G^{pq}_{lj\sigma}(\mathbf{R}, \tau) \delta_{\sigma\sigma'} e^{i(\mathbf{k}\cdot\mathbf{R} - \nu_n \tau)} \]

where \(i\nu_n = 2n\pi/\beta\) are bosonic Matsubara frequencies, and the integration over imaginary time \(\tau\) performs the convolution of two fermionic Green's functions. The Kronecker delta \(\delta_{\sigma\sigma'}\) indicates that the electron and hole must have the same spin in the non-interacting bubble.

The PolLat class (child of BLatDyn) computes this quantity. It accepts the interacting Green's function \(G\) and produces the bosonic response function that describes how the electron density responds to screened perturbations.

Screened Coulomb Interaction

The screened interaction \(W\) accounts for how the bare Coulomb potential \(V\) is reduced (screened) by the polarization of the surrounding electron gas. It is obtained by the Dyson equation:

\[ W^{p\sigma;q\sigma'}_{ijkl}(\mathbf{k}, i\nu_n) = V^{p\sigma;q\sigma'}_{ijkl}(\mathbf{k}) + \sum_{rs} \sum_{i'j'k'l'} V^{p\sigma;r\sigma'}_{ii'j'l}(\mathbf{k})\, P^{r\sigma;s\sigma'}_{i'k'l'j'}(\mathbf{k}, i\nu_n)\, W^{s\sigma;q\sigma'}_{k'jkl'}(\mathbf{k}, i\nu_n) \]

Formally, this represents an infinite resummation of bubble diagrams where the interaction line is repeatedly dressed by polarization insertions. The matrix equation must be solved for each momentum \(\mathbf{k}\) and bosonic frequency \(i\nu_n\).

The WLat class (child of BLatDyn) constructs \(W\) from \(P\) and \(V\).

GW Correlation Self-Energy

The GW correlation self-energy captures dynamical correlations through the frequency-dependent screened interaction. The correlation part is given by:

\[ \Sigma^{C,GW\, pq}_{ij\sigma}(\mathbf{k}, i\omega_n) = -\int_0^\beta d\tau \sum_{\mathbf{R}} \sum_{kl} G^{qp}_{lk\sigma}(\mathbf{R}, \tau) \times W^{C\, p\sigma q\sigma'}_{ijkl}(\mathbf{R}, \tau) \delta_{\sigma\sigma'} e^{i(\mathbf{k}\cdot\mathbf{R} - \omega_n \tau)} \]

where \(W^C = W - V\) is the dynamical part of the screened interaction. The subtraction isolates the dynamical screening contribution, avoiding double-counting since static Coulomb interaction effects are already incorporated through \(\Sigma^F\).

The SigmaGWC class (child of FLatDyn) computes this self-energy by convolving the Green's function \(G\) with the correlation part of the screened interaction \(W^C\).

Self-Consistent GW Loop

The full GW self-consistent cycle proceeds as:

  1. Non-interacting setup: Construct \(H_0\) (NIHamiltonian) and \(V\) (VBare). Build \(G_0\) (GreenBare).

  2. Initialize Green's function: Set \(G = G_0\) for the first iteration, or use the previous iteration's \(G\).

  3. Polarizability: Compute \(P = GG\) (PolLat).

  4. Screened interaction: Solve the bosonic Dyson equation \(W = V + VPW\) (WLat).

  5. Correlation self-energy: Compute \(\Sigma^{C,GW} = -GW^C\) (SigmaGWC).

  6. Static self-energies: Compute \(\Sigma^H\) (SigmaHartree) and \(\Sigma^F\) (SigmaFock) from the updated density.

  7. Dyson equation: Combine all self-energy contributions via the Dyson equation to obtain the updated \(G\) (GreenInt): \(\Sigma = \Sigma^H + \Sigma^F + \Sigma^{GW}\), then \(G = G_0 + G_0 \Sigma G\).

  8. Density and chemical potential: Extract \(n = -G(\tau = \beta^-)\) and adjust \(\mu\) to maintain the target filling.

  9. Convergence check: If converged, proceed to post-processing; otherwise, return to step 3 with the new \(G\).

Post-Processing: Quasiparticle Properties

After the GW loop converges, two additional classes extract quasiparticle properties from the frequency-dependent self-energy:

Renormalization Factor (ZFactor)

The ZFactor class extracts the quasiparticle renormalization factor from the converged GW self-energy. It receives \(\Sigma(\mathbf{k}, i\omega_n)\) stored in DLR representation and computes the inverse renormalization factor:

\[ Z(\mathbf{k})^{-1} = 1 - \frac{\partial \Sigma(\mathbf{k}, i\omega_n)}{\partial i\omega_n} \bigg|_{\omega=0} \]

The \(\mathbf{k}\)-resolved renormalization factor \(Z(\mathbf{k})\) encodes the dynamical mass enhancement and spectral weight transfer from coherent quasiparticle peaks to incoherent satellite structures.

Static Self-Energy (SigmaStc)

The SigmaStc class computes the static limit of the self-energy \(\Sigma(\mathbf{k}, \omega=0)\) from the full frequency-dependent GW result. It receives \(\Sigma(\mathbf{k}, i\omega_n)\) in DLR representation and evaluates its zero-frequency limit. This static self-energy captures the shifts in quasiparticle energies.

Quasiparticle Hamiltonian

Together, these quantities define the quasiparticle Hamiltonian:

\[ H_{QP}(\mathbf{k}) = \sqrt{Z(\mathbf{k})} \left( H_0(\mathbf{k}) + \Sigma(\mathbf{k}, \omega=0) \right) \sqrt{Z(\mathbf{k})} \]

where \(H_0(\mathbf{k})\) is the non-interacting Hamiltonian, \(\Sigma(\mathbf{k}, \omega=0)\) is the static self-energy from SigmaStc, and \(Z(\mathbf{k})\) is the renormalization factor from ZFactor. The symmetric placement of \(\sqrt{Z}\) ensures Hermiticity when \(Z(\mathbf{k})\) is a matrix in orbital space. Diagonalizing \(H_{QP}(\mathbf{k})\) yields the renormalized quasiparticle band structure, incorporating both the static self-energy shift and the dynamical mass enhancement.

References

  • L. Hedin, Phys. Rev. 139, A796 (1965).
  • M. S. Hybertsen & S. G. Louie, Phys. Rev. B 34, 5390 (1986).
  • R. Godby, M. Schluter & L. Sham, Phys. Rev. B 37, 10159 (1988).
  • S. Massidda et al., Phys. Rev. Lett. 74, 2323 (1995).
  • W. G. Aulbur, L. Jonsson & J. W. Wilkins, in Solid State Physics Vol. 54 (Academic Press, 2000).
  • P. Fulde, in Semiconductors and Insulators (Springer, 1995).
  • J. Kaye, K. Chen & H. U. Strand, Comput. Phys. Commun. 280, 108458 (2022).
  • J. Kaye, K. Chen & O. Parcollet, Phys. Rev. B 105, 235115 (2022).