The GW Approximation

In Green, the $GW$ approximation implemented is the fully self-consistent $GW$ approximation that implements Hedin’s ¹ GW approximation with its full frequency dependence and self-consistency on the imaginary axis, making the solution thermodynamically consistent and conserving.² Note that there are several variants of the $GW$ approximation that correspond to different equations and additional approximations, including non-selfconsistent, partially self-consistent, quasiparticle approximated and quasiparticle self-consistent variants.

In the $GW$ approximation ¹, the correlated self-energy is approximated as the sum of an infinite series of RPA-like `bubble’ diagrams. Implementation details of the code in Green are provided in our implementation paper.³

On the imaginary-time axis, ${(\Sigma^{GW})}^{\bf{k}}(\tau)$ reads $$ {(\Sigma^{GW})}^{\mathbf{k}}_{i\sigma,j\sigma}(\tau) = -\frac{1}{N_{k}}\sum_{\mathbf{q}}\sum_{ab} G^{\mathbf{k-q}}_{a\sigma,b\sigma}(\tau)\tilde{W}^{\mathbf{k},\mathbf{k-q},\mathbf{k-q},\mathbf{k}}_{ i a b j}(\tau) $$ where $\tilde{W}$ is the effective screened interaction tensor, defined as the difference between the full dynamically screened interaction $W$ and the bare interaction $\boldsymbol{U}$, i.e. $\tilde{W} = W - U$. Here and following, the indices $\set{i,j, k, l, a, b}$ are orbital indices, $\set{k,q}$ are crystal momentum, and $N_{k}$ is the number of momentum considered for a finite cluster. In the $GW$ approximation, the screened interaction $W$ is expressed in the frequency space as ¹ $$ \begin{aligned} &W^{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}_{\ i\ \ j \ k\ \ l}(i\Omega_{n}) = U^{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}_{\ i\ \ j \ k\ \ l}\nonumber\\ &+\frac{1}{N_{k}}\sum_{\mathbf{k}_{5}\mathbf{k}_{6}\mathbf{k}_{7}\mathbf{k}_{8}}\sum_{abcd}U^{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{5}\mathbf{k}_{6}}_{\ i\ \ j \ a\ \ b} \mathit{\Pi}^{\mathbf{k}_{5}\mathbf{k}_{6}\mathbf{k}_{7}\mathbf{k}_{8}}_{\ a\ b \ c\ \ d}(i\Omega_{n})W^{\mathbf{k}_{7}\mathbf{k}_{8}\mathbf{k}_{3}\mathbf{k}_{4}}_{\ c\ d \ \ k\ \ l}(i\Omega_{n}) \end{aligned} $$ where $\boldsymbol{\mathit{\Pi}}$ is the non-interacting polarization function $$ \mathit{\Pi^{\mathbf{k}_{1}\mathbf{k}_{2}\mathbf{k}_{3}\mathbf{k}_{4}}_{\ a\ b \ c\ \ d}(\tau)} = \sum_{\sigma}G^{\mathbf{k}_{1}}_{d\sigma,a\sigma}(\tau)G^{\mathbf{k}_{2}}_{b\sigma ,c\sigma}(-\tau)\delta_{\mathbf{k}_{1}\mathbf{k}_{4}}\delta_{\mathbf{k}_{2}\mathbf{k}_{3}}. $$ Green also provides an implementation of the GW approximation with exact two-component formalism with one-electron approximation (X2C-1e) for solving relativistic problems, such as those with spin-orbit coupling.⁴