Interacting electrons and bosons in the doubly screened $G\tilde{W}$ approximation: A time-linear scaling method for first-principles simulations

We augment the time-linear formulation of the Kadanoff-Baym equations for systems of interacting electrons and quantized phonons or photons with the $G\tilde{W}$ approximation, the Coulomb interaction $\tilde{W}$ being dynamically screened by both electron-hole pairs {\em and} bosonic particles. We also show how to combine different approximations to include simultaneously multiple correlation effects in the dynamics. The final outcome is a versatile framework comprising $2^{12}$ distinct diagrammatic methods, each scaling linearly in time and preserving all fundamental conservation laws. The dramatic improvement over current state-of-the-art approximations brought about by $G\tilde{W}$ is demonstrated in a study of the correlation-induced charge migration of the glycine molecule in an optical cavity.