Electron-Phonon Coupling from $\textit{Ab Initio}$ Linear-Response Theory within the $GW$ Method: Correlation-Enhanced Interactions and Superconductivity in Ba$_{1-x}$K$_x$BiO$_3$

We present a new first-principles linear-response theory of changes due to perturbations in the quasiparticle self-energy operator within the $GW$ method. This approach, named $GW$ perturbation theory ($GW$PT), is applied to calculate the electron-phonon ($e$-ph) interactions with the full inclusion of the $GW$ non-local, energy-dependent self-energy effects, going beyond density-functional perturbation theory. Avoiding limitations of the frozen-phonon technique, $GW$PT gives access to $e$-ph matrix elements at the $GW$ level for all phonons and scattering processes, and the computational cost scales linearly with the number of phonon modes (wavevectors and branches) investigated. We demonstrate the capabilities of $GW$PT by studying the $e$-ph coupling and superconductivity in Ba$_{0.6}$K$_{0.4}$BiO$_3$. We show that many-electron correlations significantly enhance the $e$-ph interactions for states near the Fermi surface, and explain the observed high superconductivity transition temperature of Ba$_{0.6}$K$_{0.4}$BiO$_3$ as well as its doping dependence.