Coulomb and electron-phonon interactions in metals

An accurate and consistent theory of phonons in metals requires that all long-range Coulomb interactions between charged particles (electrons and ions) be treated on equal footing. So far, all attempts to deal with this nonperturbative system were relying on uncontrolled approximations in the absence of small parameters. In this paper, we develop the diagrammatic Monte Carlo approach for a two-component Coulomb system that obtains the solution to this fundamental problem in an approximation-free way by computing vertex corrections from higher-order skeleton graphs. The feasibility of the method is demonstrated by calculating the spectrum of longitudinal acoustic phonons in a simple cubic lattice, determining their sound velocity, and obtaining the phonon spectral densities by analytic continuation of the Matsubara-Green's functions. Final results are checked against the lowest-order fully self-consistent $GW$ approximation in both adiabatic and nonadiabatic regimes.