Equilibrium and out-of-equilibrium realistic phonon self-energy free from overscreening

In model Hamiltonians, like Fröhlich's, the electron-phonon interaction is assumed to be screened from the beginning. The same occurs when this interaction is obtained by using the state-of-the-art density functional perturbation theory as starting point. In this work I formally demonstrate that these approaches are affected by a severe overscreening error. By using an out-of-equilibrium many-body technique I discuss how to merge the many-body approach with density functional perturbation theory in order to correct the overscreening error. A symmetric statically screened phonon self-energy is obtained by downfolding the exact Baym-Kadanoff equations. The statically screened approximation proposed here is shown to have the same long-range spatial limit of the exact self-energy and to respect the fluctuation-dissipation theorem. The doubly screened approximation, commonly used in the literature, is shown, instead, to be overscreened, to violate several many-body properties and to have a wrong spatial long-range decay. The accuracy of the proposed approximation is tested against the exact solution of an extended model Fröhlich Hamiltonian and it is applied to a paradigmatic material: MgB2. I find that the present treatment enhances the linewidths by 57% with respect to what has been previously reported for the anomalous E2g mode. I further discover that the A2u mode is also anomalous (its strong coupling being completely quenched by the overscreened expression). The present results deeply question methods based on state-of-the-art approaches and impact a wide range of fields such as thermal conductivity, phononic instabilities, and nonequilibrium lattice dynamics.