The Holstein-Hubbard model as a laboratory for understanding electron-phonon effects in correlated metals

The standard theory of electron-phonon interactions in metals obtains an electron-phonon coupling from linearizing the Kohn-Sham potential in atomic displacements and computes a scattering amplitude by leading order perturbation theory. In materials where the physics is dominated by strong electron correlations, the Kohn-Sham potential is not a good approximation to the low energy electronic energetics and many-body renormalizations change the electronic energy scales, potentially invalidating the Migdal approximation. To investigate these effects, we compute and analyze the electron self energy of the Holstein-Hubbard model using the single-site dynamical mean field approximation. We delineate parameter regimes where the self energy can be reliably deconstructed into electron-electron and electron-phonon parts and in these regimes show that the effective electron-phonon coupling strength is strongly renormalized by correlations. Our results provide insight into the long-standing mystery of the relative weakness of electron-phonon effects in the resistivity of high-T_c cuprates and provide guidance towards the development of a general theory of electron-phonon coupling in correlated materials.