Interplay between electron-phonon and Coulomb interactions in the honeycomb lattice

We study the impact of electron-phonon interactions on the many-body instabilities of electrons in the honeycomb lattice and their interplay with local and non-local short-ranged Coulomb interactions at charge neutrality. Therefore, we consider the in-plane optical phonon branches giving the most important contribution to the electron-phonon coupling and calculate the effective phonon-mediated electron-electron interaction by integrating out the phonon modes. The ordering tendencies are studied by means of a momentum-resolved functional renormalization group approach allowing for an unbiased investigation of the appearing instabilities. In the case of an exclusive and supercritical phonon-mediated interaction, we find a nematic ground state being favored over the s-wave superconducting state conjectured from a simple mean-field treatment. We further discuss the influence of phonon-mediated interactions on the instabilities induced by onsite, nearest neighbor and next-nearest neighbor density-density interactions. We find an extension of the parameter regime of the spin density wave order going along with an increase of the critical scales where ordering occurs.