Instabilities on graphene's honeycomb lattice with electron-phonon interactions

We study the impact of electron-phonon interactions on the many-body instabilities of electrons on the honeycomb lattice and their interplay with repulsive local and nonlocal Coulomb interactions at charge neutrality. To that end, we consider in-plane optical phonon modes with wave vectors close to the Gamma point as well as to the K, -K points and calculate the effective phonon-mediated electron-electron interaction by integrating out the phonon modes. 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 Kekule and a nematic bond ordering tendency being favored over the s-wave superconducting state. The competition between the different phonon-induced orderings clearly shows a repulsive interaction between phonons at small-and large-wave-vector transfers. We further discuss the influence of phonon-mediated interactions on electronically driven instabilities induced by on-site, 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 and a suppression of competing orders.