A study of momentum dependent electron-phonon-couplings in Ab-initio Cuprate Hamiltonians

We use the functional renormalization group (fRG) to study the impact of strong electron-phonon couplings on effective Cuprate Hamiltonians with up to three bands. In the single band case, we reproduce the transition from an antiferromagnet (AF) to a charge density wave (CDW) seen in the Hubbard-Holstein model. For electron-phonon couplings without momentum dependence, we show the transition from a charge density wave to a s-superconductivity as a function of the phonon frequency. We then look at changes to the phase diagram due to the momentum dependent A_1g and B_1g phonon modes of the Cuprate systems. The two phonon modes enhance extended s-wave and d_x²-y² superconducting fluctuations along the AF-CDW transition line. The modes also shift the transition into the CDW phase as we increase the electron-phonon coupling. The effects of the A_1g mode are unchanged in the two band model while B_1g modes with frequency less than the separation between d_z² and d_x²-y² orbitals appear to drive momentum dependent charge orders. Finally, we consider the role of the frequency of the phonon modes and doping in the three band case.