Electron-phonon Vertex in the Two-dimensional One-band Hubbard Model

Using Quantum Monte Carlo techniques, we study the effects of electronic correlations on the effective electron-phonon (el-ph) coupling in a two-dimensional one-band Hubbard model. We consider a momentum-independent bare ionic el-ph coupling. In the weak- and intermediate-coupling regimes, we find that the on-site Coulomb interaction $U$ acts to effectively suppress the ionic el-ph coupling at all electron- and phonon- momenta. In this regime, our numerical simulations are in good agreement with the results of perturbation theory to order $U^2$. However, entering the strong-coupling regime, we find that the forward scattering process stops decreasing and begins to increase as a function of $U$, leading to an effective el-ph coupling which is peaked in the forward direction. This increase comes from the irreducible vertex corrections, which depend crucially on the renormalized electronic structure of the strongly correlated system.