Effective electron-electron and electron-phonon interactions in the   Hubbard-Holstein model
  We investigate the interplay between the electron-electron and the electron-phonon interaction in the Hubbard-Holstein model. We implement the flow-equation method to investigate within this model the effect of correlation on the electron-phonon effective coupling and, conversely, the effect of phonons in the effective electron-electron interaction. Using this technique we obtain analytical momentum-dependent expressions for the effective couplings and we study their behavior for different physical regimes. In agreement with other works on this subject, we find that the electron-electron attraction mediated by phonons in the presence of Hubbard repulsion is peaked at low transferred momenta. The role of the characteristic energies involved is also analyzed. 
G. Aprea, C. Di Castro, M. Grilli, and J. Lorenzana