The effect on the stability of the superconducting phase due the addition of an electron–phonon interaction to a repulsive Hubbard model is studied. Our Hubbard–Fröhlich Hamiltonian includes electron hoping, the on-site Coulomb repulsion, vibrating ions (phonons) and the electron–phonon interaction. A Lang–Firsov transformation is used to integrate out the phonon degrees of freedom. The transformation reduces the model to simple a Hubbard Hamiltonian with an additional long-range electron–electron attraction. A variational Monte Carlo technique, with a projected BCS trial function, is used to investigate the ground state energies of our transformed Hubbard–Fröhlich Hamiltonian. For various electron densities, with a d-wave superconducting order parameter, it is found that the inclusion of the electron-phonon interaction significantly enhances the condensation energy (the energy required to break paired electrons). We show that increasing the strength of the electron-phonon interaction increases the condensation energy. It is also found that even with an infinite on-site repulsion, where the resonating valence bond state cannot exist, the EPI does still lead to a d-wave superconducting state. In addition we examine, analytically, the coexistence of ferromagnetism and superconductivity. Allowing different masses for spin-up and spin-down electrons in a BCS-type Hamiltonian two new branches in the energy spectrum are found. Including a spatially varying order parameter a new expression for the pairing amplitude of finite momentum pairs is derived
