Effective electron-phonon coupling and polaronic transition in the presence of strong correlation

We study the Hubbard-Holstein model using slave-boson mean-field and a variational Lang-Firsov transformation. We identify weak and strong e-ph coupling solutions, whose stability depends both on the bare e-ph coupling and on the correlation strength. At mean field level the evolution from weak to strong electron-phonon coupling occurs via a first-order polaronic transition if the adiabatic parameter is below a critical value. In the strongly correlated regime and in the adiabatic limit, the region in which the weak-coupling solution is stable is sizeably enlarged with respect to the weakly correlated system and the Mott metal-insulator transition is found to be robust with respect to e-ph interaction.