  The evolution of the properties of a finite density electronic system as the
electron-phonon coupling is increased are investigated in the
  Holstein model using the Dynamical Mean-Field Theory (DMFT).
  We compare the spinless fermion case, in which only isolated polarons can be
formed, with the spinful model in which the polarons can bind and form
bipolarons. In the latter case, the bipolaronic binding occurs through a
metal-insulator transition. In the adiabatic regime in which the phonon energy
is small with respect to the electron hopping we compare numerically exact DMFT
results with an analytical scheme inspired by the Born-Oppenheimer procedure.
Within the latter approach,a truncation of the phononic Hilbert space leads to
a mapping of the original model onto an Anderson spin-fermion model. In the
anti-adiabatic regime (where the phonon energy exceeds the electronic scales)
the standard treatment based on Lang-Firsov canonical transformation allows to
map the original model on to an attractive Hubbard model in the spinful case.
The separate analysis of the two regimes supports the numerical evidence that
polaron formation is not necessarily associated to a metal-insulator
transition, which is instead due to pairing between the carriers. At the
polaron crossover the Born-Oppenheimer approximation is shown to break down due
to the entanglement of the electron-phonon state.
