Polarons and Bipolarons in Holstein and Holstein $t-J$ models by   Dynamical Mean Field Theory
  The results of a comparative study of electronic spectral properties as well as phonon Probability Distribution Function (PDF) are reported.   The study is done using Dynamical Mean Field Theory for the half-filled spinless and spinful Holstein model and for a single hole in Holstein t-J model.   Qualitative changes in phonon PDF signals a polaronic crossover while an electronic Metal Insulator Transition (MIT) due to pairing into a bipolaronic state can be seen from a gap in the electronic DOS or by the vanishing of the quasi particle spectral weight. MIT can be observed only at sufficiently high density provided the Coulomb repulsion is neglected. In the limit of low density the carrier show a polaronic behavior trough the development of a definite polarization shifting the average value of the phonon PDF. At large densities occupied and empty sites result an a bimodal the phonon PDF.   At the polaron crossover fluctuation of phonon coordinates tends to be larger than those at any other coupling. In this regime Born-Oppenheimer approximation is shown to fail also when phonon frequency is much less then electron bandwidth. For a single hole in the t-J model a further source of localization is due to magnetic superexchange energy which tends to localize the spin-defect spreading due to the presence of the hole. 
S.Ciuchi, M.Capone, E.Cappelluti and G. Sangiovanni