Interaction of strongly correlated electrons and acoustical phonons
  We investigate the interaction of correlated electrons with acoustical phonons using the extended Hubbard-Holstein model in which both, the electron-phonon interaction and the on-site Coulomb repulsion are considered to be strong. The Lang-Firsov canonical transformation allows to obtain mobile polarons for which a new diagram technique and generalized Wick's theorem is used. This allows to handle the Coulomb repulsion between the electrons emerged into a sea of phonon fields (\textit{phonon clouds}). The physics of emission and absorption of the collective phonon-field mode by the polarons is discussed in detail. Moreover, we have investigated the different behavior of optical and acoustical phonon clouds when propagating through the lattice. In the strong-coupling limit of the electron-phonon interaction, and in the normal as well as in the superconducting phase, chronological thermodynamical averages of products of acoustical phonon-cloud operators can be expressed by one-cloud operator averages. While the normal one-cloud propagator has the form of a Lorentzian, the anomalous one is of Gaussian form and considerably smaller. Therefore, the anomalous electron Green's functions can be considered to be more important than corresponding polarons functions, i.e., pairing of electrons without phonon-clouds is easier to achieve than pairing of polarons with such clouds. 
V.A.Moskalenko, P.Entel, D.F.Digor