Electron-phonon interactions in the presence of strong correlations

We investigate the effect of strong electron-electron repulsion on the electron-phonon interaction from a Fermi-liquid point of view. In particular, we show that the strong interaction is responsible for vertex corrections, which are strongly dependent on the ${\mathit{v}}_{\mathit{F}}$q/ω ratio, where ${\mathit{v}}_{\mathit{F}}$ is the Fermi velocity and q and ω are the transferred momentum and frequency, respectively. These corrections generically lead to a strong suppression of the effective coupling between quasiparticles mediated by a single phonon exchange in the ${\mathit{v}}_{\mathit{F}}$q/ω≫1 limit. However, such effect is not present when ${\mathit{v}}_{\mathit{F}}$q/ω≪1. Analyzing the stability criterion for the compressibility, which involves the effective interactions in the dynamical limit, we show that a sizable electron-phonon interaction can push the system towards a phase separation instability. A detailed analysis is then carried out using a slave-boson approach for the infinite-U three-band Hubbard model describing the basic structure of a ${\mathrm{CuO}}_2$ plane in copper oxides. In the presence of a coupling between the local hole density and a dispersionless optical phonon, we explicitly confirm the strong dependence of the hole-phonon coupling on the transferred momentum versus frequency ratio. We also find that the exchange of phonons leads to an unstable phase with negative compressibility already at small values of the bare hole-phonon coupling. Close to the unstable region, we detect Cooper instabilities both in s- and d-wave channels supporting a possible connection between phase separation and superconductivity in strongly correlated systems.