| We use the dynamical mean-field method to develop a theory of electron-phonon interactions in correlated systems. The theory is valid in the adiabatic regime of phonon frequency much smaller than electron bandwidth or electron-electron interaction scale. It generalizes the conventional Migdal-Eliashberg theory by including lowest-order quantum lattice fluctuations and by allowing for arbitrary local electronic correlations. We derive two complementary and numerically tractable formulations of the theory, valid for temperatures much smaller and comparable to phonon frequency, respectively. We determine the limits of validity of each method and the relevant expansion parameters. In particular, we show that the system becomes unstable towards strong lattice distortions at a critical value of the electron-phonon coupling. We analyze this polaronic instability in great detail, using both analytic and numeric (quantum Monte Carlo) techniques. We apply the theory to calculate, among other things, isotope effects on electron effective mass, ferromagnetic transition temperature, electron spectral density, and resistivity. |
