Holstein model in infinite dimensions at half-filling

The normal state of the Holstein model is studied at half-filling in infinite dimensions and in the adiabatic regime. The dynamical mean-field equations are solved using perturbation expansions around the extremal paths of the effective action for the atoms. We find that the Migdal-Eliashberg expansion breaks down in the metallic state if the electron-phonon coupling $\lambda$ exceeds a value of about 1.3 in spite of the fact that the formal expansion parameter $\lambda \omega_0/E_F$ ($\omega_0$ is the phonon frequency, $E_F$ the Fermi energy) is much smaller than 1. The breakdown is due to the appearance of more than one extremal path of the action. We present numerical results which illustrate in detail the evolution of the local Green's function, the self-energy and the effective atomic potential as a function of $\lambda$.