Breakdown of the Migdal-Eliashberg theory in the strong-coupling adiabatic regime

In view of some recent works on the role of vertex corrections in the electron-phonon system, we readdress the important question of the validity of the Migdal-Eliashberg theory. Based on the solution of the Holstein model and 1/λ strong-coupling expansion, we argue that the standard Feynman-Dyson perturbation theory by Migdal and Eliashberg with or without vertex corrections cannot be applied if the electron-phonon coupling is strong (λ ⩾ 1) at any ratio of the phonon, ω and Fermi, E F energies. In the extreme adiabatic limit (ω ≪ E F ) of the Holstein model electrons collapse into self-trapped small polarons or bipolarons due to spontaneous translational-symmetry breaking at λ ≃ 1. With the increasing phonon frequency the region of the applicability of the theory shrinks to lower values of λ < 1.