Limits of Validity for Migdal-Eliashberg Theory: Role of Polarons and Bipolarons

It is widely believed that in the adiabatic limit the Fermi liquid state of an electron-phonon system, described by Migdal-Eliashberg theory, remains stable until the dressed phonon softens. Our variational and analytic analysis of the prototypical Holstein model shows that, in a wide range of fillings both in 3D and 2D, a polaronic/bipolaronic state emerges before phonon softening; at small filling in 3D this happens already at weak coupling. We show that a polaronic/bipolaronic state emerges, upon increasing coupling, via an intermediate pseudogap-type mixed state, in which some fermions retain Fermi liquid behavior, yet Luttinger's theorem is violated. At even larger couplings the density of states gradually approaches its form in the atomic limit.