Breakdown of the Migdal-Eliashberg theory: role of polarons/bipolarons [(Precision Many-Body Physics Focus Sessions)

Migdal-Eliashberg theory (MET) describes electrons interacting with phonons in the adiabatic limit when the phonon Debye frequency is much smaller than the Fermi energy. A conventional belief is that MET holds even at strong coupling, when electron self-energy is large, and breaks down only near the point where the dressed phonon spectrum softens to near zero. We analyze numerically and analytically a different option---collapse to a polaronic/bipolaronic ground state. The last scenario has never been analyzed in precise quantitative terms for a generic electron density. Using variational considerations, we establish rigorous upper bounds on the coupling, at which the Fermi liquid state transforms into the bipolaron/polaron state. We show that at small and near-maximum densities, this happens well before a dressed phonon softens. This is true both in 2D and 3D systems; in the latter the upper bound on the critical coupling

tends to zero in the limit of small or near-full density. We present analytical reasoning for this behavior based on hints extracted from exact diagrammatic treatment of the on-site Holstein model for the spin polarized case and argue that polarons are produced by fermions with energies comparable to the bandwidth,

i.e., polaron formation is outside of realm of MET. Closer to half-filling, the leading instability upon increasing coupling is towards a charge-density-wave state, (CDW), while the

polaron/bipolaron state develops at a larger coupling out of a CDW-ordered state and inherits a CDW order.