Breakdown of the Migdal-Eliashberg theory for electron-phonon systems. Role of polarons/bi-polarons

TL;DR

This paper critically examines the limits of Migdal-Eliashberg theory in electron-phonon systems, revealing that polaronic and bipolaronic states form before phonon softening, especially at various densities, challenging conventional assumptions.

Contribution

It provides a quantitative analysis showing polaron/bipolaron formation occurs outside MET's validity range, with bounds on coupling strength and density-dependent behavior in 2D and 3D systems.

Findings

Polaron/bipolaron states form at lower coupling than phonon softening predicts.

In 3D, the upper bound on coupling tends to zero at low or full density.

Polaron formation is outside the regime of Migdal-Eliashberg theory.

Abstract

The 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 $\lambda$, at which a FL 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 $\lambda$ 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 the realm of MET. Closer to half-filling, the leading instability upon increasing $\lambda$ is towards a charge-density-wave state (CDW), and there exists a strong coupling regime of MET near this instability, while the polaron/bipolaron state develops at larger $\lambda$ out of a CDW-ordered state and inherits a CDW order over some range of coupling.