Breakdown of the Migdal-Eliashberg theory and a theory of lattice-fermionic superfluidity

We show that the Migdal-Eliashberg theory loses validity at a finite value $\lambda_c$ of the electron-phonon coupling $\lambda$ regardless of the underlying model Hamiltonian. The value of $\lambda_c$ is approximately between 3.0 and 3.7. The new phase that emerges at $\lambda>\lambda_c$ breaks the lattice translational symmetry. Depending on the filling fraction and crystal symmetry, it is an insulator or a Fermi liquid. Its characteristic feature is a gap or a pronounced depression of the fermionic density of states near the Fermi level. We establish the breakdown from within the Migdal-Eliashberg theory by demonstrating that the normal state specific heat is negative for $\lambda\ge 3.7$ and the quasiparticle lifetime vanishes in the strong coupling limit. At fixed $\lambda>\lambda_c$, the transition to the new phase occurs at a critical temperature higher than the superconducting transition temperature. In addition, there is a first order phase transition between the new phase and the superconducting state as we vary $\lambda$ across $\lambda_c$ at fixed temperature. We put forward a new theory - lattice-fermionic theory of superfluidity - that bridges the gap between the Migdal-Eliashberg approach and the physics at stronger coupling. At small $\lambda$, our theory reduces to the Migdal-Eliashberg theory and, past $\lambda_c$, it describes the new phase and a range of other phenomena.