We exploit the fact that the Holstein model of the electron-phonon problem can be treated without approximation using fermion-minus-sign-free determinent quantum Monte Carlo methods to establish results that can be compared quantitatively and unambiguously with approximate methods based on Migdal-Eliashberg (ME) theory. In the relevant limit in which the phonon frequencies are small compared to the Fermi energy (strong retardation), we find that ME theory is extremely accurate up to moderate values of the dimensionless electron-phonon coupling λ, and then breaks down relatively suddenly beyond a characteristic value, λ*~1, beyond which polaron physics is significant. One consequence of this is that – in contrast with earlier beliefs based on ME theory – the superconducting T_c(λ) has its maximum value at λ ≈λ*. This implies that there is an upper bound on T_c from the electron phonon mechanism T_c ≤ A w_max, where w_max is the maximum phonon energy and we estimate that A ≈ 1/10.

References
[1]. I. Esterlis, B.Nosarzewski, E.W.Huang, B. Moritz, T. P. Devereaux, D. J. Scalapino, and S. A. Kivelson, “Breakdown of Migdal-Eliashberg theory; a determinant quantum Monte Carlo study,” Phys. Rev. B 97, 140501 (2018).
[2] I.Esterlis, S.A.Kivelson, and D.J.Scalapino, “A bound on the superconducting transition temperature,” npj Quantum Materials 3, 59 (2018).
[3] I. Esterlis, S. A. Kivelson, and D. J. Scalapino, “Pseudogap crossover in the
electron-phonon system,” Phys. Rev. B 99, 174516 (2019).