## Abstract

We analyze the validity of Eliashberg theory of phonon-mediated superconductivity in 2D systems in light of recent extensive Monte-Carlo studies of the Holstein model. Conventional wisdom says that Eliashberg theory is applicable as long as vertex corrections remain small. For small ratio of the phonon energy Ω_{0} and the Fermi energy E_{F}, this condition is supposed to hold even when the dimensionless electron–phonon coupling λ is larger than one, i.e., in the strong coupling regime. A comparison between various quantities computed in the Migdal approximation and those computed by Quantum Monte Carlo prove that this belief is wrong, and we identify analytically some of the ways in which this breakdown occurs for various “normal state” properties at λ=λ_{cr}, where λ_{cr}=O(1). The breakdown occurs at temperatures high enough that neither superconducting nor charge–density wave correlations extend over any significant range of distances, so it cannot be associated with the onset of an instability toward any of the relevant ordered ground-states — rather it is associated with the local physics of classical bipolaron formation. Still, we show that certain properties, including the superconducting T_{c} and the superconducting gap structure below T_{c}, can be accurately inferred from the strong-coupling limit of Eliashberg theory at λ≤λ_{cr}.

Original language | English (US) |
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Article number | 168190 |

Journal | Annals of Physics |

Volume | 417 |

DOIs | |

State | Published - Jun 2020 |

### Bibliographical note

Funding Information:We thank B. Altshuler, E. Berg, R. Combescot, R. Fernandes, A. Finkelstein, A. Klein, G. Kotliar, S. Lederer, L. Levitov, D. Maslov, A. Millis, V. Pokrovsky, N. Prokofiev, S. Raghu, M. Randeria, S. Sachdev, D. Scalapino, Y. Schattner, J. Schmalian, B. Svistunov, E. Yuzbashyan, Y. Wang, Y. Wu, and J. Zaanen for useful discussions. The work by AVC was supported by the Office of Basic Energy Sciences, U.S. Department of Energy , under award DE-SC0014402 . SAK was supported, in part, by NSF grant # DMR-1608055 at Stanford. IE acknowledges support from the Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering .

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© 2020