Applicability of Eliashberg theory for systems with electron-phonon and electron-electron interaction: A comparative analysis

The applicability of Eliashberg theory, for fermions interacting with fluctuations of a soft collective boson near a quantum-critical point (QCP) in two dimensions, remains a subject of debate. The velocity of the soft boson is comparable to the Fermi velocity. Hence, Migdal's justification of Eliashberg theory is formally not applicable. The authors argue here that some elements of Migdal's reasoning remain valid because the collective boson is overdamped, yet vertex corrections to Eliashberg theory are O (1). At the same time, they show that these corrections are numerically small, at least near an Ising-nematic QCP.