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

We present a comparative analysis of the validity of Eliashberg theory for the cases of fermions interacting with an Einstein phonon and with soft nematic fluctuations near an Ising-nematic/Ising-ferromagnetic quantum-critical point (QCP). In both cases, Eliashberg theory is obtained by neglecting vertex corrections. For the phonon case, the reasoning to neglect vertex corrections is the Migdal ``fast electron/slow boson'' argument because the phonon velocity is much smaller than the Fermi velocity, $v_F$. The same argument allows one to compute the fermionic self-energy within Eliashberg theory perturbatively rather than self-consistently. For the nematic case, the velocity of a collective boson is comparable to $v_F$ and this argument apparently does not work. Nonetheless, we argue that while two-loop vertex corrections near a nematic QCP are not small parametrically, they are small numerically. At the same time, perturbative calculation of the fermionic self-energy can be rigorously justified when the fermion-boson coupling is small compared to the Fermi energy. Furthermore, we argue that for the electron-phonon case Eliashberg theory breaks down at some distance from where the dressed Debye frequency would vanish, while for the nematic case it holds all the way to a QCP. From this perspective, Eliashberg theory for the nematic case actually works better than for the electron-phonon case.