Obstructions to odd-frequency superconductivity in Eliashberg theory

We present a necessary condition for odd-frequency (odd-f) superconductivity (SC) to occur in a large class of materials described by Eliashberg theory. We use this condition to prove a no-go theorem ruling out the occurrence of odd-f SC in standard one-band superconductors with pairing interactions mediated by phonon exchange. We also present a corresponding no-go theorem for superconductors with interactions mediated by spin-fluctuations. Our results explain why odd-f SC is rare in conventional materials, and they open up the possibility for a search for materials with interactions designed so as to allow for odd-f SC.