We study the pairing of fermions by an interaction consisting of a Hubbard repulsion, mimicking a screened Coulomb potential, and a dynamical phonon-mediated attraction. For such interaction, the gap equation allows even- and odd-frequency solutions ${\mathrm{\Delta }}_e$ and ${\mathrm{\Delta }}_o$. We show that odd-frequency pairing does not develop within the Eliashberg approximation due to over-critical pair breaking from the self-energy. When vertex corrections are included, the pairing interaction gets stronger, and ${\mathrm{\Delta }}_o$ can develop. We argue that even in this case keeping the self-energy is still a must as it cancels out the thermal piece in the gap equation. We further argue that ${\mathrm{\Delta }}_o$ is not affected by Hubbard repulsion and for strong repulsion is comparable to a reduced ${\mathrm{\Delta }}_e$. The resulting superconducting state is a superposition ${\mathrm{\Delta }}_e\pm i{\mathrm{\Delta }}_o$, which spontaneously breaks the time-reversal symmetry, despite that the pairing symmetry is an ordinary $s$ wave.

• Received 5 June 2022
• Revised 1 September 2022
• Accepted 16 September 2022

DOI:https://doi.org/10.1103/PhysRevB.106.104515