Odd-frequency pairing and time-reversal symmetry breaking for repulsive interactions

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 $\Delta_e$ and $\Delta_o$. 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 $\Delta_o$ 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 $\Delta_o$ is not affected by Hubbard repulsion and for strong repulsion is comparable to a reduced $\Delta_e$. The resulting superconducting state is a superposition $\Delta_e \pm i \Delta_o$, which spontaneously breaks the time-reversal symmetry, despite that the pairing symmetry is an ordinary $s$-wave.