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

Abstract

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 ${\mathrm{\ensuremath{\Delta}}}_{e}$ and ${\mathrm{\ensuremath{\Delta}}}_{o}$. 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 ${\mathrm{\ensuremath{\Delta}}}_{o}$ 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 ${\mathrm{\ensuremath{\Delta}}}_{o}$ is not affected by Hubbard repulsion and for strong repulsion is comparable to a reduced ${\mathrm{\ensuremath{\Delta}}}_{e}$. The resulting superconducting state is a superposition ${\mathrm{\ensuremath{\Delta}}}_{e}\ifmmode\pm\else\textpm\fi{}i{\mathrm{\ensuremath{\Delta}}}_{o}$, which spontaneously breaks the time-reversal symmetry, despite that the pairing symmetry is an ordinary $s$ wave.