Interactions for odd-ω-gap singlet superconductors

A class of singlet superconductors with a gap function Δ(k,${\mathrm{\omega }}_{\mathit{n}}$) that is odd in both momentum and Matsubara frequency was proposed recently [Phys. Rev. B 45, 13 125 (1992)]. To show an instability in the odd gap channel, a model phonon propagator was used with the p-wave interaction strength larger than the s-wave strength. We argue that the positive scattering matrix element entering the Eliashberg equations leads to a constraint on the relative strength of p- and s-wave interactions, which inhibits odd pairing. However, a general spin-dependent electron-electron interaction can satisfy all constraints and produce the odd singlet gap. A possibility that may lead to an odd gap is a strongly antiferromagnetically correlated system, such as a high-${\mathit{T}}_{\mathit{c}}$ material.