Odd frequency pairing of superconductivity in quantum critical metals

Odd frequency pairing superconductivity is an unconventional pairing state where the order parameter (the gap function) △(ω) is an odd function of frequency (△(-ω)=-△(ω)). We analyze a possibility of odd frequency pairing for a model of fermions near a metallic quantum-critical point. In this model, the effective dynamical 4-fermion interaction, mediated by a critical boson, V(Ω_m)∝1/|Ω_m|^γ, with γ>0, gives rise to both pairing and non-Fermi liquid behavior in the normal state. We assume that the interaction in the particle-particle channel is smaller by 1/N than that in the particle-hole channel, and adopt the Eliashberg formalism to analyze the interplay between the pairing and non-Fermi liquid. We show that the gap equaton allows solutions that are either even or odd in frequency. We show that odd frequency pairing develops as long as N is smaller than some critical value N_cr, which is a function of γ. For a generic γ<1, the onset temperature and the gap amplitude at T=0 for the odd frequency solution are much smaller than for the corresponding even frequency pairing, but the two become comparable at γ→1. In this limit, even frequency and odd frequency solutions may coexist.