Odd frequency pairing in quantum critical metals.

We consider superconductivity pairing at a quantum critical point(QCP). Close to such a QCP, the fermion interaction becomes dynamic and gives rise to non-Fermi liquid behavior as well as pairing instability, which can be described by a general model where the pairing interaction scales as V(Ω_n)∼1/|Ω_n|^γ. Based on this model, we investigate the case when 0<γ<1 and find there exists an infinite number of odd frequency solutions [Δ(-ω_m)=-Δ(ω_m)] to the superconductivity gap equations, with each of them having a unique gap structure Δ_n(ω_m) and onset pairing temperature T_p,n (n=0,1,2,...). We study the one with the largest pairing temperature T_p,0, and find in zero temperature limit, the gap function Δ(ω_m) scales as |ω_m|^asgn(ω_m) with a<1 at small frequencies. On real axis, this scaling implies that the quasiparticle density of states N(ω) tends to 0 at ω=0. This differs from previous studies on odd frequency pairing in normal metal/superconductor or ferromagnet/superconductor heterostructures where the odd frequency pairing is associated with a peak in N(ω) at ω=0. We also investigate the temperature evolution of N(ω) for odd frequency pairing, and find it resembles a conventional BCS superconductor, i.e. as temperature increases, the gap in N(ω) gradually closes till it reaches the onset temperature. For completeness, we also consider the case when γ>1. By analyzing the linearized gap equation, we find there is no odd-frequency superconducting order at finite temperature if γ>1.