Non-BCS behavior of the pairing susceptibility near the onset of superconductivity in a quantum-critical metal

Abstract

We analyze the dynamical pairing susceptibility $χ_{pp} (ω_m)$ at $T=0$ in a quantum-critical metal, where superconductivity emerges out of a non-Fermi liquid ground state once the pairing interaction exceeds a certain threshold. We obtain $χ_{pp} (ω_m)$ as the ratio of the fully dressed dynamical pairing vertex $Φ(ω_m)$ and the bare $Φ_0 (ω_m)$ (both infinitesimally small). For superconductivity out of a Fermi liquid, the pairing susceptibility is positive above $T_c$, diverges at $T_c$, and becomes negative below it. For superconductivity out of a non-Fermi liquid, the behavior of $χ_{pp} (ω_m)$ is different in two aspects: (i) it diverges at the onset of pairing at $T=0$ only for a certain subclass of bare $Φ_0 (ω_m)$ and remains non-singular for other $Φ_0 (ω_m)$, and (ii) below the instability, it becomes a non-unique function of a continuous parameter $ϕ$ for an arbitrary $Φ_0 (ω_m)$. The susceptibility is negative in some range of $ϕ$ and diverges at the boundary of this range. We argue that this behavior of the susceptibility reflects a multi-critical nature of a superconducting transition in a quantum-critical metal when immediately below the instability an infinite number of superconducting states emerges simultaneously with different amplitudes of the order parameter down to an infinitesimally small one.