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

Abstract

We analyze the dynamical pairing susceptibility ${\chi }_{pp}({\omega }_m)$ 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 ${\chi }_{pp}({\omega }_m)$ as the ratio of the fully dressed dynamical pairing vertex $\Phi ({\omega }_m)$ and the bare ${\Phi }_0({\omega }_m)$ (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 ${\chi }_{pp}({\omega }_m)$ is different in two aspects: (i) it diverges at the onset of pairing at $T=0$ only for a certain subclass of bare ${\Phi }_0({\omega }_m)$ and remains non-singular for other ${\Phi }_0({\omega }_m)$, and (ii) below the instability, it becomes a non-unique function of a continuous parameter $\varphi$ for an arbitrary ${\Phi }_0({\omega }_m)$. The susceptibility is negative in some range of $\varphi$ 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.

14 pages, 5 figures