Superconductivity in Strongly Correlated Systems

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 (infinitesimally small) one $\Phi_0 (\omega_m)$. We show that this susceptibility is qualitatively different the case when superconductivity emerges out of a Fermi liquid. There, the pairing susceptibility is positive above the transition, diverges at the transition, and becomes negative below it. In our quantum-critical case, for a static $\Phi_0$, $\chi_{pp} (\omega_m)$ remains positive and non-singular all the way up to a pairing instability, and becomes an oscillating function of $\omega_m$ immediately below the instability. The amplitude of oscillations depends on a continuous parameter $\phi$ and diverges for some $\phi$. We argue that this highly non-BCS behavior reflects multi-critical nature of a superconducting onset point in a quantum-critical metal: below the onset an infinite number of superconducting states emerges simultaneously, with different amplitudes of the pairing vertex down to an infinitesimally small one. We also discuss how the pairing susceptibility behaves for a generic dynamical $\Phi_0 (\omega_m)$.