Superconductivity at the onset of spin-density-wave order in a metal

We revisit the issue of superconductivity at the quantum-critical point between a 2D paramagnet and a spin-density-wave (SDW) metal with ordering momentum $(\pi,\pi)$. This problem is highly non-trivial because the system at criticality displays a non-Fermi liquid behavior and because the effective coupling constant $\lambda$ for the pairing is generally of order one, even when the actual interaction is smaller than fermionic bandwidth. Previous study [M. A. Metlitski, S. Sachdev, Phys.Rev.B 82, 075128 (2010)] has found that the leading renormalization of the pairing vertex contains $\log^2$, like in color superconductivity. We analyze the full gap equation and argue that summing up $\log^2$ term does not lead to a pairing instability. Yet, superconductivity has no threshold, even if $\lambda$ is set to be small: the subleading $\log$ terms give rise to BCS-like $T_c \propto e^{-1/\lambda}$. We argue that the analogy with BCS is not accidental as superconductivity at a QCP is a Fermi liquid phenomenon -- it comes from fermions which retain Fermi liquid behavior at criticality. We compute $T_c$ for the actual $\lambda$ and find consistency with the numerical results.