We revisit the issue of superconductivity at the quantum-critical point (QCP) between a 2D paramagnet and a spin-density-wave metal with ordering momentum ($\pi$, $\pi$). This problem is highly nontrivial 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 and S. Sachdev, Phys. Rev. B 82, 075128 (2010)] has found that the renormalizations of the pairing vertex are stronger than in BCS theory and hold in powers of ${log}^2(1/T)$. We analyze the full gap equation and argue that summing up of the leading logarithms does not lead to a pairing instability. Yet, we show that superconductivity has no threshold and appears even if $\lambda$ is set to be small, because subleading logarithmical renormalizations diverge and give rise to a BCS-like result $\log 1/T_c\propto 1/\lambda$. We argue that the analogy with BCS is not accidental as at small $\lambda$ superconductivity at a QCP predominantly comes from fermions that retain Fermi-liquid behavior at criticality. We compute $T_c$ for the actual $\lambda \sim O(1)$, and find that both Fermi-liquid and non-Fermi-liquid fermions contribute to the pairing.

• Received 8 October 2012

DOI:https://doi.org/10.1103/PhysRevLett.110.127001