BCS-like formula for $T_c$ does not necessarily imply BCS pairing mechanism: the case of magnetically-mediated quantum-critical pairing

Abstract

In the BCS theory of superconductivity, an instability towards pairing develops at arbitrary weak dimensionless coupling $\lambda$ due to a divergence of logarithmic perturbative series for the pairing susceptibility (Cooper logarithms) at $T_c\sim {\omega }_0{e}^{-1/\lambda }$, where ${\omega }_0$ is an energy cutoff. On the contrary, in many models of superconductivity out of a non-Fermi liquid, Cooper logarithm is absent and superconductivity emerges only when $\lambda$ exceeds a certain threshold. We argue that there are situations when there is no threshold and at weak coupling the formula for $T_c$ is BCS-like, yet the origin of the pairing instability is fundamentally different from that in the BCS scenario. As an example, we revisit superconductivity in a metal at the onset of $(\pi ,\pi )$ spin-density-wave order. Earlier studies of this problem found no threshold and a BCS-like expression for $T_c$ at weak coupling. We argue that, despite this, the pairing is not caused by the Cooper logarithm and in many respects is qualitatively similar to that in non-Fermi liquids.

13 pages, 5 figures, published version