BCS superconductivity in quantum critical metals

We consider the superconducting transition in fermionic quantum critical systems. Assuming the validity of Migdal theorem, the gap equation can be written in terms of the retarded pair susceptibility. Instead of the usual BCS form, the pair susceptibility is now subject to scale invariance. The gap and transition temperature is thus of the algebraic form, totally different from the exponential behavior in BCS theory. Consequently, with reasonably small glue strength, we can get very large gap and transition temperature comparable to those discovered in cuprates. The ratio of the gap to retardation gets boosted by increasing retardation. We also find the upper critical field has a different scaling with the critical temperature. With a non-Lorentzian dynamical exponent, the upper critical field is greatly enhanced when approaching the critical point, though the critical temperature only changes modestly, in agreement with recent experiments on heavy fermions.