Quantum critical point shifts under superconductivity: the pnictides and the cuprates

We compare the position of an ordering transition in a metal to that in a superconductor. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |Delta|, where Delta is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the ~ |Delta|^2 shift predicted by theories of competing orders which ignore Fermi surface effects. For Ising-nematic order, the shift from Fermi surface effects remains of order |Delta|^2. We discuss implications of these results for the phase diagrams of the cuprates and the pnictides. We conclude that recent observations imply that the Ising-nematic order is tied to the square of the SDW order in the pnictides, but not in the cuprates.