Effect of Spin Fluctuations on Phonon-Mediated Superconductivity in the Vicinity of a Quantum Critical Point

We consider an s-wave superconductor in the vicinity of a second-order ferromagnetic (FM) or spin-density-wave (SDW) quantum critical point (QCP), where the superconductivity and magnetism arise from separate mechanisms. The quantum critical spin fluctuations reduce the superconducting T_c. Near a FM QCP, we find that T_c falls to zero as 1/|ln kappa| in 3D and as kappa in 2D, where kappa ~ |J-J_c|^nu is the inverse correlation length of the spin fluctuations, and measures the distance |J-J_c| from the quantum critical point. SDW quantum critical fluctuations, on the other hand, suppress T_c to zero as sqrt(kappa) in 2D, and suppress T_c only to a finite value in 3D, producing a cusp of the form (const + |J-J_c|^nu).