Phase stiffness in an antiferromagnetic superconductor

We analyze the suppression of the phase stiffness in a superconductor by antiferromagnetic order. The analysis is based on a general expression for the phase stiffness in a mean-field state with coexisting spin-singlet superconductivity and spiral magnetism. N\'eel order is included as a special case. Close to half filling, where the pairing gap is much smaller than the magnetic gap, a simple formula for the phase stiffness in terms of magnetic quasiparticle bands is derived. The phase stiffness is determined by charge carriers in small electron or hole pockets in this regime. The general analysis is complemented by a numerical calculation for the two-dimensional Hubbard model with nearest- and next-to-nearest-neighbor hopping amplitudes at a moderate interaction strength. The resulting phase stiffness exhibits a striking electron-hole asymmetry. In the ground state, it is larger than the pairing gap on the hole-doped side and smaller for electron doping. Hence, in the hole-doped regime near half filling, the ground-state pairing gap sets the scale for the Kosterlitz-Thouless temperature ${T}_c^{\mathrm{KT}}$, while in the slightly electron-doped regime, ${T}_c^{\mathrm{KT}}$ is determined essentially by the ground-state phase stiffness.