Spin-Ordering Quantum Transitions of Superconductors in a Magnetic Field

We argue that recent neutron scattering measurements by Lake et al. [Science 291, 1759 (2001)] of the spin excitation spectrum of ${\mathrm{La}}_{2-\delta }{\mathrm{Sr}}_{\delta }{\mathrm{CuO}}_4$ in a magnetic field can be understood in terms of proximity to a phase with co-existing superconductivity and spin density wave order. We present a general theory for such quantum transitions, and argue that their low energy spin fluctuations are controlled by a singular correction from the superflow kinetic energy, acting in the region outside the vortex cores. We propose numerous experimental tests of our theory.