Competing orders in a magnetic field: Spin and charge order in the cuprate superconductors

We describe two-dimensional quantum spin fluctuations in a superconducting Abrikosov flux lattice induced by a magnetic field applied to a doped Mott insulator. Complete numerical solutions of a self-consistent large-N theory provide detailed information on the phase diagram and on the spatial structure of the dynamic spin spectrum. Our results apply to phases with and without long-range spin-density-wave order, and to the magnetic quantum critical point separating these phases. We discuss the relationship of our results to a number of recent neutron-scattering measurements on the cuprate superconductors in the presence of an applied field. We compute the pinning of static charge order by the vortex cores in the ``spin-gap'' phase where the spin order remains dynamically fluctuating, and argue that these results apply to recent scanning-tunneling-microscopy (STM) measurements. We show that, with a single typical set of values for the coupling constants, our model describes the field dependence of the elastic-neutron-scattering intensities, the absence of satellite Bragg peaks associated with the vortex lattice in existing neutron-scattering observations, and the spatial extent of charge order in STM observations. We mention implications of our theory for NMR experiments. We also present a theoretical discussion of more exotic states that can be built out of the spin- and charge-order parameters, including spin nematics and phases with ``exciton fractionalization.''