We theoretically investigate the vortex state of the cuprate high-temperature superconductors in the presence of magnetic fields. Assuming the recently derived nonlinear $sigma$-model for fluctuations in the pseudogap phase, we find that the vortex c
ores consist of two crossed regions of elliptic shape, in which a static charge order emerges. Charge density wave order manifests itself as satellites to the ordinary Bragg peaks directed along the axes of the reciprocal copper lattice. Quadrupole density wave (bond order) satellites, if seen, are predicted to be along the diagonals. The intensity of the satellites should grow linearly with the magnetic field, in agreement with the result of recent experiments.
Single band theories of quantum criticality successfully describe a single-particle lifetime with non-Fermi liquid temperature dependence. But, they fail to obtain a charge transport rate with the same dependence unless the interaction is assumed to
be momentum independent. Here we demonstrate that a quantum critical material, with a long range mode that transmutes electrons between light and heavy bands, exhibits a quasi-linear temperature dependence for {it both} the single-particle and the charge transport lifetimes, despite the strong momentum dependence of the interaction.
