Magnetic Field Induced Charge and Spin Instabilities in Cuprate Superconductors

A $d$-wave superconductor, subject to strong phase fluctuations, is known to suffer an antiferromagnetic instability closely related to the chiral symmetry breaking in ( $2+1$)-dimensional quantum electrodynamics ( ${\mathrm{QED}}_3$). Based on this idea we formulate a “ ${\mathrm{QED}}_3$ in a box” theory of local instabilities of a $d$-wave superconductor in the vicinity of a single pinned vortex undergoing quantum fluctuations. As a generic outcome we find an incommensurate 2D spin density wave forming in the neighborhood of a vortex with a concomitant “checkerboard” pattern in the local electronic density of states, in agreement with recent neutron scattering and tunneling spectroscopy measurements.