Vortex Lattices and Crystalline Geometries

We consider $AdS_2 \times R^2$ solutions supported by a magnetic field, such as those which arise in the near-horizon limit of magnetically charged $AdS_4$ Reissner-Nordstrom black branes. In the presence of an electrically charged scalar field, such magnetic solutions can be unstable to spontaneous formation of a vortex lattice. We solve the coupled partial differential equations which govern the charged scalar, gauge field, and metric degrees of freedom to lowest non-trivial order in an expansion around the critical point, and discuss the corrections to the free energy and thermodynamic functions arising from the formation of the lattice. We describe how such solutions can also be interpreted, via S-duality, as characterizing infrared crystalline phases of conformal field theories doped by a chemical potential, but in zero magnetic field; the doped conformal field theories are dual to geometries that exhibit dynamical scaling and hyperscaling violation.