We consider ${\mathrm{AdS}}_2\times R^2$ solutions supported by a magnetic field, such as those which arise in the near-horizon limit of magnetically charged ${\mathrm{AdS}}_4$ 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 that govern the charged scalar, gauge field, and metric degrees of freedom to lowest nontrivial 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.

• Received 28 April 2013

DOI:https://doi.org/10.1103/PhysRevD.88.026002