  Extremal planar black holes of four dimensional Einstein-Maxwell theory with
a negative cosmological constant have an AdS$_2 \times \R^2$ near horizon
geometry. We show that this near horizon geometry admits a deformation to a two
parameter family of extremal geometries with inhomogeneous, spatially periodic
horizons. At a linear level, static inhomogeneous perturbations of AdS$_2
\times \R^2$ decay towards the horizon and thus appear irrelevant under the
holographic RG flow. However we have found numerically that nonlinear effects
lead to inhomogeneous near horizon geometries. A consequence of these
observations is that an arbitrarily small periodic deformation of the boundary
theory at nonzero charge density does not flow to AdS$_2 \times \R^2$ in the
IR, but rather to an inhomogeneous horizon. These results shed light on
existing numerical studies of low temperature periodically modulated black
holes and also offer a new mechanism for holographic metal-insulator crossovers
or transitions.
