discuss non-linear solutions in $AdS_2\times \mathbb{R}^3$, which is the near horizon limit of the extremal black hole.

Note added: After the first version of this paper was completed, we were informed of the work [@Sachdev:2010um], which suggested that an instability to crystalline phases might be a generic feature of phases which are describable by a bulk $AdS_2$ geometry. Such an instability would provide a natural way to understand the ground state entropy.

Maxwell-Chern-Simons Theory in Minkowski Space {#sec:MCSflat}
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In this section, we consider the Maxwell theory with the Chern-Simons term in the 5-dimensional Minkowski space. The Lagrangian is given by $$\mathcal{L}=-\frac 1 4 F_{IJ} F^{IJ} + \frac{\alpha}{3!} \epsilon^{IJKLM} A_I F_{JK} F_{LM}\;,$$ where $I,J,\ldots$ run from 0 to 4. The equations of motion are $$\partial_J F^{JI} + \frac{\alpha} 2 \epsilon^{IJKLM} F_{JK} F_{LM} =0\;.
\label{eom}$$ In particular, the time component of the above can be written as the Gauss law, $$\label{E:GaussLawFlat}
\partial_A \Pi^A =0\;,\qquad \text{where $\Pi^A = -F^{A0} + \alpha \epsilon^{ABCD} A_B F_{CD}$}\;,$$ where the indices $A,B,\ldots$ run from 1 to 4.

A constant electric field is a solution to the equations of motion. However, as shown in [@Nakamura:2009tf], there are unstable modes in the following range of momentum, $$0 < 