we introduce a new variable $\tau = r_+/\pi \alpha \mu$ for later convenience. Since $\tau=T/{\mu\alpha}$, it can be thought of as a rescaled temperature.

It is important to note that we have access to the phase transition point in this limit. In [@Nakamura:2009tf], we studied the instability of the Reissner-Nordström solution and obtained the critical temperature $T_c$ for the instability as a function of the Chern-Simons coupling $\alpha$. The result of our numerical analysis is reproduced in Figure \[fig:criticaltemperature\]. For large $\alpha$, the dimensionless combination $T_c/\mu$ grows linearly in $\alpha$. Thus, we can analyze the behavior of the system near $T=T_c$ by taking the limit of $\alpha \rightarrow
\infty$ while keeping the combination $\alpha \mu$ finite.

![Critical temperature as a function of the Chern-Simons coupling $\alpha$. The shaded region indicates a phase with a non-zero expectation value of the conserved current $\vec{J}$ which is helical and position dependent.[]{data-label="fig:criticaltemperature"}](TvsAlpha.jpg){width="7cm"}

Let us find a non-linear solution to the equations of motion that describes the spatially modulated phase in this limit. We look for a solution that has the same symmetry as that of the unstable modes found in [@Nakamura:2009tf], namely a linear combination of a translation along $x_2$ and a rotation in 