pointed out that the Maxwell theory with the Chern-Simons term in the 5-dimensional Minkowski space is tachyonic when a constant electric field is turned on. A similar mechanism leads to an instability of charged black hole in the 5-dimensional anti-de Sitter space ($AdS_5$) if the Chern-Simons coupling for the Maxwell field is sufficiently large. Interestingly, the instability modes carry non-zero momenta along the boundary of $AdS_5$. This suggests that there is a novel phase transition in the holographically dual field theory at finite chemical potential, where order parameters acquire spatially modulated expectation values.

The analysis of our previous paper was at the linearized level, and what we observed was an onset of the phase transition. To understand the nature of the new phase which emerges as the end-point of the instability, we need to examine full non-linear solutions to the equations of motion in the bulk. In this paper, we construct such solutions in the limit where the Chern-Simons coupling is large and back-reaction of the Maxwell field to the metric is negligible. This is analogous to the probe limit employed in [@Hartnoll:2008vx]. Using the solutions, we compute the expectation values of the order parameters near the phase transition temperature 