In the previous paper [arXiv:0911.0679], we showed that the
Reissner-Nordstrom black hole in the 5-dimensional anti-de Sitter space coupled
to the Maxwell theory with the Chern-Simons term is unstable when the
Chern-Simons coupling is sufficiently large. In the dual conformal field
theory, the instability suggests a spatially modulated phase transition. In
this paper, we construct and analyze non-linear solutions which describe the
end-point of this phase transition. In the limit where the Chern-Simons
coupling is large, we find that the phase transition is of the second order
with the mean field critical exponent. However, the dispersion relation with
the Van Hove singularity enhances quantum corrections in the bulk, and we argue
that this changes the order of the phase transition from the second to the
first. We compute linear response functions in the non-linear solution and find
an infinite off-diagonal DC conductivity in the new phase.Comment: 21 pages, 22 figures. v2: a note and a reference adde