We present a string theory construction of a gravity dual of a spatially modulated phase. In our earlier work, we showed that the Chern-Simons term in the 5-dimensional Maxwell theory destabilizes the Reissner-Nordstrom black holes in anti-de Sitter
space if the Chern-Simons coupling is sufficiently high. In this paper, we show that a similar instability is realized on the worldvolume of 8-branes in the Sakai-Sugimoto model in the quark-gluon plasma phase. We also construct and analyze a non-linear solution describing the end-point of the transition. Our result suggests a new spatially modulated phase in quark-gluon plasma when the baryon density is above 0.8 N_f fm^{-3} at temperature 150 MeV.
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 la
rge. 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.
