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[[CALT-68-2796]{}\
[IPMU10-0110]{}]{}

[**Holographic End-Point of\
Spatially Modulated Phase Transition**]{}

[Hirosi Ooguri and Chang-Soon Park]{}

[ *California Institute of Technology, Pasadena, CA 91125, USA,\
and\
Institute for the Physics and Mathematics of the Universe,\
University of Tokyo, Kashiwa 277-8586, Japan\
*]{}

[**Abstract**]{}

In the previous paper \[arXiv:0911.0679\], we showed that the Reissner-Nordström 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.

Introduction
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In the previous paper [@Nakamura:2009tf], together with Shin Nakamura, we 