Gravity dual of spatially modulated phase

We show that the five-dimensional Maxwell theory with the Chern-Simons term is tachyonic in the presence of a constant electric field. When coupled to gravity, a sufficiently large Chern-Simons coupling causes instability of the Reissner-Nordstr\"om black holes in anti-de Sitter space. The instability happens only at nonvanishing momenta, suggesting a spatially modulated phase in the holographically dual quantum field theory in ($3+1$) dimensions, with spontaneous current generation in a helical configuration. The three-charge extremal black hole in the type IIB superstring theory on AdS₅×S⁵ barely satisfies the stability condition.