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-Nordström 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 ${\mathrm{AdS}}_5\times S^5$ barely satisfies the stability condition.

Figure 1

Critical temperature as a function of the Chern-Simons coupling $\alpha$. The shaded region indicates a phase with a nonzero expectation value of the conserved current $\overrightarrow{J}$ which is helical and position dependent.

Figure 2

The left figure indicates unstable regions for various values of the Chern-Simons coupling $\alpha$. The right figure is for a particular choice of the Chern-Simons coupling $\alpha =1.6{\alpha }_{\mathrm{crit}}$. The critical temperature $T_C$ is the maximum temperature with unstable modes. The figure indicates the unstable range $\mathbf{a}$ for some temperature $T<T_C$. The range $\mathbf{b}$ is derived from the near-horizon analysis at $T=0$. Note that the actual range of unstable momenta is wider.

Figure 3

For a given value of the Chern-Simons coupling $\alpha$, there is a discrete set of momenta $k$ for which static solutions exist. The curves I and II indicate two of such momenta for each $\alpha$. The red curve is the lower-end of the momentum range that violates the Breitenlohner-Freedman bound near the horizon. Note that the red curve coincides with the curve II. However, there is another curve I with a lower momentum. This means that the near-horizon analysis gives a sufficient but not necessary condition for the instability. Both curves end at the same critical value of $\alpha$.

Figure 4

Left: Negative frequency squared as a function of momentum $k$ at zero temperature when $\alpha =1.6{\alpha }_{\mathrm{crit}}$. Only positive $-{\omega }^2$ is plotted. The curves starting around 1 and 3 join to represent a tachyonic dispersion relation for the unstable mode predicted by the near-horizon analysis. The curve starting below 1 is also expected to be connected with another line in the higher momentum region to form a larger bell-shaped curve, but the large momentum part is difficult to analyze numerically. The zero momentum static solution does not extend to an unstable mode. Right: Negative frequency squared as a function of momentum at temperature $T=8.7\times {10}^{-4}\mu$.