Gravity Dual of Spatially Modulated Phase

TL;DR

This work demonstrates a CS-induced instability mechanism in a five-dimensional Maxwell theory that becomes tachyonic under a constant electric background, and shows that coupling to gravity renders Reissner-Nordström–AdS$_5$ black holes unstable at finite momentum for CS coupling above a critical value $\alpha_{crit}$. The destabilization signals a spatially modulated, helical current phase in the dual ($3+1$)-dimensional CFT, with a phase transition occurring below a critical temperature $T_C(\alpha)$. Near-horizon analyses in $AdS_2$ geometries bound the instability through BF criteria, while full RN-AdS$_5$ numerics reveal the actual onset and momentum range of the instability, including nontrivial curves not captured by the near-horizon analysis. The paper also analyzes Type IIB three-charge black holes, finding that the equal-charge sector is barely stable and non-equal-charge configurations remain stable, suggesting the phenomenon is delicate but potentially realizable in suitable holographic settings. Overall, the results provide a holographic route to spatially modulated phases with helical order and illuminate the role of CS couplings in finite-density quantum critical behavior.

Abstract

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-Nordstrom black holes in anti-de Sitter space. The instability happens only at non-vanishing 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 AdS_5 x S^5 barely satisfies the stability condition.

Figures (4)

• Figure 1: Critical temperature as a function of the Chern-Simons coupling $\alpha$. The shaded region indicates a phase with a non-zero expectation value of the conserved current $\vec{J}$ 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 $\alpha=1.6 \alpha_{\rm crit}$. The critical temperature $T_C$ is the maximum temperature with unstable modes. The figure indicates the unstable range $\textbf{a}$ for some temperature $T < T_C$. The range $\textbf{b}$ 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 $\alpha=1.6 \alpha_{\rm crit}$. 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 $T=8.7\times 10^{-4} \mu$.