Hairy Black Holes in AdS$_5\times S^5$

TL;DR

We construct and analyze a novel family of charged hairy black holes in global $AdS_5$, sourced from a consistent truncation of IIB supergravity on $AdS_5\times S^5$. Using the DeTurck method, we obtain fully nonlinear solutions that bifurcate from RNAdS at the superradiant onset and extend to the BPS bound, with both spherical and planar horizons. Thermodynamically, hairy black holes dominate the microcanonical ensemble but are subdominant in the grand-canonical and canonical ensembles, exhibiting rich features such as a spiral approach to the $\,\alpha=1$ singular soliton and a smooth approach to the $\alpha=0$ BPS soliton as $T\to 0$; the planar limit connects to hairy black branes and an exact planar solution. These results illuminate AdS/CFT thermodynamics for $\,\mathcal{N}=4$ SYM on $\mathbb{R}_t\times S^3$ and set the stage for extensions to rotation and other truncations.

Abstract

We use numerical methods to exhaustively study a novel family of hairy black hole solutions in AdS$_5$. These solutions can be uplifted to solutions of type IIB supergravity with AdS$_5\times S^5$ asymptotics and are thus expected to play an important role in our understanding of AdS/CFT. We find an intricate phase diagram, with the aforementioned family of hairy black hole solutions branching from the Reissner-Nordström black hole at the onset of the superradiance instability. We analyse black holes with spherical and planar horizon topology and explain how they connect in the phase diagram. Finally, we detail their global and local thermodynamic stability across several ensembles.

Figures (21)

• Figure 1: The proposed microcanonical phase diagram by Bhattacharyya et al. (taken from Bhattacharyya:2010yg, not drawn to scale). The lower solid line is the BPS bound on which the supersymmetric soliton resides. The straight red segment represents smooth branch and the wiggly black part represents singular soliton. Hairy black holes were proposed to exist between the curve indicating the onset of the superradiant instability (solid blue) and the BPS bound. The dotted black curve is the extremal RNAdS black holes. The grey solid line shows a possible phase transition between two different types of hairy black holes, with different zero size limits.
• Figure 2: Left: Charge of the solitonic solutions $Q$ versus the vacuum expectation value of the dual operator $\langle\mathcal{O_\phi}\rangle$. The black solid line from above is the singular soliton and the red line from below is the smooth solution. The dotted gridlines show coordinates of the special solution with $\alpha=2/3$. Right: The $\alpha=2$ soliton solutions. The wedges are for constant $h_2$ which decreases from $1$. These solution curves appear to extend to $|Q|\rightarrow+\infty$. We also did not find any limiting value for $\langle\mathcal{O_\phi}\rangle$.
• Figure 3: Left: Phase diagram for the hairy black holes. The merger curve (solid black) indicates the onset of the superradiant instability. The line of extremal RNAdS solutions is shown as a dashed gray line. The BPS bound is given by $M_\mathrm{BPS}(Q)=3Q$ (dashed black). The gray dotted gridlines indicate the position of the special soliton with $\alpha = 2/3$. Right: For clarity, we plot the mass difference $\Delta M=M-M_{\mathrm{ext}}$, where $M_{\mathrm{ext}}$ is the mass of an extremal RNAdS black hole with the same charge $Q$.
• Figure 4: Left: The hairy black hole charge $Q$ versus $\langle\mathcal{O}_\phi\rangle$ for constant central scalar field density $\epsilon_0$ curves. Red line is the smooth soliton and the green line is the singular soliton. Right: Mass difference versus charge $Q$. The constant parameter $\epsilon_0$ curves extend down to $T=0.055$. The inset is a zoomed in plot around $Q=Q_c$ for some value of $\epsilon_0$.
• Figure 5: Left: Zooming in around $Q=Q_c$, and observing the transition between $T<T_2$ and $T>T_2$. The color legend is the same as in Fig. \ref{['fig:mq']}. Right: An even closer look for $Q$ near $Q_c = 0.261$. The hairy black hole isotherms terminate at charges above the special singular soliton.