Hairy black holes and solitons in global AdS 5

TL;DR

This work maps the full static phase structure of global AdS_5 Einstein-Maxwell theory with a charged massless scalar, uncovering how solitons, RN AdS black holes, and hairy black holes compete as the scalar charge e is varied. By combining analytic perturbation theory, near-horizon stability analysis, and extensive numerical construction, the authors classify phase regimes, delineate instability borders, and reveal intricate soliton branches whose existence and extremal endpoints depend sensitively on e. A non-interacting mixture model of RN AdS black holes and solitons captures the leading thermodynamics of small hairy black holes for 3<e^2<32/3, while perturbative and numerical results at larger e^2 reveal a planar scaling limit and extremal, possibly singular, endpoints for hairy solutions. The study highlights a rich interplay between superradiant and BF-type instabilities, and provides a coherent picture of how hairy hair and condensate phases emerge in global AdS, with implications for the dual CFT phase structure on S^3 and potential connections to supersymmetric AdS_5 contexts.

Abstract

We use a mix of analytic and numerical methods to exhaustively study a class of asymptotically global AdS solitons and hairy black hole solutions in negative cosmological constant Einstein Maxwell gravity coupled to a charged massless scalar field. Our results depend sensitively on the charge 'e' of the scalar field. The solitonic branch of solutions we study hit the Chandrashekhar limit at finite mass at small 'e', but extends to arbitrarily large mass at larger 'e'. At low values of 'e' no hairy black holes exist. At intermediate values of 'e' hairy black holes exist above a critical charge. At large 'e' hairy black holes exist at all values of the charge. The lowest mass hairy black holes is a smooth zero entropy soliton at small charge, but a (probably) singular nonzero entropy hairy black hole at larger charge. In a phase diagram of solutions, the hairy black holes merge with the familiar Reissner-Nordstrom-AdS black holes along a curve that is determined by the onset of the superradiant instability in the latter family.

Figures (29)

• Figure 1: Schematic phase diagram for $e^2 \leq 3$. Note that the soliton curve (green) is always above the extremal black hole (black). Pure RN AdS black holes exist for all values of masses above the extremality curve. The extremal black hole is never unstable and no hairy black hole solutions exist. The soliton exists up to a certain maximum charge, has a self-similar behaviour around a value $Q_{crit}(e^2)$ and is never the dominant phase.
• Figure 2: Schematic phase diagram for $3 \leq e^2 \leq \frac{32}{3}$. RN AdS black holes exist (red shaded) for all values of charge and mass above the extremality curve (black). Extremal black holes are unstable for $Q\geq Q_0(e^2)$ and hairy black hole solutions exist for these values of charge. These solutions exist (blue shaded) between the curve of instability of RN AdS black holes (red) and a zero temperature hairy black hole solution (blue). Hairy black holes are the dominant phase whenever they exist. The soliton (green) exists upto a maximum charge $Q = Q_{crit}$ (where a cusp structure, to be discussed only later, appears) and is never the dominant phase. (The soliton curve can be below the extremal RN AdS line for $Q>Q_0(e^2)$ but keeps above the extremal hairy black hole).
• Figure 3: Schematic phase diagram for $e^2 \geq \frac{32}{3}$. As for $e^2 \leq \frac{32}{3}$ RN AdS black holes exist (red shaded) for all charges, and all masses above extremality (black). Hairy black holes exist (blue shaded) for all values of the charge, and the masses are below the curve of instability of RN AdS black holes (red). For $Q < Q_{c_2}(e^2)$, the lowest mass hairy black hole solution is a zero entropy soliton at infinite temperature (green). For $Q > Q_{c_2}(e^2)$, the lowest mass hairy black hole is extremal with finite entropy (blue).
• Figure 4: Left:$\tilde{q}_{crit}$ as a function of $\sqrt{\frac{32}{3}}-e$ for $e^2<32/3$: note that the scale on the $x$ axis is logarithmic. The black dots correspond to our data. The red line is the fit to a linear behavior as in \ref{['eqn:qmaxvse']} in the region of small $e$. The blue line corresponds to a logarithmic fit of the form $\tilde{q}_{crit}=a\,\ln\left(\sqrt{\frac{32}{2}}-e\right)+b$ of the data near $e \sim \sqrt{\frac{32}{3}}$. This diagram suggest that ${\tilde{q}}_{crit}$ blows up logarithmically as $e$ approaches $e_{solcrit}$. Right:$Q$ as a function of $f(0)$ for $e=3.,3.1,3.2,3.29,3.5, 4.$ (from bottom to top in the plot). For $e\leq e_{solcrit}$, $Q$ tends to a finite value as $f(0)\to 0$, and for $e>e_{solcrit}$, $Q$ diverges as $f(0)\to 0$.
• Figure 5: Left: Kretschmann invariant evaluated at $r=0$ as a function of $f(0)$ for the $e=1$ solitons. As $f(0)\to 0$ the Kretschmann invariant diverges at the origin, which signals the appearance of a curvature singularity there. The red dashed line corresponds to the value of the Kretschmann invariant for pure $AdS_5$. Right: Phase diagram in the microcanonical ensemble for $e=1$. On the $y$-axis of this plot we depict $\Delta M=M-M_\textrm{ext}$, where $M_\textrm{ext}$ is the mass of the extremal RN AdS black hole with the same charge $Q$. RN AdS black holes occupy the shaded region and the soliton family of solutions is given by the black curve. This curve terminates at a naked singularity at some finite $Q$. In red we show the perturbative results of Basu:2010uz; the agreement between the perturbative calculation and our numerical results is remarkable at small values of $Q$ but they disagree at sufficiently large $Q$.