Scalarized AdS Black Holes

• Scalarized AdS black holes are solutions with emergent scalar fields that bifurcate from bald AdS backgrounds via instability mechanisms such as curvature or Maxwell-induced triggers.
• They are modeled in frameworks like scalar-tensor, Gauss–Bonnet, and Einstein–Maxwell-scalar theories, exhibiting holographic implications and phases analogous to conductor-superconductor transitions.
• Thermodynamic analyses reveal second-order phase transitions and distinct branch structures, emphasizing the role of asymptotic AdS falloff conditions and dual operator dynamics.

Scalarized AdS black holes are asymptotically Anti-de Sitter black-hole solutions with nontrivial scalar configurations supported by the bulk dynamics rather than by externally imposed scalar sources. In the literature this label covers several distinct mechanisms: curvature-induced scalarization in scalar-tensor and Gauss–Bonnet models, Maxwell-induced scalarization in Einstein–Maxwell-scalar theories, superradiant condensation of charged scalars in global AdS, and scalar-supported warped AdS$_3$ geometries. An important limiting case is provided by Schwarzschild–AdS solutions in $\mathcal N=2$ gauged supergravity with a scalar sector that is present but covariantly frozen; these are AdS black holes with scalars in the theory, but not scalarized black holes in the usual sense (Brihaye et al., 2019, Guo et al., 2021, Promsiri et al., 2023, Zou et al., 2023, Zhang et al., 7 Jun 2026, Giribet et al., 2015, Basu et al., 2010, Kimura, 2011).

1. Conceptual scope and defining distinction

In the spontaneous-scalarization setting, one begins from a bald AdS black hole—typically RNAdS-, Schwarzschild-AdS-, or topological-AdS-like—and finds that the trivial scalar configuration becomes unstable. A new branch with nonzero scalar profile then bifurcates from the bald branch. This pattern appears in asymptotically AdS models with spherical, planar, and hyperbolic horizons, and it is also realized at zero temperature for certain extremal near-horizon geometries (Zhao et al., 22 Nov 2025, Marrani et al., 2022).

The central distinction is between genuine scalar hair and a frozen scalar sector. In the flux-compactification construction of four-dimensional $\mathcal N=2$ abelian gauged supergravity with one vector multiplet and the universal hypermultiplet, the black-hole analysis imposes the covariantly constant conditions

$$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$

together with $dB=0$. The resulting solutions are Schwarzschild–AdS with

$q_0=q_1=p^0=p^1=0,$

so the electromagnetic invariant $I_1$ vanishes and no spacetime-dependent scalar profile survives. The solution is therefore an AdS black hole with frozen scalars rather than a scalarized one (Kimura, 2011).

A related but distinct case is the warped-AdS$_3$ black hole with a “scalar halo.” There the scalar is regular outside and on the non-extremal horizon, but its amplitude is not a freely tunable hair parameter; instead it is fixed by the black-hole parameters and couplings. The term “scalar halo” marks that restricted status (Giribet et al., 2015).

2. Instability mechanisms in AdS

A recurring mechanism is curvature-induced effective mass generation. In the scalar-tensor model with action

$$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$

the scalar equation

$$\square \phi + (\alpha\mathcal R+\gamma\mathcal G)\phi=0$$

implies an effective mass

$\mathcal N=2$0

Scalarization requires that $\mathcal N=2$1 drop below the Breitenlohner–Freedman bound near the horizon while remaining above the bound asymptotically. The same AdS logic appears in Gauss–Bonnet models, where the curvature invariant itself generates the effective tachyonic channel (Brihaye et al., 2019, Zou et al., 2023, Zhang et al., 7 Jun 2026).

In Einstein–Maxwell-scalar models, the driving term comes from the Maxwell sector. For the nonminimal coupling $\mathcal N=2$2 or $\mathcal N=2$3, the linearized scalar equation around RNAdS yields

$\mathcal N=2$4

Near the horizon, where $\mathcal N=2$5 is smallest, this becomes sufficiently negative to trigger a tachyonic instability when it crosses the AdS$\mathcal N=2$6 BF bound

$\mathcal N=2$7

This is the basic onset mechanism for both spherical and planar scalarized charged AdS black holes (Guo et al., 2021, Promsiri et al., 2023).

A different route is superradiant condensation in global AdS$\mathcal N=2$8. For a charged scalar mode of energy $\mathcal N=2$9, superradiance occurs when

$\mathcal N=2$0

With a massless scalar, the lowest normal mode has $\mathcal N=2$1, so the instability condition becomes

$\mathcal N=2$2

The end point is a hairy black hole consisting, at leading order, of a small RNAdS core immersed in a charged scalar condensate filling the AdS “box” (Basu et al., 2010).

These mechanisms all share one AdS-specific feature: a negative local effective mass is not sufficient by itself. The asymptotic AdS region imposes BF stability and normalizability constraints, so the scalarized branch must interpolate between a destabilized near-horizon region and a stable asymptotic vacuum (Brihaye et al., 2019, Zhang et al., 7 Jun 2026).

3. Theoretical realizations and representative solution families

One major class consists of four-dimensional EMS models with a real scalar nonminimally coupled to the Maxwell invariant. For spherical symmetry, a standard ansatz is

$\mathcal N=2$3

with RNAdS recovered at $\mathcal N=2$4. The scalarized branch then deforms $\mathcal N=2$5, $\mathcal N=2$6, $\mathcal N=2$7, and $\mathcal N=2$8 self-consistently (Guo et al., 2021).

The planar EMS construction uses

$\mathcal N=2$9

together with $$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$0 and $$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$1. The scalarized planar solutions exist only in a bounded region of the $$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$2 plane, where $$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$3, and bifurcate from a planar RN-AdS-like background (Promsiri et al., 2023).

Another major class is Einstein-scalar-Gauss–Bonnet and Einstein-Maxwell-scalar-Gauss–Bonnet gravity. In the ESGB model, the action

$$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$4

supports scalarized Schwarzschild-AdS solutions. In the EMsGB model with

$$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$5

charged RN-AdS black holes can scalarize through the Gauss–Bonnet invariant $$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$6, with the sign and magnitude of $$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$7 selecting the GB$$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$8 or GB$$D_\mu t=0,\qquad D_\mu \xi^0=0,\qquad D_\mu \tilde\xi_0=0,\qquad D_\mu \phi=0,$$9 channel (Zou et al., 2023, Zhang et al., 7 Jun 2026).

At zero temperature, extremal electrically charged black holes in $dB=0$0 can scalarize in Einstein–Maxwell theory coupled to a complex scalar written in Stueckelberg form,

$dB=0$1

with

$dB=0$2

Near criticality, the horizon physics depends only on the effective interaction

$dB=0$3

This identifies a zero-temperature scalarization channel tied to the entropy-function and attractor framework (Marrani et al., 2022).

Lower- and higher-dimensional realizations broaden the scope. In three dimensions, warped-AdS$dB=0$4 black holes with scalar halo arise in Einstein gravity with a real scalar and Horndeski-type derivative coupling

$dB=0$5

In global AdS$dB=0$6, small charged hairy black holes arise in Einstein–Maxwell theory with a charged, massless, minimally coupled scalar and are constructed perturbatively in the black-hole radius and hair amplitude (Giribet et al., 2015, Basu et al., 2010).

4. AdS asymptotics, normalizability, and holographic data

Because the scalarized solutions live in AdS, asymptotic falloffs are part of the definition of the branch. In the scalar-tensor model with $dB=0$7, the scalar behaves near the boundary as

$dB=0$8

with the exponents controlled by

$dB=0$9

The dual operator dimension is $q_0=q_1=p^0=p^1=0,$0, and setting $q_0=q_1=p^0=p^1=0,$1 identifies $q_0=q_1=p^0=p^1=0,$2 as the expectation value of the dual operator. The condition $q_0=q_1=p^0=p^1=0,$3 is simultaneously the condition that the asymptotic scalar falloff be real and that the AdS vacuum remain stable (Brihaye et al., 2019).

In the spherical and planar EMS models, the scalar is also imposed to be source free at infinity. The asymptotic expansions take the form

$q_0=q_1=p^0=p^1=0,$4

for the spherical case and

$q_0=q_1=p^0=p^1=0,$5

for the planar case, with the corresponding gauge potential approaching

$q_0=q_1=p^0=p^1=0,$6

These falloffs are consistent with normalizable AdS behavior and permit direct thermodynamic comparison with the bald branch at fixed charge or fixed potential (Guo et al., 2021, Promsiri et al., 2023).

In EMsGB gravity, the asymptotic scalar behaves as

$q_0=q_1=p^0=p^1=0,$7

with

$q_0=q_1=p^0=p^1=0,$8

Imposing the source-free condition $q_0=q_1=p^0=p^1=0,$9 makes the hair secondary. The BF condition

$I_1$0

then bounds the allowed coupling region; for $I_1$1, this requires $I_1$2 (Zhang et al., 7 Jun 2026).

The holographic interpretation is most explicit in models where the condensate is read directly from the AdS falloff. In the planar EMS setting, the radial direction is interpreted as RG flow from UV to IR, and the scalar condensate develops in the deep interior at low temperature while vanishing at the boundary. In the scalar-tensor model, the condensate $I_1$3 plays the role of a real-valued order parameter in the boundary theory (Promsiri et al., 2023, Brihaye et al., 2019).

5. Topology, extremality, and branch structure

Topology strongly affects scalarization. In the extremal Stueckelberg-Higgs model, the near-horizon geometry is

$I_1$4

and scalarization occurs only for non-planar horizons,

$I_1$5

with AdS mass windows

$I_1$6

and

$I_1$7

The hairy branch appears at a critical charge $I_1$8 when the effective interaction satisfies $I_1$9, and the order parameter scales as

$_3$0

This makes the extremal transition second order in the entropy representation (Marrani et al., 2022).

At finite temperature, topological AdS black holes also scalarize. In the extended-phase-space study with spherical, planar, and hyperbolic horizons, scalarization is a universal low-temperature phenomenon for all three topologies. The spherical case is distinguished by a scalarization domain that theoretically extends to much higher temperatures under low pressure and by a richer branch structure even when $_3$1. Increasing pressure drives the condensate behavior through first-order-style and cave-of-wind regimes and toward a supercritical regime (Zhao et al., 22 Nov 2025).

The planar EMS case provides a complementary finite-temperature example. There, the scalarized solutions occupy a bounded domain in the $_3$2 plane, bounded by the bifurcation line, the critical line, and the extremal line. Scalarized and scalar-free planar RN-AdS black holes coexist between the bifurcation line and the extremal line, and the domain grows as $_3$3 increases (Promsiri et al., 2023).

Extremality can also obstruct regular hair. In the $_3$4 scalar-tensor model, extremal black holes with $_3$5 near-horizon geometry cannot support regular scalar fields on the horizon; analytically, the near-horizon scalar derivative diverges, and numerically the backreacted scalarized branches terminate before $_3$6. This obstruction is model dependent rather than universal, since the zero-temperature extremal Stueckelberg-Higgs model does admit scalarization under the conditions above (Brihaye et al., 2019, Marrani et al., 2022).

AdS asymptotics also reshape the branch counting familiar from asymptotically flat scalarization. In charged EMsGB AdS black holes, the BF-safe window supports only the single fundamental branch $_3$7 for GB$_3$8 scalarization, while the next excited branch appears only beyond the BF-safe regime; GB$_3$9 scalarization likewise yields a single branch. This directly contradicts the infinite-branch expectation carried over from asymptotically flat cases (Zhang et al., 7 Jun 2026).

6. Thermodynamics, phase structure, and limiting cases

Thermodynamics is one of the principal diagnostics of scalarized AdS black holes. In the scalar-tensor model with $$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$0, fixing $$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$1 and including backreaction yields a holographic phase transition in which

$$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$2

and nontrivial condensates exist for $$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$3. The condensate increases as temperature decreases, and the bulk scalarization is interpreted as the dual of a second-order phase transition with a real order parameter (Brihaye et al., 2019).

For scalarized planar EMS black holes, the dynamical and thermodynamic analyses align. The quasinormal-mode calculation finds no evidence of unstable modes on the scalarized branch in the parameter range studied, while both the grand canonical and canonical ensembles favor the scalarized black hole below the transition temperature $$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$4. The transition is second order: the free energy is continuous, the heat capacity is discontinuous at $$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$5, and the horizon scalar turns on continuously from zero. The paper explicitly compares this pattern to a conductor-superconductor transition (Promsiri et al., 2023).

The spherical EMS system exhibits a richer canonical phase structure. In the microcanonical ensemble, scalarized black holes are always entropically preferred over RNAdS when both exist. In the canonical ensemble, however, sufficiently large charge produces a reentrant sequence

$$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$6

composed of a zeroth-order transition followed by a second-order one. This places scalarized AdS black holes within the broader AdS phase-transition taxonomy rather than isolating them as a single universal pattern (Guo et al., 2021).

In extremal scalarization, entropy rather than free energy is the natural thermodynamic potential. The Stueckelberg-Higgs model shows a continuous entropy and first derivative across the transition, with a discontinuity in $$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$7, again indicating second order. In global AdS$$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$8, small hairy black holes have higher entropy than the corresponding RNAdS black holes of the same mass and charge whenever both exist, and the microcanonical diagram displays a second-order transition between RNAdS and hairy phases. The lower edge of the hairy band is a regular soliton interpreted as a nonlinear Bose condensate (Marrani et al., 2022, Basu et al., 2010).

Curvature-induced scalarization in EMsGB AdS gravity leads to the same qualitative conclusion. With fixed charge, the Gibbs free energy

$$S=\int d^4x\,\sqrt{-g}\left[\frac{\mathcal R}{2}-\Lambda+\phi^2(\alpha\mathcal R+\gamma\mathcal G)-\partial_\mu\phi\,\partial^\mu\phi-\frac14 F_{\mu\nu}F^{\mu\nu}\right],$$9

of the scalarized branch lies below that of RN-AdS, and the transition is second order because the branch emerges smoothly from the bald solution without a swallowtail or discontinuous jump in $$\square \phi + (\alpha\mathcal R+\gamma\mathcal G)\phi=0$$0. By contrast, the frozen-scalar Schwarzschild–AdS solutions in $$\square \phi + (\alpha\mathcal R+\gamma\mathcal G)\phi=0$$1 gauged supergravity remain uncharged and non-supersymmetric unless they degenerate to the AdS vacuum; they exemplify a no-hair or frozen-scalar outcome, and the construction explicitly remarks that genuinely charged AdS black holes would require relaxing the covariantly constant condition and allowing nonconstant fields (Zhang et al., 7 Jun 2026, Kimura, 2011).

Taken together, these results show that “scalarized AdS black holes” are not a single model but a class of hairy AdS solutions organized by instability mechanism, asymptotic BF constraints, horizon topology, and ensemble dependence. The common structure is the emergence of a nontrivial scalar branch from a bald AdS background; the main exceptions are precisely those cases in which the scalar sector is present but dynamically frozen.