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arxiv: 2606.08507 · v2 · pith:4KO2F75Jnew · submitted 2026-06-07 · 🌀 gr-qc

Curvature-induced scalarization of charged AdS black holes

Chao-Ming Zhang , Yun Soo Myung , Lina Zhang , De-Cheng Zou , Fu-Wen Shu This is my paper

Pith reviewed 2026-06-27 18:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords scalarizationGauss-Bonnet termAdS black holesReissner-Nordstrom-AdSphase transitionBreitenlohner-Freedman boundspontaneous scalarization
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The pith

A negative cosmological constant restricts Gauss-Bonnet scalarization of charged AdS black holes to a single branch and forces second-order phase transitions back to the RN-AdS solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how a negative cosmological constant modifies the scalarization of Reissner-Nordström-AdS black holes that is driven by a Gauss-Bonnet coupling to a scalar field. Unlike the asymptotically flat case, the onset of instability is set by the Breitenlohner-Freedman bound on the effective scalar mass rather than the sign of that mass alone. For coupling values between a threshold and 2.25 the solutions exist only on the fundamental branch; for negative couplings a separate single-branch family appears. Thermodynamic analysis shows that these scalarized solutions connect to the RN-AdS black hole through a second-order phase transition.

Core claim

In the Einstein-Maxwell-scalar-Gauss-Bonnet theory with negative cosmological constant, the effective mass squared induced by the Gauss-Bonnet term must violate the Breitenlohner-Freedman bound before scalarization can occur. When this condition holds only in the interval from the threshold value up to 2.25, the scalarized AdS black holes appear exclusively on the n=0 fundamental branch. Gibbs free-energy comparisons confirm that the transition between these scalarized configurations and the RN-AdS black hole is second-order.

What carries the argument

The effective scalar mass squared sourced by the Gauss-Bonnet term, whose sign relative to the Breitenlohner-Freedman bound determines whether RN-AdS black holes become unstable to scalar hair.

If this is right

  • Only the fundamental scalarized branch exists for couplings in the allowed window below the Breitenlohner-Freedman violation threshold.
  • A second-order phase transition occurs between scalarized AdS black holes and RN-AdS black holes.
  • For negative values of the coupling constant the scalarized solutions form a single branch under GB-minus scalarization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same bound-constrained mechanism may limit the number of branches in other asymptotically AdS black-hole families coupled to curvature invariants.
  • Thermodynamic preference for the scalarized state at fixed charge and temperature could be tested by tracking horizon area or entropy across the transition.
  • Extending the analysis to rotating or higher-dimensional AdS solutions would check whether the single-branch restriction survives.

Load-bearing premise

The onset of scalarization is fixed by whether the effective mass squared violates the Breitenlohner-Freedman bound at the specific threshold value of the coupling for a given cosmological constant.

What would settle it

Numerical construction of the scalarized solutions for a coupling value inside the interval from threshold to 2.25 that yields more than one branch, or computation of the Gibbs free energy that shows a discontinuous jump rather than a second-order transition.

Figures

Figures reproduced from arXiv: 2606.08507 by Chao-Ming Zhang, De-Cheng Zou, Fu-Wen Shu, Lina Zhang, Yun Soo Myung.

Figure 1
Figure 1. Figure 1: FIG. 1: Profiles of GB term and outer horizon [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Profiles of GB term in the near-horizon and critical onset curve. (a) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Coefficient of asymptotic potential [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Effective potentials [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Effective potentials [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Effective potentials [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Characteristic exponents ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Threshold coupling constant [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Radial profiles of the scalar cloud [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (c) shows the scalar profile ϕ(r). The scalar peaks at the horizon (ϕ0 = 0.1) and decays monotonically to zero at infinity. This shows that the scalar hair is tightly bound by the near￾horizon effective potential, satisfying both regularity and normalizability conditions [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Characteristic behaviors of the GB [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Gibbs free energy [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
read the original abstract

We investigate how a negative cosmological constant affects the Gauss-Bonnet (GB) scalarization in the Einstein-Maxwell-scalar-Gauss-Bonnet theory with a scalar coupling constant $\eta$ to GB term. We focus on the instability of Reissner-Nordstr\"om-AdS (RN-AdS) black holes under a scalar perturbation governed by an effective mass $\mu^2_{\text{eff}}$ sourced by the GB term. Unlike the asymptotically flat spacetime case, the onset of scalarization is not merely determined by $\mu^2_{\text{eff}} < 0$, but it is constrained by the Breitenlohner-Freedman (BF) bound. In case that the BF bound is violated ($\eta>2.25$ with $\Lambda=-0.5$), one may find AdS-tachyoinic instability. We find that for $0<\eta<2.25$, the GB$^+$ scalarization may be performed through spontaneous scalarization, while for $\eta<0$ the GB$^-$ scalarization is found to give the single branch of scalarized AdS black holes. For the GB$^+$ scalarization in $\eta_{th}\le\eta<2.25$ with $\eta_{th}$ threshold instability, we obtain the single branch ($n=0$ fundamental branch) of scalarized AdS black holes, contradicting to infinite branches. Finally, it is observed from Gibbs free energy that a phase transition from scalarized AdS black holes to RNAdS black hole occur natually and it is confirmed to be second-order.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript studies curvature-induced scalarization of charged AdS black holes in Einstein-Maxwell-scalar-Gauss-Bonnet theory with coupling η. It claims that the onset of scalarization is governed by an effective mass μ_eff² from the GB term but is further constrained by the Breitenlohner-Freedman bound, yielding AdS-tachyonic instability only for η>2.25 (at Λ=-0.5). For GB+ scalarization in the window η_th ≤ η <2.25 the solutions consist solely of the n=0 fundamental branch (contradicting the infinite branches found in asymptotically flat spacetimes), while GB- scalarization (η<0) likewise produces a single branch; a second-order phase transition between the scalarized solutions and RNAdS black holes is inferred from the behavior of the Gibbs free energy.

Significance. If the single-branch result and the second-order transition are robust, the work would demonstrate how AdS asymptotics and the BF bound qualitatively restrict the scalarization spectrum relative to flat-space cases, providing a concrete example of how boundary conditions alter the number of bound states in the effective scalar equation.

major comments (3)
  1. [Numerical results / scalar equation section] The central claim that GB+ scalarization produces only the n=0 branch for η_th ≤ η <2.25 rests on the numerical construction of solutions to the scalar equation. The manuscript must demonstrate that the effective potential generated by μ_eff² (sourced by the GB term) does not admit additional nodal solutions; without an exhaustive scan of shooting parameters, initial guesses, or an analytic argument based on the Sturm-Liouville character of the radial equation, the absence of n>0 branches cannot be taken as established.
  2. [Instability analysis] The specific threshold η>2.25 for BF-bound violation at Λ=-0.5 is stated without derivation. The manuscript should show the explicit expression for μ_eff², the evaluation of the BF bound m_BF² = -9/4 (in AdS units), and the algebraic step that isolates η=2.25 as the critical value.
  3. [Thermodynamics / Gibbs free energy analysis] The assertion that the phase transition is second-order is based on Gibbs free energy comparison. The manuscript must exhibit the continuity of the first derivative and the discontinuity (or lack thereof) of the second derivative of the free energy across the critical η or charge, together with the corresponding thermodynamic quantities, to confirm the order.
minor comments (2)
  1. [Abstract] Typos: “natually” should be “naturally”; “tachyoinic” should be “tachyonic”.
  2. [Introduction / setup] The notation for the effective mass μ_eff² and the precise definition of the GB coupling term should be introduced with an equation number at first appearance.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which will help improve the clarity and rigor of our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Numerical results / scalar equation section] The central claim that GB+ scalarization produces only the n=0 branch for η_th ≤ η <2.25 rests on the numerical construction of solutions to the scalar equation. The manuscript must demonstrate that the effective potential generated by μ_eff² (sourced by the GB term) does not admit additional nodal solutions; without an exhaustive scan of shooting parameters, initial guesses, or an analytic argument based on the Sturm-Liouville character of the radial equation, the absence of n>0 branches cannot be taken as established.

    Authors: We agree that additional justification is required. In the revision we will add both an analytic argument based on the Sturm-Liouville properties of the radial equation (showing the effective potential supports only a nodeless ground state for the relevant η range) and a description of extended numerical scans over multiple initial conditions and shooting parameters that confirm the absence of n>0 solutions. These additions will appear in a new subsection of the numerical results section. revision: yes

  2. Referee: [Instability analysis] The specific threshold η>2.25 for BF-bound violation at Λ=-0.5 is stated without derivation. The manuscript should show the explicit expression for μ_eff², the evaluation of the BF bound m_BF² = -9/4 (in AdS units), and the algebraic step that isolates η=2.25 as the critical value.

    Authors: We will insert the full derivation in the instability analysis section. The effective mass takes the explicit form μ_eff² = −(η/r⁴)×(Gauss-Bonnet invariant evaluated on the RN-AdS background). Setting this below the BF bound m_BF² = −9/4 (for Λ = −0.5) and solving the resulting algebraic inequality directly yields the threshold η > 2.25. The intermediate steps will be shown explicitly. revision: yes

  3. Referee: [Thermodynamics / Gibbs free energy analysis] The assertion that the phase transition is second-order is based on Gibbs free energy comparison. The manuscript must exhibit the continuity of the first derivative and the discontinuity (or lack thereof) of the second derivative of the free energy across the critical η or charge, together with the corresponding thermodynamic quantities, to confirm the order.

    Authors: We will augment the thermodynamics section with explicit plots of the first and second derivatives of the Gibbs free energy versus the control parameter (η or charge). These will demonstrate continuity of the first derivative and a discontinuity in the second derivative at the critical point, together with tabulated values of the relevant thermodynamic quantities, thereby confirming the second-order character of the transition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical branch count is independent result

full rationale

The paper's central results (single n=0 branch for GB+ scalarization when η_th ≤ η <2.25, second-order phase transition via Gibbs free energy) are obtained by direct numerical integration of the coupled Einstein-scalar-Maxwell equations subject to AdS asymptotics and BF-bound boundary conditions. The effective mass μ_eff² is computed from the GB term but does not define the solution count by construction; the reported absence of higher-node solutions is a numerical outcome, not a re-labeling of the input threshold. No self-citation chain, ansatz smuggling, or fitted-input-as-prediction pattern appears in the derivation. The analysis is self-contained against external benchmarks such as the known RNAdS background and standard BF bound.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Analysis rests on the standard Einstein-Maxwell-scalar-Gauss-Bonnet action with negative cosmological constant; effective mass sourced by GB term and BF bound are domain assumptions.

free parameters (2)
  • η
    Coupling constant to GB term; ranges 0<η<2.25, η<0, and η>2.25 are explored to delineate scalarization regimes.
  • Λ
    Negative cosmological constant fixed at -0.5 for the BF-bound violation threshold.
axioms (2)
  • domain assumption Scalar perturbation governed by effective mass μ_eff² sourced by the GB term.
    Used to determine instability of RN-AdS black holes.
  • domain assumption Breitenlohner-Freedman bound constrains onset of scalarization in AdS.
    Differs from asymptotically flat case.

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discussion (0)

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Reference graph

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