Matter-Induced Spontaneous Scalarization

• The topic defines matter-induced scalarization as the process where matter generates a negative effective mass squared, triggering a tachyonic instability in scalar-tensor models.
• It outlines how thresholds for scalarization depend on coupling strengths, density, symmetry (spherical vs planar), and specific source terms like the trace of the energy-momentum tensor.
• The mechanism has broad applications, affecting neutron stars, charged stars, dark matter stars, and black holes by altering their internal dynamics and gravitational properties.

Matter-induced spontaneous scalarization is a nonperturbative instability of scalar degrees of freedom in which matter, rather than vacuum curvature alone, generates an effective mass term that destabilizes the scalar-free general-relativistic branch in sufficiently dense or strongly coupled regions. In the canonical scalar–tensor realization, one works in the Einstein frame with

$$S[g_{\mu \nu},\phi,\Psi_m] = \frac{1}{16 \pi} \int d^4x \sqrt{-g}\,\left(R-2 \nabla_\mu\phi \nabla^\mu \phi\right) + S_m[\Psi_m,a(\phi)^2 g_{\mu \nu}]\,,$$

so that the scalar obeys

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$

Linearizing around a scalar-free background with $\alpha(\phi)\approx \beta \phi$ gives $\Box \phi \approx -4\pi \beta \phi\,T$, or equivalently an effective mass squared $m_{\rm eff}^2=-4\pi\beta T$; if this becomes negative over a sufficiently extended region, the trivial solution becomes tachyonically unstable and the system settles into a finite-$\phi$ configuration. This mechanism underlies the standard Damour–Esposito-Farèse picture for neutron stars, but it also appears in gauge-coupled stars, dark-matter stars, black holes surrounded by matter, and more exotic affine or effective-field-theory settings (Rahimi et al., 2023, Doneva et al., 2017).

1. Basic mechanism and source terms

Matter-induced scalarization is defined by two ingredients. First, the theory admits a scalar-free background, typically by imposing that the linear coupling vanishes at the reference configuration, such as $\alpha(0)=0$, $\alpha_{,\phi}(0)=0$, or $f_{,\phi}(0)=0$. Second, matter generates a negative effective mass squared for scalar perturbations. In the standard scalar–tensor case this source is the trace $T$ of the matter energy–momentum tensor; in related models it can instead be a matter invariant such as $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$0 or a Ricci scalar induced by non-conformal matter or trace anomalies (Rahimi et al., 2023, Minamitsuji et al., 2021, Herdeiro et al., 2019).

For neutron-star matter one typically has $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$1, so $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$2 yields $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$3 in the linearized Damour–Esposito-Farèse regime. In charged stars with scalar–gauge coupling, the scalar equation takes the form

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$4

and linearization gives $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$5. In the purely electric case $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$6, so $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$7 produces a tachyonic instability. In non-minimally coupled $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$8 models, the scalar equation is $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$9, so the sign condition is instead $\alpha(\phi)\approx \beta \phi$0 (Minamitsuji et al., 2021, Herdeiro et al., 2019).

Framework	Linear trigger	Tachyonic condition
DEF-type scalar–tensor	$\alpha(\phi)\approx \beta \phi$1	$\alpha(\phi)\approx \beta \phi$2 with $\alpha(\phi)\approx \beta \phi$3
Scalar–gauge coupling	$\alpha(\phi)\approx \beta \phi$4	$\alpha(\phi)\approx \beta \phi$5
Non-minimal $\alpha(\phi)\approx \beta \phi$6	$\alpha(\phi)\approx \beta \phi$7	$\alpha(\phi)\approx \beta \phi$8

The instability is local in origin but global in realization. A negative $\alpha(\phi)\approx \beta \phi$9 somewhere is not sufficient by itself; the corresponding effective well must be deep and extended enough to support a bound mode satisfying regularity at the center or horizon and decay at infinity. This is why scalarization is controlled not only by coupling constants but also by compactness, equation of state, matter morphology, and external environment.

2. Thresholds, morphology, and effective-potential structure

The onset of scalarization is a spectral problem. In spherical symmetry the scalar perturbation equation can be written as a Schrödinger-type radial problem, and the critical coupling is the value at which a zero mode first appears. For a constant-density spherical star, the critical matter-coupling parameter in the notation of the shape-dependence analysis satisfies

$\Box \phi \approx -4\pi \beta \phi\,T$0

so spontaneous scalarization requires a finite threshold in coupling strength, density, or size. By contrast, for a planar $\Box \phi \approx -4\pi \beta \phi\,T$1-symmetric matter configuration satisfying the conditions used in that analysis, the critical value is

$\Box \phi \approx -4\pi \beta \phi\,T$2

so any negative $\Box \phi \approx -4\pi \beta \phi\,T$3 triggers scalarization. The key difference is that spherical symmetry enforces a regularity condition at the origin that removes the lowest even bound state, whereas planar symmetry admits a node-free zero mode already at threshold (Motohashi et al., 2018).

This morphology dependence implies that compactness alone is not a universal control parameter. In the spherical case one needs a sufficiently deep effective well; in the planar case the same matter-induced term scalarizes far more easily. The result is conceptually parallel to one-dimensional quantum mechanics: a finite spherical well on the half-line has a nonzero critical depth, while a planar well on the whole line admits a bound state for arbitrarily small depth (Motohashi et al., 2018).

A different threshold structure appears in the massive “asymmetron” model, where the scalar equation can be written as $\Box \phi \approx -4\pi \beta \phi\,T$4 with

$\Box \phi \approx -4\pi \beta \phi\,T$5

For nonrelativistic matter $\Box \phi \approx -4\pi \beta \phi\,T$6, the effective potential near $\Box \phi \approx -4\pi \beta \phi\,T$7 is

$\Box \phi \approx -4\pi \beta \phi\,T$8

so the critical density is

$\Box \phi \approx -4\pi \beta \phi\,T$9

Above $m_{\rm eff}^2=-4\pi\beta T$0, $m_{\rm eff}^2=-4\pi\beta T$1 becomes unstable and the field moves to a finite-density minimum $m_{\rm eff}^2=-4\pi\beta T$2. In the deep scalarized regime one has $m_{\rm eff}^2=-4\pi\beta T$3, so the scalarized phase weakens gravity inside dense objects (Chen et al., 2015).

A further generalization appears in scalar-connection gravity, where the scalar equation takes the form

$m_{\rm eff}^2=-4\pi\beta T$4

With $m_{\rm eff}^2=-4\pi\beta T$5 and $m_{\rm eff}^2=-4\pi\beta T$6, one recovers the Damour–Esposito-Farèse linearized dependence on $m_{\rm eff}^2=-4\pi\beta T$7. With the more general choice $m_{\rm eff}^2=-4\pi\beta T$8, $m_{\rm eff}^2=-4\pi\beta T$9, the effective mass depends separately on $\phi$0 and $\phi$1; the theory admits pure density-driven, pure pressure-driven, and mixed scalarization channels (Azri et al., 2020).

3. Stellar realizations: neutron stars, proto-neutron stars, charged stars, and dark matter stars

The canonical astrophysical setting is the neutron star. In proto-neutron stars, the scalarization threshold and strength depend sensitively on thermodynamic state and composition. Using both the $\phi$2 chiral sigma model for hot pure neutron matter and finite-temperature Brueckner–Bethe–Goldstone/Brueckner–Hartree–Fock equations of state for hot $\phi$3-stable matter, the proto-neutron-star study compares cold stars, hot stars at $\phi$4 MeV, and isoentropic stars with $\phi$5, with and without neutrino trapping. For a fixed coupling $\phi$6, increasing entropy decreases the first critical density $\phi$7, but the magnitude of the central scalar field is larger at lower entropy. Neutrino trapping raises the first critical density, shifts the maximum of $\phi$8 to higher density, and suppresses scalarization overall. At $\phi$9, there is no scalarization for $\alpha(0)=0$0; one needs $\alpha(0)=0$1 without neutrino trapping or $\alpha(0)=0$2 with trapping. In the same study, hot pure-neutron proto-neutron stars at $\alpha(0)=0$3 MeV have $\alpha(0)=0$4 in GR, while Model 1 with $\alpha(0)=0$5 reaches $\alpha(0)=0$6; Model 2 stays much closer to GR (Rahimi et al., 2023).

These results show that “matter-induced” is literal: the source term depends on $\alpha(0)=0$7, so the onset is controlled by equation of state, composition, entropy, and neutrino content, not solely by $\alpha(0)=0$8. The same paper explicitly finds that the onset of scalarization is not set solely by compactness, because detailed microphysics changes the radial profile of $\alpha(0)=0$9 even at comparable global compactness (Rahimi et al., 2023).

Charged stars furnish a distinct gauge-induced realization. In Einstein–Maxwell–scalar theory with

$\alpha_{,\phi}(0)=0$0

the conditions for spontaneous scalarization are $\alpha_{,\phi}(0)=0$1 and $\alpha_{,\phi}(0)=0$2, which here imply $\alpha_{,\phi}(0)=0$3. The charge-to-matter ratio is parameterized by

$\alpha_{,\phi}(0)=0$4

For $\alpha_{,\phi}(0)=0$5, scalarized 0-node solutions exist for $\alpha_{,\phi}(0)=0$6 over a wide range $\alpha_{,\phi}(0)=0$7. In this branch the scalar reduces the effective Coulomb repulsion, and the scalarized stars have smaller masses and radii than the corresponding Einstein–Maxwell branch at the same central density (Minamitsuji et al., 2021).

Dark-matter stars extend the same logic to self-interacting asymmetric dark matter. In scalar–tensor theory with dark-matter coupling, scalarization occurs over most of the parameter region with a negative coupling parameter $\alpha_{,\phi}(0)=0$8, and this range is broader than for conventional neutron stars. Since the dark-matter parameters considered correspond to those in which self-interacting dark matter resolves problems in galaxy formation, such stars were proposed as probes of the scalar–dark-matter coupling if compact dark-matter stars exist (Tanaka, 25 May 2026).

A broader, curvature-mediated usage appears in boson-star studies with $\alpha_{,\phi}(0)=0$9 couplings: the boson-star matter generates the curvature profile that destabilizes a second scalar field. In that setting scalarization occurs for both signs of $f_{,\phi}(0)=0$0 and $f_{,\phi}(0)=0$1; for $f_{,\phi}(0)=0$2 there is an interval in $f_{,\phi}(0)=0$3 for which boson stars can never be scalarized, whereas for $f_{,\phi}(0)=0$4 there is no restriction on $f_{,\phi}(0)=0$5. Typically two branches exist, with the scalar maximal either at the center or on a shell near the outer radius, and the former can be radially excited (Brihaye et al., 2019).

4. Black holes in matter environments

In vacuum scalar–tensor theory, stationary asymptotically flat electrovacuum black holes remain Kerr–Newman with constant scalar. Matter changes this conclusion because it induces an effective mass for the scalar. In the Einstein-frame analysis of black holes with surrounding matter, expanding the conformal factor around a decoupling point with $f_{,\phi}(0)=0$6 yields

$f_{,\phi}(0)=0$7

If $f_{,\phi}(0)=0$8 in some region, the black hole develops a tachyonic instability and scalar hair; if $f_{,\phi}(0)=0$9, rotating black holes can instead undergo superradiant instability. For Kerr black holes with surrounding matter, the paper identifies both mechanisms and shows that the positive-$T$0 regime can also lead to resonant superradiant amplification factors as large as $T$1 or more (Cardoso et al., 2013).

Dark-matter halos provide a direct matter-induced black-hole example within scalar–tensor gravity. Expanding around a constant background scalar with $T$2 gives

$T$3

so $T$4 yields tachyonic scalarization while $T$5 in the rotating case supports superradiant instability. Using cold-dark-matter and scalar-field-dark-matter halo metrics, the analysis finds that for realistic low-surface-brightness galaxy halos the scalarization condition is approximately $T$6; for astronomically large halos the dependence on halo mass and size weakens and the coupling $T$7 dominates, while larger black-hole masses suppress scalarization (Tanaka, 27 May 2025).

Matter-induced scalarization also survives in de Sitter backgrounds. In Einstein–Maxwell–scalar theory with positive cosmological constant, the coupling

$T$8

to the Maxwell invariant $T$9 gives $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$00, and for Reissner–Nordström–de Sitter backgrounds $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$01, so scalarization requires $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$02. The local condition for the tachyonic instability is unchanged by $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$03, and there exist asymptotically de Sitter scalarized charged black holes with only mild differences from their asymptotically flat counterparts. This is a useful contrast with Gauss–Bonnet-induced scalarization, where the de Sitter asymptotics can be spoiled by an asymptotic tachyon (Brihaye et al., 2019).

A different matter route to black-hole scalarization uses the non-minimal coupling $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$04. Classical electro-vacuum black holes have $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$05, but trace anomalies or non-conformal matter make $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$06. In the Einstein–Maxwell–dilaton GHS solution one has $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$07, so scalarization requires $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$08; in the quantum-corrected RN–$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$09 solution one again has $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$10, hence $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$11; in the noncommutative-geometry-inspired Schwarzschild solution one has $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$12, hence $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$13. In all three cases scalar clouds exist only for an infinite discrete set of $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$14, and the nonlinear scalarized solutions are generically entropically favoured when the entropy comparison is unambiguous (Herdeiro et al., 2019).

There is also a broader curvature-mediated variant in extended scalar–tensor–Gauss–Bonnet theory with perfect-fluid dark matter. In the GB$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$15 regime, non-rotating vacuum black holes do not scalarize, but surrounding dark matter can modify $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$16 so that scalarization appears if

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$17

This is not a direct $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$18-coupling mechanism, but it is still matter-induced in the sense that matter changes the curvature invariant that controls the tachyonic mass (Tang et al., 21 Apr 2025).

5. Alternative formulations and microphysical consequences

Matter-induced scalarization is not confined to standard metric scalar–tensor gravity. In scalar-connection gravity, the fundamental fields are an affine connection and a scalar, while the metric is generated dynamically. The action is written in terms of

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$19

and variation with respect to the connection generates an emergent metric

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$20

The scalar equation becomes

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$21

With $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$22 and $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$23, the linearized equation reproduces the DEF combination $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$24. With independent $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$25, the effective mass depends separately on $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$26 and $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$27, making pressure-driven scalarization an explicit possibility (Azri et al., 2020).

Matter-induced scalarization can also alter internal microphysics. In the toy model of a “gravitational Higgs mechanism,” the scalarized neutron star phase gives the photon a mass through a gauge-covariant derivative coupling,

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$28

For static, spherically symmetric, electrically neutral stars, the only regular electromagnetic solution is $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$29, so the star’s structure reduces to the standard Damour–Esposito-Farèse system, but the photon mass profile remains nontrivial. With $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$30, the peak photon mass in the interior reaches the GeV range in the scalarized phase. The paper presents this as a toy model showing how spontaneous scalarization can act as a Higgs-like mechanism inside neutron stars (Franchini et al., 2017).

The asymmetron model provides another model-building extension. There the scalar is massive,

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$31

and couples through

$$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$32

Dense matter drives spontaneous scalarization above $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$33, while the scalar mass suppresses long-range deviations from GR and allows the same field to behave as cold dark matter after inflation. In the deep scalarized phase $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$34, and the model was explicitly proposed so that “asymmetron” could both scalarize dense objects and account for dark matter (Chen et al., 2015).

6. Relation to curvature-induced scalarization, limitations, and open directions

The sharpest conceptual distinction in the literature is between matter-induced and curvature-induced scalarization. In the standard Damour–Esposito-Farèse picture, the scalar equation is sourced by the matter trace, and the trigger is a matter-controlled effective mass. By contrast, in extended scalar–tensor–Gauss–Bonnet theories the scalar couples to $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$35, and the effective mass depends on curvature even in vacuum. The black-hole ESTGB study emphasizes that its scalarization is “induced by the curvature of the spacetime,” not by matter, while the neutron-star ESTGB study makes the same contrast for stellar solutions (Doneva et al., 2017, Doneva et al., 2017).

This distinction is operationally important. Matter-induced scalarization is typically shut off in vacuum and activated by dense or non-conformal matter, whereas curvature-induced scalarization can persist in matter-free strong-curvature regions. At the same time, the boundary is not always sharp in practice: boson-star and dark-matter-halo examples show that matter can induce scalarization indirectly by generating the curvature profile that enters the scalar equation, even when the formal coupling is to $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$36 or $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$37 rather than directly to $$\Box \phi = -4\pi\,\alpha(\phi)\,T,\qquad \alpha(\phi)\equiv \frac{d\ln a}{d\phi}\,.$$38 (Brihaye et al., 2019, Tang et al., 21 Apr 2025).

A common oversimplification is to identify scalarization solely with compactness. The current literature does not support that simplification. Proto-neutron-star calculations show strong dependence on entropy, neutrino trapping, and composition; the shape-dependence analysis shows qualitatively different thresholds for spherical and planar sources; scalar-connection gravity allows separate density- and pressure-driven channels. Compactness is often an efficient proxy, but it is not the only control parameter (Rahimi et al., 2023, Motohashi et al., 2018, Azri et al., 2020).

Most existing analyses remain restricted to highly symmetric settings. The proto-neutron-star study is static and spherically symmetric and omits rotation, magnetic fields, hyperons, quarks, superfluidity, and dynamical cooling and deleptonization. The boson-star analysis establishes existence but does not complete the stability problem for the scalarized branches. Black-hole matter-halo analyses often rely on idealized shells, spherical halos, or separable effective masses, and the fully nonlinear endpoints of some halo-induced instabilities are still open (Rahimi et al., 2023, Brihaye et al., 2019, Cardoso et al., 2013).

The broader significance of matter-induced spontaneous scalarization is that it turns the internal state of matter into a control parameter for strong-field gravity. In compact stars it ties scalar charge to equation of state, entropy, lepton content, charge fraction, or dark-sector microphysics. Around black holes it allows otherwise bald solutions to become unstable in the presence of matter, trace anomalies, or dark-matter halos. Across these settings, the recurring structure is the same: a scalar-free branch exists, matter creates an effective mass, and beyond a threshold a tachyonic mode bifurcates into a new scalarized phase.