Recent studies have demonstrated that a scalar field nonminimally coupled to the electromagnetic field can experience a spininduced tachyonic instability near KerrNewman black holes potentially driving the formation of scalar clouds. In this paper we construct such scalar clouds for both fundamental and excited modes detailing their existence domains and wave functions. Our results indicate that a sufficiently strong coupling between the scalar and electromagnetic fields is essential for sustaining scalar clouds. Within the strong coupling regime black holes that rotate either too slowly or too rapidly are unable to support scalar clouds. Furthermore we observe that scalar cloud wave functions are concentrated near the black holes poles. These findings provide a foundation for future investigations of spininduced scalarized KerrNewman black holes.
['General Relativity and Quantum Cosmology']
Spininduced Scalar Clouds around KerrNewman Black Holes
{'Spin-induced Scalar Clouds around Kerr-Newman Black Holes': "Guangzhou Guo b,a , ∗ Peng Wang a , † Tianshu Wu a , ‡ and Haitang Yang a §  \na  \nCenter for Theoretical Physics, College of Physics,  \nSichuan University, Chengdu, 610064, China and  \nb Department of Physics, Southern University of Science and Technology, Shenzhen, 518055, China  \nRecent studies have demonstrated that a scalar field non-minimally coupled to the electromagnetic field can experience a spin-induced tachyonic instability near Kerr-Newman black holes, potentially driving the formation of scalar clouds. In this paper, we construct such scalar clouds for both fundamental and excited modes, detailing their existence domains and wave functions. Our results indicate that a sufficiently strong coupling between the scalar and electromagnetic fields is essential for sustaining scalar clouds. Within the strong coupling regime, black holes that rotate either too slowly or too rapidly are unable to support scalar clouds. Furthermore, we observe that scalar cloud wave functions are concentrated near the black hole's poles. These findings provide a foundation for future investigations of spin-induced scalarized Kerr-Newman black holes.", 'I. INTRODUCTION': "The no-hair theorem, which states that stationary black holes are uniquely characterized by their mass, angular momentum and charge [1-5], is a cornerstone of general relativity in the electrovacuum context. Testing this theorem is crucial for advancing our understanding of black hole physics and for constraining the validity of alternative gravitational theories. For example, the black-hole spectroscopy program, which analyzes quasinormal modes extracted from gravitationalwave observations, has emerged as a valuable tool for probing the Kerr nature of astrophysical compact objects [6-8].  \nSince the discovery of the first hairy black hole solution within Einstein-Yang-Mills theory [9-14], numerous counterexamples to the no-hair theorem have appeared [15-17]. In particular, black holes with scalar hair have garnered significant attention due to the potential of scalar fields to model dark energy and dark matter beyond the standard model [18]. A prime example is the presence of ultralight scalar fields outside rotating black holes, which can undergo a superradiant instability [19], leading to the formation of scalar clouds [20-24]. The signatures of these scalar clouds have been used to impose stringent constraints on the scalar field's parameter space, offering valuable insights into dark matter exploration and beyond-the-Standard-Model physics [25-30]. Moreover,  \nunder specific conditions, non-linear effects can evolve scalar clouds into stationary hairy black holes [31, 32].  \nAlternatively, non-minimal couplings between a scalar field and curvature invariants have been shown to induce a tachyonic instability in the scalar field [33, 34]. This instability can lead to spontaneous scalarization, endowing black holes with scalar hair only above a certain threshold of spacetime curvature [35-39]. Consequently, spontaneous scalarization allows scalarized black holes to acquire a non-trivial scalar configuration exclusively in regimes of strong gravity, enabling them to evade constraints derived from weak-field gravity tests. Moreover, it has been demonstrated that, within a specific parameter region, the scalar field can exhibit a tachyonic instability near Kerr black holes when the black hole's spin exceeds a certain threshold [40]. Subsequently, this spin-induced tachyonic instability has been shown to be capable of generating hairy (scalarized) black holes at sufficiently high spins [41, 42]. In addition, the dynamics of spontaneous scalarization around Kerr black holes has been examined within the contexts of the Gauss-Bonnet and ChernSimons gravities [43, 44]. For a comprehensive overview of spontaneous scalarization, we refer readers to the review presented in [45].  \nSimilarly, in specific Einstein-Maxwell-scalar (EMS) models featuring a non-minimal coupling between a scalar field and the Maxwell electromagnetic invariant, the coupling, with an appropriate sign, can induce a tachyonic instability in Reissner-Nordstrom (RN) black holes [46]. The evolutions of RN black holes into scalarized RN black holes have been studied, providing valuable insights into spontaneous scalarization. Moreover, for certain parameter regimes, scalarized RN black holes have been found to possess two photon spheres outside the event horizon [47]. This unique feature leads to distinct phenomenology, including black hole images with intricate structures [48-52] and echo signals [53, 54]. Additionally, investigations into superradiant instabilities and non-linear stability of these double photon sphere black holes have been conducted [55, 56]. For a comprehensive analysis of black holes with multiple photon spheres, we refer readers to [57].  \nInterestingly, the tachyonic instability persists even when RN black holes rotate, leading to the formation of scalarized Kerr-Newman (KN) black holes [58]. The existence of these black holes is bounded by bifurcation points, corresponding to scalar clouds supported by KN black holes. However, the presence of scalarized KN black holes is suppressed by the black hole's spin, with a maximum spin threshold beyond which such solutions cease to exist. An analysis of scalar clouds induced by the tachyonic instability around KN black holes has also been conducted, revealing that black holes with sufficiently large spin cannot support scalar clouds [59]. Conversely, if the sign of the coupling constant is reversed, a spin-induced tachyonic instability emerges in KN black holes  \nwhen they rotate sufficiently fast [60, 61]. This tachyonic instability can trigger the formation of scalar clouds around KN black holes, marking the onset of spin-induced scalarized KN black holes. The existence domain of spin-induced scalar clouds has been investigated only for a limited range of black hole parameters, particularly in the strong coupling limit. Therefore, a more comprehensive exploration of spin-induced scalar clouds is necessary for a deeper understanding of spontaneous scalarization in KN black holes, providing a foundation for constructing spin-induced scalarized KN black holes.  \nThis paper presents a comprehensive investigation of scalar clouds generated by the spin-induced tachyonic instability within the EMS model. The paper is organized as follows. In Sec. II, we introduce the EMS model and its associated scalar clouds, followed by an overview of the computational framework employed to obtain scalar clouds using the spectral method. Sec. IV presents and analyzes our numerical findings. Finally, Section IV summarizes key results and discusses their implications. Throughout this paper, we adopt units where G = c = 4 πϵ 0 = 1.", 'II. SETUP': 'This section commences with a review of the EMS model and the conditions under which it exhibits a spin-induced tachyonic instability. Subsequently, we investigate the scalar clouds generated by this tachyonic instability and present the numerical method for determining their existence domains and wave functions.', 'A. Tachyonic Instability': "A tachyonic instability emerges within the EMS model, where a scalar field Φ is non-minimally coupled to the electromagnetic field A µ through a coupling function f (Φ). Explicitly, the action is given by  \nS = 1 16 π ∫ d 4 x √ -g [ R -2 ∂ µ Φ ∂ µ Φ -f (Φ) F µν F µν ] , (1)  \nwhere F µν = ∂ µ A ν -∂ ν A µ represents the electromagnetic field strength tensor. Varying the action yields the equation of motion for Φ,  \n□ Φ = f ' (Φ) F µν F µν / 4 . (2)  \nRemarkably, the inclusion of the scalar-electromagnetic non-minimal coupling term induces the tachyonic instability in the scalar field, leading to spontaneous scalarization in black holes [46, 58].  \nFor spontaneous scalarization to occur, a scalar-free solution with Φ = 0 must exist, from which scalar hair can develop. This condition imposes f ' (0) ≡ df (Φ) /d Φ | Φ=0 = 0, resulting in the series expansion of f (Φ) around Φ = 0,  \nf (Φ) = 1 + α Φ 2 + O ( Φ 3 ) , (3)  \nwhere α is a dimensionless coupling constant quantifying the strength of the scalar-electromagnetic interaction. Without loss of generality, we set f (0) = 1.  \nWithin the EMS model, the rotating scalar-free black hole solution is a KN black hole. Expressed in Boyer-Lindquist coordinates, its metric and vector potential are given by  \nds 2 = -△ Σ ( dt -a sin 2 θdφ ) 2 + sin 2 θ Σ [( r 2 + a 2 ) dφ -adt ] 2 + Σ △ dr 2 +Σ dθ 2 , A = Qr dt -a sin 2 θdφ Σ , (4)  \nwhere  \nΣ = r 2 + a 2 cos 2 θ, △ = r 2 -2 Mr + a 2 + Q 2 . (5)  \nHere, Q is the black hole charge, and a represents the ratio of black hole angular momentum J to mass M (i.e., a ≡ J/M ). The event and Cauchy horizons are located at the roots of △ , given by r + = M + √ M 2 -a 2 -Q 2 and r -= M -√ M 2 -a 2 -Q 2 , respectively. For convenience, we introduce the dimensionless reduced black hole charge and spin, defined as  \nq ≡ Q/M , χ ≡ a/M . (6)  \nTo investigate the stability of the scalar field in the scalar-free black hole background, we adopt the probe limit, neglecting the scalar field's backreaction. Within this approximation, the scalar field obeys the equation of motion  \n( □ -µ 2 eff ) Φ = 0 , (7)  \nwhere µ 2 eff = αF µν F µν / 2 is the effective mass squared. Self-interactions of the scalar field, which have a minimal impact on the onset of spontaneous scalarization [46, 62, 63], are disregarded. For KN black holes, the effective mass squared becomes  \nµ 2 eff = -αq 2 ( ˜ r 4 -6 χ 2 ˜ r 2 cos 2 θ + χ 4 cos 4 θ ) (˜ r 2 + χ 2 cos 2 θ ) 4 M 2 , (8)  \nwhere ˜ r ≡ r/M . Given the spatial dependence of µ 2 eff , the tachyonic instability is indicated by  \nmin µ 2 eff < 0 . (9)  \nMoreover, when the tachyonic instability arises, the minimum value of µ 2 eff becomes increasingly negative for a fixed χ as the magnitude of α or q increases, signifying an amplification of the tachyonic instability with larger | α | or q values.  \nThe occurrence of the tachyonic instability has been explored for both α > 0 [58] and α < 0 [60, 61]. In the case of α > 0, the region where µ 2 eff < 0 has been shown to exist outside the event horizon of KN black holes. The spatial extent of this region diminishes with increasing black hole spin, suggesting a potential suppression of the tachyonic instability for rapidly rotating black holes. For α < 0, both spin χ and charge q must be non-zero for the tachyonic instability to arise. Specifically, the parameter space admitting the tachyonic instability is constrained by  \nχ ≥ 1 + √ 1 -2 ( 2 -√ 2 ) q 2 2 √ 2 with 0 < q ≤ q cr ≡ √ 2 √ 2 -2. (10)  \nThe global minimum value of χ imposes a lower bound,  \nχ ≥ χ cr ≡ √ 2 -1 . (11)  \nIt is noteworthy that χ 2 cr + q 2 cr = 1, indicating that the KN black hole with q = q cr and χ = χ cr is extremal.", 'B. Scalar Clouds': 'The regular bound-state solutions of Eq. (7) are interpreted as scalar clouds surrounding KN black holes. The tachyonic instability can serve as a driving mechanism for the formation of scalar clouds. Indeed, scalar clouds around KN black holes induced by the tachyonic instability have been recently investigated for the α > 0 case [59]. This paper focuses on scalar clouds in the α < 0 regime. As the formation of such scalar clouds necessitates a tachyonic instability, the bounds on χ and q imposed by Eqs. (10) and (11) constrain their existence domain in the parameter space.  \nLeveraging the axial symmetry of KN black holes, we decompose the scalar field Φ into a Fourier series in terms of frequency ω and azimuthal number m ,  \nΦ( t, r, θ, φ ) = ∫ dω 2 π e -iωt ∑ m e imφ ˜ Φ( ω, r, θ, m ) . (12)  \nFor specified ω and m , Eq. (7) reduces to a Partial Differential Equation (PDE) for ˜ Φ( ω, r, θ, m ) with respect to r and θ . As scalar clouds typically serve as seeds for constructing axisymmetric hairy black hole solutions, we focus on stationary, axisymmetric configurations by setting ω = m = 0. For brevity, we denote ˜ Φ(0 , r, θ, 0) by ϕ ( r, θ ) in subsequent discussions. Analogous to hydrogen atoms, wave functions ϕ ( r, θ ) can be characterized by a discrete set of numbers ( n, l ), where the principal quantum number n = 0 , 1 , 2 · · · and the angular momentum quantum number l = 0 , 1 , 2 · · · correspond to the number of nodes of wave functions in the radial and angular directions, respectively.  \nTo determine ϕ ( r, θ ), appropriate boundary conditions must be imposed at the event horizon and spatial infinity. Given the regularity of ϕ ( r, θ ) across the event horizon, it can be expanded in a series about r = r + ,  \nϕ ( r, θ ) = ϕ 0 ( θ ) + ( r -r + ) ϕ 1 ( θ ) + · · · . (13)  \nMoreover, the condition of asymptotic flatness necessitates ϕ ( r, θ ) vanishing as r approaches infinity,  \nlim r →∞ ϕ ( r, θ ) = 0. (14)  \nAdditionally, axial symmetry, coupled with regularity on the symmetry axis, enforces,  \n∂ θ ϕ ( r, θ ) = 0, at θ = 0 and π . (15)  \nThese boundary conditions uniquely select a discrete set of KN black holes capable of supporting scalar clouds, thereby defining existence surfaces in the ( α, χ, q ) parameter space and existence lines within these surfaces for fixed α in the ( χ, q ) plane.  \nAs demonstrated in the α > 0 case, the existence lines of scalar clouds in the ( χ, q ) parameter space delineate boundaries between regions exhibiting excessively strong tachyonic instability for stationary scalar clouds and those with insufficient instability for their formation [58, 59]. For KN black holes constrained by Eqs. (10) and (11), their minimum value of µ 2 eff approaches -∞ as α → -∞ . This observation suggests that, in the limit of α → -∞ , parameter regions defined by (10) and (11) may exhibit excessively strong tachyonic instability for stationary scalar clouds. Conversely, µ 2 eff of black holes outside these constrained regions is always positive, indicating the absence of the tachyonic instability. Consequently, the existence lines of scalar clouds coincide with the boundaries of the constrained parameter regions. Specifically, as α →-∞ , the existence lines  \nin the ( χ, q ) parameter space converge to a critical existence line, given by  \nχ = 1 + √ 1 -2 ( 2 -√ 2 ) q 2 2 √ 2 for 0 < q ≤ q cr . (16)  \nMoreover, as χ increases, the critical existence line extends from ( χ, q ) = ( χ cr , q cr ) to ( χ, q ) = ( 1 / √ 2 , 0 ) .', 'C. Numerical Scheme': 'The wave equation governing ϕ ( r, θ ) in KN black holes is separable, enabling its reduction to ordinary differential equations. However, this study employs a spectral method to directly solve the wave equation for ϕ ( r, θ ), circumventing the need for separability. Consequently, this approach offers a significant advantage for computing scalar clouds around black holes in frameworks beyond general relativity. Spectral methods, a well-established method for solving PDEs [64], approximate the exact solution through a finite linear combination of basis functions. Notably, they exhibit exponential convergence for well-behaved functions, surpassing the linear or polynomial convergence rates achieved by finite difference and finite element methods. Recent investigations have successfully applied spectral methods to the identification of scalar cloud configurations [59], the construction of black hole solutions [65-67] and the calculation of black hole quasinormal modes [68-72]. A comprehensive overview of spectral methods in this context can be found in [65].  \nTo facilitate numerical implementation, we introduce a compact radial coordinate defined as  \nx = √ r 2 -r 2 + -r + √ r 2 -r 2 + + r + , (17)  \nwhich maps the event horizon and spatial infinity to x = -1 and x = 1, respectively. Under this transformation, the boundary conditions at the event horizon and spatial infinity become  \n∂ x ϕ ( x, θ ) = 0 and ϕ (1 , θ ) = 0 , (18)  \nrespectively. Without loss of generality, we assume that wave functions ϕ ( x, θ ) possess definite parity with respect to the equatorial plane, thereby permitting the restriction of the analysis to the upper half-domain 0 ≤ θ ≤ π/ 2. For even and odd parities, the boundary condition at θ = π/ 2 is ∂ θ ϕ ( x, θ ) = 0 and ϕ ( x, θ ) = 0, respectively. At θ = 0, we have ∂ θ ϕ ( x, θ ) = 0.  \nTo apply the spectral method, the function ϕ ( x, θ ) is decomposed into a spectral expansion as  \nϕ ( x, θ ) = N x -1 ∑ i =0 N θ -1 ∑ j =0 α ij T i ( x ) Θ j ( θ ) , (19)  \nwhere N x and N θ denote the resolutions in the radial and angular coordinates, respectively, T i ( x ) represents the Chebyshev polynomial, and α ij are the spectral coefficients. The angular basis Θ j ( θ ) is dependent on the parity with respect to θ = π/ 2. Specifically, we adopt  \nΘ j ( θ ) =    cos (2 jθ ) for even parity cos [(2 j +1) θ ] for odd parity . (20)  \nThis choice ensures that ϕ ( x, θ ) automatically satisfies the boundary conditions at θ = 0 and π/ 2.  \nTo determine the spectral coefficients α ij , the spectral expansion (19) is substituted into the PDE, followed by discretization at the Gauss-Chebyshev points. This procedure transforms the PDE for ϕ ( x, θ ) into a system of algebraic equations involving α ij . However, Eq. (7) exhibits linear scaling invariance, necessitating an additional constraint to guarantee a non-trivial solution for α ij . This is achieved by setting ϕ ( x, θ ) = 1 at ( x, θ ) = ( -1 , 0). This constraint introduces an extra algebraic equation for α ij through the spectral expansion (19). To balance the number of unknowns and equations, one black hole parameter (e.g., the reduced black hole charge q ) is treated as an additional unknown. The resulting system of algebraic equations for α ij and q is then solved iteratively using the Newton-Raphson method. At each iteration, the linear system of equations is solved using the built-in LinearSolve command in Mathematica. The NewtonRaphson algorithm iterates until successive iterations converge to within a tolerance of 10 -10 . Moreover, while exploring scalar cloud solutions within the ( α, χ, q ) parameter space, the residual of the spectral approximation and the number of nodes are monitored to ensure solution accuracy, maintaining a residual tolerance of 10 -7 .  \nIn the Appendix, we perform a convergence test of scalar cloud solutions by plotting the residual error as a function of N x and N θ . The results demonstrate that the error decays exponentially until reaching a round-off plateau below 10 -7 . To balance numerical precision and efficiency, we employ ( N x , N θ ) = (28 , 5) for subsequent numerical computations of ϕ ( x, θ ).', 'III. RESULTS': "In this section, we present numerical results concerning the parameter space of KN black holes that can support scalar clouds for the fundamental and first two excited modes. We also provide representative examples of the corresponding scalar cloud wave functions.  \nWe begin by analyzing the fundamental mode of scalar clouds, characterized by nodeless wave functions with ( n, l ) = (0 , 0). The left panel of Fig. 1 displays the existence domain for fundamental clouds within the ( α, χ, q ) parameter space. KN black holes supporting fundamental clouds reside  \nFIG. 1. Left Panel: Existence surface of fundamental scalar clouds with ( n, l ) = (0 , 0) in the ( α, χ, q ) parameter space. KN black holes residing on the colored surface admit the scalar clouds. The existence lines for α = -20, -50, -10 2 , -10 3 , -10 4 , -10 5 and -10 6 are shown from right to left. As α increases, this existence surface gradually converges to the critical point B with α = α cr ≃ -13 . 398. Scalar clouds cease to exist when α > α cr . Right Panel: Existence lines for α = -20, -50, -10 2 , -10 3 , -10 4 , -10 5 and -10 6 in the ( χ, q ) space, displayed from top right to bottom left. Both endpoints of the existence lines lie on the extremal KN black hole line (black dashed line), beyond which KN black holes cannot exist (gray region). As α decreases from α cr , the existence line emerges from the critical point B and gradually stretches out. In the limit of α →-∞ , the left segment of the existence line approaches the critical existence line (black dot-dashed line) while the right segment approaches the χ -axis with q = 0. The vertical and horizontal black dotted lines represent χ = χ cr and q = q cr , respectively, with their intersection marking the critical point C . As α →-∞ , the left and right endpoints of existence lines move along the extremal line towards the critical point C and the point at ( χ, q ) = (1 , 0), respectively.  \n<!-- image -->  \non the colored surface, while existence lines for various fixed α values are also shown. Our findings reveal that as α increases, these existence lines contract and converge towards the critical point B at ( α, χ, q ) ≃ ( -13 . 398 , 0 . 77001 , 0 . 63803), indicated by a black dot. Consequently, there exists a critical value of α , α cr ≃ -13 . 398, beyond which the spin-induced tachyonic instability is insufficient to form scalar clouds. The right panel of Fig. 1 illustrates the same existence lines in the ( χ, q ) plane. The extremal KN black hole line, corresponding to the condition q 2 + χ 2 = 1, is represented by a black dashed line. KN black holes cannot exist in the gray region above this extremal line, imposing an upper limit on the black hole charge q for a given χ . The vertical and horizontal dotted lines correspond to χ = χ cr and q = q cr , respectively. The intersection of these two dotted lines determines the critical point C at ( χ, q ) = ( χ cr , q cr ), marked by a red dot, which lies on the extremal line. Additionally, the black dot-dashed line represents the critical existence line given by Eq. (16).  \nFour key characteristics are observed regarding the existence lines:  \n- · Termination on Extremal Line: Both endpoints of existence lines lie on the extremal line. We assume that the left and right endpoints of the existence line with a given α locate at ( χ, q ) = ( χ low ( α ) , q up ( α )) and ( χ, q ) = ( χ up ( α ) , q low ( α )), respectively. The existence line decreases as χ increases, implying q low ( α ) < q up ( α ) and χ low ( α ) < χ up ( α ). As χ approaches χ low ( α ) from the right or χ up ( α ) from the left, the charge of the existence line converges to the upper limit set by the extremal line. This implies that, when χ < χ low ( α ) or χ > χ up ( α ), the presence of scalar clouds would necessitate a black hole charge q exceeding its extremal limit. Consequently, scalar clouds cease to exist if χ < χ low ( α ) or χ > χ up ( α ), due to insufficient tachyonic instability.\n- · Shift Toward Smaller q with Decreasing α : For α = α cr , the existence line shrinks to the critical point B with χ low ( α cr ) = χ up ( α cr ) ≃ 0 . 77001 and q low ( α cr ) = q up ( α cr ) ≃ 0 . 63803. As α decreases from α cr , the existence lines shift towards smaller q values with increasing length. This is attributed to the enhancement of the tachyonic instability for more negative α , thereby permitting a lower q to support scalar cloud formation.\n- · Approach Critical Existence Line as α →-∞ : As α goes to -∞ , the segment of the existence line with χ low ( α ) ≤ χ ≤ 1 / √ 2 converges to the critical existence line, consistent with the preceding discussion. Additionally, our results demonstrate that the segment with 1 / √ 2 < χ ≤ χ up ( α ) approaches the q = 0 line as α →-∞ . This observation indicates that, when the coupling constant α is sufficiently strong, a tiny amount of charge can trigger the formation of scalar clouds if χ exceeds 1 / √ 2. Moreover, in this limit, the left endpoint of the existence line approaches the critical point C while the right endpoint approaches the point at ( χ, q ) = (1 , 0).\n- · Serve as Threshold Line : For a given α , the existence line can be considered a threshold line, below which KN black holes exhibit insufficient tachyonic instability to support scalar clouds due to their low q values. Conversely, KN black holes above the existence line possess excessively strong tachyonic instability to sustain stationary scalar clouds. Therefore, nonlinear effects are required to suppress this instability, potentially giving rise to stationary states, such as scalarized KN black holes.  \nThe left and right panels of Fig. 2 present density plots of the fundamental cloud existence domain in the ( α, χ ) and ( α, q ) spaces, respectively. Both panels illustrate the absence of scalar  \nFIG. 2. Left Panel: Existence domain of fundamental scalar clouds in the ( α, χ ) plane, represented by a density plot where colors indicate the values of q . The domain is confined by the upper and lower boundaries, defined by the right and left endpoints of the existence lines, respectively. These boundaries merge at the critical point B , indicating the upper limit α cr on the coupling constant required to support scalar clouds. The horizontal dashed line represents χ = χ cr , above which the existence domain is located. Right Panel: Existence domain in the ( α, q ) plane, shown as a density plot with colors corresponding to χ values. The domain is bounded by the upper and lower limits, formed by the left and right endpoints of the existence lines, respectively. The horizontal dashed line depicts q = q cr , below which the existence domain is located.  \n<!-- image -->  \ncloud solutions for α > α cr , where α cr is the α value of the critical point B . In the ( α, χ ) plane, the density plot colors represent the magnitude of q , with grey regions indicating the non-existence of scalar clouds. The existence domain is bounded by the upper limit line, χ up ( α ), and the lower limit line, χ lower ( α ). As α increases towards α cr , the region of existence for scalar clouds contracts, ultimately converging at the critical point B . In the limit of α →-∞ , the upper and lower limits approach 1 and χ cr , respectively. Similarly, the existence domain in the ( α, q ) plane is confined by the upper boundary, q up ( α ), and the lower boundary, q lower ( α ), which merge at the critical point B . In the limit of α →-∞ , the upper and lower boundaries approach q cr and 0, respectively.  \nBeyond the fundamental mode, excited modes of scalar clouds can also form around KN black holes, potentially leading to excited states of scalarized black holes. Fig. 3 illustrates existence lines in the ( χ, q ) space for excited scalar clouds with ( n, l ) = (1 , 0) and (0 , 1), which exhibit similarities to the fundamental mode. Specifically, the existence lines of excited clouds lie between the extremal and critical existence lines, with both endpoints resting on the extremal line. As α approaches α cr from below, the existence lines contract and converge to the critical point B , indicating excited clouds cannot form for α > α cr . Moreover, as α becomes more negative, the existence lines shift closer to the critical existence line. To compare the existence lines of fundamental and excited  \nFIG. 3. Existence lines in the ( χ, q ) space for excited scalar clouds with ( n, l ) = (1 , 0) ( Left Panel ) and ( n, l ) = (0 , 1) ( Right Panel ). From top right to bottom left, the coupling constant α takes the values of -10 2 , -10 3 , -10 4 and -10 5 , respectively. These existence lines closely resemble those of the fundamental scalar clouds. Representative q and χ values of the existence lines are listed in Tab. I. It is evident that the existence lines for ( n, l ) = (1 , 0) lie above those for ( n, l ) = (0 , 1), suggesting that excited scalar clouds with ( n, l ) = (1 , 0) require a stronger tachyonic instability.  \n<!-- image -->  \nTABLE I. Black hole charge q and spin χ of representative clouds on the existence lines of fundamental and excited modes for α = -10 2 , -10 3 , -10 4 and -10 5 . For a given α and χ , fundamental clouds with ( n, l ) = (0 , 0) require smaller q than excited clouds, indicating that they are more prone to formation through the tachyonic instability.  \nmodes, we provide the q and χ values of representative clouds for various α in Tab. I. It is evident that, for a given α and χ , KN black holes require the smallest q to support fundamental clouds, indicating that a stronger tachyonic instability is needed to form excited clouds. Additionally, the existence line of the ( n, l ) = (0 , 1) excited mode lies just slightly above that of the fundamental mode, suggesting that n = 0 scalar clouds are more easily generated than those with n = 1.  \nFinally, we present representative scalar cloud wave functions for ( n, l ) = (0 , 0), (1 , 0) and (0 , 1) in Fig. 4. Selecting three cloud solutions on each existence line with α = -10 3 , all wave  \nFIG. 4. Wave function ϕ ( x, θ ) of representative scalar clouds for ( n, l ) = (0 , 0) ( Top Row ), ( n, l ) = (1 , 0) ( Middle Row ) and ( n, l ) = (0 , 1) ( Bottom Row ) with α = -10 3 . For all cases, we set ϕ ( x, θ ) = 1 at ( x, θ ) = ( -1 , 0). Scalar cloud wave functions are concentrated near the black hole's poles, while black hole rotation has a tendency to spread wave functions towards the equatorial plane.  \n<!-- image -->  \nfunctions exhibit concentrations near the event horizon and the poles. For a fixed r close to the event horizon, the wave functions gradually decrease along the θ direction, reaching a minimum at the equatorial plane. As the black hole's spin increases, the concentration of wave functions tends to spread towards the equatorial plane. It is noteworthy that rapidly rotating black holes with α > 0 display scalar cloud concentrations near the equatorial plane [59]. Beyond these commonalities, ( n, l ) = (1 , 0) scalar clouds feature a radial node, resulting in a valley along the  \nθ direction within their wave functions. This valley approaches the event horizon as the spin increases. For ( n, l ) = (0 , 1) scalar clouds, their odd parity with respect to the equatorial plane causes their wave functions to vanish at θ = π/ 2.", 'IV. CONCLUSIONS': "In this paper, we have explored scalar clouds generated by the spin-induced tachyonic instability around KN black holes within the framework of the EMS model, focusing on both the fundamental and excited modes. By employing the spectral method, we have successfully identified the parameter space where such scalar clouds can exist. Our findings reveal that the existence of scalar clouds is contingent upon the interplay between the black hole's charge, spin and the coupling constant α .  \nSpecifically, we have determined that for a given α , there exists a distinct existence line in the ( χ, q ) parameter space along which scalar clouds can form. Notably, this existence line intersects the extremal line at both endpoints, implying that the tachyonic instability is insufficient to induce scalar cloud formation for black holes that rotate either too slowly or too rapidly. Additionally, our analysis reveals that the region of the parameter space where scalar clouds exist shrinks as α approaches a critical value, α cr ≃ -13 . 398. This observation suggests that scalar clouds cannot form for α values greater than α cr , a conclusion further supported by the existence domains presented in the ( α, χ ) and ( α, q ) planes.  \nPrevious studies [60, 61] have established constraints on the existence domain of scalar clouds, as expressed in Eqs. (11) and (10). These constraints were suggested to be saturated in the strong coupling limit ( α →-∞ ). Our numerical results corroborate these findings. Furthermore, we also showed that for χ > 1 / √ 2, a portion of the existence lines converge towards q = 0 in the strong coupling limit, suggesting that the formation of scalar clouds requires only a minimal amount of charge.  \nOur investigation studies the influence of the scalar field mode on scalar cloud formation. While the fundamental mode requires the least charge for formation, excited modes necessitate a stronger tachyonic instability. Additionally, we have observed that scalar cloud wave functions are concentrated near the black hole's poles, differing from the concentration near the equatorial plane in the α > 0 case. As the black hole's spin increases, the concentration of scalar clouds near the poles becomes less pronounced. This wave function behavior is consistent across different modes, although excited scalar clouds exhibit additional features such as radial nodes or odd parity with  \nFIG. 5. Logarithmic plot of the residual error as a function of N x ( Top Row ) and N θ ( Bottom Row ) for scalar clouds with ( n, l ) = (0 , 0), (1 , 0) and (0 , 1). In the top row, N θ = 5 is fixed, while in the bottom row, N x = 28 is held constant. All scalar cloud solutions share the same α and a/r 2 + , namely α = -10 3 and a/r 2 + = 0 . 8. Exponential convergence is evident, with a round-off plateau observed.  \n<!-- image -->  \nrespect to the equatorial plane.  \nSince scalar clouds mark the onset of scalarization from scalar-free black holes, the findings presented in this study provide a foundation for future research on non-linear realizations of scalar clouds, namely spin-induced scalarized KN black holes. These explorations may contribute to a deeper understanding of spontaneous scalarization. Additionally, future research could delve into the non-linear dynamics of scalar clouds and their potential implications for black hole stability and related astrophysical phenomena.", 'ACKNOWLEDGMENTS': 'We are grateful to Yiqian Chen for useful discussions and valuable comments. This work is supported in part by NSFC (Grant Nos. 12105191, 12275183, 12275184, 11875196, 12347133 and 12250410250).', 'APPENDIX: CONVERGENCE TEST': "In this appendix, we assess the convergence of our numerical code by calculating fundamental and excited scalar cloud solutions with α = -10 3 and a/r 2 + = 0 . 8 at various resolutions. The top  \nrow of Fig. 5 depicts the maximum absolute value of the residual error as a function of the radial resolution N x with N θ = 5. All scalar cloud solutions demonstrate exponential convergence, with a round-off plateau approximately at N x ≥ 30. The bottom row presents the maximum absolute value of the residual error as a function of the angular resolution N θ with N x = 28. While some outliers occur at low θ resolutions, exponential convergence is observed overall, with a convergence plateau reached for N θ ≥ 5. To maintain a residual tolerance of 10 -7 , we adopt ( N x , N θ ) = (28 , 5) in our numerical calculations.  \n- [1] Werner Israel. Event horizons in static vacuum space-times. Phys. Rev. , 164:1776-1779, 1967. doi: 10.1103/PhysRev.164.1776 . I\n- [2] B. Carter. Axisymmetric Black Hole Has Only Two Degrees of Freedom. Phys. Rev. Lett. , 26:331-333, 1971. doi:10.1103/PhysRevLett.26.331 .\n- [3] Remo Ruffini and John A. Wheeler. Introducing the black hole. Phys. Today , 24(1):30, 1971. doi: 10.1063/1.3022513 .\n- [4] Tianshu Wu and Yiqian Chen. Distinguishing the observational signatures of hot spots orbiting Reissner-Nordstrom spacetime*. Chin. Phys. C , 48(7):075103, 2024. arXiv:2402.06413 , doi: 10.1088/1674-1137/ad3c2d .\n- [5] Deyou Chen, Yiqian Chen, Peng Wang, Tianshu Wu, and Houwen Wu. Gravitational lensing by transparent Janis-Newman-Winicour naked singularities. Eur. Phys. J. C , 84(6):584, 2024. arXiv: 2309.00905 , doi:10.1140/epjc/s10052-024-12950-z . I\n- [6] Maximiliano Isi, Matthew Giesler, Will M. Farr, Mark A. Scheel, and Saul A. Teukolsky. Testing the no-hair theorem with GW150914. Phys. Rev. Lett. , 123(11):111102, 2019. arXiv:1905.00869 , doi:10.1103/PhysRevLett.123.111102 . I\n- [7] Swetha Bhagwat, Xisco Jimenez Forteza, Paolo Pani, and Valeria Ferrari. Ringdown overtones, black hole spectroscopy, and no-hair theorem tests. Phys. Rev. D , 101(4):044033, 2020. arXiv:1910.08708 , doi:10.1103/PhysRevD.101.044033 .\n- [8] Ke Wang. Retesting the no-hair theorem with GW150914. Eur. Phys. J. C , 82(2):125, 2022. arXiv: 2111.00953 , doi:10.1140/epjc/s10052-022-10049-x . I\n- [9] Hugh Luckock and Ian Moss. BLACK HOLES HAVE SKYRMION HAIR. Phys. Lett. B , 176:341-345, 1986. doi:10.1016/0370-2693(86)90175-9 . I\n- [10] Serge Droz, Markus Heusler, and Norbert Straumann. New black hole solutions with hair. Phys. Lett. B , 268:371-376, 1991. doi:10.1016/0370-2693(91)91592-J .\n- [11] P. Kanti, N.E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley. Dilatonic black holes in higher curvature string gravity. Phys. Rev. D , 54:5049-5058, 1996. arXiv:hep-th/9511071 , doi: 10.1103/PhysRevD.54.5049 .  \n- [12] Thomas P. Sotiriou and Shuang-Yong Zhou. Black hole hair in generalized scalar-tensor gravity. Phys. Rev. Lett. , 112:251102, 2014. arXiv:1312.3622 , doi:10.1103/PhysRevLett.112.251102 .\n- [13] Adolfo Cisterna and Cristi'an Erices. Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario. Phys. Rev. D , 89:084038, 2014. arXiv:1401.4479 , doi:10.1103/PhysRevD.89.084038 .\n- [14] G. Antoniou, A. Bakopoulos, and P. Kanti. Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories. Phys. Rev. Lett. , 120(13):131102, 2018. arXiv:1711.03390 , doi:10.1103/PhysRevLett.120.131102 . I\n- [15] M.S. Volkov and D.V. Galtsov. NonAbelian Einstein Yang-Mills black holes. JETP Lett. , 50:346-350, 1989. I\n- [16] P. Bizon. Colored black holes. Phys. Rev. Lett. , 64:2844-2847, 1990. doi:10.1103/PhysRevLett.64. 2844 .\n- [17] Brian R. Greene, Samir D. Mathur, and Christopher M. O'Neill. Eluding the no hair conjecture: Black holes in spontaneously broken gauge theories. Phys. Rev. D , 47:2242-2259, 1993. arXiv:hep-th/ 9211007 , doi:10.1103/PhysRevD.47.2242 . I\n- [18] Carlos A.R. Herdeiro and Eugen Radu. Asymptotically flat black holes with scalar hair: a review. Int. J. Mod. Phys. D , 24(09):1542014, 2015. arXiv:1504.08209 , doi:10.1142/S0218271815420146 . I\n- [19] Richard Brito, Vitor Cardoso, and Paolo Pani. Superradiance: New Frontiers in Black Hole Physics , volume 906. Springer, 2015. arXiv:1501.06570 , doi:10.1007/978-3-319-19000-6 . I\n- [20] Shahar Hod. Kerr-Newman black holes with stationary charged scalar clouds. Phys. Rev. D , 90(2):024051, 2014. arXiv:1406.1179 , doi:10.1103/PhysRevD.90.024051 . I\n- [21] Carolina L. Benone, Lu'ıs C. B. Crispino, Carlos Herdeiro, and Eugen Radu. Kerr-Newman scalar clouds. Phys. Rev. D , 90(10):104024, 2014. arXiv:1409.1593 , doi:10.1103/PhysRevD.90.104024 .\n- [22] Yang Huang, Dao-Jun Liu, Xiang-Hua Zhai, and Xin-Zhou Li. Scalar clouds around Kerr-Sen black holes. Class. Quant. Grav. , 34(15):155002, 2017. arXiv:1706.04441 , doi:10.1088/1361-6382/ aa7964 .\n- [23] J. Kunz, I. Perapechka, and Ya. Shnir. Kerr black holes with parity-odd scalar hair. Phys. Rev. D , 100(6):064032, 2019. arXiv:1904.07630 , doi:10.1103/PhysRevD.100.064032 .\n- [24] Nuno M. Santos and Carlos A. R. Herdeiro. Black holes, stationary clouds and magnetic fields. Phys. Lett. B , 815:136142, 2021. arXiv:2102.04989 , doi:10.1016/j.physletb.2021.136142 . I\n- [25] Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Nemanja Kaloper, and John March-Russell. String Axiverse. Phys. Rev. D , 81:123530, 2010. arXiv:0905.4720 , doi:10.1103/PhysRevD.81. 123530 . I\n- [26] Richard Brito, Shrobana Ghosh, Enrico Barausse, Emanuele Berti, Vitor Cardoso, Irina Dvorkin, Antoine Klein, and Paolo Pani. Stochastic and resolvable gravitational waves from ultralight bosons. Phys. Rev. Lett. , 119(13):131101, 2017. arXiv:1706.05097 , doi:10.1103/PhysRevLett.119.131101 .  \n- [27] Hooman Davoudiasl and Peter B Denton. Ultralight Boson Dark Matter and Event Horizon Telescope Observations of M87*. Phys. Rev. Lett. , 123(2):021102, 2019. arXiv:1904.09242 , doi:10.1103/ PhysRevLett.123.021102 .\n- [28] Yifan Chen, Jing Shu, Xiao Xue, Qiang Yuan, and Yue Zhao. Probing Axions with Event Horizon Telescope Polarimetric Measurements. Phys. Rev. Lett. , 124(6):061102, 2020. arXiv:1905.02213 , doi:10.1103/PhysRevLett.124.061102 .\n- [29] Yifan Chen, Xiao Xue, Richard Brito, and Vitor Cardoso. Photon Ring Astrometry for Superradiant Clouds. Phys. Rev. Lett. , 130(11):111401, 2023. arXiv:2211.03794 , doi:10.1103/PhysRevLett.130. 111401 .\n- [30] Yifan Chen, Xiao Xue, and Vitor Cardoso. Black Holes as Neutrino Factories. 8 2023. arXiv: 2308.00741 . I\n- [31] Carlos A. R. Herdeiro and Eugen Radu. Kerr black holes with scalar hair. Phys. Rev. Lett. , 112:221101, 2014. arXiv:1403.2757 , doi:10.1103/PhysRevLett.112.221101 . I\n- [32] Jorge F. M. Delgado, Carlos A. R. Herdeiro, and Eugen Radu. Kerr black holes with synchronized axionic hair. Phys. Rev. D , 103(10):104029, 2021. arXiv:2012.03952 , doi:10.1103/PhysRevD.103. 104029 . I\n- [33] Vitor Cardoso, Isabella P. Carucci, Paolo Pani, and Thomas P. Sotiriou. Matter around Kerr black holes in scalar-tensor theories: scalarization and superradiant instability. Phys. Rev. D , 88:044056, 2013. arXiv:1305.6936 , doi:10.1103/PhysRevD.88.044056 . I\n- [34] Vitor Cardoso, Isabella P. Carucci, Paolo Pani, and Thomas P. Sotiriou. Black holes with surrounding matter in scalar-tensor theories. Phys. Rev. Lett. , 111:111101, 2013. arXiv:1308.6587 , doi:10.1103/ PhysRevLett.111.111101 . I\n- [35] Thibault Damour and Gilles Esposito-Farese. Nonperturbative strong field effects in tensor - scalar theories of gravitation. Phys. Rev. Lett. , 70:2220-2223, 1993. doi:10.1103/PhysRevLett.70.2220 . I\n- [36] Daniela D. Doneva and Stoytcho S. Yazadjiev. New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories. Phys. Rev. Lett. , 120(13):131103, 2018. arXiv: 1711.01187 , doi:10.1103/PhysRevLett.120.131103 .\n- [37] Hector O. Silva, Jeremy Sakstein, Leonardo Gualtieri, Thomas P. Sotiriou, and Emanuele Berti. Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling. Phys. Rev. Lett. , 120(13):131104, 2018. arXiv:1711.02080 , doi:10.1103/PhysRevLett.120.131104 .\n- [38] Pedro V.P. Cunha, Carlos A.R. Herdeiro, and Eugen Radu. Spontaneously Scalarized Kerr Black Holes in Extended Scalar-Tensor-Gauss-Bonnet Gravity. Phys. Rev. Lett. , 123(1):011101, 2019. arXiv: 1904.09997 , doi:10.1103/PhysRevLett.123.011101 .\n- [39] Hengyu Xu, Yizhi Zhan, and Shao-Jun Zhang. Tachyonic instability and spontaneous scalarization in parameterized Schwarzschild-like black holes. 3 2024. arXiv:2403.19392 . I\n- [40] Alexandru Dima, Enrico Barausse, Nicola Franchini, and Thomas P. Sotiriou. Spin-induced black hole spontaneous scalarization. Phys. Rev. Lett. , 125(23):231101, 2020. arXiv:2006.03095 , doi:", '10.1103/PhysRevLett.125.231101 . I': "- [41] Carlos A. R. Herdeiro, Eugen Radu, Hector O. Silva, Thomas P. Sotiriou, and Nicol'as Yunes. Spininduced scalarized black holes. Phys. Rev. Lett. , 126(1):011103, 2021. arXiv:2009.03904 , doi:10. 1103/PhysRevLett.126.011103 . I\n- [42] Emanuele Berti, Lucas G. Collodel, Burkhard Kleihaus, and Jutta Kunz. Spin-induced black-hole scalarization in Einstein-scalar-Gauss-Bonnet theory. Phys. Rev. Lett. , 126(1):011104, 2021. arXiv: 2009.03905 , doi:10.1103/PhysRevLett.126.011104 . I\n- [43] Daniela D. Doneva and Stoytcho S. Yazadjiev. Dynamics of the nonrotating and rotating black hole scalarization. Phys. Rev. D , 103(6):064024, 2021. arXiv:2101.03514 , doi:10.1103/PhysRevD.103. 064024 . I\n- [44] Daniela D. Doneva and Stoytcho S. Yazadjiev. Spontaneously scalarized black holes in dynamical Chern-Simons gravity: dynamics and equilibrium solutions. Phys. Rev. D , 103(8):083007, 2021. arXiv: 2102.03940 , doi:10.1103/PhysRevD.103.083007 . I\n- [45] Daniela D. Doneva, Fethi M. Ramazano˘glu, Hector O. Silva, Thomas P. Sotiriou, and Stoytcho S. Yazadjiev. Spontaneous scalarization. Rev. Mod. Phys. , 96(1):015004, 2024. arXiv:2211.01766 , doi:10.1103/RevModPhys.96.015004 . I\n- [46] Carlos A.R. Herdeiro, Eugen Radu, Nicolas Sanchis-Gual, and Jos'e A. Font. Spontaneous Scalarization of Charged Black Holes. Phys. Rev. Lett. , 121(10):101102, 2018. arXiv:1806.05190 , doi:10.1103/ PhysRevLett.121.101102 . I, II A, II A\n- [47] Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon spheres and spherical accretion image of a hairy black hole. Phys. Rev. D , 104(2):024003, 2021. arXiv:2104.08703 , doi:10.1103/ PhysRevD.104.024003 . I\n- [48] Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon ring and observational appearance of a hairy black hole. Phys. Rev. D , 104(4):044049, 2021. arXiv:2105.11770 , doi:10.1103/PhysRevD. 104.044049 . I\n- [49] Guangzhou Guo, Xin Jiang, Peng Wang, and Houwen Wu. Gravitational lensing by black holes with multiple photon spheres. Phys. Rev. D , 105(12):124064, 2022. arXiv:2204.13948 , doi:10.1103/ PhysRevD.105.124064 .\n- [50] Yiqian Chen, Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Appearance of an infalling star in black holes with multiple photon spheres. Sci. China Phys. Mech. Astron. , 65(12):120412, 2022. arXiv:2206.13705 , doi:10.1007/s11433-022-1986-x .\n- [51] Yiqian Chen, Peng Wang, and Haitang Yang. Interferometric Signatures of Black Holes with Multiple Photon Spheres. 12 2023. arXiv:2312.10304 .\n- [52] Yiqian Chen, Peng Wang, and Haitang Yang. Observations of orbiting hot spots around scalarized Reissner-Nordstrom black holes. Eur. Phys. J. C , 84(3):270, 2024. arXiv:2401.10905 , doi:10.1140/ epjc/s10052-024-12635-7 . I  \n- [53] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Quasinormal modes of black holes with multiple photon spheres. JHEP , 06:060, 2022. arXiv:2112.14133 , doi:10.1007/JHEP06(2022)060 . I\n- [54] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Echoes from hairy black holes. JHEP , 06:073, 2022. arXiv:2204.00982 , doi:10.1007/JHEP06(2022)073 . I\n- [55] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Superradiance instabilities of charged black holes in Einstein-Maxwell-scalar theory. JHEP , 07:070, 2023. arXiv:2301.06483 , doi:10.1007/ JHEP07(2023)070 . I\n- [56] Guangzhou Guo, Peng Wang, and Yupeng Zhang. Nonlinear Stability of Black Holes with a Stable Light Ring. 3 2024. arXiv:2403.02089 . I\n- [57] Guangzhou Guo, Yuhang Lu, Peng Wang, Houwen Wu, and Haitang Yang. Black holes with multiple photon spheres. Phys. Rev. D , 107(12):124037, 2023. arXiv:2212.12901 , doi:10.1103/PhysRevD. 107.124037 . I\n- [58] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Kerr-Newman black holes. JHEP , 10:076, 2023. arXiv:2307.12210 , doi:10.1007/JHEP10(2023)076 . I, II A, II A, II B\n- [59] Guangzhou Guo, Peng Wang, Tianshu Wu, and Haitang Yang. Stationary Scalar Clouds around KerrNewman Black Holes. 8 2024. arXiv:2408.09243 . I, II B, II B, II C, III\n- [60] Shahar Hod. Spin-charge induced scalarization of Kerr-Newman black-hole spacetimes. JHEP , 08:272, 2022. arXiv:2206.12074 , doi:10.1007/JHEP08(2022)272 . I, II A, IV\n- [61] Meng-Yun Lai, Yun Soo Myung, Rui-Hong Yue, and De-Cheng Zou. Spin-charge induced spontaneous scalarization of Kerr-Newman black holes. Phys. Rev. D , 106(8):084043, 2022. arXiv:2208.11849 , doi:10.1103/PhysRevD.106.084043 . I, II A, IV\n- [62] Pedro G. S. Fernandes, Carlos A. R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas SanchisGual. Spontaneous Scalarisation of Charged Black Holes: Coupling Dependence and Dynamical Features. Class. Quant. Grav. , 36(13):134002, 2019. [Erratum: Class.Quant.Grav. 37, 049501 (2020)]. arXiv:1902.05079 , doi:10.1088/1361-6382/ab23a1 . II A\n- [63] Shahar Hod. Spontaneous scalarization of charged Reissner-Nordstr \\ 'om black holes: Analytic treatment along the existence line. Phys. Lett. B , 798:135025, 2019. arXiv:2002.01948 . II A\n- [64] John P Boyd. Chebyshev and Fourier spectral methods . Courier Corporation, 2001. II C\n- [65] Pedro G. S. Fernandes and David J. Mulryne. A new approach and code for spinning black holes in modified gravity. Class. Quant. Grav. , 40(16):165001, 2023. arXiv:2212.07293 , doi:10.1088/ 1361-6382/ace232 . II C, II C\n- [66] Meng-Yun Lai, De-Cheng Zou, Rui-Hong Yue, and Yun Soo Myung. Nonlinearly scalarized rotating black holes in Einstein-scalar-Gauss-Bonnet theory. 4 2023. arXiv:2304.08012 .\n- [67] Clare Burrage, Pedro G. S. Fernandes, Richard Brito, and Vitor Cardoso. Spinning Black Holes with Axion Hair. 6 2023. arXiv:2306.03662 . II C\n- [68] Aron Jansen. Overdamped modes in Schwarzschild-de Sitter and a Mathematica package for the numerical computation of quasinormal modes. Eur. Phys. J. Plus , 132(12):546, 2017. arXiv:1709.", '09178 , doi:10.1140/epjp/i2017-11825-9 . II C': '- [69] Qingyu Gan, Guangzhou Guo, Peng Wang, and Houwen Wu. Strong cosmic censorship for a scalar field in a Born-Infeld-de Sitter black hole. Phys. Rev. D , 100(12):124009, 2019. arXiv:1907.04466 , doi:10.1103/PhysRevD.100.124009 .\n- [70] Adrian Ka-Wai Chung, Pratik Wagle, and Nicolas Yunes. Spectral method for the gravitational perturbations of black holes: Schwarzschild background case. Phys. Rev. D , 107(12):124032, 2023. arXiv:2302.11624 , doi:10.1103/PhysRevD.107.124032 .\n- [71] Adrian Ka-Wai Chung, Pratik Wagle, and Nicolas Yunes. Spectral method for metric perturbations of black holes: Kerr background case in general relativity. Phys. Rev. D , 109(4):044072, 2024. arXiv: 2312.08435 , doi:10.1103/PhysRevD.109.044072 .\n- [72] Adrian Ka-Wai Chung and Nicolas Yunes. Ringing out General Relativity: Quasi-normal mode frequencies for black holes of any spin in modified gravity. 5 2024. arXiv:2405.12280 . II C'}