Quasinormal Modes of Charged Fermions and Phase Transition of Black Holes
aa r X i v : . [ g r- q c ] M a y Quasinormal Modes of Charged Fermions and Phase Transition ofBlack Holes
Rong-Gen Cai , ∗ Zhang-Yu Nie , † Bin Wang , ‡ and Hai-Qing Zhang § Key Laboratory of Frontiers in Theoretical Physics,Institute of Theoretical Physics, Chinese Academy of Sciences,P.O. Box 2735, Beijing 100190, China and Department of Physics, Fudan University, Shanghai 200433, China
We study the quasinormal modes of massless charged fermions in a Reissner-Nordstr¨om-anti-de Sitter black hole spacetime. In the probe limit, we find that theimaginary part of quasinormal frequency will become positive when the temperatureof the black hole is below a critical value. This indicates an instability of the blackhole occurs and a phase transition happens. In the AdS/CFT correspondence, thistransition can be viewed as a superconducting phase transition and the bulk fermionis regarded as the order parameter. When the coupling of the fermions and thebackground electric field becomes stronger, the critical temperature of the phasetransition becomes higher. If the interaction between the fermion and the electricfield can be ignored, namely in the case of a neutral fermion, the imaginary partof the quasinormal modes is always negative, which indicates that the black hole isstable and no phase transition occurs.
I. INTRODUCTION
Over the past years the holographic models of superconductors have attracted a lot ofattentions since the works [1, 2]. For reviews see [3]. In a simple model with the Einstein-Maxwell-complex scalar field system with a negative cosmological constant, the chargedscalar field will condensate when the temperature of the anti-de Sitter (AdS) black holeis below a critical value. Above the critical temperature, the system has the Reissner-Norstr¨om (RN)-AdS black hole solution with a trivial scalar field, while below the criticaltemperature, a hairy black hole solution is more stable with a nontrivial scalar field. Thisindicates that a phase transition happens between the RN-AdS black hole and a hairy blackhole when the temperature of the black hole arrives at the critical value. According to theAdS/CFT correspondence [4–6], the phase transition of the AdS black hole can be mapped toa superconducting phase transition on the boundary of the AdS space [2], the condensationof charged scalar field around the black hole corresponds to a condensation of the charged ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] operators in the boundary field theory.It is well known that a neutral scalar field perturbation of an asymptotically AdS space-time is stable if the mass of the scalar field satisfies the Breitenlohner-Freedman (BF)bound [7]. However, the perturbation of charged scalar field in an asymptotically AdSspacetime can cause the background to become unstable [1, 8–10]. In the holographic super-conductor models, the presence of the instability of the perturbation of the charged scalarfield just indicates the occurrence of the condensation in the boundary field theory.To reveal the stability of a black hole, a useful method is to study the quasinormal modesof some perturbations in the black hole background, for a review see [11]. Recently it wasfound that quasinormal modes of charged scalar field in some AdS black hole backgroundscan be used to disclose the relation to the superconductor transition [12]. The occurrence ofthe unstable modes was observed in consistent with the superconducting phase transition.Besides the charged scalar field, it is of great interest to examine whether some otherfields can experience the condensation in the AdS black hole background, for example thecharged fermionic field, its condensation might be related to the model of color supercon-ductor in QCD theory. The perturbation of a charged fermionic field in an asymptoticallyflat spacetime was found always with decay modes [13]. The fermionic field will either beabsorbed by the horizon or go to the spatial infinity and cannot condensate near the blackhole. It is well known that the AdS spacetime has different asymptotical behavior from thatof the asymptotically flat spacetime and this difference plays crucial role on the dynamics ofthe inner geometry. For example, the late-time tail of the perturbations with a power lawform in an asymptotically flat spacetime gives way to the exponential form in an asymptot-ically AdS spacetime [14, 15]. In this work we will concentrate on the study of the chargedfermonic field perturbation in the background of an AdS black hole.We employ a simple RN-AdS spacetime geometry while leaving fermionic fields as probes.We find that when the temperature is below a certain critical value, the imaginary part of thequasinormal modes will become positive, which indicates the occurrence of the instability ofthe black hole. This shows that at the critical temperature there appears a phase transition,the RN-AdS black hole background becomes unstable and a new stable hairy black holesolution is expected to emerge. We observe that the critical temperature grows with theincrease of the charge coupling constant. This means that the stronger the fermions coupleto the Maxwell field, the easier the instability occurs. Our result is consistent with theholographic superconducting phase transition discussed in [3][12], however in our result theorder parameter is a charged fermion instead of the charged scalar field. In this sense ourmodel might be regarded as a simple model of color superconductor in QCD theory.This paper is organized as follows. In Sec. II we briefly introduce the RN-AdS geometryand the action of the charged fermions in this background. Under an explicit representation,the Dirac equation is splitted in Sec. III to a concise form. In Sec. IV, we use the Horowitz-Hubeny approach to Fourier expand the Dirac equations for the convenience to study thequasinormal modes of the fermionic perturbation numerically. We give the numerical resultsand analysis of the quasinormal modes in Sec. V. We conclude our paper in Sec. VI.
II. THE BACKGROUND GEOMETRY AND ACTION
For a general d -dimensional RN-AdS spacetime, the metric is ds = − f ( r ) dt + dr f ( r ) + r d Ω d − ,k , (1)where f ( r ) = k + r L + Q r d − − ( r r ) d − . (2) L is the AdS radius, Q is the charge of the black hole and r is related to the black holemass M via r d − = 16 πG d M ( d − A d − , (3)where A d − = 2 π d − / Γ( d − ) is the area of a unit ( d − G d is the Newton gravi-tational constant in the d -dimensional spacetime. d Ω d − ,k in (1) is the metric of constantcurvature. If k = 0, it is the metric of a flat Euclidean space R d − ; if k >
0, it is the lineelement of a ( d − √ k ; and if k <
0, it is the metric of a hyperbolicplane with radius of curvature √− k . Without loss of generality, one can take k = 0 , and ± k = 0 case in this paper for simplicity and also for the relation to asuperconductor in a plane. For the cases with k = 0, similar results can be obtained. Inparticular, we will consider a 4-dimensional RN-AdS spacetime and let G d = 1 ds = − f ( r ) dt + dr f ( r ) + r ( dx + dy ) , (4)The temperature of the black hole now is T H = 3 r + πL − Q πr . (5)where r + is the horizon radius of the black hole.The spin connection is defined as: ω ˆ a ˆ bc = e ˆ ad ∂ c e d ˆ b + e ˆ ad e f ˆ b Γ dfc , (6)where e ˆ ab is the tetrad , Γ abc is the Christoffel connection and ω ˆ a ˆ bc = − ω ˆ b ˆ ac . The nonvanishingspin connections ω ˆ a ˆ bc for the metric (4) are: ω ˆ r ˆ tt = − ω ˆ t ˆ rt = 12 f ′ , ω ˆ x ˆ rx = − ω ˆ r ˆ xx = p f , ω ˆ y ˆ ry = − ω ˆ r ˆ yy = p f . (7)where a prime denotes the derivative with respect to r . The un-hatted letter b is the index of the background spacetime while the hatted letter ˆ a denotes theindex of the tangent space. We consider the Einstein-Maxwell-fermion system with the action S T = S g + S m , (8)where S g = 12 κ Z d x √− g (cid:18) R − L (cid:19) , (9) S m = N Z d x √− g (cid:18) − F ab F ab + i (cid:0) ¯ΨΓ a ( D a − iqA a )Ψ − m ¯ΨΨ (cid:1)(cid:19) , (10) κ is the 4-dimensional gravitational constant, R is Ricci scalar, N is a total coefficient ofthe action of matter, q is the coupling constant between the fermion field and Maxwell field.Γ a is the Dirac gamma matrix and D c = ∂ c + 12 ω ˆ a ˆ bc Σ ˆ a ˆ b , Σ ˆ a ˆ b = 14 [Γ ˆ a , Γ ˆ b ] , Γ b = e b ˆ a Γ ˆ a . (11)The Dirac equation for the fermion isΓ a ( D a − iqA a )Ψ − m Ψ = 0 . (12)In the action (8), clearly we have a simple RN-AdS black hole solution with the electricpotential A t and a vanishing fermion Ψ: A t = Q ( 1 r − r + ) , Ψ = 0 . (13)In the following, we will work in the probe limit which means that the fermionic field doesnot backreact on the metric and Maxwell field. III. THE SPLIT OF THE DIRAC EQUATION
We can write the wave function Ψ( r, x µ ) into the momentum spaceΨ( r, x µ ) = ψ ( r ) e − iωt + i~k · ~x . (14)where x µ denotes the coordinates in the boundary, x µ = ( t, x, y ), while ~k = ( k x , k y ) and ~x = ( x, y ). Under this transformation, we can write the Dirac equation (12) into p f Γ ˆ r ∂ r ψ − iω √ f Γ ˆ t ψ + i~k · Γ ˆ ~x r ψ + 14 [ f ′ √ f + 4 √ fr ]Γ ˆ r ψ − ( iq Γ a A a + m ) ψ = 0 . (15)where ~k · Γ ˆ ~x = k x Γ ˆ x + k y Γ ˆ y . We can choose the Dirac gamma matrices as [16]Γ ˆ t = (cid:18) − II (cid:19) , Γ ˆ i = σ ˆ i σ ˆ i ! (16)where I is the 2 × σ ˆ i is the Pauli matrix, explicitly, σ ˆ x = (cid:18) (cid:19) , σ ˆ y = (cid:18) − ii (cid:19) , σ ˆ r = (cid:18) − (cid:19) . (17)It is easy to decompose the fermion filed Ψ into Ψ + and Ψ − , which are the eigenvectors ofΓ , i.e. Ψ = (cid:18) Ψ + Ψ − (cid:19) , P ± Ψ = ± Ψ ± , P ± = 1 ± Γ , Γ = i Γ t Γ x Γ y Γ r . (18)Under this representation, the Dirac equation (15) can be decomposed into( p f ∂ r + 14 f ′ √ f + √ fr ) σ ˆ r ψ − + ir ( ~k · ~σ ) ψ − + i √ f ( ω + qA t ) ψ − − mψ + = 0 , (19)( p f ∂ r + 14 f ′ √ f + √ fr ) σ ˆ r ψ + + ir ( ~k · ~σ ) ψ + − i √ f ( ω + qA t ) ψ + − mψ − = 0 . (20)It is easy to see from eqs.(19) and (20) that if ω → − ω and q → − q , there is a permutationsymmetry ψ + ↔ ψ − in (19) and (20). In the following, we will take m = 0, because thechiral modes of ψ will decouple in this case, which can be easily seen from eqs. (19) and(20).For the massless fermions, we will focus on the modes of ψ + , because the modes of ψ − can be obtained accordingly if we change the sign of ω and q . Furthermore, for simplicity,we can set k y = 0 because of the symmetry of ( ~x, ~y )-plane [17]. We rewrite ψ + as ψ + = r − f − / ˜ ψ. (21)And then eq. (20) can be further simplified into σ ˆ r ˜ ψ ′ − if ( ω + qA t − √ fr k x σ ˆ x ) ˜ ψ = 0 . (22)Here ˜ ψ is a 2-component fermion wavefunction, we can decompose it as ˜ ψ = (cid:18) ψ ψ (cid:19) underthe σ ˆ r matrix. Under this representation, eq. (22) can be further decomposed into ψ ′ − if ( ω + qA t ) ψ + ir √ f k x ψ = 0 , (23) ψ ′ + if ( ω + qA t ) ψ − ir √ f k x ψ = 0 . (24)Like in the case of eqs. (19) and (20), if ω → − ω, q → − q and k x → − k x , there is also apermutation symmetry ψ ↔ ψ between eqs. (23) and (24). Therefore, we can only focuson the modes ψ by extracting ψ from eq.(23) and then inserting it into eq.(24). IV. THE QUASINORMAL MODES
To calculate the quasinormal modes of the charged fermion, we introduce the tortoisecoordinates r ∗ by dr ∗ = drf . (25)Now, the black hole horizon is located at r ∗ → −∞ and the spatial infinity is at a finitevalue of r ∗ .The ingoing boundary conditions at the horizon requires the wavefunction to have theform ψ ( r ) = e − iωr ∗ u ( r ) . (26)For simplicity of the numerical calculation, we define x = 1 /r . Thus, the black hole horizonis at x + = 1 /r + while the infinite boundary r → ∞ locates at x = 0. And the equation of u is given as S ( x )¨ u ( x ) + T ( x ) x − x + ˙ u ( x ) + U ( x )( x − x + ) u ( x ) = 0 , (27)where the overdot denotes the derivative with respect to x and S ( x ) = (cid:0) − L Q x x + 4 x + 4 xx + + 4 x (cid:1) , (28) T ( x ) = 12 (cid:0) L Q x x − x − xx + − x (cid:1) (cid:0) L Q x x + 16 iL ωx − x (cid:0) L Q x + 4 (cid:1)(cid:1) , (29) U ( x ) = 2 L x (cid:8) ω (cid:2) − L qQx − iL Q x x + 8 L qQxx + 3 ix (cid:0) L Q x + 4 (cid:1)(cid:3) − ( x − x + ) (cid:2) (cid:0) L Q x x − x − xx + − x (cid:1) k x + iqQ (cid:0) − iL qQx +2 L Q x x + 8 x + 8 x (cid:0) iL qQx + x + (cid:1) − x (cid:0) L Q x + 4 (cid:1)(cid:1)(cid:3)(cid:9) . (30)Although the complex number i occurs in the potential U ( x ) here, it was argued in [18–22]and references therein that one still can have correct quasinormal modes in the cases withsuch a complex potential if some proper boundary conditions are imposed.Next we employ the Horowitz-Hubeny method [14] to calculate the quasinormal modes ofthis charged fermion. S ( x ) , T ( x ) and U ( x ) can be Fourier expanded near x + to a finite term, i.e. , S ( x ) = P n =0 s n ( x − x + ) n , T ( x ) = P n =0 t n ( x − x + ) n and U ( x ) = P n =0 u n ( x − x + ) n .We also expand the solution u ( x ) around x + as u ( x ) = ( x − x + ) α ∞ X n =0 a n ( x − x + ) n . (31)At the leading order, the equation has two solutions α = 1 / α = iL x + − L Q x ω . It isobvious that the solution α = iL x + − L Q x ω gives an outgoing mode at the horizon. As aresult, we choose α = 1 / a n = − P n n − X k =0 [( k + α )( k + α − s n − k + ( k + α ) t n − k + u n − k ] a k , (32)where P n = ( n + α )( n + α − s + ( n + α ) t + u .In the numerical calculations, we set the initial data a = 1 whose scaling will not affectthe final quasinormal modes ω because the equation of ψ is linear. In practice, we can solvethe expansion up to a large number n = N , then compare the results got from N and theresults from n > N , if the error of the two results is of the order 10 − , we will believe thatthe modes ω N is acceptable. V. NUMERICAL RESULTS
The boundary condition on the spatial infinity is that the fermion field vanishes there,which means u (0) = N X n =0 a n (0 − x + ) n + α = 0 . (33)This equation gives the eigenvalues of ω . The quasinormal modes can be decomposed intoreal and imaginary parts: ω = Re( ω ) + i Im( ω ) . (34)If Im( ω ) <
0, the perturbations always decay exponentially and then the background blackhole is stable; if Im( ω ) >
0, however, the modes grow exponentially, which means theperturbations will make the background black hole unstable. This instability suggests aphase transition of black holes [8, 12].Following [1], in the calculation we set r + = Q = 1. In this case, changing L correspondsto the change of the temperature of the black hole. Furthermore, we set the x-directionmomentum k x = 1. For the calculation precision we Fourier expand the equation up to N = 300.To find the marginally stable perturbation where Ψ depends only on r and is infinites-imally small, taking ω = 0 in (27), we will have an equation on q and L from eq. (27).Requiring both the real and imaginary parts of the resulting equation on q and L to vanish,we have two constraints on the relation between q and L , whose numerical results are shownin the left panel of Fig.1. It indicates that only q and L at the crossing points of the two con-straint curves can give the marginally stable solution with ω = 0. The three crossing points( q, L ) shown in the left panel of Fig.1 are (1 . , . , (2 . , . . , . q increases, we see that L at the crossing points decreases. Accordingto the temperature formula (5) of the black hole, the temperature increases as L decreases.Thus the stronger the fermion field couples to the Maxwell field, the easier the instability Real Part Imaginary Part q L T H c * - FIG. 1: (Left Panel) Eq. (27) with ω = 0 gives the two constraints on q and L : the real part (blue)and imaginary part (red dashed). (Right Panel) The critical temperature T Hc versus q . occurs. This is physically reasonable. We show this relation of the critical temperature andthe coupling q in the right panel of Fig.1.The marginally stable perturbation cannot guarantee the existence of the instability butjust highly suggestive [1]. In examining the stability of the perturbation, one can release thecondition ω = 0, then in principle there does not exist any constraint relation between q and L . In this paper we are interested in the quasinormal modes of the fermion perturbation andexpect to see whether a positive imaginary part of quasinormal frequency which indicatesthe instability of the fermion perturbation will appear. Fig.2 shows the frequencies of thequasinormal modes in the case with a fixed q = 1 .
546 and different L . It can be seen thatwhen L > L c = 1 .
951 the imaginary part of the quasinormal frequency becomes positive,which means that the black hole is unstable under the perturbation when the temperatureis low enough. L = L c corresponds to the critical temperature of the black hole T Hc , when T > T Hc the RN-AdS black hole is stable under the fermonic perturbation, while the RN-AdSblack hole becomes unstable when T < T Hc and the fermonic hair is expected to condensateand attach to the black hole. The RN-AdS black hole will give way to a fermion hairedblack hole.In Table I we list the quasinormal modes shown in Fig.2. One can see that the imaginarypart of the first node becomes positive when L = 1 .
96 and L = 2, while it is negative if L = 1 .
94. This manifests that the perturbation becomes unstable when L changes from 1 . L = 1 .
96. The critical value is L = 1 . ω = 0.From the left panel of Fig. 1 we can see that when q = 0, there are no critical values for L which makes ω = 0. This implies that for the neutral fermion perturbation, the black holeis always stable and the perturbation always decays exponentially. This is indeed observedin the quasinormal modes calculation as shown in Table II, where we see that the imaginarypart of the quasinormal modes of the fermion perturbation is always negative in the case of q = 0.Fig. 3 shows the behavior of the quasinormal modes with respect to the black hole tem- L = = = - - - -
20 Re H Ω L I m H Ω L FIG. 2: The real and imaginary parts of the modes when changing L while q is fixed as q = 1 . L = 2 Re( ω ) 0 . . . . . . . . . . ω ) 0 . − . − . − . − . − . − . − . − . − . L = 1 .
96 Re( ω ) 0 . . . . . . . . . ω ) 0 . − . − . − . − . − . − . − . − . L = 1 .
94 Re( ω ) 0 . . . . . . . ω ) − . − . − . − . − . − . − . perature. We can see from Fig. 3 that in the high temperature regime, both the real partand imaginary part of the quasinormal modes have a linear relation to the temperature ofthe black hole, which is similar to the result given in [22], where the quasinormal modes of aneutral massless fermion are numerically calculated in a RN-AdS black hole with a sphericalhorizon (namely the case with k = 1 in (1)). The linear behavior observed for the fermonicperturbation agrees to the finding for scalar perturbation [14, 15]. By numerical fitting, wefind that the quasinormal frequencies behave like Re ( ω ) ≈ . T H , Im ( ω ) ≈ − . T H , (35) TABLE II: Real and imaginary parts of the quasinormal modes in the case of q = 0 with differenttemperature shown in Fig. 3. T H . . . . . . . . . . ω ) 0 . . . . . . . . . . ω ) − . − . − . − . − . − . − . − . − . − . T H is the black hole temperature given in (5). In the high temperature regime,Ref. [22] gives a relation (see eq. (8) in [22]) for the case of Schwarzschild-AdS black holewith a spherical horizon Re ( ω ) = 8 . T, Im ( ω ) = − . T. (36)We can see from (35) and (36) that their behaviors are quite similar to each other, exceptthe differences in the numerical factors caused by the value of the nonzero black hole charge Q we considered here.In the low temperature regime, on the other hand, one can see that both the real andimaginary part obviously deviate from the linear behavior. This result is also consistentwith the one of scalar fields in [14, 15]. T H R e H Ω L - - - - - - - T H I m H Ω L T H R e H Ω L - - - - - T H I m H Ω L FIG. 3: Quasinormal modes of the first node in the case of q = 0. The real part versus hightemperature (Up-Left); the imaginary part versus high temperature(Up-Right); the real part versuslow temperature (Down-Left); the imaginary part versus low temperature (Down-Right). VI. CONCLUDING REMARKS
In this paper we considered the Einstein-Maxwell-Fermion system with a negative cos-mological constant. Such a system has been intensively studied recently in the holographicfermion liquid model [16, 17, 23–25]. We studied the stability of the RN-AdS black hole1under the perturbation of the charged fermion by numerically calculating the quasinormalmodes of the perturbation in the probe limit. It is found that when the temperature of theblack hole is below a critical value, the imaginary part of the quasinormal modes will becomepositive, which means that the perturbation of the fermion will grow exponentially whichmakes the black hole unstable; while when the temperature of the black hole is above thecritical value, the perturbation will decay exponentially and the black hole is stable. Our re-sult indicates that a phase transition occurs in the system from the high temperature phase,which is described by the RN-AdS black hole, to a low temperature phase, which should bedescribed by some stable new black hole solution. Like the case with charged scalar field, weexpect that the stable new black hole should be a one with nontrivial fermion condensation.When the coupling constant q increases, the critical temperature of the black hole becomeshigher. This means that the stronger the coupling is, the much easier for the phase transitionto occur. In the case of q = 0, namely for a neutral fermion, we found that the imaginarypart of the quasinormal modes is always negative. This implies that the black hole isstable under the perturbation of the uncharged fermion and it will not cause any phasetransition of the black hole. The behavior of the quasinormal modes agrees with thatfor the scalar perturbation [14, 15] and in consistent with fermonic perturbation [22] forspherical background. In high temperature regime, both the real and imaginary parts of thequasinormal modes are linearly proportional to the temperature of the black hole. In theAdS/CFT correspondence, 1 / Im( ω ) is the time scale for the system approaching to thermalequilibrium of the boundary field theory. This means if we perturb the thermal field on theboundary, the time for it to approach a thermal equilibrium is proportional to the inverseof the temperature [14]. However the linearity breaks in the low temperature regime.In the probe limit, we found the marginal stable mode exists with ω = 0 of the chargedfermion. This strongly indicates there should exist a stable charged black hole with fermionhair in AdS space. It would be of great interest to consider the backreaction of the fermionand to find such a black hole solution in our system, which is currently under investigation. Acknowledgments
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