On analytical study of holographic superconductors with Born-Infeld electrodynamics
aa r X i v : . [ h e p - t h ] A ug On analytical study of holographic superconductors with Born-Infeldelectrodynamics
Chuyu Lai , , Qiyuan Pan , , ∗ , Jiliang Jing , † and Yongjiu Wang , ‡ Institute of Physics and Department of Physics,Hunan Normal University, Changsha, Hunan 410081, China Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education,Hunan Normal University, Changsha, Hunan 410081, China and Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo 05315-970, Brazil
Abstract
Based on the Sturm-Liouville eigenvalue problem, Banerjee et al. proposed a perturbative ap-proach to analytically investigate the properties of the (2 + 1)-dimensional superconductor withBorn-Infeld electrodynamics [Phys. Rev. D , 104001 (2013)]. By introducing an iterative proce-dure, we will further improve the analytical results and the consistency with the numerical findings,and can easily extend the analytical study to the higher-dimensional superconductor with Born-Infeld electrodynamics. We observe that the higher Born-Infeld corrections make it harder for thecondensation to form but do not affect the critical phenomena of the system. Our analytical re-sults can be used to back up the numerical computations for the holographic superconductors withvarious condensates in Born-Infeld electrodynamics. PACS numbers: 11.25.Tq, 04.70.Bw, 74.20.-z ∗ [email protected] † [email protected] ‡ [email protected] I. INTRODUCTION
As one of the most significant developments in fundamental physics in the last one decade, the anti-deSitter/conformal field theories (AdS/CFT) correspondence [1–3] allows to describe the strongly coupled con-formal field theories through a weakly coupled dual gravitational description. A recent interesting applicationof such a holography is constructing of a model of a high T c superconductor, for reviews, see Refs. [4–7] andreferences therein. It was found that the instability of the bulk black hole corresponds to a second order phasetransition from normal state to superconducting state which brings the spontaneous U(1) symmetry breaking[8], and the properties of a (2 + 1)-dimensional superconductor can indeed be reproduced in the (3 + 1)-dimensional holographic dual model based on the framework of usual Maxwell electrodynamics [9]. In orderto understand the influences of the 1 /N or 1 /λ ( λ is the ’t Hooft coupling) corrections on the holographicdual models, it is of great interest to consider the holographic superconductor models with the nonlinearelectrodynamics since the nonlinear electrodynamics essentially implies the higher derivative corrections ofthe gauge field [10]. Jing and Chen introduced the first holographic superconductor model in Born-Infeldelectrodynamics and observed that the nonlinear Born-Infeld corrections will make it harder for the scalarcondensation to form [11]. Along this line, there have been accumulated interest to study various holographicdual models with the nonlinear electrodynamics [12–24].In most cases, the holographic dual models were studied numerically. In order to back up numerical resultsand gain more insights into the properties of the holographic superconductors, Siopsis et al. developed thevariational method for the Sturm-Liouville (S-L) eigenvalue problem to analytically calculate the criticalexponent near the critical temperature and found that the analytical results obtained by this way are in goodagreement with the numerical findings [25, 26]. Generalized to study the holographic insulator/superconductorphase transition [27], this method can clearly present the condensation and critical phenomena of the systemat the critical point in AdS soliton background.More recently, Gangopadhyay and Roychowdhury extended the S-L method to investigate the propertiesof the (2 + 1)-dimensional superconductor with Born-Infeld electrodynamics by introducing a perturbativetechnique, and observed that the analytical results agree well with the existing numerical results for thecondensation operator hO − i [28]. For the operator hO + i , Banerjee et al. improved the perturbative approachand explored the effect of the Born-Infeld electrodynamics on the (2 + 1)-dimensional superconductor [29].However, comparing with the case of hO − i [28], we find that for the operator hO + i the agreement of theanalytical result with the numerical calculation is not so good, for example in the case of the Born-Infeldparameter b = 0 . . hO + i , and further improvethe analytical results and the consistency with the numerical findings. On another more speculative level, itwould be important to develop a more general analytical technique which can be used to study systematicallythe d -dimensional superconductors with Born-Infeld electrodynamics and see some general features for theeffects of the higher derivative corrections to the gauge field on the holographic dual models. In order toavoid the complex computation, in this work we will concentrate on the probe limit where the backreactionof matter fields on the spacetime metric is neglected.The plan of the work is the following. In Sec. II we will introduce the holographic superconductor modelswith Born-Infeld electrodynamics in the ( d + 1)-dimensional AdS black hole background. In Sec. III we willimprove the perturbative approach proposed in [29] and give an analytical investigation of the holographicsuperconductors with Born-Infeld electrodynamics by using the S-L method. We will conclude in the lastsection with our main results. II. HOLOGRAPHIC SUPERCONDUCTORS WITH BORN-INFELD ELECTRODYNAMICS
We begin with the background of the ( d + 1)-dimensional planar Schwarzschild-AdS black hole ds = − r f ( r ) dt + dr r f ( r ) + r d − X i =1 dx i , (1)where f ( r ) = 1 − r d + /r d with the radius of the event horizon r + . For convenience, we have set the AdS radius L = 1. The Hawking temperature of the black hole is determined by T = dr + π , (2)which will be interpreted as the temperature of the CFT.Working in the probe limit, we consider the Born-Infeld electrodynamics and the charged complex scalarfield coupled via the action S = Z d d +1 x √− g " b − r bF ! − |∇ ψ − iAψ | − m | ψ | , (3)with the quadratic term F = F µν F µν . When the Born-Infeld parameter b →
0, the model (3) reduces to thestandard holographic superconductors investigated in [9, 30].With the ansatz of the matter fields as ψ = | ψ | , A t = φ where ψ and φ are both real functions of r only, wecan arrive at the following equations of motion for the scalar field ψ and the gauge field φψ ′′ + (cid:18) dr + f ′ f (cid:19) ψ ′ + (cid:18) φ r f − m r f (cid:19) ψ = 0 , (4) φ ′′ + d − r (cid:0) − bφ ′ (cid:1) φ ′ − ψ r f (cid:0) − bφ ′ (cid:1) / φ = 0 , (5)where the prime denotes the derivative with respect to r .Applying the S-L method to analytically study the properties of the holographic superconductors withBorn-Infeld electrodynamics, we will introduce a new variable z = r + /r and rewrite the equations of motion(4) and (5) into ψ ′′ + (cid:18) − dz + f ′ f (cid:19) ψ ′ + (cid:18) φ r f − m z f (cid:19) ψ = 0 , (6) φ ′′ + 1 z (cid:20) (3 − d ) + b ( d − z r φ ′ (cid:21) φ ′ − ψ z f (cid:18) − bz r φ ′ (cid:19) / φ = 0 , (7)with f = 1 − z d . Here and hereafter the prime denotes the derivative with respect to z .In order to get the solutions in the superconducting phase, we have to impose the appropriate boundaryconditions for ψ and φ . At the event horizon z = 1 of the black hole, the regularity gives the boundaryconditions ψ (1) = − dm ψ ′ (1) , φ (1) = 0 . (8)Near the AdS boundary z →
0, the asymptotic behaviors of the solutions are ψ = ψ − r ∆ − + z ∆ − + ψ + r ∆ + + z ∆ + , φ = µ − ρr d − z d − , (9)where ∆ ± = ( d ± √ d + 4 m ) / µ and ρ are interpreted as the chemical potential and charge density in the dual field theory respectively.It should be pointed out that, provided ∆ − is larger than the unitarity bound, both ψ − and ψ + can benormalizable and they can be used to define operators in the dual field theory according to the AdS/CFTcorrespondence, ψ − = hO − i and ψ + = hO + i , respectively. Just as in Refs. [9, 30], we will impose boundarycondition that either ψ − or ψ + vanishes. In this work, we impose boundary condition ψ − = 0 since weconcentrate on the condensate for the operator hO + i . For clarity, we set hOi = hO + i and ∆ = ∆ + in thefollowing discussion. III. ANALYTICAL STUDY OF HOLOGRAPHIC SUPERCONDUCTORS WITH BORN-INFELDELECTRODYNAMICS
Here we will improve the perturbative approach proposed in [29] and use the S-L method [25] to analyticallydiscuss the properties of the d -dimensional superconductor phase transition with Born-Infeld electrodynamics.We will investigate the relation between critical temperature and charge density as well as the critical exponentof condensation operators, and examine the effect of the Born-Infeld parameter. A. Critical temperature
At the critical temperature T c , the scalar field ψ = 0. Thus, near the critical point the equation of motion(7) for the gauge field φ becomes φ ′′ + 1 z (cid:20) (3 − d ) + b ( d − z r c φ ′ (cid:21) φ ′ = 0 , (10)where r + c is the radius of the horizon at the critical point. Defining ξ ( z ) = φ ′ ( z ), we can obtain ξ ′ + 3 − dz ξ = b (1 − d ) z r c ξ , (11)which is the special case of Bernoulli’s Equation y ′ ( x ) + f ( x ) y = g ( x ) y n [31] for n = 3. Considering that theboundary condition (9) for φ , we can get the solution to Eq. (11) ξ ( z ) = φ ′ ( z ) = − λr + c ( d − z d − p d − bλ z d − , (12)which leads to the expression φ ( z ) = λr + c ζ ( z ) , (13)with ζ ( z ) = Z z ( d − z d − p d − bλ ˜ z d − d ˜ z, (14)where we have set λ = ρ/r d − c and used the fact that φ (1) = 0.Obviously, the integral in (14) is not doable exactly. Just as in Refs. [28, 29], we will perform a perturbativeexpansion of ( d − bλ . In order to simplify the following calculation, we will express the Born-Infeldparameter b as b n = n ∆ b, n = 0 , , , · · · , (15)where ∆ b = b n +1 − b n is the step size of our iterative procedure. Considering the fact that( d − bλ = ( d − b n λ = ( d − b n ( λ | b n − ) + 0[(∆ b ) ] , (16)where we have set b − = 0 and λ | b − = 0, we will discuss the following two cases (note that the variable z has a range 0 ≤ z ≤ Case 1.
If ( d − b n ( λ | b n − ) <
1, we have ζ ( z ) = ζ ( z ) ≈ Z z ( d − z d − " − ( d − b n ( λ | b n − )˜ z d − d ˜ z = (1 − z d − ) + ( d − b n ( λ | b n − )2(4 − d ) (1 − z d − ) . (17) Case 2.
If ( d − b n ( λ | b n − ) >
1, we set ( d − b n ( λ | b n − )Λ d − = 1 for z = Λ. Obviously, we find that( d − b n ( λ | b n − ) z d − < z < Λ <
1, which results in ζ ( z ) = ζ A ( z ) ≈ Z Λ z ( d − z d − " − ( d − b n ( λ | b n − )˜ z d − d ˜ z + Z √ b n ( λ | b n − )˜ z (cid:20) − d − b n ( λ | b n − )˜ z d − (cid:21) d ˜ z = − z d − + ( d − z d − d − d − + 3( d − d (4 d − d − d −
4) Λ d − + ( d − d − (cid:20) Λ d − d − − (cid:21) , (18)and ( d − b n ( λ | b n − ) z d − > < z ≤
1, which leads to ζ ( z ) = ζ B ( z ) ≈ Z z √ b n ( λ | b n − )˜ z (cid:20) − d − b n ( λ | b n − )˜ z d − (cid:21) d ˜ z = ( d − d − (cid:20) Λ d − d − (cid:0) − z − d (cid:1) + 1 z − (cid:21) . (19)It should be noted that in both cases we observe that ζ (1) = 0 from (17) and (19), which is consistent withthe boundary condition φ (1) = 0 given in (8).Introducing a trial function F ( z ) near the boundary z = 0 as ψ ( z ) ∼ hOi r ∆+ z ∆ F ( z ) , (20)with the boundary conditions F (0) = 1 and F ′ (0) = 0, from Eq. (6) we can obtain the equation of motion for F ( z ) ( T F ′ ) ′ + T (cid:0) P + λ Qζ (cid:1) F = 0 , (21)with T = z − d (1 − z d ) , P = ∆(∆ − d ) z + ∆ f ′ zf − m z f , Q = 1 f . (22)According to the S-L eigenvalue problem [32], we deduce the eigenvalue λ minimizes the expression λ = R T (cid:0) F ′ − P F (cid:1) dz R T Qζ F dz , for ( d − b n ( λ | b n − ) < , (23)and λ = R T (cid:0) F ′ − P F (cid:1) dz R Λ0 T Qζ A F dz + R T Qζ B F dz , for ( d − b n ( λ | b n − ) > . (24)Using Eqs. (23) and (24) to compute the minimum eigenvalue of λ , we can obtain the critical temperature T c for different Born-Infeld parameter b , spacetime dimension d and mass of the scalar field m from the followingrelation T c = d π (cid:18) ρλ min (cid:19) d − . (25)In the following calculation, we will assume the trial function to be F ( z ) = 1 − az with a constant a .As an example, we will study the case for d = 3 and m L = − b . Setting ∆ b = 0 .
1, for b = 0 we use Eq. (23) and get λ = 4(15 − a + 12 a )10(9 − √ π − −
12 ln 3) a + (10 √ π − −
30 ln 3) a , (26)whose minimum is λ | b = 17 .
31 at a = 0 . T c = 0 . ρ / , which is in good agreement with the numerical result T c = 0 . ρ / [9]. For b = 0 .
1, we can easily have b ( λ | b ) > b ( λ | b )] − / = 0 . λ = 1 − a + a . − . a + 0 . a , (27)whose minimum is λ | b = 33 .
84 at a = 0 . T c = 0 . ρ / , which alsoagrees well with the numerical finding T c = 0 . ρ / [11]. For b = 0 .
2, we still have b ( λ | b ) > b ( λ | b )] − / = 0 . λ = 1 − a + a . − . a + 0 . a , (28)whose minimum is λ | b = 58 .
19 at a = 0 . T c = 0 . ρ / , whichis again consistent with the numerical result T c = 0 . ρ / [11]. For other values of b , the similar iterativeprocedure can be applied to give the analytical result for the critical temperature.In Table I, we provide the critical temperature T c of the chosen parameter b with the scalar operator hOi = hO + i for the (2 + 1)-dimensional superconductor if we fix the mass of the scalar field by m L = − TABLE I: The critical temperature T c obtained by the analytical S-L method and from numerical calculation [11] forthe chosen values of the Born-Infeld parameter b in the case of 4-dimensional AdS black hole background. Here we fixthe mass of the scalar field by m L = − b = 0 . b Analytical . ρ / . ρ / . ρ / . ρ / Numerical . ρ / . ρ / . ρ / . ρ / and the step size by ∆ b = 0 .
1. From Table I, we observe that the differences between the analytical andnumerical values are within 4 . T c for the scalar operator hOi = hO + i when we fix the mass of the scalar field m L = − b by choosing the step size ∆ b = 0 .
05 and 0 . b = 0 .
025 the agreement of the analytical results derived from S-L method with the numericalcalculation is impressive. Thus, we argue that, even in the higher dimension, the analytical results derivedfrom the S-L method are in very good agreement with the numerical calculation. Furthermore, reducing thestep size ∆ b reasonably, we can improve the analytical result and get the critical temperature more consistentwith the numerical result. TABLE II: The critical temperature T c with the chosen values of the Born-Infeld parameter b and the step size ∆ b inthe case of 5-dimensional AdS black hole background. Here we fix the mass of the scalar field by m L = − b Analytical (∆ b = 0 .
05) 0 . ρ / . ρ / . ρ / . ρ / Analytical (∆ b = 0 . . ρ / . ρ / . ρ / . ρ / Numerical . ρ / . ρ / . ρ / . ρ / From Tables I and II, we point out that the critical temperature T c decreases as the Born-Infeld parameter b increases for the fixed scalar field mass and spacetime dimension, which supports the numerical computationfound in Refs. [11, 12, 21]. It is shown that the higher Born-Infeld electrodynamics corrections will make thescalar hair more difficult to be developed. On the other hand, the consistency between the analytical andnumerical results indicates that the S-L method is a powerful analytical way to investigate the holographicsuperconductor with various condensates even when we take the Born-Infeld electrodynamics into account. B. Critical phenomena
Since the condensation for the scalar operator hOi is so small when T → T c , we can expand φ ( z ) in hOi near the boundary z = 0 as φ ( z ) r + = λζ ( z ) + hOi r χ ( z ) + · · · , (29)with the boundary conditions χ (1) = 0 and χ ′ (1) = 0 [25, 33, 34]. Thus, substituting the functions (20) and(29) into (7), we keep terms up to 0( b ) [29] to get the equation of motion for χ ( z )( U χ ′ ) ′ = 2 λz − d F ζf , (30)where we have introduced a new function U ( z ) = e bλ z ζ ′ / z d − . (31)Making integration of both sides of Eq. (30), we have (cid:20) χ ′ ( z ) z d − (cid:21) (cid:12)(cid:12)(cid:12)(cid:12) z → = − λα , for ( d − b n ( λ | b n − ) < , − λ ( α A + α B ) , for ( d − b n ( λ | b n − ) > , (32)with α = Z z − d F ζ f dz, α A = Z Λ0 z − d F ζ A f dz, α B = Z z − d F ζ B f dz. (33)For clarity, we will fix the spacetime dimension d in the following discussion. Considering the case of d = 3and the asymptotic behavior (9), for example, near z → ρr (1 − z ) = λζ ( z ) + hOi r [ χ (0) + χ ′ (0) z + · · · ] . (34)From the coefficients of the z terms in both sides of the above formula, we can obtain ρr = λ − hOi r χ ′ (0) , (35)where χ ′ (0) can be easily calculated by using Eq. (32). Therefore we will know that hOi = βT ∆ c (cid:18) − TT c (cid:19) , (36)where the coefficient β is given by β = (cid:0) π (cid:1) ∆ q α , for b n ( λ | b n − ) < , (cid:0) π (cid:1) ∆ q α A + α B , for b n ( λ | b n − ) > . (37)0Obviously, the expression (36) is valid for different values of the Born-Infeld parameter and scalar field massin the case of the (2 + 1)-dimensional superconductor. For concreteness, we will focus on the case for the massof the scalar field m L = − b = 0 .
1. Since in Ref. [11] the scalar operator is given by hO + i = √ ψ + which is different from hO + i = ψ + in this work, we present the condensation value γ = √ β obtained by the analytical S-L method and from numerical calculation with the chosen values of the Born-Infeld parameter b for the (2 + 1)-dimensional superconductor in Table III. We see that the condensation value γ increases as the Born-Infeld parameter b increases for the fixed scalar field mass and spacetime dimension,which indicates the consistent picture shown in T c that the higher Born-Infeld electrodynamics correctionsmake the condensation to be formed harder. On the other hand, comparing with the analytical results shownin Table II of Ref. [29], we find that the iterative procedure indeed reduces the disparity between the analyticaland numerical results. TABLE III: The condensation value γ = √ β obtained by the analytical S-L method and from numerical calculation[11] with the chosen values of the Born-Infeld parameter b in the case of 4-dimensional AdS black hole background.Here we fix the mass of the scalar field by m L = − b = 0 . b Analytical .
80 117 .
92 137 .
22 161 . Numerical .
24 207 .
36 302 .
76 432 . As another example, let us move on to the case of d = 4. From the asymptotic behavior (9), we can expand φ when z → ρr (1 − z ) = λζ ( z ) + hOi r (cid:20) χ (0) + χ ′ (0) z + 12 χ ′′ (0) z + · · · (cid:21) . (38)Considering the coefficients of z terms in above equation, we observe that χ ′ (0) → z →
0, which isconsistent with Eq. (32). Comparing the coefficients of the z terms, we have ρr = λ − hOi r χ ′′ (0) , (39)where χ ′′ (0) can be computed by using Eq. (32). So we can deduce the same relation (36) for the (3 + 1)-dimensional superconductor with the different condensation coefficient β = π ∆ q α , for 4 b n ( λ | b n − ) < ,π ∆ q α A + α B , for 4 b n ( λ | b n − ) > . (40)In Table IV, we give the condensation value β obtained by the analytical S-L method with the chosen values ofthe Born-Infeld parameter b and step size ∆ b for the (3+1)-dimensional superconductor. In both cases we find1again that, for the fixed scalar field mass and spacetime dimension, the condensation value β increases as theBorn-Infeld parameter b increases, just as the observation obtained in the (2 + 1)-dimensional superconductorwith Born-Infeld electrodynamics. TABLE IV: The condensation value β obtained by the analytical S-L method with the chosen values of the Born-Infeldparameter b and step size ∆ b in the case of 5-dimensional AdS black hole background. Here we fix the mass of thescalar field by m L = − b b = 0 .
05 238 .
91 418 .
95 697 .
64 1195 . b = 0 .
025 238 .
91 496 .
06 1005 .
38 2303 . It should be noted that one can easily extend our discussion to the higher-dimensional superconductor andget our expression (36), although the coefficient β is different. Thus, near the critical point, the scalar operator hOi will satisfy hOi ∼ (1 − T /T c ) / , (41)which holds for various values of the Born-Infeld parameter b , spacetime dimension d and mass of the scalarfield m . It shows that the phase transition is of the second order and the critical exponent of the systemalways takes the mean-field value 1 /
2. The Born-Infeld electrodynamics will not influence the result.
IV. CONCLUSIONS
We have generalized the variational method for the S-L eigenvalue problem to analytically investigate thecondensation and critical phenomena of the d -dimensional superconductors with Born-Infeld electrodynamics,which may help to understand the influences of the 1 /N or 1 /λ corrections on the holographic superconductormodels. We found that the S-L method is still powerful to disclose the properties of the holographic super-conductor with various condensates even when we take the Born-Infeld electrodynamics into account. Usingthe iterative procedure in the perturbative approach proposed by Banerjee et al. [29], we further improvedthe analytical results and the consistency with the numerical findings for the (2 + 1)-dimensional supercon-ductor. Furthermore, extending the investigation to the higher-dimensional superconductor with Born-Infeldelectrodynamics, we observed again that the analytical results derived from this method with a reasonablestep size are in very good agreement with those obtained from numerical calculation. Our analytical resultshows that the Born-Infeld parameter makes the critical temperature of the superconductor decrease, whichcan be used to back up the numerical findings as shown in the existing literatures that the higher Born-Infeld2electrodynamics corrections can hinder the condensation to be formed. Moreover, with the help of this ana-lytical method, we interestingly noted that the Born-Infeld electrodynamics, spacetime dimension and scalarmass cannot modify the critical phenomena, and found that the holographic superconductor phase transitionbelongs to the second order and the critical exponent of the system always takes the mean-field value. Itshould be noted that one can easily extend our technique to the holographic superconductor models withthe logarithmic form [20] and exponential form [21] of nonlinear electrodynamics. More recently, a model ofp-wave holographic superconductors from charged Born-Infeld black holes [35] via a Maxwell complex vectorfield model [36–38] was studied numerically. It would be of interest to generalize our study to this p-wavemodel and analytically discuss the effect of the Born-Infeld electrodynamics on the system. We will leave itfor further study. Acknowledgments
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