Holographic superconductors in the AdS black hole with a magnetic charge
aa r X i v : . [ phy s i c s . g e n - ph ] N ov Holographic superconductors in the AdS black hole with a magnetic charge
M. R. Setare , ∗ D. Momeni , † R. Myrzakulov , ‡ and Muhammad Raza, § Department of Science, Payame Noor University, Bijar, Iran Eurasian International Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan Department of Computer Science, COMSATS Institute ofInformation Technology (CIIT), Sahiwal campus, Pakistan
In this work we study the analytical properties of a 2+1 dimensional magnetically chargedholographic superconductor in
AdS . We obtain the critical chemical potential µ c analyti-cally, using the Sturm-Liouville variational approach. Also, the obtained analytic result canbe used to back up the numerical computations in the holographic superconductor in theprobe limit. PACS numbers: 11.25.TqKeywords: High- T C superconductors theory I. INTRODUCTION
In this paper, we will investigate the holographic superconductors in magnetically charged planarAdS black hole. The metric of the black hole with a magnetic charge was obtained by Romans[1]. The presence of the magnetic charge shows that the black hole has different horizon structurefrom that of the uncharged Schwarzschild black hole. The main purpose in this paper is to see howthe magnetic charge affects the holographic superconductors in this asymptotic AdS black hole.This paper is organized as follows: In Sec. II, we present the metric describing a magneticallycharged planar Schwarzschild-AdS black hole. In Sec. III, we give the basic equations. In Sec. IVwe investigate the zero temperature limit and critical chemical potential. We study analyticallyholographic superconductors in the magnetically charged planar black hole background. Our resultsshow that the magnetic charge presents the different physical effects on the different condensations.Finally, in the last section we present our conclusions. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected]
II. GEOMETRY OF A MAGNETICALLY CHARGED BLACK HOLE IN
AdS The Lagrangian that gives rise to the magnetically charged black hole is described by thecommon Lagrangian of Einstein-Maxwell theory with cosmological constant Λ is given by [1] L = 1 q (cid:16) − R + 14 F µν F µν + 12 Λ (cid:17) . (1)Where q is the electrical charge. The usual Einstein-Maxwell equations read R µν = T µν , T µν = 2 F µα F αν − g µν F αβ F αβ , ∇ ; µ F µν = 0 . (2)The components of the Maxwell tensor are F = Qr , F = − H cos θ. (3)We assume that the electric charge Q and the magnetic charge H simultaneously are not zero. Themagnetically charged planar solution for action (1) is [1] ds = − f ( r ) dt + dr f ( r ) + r ( dx + dy ) . (4)An exact solution of the equations (2) is f ( r ) = r − Mr + H r . This solution describes a planar Schwarzschild-AdS black hole with a magnetic charge. As themagnetic charge H tends to zero, the space time reduces to a four dimensional Schwarzschild-AdS black hole. The only unique non vanishing component F xy = Hr . Thus the factor H canbe interpreted as the strength of the magnetic field in bulk. It is obvious that this solution isasymptotically AdS . The outer horizon locates at f ( h ) = 0. For future numerical purposes, wetake the horizon as h = 1. Thus we have M = 1 + H . Further, as we know , to have a real horizonwe must satisfy the auxiliary condition 27 M − H ≥ M − H = 0 , which gives to us the value of H = 1 . T BH = f ′ ( h )4 π . This temperature can be read as the boundary temperature in the quantum dual theory, via CFT.We mention here that the near horizon geometry of the metric, at zero temperature has a scalinginvariance. This scaling invariance characterized by a dynamical critical exponent z . III. THE CONDENSATE OF CHARGED OPERATORS
In order to investigate the holographic superconductors in the background of the planarSchwarzschild-AdS black hole with a magnetic charge, we need a scalar condensate with a chargedscalar field ψ [2]. Here we work in the probe approximation by neglecting the backreaction of thecharged scalar field ψ . It may be an interesting topic to study the holographic superconductors ofthe scalar field with magnetic charge. However, In this paper we only consider the condensate ofan external charged scalar field in the background of a black hole with a magnetic. Let us considera Maxwell field and a charged complex scalar field, S = Z √− gd x [ 1 q (cid:16) − R + 14 F µν F µν + 12 Λ (cid:17) − | D µ ψ | − m | ψ | ] . (5)We set the AdS radius L = q ≡
1. The mass of the scalar field is chosen such that it remainsbelow to the Breitenlohner-Freedman bound [3]. We take A µ = ( φ ( r ) , , , , ψ = ψ ( r ) , (6)we can obtain the equations of motion for the complex scalar field ψ and electrical scalar potential φ ( r ) in the background of the Schwarzschild-AdS black hole with a magnetic charge H given by[4] ψ ′′ + ( f ′ f + 2 r ) ψ ′ + ( φ f + 2 f ) ψ = 0 (7) , φ ′′ + 2 φ ′ r = 2 ψ φf . (8)Here a prime denotes the derivative with respect to r . Trivially, it is not possible to obtain thenontrivial analytical solutions to the nonlinear equations. IV. ZERO TEMPERATURE LIMIT AND CRITICAL CHEMICAL POTENTIAL
We analyze the field equations using the variational method . On horizon the boundary con-dition φ ( h ) = µ is the critical chemical potential reads from the value of the field on boundary(horizon), and for the scalar field ψ ′ ( h ) = υ ( h, H ) ψ ( h ) in which the function υ ( h, H ) is a functionof h, H . Asymptotic value of the field ψ leads to two distinct conformal dimensions ∆ ± = 1 ,
2. Thefirst step for investigating the analytical properties of the superconductors via variational method isin writing the field equation in a classical Sturm-Liouville (S-L) form. We present our calculationsseparately for different conformal dimensions. We rewrite the field equations (7) and (8) in a newcoordinate z as ψ ′′ + f ′ f ψ ′ + ( h φ z f + 2 h z f ) ψ = 0 (9) φ ′′ − h ψ f z φ = 0 . (10)now prime denotes the derivative with respect to the z . In these coordinates z → AdS conformal boundary. We must first find a solution to the above equations such that near theboundary φ ≈
In this case we have µ ( α ) = − . α − . α + 0 . . α − . α + 0 . . (18)The minimum with respect to the α locates at α c = − . , (19)which leads to the following values for µ c µ c = 1 . , (20)we show that the µ c obtained from variational method is in good agreement with the results of the[4]. B. Case ∆ = 1
In this case, we have µ ( α ) = − . α + 0 . α . α − . α + 0 . . (21)The minimum with respect to the α locates at α c = − . , (22)which leads to the following values for µ c µ c = 0 . . (23)Referring to the numerical results obtained in [4] shows that the µ c obtained is in good agreementwith the results. The consistency between the analytic and numerical results indicates that theS-L method is a powerful analytic way to investigate the holographic superconductor even whenwe take the backreaction into account[5]. V. CONCLUSION
In this paper using an variational calculation we obtained the minimum of the chemical potential µ c . Our analytic result can be used to back up the numerical computations in the holographicsuperconductor in the probe limit. VI. COMMENT
The model and the method which we used here are completely different from the publishedpaper [6] in which they applied numerical methods on global monopoles in the AdS background.
VII. ACKNOWLEDGMENT
We thank Prof. Alberto Salvio for usefull suggestions. [1] L. J. Romans, Nucl. Phys. B 383, 395 (1992),[arXiv:hep-th/9203018].[2] G. T. Horowitz and M. M. Roberts, Phys. Rev. D144